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[
"PCB Coil Design Producing a Uniform Confined Magnetic Field",
"PCB Coil Design Producing a Uniform Confined Magnetic Field"
] |
[
"Ieee ",
"Letters "
] |
[] |
[] |
We present a magnetic field confining coil with a sub 10 −3 field uniformity over a large fraction of the coil. The structure is entirely made out of printed circuit boards (PCB). The PCB design allows to tailor the path of wires to fit the required geometry. We measure the field uniformity with Cesium magnetometers in a field range from 1µT to 10µT. Our application uses such a coil for an atomic magnetometry based current controller.
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10.1109/lmag.2017.2701771
|
[
"https://arxiv.org/pdf/1707.06414v2.pdf"
] | 32,489,934 |
1707.06414
|
235dc1b17f55ec1715073affd4883a667e90b34a
|
PCB Coil Design Producing a Uniform Confined Magnetic Field
2017
Ieee
Letters
PCB Coil Design Producing a Uniform Confined Magnetic Field
201710.1109/LMAG.2017.27017711Index Terms-ElectromagneticsMagnetic instrumentsMagnetic measurementsMagnetic sensors
We present a magnetic field confining coil with a sub 10 −3 field uniformity over a large fraction of the coil. The structure is entirely made out of printed circuit boards (PCB). The PCB design allows to tailor the path of wires to fit the required geometry. We measure the field uniformity with Cesium magnetometers in a field range from 1µT to 10µT. Our application uses such a coil for an atomic magnetometry based current controller.
I. INTRODUCTION
M ANY applications in modern research require a very uniform magnetic field over the experimental volume. Traditionally, a pair of Helmholtz coils or a solenoid would be used. They produce a reasonably uniform field for a coil geometry which has been known and used for many years [1], [2]. Extensions of the Helmholtz coils attempt to improve on the uniformity of the magnetic field by using more than just two coils [3]. Some applications have very stringent requirements on the field uniformity [4]. There one would prefer a cos θ or a spherical coil, which typically produces very uniform fields over a large volume of the coil [5]. All of these coils have large fringe fields. They may interact with environmental factors such as nearby high permeability materials, perturbing the actual uniform volume. One may then use either a return yoke or build a field confining coil to guide the stray field and reduce the influence of the environment [6], [7]. We present simulations and measurements of a PCB based field confining coil. The PCB design exploits the precision of modern PCB manufacturing. This allows for tailored wire paths as well as to choose the wire density, i.e. the coil constant. Furthermore, the wire width and thickness can be chosen. Thus, we can adapt the total resistivity of the coil to the used current source. Additionally, the use of PCB headers as connectors makes it easy to open and close this type of field confining coil.
A. Motivation
Our groups are involved in an international collaboration searching for the neutron electric dipole moment (nEDM) [8], [9]. The experiment uses Ramsey's method of time-separated oscillating fields [10], where neutrons precess in a magnetic field H and a parallel, or anti-parallel, electric field E. These types of measurements require a very stable and uniform magnetic field H 0 . The long-term stability for H 0 is achieved with a mu-metal shield and active field stabilization [11]. We aim at improving the shielding factor in our experiment by at least a factor 10. Then, the stability will be fundamentally limited by the current source which feeds the H 0 -coil. Thus, the present current source, 10 −7 stability on 17 mA, must be improved. In order to improve on this stability, a current controller based on atomic magnetometry is presently being developed, see Fig.1 [12]. Modern atomic magnetometers easily reach sensitivities on the order of 10 −8 for one second of integration time in the shot noise limit [13], [14]. These sensitivities can be exploited by converting drifts in currents into magnetic field changes [15]. The coil in Fig.1 is able to discriminate external field perturbations from a drift in current. The latter changes the field modulus in all quadrants by the same amount, while an external field will affect each quadrant in a different way.
II. COIL DESIGN
A. Method
The coil design, shown in Fig.1, confines the field by guiding it through all four quadrants. Each quadrant contains a scalar magnetometer, which measures the averaged field modulus over the sensor volume. A uniform field is required since field gradients broaden the magnetometer's magnetic resonance lines, thus, causing a degradation in sensitivity [16]. The coil was designed using the magnetic scalar potential Φ M [17], [18]. In a currentless region, the magnetic field can be expressed as the gradient of a scalar potential Φ M
H = −∇Φ M .(1)
In this method, we first define a region representing the coil and then apply flux boundary conditions as demonstrated in Fig.2. The behavior of Φ M can be determined by solving the Laplace equation
∇ 2 Φ M = 0(2)
for the appropriate set of boundary conditions. The solution is shown on the same figure in terms of isopotentials of Φ M .
Since H is always perpendicular to the isopotentials of Φ M , it can already be seen that the field will be quite uniform in each quadrant. Next, we simulate the magnetic field produced by this coil. For this, each isopotential is interpreted as a single rectangular current loop. For the present case, this results in a cubic coil of which Fig.2 is the top view. To simulate the behavior of the magnetic field, the Biot-Savart law formulated for a straight wire segment was used [19]
H = I 4π R i × R f R i + R f R i R f (R i R f + R i · R f ) ,(3)
where R i(f ) = r − r i(f ) is the vector from the segment initial (final) point r i(f ) to the observation point r. A current loop can then be simulated by composing it of four such wire segments. A Python program was written to simulate the field produced by the entire coil geometry.
B. Simulations
The following simulation results describe a cubic coil with dimensions a o = 250 mm and a i = 1 mm, as defined in
η = H − H c H c .(4)
The result is shown as a contour plot in Fig.3. Most of the points are distributed narrowly around η = 0. In order to characterize the uniformity of the field in the ROI, the interquartile range (IQR) of the 10 4 values for η was calculated. The IQR is more robust to outliers, which allows it to cope with the edge effects in Fig.3. The value η IQR will be used to characterize the field uniformity in the ROI. The most important technique for high field uniformity is to properly connect individual current loops. The most uniform field is reached with disconnected individual loops. A simple but realistic case would be to connect the loops at the outer boundary of the coil, the solid black line in Fig.2. This leads to a residual current loop, which follows the mentioned boundary. This can be avoided by connecting the current loops along the dashed boundary in Fig.2. There, the current of one quadrant flows outwards, whereas the current of the neighboring quadrant flows inward. This leads to a compensation of both This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. Fig.4 shows the field uniformity in each quadrant. One clearly sees that a residual current loop ruins the performance of the coil. Connecting current loops along the inner boundaries of the quadrants leads to no residual current loop, resulting in an order of magnitude better uniformity.
III. PROTOTYPE
In order to verify the uniformity of this coil design, a prototype was built, see Fig.5. Three different types of PCB panels were used to build it. A total of eight triangular "top panels" form the top and bottom planes of the coil. Four "center panels" were placed between the quadrants. These panels implement the connections between the current loops, see Fig.5b. Finally four "front panels" were connected to the rest of the coil via non-magnetic PCB headers (Molex KK4455 Series). The equidistant wires are 3 mm wide and 35 µm thick, yielding a total resistivity of 50 Ω for the entire coil. Each quadrant has 50 current loops, which yields a coil constant c B = 249.4 nT mA . We shall from here on refer to the magnetic flux density B, which is the quantity that our magnetometers measure. This coil was tested in a 4-layer mu-metal shield, which saturates around 0.8 T [20]. We measured the uniformity of B using two Cesium magnetometers (CsM) based on free spin precession signals (CsM-FSP) [14]. A CsM measures the average magnitude of the magnetic flux density, B = || B||, over the active sensor volume, a 30 mm sphere held at room temperature. This measurement is robust to misalignment of the sensor along the main component of B. In contrast, a fluxgate sensor measures the flux density along the axis of the sensor. Thus, a slight misalignment of this axis has a noticeable effect on the readout of the measured component. The uniformity measurements were all done in the α quadrant. Fig.6 shows a schematic of the positions of the two CsM, which were held 5 cm, and 10 cm, apart from each other. The gradiometer returned readings of both sensors simultaneously every 100 ms. Our CsM-FSP gradiometer had a restricted range of operation from 1 µT to 10 µT, far from the saturation of the mu-metal. For each gradiometer configuration a total of 8 measurements were made as a function of the applied magnetic flux density. Between each of these measurements, the mu-metal shield and coil were opened and closed, to get an estimate of the reproducibility of the measurements. The data are represented as the difference in readings, B 1 − B 2 , as a function of the magnetic flux density in Fig.7. The slope of the linear regression of this data, s x , gives an estimate of the uniformity over the measurement volume, given as the means x in Fig.7. The reproducibility of the measurements was estimated via the standard deviation of the 8 values for s x , given in parenthesis in Fig.7. All s x are at the sub 10 −3 level of uniformity over the central volume of the α quadrant. Alternatively, these measurements may be represented in terms of the coil constant, in nT/mA, for each individual sensor. In order to better compare the values, we set the current such that the average reading of all six sensors is 1000 nT, i.e. for I = 4.010 mA. Each sensor records a magnetic flux density value, which is slightly different from the 1000 nT, see Fig.8. The variations are on the sub-nT level for a flux density of 1000 nT. This can again be interpreted as a sub 10 −3 uniformity over the central volume of the α quadrant. These results, as well as the understanding of the working principle of the coil, make it reasonable to assume a similar uniformity in all four quadrants. The simulation case, which is closest to the measurements, is the "oblique connections" case shown in Fig.4 with the wire paths shown in Fig.5b. However, building the prototype has left some small misalignments between the PCBs, which lead to an imperfect wire pattern compared to Fig.2. This could explain the worse uniformity and the flux density behavior measured in the prototype. The simulations represent the theoretical limit of uniformity which can be reached with this coil design. The results in Figs.7 and 8 show a sub 10 −3 uniformity, which is less than a factor 10 away from the "oblique connections" case in Fig.4. This makes its performance comparable to a cos θ coil [4]. The fringes outside of our prototype were measured with fluxgate sensors. A total of four fluxgate sensors were mounted at the outer side of the α quadrant. A "leaking field", of 4 nT to 200 nT, was measured for flux densities, of 1 µT to 50 µT, measured inside of the coil, which is typically more than two orders of magnitude smaller. This "leaking field" is due to the gap left by the "front panels", see Fig.5. A design which reduces the size of this gap would improve the confinement aspect. However, small local residual "leaking fields" will always be present.
IV. CONCLUSION
The presented field confining coil is a simple PCB structure, which uses PCB headers as connectors to open and close the coil. This makes it a viable solution for many applications which require uniform magnetic fields at sub 10 −3 level.
Fig. 2 .
2Coil geometry. The solid black boundaries represent the condition Φ M = 0, i.e. the field is confined in the geometry. The dashed boundaries are the flux conditions for guiding the magnetic field from one quadrant to the next. The gray lines are isopotentials of Φ M . The dotted rectangle delimits the region of interest (ROI) used in simulations.
Fig. 3 .
3Region of interest (ROI). The dots on the plot show the locations where the magnetic field was calculated using (3). The contour plot is obtained via interpolation of these points. The vertical axis is longitudinal to the main magnetic field component. The horizontal axis has its origin in the center of the coil. The edge effects are due to the wires constituting the coil.
Fig. 2 .
2The size of the coil was chosen to fit a 70 mm diameter magnetometer in each quadrant, as shown in Fig.1. Each quadrant consists of 50 current loops and the applied current is 100 mA. The simulations used a planar region of interest (ROI) in one of the quadrants of the coil, see Fig.2, where field values were evaluated on a grid of 100 by 100 points. Only values of the main field component, hereafter H, were used for uniformity characterizations. The transverse components are several orders of magnitude smaller than the longitudinal component. The values for H were then normalized to the field value at the center of the ROI, H c
Fig. 5 .
5Prototype and wire paths. a) Prototype with PCBs soldered together to form a cube. The front panels labeled α and β quadrant are connected to the rest of the coil via PCB headers and can be removed. The direction of the magnetic flux density B is shown on the α quadrant when applying a positive current. b) Schematic of the current flow on the "center panels". They are double sided PCBs, which implement the connections between current loops, the "oblique connections" case inFig. 4. The gray lines represent the wires on the backside of the panel, while the black wires are on the frontside.
Fig. 6 .
6Gradiometer locations. For a horizontal gradiometer, the dashed circles represent sensor 1, reading B 1 , and the solid circles represent sensor 2, reading B 2 . For a vertical gradiometer, the thick circles at the bottom represent sensor 1 and the thin circles on top represent sensor 2. Thus measurements were made for 3 horizontal and 2 vertical configurations.
Fig. 7 .
7Gradiometer results. For each of the 5 gradiometer configurations 8 measurements were made as a function of the magnetic flux density. The lines are linear regressions of the data points. For each configuration a valuē sx of the mean of all slopes is given. The value in parenthesis is the standard deviation of those values.
Fig. 8 .
8Coil constant uniformity. Set the average reading of all six sensors to be 1000 nT. The coil constant of each individual sensor differs by the amount shown in each circle, representing a CsM. The values in parenthesis are the standard deviation of the coil constant for that sensor.
Corresponding author: Peter A. Koss (email: [email protected])Fig. 1. Current controller concept. A low noise current source provides the base current. Changes in this current are registered as changes in the magnetic field in a dedicated coil, the source coil, which is connected in series to the nEDM H 0 -coil. A magnetometer array in the source coil is able to detect these changes. A dedicated data acquisition system (DAQ) generates the appropriate feedback response for the detected drift.current
source
nEDM
H 0 -coil
I/V
H
DAQ
feedback control
source coil
mu-metal
shield
magnetometers
IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected]. 4. Simulation results. The bottom of the coil lies at 0 cm and the top at 25 cm. Points closer than 1cm to the bottom-, BP, and top-planes, TP, are not included because there the uniformity degrades rapidly. The better uniformity close to BP and TP is a feature of this coil.perturbations and the net effect is reduced. We extended the simulations by calculating η IQR as a function of the height h of the ROI in the coil. The values of H c , used to calculate η, have a very uniform dependence on the height h. Thus,The final version of record is available at
http://dx.doi.org/10.1109/LMAG.2017.2701771
Copyright (c) 2017 0
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20
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Height in the coil, h (cm)
1e-05
1e-04
1e-03
1e-02
Field uniformity, ´I
QR
residual loop at h = 25 cm
oblique connections
disconnected loops
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
ACKNOWLEDGMENTWe acknowledge financial support from the FWO Fund for Scientific Research Flanders. Christopher Crawford acknowledges the DOE contract DE-SC0008107. We thank Allard Schnabel and Vira Bondar for useful comments on the manuscript.
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|
[] |
[
"Electron correlations and silicon nanocluster energetics",
"Electron correlations and silicon nanocluster energetics"
] |
[
"N L Matsko \nP.N. Lebedev Physical Institute\nRussian Academy of Sciences\nLeninskii prosp. 53119991MoscowRussia\n\nMoscow Institute of Physics and Technology -Dolgoprudny\n141700Moscow RegionRussia\n",
"Yu A Uspenskii \nP.N. Lebedev Physical Institute\nRussian Academy of Sciences\nLeninskii prosp. 53119991MoscowRussia\n\nMoscow Institute of Physics and Technology -Dolgoprudny\n141700Moscow RegionRussia\n",
"E V Tikhonov \nLomonosov Moscow State University\nLeninskie Gory119991MoscowRussia\n",
"V S Baturin \nP.N. Lebedev Physical Institute\nRussian Academy of Sciences\nLeninskii prosp. 53119991MoscowRussia\n\nMoscow Institute of Physics and Technology -Dolgoprudny\n141700Moscow RegionRussia\n",
"S V Lepeshkin \nP.N. Lebedev Physical Institute\nRussian Academy of Sciences\nLeninskii prosp. 53119991MoscowRussia\n\nMoscow Institute of Physics and Technology -Dolgoprudny\n141700Moscow RegionRussia\n"
] |
[
"P.N. Lebedev Physical Institute\nRussian Academy of Sciences\nLeninskii prosp. 53119991MoscowRussia",
"Moscow Institute of Physics and Technology -Dolgoprudny\n141700Moscow RegionRussia",
"P.N. Lebedev Physical Institute\nRussian Academy of Sciences\nLeninskii prosp. 53119991MoscowRussia",
"Moscow Institute of Physics and Technology -Dolgoprudny\n141700Moscow RegionRussia",
"Lomonosov Moscow State University\nLeninskie Gory119991MoscowRussia",
"P.N. Lebedev Physical Institute\nRussian Academy of Sciences\nLeninskii prosp. 53119991MoscowRussia",
"Moscow Institute of Physics and Technology -Dolgoprudny\n141700Moscow RegionRussia",
"P.N. Lebedev Physical Institute\nRussian Academy of Sciences\nLeninskii prosp. 53119991MoscowRussia",
"Moscow Institute of Physics and Technology -Dolgoprudny\n141700Moscow RegionRussia"
] |
[] |
The first-principle prediction of nanocluster stable structure is often hampered by the existence of many isomer configurations with energies close to the ground state. This fact attaches additional importance to many-electron effects going beyond density functional theory (DFT), because their contributions may change a subtle energy order of competitive structures. To analyze this problem, we consider, as an example, the energetics of silicon nanoclusters passivated by hydrogen Si 10 H 2n with 0 ≤ n ≤ 11, the structure of which varies with passivation from compact to loose-packed, similar to branching polymers. Our calculations performed by the DFT, hybrid functionals and Hartree-Fock (H-F) methods, as well as by the GW approximation (GWA), confirm a considerable sensitivity of structure prediction and isomer energy ordering to many-electron effects and show some results which may be obtained with the methods less computationally demanding than the GWA.IntroductionThe unique properties of semiconductor nanoparticles are highly promising for many applications such as optoelectronics, nanoelectronics, solar cells, biosensors, etc [1]-[5], so investigations on them rank among the most burning topics. One of the challenging problems is the atomic structure of nanoclusters and small nanoparticles, which generally differs a lot from the structure of bulk samples and varies widely with cluster size and composition. Nanoobject structure strongly affects its properties. There is a general understanding that variations in the atomic structure of nanoclusters are conditioned by surface atoms, that contribute to cluster stability less than central atoms. The minimum of the total energy triggers atom rearrangement being individual for each cluster. Because 1 arXiv:1603.01730v1 [cond-mat.mes-hall]
|
10.1063/1.4960675
|
[
"https://arxiv.org/pdf/1603.01730v1.pdf"
] | 53,475,834 |
1603.01730
|
e9ba254d84db46ed3db867abe66e054c69f74795
|
Electron correlations and silicon nanocluster energetics
N L Matsko
P.N. Lebedev Physical Institute
Russian Academy of Sciences
Leninskii prosp. 53119991MoscowRussia
Moscow Institute of Physics and Technology -Dolgoprudny
141700Moscow RegionRussia
Yu A Uspenskii
P.N. Lebedev Physical Institute
Russian Academy of Sciences
Leninskii prosp. 53119991MoscowRussia
Moscow Institute of Physics and Technology -Dolgoprudny
141700Moscow RegionRussia
E V Tikhonov
Lomonosov Moscow State University
Leninskie Gory119991MoscowRussia
V S Baturin
P.N. Lebedev Physical Institute
Russian Academy of Sciences
Leninskii prosp. 53119991MoscowRussia
Moscow Institute of Physics and Technology -Dolgoprudny
141700Moscow RegionRussia
S V Lepeshkin
P.N. Lebedev Physical Institute
Russian Academy of Sciences
Leninskii prosp. 53119991MoscowRussia
Moscow Institute of Physics and Technology -Dolgoprudny
141700Moscow RegionRussia
Electron correlations and silicon nanocluster energetics
The first-principle prediction of nanocluster stable structure is often hampered by the existence of many isomer configurations with energies close to the ground state. This fact attaches additional importance to many-electron effects going beyond density functional theory (DFT), because their contributions may change a subtle energy order of competitive structures. To analyze this problem, we consider, as an example, the energetics of silicon nanoclusters passivated by hydrogen Si 10 H 2n with 0 ≤ n ≤ 11, the structure of which varies with passivation from compact to loose-packed, similar to branching polymers. Our calculations performed by the DFT, hybrid functionals and Hartree-Fock (H-F) methods, as well as by the GW approximation (GWA), confirm a considerable sensitivity of structure prediction and isomer energy ordering to many-electron effects and show some results which may be obtained with the methods less computationally demanding than the GWA.IntroductionThe unique properties of semiconductor nanoparticles are highly promising for many applications such as optoelectronics, nanoelectronics, solar cells, biosensors, etc [1]-[5], so investigations on them rank among the most burning topics. One of the challenging problems is the atomic structure of nanoclusters and small nanoparticles, which generally differs a lot from the structure of bulk samples and varies widely with cluster size and composition. Nanoobject structure strongly affects its properties. There is a general understanding that variations in the atomic structure of nanoclusters are conditioned by surface atoms, that contribute to cluster stability less than central atoms. The minimum of the total energy triggers atom rearrangement being individual for each cluster. Because 1 arXiv:1603.01730v1 [cond-mat.mes-hall]
the experimental determination of nanocluster structure is still problematic, so the reliable firstprinciples structure prediction is among the hottest problems of nanocluster physics.
First-principles methods based on density functional theory received general recognition as a reasonably precise approach available for a wide class of materials. In particular, they are intensively used in the study of nanoclusters and nanomaterials, including stable structure prediction [6]- [8]. A relative simplicity of DFT equations and a reasonable accuracy of ground-state properties calculated allow a detailed consideration of complicated nanoobjects and nanosystems. From a mathematical point of view, the determination of cluster structure is reduced to a search for the atomic configuration realizing the global minimum of cluster energy. The search for the global minimum is especially difficult when a system has many local minima lying slightly above the global one. In this case any inaccuracy or a small systematic error can distort a subtle energy order of atomic configurations. Of course, there is a limit of accuracy, after which further improvements lose their meaning. For instance, if the energies of isomer and ground-state configurations are very close E isomer −E ground ≤ k B T eff ∼ 0.03 eV -0.04 eV (T eff is the temperature of cluster synthesis or room temperature), both configurations have comparable chances to exist and the choice of the ground state is conventional.
In first-principles calculations, an evident source of systematic errors is the exchange-correlation (xc) contribution to the total energy. This contribution varies significantly depending on approximation employed for its description from the standard LDA and GGA approximations [9], [10] to the beyond-DFT methods. The important question of the first-principles structure prediction is, whether errors introduced by an approximated description of exchange and correlations shift the energies of all competitive configurations by nearly the same quantity or the shift is individual for each configuration. In the former case, the prediction of stable cluster structure is not sensitive to exchange-correlation approximations, while in the latter one a proper description of many-electron effects is of prime significance.
To elucidate this question, we make the total energy calculations of silicon nanoclusters passivated by hydrogen with formula Si 10 H 2n (0 ≤ n ≤ 11) using different xc approximations. According to our early first-principles studies [11], [12] the equilibrium structure of these clusters varies widely from very compact (Si 10 ) to loose-packed, similar to branching polymers built of SiH 2 monomers (Si 10 H 22 ) (figure 1). Characteristic energy differences between atomic configurations also vary greatly with the hydrogen passivation. As an example, in bare Si 10 clusters the first isomer energy lies at 0.6 eV above the ground state, while in Si 10 H 22 clusters this quantity falls to 0.04 eV. This diversity of structures and energetics renders the Si 10 H 2n family very suitable to study the effect of exchangecorrelation refinements on cluster structure prediction. The calculations were performed for both the ground-state and low-energy isomer configurations using the GGA, hybrid functionals, Hartree-Fock and GW methods. In all, 31 cluster configurations corresponding to seven cluster compositions were calculated, that gave valuable information about the impact of many-electron effects on cluster . Big dark balls -silicon atoms, small pale balls -hydrogen atoms structure prediction. It could be mentioned, that hybrid functionals are often considered as a very accurate approach for nanoclusters description, especially B3LYP (see the works [13]- [17]). On the other hand these works give no reasons of the validity of such approaches.
The paper proceeds as follows. Section 2 gives the basic formulas and discusses the physical meaning of approximations relating to many-electron processes. Section 3 considers the details of computation, while Section 4 checks the precision of our GW calculations using the simplest molecules and clusters as examples. Section 5 presents the total energies of Si 10 H 2n nanoclusters in their ground-state and isomer configurations. These energies were calculated by the GGA, hybrid functional, Hartree-Fock and GW methods. The discussion of these results reveals the sensitivity of structure prediction to refinements in the description of electron exchange and correlations.
Theory
The total energy of an electron system is given by the DFT expression [9] as:
E tot [ρ] = i f i i − drρ(r)V ef f (r) + drρ(r)V ext (r) + 1 2 drdr ρ(r)ρ(r )e 2 |r − r | + E xc [ρ](1)
The first term in the right part of (1) (in square brackets) is the kinetic energy (E kin ), where i and f i are the eigenvalues of the Kohn-Sham equation and their occupation numbers. The following terms are, respectively, the energy of interaction with external field (E ext ), the Hartree energy (E H ), and the exchange-correlation energy (E xc ). The effective potential of the Kohn-Sham equation is
V eff (r) = V ext (r) + V H (r) + V xc (r),
where V H (r) = δE H /δρ(r) and V xc (r) = δE xc /δρ(r). The standard density functional theory describes the functional E xc [ρ] in a local or semi-local manner. In the DFT GGA:
E GGA xc [ρ] = drρ(r) xc (ρ(r), ∇ρ(r))(2)
whereas the DFT LDA has no dependence on density gradient. The functional (2) assumes that the density of exchange-correlation energy xc (r) is a function of electron density ρ(r) and its gradient at the same point r. This approximation is best for metals, as their screened Coulomb interaction is short-range, that makes xc interaction rather local. In semiconductors and dielectrics the screened The Hartree-Fock approximation accounts for a long-range nature of Coulomb interaction exactly, ignoring, however, its screening. This approximation provides the exact exchange energy in terms of the density matrix ρ(r, r ):
E HF x = − 1 2 drdr ρ(r, r )ρ(r , r)e 2 |r − r |(3)
but assumes that the contribution of electron correlations to energy is zero. This inadequacy is partially compensated by hybrid functionals, which take into account a long-range nature of interaction between electrons and describe approximately contribution from their correlations. The simplest hybrid functional which realizes this idea [18] is:
E hyb xc = α mix E HF + (1 − α mix )E GGA x [ρ] + E GGA c [ρ](4)
This equation can be derived within the framework of the adiabatic connection formalism (see, for review, [19] and references there in). Its main contributions may be treated as follows. The first term of (4) describes a long-range contribution to exchange energy arising from statically screened Coulomb interaction between electrons with 0 = 1/α mix . The second term is a short-range exchange contribution given in a semi-local manner, while the third term represents contribution from electron correlations, which also is semi-local as is required by a short-range nature of correlations. The hybrid functional (4) depends only on one free parameter α mix , which is 0.25 (the PBE0 [20], [21] functional) or is taken around this value. In most cases the precision of (4) exceeds that of the LDA and GGA. More sophisticated hybrid functionals B3PW91, B3LYP, HSE [22]- [25] and others, which are common in the practical use, have three and more parameters that gives them additional flexibility and higher accuracy in the description of exchange and correlation.
The GW approximation takes into account both the exact exchange and electronic correlations, including static and dynamic ones, and disregards vertex corrections. In this approximation the energies of electronic quasiparticles E i are the eigenvalues of Dyson's equation:
p 2 2m + V ext (r) + V H (r) φ i (r) + dr Σ xc (r, r , E i )φ i (r ) = E i φ i (r)(5)
In the GWA the self-energy operator (SEO) of this equation is given by its simplest expression:
Σ xc (r, r , E) = i 2π dE G(r, r , E + E )W (r, r , E )(6)
Here W (r, r , E) and G(r, r , E) are, respectively, the dynamically screened Coulomb interaction and the electron Green function. In many cases it is convenient to use the spectral representations of G(r, r , E):
G(r, r , E) = ∞ −∞ dE A(r, r , E) E − E − iδ · sgn(µ − E )(7)
where µ is the chemical potential and A(r, r , E) is the electron spectral function. In the basis of Kohn-Sham's eigenstates this function is:
A(r, r , E) = i,i ψ i (r)A i,i (E)ψ † i (r )(8)
It is frequently assumed (see, as example, [26]) that the eigenfunctions of Dayson's equation are
nearly identical to those of the Kohn-Sham equation, so A i,i (E) ≈ A i (E)δ i,i . Introducing the density of quasiparticle states A(E) = i A i (E)
, the total number of electrons is expressed as:
N = µ −∞ dEA(E)(9)
Being an approximation, the GWA satisfies all the conservation laws when its Green function is the solution of (5) [27], [28]. In this approximation the total energy can be calculated by two methods. One of them uses the Luttinger-Ward functional E LW tot [G] [31], which is similar in structure to the functional of DFT (1). This functional is variational, so E tot resulting from it is accurate even when an imperfect Green function G is used. The correlation contribution to E LW tot is given by the functional Φ c [G], which is reduced in the GWA to the series of ring diagrams corresponding to the random-phase approximation (RPA). The calculation of this series is not easy for real solids, particularly for nanoclusters, which hampers the practical use of E LW tot [G]. The other method employs the Galitskii-Migdal (GM) formula [29], [30] which gives the total energy of electrons in terms of the electron quasiparticle spectrum. This formula is derived from the equation of motion for electrons and therefore has no analogs in DFT which is merely a static theory.
The exchange-correlation contribution to energy is given by the GM formula as:
E GM xc = 1 2 µ −∞ dEA(E) · E − i f i i + drρ(r)V xc (r)(10)
The assumption that the Dyson and Kohn-Sham equations have nearly identical eigenfunctions φ i (r) ≈ ψ i (r) leads to the equality of their energy contributions E kin , E ext , and E H . By this means, a higher precision of E tot in the GWA as compared to DFT is determined by the difference between E GM xc (10) and E xc [ρ] (2). The actual GW computation meets with two involved questions. The first one is associated with the level of the self-consistency in the solution of Dyson's equation. In this way G 0 W 0 is the simplest scheme. G 0 is usually picked as the DFT, Hartree-Fock or hybrid functional Green
function, W 0 (r, r , t) = W [G 0 ] and Σ 0 (r, r , t) = iG 0 (r, r , t)W 0 (r, r , t). Quasiparticle (QP) energies goes from expression E QP = E DFT QP − < ψ|V xc |ψ > +Σ(E DFT QP )
. The QP spectrum found at this step is much more precise than Kohn-Sham's one. In particular, the HOMO-LUMO gap of semiconductor nanoobjects calculated in the G 0 W 0 approximation is rather close to the experimental gap, while the DFT gap is two-three times narrower [32], [33]. Elementary improvement can be obtained by geting QP energies as E QP = E DFT QP − < ψ|V xc |ψ > +Σ(E QP ) and making iterations of the substitutions E QP to Σ for next step until convergence is achieved (self-consistency in the eigenvalues or ev-scGW).
Further improvement can be obtained as follows. As the dynamically screened Coulomb interaction W(r, r , E) is not sensitive to the variations of G, an iterative solution of Dyson's equation is frequently obtained with the SEO Σ G (r, r , t) = iG(r, r , t)W 0 (r, r , t), where only the Green's function is iterated to the self-consistency at the fixed W 0 (the scGW 0 ). Fully self-consisted scGW is obtained when Dyson equation is iterated both by G and W. In principle full self-consistency eliminates errors of the start point calculations and leads to the satisfaction of conservation laws for the total energy, momentum and particles number. In works [34,35,36] it was noticed that fully self-consisted GW improves the G 0 W 0 total energy and ionization potentials, significantly improving the DFT and hybrid functional results. However both approximations (G 0 W 0 and GW 0 ) greatly decrease the body of computation and yet provide precise quasiparticle spectra. It has been observed that sometimes the simpler G 0 W 0 and GW 0 schemes provide even better spectra than the fully self-consistent GW calculation [37,38,39]. This fact is explained by partial cancelation between vertex correction diagrams and the self-consistency effects [40]. According to (10), exact quasiparticles spectra give precise total energies. For this reason the present study uses invariably the G 0 W 0 approximation, while the GW 0 is applied only for few simple nanoobjects.
The second involved question is connected with the satellite structure in A(E), which arises from dynamical interaction between electrons due to plasmon exchange. These plasmon satellites manifest themselves as the peaks of A(E) positioned at multiples of the plasmon energy below each quasiparticle level E i . An often-used method of satellite description is the cumulant expansion, in which the Green function for an occupied state i is taken as
G i (t) = iexp{−iE i t + C i (t)}, where E i
is the quasiparticle energy and C i (t) is the cumulant. This approach precisely reproduces plasmon satellites in the experimental spectra of electron photoemission [41], [42]. It is noticeable that the gravity center of the spectral density dEA(E) E taken over occupied states remains invariant to satellite formation [43]. This invariance implies an upward shift of quasiparticle energies, which balances the formation of low-energy satellite structure. When G 0 W 0 calculation is restricted to the quasiparticle states (ignoring satellites), this upward shift gives an illusion of decreasing cohesion [44].
To simply circumvent this difficulty, a model spectral function can be employed. The model describes an electron at the level i, which interacts with a plasmon having the plasma energy E pl .
The spectral function of this model can be done analytically, in terms of E pl and the renormalization factor Z i [26]. Following this research and assuming that the satellite series is infinite, the total energy correction to DFT can be given as:
E GM tot − E DFT tot = 1 2 i f i (E i − i ) − N E pl |lnZ| + drρ(r)V xc − E xc [ρ](11)
Here E i is the quasiparticle energy calculated by the G 0 W 0 , N is the total number of electrons in a nanocluster, and lnZ = Σ i f i · lnZ i /N , where 0 < Z i ≤ 1. In the homogeneous electron gas two values E pl and |lnZ| ≈ 1 − Z have opposite trends: with the growth of ρ the plasma energy increases as ρ 1 2 , while |lnZ| decreases, approaching zero at ρ → ∞. Because electron density averaged over a cluster varies only slightly from one low-energy configuration to other, we expect that the second term of (11) (the satellite contribution) is nearly invariant to atom rearrangements and affects very weakly the structures competition. This point is examined closer in Sections 3 and 4.
It is also of interest to examine correlation between nanoclusters' polarizabilities and energetics. Dielectric properties of the system reflect its interaction with external and internal electric field, electron screening, and affect system energy as well. According to the adiabatic connection fluctuation-dissipation theorem (ACFDT) [45,46,47] the correlation energy can be expressed as:
E C = − 1 0 dλ dω 2π T r{v[χ λ (iω) − χ 0 (iω)]}(12)
where λ is dimensionless coupling constant (λ=0 for the case of noninteraction electron system and λ=1 corresponds to the real physical system), χ is the electron response function, v denotes Coulomb interaction. It can be seen, that system correlation energy increases when χ λ , being negative, increases its absolute value. In the GWA SEO (6), screened Coulomb W can be rewritten as −1 (q, ω)×v(q) , where −1 (q, ω) is an inverse dielectric function, thus in GWA dielectric properties of the electron system explicitly affect computation results. According to these simple reasons it seems reasonable to expect, that for nanocluster isomers with defined chemical formula the structures with greater |χ λ (iω)| have, in general, higher total energy and thus they are energetically less favorable.
Although and χ functions depend on frequency, the static polarizability can be roughly considered as representing an approximate system dielectric response. Calculation of the static polarizability is implemented in many DFT packages and is much less time consuming than calculations on the GWA level. The consideration of the relationship between Si 7 and Si 10 H 2n isomers total energies and their static polarizabilities will be examined in Sections 4 and 5.
Computational methods
Our density functional calculations were performed with the DFT GGA xc functional using the Quantum Espresso (QE) [48] and VASP [49]- [52] codes. The QE calculations were made using PBE pseudopotential and a plane wave basis set having the cutoff energy of 50 Ry, while VASP's ones were done with the Projector Augmented Waves (PAW) basis set having the cutoff energy of 500 eV with appropriate pseudopotential [53,54]. Computations were performed for the supercell geometry with the vacuum layer of 13 Å between nanoobject replicas (section 4 argues this layer choice). The atomic structure of considered molecules and nanoclusters was found by the QE calculation, in the process of which the positions of atoms were relaxed until resulting atomic forces became less than 10 −4 Ry/Å. In the case of silicon clusters a nontrivial determination of cluster geometry was made with the evolutionary algorithm realized in the USPEX code [55,56], as has been described in our previous publications [11,12]. Both the Hartree-Fock and hybrid functional (PBE0 and B3LYP) calculations were performed using the QE code with the parameters described above. In this paper for the nanocluster isomers energy calculations using GM formula we will neglect plasmonic modifications in the spectral function. A(ω) will be considered as a number of the quasi- This estimation is very rough and precision of our calculations for the test cases will be examined in the next section.
In our work we also perform an analysis of the polarizabilities of the studied silicon nanoclusters.
Polarizability values α were calculated using VASP and averaged over directions. Polarizabilities are measured in the relative units, where the lowest polarizability among isomers with given formula is defined as 1. Since total energy of a system is a value determined up to a constant shift and we were interested only in the relative energies of the nanoobjects under study, further discussion will be concerned only with energy differences. All isomer energies will be counted from the ground state structure. Comparison of the BerkeleyGW schemes shows that self-consistency in the eigenvalues improves results, but G 0 W 0 results are also better than DFT and hybrid functional ones For the DFT nanocluster and molecular calculations, size of vacuum layer needed to converge is usually referred to 7-10 angstroms (for the systems with zero electrical charge) [61], [62]. Specified value of vacuum layer makes influence of the system's copies from other supercells negligible. Because of the dynamical nature of the xc interaction, GW should be sensitive to the induced dipole-dipole interactions or dispersion interaction. This interaction could be significant at distances of about 10 Å. Besides, in practical applications nanoclusters could be embedded in a matrix and form a periodic structure, where period value would affect system properties. It is of interest to study the dependence of the system energetics convergence (in particular within GW approach) on the vacuum layer and the supercell size. In case of the 10,5 Å supercell and larger, the structure with the isomer number 1 has the lowest energy for all numerical schemes. Situation dramatically changes for the 8,5 Å supercell case. DFT energy sequence remains almost similar and the structure 1 is still the lowest isomer. But in the GW schemes structure 2 becomes ground state isomer, with energy much lower than other isomers.
Calculations precision and the influence of the environment
System polarizabilities analysis shows, that structure 2 in the 8,5 Å supercell acquires α much less than other structures, that correlates with relative structure stability. Such behaviour can be associated with the increase in the interaction of the clusters in neighboring supercells. In the case of B3LYP for the 8,5 Å supercell, structure 2 is also low energy isomer close to the structure 1 (ground state one). PBE0 scheme mainly represents DFT results. GPP G 0 W 0 and frequency-dependent scGW 0 approaches show identical isomer energy ordering. For the supercells larger than 8,5 Å, isomers' polarizabilities have small differences and do not give noticeable contribution to the energy ordering of clusters. Table 2 shows that for the GW approach the increase in the supercell size leads not only to a monotonous convergence of the relative energies of the isomers. It could be seen, that structures value. This behavior can be explained as follows: for the Si 10 H 2n clusters with n < 9 an inner Si core can be localized; increasing n we get clusters of loose structure with no inner part (see figure 1). In case of Si 10 H 20 and Si 10 H 22 isomers the structure is branched, rather one-dimensional for each branch.
In the works [63,64] it was pointed out, that in small clusters microscopic dielectric properties at a few atomic distances away from the surface are almost identical to the bulk ones, whereas surface is one of the main factors that significantly change cluster polarizability. Thus Si 10 H 2n clusters with evident inner part show standard relation between system polarizability and energetics, while branched structures do not exhibit such obvious dependency.
Conclusions
DFT, hybrid functionals, Hartree-Fock, Galitskii-Migdal GW approximations were applied for the silicon-hydrogen nanoclusters' total energy computations. Precision of the methods was tested for the cases of Li 2 , N 2 molecule dissociation and ethyl-dimethyl ether isomer energy difference. GM GW
gives the most precise results of all methods examined, introducing a correction for DFT method, being, in turn, the starting point for the GW computation. GW energy calculations also demonstrate significantly higher sensitivity to the nanocluster environment, requiring vacuum layer to converge two times more than DFT. Moreover a non-monotonic dependence of the isomers energy distribution on supercell size was found.
Total energy calculations of the Si 7 and Si 10 H 2n isomers show, that correct account of the electron correlation effects is of great importance in the nanocluster systems with a big variety of structures close in energy. GWA demonstrates a notable change in the isomers' energy ordering and gives a correction to the energy of the order of tenths of eV for the isomers studied, comparing to other methods applied. Such correction will be significant for the energy ranking of the competitive structures up to a temperature of about a thousand degrees of Kelvin. It was established that standard GGA and hybrid functional methods may introduce noticeable errors into the total energy calculations for the nanoclusters consisting of tens of atoms. It is especially important for ground state structures prediction when isomers have energy differences of an order of tenths of eV or less.
It was found that, in general, the compact nanoclusters isomers with lower mean polarizability are more stable. In the branched, loose-packed structures such correlation vanishes. This provides enough reason to expect that the minimal polarizability principle can be valid criterion for isomer stability when energy ranking large nanoclusters systems, difficult for GW calculations.
Figure 1 :
1Si 10 , Si 10 H 16 , Si 10 H 22 ground state structures (upper line) and closest isomers (bottom line)
Coulomb interaction decreases very slowly with distance, approximately as e 2 /( 0 |r − r |), where 0 is the static dielectric constant. For this reason the use of local and semi-local approximations (the DFT LDA and GGA) is not well-justified for dielectric materials.
For the GWA calculations two packages were used: BerkeleyGW[57]-[59] and VASP. In both cases for all calculations the start point were DFT eigenfunctions and eigenvalues calculated in VASP for the VASP GW 0 and in QE for the case of BerkeleyGW G 0 W 0 and ev-scGW calculations.The VASP GW algorithm performs a direct inversion of the dielectric matrix on a frequency grid.The BerkeleyGW package can employ both the direct inversion of the dielectric matrix and the Generalized Plasmon Pole (GPP) model[32] (GPP accelerates calculations and reduces computer memory demands). The nonuniform frequency grid for the direct inversion of the dielectric matrix in the VASP computations consisted of 50 points and of 200 points for the BerkeleyGW, the dielectric matrix was cut off at 300 eV. In the BerkeleyGW GPP dielectric matrix was cut off at 6 Ry in the momentum space. When computing the self-energy operator of the GWA, we performed summation over all occupied and 600 unoccupied electron states.GW 0 and ev-scGW require notable extra resources, therefore these schemes were applied only for small nanoobjects -the Li 2 , N 2 , ethyl and dimethyl ether molecules. BerkeleyGW dielectric matrix direct inversion computations were applied only for the study of the plasmon satellites in the ethyl and dimethyl ether molecules and the Si 7 clusters. The GW calculations of Si 10 H 2n clusters were performed with the BerkeleyGW G 0 W 0 GPP approximation.
Figure 2 :W 0 Figure 3 :W 0 Figure 4 :
20304number 2 and 3 alternately change places on the energy scale when the supercell changes from 8.5 to 10.5 Å, from 10.5 to 13.2 Å, from 13.2 to 18.5 Å. Only for the supercell 18.5 Å this alternation stops and for 23.8 Å supercell it is possible to say that convergence is achieved. Such behavior indicates a complex nature of the decrease in screened Coulomb interaction with distance in nanoclusters.For the supercell more than 13.2 Å (about 9 Å of vacuum layer) supercell size change causes DFT energies deviations less than 0.2% and less than 1% for the PBE0 and B3LYP. GW relative energies show slower convergence, the convergence of energy at a level of accuracy within one percent requires an increase in vacuum layer up to 15 Å. The results show that hybrid functionals vacuum layer to convergence is half as much again DFT, GW requires vacuum layer that is twice as large as DFT.5 Electron correlation effects in Si 10 H 2n and isomer energy distribution Si 10 (left) and Si 10 H 6 (right) clusters' isomers relative energies in DFT and G 0 Si 10 H 16 (left) and Si 10 H 22 (right) clusters' isomers relative energies in DFT and G 0 Si 10 H 12 cluster isomers relative energies in DFT, PBE0, HF, B3LYP and G 0 W 0 .
Figure 5 :
5Si 10 H 20 cluster isomers relative energies in DFT, PBE0, HF, B3LYP and G 0 W 0 . The right graph for the G 0 W 0 results has different scale. new results. For the Si 10 H 6 and Si 10 H 12 isomers (figures 2 and 4) GW ground state structures also differ from DFT and hybrid functionals. In most cases the energy spread of the isomers in GWA is expanded comparing to DFT.
Figure 6
6presents values of the polarizability α and total energy from DFT and Galitskii-Migdal G 0 W 0 calculations. Each part on figure 6 presents results for the isomers with given formula: Si 10 , Si 10 H 6 , Si 10 H 12 , Si 10 H 16 , Si 10 H 20 and Si 10 H 22 . As can be seen, Si 10 , Si 10 H 6 , Si 10 H 12 and Si 10 H 16 clusters in GM GWA mainly have lower total energy (more stable) for the isomers with lower α. The DFT calculations do not exhibit such apparent energy-polarizability correlation. One can note violation of this rule for the Si 10 H 6 isomers: for the GW energy-polarizability curve structure with α=1 (point marked as 1) has energy of 0.3 eV higher than structure with α=1.012 (mark 2). Calculations of the Si 10 H 2n clusters' dipole moments show, that the structure marked as 1 has dipole moment of 1.2 atomic units, the largest value of all other clusters examined. Other clusters have dipole moments of 4-100 times less. Apparently such a large dipole moment affects the energy of the structure 1, decreasing its stability. The situation for the energy-polarizability correlation changes with the increase in the hydratation rate. For the Si 10 H 20 and Si 10 H 22 clusters there is no correlation between α and GM total energy
Figure 6 :
6Si 10 , Si 10 H 6 , Si 10 H 12 , Si 10 H 16 , Si 10 H 20 , Si 10 H 22 isomers' polarizability and total energy from DFT (rhombuses, solid line) and GM GWA (boxes, dashed line).
particle peaks. Our calculations show, that plasmon satellites carry about 15% of the valence spectral function weight in case of Si 7 isomers and about 10% in case of dimethyl ether molecule. Thus lnZ in mentioned cases is about 0.15 and 0.1 respectively (formula 11). For the nanocluster isomers of a given formula, E pl can be treated similar with high accuracy. In case of the Si 7 nanoclusters, shift of the satellites from QP peaks is the same for all isomers with a precision better than 2%. For the Si 10 H 2n isomers it means that plasmonic corrections to the relative energy should be less than 0.4 eV.
Table 1 Table 1 .
11presents comparison of the total energies from the experiment, DFT, hybrid functional, Hartree-Fock and GM GW calculations. First and second columns present data on Li 2 and N 2 molecules dissociation energy. Third column contains information on energy difference between two C 2 H 6 O molecule configurations (ethyl and dimethyl ether). By reason of the difficulties of the spin polarized BerkeleyGW computations and excessive memory requirements in the VASP GW, corresponding fields for the N 2 and Li 2 are respectively blank. Results for the experimental, DFT, Hartree-Fock, hybrid functional and GW total energy calculations. PBE, PBE0, B3LYP and H-F calculations were made in QE package; PAW and scGW 0 -in VASP; G 0 W 0 and ev-scGW -in BerkeleyGW. The data in table 1 show that obtained GW energies tend to modify DFT values towards the experimental ones, in case of Li 2 and N 2 dissociation this modification is little excessive. GW energies exhibit better agreement with experiment than DFT results, they are based on. Accurate calculations of the given systems require large vacuum layer, leading to the dramatic increase of the computation cost in case of the plane wave basis set. Thus the use of high parameters was restricted, especially to VASP GW 0 . PBE0 and B3LYP show rather bad energy values for the systems studied (and the worst results in case of H-F). For the Li 2 and N 2 dissociation PBE0 and B3LYP give significantly inaccurate values. In case of the ethyl-dimethyl ether molecules energy difference, PBE0 gives very good result,Li 2 dissociation en-
ergy, eV
N 2 dissociation en-
ergy, eV
ethyl -dimethyl
ether energy differ-
ence, eV
Experiment
1.03
9.8
0.526
PBE / PAW
1.375 / 1.304
10.15 / 10.24
0.493 / 0.475
PBE0
1.87
9.45
0.529
B3LYP
1.71
9.09
0.479
QE HF
2.64
4.88
0.474
G 0 W 0
0.697
0.504
ev-scGW / scGW 0
0.72 /
/ 9.71
0.512 / 0.56
while B3LYP error is noticeably bigger than DFT one. Thus hybrid functional in a few cases may
improve results comparing to DFT, but this seems rather occasional. Summarizing the results of
the table 1 we can say that the used GM GW methods give the values closest to experimental.
Table 2
2presents energy ordering for the first four Si 7 isomers inside 8.5, 10.5, 13.2, 18.5, 23.8 Å
Table 2 .
2PBE, B3LYP, PBE0, BekkeleyGW G 0 W 0 , VASP scGW 0
energies and polarizabilities in relative units for the first four Si 7
isomers in the 8.5 Å -23.8Å cubic supercells.
isomer number
1
2
3
4
8.5 Å cubic supercell
DFT, eV
0
0.17
0.412
0.416
PBE0, eV
0
0.214
0.436
0.441
B3LYP, eV
0
0.011
0.246
0.256
G 0 W 0 , eV
0.944
0
0.82
0.833
scGW 0 , eV
1.207
0
0.995
1.024
α
1.472
1
1.452
1.215
10.5 Å cubic supercell
DFT, eV
0
0.7481
0.9878
0.7865
PBE0, eV
0
0.7491
1.0426
0.7815
B3LYP, eV
0
0.5699
0.7484
0.6072
G 0 W 0 , eV
0
0.4669
0.4473
0.5452
13.2 Å cubic supercell
DFT, eV
0
0.8067
0.9898
0.8079
PBE0, eV
0
0.8006
1.0458
0.8057
B3LYP, eV
0
0.6252
0.7504
0.6311
G 0 W 0 , eV
0
0.6419
0.6854
0.75
18.5 Å cubic supercell
DFT, eV
0
0.8091
0.9904
0.8093
PBE0, eV
0
0.8134
1.0484
0.8148
B3LYP, eV
0
0.636
0.7525
0.6387
G 0 W 0 , eV
0
0.7406
0.7109
0.7769
23.8 Å cubic supercell
DFT, eV
0
0.8092
0.9904
0.8094
PBE0, eV
0
0.8159
1.0487
0.8162
B3LYP, eV
0
0.6381
0.7529
0.6399
G 0 W 0 , eV
0
0.745
0.7017
0.7747
Figures 2-5 demonstrate the relative position of energy levels for the Si 10 , Si 10 H 6 , Si 10 H 12 , Si 10 H 16 , Si 10 H 20 , Si 10 H 22 nanocluster isomers. The size of the vacuum layer was set to 13 angstroms. Graphs 2 and 3 present DFT PBE calculations (left parts) and BerkeleyGW G 0 W 0 calculations (right parts) for the Si 10 , Si 10 H 6 , Si 10 H 16 , Si 10 H 22 nanoclusters. Graphs 4 and 5 present DFT PBE, PBE0, B3LYP,Hartree-Fock and G 0 W 0 calculations for the Si 10 H 12 and Si 10 H 20 isomers. It could be seen, that in most cases GW isomer energies change their relative ordering comparing to the DFT results. PBE0 energies mainly represent DFT ones. B3LYP and Hartree-Fock calculations exhibit some differences from DFT but also give no results consistent to the GW (even on a qualitative level). We also made PBE0 calculations with different mixing constant α mix for the xc term (see formula 4). Computations with α mix varied from 0 to 1 just reproduce the results close to PBE or H-F and give no conceptually
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|
[] |
[
"Revealing a charge-density-wave gap in the predicted weak topological insulator HoSbTe",
"Revealing a charge-density-wave gap in the predicted weak topological insulator HoSbTe"
] |
[
"J L Liu \nCenter for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n",
"R Liu ",
"M Yang \nCenter for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n\nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nCenter of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n",
"L Y Cao \nCenter for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n",
"B X Gao \nCenter for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n",
"L Wang \nCenter for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n",
"A F Fang \nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n",
"Y G Shi \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nCenter of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n",
"Z P Yin \nCenter for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n",
"R Y Chen \nCenter for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n"
] |
[
"Center for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina",
"Center for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina",
"Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"Center of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Center for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina",
"Center for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina",
"Center for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina",
"Department of Physics\nBeijing Normal University\n100875BeijingChina",
"Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"Center of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina",
"Center for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina",
"Center for Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina"
] |
[] |
HoSbTe was predicted to be a weak topological insulator, whose spin-orbit coupling (SOC) gaps are reported to be as large as hundreds of meV. Utilizing infrared spectroscopy, we find that the compound is of metallic nature from 350 K down to 10 K. Particularly, both of its itinerant carrier density and scattering rate are demonstrated to decrease with temperature cooling, which is responsible for the appearance of a broad hump feature in the temperature dependent resistivity around 200 K. More importantly, we reveal the appearance of a chargedensity-wave (CDW) gap in addition to the SOC related gap. The energy scale of the CDW gap is identified to be 364 meV at 10 K, which shift to 252 meV at 350 K. The coexistence of CDW and SOC gaps in the same compound paves a new avenue to explore more intriguing physics. arXiv:2110.02184v1 [cond-mat.str-el] 5 Oct 2021
|
10.1103/physrevb.105.075111
|
[
"https://arxiv.org/pdf/2110.02184v1.pdf"
] | 238,354,082 |
2110.02184
|
9398b2dcc6d6150bc12cb377e45878ec11387186
|
Revealing a charge-density-wave gap in the predicted weak topological insulator HoSbTe
J L Liu
Center for Advanced Quantum Studies
Department of Physics
Beijing Normal University
100875BeijingChina
R Liu
M Yang
Center for Advanced Quantum Studies
Department of Physics
Beijing Normal University
100875BeijingChina
Beijing National Laboratory for Condensed Matter Physics
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
Center of Materials Science and Optoelectronics Engineering
University of Chinese Academy of Sciences
100049BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
L Y Cao
Center for Advanced Quantum Studies
Department of Physics
Beijing Normal University
100875BeijingChina
B X Gao
Center for Advanced Quantum Studies
Department of Physics
Beijing Normal University
100875BeijingChina
L Wang
Center for Advanced Quantum Studies
Department of Physics
Beijing Normal University
100875BeijingChina
A F Fang
Department of Physics
Beijing Normal University
100875BeijingChina
Y G Shi
Beijing National Laboratory for Condensed Matter Physics
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
Center of Materials Science and Optoelectronics Engineering
University of Chinese Academy of Sciences
100049BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
Z P Yin
Center for Advanced Quantum Studies
Department of Physics
Beijing Normal University
100875BeijingChina
R Y Chen
Center for Advanced Quantum Studies
Department of Physics
Beijing Normal University
100875BeijingChina
Revealing a charge-density-wave gap in the predicted weak topological insulator HoSbTe
HoSbTe was predicted to be a weak topological insulator, whose spin-orbit coupling (SOC) gaps are reported to be as large as hundreds of meV. Utilizing infrared spectroscopy, we find that the compound is of metallic nature from 350 K down to 10 K. Particularly, both of its itinerant carrier density and scattering rate are demonstrated to decrease with temperature cooling, which is responsible for the appearance of a broad hump feature in the temperature dependent resistivity around 200 K. More importantly, we reveal the appearance of a chargedensity-wave (CDW) gap in addition to the SOC related gap. The energy scale of the CDW gap is identified to be 364 meV at 10 K, which shift to 252 meV at 350 K. The coexistence of CDW and SOC gaps in the same compound paves a new avenue to explore more intriguing physics. arXiv:2110.02184v1 [cond-mat.str-el] 5 Oct 2021
INTRODUCTION
Topological materials with nontrivial electronic bands have drawn tremendous interests in condensed matter physics due to their exotic physical properties and their potential applications [1][2][3]. Huge progress has been made in the field and a large number of topological materials have been recognized. However topological materials with strong electronic correlation are much less explored and might host more interesting physics [4][5][6][7][8][9]. The WHM (W=Zr, Hf, or Lanthanides, H=Si, Ge, Sn, or Sb, and M=O, S, Se, or Te) family, initially noticed by theorist in the search for two-dimensional topological insulators [10], can serve as a good platform to study the interactions between topology, magnetism and other emergent instabilities.
Most of the WHM compounds are reported to be Dirac nodal line semimetals [11][12][13][14][15][16][17] or weak topological insulators [10,18]. However when W are Lanthanides with f electrons, magnetic orderings and Kondo effect could play a role in the system. For example, LaSbTe was suggested to be a topological insulator [19] or nodal-line semimetal [13] by different experiments. Its counterpart CeSbTe with f electrons from Ce is found to be a low-carrier-density Kondo semimetal with a Kondo temperature of 10 K [20], and its electronic structure could be severely modified by long range magnetic orders [6]. Meanwhile GdSbTe is demonstrated to be a nodalline semimetal with a robust Dirac-like band structure across the AFM transition at 13 K [21].
Among these WHM materials HoSbTe was theoretically predicted to be a weak topological insulator [22], which is corroborated afterwards by angle-resolved photoemission spectroscopy (ARPES) experiments exhibiting energy gaps much larger than 200 meV along certain momentum directions [23]. However the temperature dependent resistivity of HoSbTe exhibits a broad hump-like feature at around 200 K, indicating a bad-metal-like state at low temperatures, the underlying physics of which is yet to be revealed [22]. The 4 f electrons of Ho atoms are believed to be generally localized and far away from the Fermi level, but an extremely large Sommerfeld coefficient γ ∼ 382.2 mJ/mol −1 /K 2 is obtained by a specific heat measurement [22], which is much larger than that of CeSbTe (γ = 41 mJ/mol −1 /K 2 ) [20], and comparable to many typical heavy fermion materials like CeCoIn5 [24]. This seems to infer a very large Kondo hybridization strength and thus the itinerancy of the f electrons in HoSbTe.
In order to resolve the existing enigma and further investigate the role played by 4 f electrons, we performed infrared spectroscopy measurements on single crystalline HoS-bTe samples. Our results suggest that the hump feature in resistivity could be well explained by the competing effect of carrier density and scattering rate. In addition to the expected SOC gap we observe for the first time another temperature dependent gap whose value varies from 364 meV at 10K to 252 meV at 250K. We propose that this newly observed gap is caused by CDW ordering similar to those in CeSbTe [25].
EXPERIMENTAL
Single crystalline HoSbTe samples were synthesized by Sbflux method [22]. The temperature dependent resistivity ρ(T ) and magnetization χ(T ) up to 350 K were measured in a Quantum Design physical property measurement system. Infrared spectroscopic studies were performed with a Bruker IFS 80 V in the frequency range from 30 to 50 000 cm −1 , on the as-grown shiny surfaces of HoSbTe which is the ab plane of this quasi-two dimensional compound. In the measurement of frequency dependent reflectivity R(ω), either gold or aluminum coating techniques are adopted in order to eliminate the impact of microscopic surface textures of the single crystalline compound. The real part of the optical conductivity σ 1 (ω) are derived through Kramers-Kronig transformation of the reflectivity R(ω), which is extrapolated by a Hagen-Rubens relation to zero at the low frequency end, and by ω −1.5 from 50 000 cm −1 to 800 000 cm −1 , and then by ω −4 for frequency higher than 800 000 cm −1 .
Theoretical calculations were carried out using the linearized-augmented plane-wave method implemented in WIEN2K [26].
The Perdew-Burke-Ernzerhof version of generalized-gradient-approximation (GGA) to the exchange correlation functional [27] was used. To move the f orbitals of Ho atoms away from Fermi level, GGA+U method was adopted in our calculations, and we applied an effective Hubbard U e f f = U − J H = 7 eV to the Ho f orbitals. Because the AFM transition temperature of HoSbTe (T N ∼ 4 K) is very low, we ignore the magnetic order and force the calculations to non-magnetic solution. In self-consistent calculation we used 27×27×12 k-mesh grid, and in order to clearly show Fermi surfaces sections, 500×500 k-mesh grid was used in each specified kz plane. Spin-orbit coupling (SOC) was taken into account in our calculation. Fig. 1(a) and (b) display the temperature dependent resistivity ρ(T ) and magnetization χ(T ), respectively. Although we have measured ρ(T ) and χ(T ) up to a higher temperature, there is not much difference with the previous report [22]: ρ(T ) shows a broad hump around 200 K and χ(T ) follows the Curie-Wiess law down to at least 100 K; an antiferromagnetic phase transition occurs at around 4 K. In order to check if there are any other phase transitions that may lead to the "insulator to metal" change inferred by resistivity, the first-order derivatives dρ(T )/dT and dχ(T )/dT are calculated and displayed in the inset of Fig. 1(a) and (b), respectively. The smooth deriva-tive curves at relatively high temperatures obviously do not support existence of such transitions.
RESULTS AND DISCUSSION
The frequency dependent reflectivity R(ω) at several temperatures is shown in Fig. 2 (a). In the far-infrared (IR) region, metallic responses in the whole temperature range are clearly evidenced by the increasing of R(ω) with decreasing energy and its approaching to unity at zero frequency. Although HoS-bTe was predicted to be a weak topological insulator, it could still be a metal since the Fermi level may not lie in the topological gaps and may pass through other trivial energy bands. A diamond-shaped Fermi surface (FS) around Γ is indeed observed by ARPES experiments on HoSbTe [23]. However this is somewhat inconsistent with the insulating behavior of ρ(T ) above 200 K, which will be discussed in the following.
In the mid-IR region, a platform-like feature can be clearly seen in R(ω) at 350 K, which is gradually suppressed at lower temperatures and ultimately replaced by two dip structures as indicated by the red arrows in Fig. 2(a). Such kind of low energy absorptions in R(ω) are usually linked to the loss of density of state at the Fermi level. The growing spectral weights of the two dips with temperature cooling further imply that they might be related to some kind of gap openings. For even higher frequencies, R(ω) decreases with increasing frequency up to 18 000 cm −1 and exhibits only mild temperature dependence. A few weak humps could be resolved in this frequency range.
The evolution of the electronic structure could be more straightforwardly reflected by the optical conductivity σ 1 (ω), which is directly related to the joint density of states for electron transitions from occupied to unoccupied states. As displayed in Fig. 2(b), the direct current value of σ 1 (ω → 0) first reduces slightly with decreasing temperature above 250K, then increases rapidly down to 10 K, roughly matching with the result of ρ(T ). The presence of the Drude components centered at zero frequency is another evidence of the metallic nature of HoSbTe. In correspondence with the dip structures in R(ω), two extremely pronounced peaks could be identified in the optical conductivity σ(ω) at 10 K, where a crossover can be observed at around 2400 cm −1 between the two peaks. As temperature increases, the two individual peaks tend to get closer to each other and seem to turn into a very broad single peak above room temperature. Considering the two-dip structure in R(ω) can persist up to 300 K, we believe the broad peak in σ 1 (ω) is actually composed by two independent peaks.
To better analyze the evolvement of the two peaks, we plot the renormalized optical conductivity spectra in Fig.2(c). It is clearly shown that the energy scale of both peaks shift to higher frequency as temperature decreasing. Supposing they are originated from SOC gaps as revealed by the ARPES experiments or trivial interband transitions, the spectral weight of them should undergo a subtle decrease upon temperature cooling, due to the reduction of thermal excitations. While the lower energy peak is slightly suppressed as expected, the higher energy one gets much more pronounced when temperature decreases, suggesting a totally different origination. We further present the spectral weight SW= a function of frequency and temperature in Fig.2(d). At the low frequencies end, the SW increases as temperature cooling due to the narrowing of the Drude component. Then the higher temperature SW increases and surpasses the lower temperature value roughly above 700 cm −1 . At even higher energies, SW at different temperatures almost merge together above 5000 cm −1 , complying with the conservation law. Additionally, the renormalized SW reaches a minimum at around 1200 cm −1 at 10 K, as can be seen in the inset of Fig.2(d), indicating the SW transfer from frequencies lower than 1200 cm −1 to higher energy region. Notably, these behaviors are similar with some CDW materials, such as CeTe 3 [28].
In order to unravel the individual mechanism of the two peaks, we performed first-principals calculations on the optical conductivity σ 1−cal (ω) and band structure of HoSbTe, as shown in Fig.3(b) and (d) respectively. Surprisingly, there are some distinct differences between the calculated and experimental optical conductivity: the overall calculated spectra are much higher than the experimental data, partially due to the high-energy extrapolation of R(ω); σ 1−cal (ω) only shows a single peak in the mid-IR region at around 1450 cm −1 , which corresponds to interband transition across an SOC associated gap, as indicated by the light green arrow in Fig.3(d). These disagreements clearly indicate a yet unknown reconstruction of the electronic structure. Nevertheless, as the lowest energy interband transition, the SOC gap in σ 1−cal (ω) locates very close to the lower energy peak in the experimental data, pointing to a similar origination.
We further analyze the optical conductivity quantitatively by decomposing it into several different terms according to the Drude-Lorentz model:
σ 1 (ω) = ω 2 p 4π γ D ω 2 + γ 2 D + j S 2 j 4π γ j ω 2 (ω 2 j − ω 2 ) 2 + ω 2 γ 2 j ,(1)
where ω p and γ D are the plasma frequency and scattering rate of the itinerant carriers, while ω j , γ j and S j are the resonant frequency, damping and strength of the jth Lorentz oscillators, respectively. The first term at the right side of equation (1) is the Drude component modeling intraband transitions of free electrons while the second Lorentz term describes interband transitions across energy gaps. A presentative example of the fitting results of σ 1 (ω) at 10 K are plotted in Fig.4, where one Durde and seven Lorentz terms are employed. Part of the fitting parameters are listed in Table I. Among the fitting parameters, the plasma frequency ω p is related to the itinerant carrier density n and effective mass m * by ω 2 p = 4πne 2 /m * . The variation of ω p with temperature is shown in Fig.5 (a), where ω p decreases monotonically from 19 175 cm −1 at 300 K to 11 311 cm −1 at 10 K, indicating either a continuous losing of the free carrier density or a constant enhancement of the effective mass. It is worth to remark that the data of 350 K seems to exhibit a deviation from the above mentioned trend. This is because the Drude pattern of σ 1 (ω) at 350 K is very broad and highly overlap with the first Lorentz term. Therefore, an extremely huge error bar is generated in the fitting procedure. The temperature dependent (2) 269 1647 1276(2) 19455(16) 2934 2173(1) 24967 100K 12417(2) 305 1492 1224(1) 19830(14) 2700 2183(1) 24163(14) 200K 12983 (2) 455 1405 1504(1) 21539(15) 2523 2177(1) 22211(15) 250K 14698 (7) 752 1363 1465(2) 18848(23) scattering rate γ D , which is the width at half maximum of the Drude spectra, is plotted in Fig.5 (a) as well. Notably, γ D evolves in a very similar fashion with ω p . It is well known that the electrical conductivity of a material is proportional to its density of free carriers n and relaxation time τ = 1/γ D . Here for HoSbTe, τ obviously increases with lowering temperature, in favor of a diminishing resistivity. As a consequence, how the free carrier density n evolve with temperature is crucial to explain the hump feature in ρ(T ), which will be further elaborated later.
T ω P γ D ω 1 γ 1 S 1 ω 2 γ 2 S 2 10K 11311
The two prominent peaks of σ 1 (ω) in the mid-IR region could be well fitted by two Lorentz oscillators, corresponding to transitions across two different gaps. The obtained resonant frequencies, as displayed in Fig.5 (b), can be considered as the excitation gap energies, which are identified to be 2∆ 1 204 meV and 2∆ 2 364 meV at 10 K, respectively. In a previous report [23], ARPES measurements have revealed SOC gaps as large as hundreds of meV, which seems to be consistent with both 2∆ 1 and 2∆ 2 . As discussed above, although this SOC scenario or trivial interband transitions can work for the lower energy gap, they can not explain the drastic decrease of 2∆ 2 to 252 meV at 350 K and the increase of its SW at lower temperatures. Taking the closeness between the calculated SOC gap and 2∆ 1 into consideration, we propose this lower energy gap to be an SOC gap, while the higher energy gap requires an alternative mechanism which will be discussed in the following.
In consideration of the exceptional large Sommerfeld coefficient γ [22], there is a possibility that the higher energy gap arises from the Kondo effect, in which case the interaction between located 4 f electrons and conduction bands would generate a mid-IR peak. The Drude spectral weight as well as the plasma frequency ω p will be severely suppressed owing to the enhancement of effective mass m * . This is in excellent agrement with our spectroscopic results. Even the drop of ρ(T ) below 200 K could be well justified because the initial localized f electrons will be enclosed by the Fermi surface and contribute carrier density as a result of Kondo screening, which reduces the scattering rate of itinerant carriers at the same time. However, even if we adopt a constant carrier density, the enhancement of m * from 350 K to 10 K is estimated to be only about m * 10K /m * 350K = ω 2 p(350K) /ω 2 p(10K) 2.55, which disagrees violently with the huge Sommerfeld coefficient γ = 382.2 mJ/mol −1 /K 2 [22]. Besides, the sustaining of 2∆ 2 up to 350 K would indicate an even higher Kondo temperature T K , below which the magnetic susceptibly is supposed to deviate from the Curie-Weiss law along with a downward bending resistivity. Such characters is obviously absent in our measurements. Hence we have ruled out the existence of Kondo effect in this system .
Another option for 2∆ 2 is a density-wave gap, since the transfer of spectral weight from the free electron Drude response to higher energy peaks agrees perfectly with the electrodynamic prediction of density-wave orderings with a case-I coherent factor. Notably, a charge density wave gap about 0.3 eV is observed in the Dirac semimetal CeSbTe [25], driven by electron-phonon coupling due to FS nesting. The FSs of CeS-bTe are constituted by a small enclosed hole pocket at Γ and two 2D rhombus-shaped sheets, the latter of which contain largely parallel segments that could be connected by a single wave vector, in favor of FS nesting and thus the development of density wave orderings. For comparison, we plot the FS section of HoSbTe at several specified kz planes, as shown in Fig.6. It also contains two rhombus-shape sheets at kz=0, but the two sheets are closer to each other and disconnected along the Γ − M direction due to SOC gaps. When kz increases, they gradually shrinks and eventually disappears as the SOC gaps shift closer to the Fermi level. As a result, the two sheets are indistinguishable within the energy resolution limitation of ARPES measurements [23]. In reality, however, it is most likely the nearly perfect nesting of these two sheets induces a density wave instability, and the corresponding single particle energy gap is captured by our infrared spectroscopy. As there are no additional signatures in support of spin-density-wave instabilities, like AFM orderings above 10 K, we speculate that 2∆ 2 is actually a charge density wave gap. Since the two rhombus-shape are so close to each other, an extremely small nesting wave vector is expected. Assuming that m * is temperature independent and equal to the electron rest mass m e , n is estimated to be about 3.62 × 10 21 cm −3 at 350 K, which reduce to 1.42 × 10 21 cm −3 at 10 K, meaning a 60.8% loss of the FS due to the gap enlargement. The residual FS at the lowest temperature manifests that the CDW gaps are partially opened. As addressed above, the scattering rate γ D also decreases with temperature lowering. Consequently, the broad hump feature appeared in ρ(T ) is probably caused by the competition of the simultaneous reduction of n and γ D , which contribute oppositely to the resistivity. This phenomena is also observed in the three dimensional Dirac semimtal ZrTe 5 [29].
Although we can principally ascertain the occurrence of a CDW ordering in HoSbTe, we have not been able to determine its phase transition temperature and ordering wave vector from our current results. The large gap energies at 350 K suggest that the CDW transition takes place at a much higher temperature, which is inaccessible for our current measurement. In the weak coupling mean field theory framework, the gap energy 2∆ = 3.52k B T C for strict 1D systems, where T C is the phase transition temperature. By this relation the CDW T C of HoSbTe would be over 2000 K with gap value 2∆ 2 = 364 meV. However T C will be severely suppressed when the effect of fluctuations is taken into account, especially in strongly anisotropic materials. Therefore the 2∆/k B T C ratio is typically much larger than 3.52. For example, another Ho-based material HoTe 3 undergoes a CDW transition at 288 K with an energy gap of 380 meV [30], very close to the energy scale of 2∆ 2 . We therefore conjecture that the transition temperature of HoSbTe is probably of the same order of magnitude with HoTe 3 .
Finally we would like to compare HoSbTe with LaSbTe and CeSbTe. Although all of the three compounds bear strikingly similar FSs [13,23,25], their physical properties are much different from each other. Our results on HoSbTe and previous reports on CeSbTe demonstrate that these two materials have CDW instabilities, while LaSbTe is reported to be a nodal line semimetal [13] with no CDW observed by so far. For CeSbTe the CDW gap is estimated to be about 0.3 eV with a transition temperature above 338 K and a weak Kondo effect is also identified [25]. We observe a CDW gap of about 364 meV in HoSbTe with no Kondo effect. This rich phase diagram supplies an ideal platform to investigate the influence of chemical pressure on the CDW ordering and its interplay with Kondo effect by substituting LnSbTe with different lanthanide elements.
CONCLUSION
In summary, we have investigated the charge carrier dynamics of a predicted topological insulator compound HoS-bTe. The temperature dependent reflectivity and optical conductivity both exhibit metallic behaviors from 350 K to 10 K, with a decreasing carrier density and scattering rate, the competing contribution of which gives rise to the hump feature in resistivity. We identify an expected SOC gap of around 204 meV, and an additional temperature dependent gap whose value shifts from 364 meV at 10 K to 252 meV at 350 K. This enormous change along with its sizable increase of spectral weight upon cooling are clearly incompatible with SOC band gaps or trivial interband transitions. As Kondo hybridization is also excluded as the origin of this gap, due to a minor enhancement of the effective mass, we ultimately propose it to stem from a CDW ordering. This complex system together with its LnSbTe cousins provide a promising arena to explore the entanglement of CDW, Kondo effect and topological orders.
FIG. 1 .
1The in-plane (a) resistivity and (b) magnetic susceptibility as a function of temperature. The magnetic susceptibility is measured with H = 1000 Oe. The first order derivatives of resistivity and magnetic susceptibility are displayed in the inset of (a) and (b), respectively.
FIG. 2 .
2The frequency dependent (a) reflectivity R(ω) and (b) optical conductivity σ 1 (ω) at several temperatures up to 10 000 cm −1 . The two insets show R(ω) and σ 1 (ω) in an expanded range up to 50 000 cm −1 , respectively. (c) The renormalized optical conductivity as a function of frequency up to 5000 cm −1 . (d ) The spectral weight as a function of frequency. The inset shows renormalized SW at different temperatures.
FIG. 4 .
4structure of HoSbTe. A unit cell is indicated by black lines. (b) The calculated conductivity σ 1−cal and experimental data at 10 K. (c) Brillouin zone of HoSbTe. (d) The calculated bulk band structure with orbital characters for HoSbTe. The light green arrow marks the optical interband transition that give rise to the mid-IR peak in the optical conductivity. Te 5p, Sb 5p and Ho 4 f orbital characters are presented by red, blue, and green circle, respectively. The Drude-Lorentz model fitting result of optical conductivity σ 1 (ω) at 10 K, where one Drude term and seven Lorentz terms are used.
FIG. 5 .
5I. The fitting parameters at different temperatures. ω P is the plasma frequency and γ D =1/τ D is the scattering rate of the Drude term. The center frequency ω j , width γ j =1/τ j and square root of the oscillator strength S j of the first two Lorentz components are listed. The error bars of all fitting parameters are included. (a) The temperature dependent plasma frequency ω p and scattering rate γ D . (b) The energy scales of the two gaps as a function of temperature.
FIG. 6 .
6Fermi surface sections in (a) kz = 0, (b) kz = π/3c, (c) kz = 2π/3c and (d) kz = π/c plane of HoSbTe. Red, green and blue curves represent Fermi surfaces formed by different bands which are divided by the order of energy. The black lines mark the boundaries of Brillouin zone.
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|
[] |
[
"Linear Network Coding for Two-Unicast-Z Networks: A Commutative Algebraic Perspective and Fundamental Limits",
"Linear Network Coding for Two-Unicast-Z Networks: A Commutative Algebraic Perspective and Fundamental Limits"
] |
[
"Mohammad Fahim [email protected] \nDepartment of Electrical Engineering\nPennsylvania State University\n\n",
"Viveck R Cadambe [email protected]. \nDepartment of Electrical Engineering\nPennsylvania State University\n\n"
] |
[
"Department of Electrical Engineering\nPennsylvania State University\n",
"Department of Electrical Engineering\nPennsylvania State University\n"
] |
[] |
We consider a two-unicast-Z network over a directed acyclic graph of unit capacitated edges; the two-unicast-Z network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted (interfering) message. In this paper, we settle open questions on the limits of network coding for two-unicast-Z networks by showing that the generalized network sharing bound is not tight, vector linear codes outperform scalar linear codes, and non-linear codes outperform linear codes in general. We also develop a commutative algebraic approach to deriving linear network coding achievability results, and demonstrate our approach by providing an alternate proof to the previous result of Wang et. al. regarding feasibility of rate (1, 1) in the network.
|
10.1109/isit.2017.8006714
|
[
"https://arxiv.org/pdf/1705.02704v1.pdf"
] | 7,085,858 |
1705.02704
|
b4252308740abfc23b3ab9acbbe14b9d0c0217fa
|
Linear Network Coding for Two-Unicast-Z Networks: A Commutative Algebraic Perspective and Fundamental Limits
Mohammad Fahim [email protected]
Department of Electrical Engineering
Pennsylvania State University
Viveck R Cadambe [email protected].
Department of Electrical Engineering
Pennsylvania State University
Linear Network Coding for Two-Unicast-Z Networks: A Commutative Algebraic Perspective and Fundamental Limits
1
We consider a two-unicast-Z network over a directed acyclic graph of unit capacitated edges; the two-unicast-Z network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted (interfering) message. In this paper, we settle open questions on the limits of network coding for two-unicast-Z networks by showing that the generalized network sharing bound is not tight, vector linear codes outperform scalar linear codes, and non-linear codes outperform linear codes in general. We also develop a commutative algebraic approach to deriving linear network coding achievability results, and demonstrate our approach by providing an alternate proof to the previous result of Wang et. al. regarding feasibility of rate (1, 1) in the network.
I. INTRODUCTION
There is significant interest in multiple unicast network coding and index coding in recent times. In addition to capturing the essence of network communication, there are interesting connections between special instances of the multiple unicast network communication problem and several emerging applications including topological interference management in wireless networks [6], codes for caching and content distribution [9] and regenerating and locally recoverable codes for distributed storage [1], [10]. In this paper, we study the most simple multiple unicast communication scenario, in terms of message structure, whose capacity is unknown: the two-unicast-Z network.
The two-unicast-Z network, like the two-unicast network, has two independent message sources and two destinations, each destination respectively requiring to decode one of the two message sources. One of the two destinations, say the second destination, has apriori side information of the unintended (first) message source (See Fig. 1). Like the Z-interference channel in wireless communications, the two sources of the network interfere at only one destination. The study of twounicast-Z networks is important, because, like index coding and other simplified variants, insights obtained through code development for two-unicast-Z networks can potentially influence code design for more general multiple unicast networks and its related applications. A low-complexity linear network coding algorithm for two-unicast-Z networks has been developed in [13].
It is shown in [12] that the generalized network sharing bound is tight for a specific class of two-unicast-Z networks. However, unlike the two-unicast network [7] where (a) linear network coding is insufficient for capacity, (b) vector linear codes outperform scalar linear codes, and (c) the generalized network sharing (GNS) cut set bound is not tight in general, it is not known whether non-linear network coding, vector linear codes, or bounds stronger than the GNS bound are required to characterize capacity for the two-unicast-Z network. In particular, because twounicast-Z networks are a special case of two-unicast networks, the results of two-unicast networks do not naturally extend to two-unicast-Z networks. In this paper, we resolve these open questions for two-unicast-Z networks. In particular, we show that for two-unicast-Z networks, vector linear codes outperform scalar linear codes, non-linear codes outperform linear codes, and that the GNS bound is not tight. A second contribution of this paper is the development of a commutative algebraic perspective of linear network coding. We describe our perspective through an alternate proof, for two-unicast-Z networks, of the result of [11], which establishes the feasibility rate (1, 1) for two-unicast networks. In particular, the references show that rate tuple (1,1) is achievable if and only if the generalized network sharing cut set bound [7] is at least 2, and the individual source destination pairs have a cut of at least 1. Our alternate proof, however, encompasses new ideas and methods. Our starting point is the algebraic framework of network coding [8], where scalar linear solvability over a network was cast as a polynomial solvability problem. We develop a network decomposition framework (Sec. III) and combine this framework with tools from commutative algebra (Sec. II) to derive the achievability proof. We present our achievability proofs in Sections IV and V. Proofs of our impossibility results describing the limits of linear network coding are provided in Sections VI and VII.
II. SYSTEM MODEL Consider a directed acyclic graph (DAG) G = (V, E), where V denotes the set of vertices and E denotes the set of edges. We allow multiple edges between vertices, hence, E ⊂ V × V × Z + , where Z + denotes the set of positive integers. For an edge e = (u, v, i) ∈ E, we denote Head(e) = v and Tail(e) = u. For a given vertex v ∈ V, we denote In(v) = {e ∈ E : Head(e) = v} and Out(v) = {e ∈ E : Tail(e) = v}.
A path p is a sequence of edges (e m1 , e m2 , . . . , e ml ) where Head(e mi ) = Tail(e mi+1 ) for i = 1, 2, . . . , l − 1. Let E 1 , E 2 , E 3 ⊆ E. We denote E 1 → E 2 if, for some e i ∈ E i , i = 1, 2, there is a path from e 1 to e 2 . We also denote E 1 → E 2 \E 3 if for some e i ∈ E i , i = 1, 2, there is a path from e 1 to e 2 that does not contain any of the edges in E 3 . If E 1 = {e 1 }, E 2 = {e 2 }, E 3 = {e 3 } are singletons, then we simply write e 1 → e 2 or e 1 → e 2 \e 3 as the case may be. Because G is a DAG, there is a topological ordering Ord : E → Z + on the edges of the graph with the property that e 1 → e 2 ⇒ Ord(e 1 ) < Ord(e 2 ).
Algebraic Framework for Linear Network Coding: We set up scalar linear network coding schemes based on the algebraic framework of [8]. Let K denote the algebraic closure of the field F 2 . Let e 1 , e 2 , . . . , e |E| denote the edges of E in topological order, i.e., i < j ⇔ Ord(e i ) < Ord(e j ) . The local coding matrix is an upper triangular matrix F G whose element in the i-th row and j-th column F G i,j is given as,
where β ei,ej is a free variable. Wherever the graph G being considered is clear, we will simply omit the superscript and simply express the local coding matrix as F. We denote the set whose elements are the (non-zero) entries of F asF. We denote the polynomial ring with field K and variables being the (non-zero) entries of F as K [F]. A linear network code is specified by a |E| × |E| matrix F * with entries in K. The network extended transfer matrix, for this specific network code, is simply obtained by evaluating the corresponding polynomials, H(F * ). For a path p = (e m1 , e m2 , . . . , e ml ), the weight of the path is a function that maps the path to an element of K[F] defined as w(p) = l−1 i=1 β em i ,em i+1 . For two edges e i , e j , let P(e i , e j ) denote the set of all paths from e i to e j . Let H i,j (F) = p∈P(ei,ej) w(p). The network extended transfer matrix H(F) is a |E| × |E| matrix whose element in the i-th row and the j-th column is H i,j . Note that every element of H(F) lies in the polynomial ring K [F]. It can be shown that H(F) = (I − F) −1 , where I is the |E| × |E| identity matrix in K [8].
Two-Unicast-Z Network: We depict a two-unicast-Z network in Fig. 1.
Definition 2.1 (Two-Unicast-Z Network): A (G, S 1 , T 1 , S 2 , T 2 ) two unicast-Z network consists of a graph G = (V, E), and sets S 1 , S 2 , T 1 , T 2 ⊆ E. For i ∈ {1,
2} the sets S i , T i are respectively referred to the edges of the i-th source and destination respectively. Consider the scalar linear network coding fraomework for a two-unicast-Z network with sources S i , i = 1, 2 and destinations T i , i = 1, 2. For source edge set S i , i = 1, 2 and destination edge set T j , j = 1, 2, the transfer matrix H i,j is a |S i |×|T j | matrix with entries in K(F) whose rows (columns) are the rows (columns) of H corresponding rows corresponding to S i (T j ). We denote the |S i | × |T j | transfer matrix from source i to destination j as G i,j (F).
We now define the notion of achievability of rate (m, n). It is worth noting that for a scalar linear achievable coding scheme, there is no loss in generality in assuming that |S 1 | = |T 1 | = m and |S 2 | = |T 2 | = n. To understand our definition of achievability of (m, n) it is useful to imagine source vectors X 1 , X 2 with entries in K of dimensions 1 × m and 1 × n respectively. The goal of a network coding scheme is to convey these vectors to their respective desintations. For a specific linear network coding scheme with local coding co-efficients F * ∈ K |E|×|E| , the vectors Y 1 , Y 2 received respectively by the two receivers in a two-unicast-Z network can be written as
Y 1 = X 1 G 1,1 (F * ) + X 2 G 2,1 (F * ) (2) Y 2 = X 2 G 2,2 (F * ).(3)
We have not written the effect of X 1 at receiver 2, since the receiver can subtract the effect of X 1 from the side information that it posses. We refer to the linear coding scheme F * as an achievable scheme if X 1 , X 2 are recoverable from Y 1 , Y 2 respectively. For successful recovery of the two sources from the respective destinations, we require
det (G i,i (F * )) = 0, i = 1, 2, G 2,1 (F * ) = 0 m×n(4)
Note that the above conditions are necessary and sufficient since we restricted
|S 1 | = |T 1 | = m, |S 2 | = |T 2 | = n.
We now define our notion of achievability formally. Definition 2.2 (Achievability of rate (m, n)): In a (G, S 1 , T 1 , S 2 , T 2 ) two-unicast-Z network with |S 1 | = |T 1 | = m, |S 2 | = |T 2 | = n, the rate (m, n) is said to be achievable via scalar linear network coding, if there exists a linear network coding scheme F * ∈ K |E|×|E| such that (4) holds.
In Sec. III, we describe the notion of achievability in the language of commutative algebra. The main topic of discussion of this paper is the achievability of rate (1,1). For the remainder of this document, we will simply assume that m = 1, n = 1.
Remark 2.1 (Notation): In the remainder of this paper, we consider a two-unicast-Z network with S 1 = {s 1 }, T 1 = {t 1 }, S 2 = {s 2 } and T 2 = {t 2 }. We will assume that the min cut between S 1 and T 1 is at least 1 and the min-cut between S 2 and T 2 is at least 1. Because of the max-flow min-cut theorem, this is equivalent to stating that G 1,1 (F)G 2,2 (F) = 0. We drop the dependence on F with the understanding that, unless otherwise specified, all network transfer polynomials lie in the ring K(F). In instances where we refer to a specific network code, F * ∈ K |E|×|E| , we specify this explicitly; in this latter case, the network coding co-efficients lie in the field K.
Remark 2.2: It is worth noting that we have chosen the field of operation K as the algebraic closure of F 2 in the above definitions. The algebraic closure of F 2 consists of every finite extension of F 2 as a sub-field. It is therefore useful to note that as per Definition 2.2, a rate (m, n) is achievable if and only if there is some finite extension of F 2 over which the rate is achievable. It is also worth noting that there is, in general, a loss of generality in restricting to extensions of F 2 , since there exist networks where the notion of solvability depends on the characteristic of the field [3]. However, the field characteristic does not influence the results of this paper, so we restrict ourselves to extensions of F 2 in this document.
Remark 2.3 (Notation): Let s, t, s i , t i , i ∈ {1,
2} be edges, and let U 1 , U 2 be two sets of edges.
• p:s→t w(p) denotes the sum of the weights of all possible s → t paths.
• p:s→t via U1\U2 w(p) denotes the sum of the weights of all possible s → t paths such that each of these s → t paths goes through at least on edge in U 1 and does not go through any edge in U 2 .
• p1:s1→t1 p2:s2→t2 w(p 1 )w(p 2 ) denotes the sum of the weights of all possible (p 1 , p 2 ) pairs of paths such that p 1 is a s 1 → t 1 path and p 2 is a s 2 → t 2 path.
A. Commutative Algebra Background
In this section we describe some elementary concepts of commutative algebra [2], and state an elementary, central, result: Hilbert's Nullstellensatz theorem. Afterwards, we state and describe conditions equivalent to (4) for achievability of rate (1, 1) as a corollary to Hilbert's Nullstellensatz. We begin with some definitions. Definition 2.3 (Ideals): Let K be a field. A subset I of the polynomial ring K[x 1 , x 2 , · · · , x n ] is an ideal if it satisfies: (i) 0 ∈ I.
(ii) If f , g ∈ I, then f + g ∈ I.
(iii) If f ∈ I and h ∈ K[x 1 , x 2 , · · · , x n ], then hf ∈ I. Definition 2.4 (Ideals generated by polynomials): Let K be a field, and let f 1 , f 2 , · · · , f m be polynomials in the polynomial ring K[x 1 , x 2 , · · · , x n ]. The ideal generated by polynomials f 1 , f 2 , · · · , f m in K[x 1 , x 2 , · · · , x n ] is denoted as < f 1 , f 2 , · · · , f m > and defined as
< f 1 , f 2 , · · · , f m >= m i=1 h i f i : h 1 , h 2 , · · · , h m ∈ K[x 1 , x 2 , · · · , x n ] .
Definition 2.5 (Affine varieties): Let K be a field, and let f 1 , f 2 , · · · , f m be polynomials in the polynomial ring K[x 1 , x 2 , · · · , x n ]. The affine variety denoted by V(f 1 , f 2 , · · · , f m ) ⊆ K n is defined to be its set of "roots", that is, a 2 , · · · , a n ) ∈ K n : f i (a 1 , a 2 , · · · , a n ) = 0 ∀i ∈ [m]}. Definition 2.6 (Ideals of varieties): Let K be a field, K[x 1 , x 2 , · · · , x n ] be its associated polynomial ring, and let V ⊂ K n be an affine variety. The ideal of the variety V is denoted as I(V ) and defined as
V(f 1 , f 2 , · · · , f m ) = {(a 1 ,I(V ) = {f ∈ K[x 1 , x 2 , · · · ,
x n ] : f (a 1 , a 2 , · · · , a n ) = 0 ∀(a 1 , a 2 , · · · , a n ) ∈ V } . Remark 2.4 (A reversing-inclusion property [2]): Let K be a field. Let V and W be affine varieties in K n . Then, V ⊆ W if, and only if, I(V ) ⊇ I(W ). Theorem 2.1: (Hilbert's Nullstellensatz). Let K be an algebraically closed field and f, f 1 , f 2 , · · · , f m ∈ K[x 1 , x 2 , · · · , x n ]. The polynomials f, f 1 , f 2 , · · · , f m are such that f ∈ I(V(f 1 , f 2 , · · · , f m )) if, and only if, there exists a positive integer L such that f L ∈< f 1 , f 2 , · · · , f m >.
In the following corollary, we use Hilbert's Nullstellensatz to describe a condition equivalent to (4) for achievability of rate (1, 1).
Corollary 2.2: The rate (1, 1) is not achievable in a two-unicast-Z network using scalar linear coding if, and only if, for some positive integer L, there exist a polynomial P such that
G 2,1 P = (G 1,1 G 2,2 ) L .(5)
Proof: First, suppose that there exists a polynomial P such that G 2,1 P = (G 1,1 G 2,2 ) L , for some positive integer L. In order to achieve the rate pair (1,1), we need to satisfy the conditions in (4), that is, to set G 2,1 = 0 such that G 1,1 = 0 and G 2,2 = 0. However, from (5), setting G 2,1 = 0 gives G 1,1 = 0 or G 2,2 = 0. Hence, the first condition in (4) cannot be satisfied and rate (1, 1) is not achievable in the network using scalar linear coding. For the other direction, suppose that the rate pair (1, 1) is not achievable in the network using scalar linear coding. Thus the conditions in (4) cannot be satisfied simultaneously. Therefore, whenever G 2,1 = 0, we have G 1,1 = 0 or G 2,2 = 0 . In other words,
V G 1,1 ∪ V G 2,2 ⊇ V G 2,1 . (6) Since V G 1,1 ∪ V G 2,2 = V G 1,1 G 2,2 [2]
, substituting in (6) gives
V G 1,1 G 2,2 ⊇ V G 2,1 .(7)
Then, from Remark 2.4,
I V G 1,1 G 2,2 ⊆ I V G 2,1 .(8)
Since G 1,1 G 2,2 ∈ I V G 1,1 G 2,2 , from (8), we get
G 1,1 G 2,2 ∈ I V G 2,1(9)
The last equation satisfies the hypothesis of Theorem 2.1 (Hilbert's Nullstellensatz), hence there exists an integer L ≥ 1 such that (G 1,1 G 2,2 ) L ∈ G 2,1 . In other words, there exists a polynomial P such that G 2,1 P = (G 1,1 G 2,2 ) L , for some positive integer L.
III. NETWORK TRANSFER MATRIX DECOMPOSITION
In this section, we develop a network transfer matrix decomposition method that is central to our achievability proof. While our method is more generally applicable, we present our decomposition for the case of a two-unicast-Z network with two GNS edges C GN S = {e 1 , e 2 } where Ord(e 1 ) < Ord(e 2 ). This network decomposition is formulated in Lemma 3.2. Afterwards, in section III-A, we state and describe a condition, in Lemma 3.3, that is equivalent to the condition stated in Corollary 2.2 for achievability of rate (1, 1). The described new condition has some useful properties as will be shown.
First, we give the following definitions and remarks. Definition 3.1 (Left-side of the network): Consider a (G, S 1 , T 1 , S 2 , T 2 ) two-unicast-Z network with GNS cut set C GN S = {e 1 , e 2 }, Ord(e 1 ) < Ord(e 2 ). The left-side side of the network is defined as the subgraph G 1 = (V 1 , E 1 ) ⊆ G, where E 1 = {e ∈ E : e belongs to some S 1 ∪ S 2 → C GN S path} and V 1 = {v ∈ V : v is the head or tail of edge e, for some e ∈ E 1 }.
Definition 3.2 (Right-side of the network): Consider a (G, S 1 , T 1 , S 2 , T 2 ) two-unicast-Z network with GNS cut set C GN S = {e 1 , e 2 }, Ord(e 1 ) < Ord(e 2 ). The right-side side of the network is defined as the subgraph G 2 = (V 2 , E 2 ) ⊆ G, where E 2 = {e ∈ E : e belongs to some C GN S → T 1 ∪ T 2 path} and V 2 = {v ∈ V : v is the head or tail of edge e, for some e ∈ E 2 }. , let S = {s 1 , s 2 , · · · , s m } and T = {t 1 , t 2 , · · · , t n } be any two subsets of E. The transfer matrix M (S ,T ) is defined as the m × n matrix whose element at the index (i, j) is
p:s i →t j w(p).
Definition 3.4 (Restricted Transfer matrix): Consider a DAG G = (V, E), let S = {s 1 , s 2 , · · · , s m }, T = {t 1 , t 2 , · · · , t n }, U = {u 1 , u 2 , · · · , u k } be any three subsets of E. The transfer matrix M U (S ,T ) is defined as the m × n matrix whose element at the index (i, j) is
p:s i →t j via U w(p).
Definition 3.5 (Network transfer matrix): Consider a (G, S 1 , T 1 , S 2 , T 2 ) two-unicast-Z network. The network transfer matrix is defined as M (S1∪S2,T1∪T2) . Definition 3.6 (Restricted network transfer matrix): Consider a (G, S 1 , T 1 , S 2 , T 2 ) two-unicast-Z network. Let U ⊆ E, the restricted network transfer matrix with respect to U is defined as M U (S1∪S2,T1∪T2) . Definition 3.7 (Terminals-excluded transfer matrix): Consider a DAG G = (V, E), let S = {s 1 , s 2 , · · · , s m } and T = {t 1 , t 2 , · · · , t n } be any two subsets of E. For any i, j ∈ {1, · · · , n} such that i < j, Ord(t i ) < Ord(t j ). The terminals-excluded transfer matrix M (S ,T ) is defined as the m × n matrix whose element at the index (i, j) is p:s i →t j \{t k }k<j w(p).
Definition 3.8 (Sources-excluded transfer matrix): Consider a DAG G = (V, E), let S = {s 1 , s 2 , · · · , s m } and T = {t 1 , t 2 , · · · , t n } be any two subsets of E. For any i, j ∈ {1, · · · , m} such that i < j, Ord(s i ) < Ord(s j ). The sources-excluded transfer matrix M (S ,T ) is defined as the m × n matrix whose element at the index (i, j) is p:s i →t j \{s k }k>i w(p). Definition 3.9 (Coupling matrix): Consider a DAG G = (V, E), let U = {u 1 , u 2 , · · · , u m } be any subset of E, let the edges of the set be ordered such that Ord(u j ) > Ord(u i ) if j > i. The coupling matrix Λ U is defined as the m × m upper triangular matrix with all ones entries diagonal whose (i, j)-th element is p:ui→uj w(p). Lemma 3.1: Consider a (G, S 1 , T 1 , S 2 , T 2 ) two-unicast-Z network and let U ⊆ E. M U (S1∪S2,T1∪T2) = M (S1∪S2,U ) Λ U M (U ,T1∪T2) .
Proof: A pictorial description of the lemma is shown in Fig. 2.
For simplicity, let U = {u 1 , u 2 } where Ord(u 1 ) < Ord(u 2 ). Let s and t be two edges ∈ E, then p:s→t via U w(p) can be decomposed as: In addition, p:u1→t w(p) can be expressed as:
p:u1→t w(p) = p:u1→t \u2 w(p) + p:u1→t via u2 w(p) = p:u1→t \u2 w(p) + p:u1→u2 w(p) p:u2→t w(p)(11)
Then, we have
p:u1→t w(p) p:u2→t w(p) = 1 p:u1→u2 w(p) 0 1 p:u1→t\u2 w(p) p:u2→t w(p) (12)
Hence, (10) can be written as:
p:s→t via U w(p) = p:s→u1 w(p) p:s→u2\u1 w(p) 1 p:u1→u2 w(p) 0 1 p:u1→t\u2 w(p) p:u2→t w(p) (13) From the last equation, it is clear that M U (S1∪S2,T1∪T2) = M (S1∪S2,U ) Λ U M (U ,T1∪T2) .
Remark 3.3 (Notation):
In the remainder of this paper, we consider U = C GN S , where C GN S is a GNS cut set in the (G, S 1 , T 1 , S 2 , T 2 ) two-unicast-Z network of size two. Specifically, C GN S = {e 1 , e 2 } where Ord(e 1 ) < Ord(e 2 ). In addition, for simplicity, we write M, M 1 , Λ, and M 2 to denote M U (S1∪S2,T1∪T2) , M (S1∪S2,U ) , Λ U , and M (U ,T1∪T2) , respectively. Explicit expressions for matrices M, M 1 , M 2 , Λ are shown at the top of this page. Recall that F is the local coding matrix of the network andF is the set whose elements are the
M = p:s1→t1 w(p) p:s1→t2 via CGNS w(p) p:s2→t1 w(p) p:s2→t2 w(p) , Λ = 1 p:e1→e2 w(p) 0 1 , M 1 = p:s1→e1 w(p) p:s1→e2\e1 w(p) p:s2→e1 w(p) p:s2→e2\e1 w(p) , M 2 = p:e1→t1\e2 w(p) p:e1→t2\e2 w(p) p:e2→t1 w(p) p:e2→t2 w(p) .(14)
(non-zero) entries of F.
Lemma 3.2 (Network Decomposition Lemma): Consider a (G, S 1 , T 1 , S 2 , T 2 ) two-unicast-Z network with GNS cut set C GN S = {e 1 , e 2 }, Ord(e 1 ) < Ord(e 2 )
. Let G 1 and G 2 be the left-side and right-side of G, respectively, with respect to C GN S . Then We show a contradiction that precludes the existence of β ea,eb . Notice that m 1 is of the form w(p 1 ) where p 1 is some s i → e 1 path, for some i ∈ {1, 2}. In addition, from (14), m 2 is of the form w(p 2 ) where p 2 is some e 2 → t j path, or some e 1 → t j \e 2 path, for some j ∈ {1, 2}. Because β ea,eb occurs in w(p 1 ), edges e a , e b occur in path p 1 which begins at s i and ends at e 1 . Therefore, the topological order of e a is strictly smaller than the topological order of e 1 . Moreover, because β ea,eb occurs in w(p 2 ) where p 2 is some e 2 → t j path with Ord(e 2 ) > Ord(e 1 ), or some e 1 → t j \e 2 path, for some j ∈ {1, 2}, edges e a , e b occur in path p 2 which begins at e 2 and ends at t j or begins at e 1 and ends at t j . Therefore, the topological order of e a is at least the topological order of e 1 . Since the topological order of e a cannot be strictly smaller than the topological order of e 1 and at least the topological order of e 1 simultaneously, we conclude that such a β ea,eb variable cannot occur, contradicting our previous assumption.
(a) M = M 1 ΛM 2 . (b) In graph G 1 , the network transfer matrix from the source {s 1 , s 2 } to edges {e 1 , e 2 } is M 1 Λ. In graph G 2 , the network transfer matrix from {e 1 , e 2 } to {t 1 , t 2 } is ΛM 2 (c) Let F 1 ⊂F be the set of variables in p:si→e1 w(p)), i ∈ {1, 2}, and let F 2 ⊂F be the set of variables in M 2 , we have F 1 ∩ F 2 = φ. (d) Let F 1 ⊂F be the set of variables in M 1 , and let F 2 ⊂F be the set of variables in p:e2→ti w(p), i ∈ {1, 2}, we have F 1 ∩ F 2 = φ. (e) Let
(d) Here, we aim to show that the indeterminate variables that occur in polynomials of M 1 do not occur in the polynomials of 2 and vice-versa. If possible, let β ea,eb -the local coding co-efficient from edge e a to edge e b be a variable that occurs in both m 1 and m 2 . We show a contradiction that precludes the existence of β ea,eb .
Notice that, from (14), m 1 is of the form w(p 1 ) where p 1 is some s i → e 1 path, or some s i → e 2 \e 1 path, for some i ∈ {1, 2}. In addition, m 2 is of the form w(p 2 ) where p 2 is some e 2 → t j path, for some j ∈ {1, 2}. Because β ea,eb occurs in w(p 1 ) where p 1 is some s i → e 1 path with Ord(e 1 ) < Ord(e 2 ), or some s i → e 2 \e 1 path, for some i ∈ {1, 2}, edges e a , e b occur in path p 1 which begins at s i and ends at e 1 or begins at s i and ends at e 2 , i ∈ {1, 2}. Therefore, the topological order of e b is at most the topological order of e 2 . Moreover, because β ea,eb occurs in w(p 2 ), edges e a , e b occur in path p 2 which begins at e 2 and ends at t i , i ∈ {1, 2}. Therefore, the topological order of e b is strictly larger than the topological order of e 2 . Since the topological order of e b cannot be at most the topological order of e 2 and strictly larger than the topological order of e 2 simultaneously, we conclude that such a β ea,eb variable cannot occur, contradicting our previous assumption.
(e) Here, we aim to show that the indeterminate variables that occur in polynomials of M 1 do not occur in the polynomials of M 2 and vice-versa. Let m 1 be a monomial that occurs in some polynomial in M 1 . Similarly, let m 2 be a monomial that occurs in some polynomial in M 2 . It suffices to show that the variables in m 1 do not occur in the variables of m 2 and vice-versa. If possible, let β ea,eb -the local coding co-efficient from edge e a to edge e b be a variable that occurs in both m 1 and m 2 . We show a contradiction that precludes the existence of β ea,eb . We have the following cases.
Case 1:
p 1 is some s i → e 1 , i ∈ {1, 2}.
In this case, from part (c), m 1 and m 2 do not share any variables.
Case 2:
p 2 is some e 2 → t i , i ∈ {1, 2}.
In this case, from part (d), m 1 and m 2 do not share any variables.
Case 3: p 1 is a s i → e 2 \e 1 path and p 2 is a e 1 → t j \e 2 path, for some i, j ∈ {1, 2}. In this case, we show a contradction that precludes the existence of β ea,eb . Notice that m 1 is of the form w(p 1 ) where p 1 is some s i → e 2 → e 1 path, for some i ∈ {1, 2}. In addition, m 2 is of the form w(p 2 ) where p 2 is some e 1 → t j \e 2 path, for some j ∈ {1, 2}. Because β ea,eb occurs in w(p 2 ) where p 2 is some e 1 → t j \e 2 path, for some j ∈ {1, 2}, edges e a , e b occur in path p 2 which begins at e 1 and ends at t j . Therefore there exists an e 1 → e a path. In addition, because β ea,eb occurs in w(p 1 ) where p 1 is some s i → e 2 \e 1 path, for some i ∈ {1, 2}, edges e a , e b occur in path p 1 which begins at s i and ends at e 2 . Therefore there exists an e a → e 2 path. The concatenation of the e 1 → e a path and the e a → e 2 path gives a e 1 → e 2 path via e a , a contradiction to the hypothesis that there are no e 1 → e 2 paths in the graph. Hence, we conclude that such a β ea,eb variable cannot occur.
A. Consequence of Network Decomposition
In the following, we describe an equivalent condition to the condition stated in Corollary 2.2 for achievability of rate (1,1). This equivalent condition is stated in Lemma 3.3 and makes advantage of Lemma 3.2 in order to get some favorable properties. These properties are discussed at the end of this section and will help in developing the rate (1, 1) feasibility proofs in sections IV and V. Lemma 3.3: The rate (1, 1) is not achievable using scalar linear coding if, and only if, for some positive integer L, there exist a polynomial P such that
P p:s2→t1 w(p) = det(M) L .(15)
Proof: First, suppose that there exists a polynomial P such that p:s2→t1 w(p)P = det(M) L , for some integer L. We will show that rate (1, 1) is not achievable. In order to achieve the rate pair (1,1), we need to find a linear coding scheme, denote it F * , that satisfies the conditions in (4).
Note that, for this achievable scheme, we have
p:s1→t1 w(p) = G 1,1 (F * ), p:s2→t2 w(p) = G 2,2 (F * ),
and p:s2→t1 w(p) = G 2,1 (F * ). Therefore, the three conditions of achievability in (4) maps to the following three conditions
p:si→ti w(p) = 0, i = 1, 2, p:s2→t1 w(p) = 0.(16)
From (16), In order to achieve (1,1), we need to set For the other direction, suppose that the rate pair (1, 1) is not achievable using scalar linear coding. We will show that there exists a polynomial P such that (15) is true for some positive integer L. Since the rate pair (1, 1) is not achievable using scalar linear coding, we conclude from Corollary 2.2 that there exists a polynomial P such that
G 2,1 P = (G 1,1 G 2,2 ) L ,(17)
for some positive integer L. Notice that
w(p) p:s2→t2 w(p) − p:s2→t1 w(p) p:s1→t2 via CGNS w(p) L (1) = p:s1→t1 w(p) p:s2→t2 w(p) L + p:s2→t1 w(p)P (2) = p:s2→t1 w(p)P + p:s2→t1 w(p)P = p:s2→t1 w(p) P + P (3) = p:s2→t1 w(p)P,(19)
where (1) follows from the binomial expansion and an appropriate polynomial P , (2) follows from (18), and (3) follows from defining P = P + P .
Remark 3.4:
The right hand side of
p:s2→t1 w(p)P = det(M) L is non-zero.
The remark follows by observing that det(M) = det(M 1 ) det(M 2 ). If det(M) = 0, then det(M i ) = 0, i = 1 or 2. Therefore, by max-flow min-cut theorem, there is a single edge cut set in (G 1 , S 1 , e 1 , S 2 , e 2 ) or (G 2 , e 1 , T 1 , e 2 , T 2 ). However, a single edge cut set in (G 1 , S 1 , e 1 , S 2 , e 2 ) or (G 2 , e 1 , T 1 , e 2 , T 2 ) gives a single edge GNS cut set in (G, S 1 , T 1 , S 2 , T 2 ), a contradiction to the fact that the network (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut set of size two.
B. Notations and Observations
In the following, we introduce some notations which will be used for the rest of the paper. In addition, based on these notations, we give some observations on the advantage of the achievability condition derived in Lemma 3.3.
For i ∈ {1, 2}, let u i denote an In(e i ) × 1 vector of indeterminate variables representing the local coding co-efficients from the edges incoming into e i to e i . Specifically, denoting In(e i ) = {e i,1 , e i,2 , . . . , e i,|In(ei)| }, the vector u i is equal to (β ei,1,ei , β ei,2,ei , . . . , β ei,|In(e i |,ei ). We now aim to express the polynomials in M, M 1 , Λ as polynomials in u 1 , u 2 . We write
M 1 = a 1 u 1 a 2 u 2 b 1 u 1 b 2 u 2 , Λ = 1 λ 12 u 2 0 1 ,(20)
where a i , with i ∈ {1, 2}, is the 1 × In(e i ) vector of transfer polynomials from s 1 to In(e i ) containing paths that do not go through
C GN S − {e i }. Specifically, a i = (a i,1 , a i,2 , · · · , a i,|In(ei)| ), where a i,j = p:s1→ei,j\CGNS−{ei} w(p), j ∈ {1, · · · , |In(e i )|}. The row vectors b 1 and b 2 are defined similarly but with respect to s 2 . λ 12 = (λ 12,1 , λ 12,2 , · · · , λ 12,|In(e2)| ) is a 1 × In(e 2 ) vector where λ 12,|In(e2)| = p:e1→e2,j w(p), j ∈ {1, · · · , |In(e 2 )|}. Finally, we write M 2 = µ 11 µ 12 µ 21 µ 22 ,(21)
where µ ij = p:ei→tj\CGNS−{ei} w(p), i, j ∈ {1, 2}. Now, (15) can be written as
b 1 u 1 (µ 11 + λ 12 u 2 µ 21 ) + b 2 u 2 µ 21 P = µ 11 µ 22 − µ 12 µ 21 L a 1 u 1 b 2 u 2 − b 1 u 1 a 2 u 2 L .(22)
The main utility of Lemma 3.3 is that it "homogenizes" the right hand side of Corollary 2.2 with respect to variables u i , i = 1, 2. To see this more clearly, we state some basic definitions related to the degree of multi-variate polynomials.
For any field K and any set of indeterminates x 1 , x 2 , . . . , x n , we denote the rational field containing K[x 1 , x 2 , . . . , x n ] as K(x 1 , x 2 , . . . , x n ). Let us denote by K, the rational field K(F − {u i , i = 1, 2}). For i ∈ {1, 2}, we will also denote by K (i) , the polynomial ring
K(u j )[u i ] where j ∈ {1, 2} − {i}.
Note that for a network coding co-efficient polynomial p, the quantity sumdeg Note that in contrast, the right hand side of (5) of Corollary 2.2 does not necessarily satisfy this property. Lemma 3.3 will be used to show Theorem 4.3. In particular, we will show that if the graph in a two-unicast-Z network satisfies certain properties, then it is not possible to find polynomial P satisfying (15) because the degree of the left hand side and the degree of the right hand side will be inconsistent.
IV. AN INTERMEDIATE STEP
In this section, we first give a background on orderings on monomials. This background is used in establishing the two lemmas: Lemma 4.1 and Lemma 4.2. Afterwards, we use these lemmas to prove Theorem 4.3. Afterwards, Theorem 4.3 is used to give an intermediate result, in Corollary 4.4, which will be a stepping stone towards deriving the feasibility of rate (1, 1) in Theorem 5.2.
A. Background on Orderings on Monomials
A background on orderings on monomials is inrtroduced in this section [2]. x ti i in the polynomial ring K[x 1 , · · · , x n ], the multi-degree of this monomial is multideg where, for all i ∈ {1, · · · , N }, a i ∈ K and m i is a monomial in K[x 1 , · · · , x n ] and let > be a monomial order. Then, the multi-degree of p is multideg
K[x1,··· ,xn] (m) = (t 1 , t 2 , · · · , t n ).K[x1,··· ,xn] (p) = max i∈{1,··· ,N } multideg K[x1,··· ,xn] (m i ) ,
where the maximum is taken with respect to >. (m) = (t 1 , t 2 , · · · , t n ), the sum-degree of this monomial is sumdeg Consider two vectors α = (α 1 , α 2 , · · · , α n ) and β = (β 1 , β 2 , · · · , β n ) ∈ Z n . A graded lexicographic order is defined as follows. Let α = ( n i=1 α i , α 1 , α 2 , · · · , α n ) and β = ( n i=1 β i , β 1 , β 2 , · · · , β n ) ∈ Z n . We say α > gralex β if, in the vector difference α − β ∈ Z n , the leftmost nonzero entry is positive. Also, we say α = gralex β, if all the entries in α − β ∈ Z n are zeros.
K[x1,··· ,xn] (m) = n i=1 t i .
Definition 4.6 (Reverse graded lexicographic order): Consider two vectors α = (α 1 , α 2 , · · · , α n ) and β = (β 1 , β 2 , · · · , β n ) ∈ Z n . A reverse graded lexicographic order is defined as follows. Let α = ( n i=1 α i , α 1 , α 2 , · · · , α n ) and β = ( n i=1 β i , β 1 , β 2 , · · · , β n ) ∈ Z n . We say α > revgralex β if, in the vector difference α − β ∈ Z n , the leftmost nonzero entry is negative. Also, we say α = revgralex β, if all the entries in α − β ∈ Z n are zeros. x αi i and n i=1 y βi i be two monomials. We say
n i=1 x αi i ≥ gralex n i=1 x βi i if multideg k[x1,··· ,xn] n i=1 x αi i ≥ gralex multideg k[x1,··· ,xn] n i=1
x βi i . x αi i and n i=1 y βi i be two monomials. We say
n i=1 x αi i ≥ revgralex n i=1 x βi i if multideg k[x1,··· ,xn] n i=1 x αi i ≥ revgralex multideg k[x1,··· ,xn] n i=1
x βi i .
B. An Intermediate
Step for The Feasibility of Rate (1, 1) In this section, we prove the feasibility of rate (1, 1) using scalar linear coding for a specific class of two-unicast-Z networks in Corollary 4.4. First, we start with the following two lemmas. Proof: In this proof, we will consider a graded lexicographic order on monomials. Notice that sumdeg
K (i) (α) ∈ {0
, 1} since we conisder directed acyclic graphs which implies that no path can go through e 1 (e 2 ) more than once. Let P = P (ui) + P (ui) where sumdeg
K (i) (m) ≤ L − sumdeg K (i)
(α) for any monomial m in P (ui) , and
sumdeg K (i) (m) > L − sumdeg K (i)
(α) for any monomial m in P (ui) . We can rewrite αP = (det(M)) L as
αP (ui) + αP (ui) = (det(M)) T .(23)
Since sumdeg
K (i) (α) ∈ {0, 1} and sumdeg K (i) (m) ≤ L − sumdeg K (i) (α) for any monomial m in P (ui) , we have sumdeg K (i)
(m) ≤ L for any monomial m in αP (ui) . Notice that if P (ui) = 0, then we have P = P (ui) and αP (ui) = (det(M)) L and we are done. For the rest of the proof, we suppose that P (ui) = 0. Because sumdeg (b) For brevity, we will prove this part for the scenario in which G has a s 2 − t 1 path that goes through e 1 . The proof for i = 2 will be similar.
K (i) (α) ∈ {0, 1} and sumdeg K (i) (m) > L − sumdeg K (i)(
Notice that since G is acyclic, no s 2 → t 1 path can go through e 1 twice. Therefore, at most one element of u 1 can appear in any monomial in α. This means that sumdeg
K (1) (α) is at most one.
Consider a graded lexicographic ordering on monomials in the polynomial ring K (1) . Because sumdeg K (1) (det(M) L ) = L and αP = (det(M)) L , we have sumdeg
K (1) (αP) = L. Since sumdeg K (1) (αP ) = sumdeg K (1) (α) + sumdeg K (1) (P ), sumdeg K (1) (P ) = L − sumdeg K (1) (α).
Since there is a s 2 → t 1 path via e 1 in G, there exists a monomial in α with one element of u 1 . In general, any monomial in α contains at most one element of u 1 due to the graph being acyclic. Therefore, recalling that in the graded lexicographic ordering, the sum-degree of the polynomial is the sum-degree of monomial with maximum sum-degree in this polynomial, sumdeg
K (1) (α) = 1. Therefore, sumdeg K (1) (P ) = L − 1.
Again, since we use graded lexicographic ordering, we conclude from sumdeg (c) For brevity, we will prove this part for the scenario in which G has a s 2 − t 1 path that does not go through e 1 . The proof for i = 2 will be similar.
Consider a reverse graded lexicographic ordering on monomials in the polynomial ring K (i) . Because sumdeg (αP) = L. Since sumdeg
K (1) (αP ) = sumdeg K (1) (α) + sumdeg K (1) (P ), sumdeg K (1) (P ) = L − sumdeg K (1) (α).
Since there is a s 2 → t 1 path not via e 1 in G and recalling that in the reverse graded lexicographic ordering, the sum-degree of the polynomial is the sum-degree of monomial with minimum sum-degree in this polynomial, sumdeg Now we state the following theorem which will be used in Corollary 4.4 to establish the feasibility of rate (1, 1) using scalar linear coding for a specific class of two-unicast-Z networks. Theorem 4.3: Consider a (G, {s 1 }, {t 1 }, {s 2 }, {t 2 }) two-unicast-Z network such that there is a path from s 1 to t 1 , there is a path from s 2 to t 2 , and G has a minimum GNS cut set {e 1 , e 2 } of size two. If there is a s 2 → t 1 via e i path and a s 2 → t 1 \e i path for some i ∈ {1, 2}, then the rate (1, 1) is achievable in the network using scalar linear coding.
Proof: To keep the notation simple and clear, we show the theorem for i = 1. Consider a twounicast-Z network which satisfies the hypothesis of the theorem with i = 1, i.e., there are s 2 → t 1 via e 1 and s 2 → t 1 \e 1 paths in the graph. Suppose that the rate (1, 1) is not achievable in the two-unicast-Z network. Then, the hypothesis of Lemma 3.3 is satisfied. Therefore (15) holds. Because (15) holds, the hypothesis of Lemma 4.1 holds. As a consequence of the lemma, there exists a polynomial P such that for any monomial m in αP , we have sumdeg (m) = L, for any monomial m in P , which contradicts our previous conclusion based on statement (b) of the lemma. Therefore, P does not contain any nonzero monomials, i.e. P = 0. Thus, from Lemma 4.2, we have det(M) = 0. Since det(M) = det(M 1 ) det(Λ) det(M 2 ), and det(Λ) = 1 since Λ is an upper triangular matrix with an all ones diagonal, we have det (M 1 ) = 0 or det(M 2 ) = 0 which implies, by max-flow min-cut therorem, G 1 or G 2 has a cut set of size less than 2; any cut of G 1 or G 2 is also a GNS cut set in G. Therefore, G has a GNS cut set of size smaller than 2, which violates the theorem hypothesis. Now, we give the following intermediate result for the feasibility of rate (1,1). Corollary 4.4: Consider a (G, {s 1 }, {t 1 }, {s 2 }, {t 2 }) two-unicast-Z network such that there is a path from s 1 to t 1 , there is a path from s 2 to t 2 , and G has a minimum GNS cut set {e 1 , e 2 } of size two. If the network has at least two of the following path types,
• s 2 → t 1 via e 1 \e 2 path. • s 2 → t 1 via e 2 \e 1 path. • s 2 → t 1 via {e 1 , e 2 } path.
then, the rate (1, 1) is achievable in the network using scalar linear coding.
The proof follows directly from Theorem 4.3.
V. FEASIBILITY OF RATE (1, 1): THE ALTERNATE PROOF
The main contribution in this section is presenting an alternate proof for the feasibility of rate (1,1) in two-unicast-Z networks using scalar linear coding in Theorem 5.2.
Before we start the proof of Theorem 5.2, we give the following definition and lemma. Definition 5.1 (Inner edges): Consider a graph G = (V, E) where the edges in E follow some topological ordering. An inner edge in G is any edge whose topological order lies strictly between the smallest and the largest topological order in the graph.
Lemma 5.1: Consider a (G, {s 1 }, {t 1 }, {s 2 }, {t 2 }) two-unicast-Z network such that there is a path from s 1 to t 1 , there is a path from s 2 to t 2 , and G has a minimum GNS cut set of size at least two. If there does not exist a GNS cut in the network such that it only consists of inner edges, then the rate (1, 1) is achievable in the network using routing.
The proof of this lemma follows from some simple graph theoretic arguments. Now, we begin the proof of the theorem. Theorem 5.2: Consider a (G, {s 1 }, {t 1 }, {s 2 }, {t 2 }) two-unicast-Z network such that there is a path from s 1 to t 1 and there is a path from s 2 to t 2 . If G has a minimum GNS cut set of size at least two, then the rate (1, 1) is achievable in the network using scalar linear coding.
Proof: In our proof we assume, without loss of generality, that G has a minimum GNS cut C GN S = {e 1 , e 2 } of size two. If the original graph has a minimum GNS cut of size larger than two, we can keep removing edges till we get a graph G with minimum GNS cut set of size two.
In our proof, we assume that the graph G has some s 2 → t 1 path, i.e. interference at t 1 . Otherwise, the graph G has two edge disjoint s 1 → t 1 and s 2 → t 2 paths, and the rate (1,1) achievability directly follows using routing.
If we cannot pick a GNS cut set in the network such that it only consists of inner edges, then, by Lemma 5.1, the rate (1, 1) is achievable by routing. Therefore, we restrict our analysis to networks where a GNS cut that consists only of inner edges can be obtained. For these networks, we always pick a GNS cut of inner edges. This implies that both G 1 and G 2 are strictly contained in the original graph G, i.e. the number of edges in G 1 and the number of edges in G 2 are strictly smaller than |E|. In our proof, we use induction on the number of edges of the graph. Specifically, the induction hypothesis assumes that the theorem statement holds for any subgraph of G with number of edges less than the number of edges of G. From Corollary 4.4, it remains to prove the theorem only for three cases:
Case A, every s 2 → t 1 path goes through e 1 and not e 2 , Case B, every s 2 → t 1 path goes through e 2 and not e 1 , and Case C, every s 2 → t 1 path goes through both e 1 , e 2 .
For all these three cases, we prove the achievability by contradiction. That is, we assume, for contradiction, that (1, 1) is not achievable in the network using scalar linear coding. Therefore, from Lemma 3.3, there exists a polynomial P such that (15) and equivalently (22) are true. Afterwards, we show that assuming that (1, 1) is not achievable in the network using scalar linear coding contradicts one of the assumption hypotheses of the theorem. Hence, (1, 1) is achievable in the network using scalar linear coding. Now, we go through the different three cases. Case A: every s 2 → t 1 path goes through e 1 and not e 2 .
In this case, there are neither s 2 → t 1 via e 2 \e 1 paths, nor s 2 → t 1 via e 1 and e 2 paths. That is, from (22), p:s2→t1 via e2\e1 w(p) = b 2 u 2 µ 21 = 0 and p:s2→t1 via e1 and e2 w(p) = b 1 u 1 λ 12 u 2 µ 21 = 0. Therefore, (22) can be written as
µ 11 b 1 u 1 P = µ 11 µ 22 − µ 12 µ 21 L a 1 u 1 b 2 u 2 − b 1 u 1 a 2 u 2 L .(24)
Since b 2 u 2 µ 21 = 0 and b 1 u 1 λ 12 u 2 µ 21 = 0, and observing that Otherwise, (G, S 1 , T 1 , S 2 , T 2 ) has e 1 as a single edge GNS cut, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Now, we can use the inductive assumption on (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }). Since (1, 1) is not achievable in (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }) using scalar linear coding, by inductive assumption there is a single edge GNS cut in (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }), call it e . Since, there are no e 2 → t 1 paths in (G, S 1 , T 1 , S 2 , T 2 ), e is a single edge GNS cut set in (G, S 1 , T 1 , S 2 , T 2 ) as well, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Hence, (1, 1) is achievable in (G, S 1 , T 1 , S 2 , T 2 ) using scalar linear coding.
Case A2) Let (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }) define the right network. (24) can be written as
µ 11 P = (µ 12 µ 21 − µ 11 µ 22 ) L b 1 u 1 L−1 a 2 u 2 L .(27)
Let F 1 ⊂F be the set of variables in M 1 , i.e. the set of variables in a 1 u 1 , b 2 u 2 , b 1 u 1 , a 2 u 2 , and let F 2 ⊂F be the set of variables in M 2 , i.e. the set of variables in µ ij , i, j ∈ {1, 2}. Since, in this case, there are no e 1 → e 2 paths, from Lemma 3.2 part (e) we conclude that F 1 ∩ F 2 = φ. This implies that the variables in the polynomial µ 11 are disjoint from the variables in the polynomial b 1 u 1 L−1 a 2 u 2 L . Therefore, P = b 1 u 1 L−1 a 2 u 2 L P for some polynomial P . Now, (27) can be written as
µ 11 P = (µ 12 µ 21 − µ 11 µ 22 ) L = (det(M 2 )) L = (det(ΛM 2 )) L .(28)
Recall from Lemma 3.2 that ΛM 2 is the network transfer matrix for (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }). Moreover, notice that In order to use the inductive assumption, we first need to check that (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }) possesses the same structure described in the theorem statement. That is, there are e 2 → t 1 and e 1 → t 2 paths in (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }). Indeed, there must be an e 1 → t 2 path. Otherwise, there are no s 2 → t 2 paths in (G, S 1 , T 1 , S 2 , T 2 ) because every s 2 → t 2 path goes through the GNS edges e 1 or e 2 and since there are no s 2 → e 2 paths, s 2 connects to t 2 only through e 1 . Therefore, an e 1 → t 2 path exists in (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }). Moreover, there exists an e 2 → t 1 path in (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }). Otherwise, (G, S 1 , T 1 , S 2 , T 2 ) has e 1 as a single edge GNS cut, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Now, we can use the inductive assumption on (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }). Since (1, 1) is not achievable in (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }) using scalar linear coding, by inductive assumption there is a single edge GNS cut in (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }), call it e . Since, there are neither s 2 → e 2 nor e 1 → e 2 paths in (G, S 1 , T 1 , S 2 , T 2 ), e is a single edge GNS cut set in (G, S 1 , T 1 , S 2 , T 2 ) as well, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Hence, (1, 1) is achievable in (G, S 1 , T 1 , S 2 , T 2 ) using scalar linear coding.
Case B: every s 2 → t 1 path goes through e 2 and not e 1 .
In this case, there are neither s 2 → t 1 via e 1 \e 2 paths, nor s 2 → t 1 via e 1 and e 2 paths. That is, from (22), p:s2→t1 via e1\e2 w(p) = b 1 u 1 µ 11 = 0 and p:s2→t1 via e1 and e2 w(p) = b 1 u 1 λ 12 u 2 µ 21 = 0. Therefore, (22) can be written as
µ 21 b 2 u 2 P = µ 11 µ 22 − µ 12 µ 21 L a 1 u 1 b 2 u 2 − b 1 u 1 a 2 u 2 L .(29)
In this case, there are neither s 2 → t 1 via e 1 \e 2 paths, nor s 2 → t 1 via e 1 and e 2 paths. That is, from (22), b 1 u 1 µ 11 = 0 and b 1 u 1 λ 12 u 2 µ 21 = 0. From the last two equations, and observing that µ 21 = 0 since there exists a s 2 → t 1 via e 2 path, we conclude that case B can occur only due to one of the following scenarios: B1) s 2 → e 1 path does not exist (b 1 u 1 = 0). B2) e 1 → t 1 and e 1 → e 2 paths do not exist, (µ 11 = 0, λ 12 u 2 = 0).
Case B1) Let (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }) define the left network.
(29) can be written as
µ 21 P = (b 2 u 2 ) L−1 (a 1 u 1 ) L det(M 2 )) L(30)
Let F 1 ⊂F be the set of variables in M 1 , i.e. the set of variables in a 1 u 1 , b 2 u 2 , b 1 u 1 , a 2 u 2 . Recall that µ 21 = p:e2→t1 w(p), and let F 2 ⊂F be the set of variables in
p:e2→t1 w(p). From Lemma 3.2 part (d), F 1 ∩ F 2 = φ.
This implies that the variables in the polynomial µ 21 are disjoint from the variables in the polynomial (b 2 u 2 ) L−1 (a 1 u 1 ) L . Therefore, P = (b 2 u 2 ) L−1 (a 1 u 1 ) L P for some polynomial P . Now, (30) can be written as . Indeed, there must be an e 1 → t 1 path. Otherwise, (G, S 1 , T 1 , S 2 , T 2 ) has e 2 as a single edge GNS cut, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Moreover, there exists an e 2 → t 2 path in (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }). Otherwise, there are no s 2 → t 2 paths in(G, S 1 , T 1 , S 2 , T 2 ) because every s 2 → t 2 path goes through the GNS edges e 1 or e 2 and since there are no s 2 → e 1 paths, s 2 connects to t 2 only through e 2 . Therefore, an e 2 → t 2 path exists in (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }). Now, we can use the inductive assumption on (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }). Since (1, 1) is not achievable in (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }) using scalar linear coding, by inductive assumption there is a single edge GNS cut in (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }), call it e . Since, there are no s 2 → e 1 paths in (G, S 1 , T 1 , S 2 , T 2 ), e is a single edge GNS cut set in (G, S 1 , T 1 , S 2 , T 2 ) as well, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Hence, (1, 1) is achievable in (G, S 1 , T 1 , S 2 , T 2 ) using scalar linear coding.
µ 21 P = det(M 2 )) L = det(ΛM 2 )) L .(31)
Case B2) Let (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }) define the right network. (29) can be written as
b 2 u 2 P = µ L−1 21 µ L 12 a 1 u 1 b 2 u 2 − b 1 u 1 a 2 u 2 L .(32)
Let Since, there are no e 1 → t 1 paths in (G, S 1 , T 1 , S 2 , T 2 ), e is a single edge GNS cut set in (G, S 1 , T 1 , S 2 , T 2 ) as well, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Hence, (1, 1) is achievable in (G, S 1 , T 1 , S 2 , T 2 ) using scalar linear coding.
Case C: every s 2 → t 1 path goes through both e 1 , e 2 . Notice that in this case there are no s 2 → e 2 \e 1 paths in (G, S 1 , T 1 , S 2 , T 2 ). Otherwise, the concatenation of the s 2 → e 2 \e 1 path with the e 2 → t 1 path contained in any s 2 → t 1 via e 1 and e 2 path gives a s 2 → t 1 via e 2 \e 1 path in (G, S 1 , T 1 , S 2 , T 2 ), a contradiction to the fact that every s 2 → t 1 path in (G, S 1 , T 1 , S 2 , T 2 ) goes through both e 1 and e 2 . Similarly, there are no e 1 → t 1 \e 2 paths in (G, S 1 , T 1 , S 2 , T 2 ). Otherwise, the concatenation of the e 1 → t 1 \e 2 path with the s 2 → e 1 path contained in any s 2 → t 1 via e 1 and e 2 path gives a s 2 → t 1 via e 1 \e 2 path in (G, S 1 , T 1 , S 2 , T 2 ), a contradiction to the fact that every s 2 → t 1 path in (G, S 1 , T 1 , S 2 , T 2 ) goes through both e 1 and e 2 .
Since there are neither s 2 → e 2 \e 1 nor e 1 → t 1 \e 2 paths in (G, S 1 , T 1 , S 2 , T 2 ), we have b 2 u 2 = µ 11 = 0. Therefore, (22) can be written as
µ 21 b 1 u 1 λ 12 u 2 P = µ 12 µ 21 L b 1 u 1 a 2 u 2 L .(34)
After removing common factors, we get
λ 12 u 2 P = µ L 12 µ L−1 21 b 1 u 1 L−1 a 2 u 2 L .(35)
Notice that, using topolgical order arguments, we conclude that the variables in the polynomials µ 21 and b 1 u 1 are disjoint from the variables in the polynmoial λ 12 u 2 . Therefore, P = µ L−1
21 b 1 u 1 L−1 P
for some polynomial P . Now, (35) can be written as
λ 12 u 2 P = (µ 12 a 2 u 2 ) L .(36w(p),(40)
for some polynomialP . Therefore,
p:s1→e2 w(p) p:e1→t2 w(p) L = p:s1→e2 \e1 w(p) p:e1→t2 \e2 w(p) L + P p:e1→e2 w(p),(41)
for some polynomial P . Substituting from (37) in (41), we get In the following, we show that e is a single edge GNS cut set in (G, S 1 , T 1 , S 2 , T 2 ) as well. Since e is a single edge GNS cut in (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }), e is a s 1 − e 2 cut, an e 1 − t 2 cut and an e 1 − e 2 cut. Now, we show that e is a s 1 − t 1 cut. Since there are no e 1 → t 1 \e 2 paths in (G, S 1 , T 1 , S 2 , T 2 ), every s 1 → t 1 path in (G, S 1 , T 1 , S 2 , T 2 ) goes through e 2 . Since every s 1 → t 1 path in (G, S 1 , T 1 , S 2 , T 2 ) goes through e 2 , and e is a s 1 − e 2 cut and e 1 − e 2 cut, we conclude that e is a s 1 − t 1 cut as well. Similarly, we show that e is a s 2 − t 2 cut. Since there are no s 2 → e 2 \e 1 paths in (G, S 1 , T 1 , S 2 , T 2 ), every s 2 → t 2 path in (G, S 1 , T 1 , S 2 , T 2 ) goes through e 1 . Since every s 2 → t 2 path in (G, S 1 , T 1 , S 2 , T 2 ) goes through e 1 , and e is a e 1 − t 2 cut and e 1 − e 2 cut, we conclude that e is a s 2 − t 2 cut as well. Now, we show that e is a s 2 − t 1 cut. Since in this case every s 2 → t 1 path goes through both e 1 and e 2 , and e is an e 1 − e 2 cut, we conclude e is a s 2 − t 1 cut as well. Since e is a s 1 − t 1 cut, a s 2 − t 2 cut and a s 2 − t 2 cut, we conclude that e is a single edge GNS cut in (G, S 1 , T 1 , S 2 , T 2 ) as well, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Hence, (1, 1) is achievable in (G, S 1 , T 1 , S 2 , T 2 ) using scalar linear coding. In this section, we give a two-unicast-Z network construction where the GNS bound is not tight. The same network also shows that vector linear codes outperform scalar linear codes. The network, which has GNS bound of 3, is depicted in Fig. 3; we show that its sum-rate can be no larger than 2.5. Consider a scheme with n uses of the network and probability of error bounded by , let X 1j denote the symbol sent along edge s 1j , j = 1, 2, and X 2 denote the symbol sent by s 2 . Let Y 1j denote the symbol received along t 1j , j = 1, 2, and Y 2 denote the symbol received along t 2 , and let U i denote the symbol sent along e i , i = 1, 2, 3. Notice that there is no loss of generality in assuming that U 1 = X 11 , U 2 = Y 12 , U 3 = Y 2 . For the first source, we have
VI. INSUFFICIENCY OF SCALAR LINEAR NETWORK CODES AND EDGE CUT BOUNDS
nR 1 − ≤ I(X 11 , X 12 ; Y 11 , U 2 ) (1) = I(X 11 ; Y 11 ) + I(X 11 ; U 2 |Y 11 ) + I(X 12 ; Y 11 , U 2 |X 11 ),
where (1) follows from the application of the chain rule of mutual information.
Similarly, for the second source, we have
nR 2 −(2)
≤ I(X 2 ; U 3 |X 11 , X 12 ).
We also have
I(X 11 ; Y 11 , U 2 )(3)
= I(X 11 ; Y 11 ) + I(X 11 ; U 2 |Y 11 )
≤ H(X 11 )
≤ n, where (3) follows from the chain rule of mutual information, (4) follows from the fact that I(X 11 ; Y 11 , U 2 ) = H(X 11 ) − H(X 11 |Y 11 , U 2 ) where H(X 11 |Y 11 , U 2 ) is non-negative, and (5) follows from the edge capacity constraint. In addition, we can write I(X 11 ; Y 11 ) + I(X 12 ; Y 11 , U 2 |X 11 ) = I(X 11 ; Y 11 ) + I(X 12 ; U 2 |X 11 ) + I(X 12 ; Y 11 |X 11 , U 2 )
= I(X 11 ; Y 11 ) + I(X 12 ; Y 11 |X 11 , U 2 )
= I(X 11 ; Y 11 ) + I(X 12 , U 2 , X 2 ; Y 11 |X 11 ) − I(U 2 ; Y 11 |X 11 ) − I(X 2 ; Y 11 |X 11 , U 2 , X 12 )
≤ I(X 11 ; Y 11 ) + I(U 2 ; Y 11 |X 11 ) + I(X 12 ; Y 11 |X 11 , U 2 ) + I(X 2 ; Y 11 |X 11 , U 2 , X 12 )
= I(X 11 , U 2 , X 12 , X 2 ; Y 11 ) ≤ H(Y 11 ) (8) ≤ n,
where (6) follows since X 12 and U 2 are independent, and (7) follows from the application of mutual information chain rule on I(X 12 , U 2 , X 2 ; Y 11 |X 11 ). Moreover, we have I(X 11 ; U 2 |Y 11 ) + I(X 12 ; Y 11 , U 2 |X 11 ) = I(X 11 ; U 2 |Y 11 ) + I(X 12 ; Y 11 |X 11 ) + I(X 12 ; U 2 |X 11 , Y 11 ) = I(X 11 , X 12 ; U 2 |Y 11 ) + I(X 12 ; Y 11 |X 11 ) (9) ≤ I(X 11 , X 12 ; U 2 |Y 11 ) + I(X 12 ; U 3 |X 11 )
≤ n + I(X 12 ; U 3 |X 11 ) = n + I(X 12 , X 2 ; U 3 |X 11 ) − I(X 2 ; U 3 |X 11 , X 12 )
≤ 2n − I(X 2 ; U 3 |X 11 , X 12 ), where (9) follows from the data processing inequality since Y 11 is a function of U 3 . Finally, performing 2 × (1) + (2) + (5) + (8) + (10) and letting n → ∞ gives 2R 1 + R 2 ≤ 4. In conjunction with, the cutset bounds for R 1 , R 2 , we infer that R 1 + R 2 ≤ 2.5.
The rate (1.5, 1) can be achieved via vector linear network coding. An achievability scheme for the rate (1.5, 1) in the network shown in Fig. 3 using vector linear coding is shown in Fig. 4.
VII. INSUFFICIENCY OF LINEAR CODES
In this section, we show that non-linear codes outperform linear codes in two-unicast-Z networks. Our approach is inspired by the method of [7]. We consider an arbitrary m-unicast network B and construct a two-unicast-Z network, where achievability of rate (m, m) in the two-unicast-Z network necessarily requires achieving rate (1, 1, . . . , 1) in the m-unicast network B. Because linear codes are, in general, insufficient to achieve (1, 1, . . . , 1) in m-unicast networks for m ≥ 10 [4], [5], [7], our construction implies that linear codes are insufficient to achieve rate (m, m) in two-unicast-Z networks for m ≥ 10. Our construction is shown in Fig. 5; for simplicity, we describe our method for the special case of m = 2. The random variables representing the symbol carried by each edge are defined as shown in Fig. 5.
In the following, we exhibit the proof idea used to show that rate (1, 1) is achievable in the two-unicast network B if, and only if, rate (2, 2) is achievable in the overall two-unicast-Z network. If (1, 1) is achievable in the two-unicast network B, then simply setting Y i = X 1i + X 2i and V i = Z i − X 2i suffices to achieve (2,2) in the two-unicast-Z network. For the other direction, we show that it is necessary for any zero-error achievable scheme with rate (2, 2) in the original network, there must exist one-to-one mappings from Y i to Z i , which implies that (1, 1) must be achievable in B. The intuition for this result is that, to ensure that X 11 , X 12 can be decoded from (V 1 , V 2 ),, and since terminal 1 has no side information on the second source, both V 1 and V 2 must be functions of only X 11 and X 12 . In other words, it is necessary to completely cancel the effect of X 22 from Z 1 , and completely cancel the effect of X 21 from Z 2 . Since node g 1 only has X 21 as side information, it must be the case that Z 1 does not depend on X 22 and, similarly, Z 2 does not depend on X 21 . Since Y 2 depends on X 22 , and Y 1 depends on X 21 , we require that Z i is simply a function of Y i for i = 1, 2. The rate of the messages dictates that these functions must be one-to-one. Now, we formally state our result of the insufficiency of linear codes in the two-unicast-Z networks in the following theorem, Theorem 7.1: Linear codes are insufficient to achieve the capacity of two-unicast-Z networks.
Proof: For arbitrary m, we give a construction in which the achievability of (m, m) in the twounicast-Z network requires achieving (1, 1, · · · , 1) in the m-unicast network. Since linear codes are insufficient to achieve (1, 1, · · · , 1) in m-unicast networks, our construction implies that linear codes are insufficient to achieve (m, m) in two-unicast-Z networks. Our construction is shown in Fig. 5, for simplicity, we use m = 2.
In order to proof the theorem, we prove that rate (1, 1) is achievable in the two-unicast network B if, and only if, rate (2, 2) is achievable in the overall two-unicast-Z network.
For the first direction, if (1, 1) is achievable in the two-unicast network B, then simply setting Y i = X 1i + X 2i and V i = Z i − X 2i suffices to achieve (2,2) in the two-unicast-Z network.
For the other direction, let (2, 2) be achievable in the two-unicast-Z network, this implies that terminal 1 is satisfied, that is, there exists a one-to-one mapping f between X 11 , X 12 and V 1 , V 2 , i.e. (V 1 , V 2 ) = f (X 11 , X 12 ) = (f 1 (X 11 , X 12 ), f 2 (X 11 , X 12 )). In addition, let h be the mapping between the inputs and the outputs of the two-unicast network B, that is, (Z 1 , Z 2 ) = h(Y 1 , Y 2 ) = (h 1 (Y 1 , Y 2 ), h 2 (Y 1 , Y 2 )), hence to prove that (1, 1) is achievable in B, we need to prove that Z i = h i (Y 1 , Y 2 ) = h i (Y i ) and h i is one-to-one, for i ∈ {1, 2}.
In the following, we prove that h i (Y 1 , Y 2 ) = h i (Y i ) for i ∈ {1, 2}. For i ∈ {1, 2}, using the nodes encoding function and the mapping h, V i can be written as V i = g i (h i (f 1 (X 11 , X 21 ), f 2 (X 12 , X 22 )), X 2i ), however, since we already know that V i = f i (X 11 , X 12 ), the output of g i does not contain X 21 , X 22 . Since the output of g i does not contain X 21 , X 22 and g i does not have X 2j , where j ∈ {1, 2} and j = i as one of its two inputs, the output of h i (f 1 (X 11 , X 21 ), f 2 (X 12 , X 22 )) does not contain X 2j , where j ∈ {1, 2} and j = i. In other words, h i (f 1 (X 11 , X 21 ), f 2 (X 12 , X 22 )) is not a function of X 2j , where j ∈ {1, 2} and j = i. Thus, either f j (X 1j , X 2j ) is not a function of X 2j or f j (X 1j , X 2j ) is a function of X 2j and h i (f 1 (X 11 , X 21 ), f 2 (X 12 , X 22 )) is not a function of f j (X 1j , X 2j ). However, we cannot have that f j (X 1j , X 2j ) is not a function of X 2j , otherwise terminal 2 is not satisfied which contradicts the hypothesis that (2, 2) is achievable in the two-unicast-Z network. Thus, we conclude that h i (f 1 (X 11 , X 21 ), f 2 (X 12 , X 22 )) is not a function of f j (X 1j , X 2j ) = Y j for i, j ∈ {1, 2}, i = j, i.e.
Z i = h i (Y 1 , Y 2 ) = h i (Y i ) for i ∈ {1, 2}.
Now, it remains to prove that, for i ∈ {1, 2}, the mapping h i between Y i and Z i is one-to-one. We will do the proof for i = 1. For i = 2, the proof follows similarly. Let h 1 be not one-to-one, then there exist distinct Y
2 ). In other words, the mapping between the second source and the second terminal of the two-unicast-Z network is not one-to-one, i.e. the second terminal cannot always recover the message transmitted by the second source, a contradiction to the hypothesis that (2, 2) is achievable in the two-unicast-Z network. Thus, h 1 is one-to-one.
For X
21 = X(1)
21 and X
11 = X(1)
11 , we already know that we have Z
1 = Z(1)
1 and Z
2 = Z(1)
2 . In addition, we have (X
2 ). This means that there exists two distinct pairs (X
2 ). In other words, the mapping between the first source and the first terminal of the two-unicast-Z network is not one-to-one, i.e. the first terminal cannot always recover the message transmitted by the first source, a contradiction to the hypothesis that (2, 2) is achievable in the two-unicast-Z network. Thus, h 1 is one-to-one.
VIII. CONCLUSION
In this paper, we show that the generalized network sharing bound is not tight for two-unicast-Z networks. In addition, we show that for the two-unicast-Z networks vector linear codes outperform scalar linear codes in general. Another contribution of this paper is introducing a commutative algebraic approach to deriving linear network coding achievability results. This commutative algebraic approach is demonstrated by providing an alternate proof to the result of Wang et. al. regarding the achievability of rate (1, 1) in the network. Future directions to this work include exploring further the power of the developed commutative algebraic approach in deriving new feasibility results for different networks.
F
G i,j = β ei,ej if Head(e i ) = Tail(e j ) 0 otherwise,
Definition 3. 3 (
3Transfer matrix): Consider a DAG G = (V, E)
Fig. 2 .
2Network decomposition with respect to the subset U where S = S1 ∪ S2 and T = T1 ∪ T2
F 1
1⊂F be the set of variables in M 1 , and let F 2 ⊂F be the set of variables in M 2 . If there are no e 1 → e 2 paths in G, then F 1 ∩ F 2 = φ.Proof: (a) The proof of this part follows from Lemma 3.1, where U = C GN S . (b) The proof of this part follows from Lemma 3.1. (c) Here, we aim to show that the indeterminate variables that occur in polynomials of {1, 2}, do not occur in the polynomials of M 2 and vice-versa. Let m 1 be a monomial that occurs in some polynomial in p:si→e1 w(p), i ∈ {1, 2}. Similarly, let m 2 be a monomial that occurs in some polynomial in M 2 . It suffices to show that the variables in m 1 do not occur in the variables of m 2 and vice-versa. If possible, let β ea,eb -the local coding co-efficient from edge e a to edge e b be a variable that occurs in both m 1 and m 2 .
p), i ∈ {1, 2} and vice-versa. Let m 1 be a monomial that occurs in some polynomial in M 1 . Similarly, let m 2 be a monomial that occurs in some polynomial in p:e2→ti w(p), i ∈ {1, 2}. It suffices to show that the variables in m 1 do not occur in the variables of
.
Now, we will show that this setting cannot occur. To show that, first, set the achievability conditions in (16). Therefore, rate (1, 1) is not achievable using scalar linear coding.
degree of polynomial p with respect to the indeterminates in u i alone. Based on this notation, we can make the following important observation: For every monomial m in det(M) L , we have sumdegK (i) (m) = L, i = 1, 2. In effect, the above equation means that every monomial on the left hand side of (15) of Lemma 3.3 should also have a sum-degree of L with respect to the variables in u i alone, for each i = 1, 2.
Definition 4. 1 (
1Multi-degree of monomials): For any monomial m = n i=1
Definition 4. 2 (
2Multi-degree of polynomials): Let p = N i a i m i be a nonzero polynomial in the polynomial ring K[x 1 , · · · , x n ]
Definition 4. 3 (
3Sum-degree of monomials): For any monomial m = n i=1 x ti i in the polynomial ring K[x 1 , · · · , x n ] with multideg K[x1,··· ,xn]
Definition 4. 4 (
4Sum-degree of polynomials): Let p = N i a i m i be a nonzero polynomial in the polynomial ring K[x 1 , · · · , x n ] where, for all i ∈ {1, · · · , N }, a i ∈ K and m i is a monomial in K[x 1 , · · · , x n ]. Then, the sum-degree of p is sumdeg K[x1,··· ,xn] (p) = max i∈{1,··· ,N } sumdeg K[x1,··· ,xn] (m i ) .
Definition 4. 5 (
5Graded lexicographic order):
Definition 4. 7 (
7Monomial graded lexicographic order): A monomial graded lexicographic order is defined as follows. Let n i=1
Definition 4.8 (Monomial reverse graded lexicographic order): A monomial reverse graded lexicographic order is defined as follows. Letn i=1
If there exists a polynomial P ∈ K[F] such that αP = (det(M)) L , then for any monomial m in αP , we have sumdegK (i) (m) ≤ L.
α) for any monomial m in P (ui) ) > L − 1 for any monomial m ∈ αP(ui) . Therefore, for any monomial m in αP(ui) We have αP (ui) = det(M ) L − αP(ui) . Now, every monomial m on the left hand side of the last equation has sumdegK (i) (m) ≥ L. For the right hand side, sumdeg K (i) (det(M ) L − αP (ui) ) = L. Since the right hand side and the left hand side must have the same sum-degree, every monomial in αP (ui) has sum-degree in K (i) equal to L. We now describe a consequence of Lemma 4.1. Lemma 4.2: Let P be a polynomial in K[F] such that αP = (det(M)) L , where α = p:s2→t1 w(p), and for any monomial m in αP , we have sumdeg K (i) (m) ≤ L. Then, the following properties hold: (a) For every monomial m in P , sumdeg K (i) (m) ≤ L, (b) If there is at least one s 2 → t 1 path via e i , i ∈ {1, 2}, then, for every monomial m in P , sumdeg K (i) (m) ≤ L − 1, (c) If there is at least one s 2 → t 1 path that does not go through the edge e i , i ∈ {1, 2}, then, for every monomial m in P , (M) L ) = L, and considering a graded lexicographic ordering on monomials.
) ≤ L − 1 for any monomial m in P .
(M) L ) = L and αP = (det(M)) L , we have sumdeg K (1)
(
P ) = L. Again, since we use reverse graded lexicographic ordering, ) ≥ L for any monomial m in P . Combining the fact that sumdeg K(1) (m) ≥ L for any monomial m in P with part (a), we conclude that for every monomial m
) ≤ L. . Therefore the hypothesis of Lemma 4.2 holds. Now, because there is a s 2 → t 1 via e 1 path, statement b) of Lemma 4.2 implies that sumdeg K (1) (m) ≤ L − 1 for any monomial m in P . However, because there is an s 2 → t 1 \e 1 path, statement (c) of Lemma 4.2 implies that sumdeg K (1)
p) = b 1 u 1 = 0 because there exists a s 2 → t 1 via e 1 path, we conclude that case A can occur only due to one of the following scenarios:A1) e 2 → t 1 paths do not exist (µ 21 = 0), or A2) s 2 → e 2 and e 1 → e 2 paths do not exist, (b 2 u 2 = 0, λ 12 u 2 = 0).Case A1) Let (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }) define the left network.(24) can be written asb 1 u 1 P = µ L−1 11 µ L 22 a 1 u 1 b 2 u 2 − b 1 u 1 a 2 u 2 L .(25)Recall, from (20), that b 1 u 1 =p:s2→e1 w(p). Let F 1 ⊂F be the set of variables in p:s2→e1 w(p), and let F 2 ⊂F be the set of variables in M 2 , i.e. the set of variables inµ ij , i, j ∈ {1, 2}. From Lemma 3.2 part (c), F 1 ∩ F 2 = φ.This implies that the variables in the polynomial b 1 u 1 are disjoint from the variables in the polynomial µ L−1 11 µ L 22 . Therefore, P = µ L−1 11 µ L 22 P for some polynomial P . Now, (25) can be written asb 1 u 1 P = a 1 u 1 b 2 u 2 − b 1 u 1 a 2 u 2 L = (det(M 1 )) L = (det(M 1 Λ)) L .(26)Recall from Lemma 3.2 that M 1 Λ is the network transfer matrix for (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }). Since b 1 u 1 = p:s2→e1 w(p), we conclude from (26) along with Lemma 3.3 that (1,1) is not achievable in (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }) using scalar linear coding. Now, we aim to use the inductive assumption on (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }). In order to use the inductive assumption, we first need to check that (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }) possesses the same structure described in the theorem statement. That is, there are s 1 → e 1 and s 2 → e 2 paths in (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }). Indeed, there must be a s 1 → e 1 path. Otherwise, there are no s 1 → t 1 paths in (G, S 1 , T 1 , S 2 , T 2 ) because every s 1 → t 1 goes through the GNS edges e 1 or e 2 and since there are no e 2 → t 1 paths, s 1 connects to t 1 only through e 1 . Therefore, a s 1 → e 1 path exists in (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }). Moreover, there exists a s 2 → e 2 path in (G 1 , {s 1 }, {e 1 }, {s 2 }, {e 2 }).
.
However, there are no e 1 → e 2 paths in the network. Therefore, p) = µ 11 . Hence, we conclude from (28) along with Lemma 3.3 that (1, 1) is not achievable in (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }) using scalar linear coding. Now, we aim to use the inductive assumption on (G 2 , {e 2 }, {t 1 }, {e 1 }, {t 2 }).
Recall from Lemma 3.2 that ΛM 2 is the network transfer matrix for (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }). Since µ 21 = p:e2→t1 w(p), we conclude from (31) along with Lemma 3.3 that (1,1) is not achievable in (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 })using scalar linear coding. Now, we aim to use the inductive assumption on (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }). In order to use the inductive assumption, we first need to check that (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 }) possesses the same structure described in the theorem statement. That is, there are e 1 → t 1 and e 2 → t 2 paths in (G 2 , {e 1 }, {t 1 }, {e 2 }, {t 2 })
F 1
1⊂F be the set of variables in M 1 , i.e. the set of variables in a 1 u 1 , b 2 u 2 , b 1 u 1 , a 2 u 2 , and let F 2 ⊂F be the set of variables in M 2 , i.e. the set of variables in µ ij , i, j ∈ {1, 2}. Since, in this case, there are no e 1 → e 2 paths, from Lemma 3.2 part (e) we conclude thatF 1 ∩ F 2 = φ.This implies that the variables in the polynomial b 2 u 2 are disjoint from the variables in the polynomial µ L−1 21 µ L 12 . Therefore, P = µ L−1 21 µ L 12 P for some polynomial P . Now, (32) can be writtenas b 2 u 2 P = a 1 u 1 b 2 u 2 − b 1 u Recall from Lemma 3.2 that M 1 Λ is the network transfer matrix for (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }). p) = b 2 u 2 .Hence, we conclude from(33)along with Lemma 3.3 that (1, 1) is not achievable in (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }) using scalar linear coding. Now, we aim to use the inductive assumption on (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }). In order to use the inductive assumption, we first need to check that (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }) possesses the same structure described in the theorem statement. That is, there are s 1 → e 2 and s 2 → e 1 paths in (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }). Indeed, there must be a s 1 → e 2 path. Otherwise, there are no s 1 → t 1 paths in (G, S 1 , T 1 , S 2 , T 2 ) because every s 1 → t 1 path goes through the GNS edges e 1 or e 2 and since there are no e 1 → t 1 paths, s 1 connects to t 1 only through e 2 . Therefore, a s 1 → e 2 path exists in (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }). Moreover, there exists a s 2 → e 1 path in (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }). Otherwise, (G, S 1 , T 1 , S 2 , T 2 ) has e 2 as a single edge GNS cut, a contradiction to the hypothesis that (G, S 1 , T 1 , S 2 , T 2 ) has a minimum GNS cut of size two. Now, we can use the inductive assumption on (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }). Since (1, 1) is not achievable in (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }) using scalar linear coding, by inductive assumption there is a single edge GNS cut in (G 1 , {s 1 }, {e 2 }, {s 2 }, {e 1 }), call it e .
Now, we consider the two unicast-Z network (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }), where G 3 = (V 3 , E 3 ) ⊂ G, E 3 = {e ∈ E : e belongs to some s 1 ∪ e 1 → e 2 ∪ t 2 path} − {s 2 , t 1 } and V 3 = {v ∈ V : v is the head or tail of edge e, for some e ∈ E 3 }. Notice that for the network (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }), is not achievable in (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }) using scalar linear coding. Now, we aim to use the inductive assumption on (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }). In order to use the inductive assumption, we first need to check that (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }) possesses the same structure described in the theorem statement. That is, there are s 1 → e 2 and e 1 → t 2 paths in (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }). Indeed, there must be a s 1 → e 2 path. Otherwise, there are no s 1 → t 1 paths in (G, S 1 , T 1 , S 2 , T 2 ) because every s 1 → t 1 path goes through the GNS edges e 1 or e 2 and since there are no e 1 → t 1 \e 2 paths, s 1 connects to t 1 through e 2 . Therefore, a s 1 → e 2 path exists in (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }). Moreover, there exists a e 1 → t 2 path in (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }). Otherwise, there are no s 2 → t 2 paths in (G, S 1 , T 1 , S 2 , T 2 ) because every s 2 → t 2 path goes through the GNS edges e 1 or e 2 and since there are no s 2 → e 2 \e 1 paths, s 2 connects to t 2 through e 1 . Therefore, a e 1 → t 2 path exists in (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }). Now, we can use the inductive assumption on (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }). Since (1, 1) is not achievable in (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }) using scalar linear coding, by inductive assumption there is a single edge GNS cut in (G 3 , {s 1 }, {e 2 }, {e 1 }, {t 2 }), call it e .
Fig. 3 .
3A two-unicast-Z network where GNS bound is 3 and the maximum achievable sum-rate is 2.5; the network requires vector linear codes.
Fig. 4 .
4An achievability scheme for rate (1.5, 1) using vector linear codes.
Fig. 5 .
5A two-unicast-Z network where rate (2, 2) is achievable if and only if rate (1, 1) is achievable over two-unicast network B.
. Since V i = g i (X 2i , Z i ), (X
Remark 3.1: Let s and t be edges. Suppose that all s → t paths contain some edge e, thenp:s→t
w(p)
can be factorized as follows
p:s→t
w(p) =
p:s→e
w(p)
p:e→t
w(p).
Remark 3.2: Let s 1 , s 2 , t 1 , t 2 be edges,
p1:s1→t1
p2:s2→t2
w(p 1 )w(p 2 ) =
p:s1→t1
w(p)
p:s2→t2
w(p).
)Recall that λ 12 u 2 =
p:e1→e2
w(p), a 2 u 2 =
p:s1→e2 \e1
w(p) and µ 12 =
p:e1→t2 \e2
w(p). Therefore, (36)
can be rewritten as
p:e1→e2
w(p)P =
p:s1→e2 \e1
w(p)
p:e1→t2 \e2
w(p)
L .
(37)
We can write
p:s1→e2
w(p) =
p:s1→e2 via e1
w(p) +
p:s1→e2 \e1
w(p)
=
p:s1→e1
w(p)
p:e1→e2
w(p) +
p:s1→e2 \e1
w(p).
(38)
Also, we have
p:e1→t2
w(p) =
p:e1→t2 via e2
w(p) +
p:e1→t2 \e2
w(p)
=
p:e1→e2
w(p)
p:e2→t2
w(p) +
p:e1→t2 \e2
w(p).
(39)
From (38) and (39), we have
p:s1→e2
w(p)
p:e1→t2
w(p) =
p:s1→e2 \e1
w(p)
p:e1→t2 \e2
w(p) +P
p:e1→e2
Asymptotic interference alignment for optimal repair of mds codes in distributed storage. V R Cadambe, S A Jafar, H Maleki, K Ramchandran, C Suh, IEEE Transactions on Information Theory. 595V. R. Cadambe, S. A. Jafar, H. Maleki, K. Ramchandran, and C. Suh. Asymptotic interference alignment for optimal repair of mds codes in distributed storage. IEEE Transactions on Information Theory, 59(5):2974-2987, 2013.
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|
[] |
[
"Gravitational waves from a particle in circular orbits around a rotating black hole to the 11th post-Newtonian order",
"Gravitational waves from a particle in circular orbits around a rotating black hole to the 11th post-Newtonian order"
] |
[
"Prog ",
"Theor "
] |
[] |
[
"Exp. Phys"
] |
We compute the energy flux of the gravitational waves radiated by a particle of mass µ in circular orbits around a rotating black hole of mass M up to the 11th post-Newtonian order (11PN), i.e. v 22 beyond the leading Newtonian approximation where v is the orbital velocity of the particle. By comparing the PN results for the energy flux with high-precision numerical results in black hole perturbation theory, we find the region of validity in the PN approximation becomes larger with increasing PN order. If one requires the relative error of the energy flux in the PN approximation to be less than 10 −5 , the energy flux at 11PN (4PN) can be used for v 0.33 (v 0.13). The region of validity can be further extended to v 0.4 if one applies a resummation method to the energy flux at 11PN. We then compare the orbital phase during a two-year inspiral from the PN results with the high-precision numerical results. We find that for late (early) inspirals when q ≤ 0.3 (q ≤ 0.9), where q is the dimensionless spin parameter of the black hole, the difference in the phase is less than 1 (10 −4 ) rad and hence these inspirals may be detected in the data analysis for space detectors such as eLISA/NGO by the PN templates. We also compute the energy flux radiated into the event horizon for a particle in circular orbits around a non-rotating black hole at 22.5PN, i.e. v 45 beyond the leading Newtonian approximation, which is comparable to the PN order derived in our previous work for the energy flux to infinity at 22PN.
|
10.1093/ptep/ptv012
|
[
"https://arxiv.org/pdf/1412.5689v2.pdf"
] | 118,585,350 |
1412.5689
|
ea713a411f70b44fe47bfde200772364c8029305
|
Gravitational waves from a particle in circular orbits around a rotating black hole to the 11th post-Newtonian order
30 pages
Prog
Theor
Gravitational waves from a particle in circular orbits around a rotating black hole to the 11th post-Newtonian order
Exp. Phys
201530 pages10.1093/ptep/ptv012arXiv:1412.5689v2 [gr-qc]
We compute the energy flux of the gravitational waves radiated by a particle of mass µ in circular orbits around a rotating black hole of mass M up to the 11th post-Newtonian order (11PN), i.e. v 22 beyond the leading Newtonian approximation where v is the orbital velocity of the particle. By comparing the PN results for the energy flux with high-precision numerical results in black hole perturbation theory, we find the region of validity in the PN approximation becomes larger with increasing PN order. If one requires the relative error of the energy flux in the PN approximation to be less than 10 −5 , the energy flux at 11PN (4PN) can be used for v 0.33 (v 0.13). The region of validity can be further extended to v 0.4 if one applies a resummation method to the energy flux at 11PN. We then compare the orbital phase during a two-year inspiral from the PN results with the high-precision numerical results. We find that for late (early) inspirals when q ≤ 0.3 (q ≤ 0.9), where q is the dimensionless spin parameter of the black hole, the difference in the phase is less than 1 (10 −4 ) rad and hence these inspirals may be detected in the data analysis for space detectors such as eLISA/NGO by the PN templates. We also compute the energy flux radiated into the event horizon for a particle in circular orbits around a non-rotating black hole at 22.5PN, i.e. v 45 beyond the leading Newtonian approximation, which is comparable to the PN order derived in our previous work for the energy flux to infinity at 22PN.
in Sec. 4. Section 5 is devoted to a summary and discussions. Finally in the appendices we describe supplemental equations to practically compute the formulas in Sec. 2. Throughout this paper, we use geometrized units with c = G = 1.
Basic formulation
We solve the Teukolsky equation to calculate gravitational radiation from a particle orbiting around a Kerr black hole. In the Teukolsky formalism, the gravitational perturbation of the Kerr black hole is described by a master variable. If we consider the outgoing radiation to infinity, the master variable, the Newman-Penrose quantity Ψ 4 , is related to the amplitude of the gravitational wave at infinity by
Ψ 4 → 1 2 (ḧ + − iḧ × ) for r → ∞.(1)
The Teukolsky equation can be solved by the decomposition of Ψ 4 as
Ψ 4 = 1 (r − ia cos θ) 4 ℓ,m ∞ −∞ dω e −iωt e imϕ √ 2π −2 S a ω ℓm (θ) R ℓmω (r),(2)
where a is the angular momentum of the black hole and the angular function −2 S aω ℓm (θ) is the spin-weighted spheroidal harmonic with spin weight s = −2, normalized as π 0 | −2 S aω ℓm (θ)| 2 sin θ dθ = 1.
The decomposition of Ψ 4 Eq. (2) leads to the separation of the Teukolsky equation into radial and angular parts,
∆ 2 d dr 1 ∆ d dr − − K 2 + 4 i (r − M ) K ∆ + 8 i ωr + λ R ℓmω (r) = T ℓmω (r),(4)
where λ is the eigenvalue of −2 S aω ℓm , ∆ = r 2 − 2M r + a 2 , K = (r 2 + a 2 )ω − ma and T ℓmω is the source term of the particle.
To solve Eq. (5), we define two independent homogeneous solutions of the radial Teukolsky equation as
R in ℓmω = B trans ℓmω ∆ 2 e −ikr * for r * → −∞, r 3 B ref ℓmω e iωr * + r −1 B
inc ℓmω e −iωr * for r * → +∞, R up ℓmω = C up ℓmω e ikr * + ∆ 2 C ref ℓmω e −ikr * for r * → −∞, r 3 C trans ℓmω e iωr * for r * → +∞,
where k = ω − ma/(2M r + ) and r * is the tortoise coordinate defined as
r * = r + 2M r + r + − r − ln r − r + 2M − 2M r − r + − r − ln r − r − 2M ,(7)
with r ± = M ± √ M 2 − a 2 .
3/30
Using the two independent solutions Eq. (6), with the Green function method one can construct a solution of the radial Teukolsky equation that is purely outgoing at infinity and purely incoming at the horizon
R ℓmω (r) = 1 W ℓmω R up ℓmω (r) r r+ dr ′ R in ℓmω T ℓmω ∆ 2 + R in ℓmω (r) ∞ r dr ′ R up ℓmω T ℓmω ∆ 2 ,(8)
where the Wronskian W ℓmω is given as
W ℓmω = 2 i ω C trans ℓmω B inc ℓmω .(9)
Then, the solution has the asymptotic form at the horizon as
R ℓmω (r → r + ) = B trans ℓmω ∆ 2 e −ikr * 2iωC trans ℓmω B inc ℓmω ∞ r+ dr ′ R up ℓmω T ℓmω ∆ 2 ≡ Z H ℓmω ∆ 2 e −ikr * ,(10)
while the solution has the following asymptotic form at infinity as
R ℓmω (r → ∞) = r 3 e iωr * 2iωB inc ℓmω ∞ r+ dr ′ R in ℓmω T ℓmω ∆ 2 ≡ Z ∞ ℓmω r 3 e iωr * .(11)
Using the formula of the source term T ℓmω [2,3], Z ∞,H ℓmω are expressed as ,
I ∞ ℓmω = R in ℓmω {A nn0 + Am n0 + Amm 0 } − dR in ℓmω dr {Am n1 + Amm 1 } + d 2 R in ℓmω d 2 r Amm 2 r=r(t),θ=θ(t) ,(13)
and where A nn0 and other terms are given in Appendix A. When the particle follows bound geodesics of Kerr spacetime, there exist three fundamental frequencies for the orbits [34] and hence the frequency spectrum of T ℓmω becomes discrete. In the case of circular orbits,Z ∞,H ℓmω in Eq. (12) takes the form
Z ∞,H ℓmω =Z ∞,H ℓmω δ(ω − m Ω),(14)
where Ω = v r /(r 0 (1 + qv 3 r )) is the angular frequency of the particle, r 0 is the orbital radius, q = a/M , and v r = M/r 0 is the orbital velocity. 4/30 The time-averaged gravitational wave luminosity at infinity is then given by [35]
dE dt ∞ = ∞ ℓ=2 ℓ m=−ℓ |Z ∞ ℓmω | 2 4πω 2 ≡ dE dt N ∞ ℓ=2 ℓ m=−ℓ η ∞ ℓm ,(15)
where · · · represents the time average, ω = mΩ, and (dE/dt) N is the Newtonian quadrupole formula defined by
dE dt N = 32 5 µ M 2 v 10 ,(16)
with v ≡ (M Ω) 1/3 . Similarly, the time-averaged gravitational wave luminosity at the horizon becomes [35] dE
dt H = ∞ ℓ=2 ℓ m=−ℓ α ℓmω |Z H ℓmω | 2 4πω 2 ≡ dE dt N v 5 ∞ ℓ=2 ℓ m=−ℓ η H ℓm ,(17)
where α ℓmω = 256(2M r + ) 5 k(k 2 + 4ǫ 2 )(k 2 + 16ǫ 2 )ω 3 |C| 2 , Finally, the gravitational waveforms are given in terms ofZ ∞ ℓmω as
h + − i h × = − 2 r ℓ,mZ ∞ ℓmω ω 2 e imϕ √ 2π −2 S aω ℓm (θ) e iω(r * −t) .(19)
In this paper, using Eqs. (15), (17) and (19) we compute the gravitational energy flux and waveforms in the post-Newtonian approximation, i.e., in the expansion with respect to v = (M Ω) 1/3 . For this purpose, it is necessary to compute the asymptotic amplitudes Z ∞,H ℓmω , which involve calculations of the angular Teukolsky function −2 S aω ℓm (θ) and the radial Teukolsky functions R in,up ℓmω (r). To this end, in Appendices B and C, we give a short review of the calculations of the series expansions of −2 S aω ℓm (θ) and R in,up
ℓmω (r) in terms of ǫ ≡ 2M ω = O(v 3 ) and z = ωr = O(v).
11PN results for the time-averaged energy flux
In this paper, we derive the 11PN formula for the energy flux in the case of a test particle in a circular orbit around the equatorial plane of a Kerr black hole. Since the expressions are very long, we exhibit the 7.5PN expression for the energy flux at infinity in Sec. 3.1 and the next new 7PN terms in the energy flux into the horizon in Sec. 3.2. We also compute the energy flux into the event horizon for a particle in a circular orbit around a Schwarzschild black hole up to 22.5PN beyond the Newtonian approximation. The complete expressions for the energy flux will be publicly available online [
+ − 8 45 q + 2 15 q 3 Ψ (0,1) A (q) + − 5632 315 q − 5632 105 q 3 Ψ (0,2) A (q) + 109028 315 π 2 + 1149838672589069 235994358600 − 18591892 6615 ln 2 − 15500876 11025 γ q − 34227289 52920 π q 2 + 105412967 238140 q 3 ln v − 7750438 11025 q(ln v) 2 v 15 ,(20)
where κ = 1 − q 2 , γ is the Euler constant, ζ(n) is the zeta function, (20) are the new terms derived by the post-Newtonian approximation in this paper. 1 Among these terms, the O(v 9 ) − O(v 11 ) terms agree with the analytic expressions in Ref. [36], which determined the post-Newtonian coefficients of the energy flux up to 20PN by fitting with very accurate, one part in 10 600 , numerical calculation of the energy flux. For the 6PN and higher PN order energy flux at infinity, in Ref. [36], some of the post-Newtonian coefficients are not extracted as analytic values but as numerical values. This is not only because it is difficult to numerically extract analytic coefficients for combinations of transcendental numbers such as π, Euler's constant, and logarithms of prime numbers, but also because numerical fitting of post-Newtonian coefficients is done by presenting these coefficients as a polynomial in q although irrational functions in q such as polygamma functions and logarithms appear from 6.5PN onward as shown in Eq. (20). Further, by performing a small q expansion of our 11PN expression, we also find that our 11PN energy flux to infinity is consistent with the one in Ref. [36] up to 11PN. 2 From Eq. (20), we find the coefficient in q (ln v) 0 at 6PN is given by
Ψ (n,m) A (q) = 1 2 Ψ (n) 3 + i m q 1 − q 2 + Ψ (n) 3 − i m q 1 − q 2 , Ψ (n,m) B (q) = 1 2 i Ψ (n) 3 + i m q 1 − q 2 − Ψ (n) 3 − i m q 1 − q 2 , and Ψ (n) (z) is the polygamma function. The O(v 9 ) − O(v 15 ) terms in Eq.− 270159823411 558835200 π + 81878 315 π γ + 54514 105 π ln 2.(21)
The above analytic value is consistent with the numerical value of the coefficient in q (ln v) 0 at 6PN energy flux to infinity in Ref. [36], 83.16039023577041 . . ..
Horizon flux
The next new 7PN terms for the energy flux into the horizon are given by dE dt
q 3 κ Ψ (1,2) B (q) + 6151 252 + 8 3 κ q + 2 κ − 1437 112 q 3 + − 26 3κ − 87 28 q 5 + 4 q 7 κ Ψ (0,1) B (q) + 8 3 q 2 − 2 q 4 Ψ (0,1) B (q) 2 + 24 κ + 24 + 296 + 272 κ q 2 + − 56 κ + 96 q 4 − 48 q 6 κ + 32 + 64 κ q + 96 + 192 κ q 3 Ψ (0,2) B (q) Ψ (1,2) A (q) + 64 3 q + 64 q 3 Ψ (0,2) B (q) 3 + − 1712 105κ − 1712 105 + − 6848 35κ − 60712 315 q 2 + 1712 15κ − 3146 105 q 4 + 3424 35 q 6 κ + − 6848 105 q − 6848 35 q 3 Ψ (0,2) B (q) ln v,(22)
where dE dt (9) H is defined through
dE dt H = dE dt N v 5 2NPN−5 n=0 dE dt (n) H v n ,
and N PN is the PN order, i.e. N PN = 11 when the PN order is 11PN. Notice that the energy flux down into the horizon starts at O(v 5 ) (O(v 8 )), i.e. 2.5PN (4PN), beyond the quadrupole formula when q = 0 (q = 0) [32,37]. Again, performing a small q expansion of our 11PN expression, we find our 11PN energy flux to the horizon is consistent with the one in Ref. [36] up to 11PN. For the case of a particle in a circular orbit around a Schwarzschild black hole, we also derive the energy flux down the event horizon at 22.5PN, which is consistent with the one in Ref. [36] up to 22.5PN.
Comparisons between 11PN results and numerical results
To investigate the accuracy of the energy flux in the post-Newtonian approximation, we compare PN results with numerical results, based on a method in Refs. [38,39]. With this 9/30 numerical method, one can investigate gravitational waves with an accuracy of about 14 significant figures in double precision calculations. Hence one can use the numerical results to estimate the accuracy in the PN results by a comparison. For the comparison in this section, we need to compute the energy flux using Eqs. (15) and (17). To numerically compute the energy flux, we set the maximum value of ℓ to 15, which gives the relative error in the energy flux better than 10 −5 for the comparisons in this section. For the energy flux at 11PN, we need to compute ℓ up to 13 (5) for the energy flux to infinity (the horizon).
In Sec. 4.1, comparisons for the the energy flux are done for several values of the spin of the Kerr black hole. In Secs. 4.2 and 4.3, the same comparisons are done using resummation techniques, the factorized resummation introduced in Ref. [40] and the exponential resummation in Ref. [41], for the post-Newtonian approximation to the energy flux. We will see how resummation methods improve the performance in the post-Newtonian approximation for the energy flux. Finally, in Sec 4.4, we compare the total cycle of orbits during a two-year inspiral for representative binaries in the eLISA frequency band. . From these figures, one will find that the relative error becomes smaller with increasing PN order for v ≤ 0.3, except for accidental agreements for certain values of the velocity. However, the relative error around ISCO does not necessarily become smaller with increasing PN order when q > 0.3. The relative error for 11PN is smaller than 10 −5 when v 0.33, irrespective of the values of q investigated in the paper. Figure 3 shows the relative error in the energy flux down the horizon from numerical results and PN approximations as a function of the orbital velocity up to ISCO in the case of the Schwarzschild black hole. The agreement between the numerical energy flux and post-Newtonian energy flux becomes better when the PN order is higher even around ISCO. The relative error in the 22.5PN energy flux into the horizon around ISCO is about 10 −5 , which is comparable to the one for the 22PN energy flux to infinity in Ref. [27].
Energy flux: Taylor expanded PN approximation
Energy flux: Factorized resummation to PN approximation
In this section, we compare the total energy flux from numerical results with PN results using a factorized resummation. The factorized resummation was introduced to improve the convergence in the PN energy flux to infinity for a test particle moving in Schwarzschild spacetime [40,42,43] and Kerr spacetime [31]. The factorized resummation was then extended to the PN energy flux down the horizon of the Schwarzschild black hole in Ref. [44] and the Kerr black hole in Ref. [45].
Factorization of the energy flux at infinity.
In the factorized resummation of the energy flux at infinity, we decompose the multipolar gravitational waveforms into five factors as
h ℓm = h (N,ǫp) ℓmŜ (ǫp) eff T ℓm e iδℓm (ρ ℓm ) ℓ ,(23)
where ǫ p denotes the parity of the multipolar waveforms, h
|1-dEdt PN /dEdt Numerical | v q=0.998 4PN 5PN 6PN 7PN 8PN 9PN 10PN 11PN Fig. 1
Absolute values of the relative error in the total energy flux from numerical results and PN approximations as a function of the orbital velocity, v = (M/r 0 ) 1/2 [1 + q (M/r 0 ) 3/2 ] −1/3 , up to ISCO for q = 0.1, 0.3, 0.5, 0.7, 0.9 and 0.998. The relative error for 11PN is smaller than 10 −5 when v 0.33. formalism, T ℓm resums the leading logarithms of the tail effects,δ ℓm is the supplemental phase factor and ρ ℓm is the ℓth root of the amplitude of the waveforms, which takes care of a term linear in ℓ at 1PN in the waveforms and means that a better convergence in the factorized waveforms might be expected (for more details see, e.g., Refs. [40,42,43]).
The first factor h (N,ǫp) ℓm takes the form where φ is the orbital phase and n
h (N,ǫp) ℓm = GM ν c 2 r n (ǫp) ℓm c ℓ+ǫp (ν) v ℓ+ǫp Y ℓ−ǫp,−m π 2 , φ ,(24)|1-dEdt PN /dEdt Numerical | v q=-0.90 4PN 5PN 6PN 7PN 8PN 9PN 10PN 11PN(ǫp) ℓm are n (0) ℓm =(im) ℓ 8π (2ℓ + 1)!! (ℓ + 1)(ℓ + 2) ℓ(ℓ − 1) , (25a) n (1) ℓm = − (im) ℓ 16πi (2ℓ + 1)!! (2ℓ + 1)(ℓ + 2)(ℓ 2 − m 2 ) (2ℓ − 1)(ℓ + 1)ℓ(ℓ − 1) ,(25b)
and c ℓ+ǫp (ν) are functions of the symmetric mass ratio ν ≡ µ M/(M + µ) 2 , defined by
c ℓ+ǫp (ν) = 1 2 − 1 2 √ 1 − 4ν ℓ+ǫp−1 + (−) ℓ+ǫp 1 2 + 1 2 √ 1 − 4ν ℓ+ǫp−1 .(26)
12/30 The second factorŜ
(ǫp) eff in Eq. (23) is defined bŷ S (ǫp) eff = Ẽ for ǫ p = 0 (ℓ + m = even) , vL z /M for ǫ p = 1 (ℓ + m = odd) ,(27)
whereẼ andL z are the specific energy and the angular momentum of the particle, given bỹ
E = 1 − 2v 2 r + qv 3 r 1 − 3v 2 r + 2qv 3 r ,L z = r 0 v r (1 − 2qv 3 r + q 2 v 4 r ) 1 − 3v 2 r + 2qv 3 r .(28)
The third factor T ℓm in Eq. (23) is defined by
T ℓm = Γ(ℓ + 1 − 2imM Ω) Γ(ℓ + 1) e mπM Ω e 2imM Ω ln(2mΩr0s) ,(29)
where r 0s = 2M/ √ e is introduced to reproduce the test-particle limit waveforms [31].
The fourth and fifth factors in Eq. (23),δ ℓm and ρ ℓm , can be derived by comparing the multipolar waveforms Eq. (23) with those obtained from Eq. (19). For the comparison, it is useful to express waveforms Eq. (19) in terms of the −2 spin-weighted spherical harmonics
−2 Y lm (θ, ϕ) ≡ −2 P lm (θ) e imϕ / √ 2π [31] h + − i h × = − 2 r ℓ,mZ ∞ ℓmω ω 2 e imϕ √ 2π −2 S aω ℓm (θ) e iω(r * −t) , ≡ − 2 r l,mC ∞ lmω ω 2 e imϕ √ 2π −2 P lm (θ) e iω(r * −t) ,(30)
where −2 P lm (θ) is defined as
−2 P lm (θ) = (−1) m (l + m)!(l − m)!(2l + 1) 2(l + 2)!(l − 2)! sin 2l θ 2 13/30 × l+2 r=0 l + 2 r l − 2 r − 2 − m (−1) l−r+2 cot 2r−2−m θ 2 . (31) From Eq. (30), one can computeC ∞ lmω fromZ ∞ ℓmω C ∞ lmω = 2π 0 dϕ π 0 sin θ dθ ℓ ′ ℓ ′ m ′ =−ℓ ′Z ∞ ℓ ′ m ′ ω ′ −2 S aω ′ ℓ ′ m ′ (θ) −2 P lm (θ) 2π e i (m ′ −m) ϕ , = π 0 sin θ dθ ℓ ′Z ∞ ℓ ′ mω −2 S aω ℓ ′ m (θ) −2 P lm (θ),(32)
where we used the orthogonality condition of the −2 spin-weighted spherical harmonics,
2π 0 dϕ π 0 sin θ dθ −2 Y lm (θ, ϕ) −2Ȳl ′ m ′ (θ, ϕ) = δ ll ′ δ mm ′ ,(33)
andX is the complex conjugate of X. Note that in Eq. (32) the mixing ofZ ∞ ℓmω happens among the same m and different ℓ modes [31]. Note that the infinite summation over ℓ ′ in Eq. (32) can be truncated at a certain ℓ ′ for a given post-Newtonian order sinceZ ∞ ℓ ′ mω = O(v ℓ ′ +2+ǫp ) (see, e.g., Ref. [30]).
Once we obtainC ∞ ℓmω from Eq. (32), it is straightforward to computeδ ℓm and ρ ℓm from the following relation between h ℓm andC ∞ ℓmω [14,30]
h ℓm = sin Θ dΘ dΦ (h + − i h × ) −2Ȳℓm (Θ, Φ), = − 2 r ℓ ′ ,m ′C ∞ ℓ ′ m ′ ω ′ e im ′ Ωr * (m ′ Ω) 2 sin Θ dΘ dΦ e −im ′ (Ω t−Φ) −2 Y ℓ ′ m ′ (Θ, ϕ) −2Ȳℓm (Θ, Φ), = − 2 rC ∞ ℓmω e i mΩ(r * −t) e i m ϕ (mΩ) 2 .(34)
Using the factorized waveforms h ℓm , Eq. (23), computed from Eqs. (32) and (34), the time-averaged energy flux to infinity is computed as
dE dt ∞ = 1 16π ∞ ℓ=2 ℓ m=−ℓ (mM Ω) 2 r M h ℓm 2 .(35)
Factorization of the energy flux down the horizon.
For the factorized resummation of the energy flux down the horizon, the modal energy flux is decomposed as [44,45]
η H ℓm = 1 − 2v 3 r + a η N,H ℓm (Ŝ (ǫp) eff ) 2 (ρ H ℓm ) 2ℓ ,(36)
where the factor (1 − 2v 3 r + /a) is motivated by the factor k = ω − ma/(2M r + ) = m (Ω − a/(2M r + )) in Eq. (18), which is responsible for the sign of the modal energy flux to the horizon. The second factor η N,H ℓm represents the leading term in the modal energy flux into the horizon and takes the form
η N,H ℓm = v 4(ℓ−2)+2ǫp n (H,ǫp) ℓm c H ℓm (q) ,(37)
where n (H,0) ℓm = − 5 32
(ℓ + 1)(ℓ + 2) ℓ(ℓ − 1) 2ℓ + 1 [(2ℓ + 1)!!] 2 (ℓ − m)! [(ℓ − m)!!] 2 (ℓ + m)! [(ℓ + m)!!] 2 , 14/30 n (H,1) ℓm = − 5 8ℓ 2 (ℓ + 1)(ℓ + 2) ℓ(ℓ − 1) 2ℓ + 1 [(2ℓ + 1)!!] 2 [(ℓ − m)!!] 2 (ℓ − m)! [(ℓ + m)!!] 2 (ℓ + m)! ,(38a)
and c ℓ+ǫp (ν)
c H ℓm (q) = 1 q ℓ k=0 k 2 + m 2 − k 2 q 2 , = q m 2 1 − q 2 ℓ 1 − imq 1 − q 2 ℓ 1 + imq 1 − q 2 ℓ ,(39)
where (z) n = Γ(z + n)/Γ(z).
The definition for the third factorŜ Figures 4 and 5 show the relative error in the total energy flux from numerical results and PN approximations as a function of the orbital velocity up to ISCO using the factorized resummation to PN approximations [40]. From these figures, one will find that the relative error becomes smaller as PN order becomes higher for v ≤ 0.3, except for accidental agreements for certain values of the velocity. However, the relative error around ISCO does not necessarily become smaller for higher PN orders when q > 0.3. The relative error for 11PN is smaller than 10 −5 when v 0.4, irrespective of the values of q investigated in the paper. We note that the region of the velocity, v 0.4, is larger than the one using the Taylor expanded PN energy flux, v 0.33.
Comparisons with numerical results.
Energy flux: Exponential resummation to PN approximation
In this section, we compare the total energy flux from numerical results with PN results using the exponential resummation [41,46]. In the exponential resummation, the modal energy fluxes to infinity, η ∞ ℓm , and the horizon, η H ℓm , are decomposed as
η ∞ ℓm = 1 1 − 3v 2 r + 2qv 3 r η N,∞ ℓm exp [ln (η ∞ ℓm )] , η H ℓm = 1 − 2v 3 r+ a 1 − 3v 2 r + 2qv 3 r η N,H ℓm exp ln η H ℓm ,(40)
where η N,∞ ℓm and η N,H ℓm are the leading terms for η ∞ ℓm and η H ℓm respectively, the denominator (1 − 3v 2 r + 2qv 3 r ) is the square of the denominator ofŜ
η N,∞ ℓm = 5 256π 2 m 2 n (ǫp) ℓm 2 c ℓ+ǫp (ν) 2 v 2(ℓ−2)+2ǫp P ℓ−ǫp,−m π 2 2 .(41)
The factorsη ∞ ℓm andη H ℓm in the exponential resummation, Eq. (40), can be derived by comparing with the Taylor expanded modal energy fluxes η ∞ ℓm and η H ℓm . Figures 6 and 7 show the relative error in the total energy flux from numerical results and PN approximations as a function of the orbital velocity up to ISCO using exponential 15/30 resummation to PN approximations [41]. From these figures, one will find that the relative error becomes smaller as the PN order becomes higher for v ≤ 0.3, except for accidental agreements for certain values of the velocity. However, the relative error around ISCO does not necessarily become smaller at higher PN orders when q > 0.3. The relative error for 11PN is smaller than 10 −5 when v 0.4, irrespective of values of q investigated in the paper. Again, we note that the region of the velocity, v 0.4, is larger than the one using the Taylor expanded PN energy flux, v 0.33. 16
|1-dEdt PN /dEdt Numerical | v q=0.90 4PN-ρ 5PN-ρ 6PN-ρ 7PN-ρ 8PN-ρ 9PN-ρ 10PN-ρ 11PN-ρ|1-dEdt PN /dEdt Numerical | v q=0.998 4PN-ρ 5PN-ρ 6PN-ρ 7PN-ρ 8PN-ρ 9PN-ρ 10PN-ρ 11PN-ρ|1-dEdt PN /dEdt Numerical | v q=-0.30 4PN-ρ 5PN-ρ 6PN-ρ 7PN-ρ 8PN-ρ 9PN-ρ 10PN-ρ 11PN-ρ|1-dEdt PN /dEdt Numerical | v q=-0.90 4PN-ρ 5PN-ρ 6PN-ρ 7PN-ρ 8PN-ρ 9PN-ρ 10PN-ρ 11PN-ρ
Phase difference during the two-year inspiral
We compare the orbital phase from PN results with numerical results during two-year inspirals to estimate the applicability of the PN results in the data analysis. For the comparison, we choose two representative systems of EMRIs in the eLISA frequency band, System-I and System-II, following Refs. [26,27,47,48]. System-I is an early inspiral of an EMRI with masses (M, µ) = ( For the calculation of the orbital phase, we define the phase as Ψ ℓm (t) = m t 0 Ω(t ′ )dt ′ , where Ω(t) = M 1/2 /r(t) 3/2 /(1 + qM 3/2 /r(t) 3/2 ) is the angular frequency of the particle and r(t) is the orbital radius as a function of time. The orbital radius r(t) is derived as r(t) = t (dr/dt ′ ) dt ′ = t (∂r/∂Ẽ) ( Fig. 2 but using exponential resummation to the energy flux in the post-Newtonian approximation. The relative error for 11PN is less than 10 −5 when v 0.4, whose region is larger than v 0.33 for the Taylor expanded energy flux in Fig. 2. time to perform the numerical integration is less than a second if we use the cubic spline interpolation. Figures 8 and 9 show absolute values of the difference in the orbital phase for the dominant ℓ = m = 2 mode between the PN and the numerical results during two-year inspirals for several values of the spin of the black hole. As for the PN approximations, we show results using the factorized resummation in Sec. 4.2, which are better than those using the Taylor expanded PN energy flux and comparable to those using the exponential resummation. The dephases between the 11PN results and numerical results after the two-year inspiral are less than 10 −4 rad for System-I. However, the dephases after the two-year inspiral become larger than a radian for System-II when q > 0.3. Thus, one has to derive higher PN order results 19/30 for the energy flux to achieve a dephase of less than a radian for System-II when q > 0.3, which represents a stronger-field situation than the one for System-I.
Summary
We have investigated gravitational waves from a particle moving in circular orbits in Kerr spacetime using the post-Newtonian approximation and computed the energy flux up to 11PN. We have also computed the energy flux down the event horizon for a particle in circular orbits around a Schwarzschild black hole at 22.5PN beyond the Newtonian approximation to fill the gap in the PN order between the energy flux at infinity, currently known at 22PN, and the event horizon, previously known at 6.5PN beyond Newtonian approximation.
To investigate how higher PN order expressions improve the applicability to data analysis of eLISA/NGO, comparisons between PN results and high-precision numerical results in black hole perturbation theory have been done. We first compared PN energy flux to numerical energy flux and found that the region of validity in the PN energy flux becomes larger as the PN order becomes higher. If the relative error of the energy flux in the PN approximation should be less than 10 −5 , the energy flux at 11PN satisfies this requirement for 20/30 v 0.33, which clearly shows an improvement from v 0.13 in an earlier work at 4PN [24]. The region of validity in the 11PN energy flux can become larger, v 0.4, if one uses resummation techniques such as factorized resummation [40] and exponential resummation [41]. 3 The region of validity might become further larger if one takes account of the structure of homogeneous solutions of the Teukolsky equation more carefully [46].
Finally, we compared the orbital phase during the two-year inspiral using the factorized resummed PN flux and the high-precision numerical flux. We found that the dephase is less than 1 (10 −4 ) rad for late (early) inspirals when q ≤ 0.3 (q ≤ 0.9). This implies that the 11PN factorized resummed flux may be used to detect early inspirals in the data analysis of eLISA/NGO. To detect gravitational waves from late inspirals when q > 0.3, however, it is necessary to obtain higher PN order expressions than 11PN. From numerical calculations in black hole perturbation theory, it is estimated that we may need to compute at least up to ℓ = 30, i.e. 28PN, to obtain the relative error of 10 −5 in the energy flux at ISCO for q = 0.9 [27]. If it is not possible to perform such a high PN order calculation, it may be necessary to use other approaches that compute unknown PN coefficients by numerical fitting [36,47,48].
where ξ = a ω and s E ℓm (ξ) = λ + s(s + 1) − a 2 ω 2 + 2 a m ω.
When ξ = 0, the solutions s S aω ℓm (x) in Eq. (B1) reduce to the spin-weighted spherical harmonics and the eigenvalue s E ℓm (ξ) becomes ℓ(ℓ + 1) [35]. Thus, it might be useful to express the spin-weighted spheroidal harmonics in a series of the spin-weighted spherical harmonics [3,24,51,52].
Taking account of singularities at x = ±1 and ∞ in the differential equation (B1), it is also possible to expand the spin-weighted spheroidal harmonics in a series of Jacobi polynomials [38,53]. For this purpose, we introduce new functions s U ℓm (x) and s V ℓm (x) through
s S aω ℓm (x) = e ξx 1 − x 2 α 2 1 + x 2 β 2 s U ℓm (x),(B2)
and
s S aω ℓm (x) = e −ξx 1 − x 2 α 2 1 + x 2 β 2 s V ℓm (x),(B3)
where α = |m + s| and β = |m − s|. Note that Eqs. (B2) and (B3) imply
s V ℓm (x) = exp(2ξx) s U ℓm (x).(B4)
Substituting Eqs. (B2) and (B3) into Eq. (B1), s U ℓm (x) and s V ℓm (x), respectively, satisfy the differential equations as
(1 − x 2 ) s U ′′ ℓm (x) + [β − α − (2 + α + β)x] s U ′ ℓm (x) + s E ℓm (ξ) − α + β 2 α + β 2 + 1 s U ℓm (x) =ξ −2(1 − x 2 ) s U ′ ℓm (x) + (α + β + 2s + 2)x s U ℓm (x) −(ξ + β − α) s U ℓm (x)] ,(B5)
and
(1 − x 2 ) s V ′′ ℓm (x) + [β − α − (2 + α + β)x] s V ′ ℓm (x) + s E ℓm (ξ) − α + β 2 α + β 2 + 1 s V ℓm (x) =ξ 2(1 − x 2 ) s V ′ ℓm (x) − (α + β − 2s + 2)x s V ℓm (x) −(ξ − β + α) s V ℓm (x)] .(B6)
When ξ = 0, Eqs. (B5) and (B6) reduce to the differential equation for Jacobi polynomials P (α,β) n (x)
(1 − x 2 ) P (α,β) n ′′ (x) + [β − α − (α + β + 2)x] P (α,β) n ′ (x) + n(n + α + β + 1) P (α,β) n (x) = 0,(B7)
provided the eigenvalue s E ℓm (ξ) in Eqs. (B5) and (B6) becomes ℓ(ℓ + 1), where ℓ = n + (α + β)/2 = n + max(| m |, | s |). Here the Jacobi polynomials are defined by the Rodrigue's 23/30 formula
P (α,β) n (x) = (−1) n 2 n n! (1 − x) −α (1 + x) −β d dx n (1 − x) α+n (1 + x) β+n . (B8)
If we expand s U ℓm (x) and s V ℓm (x) as infinite series of Jacobi polynomials,
s U ℓm (x) = ∞ n=0 s A (n) ℓm (ξ) P (α,β) n (x),(B9)
and
s V ℓm (x) = ∞ n=0 s B (n) ℓm (ξ) P (α,β) n (x),(B10)
we obtain three-term recurrence relations for the expansion coefficients s A (n) ℓm (ξ) and s B (n) ℓm (ξ), respectively, as
α (0) s A (1) ℓm (ξ) + β (0) s A (0) ℓm (ξ) = 0, α (n) s A (n+1) ℓm (ξ) + β (n) s A (n) ℓm (ξ) + γ (n) s A (n−1) ℓm (ξ) = 0, (n ≥ 1),(B11)
with α (n) = 4ξ(n + α + 1)(n + β + 1)(n + (α + β)/2 + 1 − s) (2n + α + β + 2)(2n + α + β + 3) ,
β (n) = s E ℓm (ξ) + ξ 2 − n + α + β 2 n + α + β 2 + 1 + 2ξs(α − β)(α + β) (2n + α + β)(2n + α + β + 2)
,
γ (n) = − 4ξn(n + α + β)(n + (α + β)/2 + s) (2n + α + β − 1)(2n + α + β) ,(B12)andα (0) s B (1) ℓm (ξ) +β (0) s B (0) ℓm (ξ) = 0, α (n) s B (n+1) ℓm (ξ) +β (n) s B (n) ℓm (ξ) +γ (n) s B (n−1) ℓm (ξ) = 0, (n ≥ 1) (B13)
with α (n) = − 4ξ(n + α + 1)(n + β + 1)(n + (α + β)/2 + 1 + s) (2n + α + β + 2)(2n + α + β + 3) ,
β (n) = s E ℓm (ξ) + ξ 2 − n + α + β 2 n + α + β 2 + 1 + 2ξs(α − β)(α + β) (2n + α + β)(2n + α + β + 2) , γ (n) = 4ξn(n + α + β)(n + (α + β)/2 − s) (2n + α + β − 1)(2n + α + β) .(B14)
Note that, for deriving Eq. (B11) and Eq. (B13), we use recurrence relations for Jacobi polynomials [53]. From the behavior of the three-term recurrence relation Eq. (B11) for sufficiently large n, there may be two independent solutions in Eq. (B11) as
A (n) (1) ∼ const.(−ξ) n Γ(n + (α + β + 3)/2 − s) ,(B15)
A (n)
(2) ∼ const.ξ n Γ(n + (α + β + 1)/2 + s).
(B16) 24/30 According to the theory of three-term recurrence relations [54], A
(1) is a minimal solution and A (1) requires that the eigenvalue s E ℓm (ξ) satisfies a certain transcendental equation, which is expressed in terms of continued fractions.
In order to obtain the equation that determines the eigenvalue s E ℓm (ξ), it is convenient to introduce the following quantities:
R n ≡ A (n) (1) A n−1 (1) , L n ≡ A (n) (1) A n+1(1)
.
(B17)
Using the three-term recurrence relation Eq. (B11) we can express R n as an infinite continued fraction,
R n = − γ (n) β (n) + α (n) R n+1 = − γ (n) β (n) − α (n) γ (n+1) β (n+1) − α (n+1) γ (n+2) β (n+2) − · · · ,(B18)
and L n as a finite continued fraction,
L n = − α (n) β (n) + γ (n) L n−1 = − α (n) β (n) − α (n−1) γ (n) β (n−1) − α (n−2) γ (n−1) β (n−2) − · · · α (1) γ (2) β (1) − α (0) γ (1) β (0) . (B19)
The expression for R n is valid if this infinite continued fraction converges. Noting the properties of the three-term recurrence relations (see p. 35 in Ref. [54]), it can be proved that the continued fraction Eq. (B18) converges if the eigenvalue s E ℓm (ξ) is finite. We obtain the equation to determine the eigenvalue s E ℓm (ξ) dividing Eq. (B11) by the expansion coefficients s A (n) ℓm (ξ)
β (n) + α (n) R n+1 + γ (n) L n−1 = 0,(B20)
where R n+1 and L n−1 are defined by the continued fractions Eqs. (B18) and (B19), which are convergent for finite values of s E ℓm (ξ). There are many roots in Eq. (B20) for given n, m, s and ξ. These roots are associated with the same m, s, and ξ but with different ℓ. In order to find the root for a given ℓ, it is useful to choose n = n ℓ ≡ ℓ − (α + β)/2 in Eq. (B20) since in the limit ξ → 0 all the terms in Eq. (B20) become O(ξ 2 ). This means that the choice naturally gives the leading term of a series expansion of the eigenvalue s E ℓm (ξ) in terms of ξ as ℓ(ℓ + 1). When | ξ | is not large, we can obtain the analytic expression of s E ℓm (ξ) in a series of ξ as
s E ℓm (ξ) = ℓ(ℓ + 1) − 2s 2 m ℓ(ℓ + 1) ξ + [H(ℓ + 1) − H(ℓ) − 1] ξ 2 + O(ξ 3 ),(B21)
where
H(ℓ) = 2(ℓ 2 − m 2 )(ℓ 2 − s 2 ) 2 (2ℓ − 1)ℓ 3 (2ℓ + 1) .(B22)
For the numerical calculation to determine s E ℓm (ξ), one can use the analytic expression of s E ℓm (ξ) above as an initial value to find the root in Eq. (B20).
25/30
Once we obtain the eigenvalue, using Eqs. (B18) and (B19) we can compute all the coefficients s A (n) ℓm (ξ) from s A (ñ) ℓm (ξ) for a givenñ. The coefficient for n = n ℓ = l − (α + β)/2 is usually the largest term. The ratio of other terms to the dominant term, i.e. s A (n) ℓm (ξ)/ s A (nℓ) ℓm (ξ), can be determined using Eqs. (B18) and (B19) for 0 < n < n ℓ and n > n ℓ , respectively.
We can also deal with the coefficients s B (n) ℓm (ξ) in a similar way. Notingβ (n) = β (n) and α (n)γ(n+1) = α (n) γ (n+1) , we obtain the same equation (B20) for s A (n) ℓm (ξ) to determine the eigenvalue s E ℓm (ξ). Thus, the minimal solution of the three-term recurrence relation for s B (1) } is the minimal solution, we have
B (n) (1) B (n−1) (1) = −γ (n) β (n) −α (n)γ(n+1) β (n+1) −α (n+1)γ(n+2) β (n+2) − · · · ,(B23)B (n) (1) B (n+1) (1) = −α (n) β (n) −α (n−1)γ(n) β (n−1) −α (n−2)γ(n−1) β (n−2) − · · ·α (1)γ(2) β (1) −α (0)γ(1) β (0) .(B24)
From these equations, we can determine the ratios of all the coefficients, s B (n)
ℓm (ξ)/ s B (nℓ)
ℓm (ξ). Now we come to the problem of the normalization of the two unknown coefficients A (nℓ) (1) and B
(nℓ) (1) . Since Eq. (B4) must be satisfied for any value of x, we may substitute x = 1 in Eq. (B4) to obtain
s B (nℓ) ℓm (ξ) ∞ n=0 s B (n) ℓm (ξ) s B (nℓ) ℓm (ξ) n + α n = exp(2ξ) s A (nℓ) ℓm (ξ) ∞ n=0 s A (n) ℓm (ξ) s A (nℓ) ℓm (ξ) n + α n .(B25)1 −1 dx 1 − x 2 α 1 + x 2 β ∞ n1=0 s A (n1) ℓm (ξ)P (α,β) n1 (x) ∞ n2=0 s B (n2) ℓm (ξ)P (α,β) n2 (x) = 1. (B26)
Using the orthogonality condition of the Jacobi polynomials, we have
1 −1 dx 1 − x 2 α 1 + x 2 β P (α,β) n1 (x)P (α,β)
n2 (x) = 2 Γ(n + α + 1)Γ(n + β + 1)δ n1,n2 (2n + α + β + 1)Γ(n + 1)Γ(n + α + β + 1)
.
(B27) Then, Eq. (B26) reduces to ∞ n=0 s A (n) ℓm (ξ) s A (nℓ) ℓm (ξ) s B (n) ℓm (ξ) s B (nℓ) ℓm (ξ)
2 Γ(n + α + 1)Γ(n + β + 1) (2n + α + β + 1)Γ(n + 1)Γ(n + α + β + 1) ℓm (ξ) is made by the requirement that s S aω ℓm (x) reduces to the spin-weighted spherical harmonics in the limit ξ → 0. 26/30 C. Homogeneous solutions of the radial Teukolsky equation
In this paper, we use a formalism developed by Mano, Suzuki, and Takasugi (MST) to compute the homogeneous solutions of the radial Teukolsky equation [28,29]. In the formalism, analytic expressions of homogeneous solutions are given using two kinds of series expansions in terms of hypergeometric functions and Coulomb wave functions, which are, respectively, convergent at the horizon and infinity. One can obtain analytic expressions of the asymptotic amplitudes of the homogeneous solutions by analytic matching of the two kinds of solutions in the overlapping region of convergence. Compared to numerical integration methods to solve the Teukolsky equation, the formalism is quite powerful for very accurate numerical calculations of gravitational waves [36,38,39,55]. The formalism is also very powerful for the performance of post-Newtonian expansions of gravitational waves at higher orders since the series expansion of homogeneous solutions is closely related to the low-frequency expansion. Applying the formalism to the post-Newtonian approximation in black hole perturbation theory, the energy flux going down the horizon was calculated up to 6.5PN for a particle in a circular and equatorial orbit around a Kerr black hole [32] and the 2.5PN energy flux to infinity was computed for a particle in a slightly eccentric and inclined orbit around the Kerr black hole [56,57]. More recently, we applied the formalism to obtain the 5.5PN waveforms for a particle in a circular orbit around a Schwarzschild black hole [30] and the 4PN waveforms for a particle in a circular and equatorial orbit around the Kerr black hole [31], which, respectively, confirmed the 5.5PN energy flux in Ref. [23] and the 4PN energy flux in Ref. [24]. In Refs. [26,27], we extended the formalism to obtain very high PN expressions for the energy flux to infinity for the particle in a circular orbit around the Schwarzschild black hole. For more details of the formalism, we refer the reader to a recent review, Ref. [2].
In the MST formalism, one can expand a homogeneous solution of the radial Teukolsky equation in a series of Coulomb wave functions as
R ν C =ẑ −1−s 1 − ǫκ z −s−i(ǫ+τ )/2 ∞ n=−∞ (−i) N (ν + 1 + s − iǫ) n (ν + 1 − s + iǫ) n a ν n F n+ν (−is − ǫ,ẑ), (C1)
whereẑ = ω(r − r − ), τ = (ǫ − m q)/κ, (a) n = Γ(a + n)/Γ(a) and F N (η, z) is a Coulomb wave function defined by
F N (η,ẑ) = e −iẑ 2 NẑN +1 Γ(N + 1 − iη) Γ(2N + 2) Φ(N + 1 − iη, 2N + 2; 2iẑ),(C2)
where Φ(α, β; z) is the confluent hypergeometric function, regular at z = 0 (see Sec. 13 in Ref. [58]). Note that the so-called renormalized angular momentum ν is introduced in the homogeneous solution in a series of Coulomb wave functions, Eq. (C1). ν is a generalization of ℓ, which has a property such that ν → ℓ as ǫ → 0, and is determined through conditions that the series of Coulomb wave functions, Eq. (C1), converges and actually represents a homogeneous solution of the radial Teukolsky equation.
Substituting the homogeneous solution, Eq. (C1), into the radial Teukolsky equation (5) with T ℓmω = 0, one obtains the following three-term recurrence relation for the expansion coefficients a ν n :
α ν n a ν n+1 + β ν n a ν n + γ ν n a ν n−1 = 0, 27/30 where α ν n = iǫκ(n + ν + 1 + s + iǫ)(n + ν + 1 + s − iǫ)(n + ν + 1 + iτ ) (n + ν + 1)(2n + 2ν + 3) , (C4a) β ν n = −λ − s(s + 1) + (n + ν)(n + ν + 1) + ǫ 2 + ǫ(ǫ − mq) + ǫ(ǫ − mq)(s 2 + ǫ 2 ) (n + ν)(n + ν + 1) , (C4b) γ ν n = − iǫκ(n + ν − s + iǫ)(n + ν − s − iǫ)(n + ν − iτ ) (n + ν)(2n + 2ν − 1) .
The series of Coulomb wave functions, Eq. (C1), converges and represents a homogeneous solution of the radial Teukolsky equation if ν satisfies the following equation:
R ν n L ν n−1 = 1,(C5)
where R ν n and L ν n are defined in terms of infinite continued fractions,
which can be derived from the three-term recurrence relation, Eq. (C3). Observe that one can obtain two kinds of expansion coefficients, a ν n , from two kinds of the continued fractions, R ν n and L ν n . If ν is chosen to satisfy Eq. (C5) for a certain n, the two kinds of the expansion coefficients coincide and the series of Coulomb wave functions, Eq. (C1), converges for r > r + .
Since α −ν−1 −n = γ ν n and β −ν−1 −n = β ν n in Eq. (C4), one finds that a −ν−1 −n satisfies the same recurrence relation Eq. (C3) as a ν n . Then it can be shown that R −ν−1 C is also a homogeneous solution of the Teukolsky equation, which converges for r > r + .
Matching the solution in a series of Coulomb wave functions, which converges for r > r + , with the one in a series of hypergeometric functions, which converges for r < ∞, one can obtain the incoming solution of the radial Teukolsky equation, R in ℓmω , which converges in the entire region as
R in ℓmω = K ν R ν C + K −ν−1 R −ν−1 C ,(C8)
where K ν = e iǫκ (2ǫκ) s−ν−N 2 −s i N Γ(1 − s − iǫ − iτ )Γ(N + 2ν + 2) Γ(N + ν + 1 − s + iǫ)Γ(N + ν + 1 + iτ )Γ(N + ν + 1 + s + iǫ) × ∞ n=N (−1) n Γ(n + N + 2ν + 1) (n − N )! Γ(n + ν + 1 + s + iǫ)Γ(n + ν + 1 + iτ ) Γ(n + ν + 1 − s − iǫ)Γ(n + ν + 1 − iτ ) a ν n × N n=−∞ (−1) n (N − n)!(N + 2ν + 2) n (ν + 1 + s − iǫ) n (ν + 1 − s + iǫ) n a ν n −1 ,
and N is an arbitrary integer. The factor K ν is a constant to match the solutions in the overlap region of convergence, and is independent of the choice of N . By comparing R in ℓmω in Eq. (6) to Eq. (C8) in the limit r * → ±∞, one derives analytic expressions for the asymptotic amplitudes B inc ℓmω = ω −1 K ν − ie −iπν sin π(ν − s + iǫ) sin π(ν + s − iǫ) K −ν−1 A ν + e −iǫ(ln ǫ− 1−κ 2 ) ,
B ref ℓmω = ω −1−2s [K ν + ie iπν K −ν−1 ]A ν − e iǫ(ln ǫ− 1−κ 2 ) ,(C10a)
where A ν + = 2 −1+s−iǫ e − πǫ 2 e π 2 i(ν+1−s) Γ(ν + 1 − s + iǫ)
Γ(ν + 1 + s − iǫ) +∞ n=−∞ a ν n ,(C11a)
A ν − = 2 −1−s+iǫ e − πǫ 2 e −π 2 i(ν+1+s) +∞ n=−∞ (−1) n (ν + 1 + s − iǫ) n (ν + 1 − s + iǫ) n a ν n .
For obtaining Eq. (C10), it is useful to note that the asymptotic form of r * in the limit r * → ±∞ takes ωr * →ẑ + ǫ lnẑ − ǫ ln ǫ for r → ∞,
kr * → ǫ + τ 2 ln r + − r 2M κ + κ ǫ + τ 2 + κ (ǫ + τ ) 1 + κ ln κ for r → r + .
As for the other homogeneous solution R up ℓmω in Eq. (6), we decompose the homogeneous solution in a series of Coulomb wave functions R ν C as
R ν C = R ν + + R ν − ,(C13)
where R ν + = 2 ν e −πǫ e iπ(ν+1−s) Γ(ν + 1 − s + iǫ) Γ(ν + 1 + s − iǫ) e −iz z ν+i(ǫ+τ )/2 (z − ǫκ) −s−i(ǫ+τ )/2 × ∞ n=−∞ i n a ν n (2z) n Ψ(n + ν + 1 − s + iǫ, 2n + 2ν + 2; 2iz),
R ν − = 2 ν e −πǫ e −iπ(ν+1+s) e iz z ν+i(ǫ+τ )/2 (z − ǫκ) −s−i(ǫ+τ )/2 ∞ n=−∞ i n × (ν + 1 + s − iǫ) n (ν + 1 − s + iǫ) n a ν n (2z) n Ψ(n + ν + 1 + s − iǫ, 2n + 2ν + 2; −2iz) . (C15)
For the decomposition, we used an analytic property of the confluent hypergeometric function (see p. 259 in Ref. [59]):
Φ(α, γ; x) = Γ(γ) Γ(γ − α) e iαπσ Ψ(α, γ; x) + Γ(γ) Γ(α) e iπ(α−γ)σ e x Ψ(γ − α, γ; −x),(C16)
where Ψ is the irregular confluent hypergeometric function and σ = sgn[Im(x)] is assumed. Since Ψ(α, β, x) → x −α in the limit | x |→ ∞ (see Sec. 13 in Ref. [58]), one finds R ν + = A ν + z −1 e −i(z+ǫ ln z) , R ν − = A ν − z −1−2s e i(z+ǫ ln z) for r → ∞.
Noting the functions R ν + and R ν − have factors e −iz and e iz , respectively, one finds that R ν + (R ν − ) is an incoming (outgoing) wave solution at infinity. Then the upgoing solution R up ℓmω is given by
R up ℓmω = R ν − .(C18)
Again noting the asymptotic form of r * in the limit r * → ±∞ in Eq. (C12) and comparing R up ℓmω in Eq. (6) to Eq. (C18) in the limit r * → +∞, one finds the asymptotic amplitude 29/30 C trans ℓmω as C trans ℓmω = ω −1−2s A ν − e i(ǫ ln ǫ− 1−κ 2 ǫ) .
(C19)
− a 2 ω 2 sin 2 θ − (m + s cos θ) 2 sin 2 θ −2 a ωs cos θ + s + 2 m a ω + λ −2 S aω ℓm (θ) = 0,
iωt−imφ(t) I ∞ ℓmω [r(t), θ(t)],
withǫ =√ M 2 − a 2 /(4M r + ) and|C| 2 = (λ + 2) 2 + 4 a ω m − 4 a 2 ω 2 λ 2 + 36 a ω m − 36 a 2 ω 2 +(2 λ + 3) (96 a 2 ω 2 − 48 a ω m) + 144 ω 2 (M 2 − a 2 ).
Figures 1
1and 2 show the relative error in the total energy flux from numerical results and PN approximations as a function of the orbital velocity up to the innermost stable circular orbit (ISCO)
(
N,ǫp) ℓm represents the Newtonian contribution to waveforms,Ŝ(ǫp)eff an effective source term for partial waves in the perturbation
Fig. 2
2Same as Fig. 1 but for q = −0.01, −0.05, −0.1, −0.3, −0.5 and −0.9. The relative error for 11PN is smaller than 10 −5 when v 0.33.
Fig. 3 (
3Left) Absolute values of the difference in the energy flux down the horizon from numerical results and PN approximation as a function of the orbital velocity up to ISCO for q = 0. (Right) Same as the left figure but for the total energy flux, which includes fluxes to infinity and the horizon.
the same as in Eq.(27), which is used for the resummed multipolar waveforms Eq.(23). The fourth factor ρ H ℓm is the 2ℓth root of the residual amplitude of the modal energy flux and can be derived by comparing the Taylor expanded modal energy flux η H ℓm with the factorized modal energy flux Eq.(36).
Eq. (23), and the factor (1 − 2v 3 r + /a) is motivated by the factor k = ω − ma/(2M r + ) = m/M (v 3 − a/(2r + )) in Eq. (18), which is again responsible for the sign of the modal energy flux to the horizon. Similarly to η N,H ℓm defined in Eq. (37), the explicit expression for η N,∞ ℓm can be derived from factors n (ǫp) ℓm and c ℓ+ǫp (ν) in the Newtonian contribution to waveforms h (N,ǫp) ℓm , Eq. (24), as
Fig. 4
4Same asFig. 1but using factorized resummation to the energy flux in the post-Newtonian approximation. The relative error for 11PN is less than 10 −5 when v 0.4, whose region is larger than v 0.33 for the Taylor expanded energy flux inFig. 1.
Fig. 5
5Same asFig. 2but using factorized resummation to the energy flux in the post-Newtonian approximation. The relative error for 11PN is less than 10 −5 when v 0.4, whose region is larger than v 0.33 for the Taylor expanded energy flux inFig. 2.
10 5 , 10)M ⊙ , i.e. µ/M = 10 −4 , which reaches r 0 ≃ 16M after the two-year inspiral. System-II is a late inspiral of an EMRI with masses (M, µ) = (10 6 , 10)M ⊙ , i.e. µ/M = 10 −5 , which reaches ISCO after the two-year inspiral. Although the initial values for orbital radius, velocity, and GW frequency depend on the spin of the Kerr black hole, System-I inspirals from r 0 ≃ 29M to r 0 ≃ 16M with associated velocities v ∈ [0.2, 0.25] and frequencies f GW ∈ [4 × 10 −3 , 10 −2 ]Hz, while System-II explores orbital
Fig. 6
6Same asFig. 1but using exponential resummation to the energy flux in the post-Newtonian approximation. The relative error for 11PN is less than 10 −5 when v 0.4, whose region is larger than v 0.33 for the Taylor expanded energy flux inFig. 1.separation in the range r 0 /M ∈ [r ISCO , 11M ], velocities v ∈ [0.3, v ISCO ] and frequencies f GW ∈ [10 −3 , f ISCO GW ]Hz. The orbital phase for the System-I (System-II) after the two-year inspiral is about 10 6 (5 × 10 5 ) rad. Moreover, System-I (System-II) sweeps the high-(low-) frequency region of the eLISA frequency band.
Fig. 7
7Same as
Fig. 8
8Absolute values of the dephasing during the two-year inspiral between the factorized PN and the numerical results for the dominant ℓ = m = 2 mode as a function of time in months when q = 0.1, 0.3, 0.5 and 0.9. These panels show the dephases for System-I with masses (M, µ) = (10 5 , 10)M ⊙ , which inspirals from r 0 ≃ 29M to r 0 ≃ 16M with associated frequencies f GW ∈ [4 × 10 −3 , 10 −2 ]Hz. These inspirals represent the early inspiral phase in the eLISA band. The dephases between the 11PN results and numerical results after the two-year inspiral are less than 10 −4 rad.
Fig. 9
9Same as Fig. 8 but for System-II with masses (M, µ) = (10 6 , 10)M ⊙ , orbital radius in the range r 0 /M ∈ [r ISCO , 11] and frequencies in the range f GW ∈ [10 −3 , f ISCO GW ]Hz. These inspirals represent the late inspiral phase in the eLISA band. The dephases between the 11PN results and numerical results for q ≤ 0.3 (q > 0.3) after the two-year inspiral are less (larger) than a radian.
for the series expansion Eq. (B9) to obtain a solution that converges uniformly. This choice of A (n)
ξ), Eq. (B13), should give the same eigenvalue s E ℓm (ξ) for s A
From
Eq. (B25), we obtain an equation for B (ξ), if neither of the infinite series vanishes. An equation for s A (nℓ) ℓm (ξ) s B (nℓ) ℓm (ξ) is derived from the orthogonality condition of the spin-weighted spheroidal harmonics Eq. (3):
From
Eqs. (B25) and (B28). we can obtain the squares of s A (nℓ) ℓm (ξ) and s B (nℓ) ℓm (ξ). The final determination of the signs of s A (nℓ) ℓm (ξ) and s B (nℓ)
B trans ℓmω , B inc ℓmω , and B ref ℓmω in Eq. (6) as B trans ℓmω = ǫκ ω 2s e iκ(ǫ+τ )( 1 2 + ln κ 1+κ )
where I
whereH ℓmω = R up ℓmω {A nn0 + Am n0 + Amm 0 } −dR up
ℓmω
dr
{Am n1 + Amm 1 } +
d 2 R up
ℓmω
d 2 r
Amm 2
r=r(t),θ=θ(t)
The 7.5PN energy flux to infinity is given by33].
5/30
3.1. Infinity flux
dE
dt ∞
=
dE
dt N
1 + {q-independent terms} −
11
4
qv 3 +
33
16
q 2 v 4 −
59
16
qv 5
+ −
65
6
πq +
611
504
q 2 v 6 +
162035
3888
q +
65
8
πq 2 −
71
24
q 3 v 7
+ −
359
14
πq +
22667
4536
q 2 +
17
16
q 4 v 8
+ −
9828207709
52390800
+
40939
315
ln 2 −
43
3
π 2 +
6841
105
γ +
6841
105
ln v q
+
8447
672
π q 2 −
112025
4536
q 3 v 9
+
23605
144
π q +
93301799461
628689600
−
27499
420
ln 2 +
43
4
π 2 −
4601
140
γ −
4601
140
ln v q 2
−
45
4
π q 3 +
731
126
q 4 v 10
+ −
244521688471
272432160
−
1280791
10584
ln 2 −
671
12
π 2 +
128459
7560
γ +
486243
3136
ln 3
+
128459
7560
ln v q +
34211
1512
π q 2 −
257407
9072
q 3 +
33
8
π q 4 −
1
8
q 5 v 11
+ −
270159823411
558835200
π +
81878
315
π γ +
54514
105
π ln 2 +
81878
315
π ln v q
+
13501670684927
28605376800
−
22901
196
γ +
24763
756
π 2 −
537013
3780
ln 2
−
142155
1568
ln 3 −
22901
196
ln v q 2 −
67426
567
π q 3 +
24397
1008
q 4 v 12
+
256
15
ln κ −
5160697541
6735960
γ −
16342453091
47151720
ln 2 −
67221333
86240
ln 3
+
526805
2916
π 2 +
1290587071610633
606842636400
q
+ −
18297
70
π ln 2 −
27499
210
π γ +
1241484285313
2235340800
π q 2
+ −
416537225257
1414551600
−
263
18
π 2 +
31467
140
ln 2 +
256
5
ln κ +
174659
1260
γ q 3
+
60869
2016
π q 4 +
11311
1008
q 5 +
256
15
q +
256
5
q 3 Ψ
(0,2)
A
(q)
+ −
5045737157
6735960
q −
27499
210
π q 2 +
239171
1260
q 3 ln v v 13
+ −
2628337
15120
π ln 2 +
1458729
1568
π ln 3 +
8005397
21168
π γ −
748453609847597
152562009600
π q
+ −
921209351
7858620
γ −
5442958727
7858620
ln 2 +
206753
3402
π 2 +
6830001
34496
ln 3
+
3708535586671457
2831932303200
q 2 −
238825
1008
π q 3
6/30
+ −
1391
84
γ −
4601
140
ln 2 +
65
12
π 2 +
305698147
1270080
q 4 −
1
4
π q 5 +
827
336
q 6
+
8005397
21168
π q −
921209351
7858620
q 2 −
1391
84
q 4 ln v v 14
+ −
18591892
6615
γ ln 2 +
1316293890558029
235994358600
ln 2 +
1154100056435789
235994358600
γ
+
109028
315
π 2 γ +
76
5
π 4 +
1195923689
15717240
π 2 −
7750438
11025
γ 2 −
30978854
11025
(ln 2) 2
−
632
35
ln κ +
6197998046875
5479778304
ln 5 +
130748
189
π 2 ln 2 −
36461702637
157853696
ln 3
+
109028
105
ζ (3) −
2049619936309873009541
1807162200133290000
q
+
418783053049409
152562009600
π −
426465
784
π ln 3 −
13188979
17640
π ln 2 −
34227289
52920
π γ q 2
+
1079649
3136
ln 3 +
166255711
238140
ln 2 −
681171487193473
257448391200
−
683477
3402
π 2
+
118154591
238140
γ −
5618
105
ln κ q 3 +
1790339
13608
π q 4 −
10080445
81648
q 5
dẼ/dt ′ ) dt ′ , where dẼ/dt is computed by the energy balance equation, dẼ/dt = − dE/dt ∞ − dE/dt H .To save computation time, we apply cubic
spline interpolation [49] to perform the integration using 10 3 data points for (v, dẼ/dt ),
i.e. (v, dr/dt), in the range from v = 0.01 to v = v ISCO [26, 27, 47, 50]. The computation
18/30
In Ref.[36], the relative error in the fitting formula of the 20PN energy flux using a series expansion in q is less than 10 −4 for Ω 0.8Ω ISCO when q = 0.5 and Ω 0.35Ω ISCO when q = 0.9, where Ω ISCO is the angular frequency of the particle at ISCO. The region of Ω in Ref.[36] is comparable to the one for our 11PN results using resummation techniques, which is estimated as Ω 0.78Ω ISCO when q = 0.5 and Ω 0.37Ω ISCO when q = 0.9.21/30
AcknowledgementsIt is our pleasure to thank Bala Iyer for his continuous encouragement and useful comments on the manuscript. We also thank Abhay Shah for sharing his results before submitting Ref.[36], which was very helpful for correcting errors in our results. This work was supported by the European Union FEDER funds, the Spanish Ministry of Economy and Competitiveness (Projects No. FPA2010-16495 and No. CSD2007-00042), the Conselleria d'Economia Hisenda i Innovacio of the Govern de les Illes Balears and the European Union's FP7 ERC Starting Grant "The dynamics of black holes: testing the limits of Einstein's theory" grant agreement no. DyBHo-256667. Some analytic calculations were carried out on HA8000/RS440 at Yukawa Institute for Theoretical Physics in Kyoto University.A. Source term of the Teukolsky equation. A nn0 and etc. in Eq.(13)are defined aswhere L † σ = ∂ θ − m/sin θ + aω sin θ + σ cot θ, ρ = 1/(r − ia cos θ) andwithṫ = dt/dτ and Σ = r 2 + a 2 cos 2 θ.B. Spin-weighted spheroidal harmonicsUsing x = cos θ, the angular Teukolsky equation (4) takes the form1 − x 2 − 2sξx + s E ℓm (ξ) s S aω ℓm (x) = 0, (B1)
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. N Yunes, A Buonanno, S A Hughes, Y Pan, E Barausse, M C Miller, W Throwe, Phys. Rev. D. 8344044N. Yunes, A. Buonanno, S. A. Hughes, Y. Pan, E. Barausse, M. C. Miller and W. Throwe, Phys. Rev. D 83, 044044 (2011).
W H Press, S A Teukolsky, W T Vetterling, B P Flannery, Numerical Recipes in C. Cambridge, EnglandCambridge University PressW. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, England, 1992).
. S A Hughes, Phys. Rev. D. 6464004S. A. Hughes, Phys. Rev. D 64, 064004 (2001).
. W H Press, S A Teukolsky, 649. 30/30Astrophys. J. 185W. H. Press and S. A. Teukolsky, Astrophys. J. 185 (1973), 649. 30/30
. S A Hughes, Phys. Rev. D. 6184004S. A. Hughes, Phys. Rev. D 61, 084004 (2000).
. E D Fackerell, R G Crossman, J. Math. Phys. 91849E. D. Fackerell and R. G. Crossman, J. Math. Phys. 9, 1849 (1977).
. W Gautschi, S I A , Review. 924W. Gautschi, S. I. A. M. Review 9 (1967) 24.
. R Fujita, W Hikida, H Tagoshi, Prog. Theor. Phys. 121843R. Fujita, W. Hikida and H. Tagoshi, Prog. Theor. Phys. 121, 843 (2009).
. N Sago, T Tanaka, W Hikida, K Ganz, H Nakano, Prog. Theor. Phys. 115873N. Sago, T. Tanaka, W. Hikida, K. Ganz and H. Nakano, Prog. Theor. Phys. 115, 873 (2006).
. K Ganz, W Hikida, H Nakano, N Sago, T Tanaka, Prog. Theor. Phys. 1171041K. Ganz, W. Hikida, H. Nakano, N. Sago and T. Tanaka, Prog. Theor. Phys. 117, 1041 (2007).
Handbook of Mathematical Functions. M. Abramowitz and I. A. StegunNew YorkDoverM. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions (Dover, New York, 1972).
I. 31/30Higher Transcendental functions. A. ErdélyiFloridaKriegerA. Erdélyi (ed.), Higher Transcendental functions (Krieger, Florida, 1981), Vol. I. 31/30
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"N Giglietto \nDipartimento Interateneo di Fisica di Bari and INFN\nVia Orabona70126BariItaly\n",
"S Stramaglia \nDipartimento Interateneo di Fisica di Bari and INFN\nVia Orabona70126BariItaly\n"
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"Dipartimento Interateneo di Fisica di Bari and INFN\nVia Orabona70126BariItaly",
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About a century ago, cosmic rays were identified as being a source of radiation on Earth. The proof came from two independent experiments. The Italian physicist Domenico Pacini observed the radiation strength to decrease when going from the surface to a few meters underwater (both in a lake and in a sea). At about the same time, in a balloon flight, the Austrian Victor Hess found the ionization rate to increase with height. The present article attempts to give an unbiased historical account of the discovery of cosmic rays -and in doing so it will duly account for Pacini's pioneering work, which involved a technique that was complementary to, and independent from, Hess'. Personal stories, and the pre-and post-war historical context, led Pacini's work to slip into oblivion.
| null |
[
"https://arxiv.org/pdf/1002.2888v2.pdf"
] | 119,251,886 |
1002.2888
|
a32ee2f3bc5dee47e76ac2566b78f2c82dbb551f
|
Domenico Pacini, the forgotten pioneer of the discovery of cosmic rays
A De Angelis
Dipartimento di Fisica dell'Università di Udine and INFN
LIP/IST
INAF Trieste
Via delle Scienze33100Udine, LisboaItaly;, Italy, Portugal
N Giglietto
Dipartimento Interateneo di Fisica di Bari and INFN
Via Orabona70126BariItaly
S Stramaglia
Dipartimento Interateneo di Fisica di Bari and INFN
Via Orabona70126BariItaly
Domenico Pacini, the forgotten pioneer of the discovery of cosmic rays
About a century ago, cosmic rays were identified as being a source of radiation on Earth. The proof came from two independent experiments. The Italian physicist Domenico Pacini observed the radiation strength to decrease when going from the surface to a few meters underwater (both in a lake and in a sea). At about the same time, in a balloon flight, the Austrian Victor Hess found the ionization rate to increase with height. The present article attempts to give an unbiased historical account of the discovery of cosmic rays -and in doing so it will duly account for Pacini's pioneering work, which involved a technique that was complementary to, and independent from, Hess'. Personal stories, and the pre-and post-war historical context, led Pacini's work to slip into oblivion.
Introduction
It is generally well known, since Faraday's early observations, that electroscopes spontaneously discharge. This phenomenon remained unexplained until the beginning of the XX century: its explanation paved the way to one of mankind's revolutionary scientific discoveries: cosmic rays.
Since the early XX century cosmic rays were used to probe and understand the constituents of matter. Indeed many early discoveries in particle physics (antimatter, mesons, muons, baryons, ...) were made while studying cosmic rays. Cosmic rays are still being used in the framework of fundamental physics, as well as to investigate astrophysical properties of their sources.
In 1896 the French physicist Henri Becquerel discovered the instability of some chemical elements. Some years later Marie and Pierre Curie discovered that Radium showed that same behavior: such transmutation processes were then called "radioactive decays". In the presence of a radioactive material, a charged electroscope promptly discharges. It was concluded then that some elements were able to emit charged particles, that in turn were responsible for discharging the electroscope. An electroscope's discharge rate was then used to gauge the level of radioactivity.
The spontaneous discharge observed in electroscopes made it evident that in insulated environments, too, a background radiation did exist. The obvious questions concerned the nature of such radiation, and whether it was of terrestrial or extra-terrestrial origin. It was generally believed that its origin was likely related to radioactive materials, hence its terrestrial origin was a commonplace assumption. An experimental proof, however, seemed hard to achieve.
At the very beginning of the XX century, several scientists made experiments about penetrating radiation, trying to understand its origin and nature 1 .
Around 1910, the Austrian Victor Hess and the Italian Domenico Pacini simultaneously and independently carried out two different, ingenious, and complementary research lines that would eventually clarify the origin of the yet mysterious ionizing background radiation.
Pacini made several measurements to establish the variations of an Ebert electroscope's discharge rate as a function of the environment: he placed the electroscope on the ground, on the sea a few km off the coast, and a few meters underwater. He reported those measurements, the ensuing results, and their interpretation in a note titled "La radiazione penetrante alla superficie ed in seno alle acque" ("Penetrating radiation at the surface of and in water") [3]. In that paper Pacini wrote: "Observations carried out on the sea during the year 1910 [4] led me to conclude that a significant proportion of the pervasive radiation that is found in air had an origin that was independent of direct action of the active substances in the upper layers of the Earth's surface." What was he lacking, at that time, before he could reach a firm conclusion about the extraterrestrial origin of the ionizing background radiation? Only in 1911 did Pacini develop his experimental technique for underwater measurements, that allowed him to measure a significant decrease in the discharge rate when the electroscope was placed underwater. "The apparatus ... was enclosed in a copper box so that it could immerse in depth. ... From June 24 to June 30 observations were performed with the instrument at the surface, and with the instrument immersed in water, at a depth of 3 meters. ...
[It] appears from the results of the work described in this Note that a sizable cause of ionization exists in the atmosphere, originating from penetrating radiation, independent of the direct action of radioactive substances in the soil."
Who was Domenico Pacini and, while Victor Hess [5] is honored as the discoverer of cosmic rays, why did Pacini's contemporary (or even earlier) discovery go unnoticed and was soon forgotten (notably in Italy)? Personal stories and historical events contributed to this outcome. Domenico Pacini ( Figure 1) was born on February 20, 1878, in Marino, near Rome. He graduated in Physics in 1902 at the Faculty of Sciences of Rome University. There, for the next three years, he worked as an assistant to Professor Pietro Blaserna while also studying electric conductivity in gaseous media under the supervision of Alfonso Sella. In 1904 he set out to study the infamous N-rays: he performed an experiment, the (null) results of which were communicated in a letter to Nature [6] as "careful experiments made ... with the object of observing the effects of n-rays described by M. Blondlot and other investigators." Though "observations were made under very favourable conditions," he "was unable to detect any increase of luminosity of a phosphorescent screen caused by unknown rays from strained or tempered steel, an Auer lamp, a Nernst lamp, sound vibrations, or a magnetic field, though various French observers have affirmed that in each of these cases N-rays are emitted which produce an effect upon the screen". In 1906 Pacini was appointed assistant at Italy's Central Bureau of Meteorology and Geodynamics, heading the department that was in charge of studying thunderstorms and electric phenomena in the atmosphere. (Most of the department's experimental work was carried out near Castelfranco Veneto, near Padova.) Pacini's held that position until 1927, when he was upgraded to Principal Geophysicist. After several more years of work as an assistant professor in Rome, finally in 1928 he was appointed full professor of Experimental Physics at the University of Bari, where he was incharged of setting up the studies of Physics within the Faculty of Medicine. While in The long way that led Pacini to the hypothesis of cosmic rays started from his studies on electric conductivity in gaseous media that he performed at the University of Rome during the early years of the XX century. While working at the Central Bureau of Meteorology, he became interested in the problem of the ionization of air. During 1907-1912, he performed several measurements on the air's conductivity on the ground (at different elevations, including at sea level), on the sea, and (later in 1911) underwater [4,3,7]. Those measurements, performed with electroscopes (Figure 1), were aimed at checking whether the radioactivity within the Earth's crust was sufficient to explain the ionization effects (about 13 ions per second per cubic centimeter of air) that had been measured on the Earth's surface. Pacini concluded that the Earth's radioactivity alone was not sufficient to explain the observations. Pacini made several measurements on an Italian Navy ship (the cacciatorpediniere "Fulmine", Figure 1). First he concluded that the ionization above the sea, at sea level, and far from the coast, was consistent with that measured at ground level on land. From 1910 on, Pacini proposed a new experimental technique that proved very successful and was to became important for later developments of physics: he measured the radiation intensity in water, at a depth of 3 meters, in the Genoa Gulf and in the Bracciano Lake (near Rome), proving that the radiation was significantly smaller underwater than on the ground. To explain his results, that marked the beginning of the underground/underwater technique for cosmic-ray studies (that has been implemented so many times up to this day), Pacini proposed the existence of a radiation of extraterrestrial origin -later to be called "cosmic rays". A few months later Victor Hess, using balloon flights, confirmed those results with measurements that eventually earned him the Nobel Prize for Physics in 1936, two years after Pacini's death.
In his paper containing the interpretation of such results [3], published a few months before the paper by Hess, Pacini was evidently aware of Hess' results as he did quote them correctly. Some excerpts from mail exchanges that occurred between the two scientists in 1920 are very illuminating on I was already aware of some of these works from summaries that had been reported to me during the war. [But] the paper entitled "Die Frage der durchdring. Strahlung ausserterrestrischen Ursprunges" ("The problem of penetrating radiation of extraterrestrial origin") was unknown to me. While I have to congratulate you on the clarity in which this important matter is explained, I have to remark, unfortunately, that the Italian measurements and observations, which take priority as far as the conclusions that you, Gockel and Kolhörster draw, are missing; and I am so sorry about this, because in my own publications I never forgot to mention and cite anyone...". The answer by Hess, dated March 17, 1920, was: "Dear Mr. Professor, your very valuable letter dated March 6 was to me particularly precious because it gave me the opportunity to re-establish our links that unfortunately were severed during the war. I could have contacted you before, but unfortunately I did not know your address. My short paper "Die Frage der durchdring. Strahlung aussert-errestrischen Ursprunges" is a report of a public conference, and therefore has no claim of completeness. Since it reported the first balloon measurements, I did not provide an in-depth explanation of your sea measurements, which are well known to me. Therefore please excuse me for my unkind omission, that was truly far from my aim ...". On April 12, 1920, Pacini in turn replied to Hess: "... [W]hat you say about the measurements on the penetrating radiation performed on balloon is correct; however the paper "Die Frage der durchdring. Strahlung ausserterrestrischen Ursprunges" lingers quite a bit on measurements of the attenuation of this radiation made before your balloon flights, and several authors are cited whereas I do not see any reference to my relevant measurements (on the same matter) performed underwater in the sea and in the Bracciano Lake, that led me to the same conclusions that the balloon flights have later confirmed."
Edoardo Amaldi had no doubt that Domenico Pacini was indeed the discoverer of cosmic rays. This is reported in a letter that E. Amaldi wrote on July 14, 1941, to the Director of the Physics Institute of Rome University, Antonino Lo Surdo (the letter belongs to the Amaldi Archive at the "La Sapienza" University of Rome [8]). E. Amaldi's letter was motivated by an article that had earlier appeared on a local newspaper, in which it was stated that nuclear physics and cosmic ray physics were "Jewish sciences". Here is the relevant quote from E. Amaldi's answer: "... this statement appear so strange to anyone who knows, as you certainly do, that the Italian Domenico Pacini was the discoverer of the Cosmic Rays and only afterwards did the German Hess, Kolhörster, etc. follow ...".
The route to cosmic rays, opened by Pacini, continued in the following years and, after the second world war, Italy took up a leading role in the field. Back in those years most of the research was performed at laboratories placed on mountain tops. In Italy the most important one was located at Testa Grigia on the Plateau Rosa, at an altitude of 3500 meters on the Matterhorn. A team from the University of Padova established an observatory at the Passo Fedaja on the Dolomites near Belluno, at an elevation of approximately 2000 meters: thanks to the Società Adriatica di Elettricità (SADE), the observatory had enough electric power to run a large magnet for experiments that used counters and cloud chambers.
The complementary work by Pacini and Hess were seminal in starting, respectively, underwater and upper-atmosphere/space studies of cosmic rays.
Is it possible to say now, as Edoardo Amaldi did once, that "the Italian Domenico Pacini was the discoverer of the Cosmic Rays [and was followed] by the Germans Hess, Kolhörster etc."? A great discovery is in general the result of joint efforts by many people. It is certainly true that Pacini wrote, in as early as 1910 [4], that the action of active substances in the soil was not sufficient to explain the observed properties of the penetrating radiation; and that he was the first to publish such a statement. However, a whole community of researchers was already involved in that field. Pacini's work was carried out in difficult conditions because of lack of resources available to him, because of lack of scientific freedom during the crucial years when he was working at the Central Bureau of Meteorology and Geodynamics, and finally because of the substantial indifference his work was met with by the Italian academic world -a fact that still today is sadly evident to anyone who treads non-mainstream scientific paths.
We conclude by remarking that, as described in the Bulletin of the Società Aeronautica Italiana, at the beginning of the XX century an Italian balloon, named after Jules Verne, did reach a height of about 5000 meters. This proves that in Italy, too, a balloon-borne experiment would have been technically feasible in those years. An open-minded national research agency -such as it did not exist in Italy in those days -can really be the key to success in experimental Physics. This may be a lesson Domenico Pacini has taught us, for the present time as well as for future times.
Figure 1 :
1Domenico Pacini. Bari, his research interests mainly focused on the diffusion processes of light in the atmosphere. Domenico Pacini died of pneumonia in Rome on May 23, 1934, shortly after his marriage.
Figure 2 :
2Three electroscopes actually used by Pacini for his measurements. In foreground, the Ebert's electroscope.
Figure 3 :
3The cacciatorpediniere "Fulmine", used by Pacini for his measurements on the sea. the issue of the priority in discovering cosmic rays. (Such exchanges were reported in the Pacini commemoration held in Bari in 1935.) On March 6, 1920, Pacini wrote to Hess: "...I had the opportunity to study some of your papers about electrical-atmospherical phenomena that you submitted to the Principal Director of the Central Bureau of Meteorology and Geodynamics.
Cline in 1910 summarizes[1] the status of the art: experiments were mainly oriented to measure the daily variations or seasonal variations. Cline cited the work by the Italian Domenico Pacini[2] about the daily variations of the radiation measured on the sea at Sestola, in Italy. Pacini's measurement was remarked in Cline's paper as a first evidence of the atmosphere being the main responsable of the penetrating radiation, excluding the Sun as the main origin.
AcknowledgementsWe are grateful to the University of Bari, and in particular to Professor A. Garuccio, for supporting the research of documents regarding Domenico Pacini; to the Dipartimento Interateneo di Fisica of Bari for organizing the Domenico Pacini memorial day that was held in Bari on April 17, 2007; to Professors F. Guerra and N. Robotti for uncovering relevant material in the Amaldi Archive at Rome's "La Sapienza" University and in the Bracciano museum; to Roberto Garra for uncovering material in "Collegio Romano"; to Comandante E. Bagnasco, A. Lombardi from Associazione Culturale Italia, C. D'Adamo from Regia Marina Italiana for historical pictures and information on the ships by the Italian Navy; and to Professors L. Guerriero, E. Menichetti, P. Spinelli, L. Cifarelli, P. Carlson and M. Persic for help, support, discussions and suggestions.
. G A Cline, Phys. Rev., Series I. 3035Cline, G.A. (1910). Phys. Rev., Series I 30, 35.
. D Pacini, Rend. Acc. Lincei. 18123Pacini, D. (1909). Rend. Acc. Lincei, 18, 123.
La radiazione penetrante alla superficie ed in seno alle acque. D Pacini, arXiv:1002.1810v1Nuovo Cim. Penetrating radiation at the surface of and in water. physics.hist-phPacini, D. (1912). La radiazione penetrante alla superficie ed in seno alle acque, Nuovo Cim., VI/3, 93, translated and commented by A. De Angelis, "Penetrating radiation at the surface of and in water", (Preprint arXiv:1002.1810v1 [physics.hist-ph]).
La radiazione penetrante sul mare. D Pacini, Ann. Uff. Centr. Meteor. Pacini, D., (1910). La radiazione penetrante sul mare, Ann. Uff. Centr. Meteor., XXXII, parte I;
La radiation pénétrante sur la mer. D Pacini, 307Le Radium, VIIIPacini, D., (1911). La radiation pénétrante sur la mer, Le Radium, VIII, 307.
. V Hess, Phys. Zeit. 131084Hess, V. (1913). Phys. Zeit., 13, 1084.
. Nature. 107Nature (1904) LXX, 107.
Misure di ionizzazione dell'aria su terraferma ed in mare. D Pacini, Nuovo Cim. 5Pacini, D., (1908). "Misure di ionizzazione dell'aria su terraferma ed in mare, Nuovo Cim., V/15, 5;
Sulla radioattività indotta dell'atmosfera nel golfo ligure. D Pacini, Nuovo Cim. 24Pacini, D., (1908). "Sulla radioattività indotta dell'atmosfera nel golfo ligure, Nuovo Cim., V/15, 24;
Questioni di elettricità atmosferica. D Pacini, Nuovo Cim. Pacini, D., (1910). "Questioni di elettricità atmosferica, Nuovo Cim., V/19, 449;
Sui prodotti del radio e del torio nell'atmosfera. D Pacini, Nuovo Cim. Pacini, D., (1910). "Sui prodotti del radio e del torio nell'atmosfera, Nuovo Cim., V/19, 345;
Osservazioni di elettricità atmosferica eseguite in occasione del passaggio della cometa di Halley. D Pacini, Ann. Uff. Centr. Meteor. XXXII, parte IPacini, D., (1910). "Osservazioni di elettricità atmosferica eseguite in occasione del passaggio della cometa di Halley, Ann. Uff. Centr. Meteor., XXXII, parte I.
La scoperta dei raggi cosmici. F Guerra, N Robotti, Domenico Pacini', Lezione alla Scuola di Dottorato di OtrantoGuerra, F. and Robotti, N. (2007). La scoperta dei raggi cosmici: Domenico Pacini', Lezione alla Scuola di Dottorato di Otranto.
|
[] |
[
"Implementing the De-thinning Method for High Energy Cosmic Rays Extensive Air Showers Simulations",
"Implementing the De-thinning Method for High Energy Cosmic Rays Extensive Air Showers Simulations"
] |
[
"Alex Estupiñan \nEscuela de Física\nUniversidad Industrial de Santander\nBucaramangaColombia\n",
"Hernan Asorey \nEscuela de Física\nUniversidad Industrial de Santander\nBucaramangaColombia\n\nLaboratorio Detección de Partículas y Radiación\nCentro Atómico Bariloche &\n\n",
"Luis A Núñez \nEscuela de Física\nUniversidad Industrial de Santander\nBucaramangaColombia\n",
"Instituto Balseiro ",
"San Carlos ",
"Argentina Bariloche "
] |
[
"Escuela de Física\nUniversidad Industrial de Santander\nBucaramangaColombia",
"Escuela de Física\nUniversidad Industrial de Santander\nBucaramangaColombia",
"Laboratorio Detección de Partículas y Radiación\nCentro Atómico Bariloche &\n",
"Escuela de Física\nUniversidad Industrial de Santander\nBucaramangaColombia"
] |
[] |
To simulate the interaction of cosmic rays with the Earth atmosphere requires highly complex computational resources and several statistical techniques have been developed to simplify those calculations. It is common to implement the thinning algorithms to reduce the number of secondary particles by assigning weights to representative particles in the evolution of the cascade. However, since this is a compression method with information loss, it is required to recover the original flux of secondary particles without introduce artificial biases. In this work we present the preliminary results of our version of the de-thinning algorithm for the reconstruction of thinned simulations of extensive air showers initiated by cosmic rays and photons in the energy range 10 15 < E/eV < 10 17 .
|
10.1016/j.nuclphysbps.2015.10.140
|
[
"https://arxiv.org/pdf/1503.07960v1.pdf"
] | 118,621,024 |
1503.07960
|
3af7bbc2e576ba03d5b364b5779b0caa4e143e89
|
Implementing the De-thinning Method for High Energy Cosmic Rays Extensive Air Showers Simulations
March 30, 2015
Alex Estupiñan
Escuela de Física
Universidad Industrial de Santander
BucaramangaColombia
Hernan Asorey
Escuela de Física
Universidad Industrial de Santander
BucaramangaColombia
Laboratorio Detección de Partículas y Radiación
Centro Atómico Bariloche &
Luis A Núñez
Escuela de Física
Universidad Industrial de Santander
BucaramangaColombia
Instituto Balseiro
San Carlos
Argentina Bariloche
Implementing the De-thinning Method for High Energy Cosmic Rays Extensive Air Showers Simulations
March 30, 2015thinningde-thinningExtensive Air ShowersCosmic Rays
To simulate the interaction of cosmic rays with the Earth atmosphere requires highly complex computational resources and several statistical techniques have been developed to simplify those calculations. It is common to implement the thinning algorithms to reduce the number of secondary particles by assigning weights to representative particles in the evolution of the cascade. However, since this is a compression method with information loss, it is required to recover the original flux of secondary particles without introduce artificial biases. In this work we present the preliminary results of our version of the de-thinning algorithm for the reconstruction of thinned simulations of extensive air showers initiated by cosmic rays and photons in the energy range 10 15 < E/eV < 10 17 .
Introduction
The Earth atmosphere interacts continuously with a flux of particles of galactic and extragalatic origin. The interaction of one of these cosmic rays with an atomic element in the atmosphere produces a cascade of particles (the so called Extensive Air Shower, EAS), in which the number of particles could reach billions at the maximum of the development of the cascade.
To understand and simulate these processes requires highly complex computational resources. To reduce the number of particles that have to be followed at the highest energies, it is common the usage of the thinning algorithm (see for example [1]), a statistical method that reduce the number of secondary particles by assigning weights to representative particles in the evolution of the cascade. However, since this is a compression method with loss of information, it is required to recover the original flux of secondary particles without introduce artificial biasses.
The so called de-thinning method [2] is one of the existent methods designed to deal with this information loss. Since the EAS is an stochastic process, in this work we will compare the distribution of particles at ground level for several non-thinned showers, with the corresponding reconstructed thinned (with two different levels of thinning, ε th = 10 −7 and ε th = 10 −8 ) and de-thinned showers obtained from exactly the same set of initial parameters. All the simulations were made by using CORSIKA air shower simulation program [3] and several analysis routines developed within the LAGO (Latin American Giant Observatory) project simulation chain [4,5].
Thinning Method
The implementation of this algorithm is applied during the shower simulation over the secondary particles when this condition is fulfilled:
E 0 ε th > n j=1 E j ,(1)
where E j is the energy of the secondary particle, E 0 is the energy of the primary particle and ε th = E j /E 0 is defined as the level of thinning.
In this case, only one secondary particle i survives probability:
P i = E i / n j=1 E j .(2)
Otherwise, if the total sum of the energy of the n secondary particles is greater than the thinning energy threshold, i.e.:
E 0 ε th < n j=1 E j ,(3)
then the secondary particle with energy below the thinning threshold will survive with a probability:
P i = E i /E 0 ε th .(4)
In both cases, the particles that survive have their weight multiplied by a factor of w i = P −1 i .
De-thinning Method
The main questions to answer for any reconstruction method of thinned showers are:
• How do you determine the accuracy of the survival sample of secondary particles?
• How do you use the thinned sample to completely rebuild the original shower avoiding the introduction of artificial biases?
• What is the maximum value of ε th , for which the sample is a good representative of the original shower for a particular type of reconstruction method?
Several techniques has been developed to asses these questions. Our implementation of the de-thinning algorithm is based in the original development of Stokes et al. [2], and consist in the successive application of the following steps over the thinned sample of secondary particles:
1. choose a vertex point on the trajectory of the weighted particle in the way given in the next paragraph;
2. centred in the weighted particle path, chose a random direction of propagation for the new particles (also called "daughter" particles) by using a 2D gaussian distribution with zero mean and a fixed standard deviation;
3. project the daughter particle to ground level, and calculate the travel distance in units of atmospheric depth;.
4. determine the energy of the inserted particle using a gaussian random distribution with mean E i and σ = 0.1E i , where E i correspond to the energy of the weighted particle;
5. determine the probability of atmospheric absorption of the inserted particle by using a fixed atmospheric interactions length, and decide if it will reach the ground;
6. depending on the daughter particle trajectory and its starting point, calculate the time of flight of the inserted particle; Figure 1: Particle density as a function of the core distance for a vertical proton of energy 5 × 10 6 GeV. We use the non-thinned shower (red line) as the reference, and compare it with the reconstructed density profile obtained after the application of our de-thinning method over the same showers but with thinning levels of ε th = 10 −7 (dark green line) and 10 −8 (blue line). We optimize our method to get good results on intermediate distances to the core, where typical detectors used in many astroparticles observatories are triggered but not saturated.
7. Repeat steps two to six, (w − 1) times to build a complete set of secondary unweighted particles.
To assure temporal consistency of the particles reaching ground level, the distance between the vertex where the sub-shower of the de-thinned particles begins and the ground should be less than D max , which is given by:
D max = c 2 (t i − t 0 ) 2 − | x i − x 0 | 2 2(c(t i − t 0 ) − ( x i − x 0 ) ·p i )(5)
where c is the speed of light. Any shorter separation will generate de-thinned particles temporally consistent with the development of the EAS. We have implemented the above de-thinning algorithms, and introduce some improvements in the overall algorithm, such as:
• the standard deviation used to determine the daughter trajectory depended on the type (electromagnetic, muon or hadron) and the energy E i of the weighted particle;
• location dependent atmospheric model to determine the atmospheric depth as a function of the altitude and trajectory of the weighted particle;
• the atmospheric interaction length used to determine the probability of reaching ground level also depends on the type of secondary particle;
Our de-thinning code was implemented in python 2.7 1 and will be licensed under GPLv3 2 .
Results
Our main results for this first approach to this problem are displayed in figures 1 and 2, where it is possible to compare the secondary particle density and the average energy as a function of the distance to the shower core. As explained above, we simulate showers at fixed energies and arrival directions for different primaries and different values of thinning levels: ε th = 0 (non-thinned, used as reference) and ε th = 10 −7 and ε th = 10 −8 . The set of parameters used for each simulation, including the CORSIKA random generators seeds, were exactly the same to allow proper comparisons between the de-thinned and the non-thinned showers. Figure 2: Average energy of secondaries as a function of the radial distance for a shower initiated by a vertical proton with energy E 0 = 5×10 6 GeV without thinning (blue circles), and de-thinned with original thinned levels of ε th = 10 −7 (magenta thin circles) and ε th = 10 −8 (blue circles).
With this implementation, we obtained very good results with a significant improvement in the usage of computing resources, which is specially useful for simulate large numbers of high energy showers in remote simulation clusters or GRID-CORSIKA implementations, as the cloud storage and network transfers of the large outputs obtained without thinning could be extremely difficult. The outcomes of the computing times and the storage for the different configurations tested are presented in the table 1. It is clear a significant reduction in the computing times and output file sizes. Table 1: Simulation time and final secondary particles file sizes for different thinning levels (ε th ) for a shower initiated by proton of E 0 = 10 8 GeV.
www.python.org 2 www.gnu.org/copyleft/gpl.html
Conclusions and AcknowledgementsIn this work we show the first results of a python-based implementation of the so called dethinning algorithm, which is applied for the reconstruction of thinned CORSIKA simulations of the interaction of primary cosmic rays with the atmosphere. After comparing the results of the application of this algorithm over thinned showers with the corresponding non-thinned equivalent ones, we were able to tune different parameters of the reconstruction method, such as the energy dependent angular aperture of the cone for the injected new particles and specific atmospheric absorption coefficients for each type of secondary particle. When totally implemented, this code will be publicly released under the license GPLv3.The authors of this work thank the support of COLCIENCIAS "Semillero de Investigación" grant 617/2014 and one of us (LAN) to CDCHT-ULA project C-1598-08-05-A.
A sampling procedure to regenerate particles in a ground detector from a ?thinned? air shower simulation output. P Billoir, Astroparticle Physics. 305P. Billoir, A sampling procedure to regenerate particles in a ground detector from a ?thinned? air shower simulation output, Astroparticle Physics 30 (5) (2008) 270-285.
Dethinning extensive air shower simulations. B Stokes, R Cady, D Ivanov, J Matthews, G Thomson, Astroparticle Physics. 3511B. Stokes, R. Cady, D. Ivanov, J. Matthews, G. Thomson, Dethinning extensive air shower simulations, Astroparticle Physics 35 (11) (2012) 759-766.
CORSIKA : A Monte Carlo Code to Simulate Extensive Air Showers. D Heck, J Knapp, J Capdevielle, G Schatz, T Thouw, FZKA. 6019D. Heck, J. Knapp, J. Capdevielle, G. Schatz, T. Thouw, CORSIKA : A Monte Carlo Code to Simulate Extensive Air Showers, FZKA 6019 (1998) 1-98.
The LAGO Solar Project. Proceedings of the 33th International Cosmic Ray Conference ICRC 2013. the 33th International Cosmic Ray Conference ICRC 2013BrazilRío de Janeiroin pressThe LAGO Collaboration [H. Asorey], The LAGO Solar Project, in: Proceedings of the 33th International Cosmic Ray Conference ICRC 2013, Vol. in press, Río de Janeiro, Brazil, 2013, pp. 1-4.
The Latin American Giant Observatory Project. H Asorey, Proceedings of the X SILAFAE. the X SILAFAEMedellín, ColombiaH. Asorey, The Latin American Giant Observatory Project, in: Proceedings of the X SILAFAE, Medellín, Colombia, 2014.
|
[] |
[
"Chaplygin-gas Solutions of f (R) Gravity",
"Chaplygin-gas Solutions of f (R) Gravity"
] |
[
"Maye Elmardi \nAstrophysics, Cosmology and Gravity Centre\nUniversity of Cape Town\n7701RondeboschSouth Africa\n\nDepartment of Mathematics and Applied Mathematics\nUniversity of Cape Town\n7701RondeboschSouth Africa\n",
"Amare Abebe \nDepartment of Physics\nNorth-West University\n2735MafikengSouth Africa\n",
"Abiy Tekola \nLas Cumbres Observatory Global Telescope Network\n93117GoletaCAUSA\n"
] |
[
"Astrophysics, Cosmology and Gravity Centre\nUniversity of Cape Town\n7701RondeboschSouth Africa",
"Department of Mathematics and Applied Mathematics\nUniversity of Cape Town\n7701RondeboschSouth Africa",
"Department of Physics\nNorth-West University\n2735MafikengSouth Africa",
"Las Cumbres Observatory Global Telescope Network\n93117GoletaCAUSA"
] |
[] |
We explore exact f (R) gravity solutions that mimic Chaplygin-gas inspired ΛCDM cosmology. Starting with the original, generalized and modified Chaplygin gas equations of state, we reconstruct the forms of f (R) Lagrangians. The resulting solutions are generally quadratic in the Ricci scalar, but have appropriate ΛCDM solutions in limiting cases. These solutions, given appropriate initial conditions, can be potential candidates for scalar field-driven early universe expansion (inflation) and dark energy-driven late-time cosmic acceleration.
|
10.1142/s0219887816501206
|
[
"https://arxiv.org/pdf/1603.05535v4.pdf"
] | 119,261,227 |
1603.05535
|
257920870c72604e9e786f15cd0a170a8f73fb8e
|
Chaplygin-gas Solutions of f (R) Gravity
23 Nov 2016
Maye Elmardi
Astrophysics, Cosmology and Gravity Centre
University of Cape Town
7701RondeboschSouth Africa
Department of Mathematics and Applied Mathematics
University of Cape Town
7701RondeboschSouth Africa
Amare Abebe
Department of Physics
North-West University
2735MafikengSouth Africa
Abiy Tekola
Las Cumbres Observatory Global Telescope Network
93117GoletaCAUSA
Chaplygin-gas Solutions of f (R) Gravity
23 Nov 2016PACS numbers: 04.50.Kd, 04.25.Nx Chaplygin-gas Solutions of f (R) Gravity 2Cosmologyf (R) gravityChaplygin gasdark energydark matter
We explore exact f (R) gravity solutions that mimic Chaplygin-gas inspired ΛCDM cosmology. Starting with the original, generalized and modified Chaplygin gas equations of state, we reconstruct the forms of f (R) Lagrangians. The resulting solutions are generally quadratic in the Ricci scalar, but have appropriate ΛCDM solutions in limiting cases. These solutions, given appropriate initial conditions, can be potential candidates for scalar field-driven early universe expansion (inflation) and dark energy-driven late-time cosmic acceleration.
Introduction
Explaining the not-yet-so-fully-understood accelerated expansion phases of the early and current stages of the Universe as well as the flatness in the rotational curves of spiral galaxies have certainly become in the top of to-do lists in modern cosmology and astrophysics. As a result, current research into the nature of so-called dark energy and dark matter has led to reconsiderations of the gravitational physics (General Relativity, GR) and cosmic fluid (baryonic matter) models underlying the standard cosmological model.
Among a large pool of alternative theories put forward to explain early-universe inflation [1] and late-time cosmic acceleration [2][3][4][5][6][7][8] are f (R) theories of gravity (see Refs. [9][10][11][12][13][14] for more detailed reviews). Such theories were first speculated on by Buchdahl [15] but recent interest has centered on their potential candidacy as possible infrared (IR) and ultraviolet (UV) completions of GR (see, e.g., [16][17][18][19][20][21]). More recently, these gravitational alternative theories have found cosmological applications in, inter alia, the dynamical study of homogeneous cosmological models [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] and the linear growth of large-scale structures [39][40][41][42][43][44][45][46][47].
For quite sometime now, the Chaplygin gas [48] has been considered as another alternative cosmological model of a Friedmann-Lemaître-Robertson-Walker (FLRW) universe with an exotic perfect-fluid equation of state [49][50][51][52][53][54]. The model provides a cosmic expansion history consistent with a universe that transitions from a decelerating matter-dominated phase to a late-time accelerated one [52,55,56]. As a result, there have been several efforts to come up with a unified description of both dark matter and dark energy as manifestations of a single cosmic fluid that is the Chaplygin gas [52,[57][58][59].
The main objective of this work is to come up with a unified picture of both alternative cosmological scenarios discussed above. We study models of f (R) gravity which, when we impose the Chaplygin gas equations of state (EoS) to their effective pressure and energy density, produce viable exact solutions that reduce to the ΛCDM scenario in the approximate cosmological limits.
The rest of this paper is organised as follows: In Sec. (2) we give a brief overview of f (R) gravitational models and their corresponding field equations, and describe the background cosmic evolution under consideration. We give a brief description of the Chaplygin gas model and its generalized and modified EoS in Sec. (3). By comparing these EoS with the EoS for f (R) models under certain assumptions, we deduce the form of the corresponding f (R) Lagrangians. Finally in Sec. (4) we discuss the results and give our conclusions.
Natural units ( = c = k B = 8πG = 1) will be used throughout this paper, and Latin indices run from 0 to 3. The symbols ∇,∇ and the overdot . represent the usual covariant derivative, the spatial covariant derivative, and differentiation with respect to cosmic time. We use the (− + ++) spacetime signature and the Riemann tensor is defined by
R a bcd = Γ a bd,c − Γ a bc,d + Γ e bd Γ a ce − Γ f bc Γ a df ,(1)
where the Γ a bd are the Christoffel symbols (i.e., symmetric in the lower indices) defined by
Γ a bd = 1 2 g ae (g be,d + g ed,b − g bd,e ) .(2)
The Ricci tensor is obtained by contracting the first and the third indices of the Riemann tensor:
R ab = g cd R cadb .(3)
Unless otherwise stated, primes ′ etc are shorthands for derivatives with respect to the Ricci scalar
R = R a a(4)
and f is used as a shorthand for f (R).
f (R) Models
The action for f (R) gravity can be given as
A = d 4 x √ −g 1 2 f (R) + L m ,(5)
where f (R) is a general differentiable function of the Ricci scalar and L m corresponds to the matter Lagrangian. For models of gravitation with such action, applying the variational principle with respect to the metric g ab results in the generalized field equations
G ab = T m ab + T R ab ≡ T ab ,(6)
where T ab is the total energy momentum tensor (EMT) and is conserved by virtue of the Einstein tensor G ab ≡ R ab − 1 2 Rg ab being a covariantly conserved quantity. Here T m ab is the EMT of standard matter and
T R ab = 1 2 g ab (f − Rf ′ ) + ∇ b ∇ a f ′ − g ab ∇ c ∇ c f ′ + (1 − f ′ ) G ab(7)
is the curvature contribution to the EMT. Since T m ab is a conserved quantity, we see from Eq. (6) that T R ab should also be conserved. In this setting, the linearized curvature components of the thermodynamical quantities are given by
µ R ≡ T R ab u a u b = 1 2 (Rf ′ − f ) − Θf ′′Ṙ + 1 3 (1 − f ′ )Θ 2 + f ′′∇2 R ,(8)p R ≡ 1 3 (T R ab h ab ) = 1 6 (3f − Rf ′ − 2R) + f ′′R + f ′′′Ṙ2 + 2 3 Θf ′′Ṙ + 1 9 (1 − f ′ )Θ 2 − 2 3 f ′′∇2 R ,(9)q R a ≡ −T R bc u b h c a = 1 3 f ′′ Θ∇ a R − f ′′′Ṙ∇ a R − f ′′∇ aṘ − 2 3 (1 − f ′ )∇ a Θ ,(10)π R ab ≡ T R cd h c a h d b = f ′′∇ a∇b R ,(11)
and the total cosmic medium is composed of standard matter and the curvature fluid, with the total thermodynamical quantities given by
µ ≡ µ m + µ R , p ≡ p m + p R , q a ≡ q m a + q R a , π ab ≡ π R ab .(12)
The trace of the Einstein field equations (6) gives
3f ′′R + 3Ṙ 2 f ′′′ + 3ΘṘf ′′ − 3f ′′∇2 R − Rf ′ + 2f + 3p m − µ m = 0 ,(13)
and plays an important role as a constraint relation between matter and curvature. Analogous to the matter EoS p m = w m µ m , where w m is the matter EoS parameter, we can define the EoS for the curvature fluid as
p R = w R µ R .
For FLRW spacetimes, the Ricci scalar R is given by
R = 2Θ + 4 3 Θ 2 ,(14)
where Θ is the cosmic expansion parameter related to the cosmological scale factor a(t) and the Hubble parameter H(t) via the equations
Θ ≡ 3ȧ (t) a(t) = 3H(t) .(15)
Solving forΘ one can rewrite the above equation aṡ
Θ = R 2 1 − 4 3R Θ 2 .(16)
If the Ricci scalar varies slowly, i.e., if R is almost constant, the solution of this ordinary differential equation (o.d.e) takes the form
Θ = 1 2 √ 3R tanh R 3 (t − t 0 ) ,(17)
for some constant of integration t 0 that can be taken to be the time at the commencement of the inflationary phase of expansion. Solving for the cosmological scale factor gives
a(t) = a 0 cosh R 3 (t 0 − t) .(18)
For simplicity we set t 0 ≃ 0. During steady-state exponential expansion in a de Sitter spacetime (such as during inflation or late-time evolution), the approximationΘ → 0 results in
R = 4 3 Θ 2 = const , a(t) = a 0 e 1 3 Θt ,(19)
and the matter content evolves according to
µ m = µ 0 a(t) a 0 −3(1+wm) .(20)
During such an exponentially expanding cosmic evolution phase, for an initial energy density µ 0 , we see that the matter energy density decays exponentially:
µ m = µ 0 e −(1+wm)Θt .(21)
Chaplygin-gas Solutions
Chaplygin-gas fluid models are perfect-fluid models currently posing as candidates to unify dark energy and dark matter. These fluid models were originally studied [48] in the context of aerodynamics, but only recently did they see cosmological applications [49-51, 57, 59, 60]. Among the interesting features of these models is that in the FLRW framework, a smooth transition between an inflationary phase, the matter-dominated decelerating era, and then late-time accelerated de Sitter phase of cosmic expansion can be achieved [52,55].
Original and Generalized Chaplygin Gases
In the original treatment, the negative pressure associated with the Chaplygin gas models is related to the (positive) energy density through the EoS
p = − A µ α(22)
for positive constant A and α = 1. But this was later generalized [55,61] to include 0 ≤ α ≤ 1. One of the first cosmological interpretations of such a fluid model was given in [62] where for flat universes, Eq. (22) corresponds to a viscosity term that is inversely proportional to dust energy density. Ever since the discovery of cosmic acceleration, however, both the original and generalized Chaplygin gas models have been extensively studied as alternatives to dark energy and/or unified dark energy and dark matter models (see, e.g., [50, 52, 55, 58-60, 63, 64]). Now if we consider the background curvature energy density and isotropic pressure terms defined in Eqs. (8) and (9) above, in the constant-curvature limiting case, we have
µ R = 1 4 [R(f ′ + 1) − 2f ] = −p R .(23)
This equation of state, with an effective EoS parameter w R = −1, provides the condition for an exponential (accelerated) expansion with a constant Hubble parameter. The energy density µ R (with its negative pressure p R ) remains constant and can be interpreted as playing the role of the cosmological constant Λ. Considering the curvature fluid as a manifestation of the Chaplygin gas with the EoS (22), we obtain
p R = −µ R = − A µ α R ,(24)f (R) = R + C 1 R 2 − 2A 1 α+1 (26)
for an arbitrary (integration) constant C 1 . We note that the ΛCDM solution f (R) = R − 2Λ is already a particular solution with C 1 = 0 and A = Λ α+1 . In particular, if α = 0, then A = Λ, from which, going back to Eq. (24), one concludes µ R = Λ.
If we include the linearized Laplacian term in Eqs. (8) and (9) and use the eigenvalue − k 2 a 2 of the covariantly defined Laplace-Beltrami operator∇ 2 on (almost) FLRW spacetimes
∇ 2 R = − k 2 a 2 R(27)
for a comoving wavenumber k, we obtain the second-order o.d.e
B 2 Rf ′′ (R) − Rf ′ (R) + 2f (R) − R + 4A 1 α+1 = 0 ,(28)
where here we have defined
B 2 ≡ 4(2 + 3α) 3(1 + α) k 2 a 2 .(29)
The solution of Eq. (28) is given, for arbitrary constants C 2 , C 3 , by
f (R) = R+C 2 R 2 − 2RB 2 +C 3 (R 2 − 2RB 2 )Ei 1, − R B 2 + (R − B 2 )B 2 e R B 2 −2A 1 α+1 ,(30)
which should reduce to the quadratic solution (26) for negligible values of B 2 , i.e., for small first-order contributions to the energy density and pressure terms.
Modified Chaplygin Gas
Over the years, several modifications to the original and generalized Chaplygin gas models have been studied. If one considers the modified Chaplygin gas (MCG) EoS [61,[65][66][67][68][69][70][71]
p R = γµ R − A µ α R ,(31)
then the resulting f (R) model generalizes to
f (R) = R + C 4 R 2 − 2 A γ + 1 1 α+1 ,(32)
where C 4 is an arbitrary integration constant. The ΛCDM solution is a limiting case of this generalized model when C 4 = 0 and A = (γ + 1)Λ α+1 . In particular, if α = 0 = γ, then A = Λ.
Following similar arguments as in the preceding subsection, if we include the linearized Laplacian contributions to the energy density and pressure, we get Eq. (28) generalized to
B 2 Rf ′′ (R) − Rf ′ (R) + 2f (R) − R + 4 A γ + 1 1 α+1 = 0 ,(33)
the solution of which can be given by
f (R) = R+C 5 R 2 − 2RB 2 +C 6 (R 2 − 2RB 2 )Ei 1, − R B 2 + (R − B 2 )B 2 e R B 2 −2 A γ + 1 1 α+1 ,(34)
for an arbitrary integration constants C 5 and C 6 . This solution obviously generalizes Solutions (26), (30) and (32) and should reduce to the quadratic solution (26) for vanishingly small B 2 values. In [72], it has been shown that any quadratic Lagrangian leading to an isotropic, homogeneous cosmological model takes the form
f (R) = R − 2Λ − 1 6 βR 2 ,(35)
where β is an arbitrary, real constant. If we keep only the quadratic solution in (34), i.e., if we set C 6 = 0, the Lagrangian (35) corresponds to the choice
C 5 = − 1 6 β , B = 0 , A = (γ + 1)Λ α+1 .(36)
Another interesting fact worth pointing out here is that the condition for the existence of a maximally symmetric vacuum solution in f (R) gravity [72]
R 0 f ′ (R 0 ) = 2f (R 0 )(37)
leads to the quadratic solution resulting in the constraint
R 0 1 − 2C 5 B 2 − 4 A γ + 1 1 α+1 = 0 .(38)
The corresponding GR de Sitter, anti-de Sitter and Minkowski solutions R 0 = 4Λ (respectively for Λ > 0 , Λ < 0 and Λ = 0) are obtained when C 5 B 2 = 0 and A = (γ + 1)Λ α+1 .
Modified Generalized Chaplygin Gas
The so-called modified generalized Chaplygin gas (mGCG) model is described by a barotropic equation of state of the form [59,60]
p = βµ − (1 + β) A µ α .(39)
Models of f (R) gravity that satisfy the condition (23), at the sam time mimicking the mGCG, can be shown to be governed by the same equation as (25) and admit the same solutions (26), provided β = −1. On the other hand, if linearized Laplacian terms are included, then the corresponding differential equation in f (R) generalizes to
D 2 Rf ′′ (R) − Rf ′ (R) + 2f (R) − R + 4A 1 α+1 = 0 ,(40)
where we have defined
D 2 ≡ 4 [2 + 3α + 3β(1 + α)] 3(1 + α)(1 + β) k 2 a 2 .(41)
Worthy of note is that this equation and its solution
f (R) = R+C 7 R 2 − 2RD 2 +C 8 (R 2 − 2RD 2 )Ei 1, − R D 2 + (R − D 2 )D 2 e R D 2 −2 A γ + 1 1 α+1 ,(42)
reduce to their generalized counterparts of Eqs. (28) and (30) when β = 0.
Discussions and Conclusion
In this paper we have explored models of f (R) gravity that have Chaplygin gas equations of state. We started with the assumption that the effective energy density and isotropic pressure terms of Eqs. (8) and (9), sourced by the energy-momentum tensor of the curvature fluid (7), satisfy the equation of state of the Chaplygin gas, both in its original (Eq. 22) and generalized (Eq. 31) forms. We then reconstructed, for timeindependent Ricci scalar, the f(R) Lagrangians that allow such equations of state. When only background (zeroth-order) terms are considered for the energy density and pressure, we notice that the effective equation of state is w R = −1, and hence we naturally get f (R) models that are quadratic in R but have limiting ΛCDM solutions.
On the other hand, keeping the first-order (Laplacian of R) terms in the energy density and pressure of the curvature fluid results in the effective energy density picking up linearized corrections to −1. The resulting linearized second-order o.d.es in f (R) result in more generalized solutions including exponential integral terms, of which the quadratic Lagrangians are but a subset of solutions. These solutions also include ΛCDM as a limiting solution.
α+1 .(25)
AcknowledgmentsThe authors thank John Barrow and Behnam Pourhassan for their constructive comments.
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|
[] |
[
"Valuing Evaluation: Methodologies to Bridge Research and Practice",
"Valuing Evaluation: Methodologies to Bridge Research and Practice"
] |
[
"Petteri Kaskenpalo [email protected] \nSERL\nSchool of Computing and Mathematical Sciences\nAUT University\nPrivate Bag92006, 1142AucklandNew Zealand\n",
"Stephen G Macdonell [email protected] \nSERL\nSchool of Computing and Mathematical Sciences\nAUT University\nPrivate Bag92006, 1142AucklandNew Zealand\n"
] |
[
"SERL\nSchool of Computing and Mathematical Sciences\nAUT University\nPrivate Bag92006, 1142AucklandNew Zealand",
"SERL\nSchool of Computing and Mathematical Sciences\nAUT University\nPrivate Bag92006, 1142AucklandNew Zealand"
] |
[
"Proceedings of the 2nd International Workshop on Evidential Assessment of Software Technologies (EAST2012)"
] |
The potential disconnect between research and practice in software engineering (SE) means that the uptake of research outcomes has at times been limited. In this paper we seek to identify research approaches that are rigorous in terms of method but that are also relevant to software engineering practitioners. After considering the correspondence of several approaches to software systems research and practice we recommend a framework for applying grounded theory in SE research, as a means of delivering both robust and useful outcomes.
|
10.1145/2372233.2372242
|
[
"https://arxiv.org/pdf/2101.03675v1.pdf"
] | 6,787,150 |
2101.03675
|
2754d940066be1af6d1cf6195f1e44d5ab9e17ea
|
Valuing Evaluation: Methodologies to Bridge Research and Practice
ACM PressCopyright ACM Press
Petteri Kaskenpalo [email protected]
SERL
School of Computing and Mathematical Sciences
AUT University
Private Bag92006, 1142AucklandNew Zealand
Stephen G Macdonell [email protected]
SERL
School of Computing and Mathematical Sciences
AUT University
Private Bag92006, 1142AucklandNew Zealand
Valuing Evaluation: Methodologies to Bridge Research and Practice
Proceedings of the 2nd International Workshop on Evidential Assessment of Software Technologies (EAST2012)
the 2nd International Workshop on Evidential Assessment of Software Technologies (EAST2012)Lund, SwedenACM Press10.1145/2372233.2372242Full citation: Kaskenpalo, P., & MacDonell, S.G. (2012) Valuing evaluation: methodologies to bridge research and practice, inResearchpracticedesign sciencesoftware engineeringgrounded theory
The potential disconnect between research and practice in software engineering (SE) means that the uptake of research outcomes has at times been limited. In this paper we seek to identify research approaches that are rigorous in terms of method but that are also relevant to software engineering practitioners. After considering the correspondence of several approaches to software systems research and practice we recommend a framework for applying grounded theory in SE research, as a means of delivering both robust and useful outcomes.
INTRODUCTION
The desire to improve the relevance of software engineering (SE) research has been long evident. For instance, several calls for increased experimentation in SE research have been made; the growing use of systematic literature reviews is motivated in part by the goal of practice improvement; papers that traverse the literature and reflect on (among other things) the breadth and accessibility of the research methods utilized have appeared; and as recently as 2011 a special issue promoting the use of qualitative methods in empirical SE, due in part to their greater relevance to practice, was published in Empirical Software Engineering.
The focus in this paper is on research centered on the development and evaluation of SE artifacts. The design science research process (DSRP) [16] proposes an artifactcentred research methodology with the key steps of problem identification and motivation, objectives of a solution, design and development, demonstration, evaluation, and communication. At a glance, this process is analogous to the outline of many software development methods and software systems projects.
How does this process then account as a model for scientific research, and, considering the number of failed software projects that follow similar well-established development models, how might this process provide more relevant and trustworthy results?
In this paper we explore how the DSRP is proposed to achieve these goals of rigor and relevance, and then suggest a framework of using grounded theory (GT) [5] as a means of further increasing the value of artifact-based research. We discuss the key differences of developing software as a research artifact as opposed to doing so for a typical commercial goal, and by linking these two reasons for developing software, we explore opportunities for improving the relevance of research results to software practitioners.
We map several different life-cycle models, including those proposed previously for DS-based research along with well-established software and system models, and combine them into a reference model that enables us to discuss the steps of these processes together. We call this the Software Artifact-centred Research reference Process (SARP). Given constraints on space we focus in this paper on two steps in this process -problem identification and evaluation. For each step we discuss the requirements outlined in the literature for design science-based research, we outline the key activities that would happen in a non-research oriented software development process, we discuss the relevant GT processes and their benefits from both research and practice points of view, and in this we way establish a traceable trail of relevance throughout the process.
In the next section we review and compare some proposed DS research processes, we outline our reference model, we discuss the relationship, roles and contribution of the different academic computing disciplines and their matching professional roles in this model and in building theory. We then outline key GT principles and processes, followed by a discussion of how grounded theory can support the problem identification and evaluation steps of the SARP framework. Finally we conclude with a summary, and we outline intended future work.
DESIGN SCIENCE RESEARCH
The design science research process has been outlined by Peffers et al. [16] as a conceptual process and a mental model for the production and presentation of design science research in information systems (IS). Their model provides a roadmap for an artifact-centred research process and is consistent with earlier DS literature [1,10,14,19] as well as the software engineering research methodology (SERM) proposed by Gregg et al. [8].
While the terminology used in these methodologies differs -for example, the DSRP includes steps of problem identification and motivation, objectives of a solution, design and development, demonstration, evaluation, and communication whereas the SERM comprises the phases of conceptualization, formalization and development -on closer inspection both methodologies are very similar and include similar expectations and goals.
The above cited authors also have a slightly different view of the nature of the involved artifact. The DSRP is said to represent a suitable methodology for research around any type of system-related artifact. Nunamaker et al. [14] include systems development as a key component of the process. Walls et al. [19] define Information Systems Design Theory and its products as the artifact, but also expect an actual system to be developed as a means of validation. The SERM process is suggested to involve an artifact as a product of a tangible nature, which takes the form of a software prototype or a formal specification of a solution. While it seems clear that a formal model of a solution would likely be sufficient for proving the validity of new concepts and theories to other researchers, it would be more difficult to use this as a means of communicating outcomes to practitioners, and impossible to "observe and measure how well the artifact supports the solution to the identified problem" [16] in a practical situation. However it has long been held that systems research should study the application of information technology (IT) in society [11], so it is crucial to assess such research in a practical setting (and preferably more than one), and to present outcomes in ways that are accessible and beneficial to practitioners [13]. For this reason, in addition to artifact-as-research views, we have considered the DSDM and V-model software development processes and the ITIL application management process [15] in the SARP framework (see Figure 1).
While the DSRP and SERM methodologies suggest a process of performing research based on developing a solution and then evaluating its suitability for addressing the identified problem, they also outline the need for this solution to be grounded in theory and to extend current knowledge boundaries [8]. However, these methodologies do not explicitly assist a researcher in doing this. The need for building a theory as part of computing research methodologies has been discussed in the literature [1,9,18]. Overall this should add rigor to research and assist in the evaluation of results in a broader context, ensuring a contribution to more general theories in the discipline. While this is often discussed in the context of information systems research, this need can also be identified in DSbased research in software engineering [6] and computer science [17].
We accept both the need for theory building and having relevance to practice as fundamental requirements for a rigorous and useful research methodology in SE. In this paper, we focus on a process that involves software development and a software prototype artifact, and we suggest a grounded theory (GT)-based approach in conducting software artifact-based research. We describe how this can integrate with SARP and how it should enable the results to be utilised by practising professionals.
ROLE OF GROUNDED THEORY
The principle of using Grounded Theory with the DS research process has been introduced by Adams and Courtney [1] as a theory-building technique, as a part of a multimethodical research framework integrating DS, GT and action research (AR), and by Fernandez and Lehmann [4] as a way to improve the rigor and relevance in information systems studies when investigating the role of such systems in organisational change via case studies.
Before we move on to consider the use of Grounded Theory with the DSRP, we briefly describe some of the key principles of the GT methodology and point the reader to material that provides a summary of the concepts and processes. Grounded theory has been discussed in numerous articles and textbooks in detail and to different degrees of clarity. Glaser [5, para.7] reminds us "that classic GT is simply a set of integrated conceptual hypotheses systematically generated to produce an inductive theory about a substantive area", and continues [5, para.41]:
"GT is not findings, not accurate facts and not description.
It is just straightforward conceptualization integrated into theory -a set of plausible, grounded hypotheses. It is just that -no more -and it is readily modifiable as new data come from whatever source -literature, new data, collegial comments, etc. The constant comparative method weaves the new data into the subconceptualization."
and summarizes the goal of GT being a "conceptual theory abstract of time, place and people" [5, para.42].
The use of grounded theory as an overarching methodology to study data from multiple sources helps in building a consistent theory as well as to moderate any bias the researcher's personal experience may introduce into the study by building the theory about the domain and via documented observations and developing conceptual coding of the analysed data. GT can use any data for building the conceptual theory. Evidence sources such as those suggested by Yin [20] for case study-based research are all suitable: documentation, archival records, interviews, direct observation, participant observation and physical artifacts. Sources can also include research literature reviews and theoretical or practical exploratory scenarios, software artifacts, system log-files, and system test records.
The GT process starts with open coding where the data is coded in order to identify substantive codes. This stage is literally open, as all available data is analysed. The aim is to relate coded incidents in the data to patterns that can be identified as categories and relationships between them. If these model a pattern, then new incidents in the data will fit these, and as the theory matures new incidents fit the existing categories better and better. Through the constant comparative method, coded incidents are constantly compared with other incidents to establish similarities in and differences between the incidents. The similarities support the identified categories, and new categories or properties for the existing categories are generated in order to better model the variation of incidents and their conditions. "Open coding allows the analyst to see the direction in which to take the study by theoretical sampling before he/she has become selective and focused on a particular problem. Thus, when he/she does begin to focus, he/she is sure of relevance." [5, para.49] In grounded theory, data sources are chosen as they are needed rather than before the research begins, and it is an ongoing part of the research process to select suitable data sources according to the developing categories and the emerging theory. This is referred to as theoretical sampling. This approach is also in line with the three reasons suggested by Benbasat et al. for a case-based research strategy for studying such systems:
"First, the researcher can study information systems in a natural setting, learn about the state of the art, and generate theories from practice. Second, the case method allows the researcher to answer "how" and "why" questions, that is, to understand the nature and complexity of the processes taking place. Questions such as, "How does a manager effectively introduce new information technologies?" are critical ones for researchers to pursue. Third, a case approach is an appropriate way to research an area in which few previous studies have been carried out." [2] For the remainder of this paper, we use the definitions of the GT procedures as presented in [5], and refer to these as we discuss two of the DSRP steps.
THE FRAMEWORK
The reviewed DS research processes suggest clear steps for performing artifact-based research, but provide limited coverage of the actual methods that could be used during the process. As each of the stages produces more information from different viewpoints about the topic of the research, this information should be analysed and managed in a structured manner. Classic Grounded Theory [5] provides a suitable methodology to accompany the SARP steps for this purpose. We contend that a researcher is thus provided with a rigorous process and procedures, as the conceptual theory generation around the concern of the study, including the emergence of any problems that are to be addressed by the solution, is combined with the steps of practical implementation of the discovered/designed solution and its communication and evaluation. An overview of this integration is depicted in Figure 2.
This framework differs from, for example, the design science research framework presented by March and Smith [12] in that their framework consists of building, evaluating, theorizing and justifying as research activities, and our methodology emphasises the requirement of hypothesis-building or theorizing prior to taking other steps. Our approach builds on the theorizing as a required step that guides all the other activities. This way a greater emphasis is directed towards the reasons why the technology is required as well as towards assessing the resulting artifact with the relevant evaluation criteria, including any social and non-technical motivations for the work.
In the following two sub-sections we consider in more detail the role of GT in each of the DSRP research steps of problem identification and evaluation.
Problem Identification
Where do we start? How do we find a problem? What is the research question? Most methodologies and frameworks fall short in answering these questions or providing support in terms of mechanisms for addressing this fundamental part of a research process. Peffers et al. [16, p.89] note the following:
"Define the specific research problem and justify the value of a solution. Since the problem definition will be used to develop an effective artifactual solution, it may be useful to atomize the problem conceptually so that the solution can capture the problems complexity." We argue that this complexity must be captured in order to provide grounding for the claim that the solution indeed addresses a valid problem (rigor), to understand the applicability of the solution to other similar problems (relevance), and to provide a grounded foundation for subsequent steps in the research process.
For this reason the problem identification and motivation stages are crucial. At this stage the focus is on developing a conceptual model that represents the domain under analysis and is grounded in the evidence available in the data. As concepts emerge, hypotheses are developed that explain their relationships with other concepts. This integration develops the grounded conceptual theory throughout the research process, and guides the selective data collection and the saturation of the theory.
In effect, design science-based research aims to satisfy two goals, those of research and those of design. Eekels and Roozenburg [3, p.200] summarize these two goals and their starting points:
• "the starting problem in the research cycle is a discrepancy between the facts and our set of truthstatements concerning these facts. The purpose of the process is adaption of the truth-statements (our knowledge) to the facts";
• "the starting problem in the design cycle is a discrepancy between the facts and our set of valuepreferences concerning these facts. The purpose of the process is the adaption of the facts (through applications of the designed objects) to our value preferences";
in other words, the discovery of new knowledge through the research by examination of the real world, or the alteration of the real world, through the design, into one of the potentially realizable worlds. Which one of these starting points then should we adopt for a design sciencebased SE research project?
We suggest that either one is a potential starting candidate. Taking the research perspective, the researcher would start by identifying the problem and then explore solution candidates that could solve the problem. Alternatively, using the design view as a starting point, the researcher may suspect that a new technology, a new way of doing things, or the application of existing technology in a new situation could be beneficial and worth researching. With this approach the researcher is similarly challenged to discover real-world problems for which the system would provide a solution.
Either way, this is the beginning stage of applying GT in the process. The GT data analysis should be used immediately, and be applied to all available data with the focus on conceptualisation using the constant comparative method. As "GT stands alone as a conceptual theory generating methodology" [5, para.45], and can be described as a structured induction process, it is not only ideal for building a profound understanding of the domain and problem/opportunity under investigation, but also for developing a generalised theory. The analysis may well reveal several concerns, some more significant than others, but through conceptualisation a central concern will emerge.
This stage is conducted with open coding, and the researcher focuses on "patterns among incidents that yield codes and to rise conceptually above detailed description of incidents" [5, para.48]. Individual case studies may provide incidental evidence towards a pattern and suggest new categories, and additional case studies may provide incidental evidence that fit or reject the previous categories, and new categories may emerge.
Likely initial data sources include case studies, stakeholder interviews and the relevant literature. If the researcher is working in an area where field evidence does not exist or is not available, she or he can analyse the literature in order to identify weaknesses with other proposed, earlier solutions. In a design-driven approach, the researcher could use theoretical scenarios that would benefit from the solution and explore their relationship with the current reality. The researcher can describe this relationship and integrate any observations into the emerging theory.
Nunamaker et al. [14] state that in the beginning of the research process the researcher should construct a conceptual framework, which should address the stating of a meaningful research question. This step is further described as follows [14, p.635]: "Researchers should first justify the significance of research questions pursued. An ideal research problem is one that is new, creative, and important in the field." While we do not address several of the other stages here it is of note that both Nunamaker et al. [14] and Peffers et al. [16] also place significant emphasis on the objectives of the solution to be produced, which naturally form the necessary criteria for evaluation later in the process.
Evaluation
The main goal of the evaluation phase is to observe and measure how well the system solves the identified problems in terms of utility, quality and efficacy [16].
Hevner et al. [10] suggest a number of possible evaluation categories including observational, analytical, experimental, testing and descriptive methods. Following from the demonstration phase, the evaluation is based on data collected during and after the demonstration, deployment and operation phases, and focuses on analysing how well the objectives of the solution have been met, by examining simulation results and any experiences gained in configuring the system for laboratory scenarios, as well as analysing the data collected from test user groups.
Data collected during these phases is treated as new data for saturating the grounded theory, and as such it is coded and merged into the theory. This can include performance logs, system error logs, test user group interviews and so on. For example, error logs should be related to the user interview data in order to be able to eliminate user bias due to coding errors.
Note that we have included deployment and operational phases in the SARP framework in order to provide broader coverage of aspects that are relevant in the evaluation of software artifacts in practice, and to serve as a reminder of issues that need to be considered when conducting field experiments with the artifact.
The way to validate the results of course depends on the nature of the research, and an analysis of the objectives guides the evaluation process. If an artifact and the research process address several aspects, the validation process will in turn need to address each aspect, and significant planning is required to conduct experiments and data gathering in order to isolate the impact of each different aspect in the results. Eekels and Roozenburg [3, p.201] remind us of the difference between a scientific discovery and pure design: "The genuine scientist strives for explanation-power of his theories; the engineer is in most cases already satisfied with prediction-power." As researchers in an applied discipline such as software engineering, we should embrace both goals equally.
SUMMARY
As a research community we have not always paid due attention to the robust evaluation of our work, sometimes relying on small-scale proofs of concept assessed with student subjects. While there are many valid reasons why such an approach has been predominant, it must contribute to the limited uptake of some research outcomes. In this paper we have introduced a framework that enables a design science research process to be carried out with the support of grounded theory in order to lead to relevant and rigorous research and practice outcomes arising from the development of software artifacts. We have briefly presented the entire approach in diagram form and have described the proposed conduct of two steps -problem identification and evaluation.
Figure 1 .
1The Software Artifact-centered Research reference Process (SARP) as the aggregation of design science, software and systems processes
Figure 2 .
2Development of Grounded Theory through project stages
We are currently making extensive use of the design science research process in our work and are looking to test out the proposed framework by using GT as the means of identifying software system needs, informing development of appropriate solutions, and evaluating their utility, quality and efficacy in both testing and in actual use. We also intend to compare the characteristics and coverage of our approach to other evaluation schemes centered on artifacts[7,13].
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|
[] |
[
"Non-adiabatic transport in a quantum dot turnstile",
"Non-adiabatic transport in a quantum dot turnstile"
] |
[
"Valeriu Moldoveanu \nNational Institute of Materials Physics\nP.O. Box MG-7Bucharest-MagureleRomania\n",
"Vidar Gudmundsson \nScience Institute\nUniversity of Iceland\nDunhaga 3IS-107ReykjavikIceland\n",
"Andrei Manolescu \nNational Institute of Materials Physics\nP.O. Box MG-7Bucharest-MagureleRomania\n"
] |
[
"National Institute of Materials Physics\nP.O. Box MG-7Bucharest-MagureleRomania",
"Science Institute\nUniversity of Iceland\nDunhaga 3IS-107ReykjavikIceland",
"National Institute of Materials Physics\nP.O. Box MG-7Bucharest-MagureleRomania"
] |
[] |
We present a theoretical study of the electronic transport through a many-level quantum dot driven by time-dependent signals applied at the contacts to the leads. If the barriers oscillate out of phase the system operates like a turnstile pump under a finite constant bias, as observed in the experiments of Kouwenhoven et al. [Phys. Rev. Lett. 67, 1626(1991]. The time-dependent currents and their averages over succesive pumping periods are computed from the Keldysh formalism for tight-binding models. The calculation considers a sudden application of the pumping potentials at t = 0 which leads to transient features of the time-dependent and averaged currents during the first pumping cycles which turn out to be important in the high-frequency regime. We show that in the transient regime the efficiency of the system as a pump is rather poor because it mainly absorbs charge from both leads in order to fill the levels located below the bias window. Under a finite bias and a low-frequency pumping signal the charge transferred across the system depends on the number of levels located within the bias window. The internal charge dynamics and the role of energy sidebands are investigated. The so called satellite peaks of the averaged current are observed also in the transient regime.
|
10.1103/physrevb.76.165308
|
[
"https://arxiv.org/pdf/0706.0968v1.pdf"
] | 119,718,779 |
0706.0968
|
eaba667b5a62a82d401a1ae5f003e413f3f4de06
|
Non-adiabatic transport in a quantum dot turnstile
7 Jun 2007
Valeriu Moldoveanu
National Institute of Materials Physics
P.O. Box MG-7Bucharest-MagureleRomania
Vidar Gudmundsson
Science Institute
University of Iceland
Dunhaga 3IS-107ReykjavikIceland
Andrei Manolescu
National Institute of Materials Physics
P.O. Box MG-7Bucharest-MagureleRomania
Non-adiabatic transport in a quantum dot turnstile
7 Jun 2007numbers: 7323Hk8535Ds8535Be7321La
We present a theoretical study of the electronic transport through a many-level quantum dot driven by time-dependent signals applied at the contacts to the leads. If the barriers oscillate out of phase the system operates like a turnstile pump under a finite constant bias, as observed in the experiments of Kouwenhoven et al. [Phys. Rev. Lett. 67, 1626(1991]. The time-dependent currents and their averages over succesive pumping periods are computed from the Keldysh formalism for tight-binding models. The calculation considers a sudden application of the pumping potentials at t = 0 which leads to transient features of the time-dependent and averaged currents during the first pumping cycles which turn out to be important in the high-frequency regime. We show that in the transient regime the efficiency of the system as a pump is rather poor because it mainly absorbs charge from both leads in order to fill the levels located below the bias window. Under a finite bias and a low-frequency pumping signal the charge transferred across the system depends on the number of levels located within the bias window. The internal charge dynamics and the role of energy sidebands are investigated. The so called satellite peaks of the averaged current are observed also in the transient regime.
We present a theoretical study of the electronic transport through a many-level quantum dot driven by time-dependent signals applied at the contacts to the leads. If the barriers oscillate out of phase the system operates like a turnstile pump under a finite constant bias, as observed in the experiments of Kouwenhoven et al. [Phys. Rev. Lett. 67, 1626(1991]. The time-dependent currents and their averages over succesive pumping periods are computed from the Keldysh formalism for tight-binding models. The calculation considers a sudden application of the pumping potentials at t = 0 which leads to transient features of the time-dependent and averaged currents during the first pumping cycles which turn out to be important in the high-frequency regime. We show that in the transient regime the efficiency of the system as a pump is rather poor because it mainly absorbs charge from both leads in order to fill the levels located below the bias window. Under a finite bias and a low-frequency pumping signal the charge transferred across the system depends on the number of levels located within the bias window. The internal charge dynamics and the role of energy sidebands are investigated. The so called satellite peaks of the averaged current are observed also in the transient regime.
I. INTRODUCTION
The ability to control the transport properties of semiconductor quantum dots by time-dependent perturbations (e.g. microwave signals or optical pulses) allows the observation of photon-assisted tunneling, charge pumping 1,2,3 and coherent Rabi oscillations. 4 Also, pump-and-probe techniques were used to estimate relaxation rates and to control spins in quantum dots. 5 A common point of these experiments is that the timedependent driving potential is applied on the system itself, i.e. on a central metalic gate defining the quantum dot.
Some time ago Kouwenhoven et al. 6 proposed a different setup, in which a quantum dot is coupled to source and drain reservoirs by oscillating tunneling barriers. Technically this is achieved by applying radio-frequency signals to the metalic gates that control the opening of the quantum dot to its surroundings. The two barriers at the contacts are varied in such a way that the system undergoes a cyclic transformation and, under a constant bias applied on the leads, an integer number of electrons is transmitted during one cycle. Therefore the system operates as a turnstile pump. An important feature of the turnstile configuration is that the pumped current has a definite direction due to the finite bias. We remind here that originally the concept of parametric charge pumping was introduced in the context of a net current generation in an unbiased system. 7 As noted in the literature, a symmetry breaking is necessary in order to get a nonvanishing current without a bias. In spite of the fact that the turnstile operation was experimentally observed some time ago, it attracted little attention in the theoretical literature (see Refs. 10,13,14,17 below). The purpose of this work is to explore the transport properties of turnstile quantum dots submitted to time-dependent signals of arbitrary amplitude and frequency.
At theoretical level the quantum pumping was discussed basically within two frameworks: the adiabatic or Floquet scattering theory 7,8,9,11 and the non-equilibrium Green-Keldysh formalism. 14,15,16,17 The scattering approach was primarily designed to describe the adiabatic pumping, in which the driving potential varies very slowly. The timescale on which the applied signal varies significantly exceeds the time needed for the electron to pass through the system. The key result of the adiabatic scattering is a current formula in terms of an instantaneous (frozen) S matrix. This matrix is computed perturbatively and only the linear term in frequency is usually retained. A rigorous mathematical treatment of adiabatic quantum pumping 12 recovered the BPT formula given in. 7 Within this framework the relation between resonant transmission and quantized pumped charge in an unbiased turnstile was analyzed. 10 An extended scattering formalism for studying both adiabatic and nonadiabatic quantum pumping was developed in Ref. 9 and uses the Floquet theory and an S matrix depending on two-energies. Recently Mahmoodian et al. 13 have computed the stationary current for a quantum wire submitted to alternating δ-like voltages. It was shown there that the current displays multiphoton peaks as a function of the Fermi momentum of the leads.
Using the Green-Keldysh formalism Q-f Sun and T. S. Lin 14 have computed the current through a single level quantum dot when rectangular or harmonic potentials are applied at the contact to the leads. As it is well known, this model is exactly solvable within the wideband limit (WBL) approximation since the Dyson equation for the retarded Green function is greatly simplified due to the leads' self-energy which within WBL is simply a delta-function. 15 Later on Wang et al. used the Keldysh approach to investigate the non-adiabatic charge pump-ing in the presence of photon-assisted tunneling. The explicit calculation was done within WBL and for an unbiased double barrier pump driven by a local sinusoidal signal. It was shown that at large frequencies a nonvanishing current is generated even with a single-parameter perturbation. Moreover, a sign change of the pumped current was reported when the Fermi level of the leads crosses the eigenvalues of the system. This feature was predicted also by Büttiker and Moskalets. 9 Further progress was achieved by L. Arrachea for tightbinding models and periodic potentials. 17 The method developed in this paper allows the calculation of the d.c. component of the pumped current once the partial Fourier transform (i.e. the Fourier transform with respect to one time only) of the Green functions is known. It was shown that the d.c. component can be related to a transmission function T (ω) which is interpreted as the difference between the probabilities of tunneling out from and into the system. The numerical simulations are performed for unbiased one-dimensional pumps and the pumping potential is described by a diagonal timedependent term added to the energy at the contact sites. For two harmonic potentials the retarded Green function is computed perturbatively for weak pumping amplitude and pumping frequency. The connection between the Floquet scattering and the Green-Keldysh function formalism for time-dependent transport was discussed in Ref. 18.
Besides the scattering theory and NEGF approach the pumping problem can be addresed via time integration of the Schrödinger equation by the Crank-Nickolson approximation, as proposed by Stefanucci et al. in a series of papers. 19,20 Their setup starts from the ground state of the unperturbed but coupled system and has therefore the advantage of introducing naturally the bias as a perturbation. We remind that the Keldysh formalism requires a partitioning of the system into 'central region' and 'leads', the perturbation being the coupling between them which is established usually adiabatically in the remote past. 22 The Keldysh approach is however appropriate for studying the transport in the turnstile configuration which requires to connect and disconnect periodically the pump from the leads.
Recently the equation of motion method was applied to compute various currents in 1D pumps coupled to finite wires with constant chemical potentials. 21 Such an assumption is questionable for finite systems, especially when one is interested in the long time behavior.
In this work we are primarily interested in the transient effects on the transport properties of a many-level quantum dot turnstile which is submitted to pumping potentials at t = 0. We believe these effects could seriously affect the transport properties of nanostructures driven by fast oscillating signals. Also, we mention that most of the previous theoretical approaches present calculations of the stationary averaged current which implies either to look at the long-time limit behavior 19 , either to consider small pumping frequencies. 17 Since in the stationary regime the transient effects are presumably washed out, one does not need to specify how and when the pumping signal is turned on. Note also that for a proper application of the Keldysh formalism it is crucial to have a well-defined equilibrium state of the decoupled system.
While in the adiabatic regime the main advantage is to express transport quantities in terms of the frozen scattering matrix up to errors of O(ω 2 ) or even O(ω 3 ) (see the higher order corrections in Ref. 8), the NEGF formalism covers the entire frequency range, allowing therefore an equal footing treatment of adiabatic and nonadiabatic pumping. The Green functions needed in the current formula are computed using a recently developed method 23 which solves the integral Dyson equation exactly by transforming it into an algebraic equation. Through this procedure the Green functions are computed taking into account all back-and-forth scattering processes.
The content of the paper is divided as follows. Section II describes the model and gives the relevant equations, as well as the considered pumping potentials; more details about the formalism should be traced back from Ref. 23. Section III is the main part of the paper and presents the numerical results and their discussion. Conclusions are summarised in Section IV.
II. THE MODEL
Within the tight-binding model which is adopted throughout this work the Hamiltonian of the system contains three terms: the semiinfinite leads (H L ), the quantum dot turnstile (H S ) and the time-dependent pumping signals H T (t):
H(t) = H S + H L + H T (t).
(1) H S has a usual tight-binding form
H S = N m=1 (ǫ m + V g )d † m d m + m,n t mn d † m d n .(2)
Here t mn are hopping terms, m, n denotes nearestneighbor summation over the system sites. ǫ m is the onsite energy and the diagonal term V g simulates a plunger gate potential applied on the system. N is the number of sites in the dot. Following the experimental setup from Ref. 6 we describe the oscillating tunneling barriers between the dot and the leads by time-dependent hopping terms (l, r denote the left and the right lead):
H T (t) = α=l,r V α (t)(c † iα d mα + h.c).(3)
Here c iα and c † iα denote the annihilation/creation operators on the i-th site of the lead α which is connected via the nearest neighbor hopping V α to the site m α of the dot. Similarly, d mα and d † mα correspond to the site m α of the dot which is coupled to the lead α. The signals applied at the contacts between the dot and the leads have a trapezoidal form and are defined with the help of a function with period 2π introduced as follows (∆ is a positive number):
f (ωt) = 1 if ωt ∈ [0, π − ∆], 1 − 1 ∆ (ωt − (π − ∆)) if ωt ∈ [π − ∆, π], 0 if ωt ∈ [π, 2π − ∆], 1 ∆ (ωt − (π − ∆)) if ωt ∈ [2π − ∆, 2π].(4)
Then the pumping potentials of period T are given by the relations:
V l (ωt) = v l f (ωt) (5) V r (ωt) = v r (1 − f (ωt)),(6)
where ω = 2π/T is the frequency and v l,r are the amplitudes of the pumping signals. It is useful to introduce the number of pumping cycles k considered in the numerical simulation. We show for clarity in Fig. 1 a train of two such pulses (i.e. k = 2) that we use to simulate the turnstile configuration. The quantum dot is coupled suddenly to the left lead at t 0 = 0 while the right contact is off. In the range [kT /2 − ∆, kT /2] the sample is simultaneously isolated from the left lead and connected to the right lead. This switching is done linearly at a slope 1/∆ (note that a larger ∆ implies a slower onset of the couplings). In the second halfperiod of each pumping cycle the right contact is open, allowing thus the charge pumping. The cycle is completed by lowering the left tunneling barrier (i.e. increasing V l (t)) and turning off the coupling to the right lead.
As it is widely known, the standard application of the Keldysh formalism leads to the following formula for the current entering the system from the left lead (here we take for simplicity one dimensional leads; a many-channel formula and more details are to be found in Ref. 23):
J l (t) = − 2e h Im( 2tL −2tL dE t 0 dse −iE(s−t) Γ l (E; t, s)(G R ll (t, s)f l (E) + G < ll (t, s))).(7)
In the above formula the retarded and lesser Green functions are given as usual in terms of Heisenberg operators
G R ll (t, t ′ ) = −iθ(t − t ′ ) {c i l (t ′ ), d † m l (t)} and G < ll (t, t ′ ) = i c † i l (t ′ )d m l (t) . f l (E)
is the Fermi function of the left lead, t L is the hopping energy on leads and Γ l is the linewidth depending on energy and time:
Γ l (E; t, s) = ρ(E)V l (t)V l (s),(8)
containing the pumping potentials at different times and the density of states at the endpoint of the semiinfinite one-dimensional lead ρ(E):
ρ(E) = θ(2t L − |E|) 4t 2 L − E 2 2t 2 L .(9)
The retarded and the lesser Green functions are computed from the Dyson and Keldysh equations:
G R (t, t ′ ) = G R 0 (t, t ′ ) + t 0 dt 1 G R (t, t 1 ) t1 0 dt 2 Σ R (t 1 , t 2 )G R 0 (t 2 , t ′ ) (10) G < (t, t ′ ) = t 0 dt 1 G R (t, t 1 ) t ′ 0 dt 2 Σ < (t 1 , t 2 )G A (t 2 , t ′ ),(11)
where G R,A 0 (t, t ′ ) are the retarded and advanced Green functions of the isolated dot and Σ R,< are the retarded and lesser self-energies. It worths mentioning that both self-energies contain the known Green functions of the semiinfinite leads but they are also quadratic functions of the pumping potentials whose time variable is different (see Eq. 8) . Therefore, their time-dependence is much more complicated than in other approaches were the pumping signals are applied to the system or to the leads and are described by diagonal terms in the Hamiltonian. In particular, the algorithm taken in Ref. 20 would we difficult to use. The time-dependent occupation number can be computed from the lesser Green function of the dot:
N (t) = Im N m=1 G < mm (t, t) = N m=1 N m (t).(12)
N m (t) are on-site occupation numbers and will be used in the next section to gain information about the internal charge dynamics during the pumping cycle. A similar formula can be written down for the current J r (t) flowing from the system towards the right lead.
As we have said, taking explicitely into account the initial instant when the pumping signals are applied leads to transient effects. 23 One consequence is that the periodaveraged currents depend on the period index k. We introduce therefore a k-indexed period-average for the currents (the k-th period covers the interval [t k−1 , t k ] and t 0 = 0):
J α,k = 1 T t k t k−1 dtJ α (t), α = l, r.(13)
Although the approach taken in this work does not include the electron-electron interaction it captures the basic known features of turnstile pumps: the quantized pumped charge in low frequency regime and the satellite peaks due to photon-assisted tunneling in high-frequency regime. Moreover, most of the results are presented at rather strong coupling to the leads when the dot is fairly open and the Coulomb blockade effects are not important. In a recent work Splettstoesser et al. 24 proposed a method for dealing with Coulomb interactions in adiabatic quantum pumps. This approach is based on the quantum Master equation and uses a perturbative expansion in the tunnel coupling. In the case of the unbiased quantum dot turnstile these authors found that one has to consider the second order term in the tunnel coupling to get a non vanishing pumping.
III. NUMERICAL SIMULATIONS
In this section we present the main numerical results and discuss the transport properties of the turnstile pump in different regimes. The bias, the frequency, the energy, the hopping constants on the leads, the coupling strengths and the gate potential will be expressed in terms of the hopping energy of the central region t D which is chosen as energy unit. The current is therefore given in units of et D /h and the time expressed in units ofh/t D . We take e =h = 1. The bias window (BW) is defined as the difference between the chemical potentials of the leads W = µ l − µ r . To make a connection to physical units one could take for example the energy unit as t D = 0.1 meV. Then the frequency unit would be ω ∼ 25 GHz and the time unit t ∼ 7 ps. The termal energy kT = 0.0001 in all numerical simulations.
We consider first a two-site one-dimensional turnstile submitted to a finite bias and modulated by the trapezoidal signal introduced in Section II. In order to simulate the conditions of the experiment performed by Kouwenhoven et al. 6 we set the bias window to W = 3.0 with respect to the chemical potential of the right lead µ r = 0 such that the highest level of the isolated dot E 1 = 1 is located in it. Fig. 2a shows the current J l (t) from the left lead towards the turnstile and the current J r (t) pumped into the right lead during two pumping cycles. The frequency is ω = 0.6 and the maximum height of the tun-neling barriers is v l = v r = 0.75. As expected, a nonvanishing current is generated in the right lead during the second halfperiod of each pumping cycle. The pumping mechanism is proved by the behavior of the occupation number N (t) which is given in Fig. 2b. At the beginning of the first cycle the system collects charge from the left lead and since the coupling to the right lead is zero there are almost two electrons in the system after one halfperiod. We shall call this halfperiod the charging halfperiod. The charge dynamics within the system is also given in Fig. 2b through the occupation numbers of the two sites N 1 and N 2 . For the clarity of discussion we have also included in the figure the two potentials applied on the leads. The first site is rapidly populated up to 0.5 and then stays at this occupation while the second site starts to be filled. The current J l decreases in this short time range.
Then we see that N 2 increases faster than N 1 and both are reaching a constant value. Due to the small frequency considered here the halfperiod of the pumping cycle exceeds the time needed for the system to be completely filled with electrons and therefore, as long as the coupling to the right lead is still turned off, the total occupation number is steplike. The current J l becomes very small in this range because as both sites are filled it is more difficult to inject charge (notice the lower slope at which N 1 increases). The pumping starts effectively in the second half of the cycle and leads to the transmission of one electron in the right lead, as observed in Ref. 6. During the pumping halfperiod the occupation of the right contact drops quickly to 0.75 but then decreases at a lower rate as the first site is depopulating. Note that in the pumping sequence N 1 drops more rapidly than N 2 . The occupation number at the end of the pumping cycle does not decrease below 1 because the lowest level of the dot E 2 = −1 is well below the bias window and cannot contribute to transport.
From the above discussion it is clear that the efficiency of the pump depends on two facts: first, the charging halfperiod should allow the complete filling of the turnstile and second, the levels within the bias window must be entirely depopulated during the pumping halfperiod. In the intermediate regime ω ∼ V l,r (not shown) the system still transfers one electron, the difference being that the occupation number decreases faster and more abruptly than in Fig. 2. The transient effects on the timedependent currents and pumped charge that appear as the pumping frequency increases are captured in Figs. 3ad. At ω = 1.25 the shape of the currents is similar to the one in Fig. 2a. However, the current pumped during the first cycle is smaller that the ones corresponding to the next cycles and the occupation number plotted in Fig. 3d shows that less charge is pumped. Also, there are no more steps in N (t). The behavior of the occupation number helps us to identify the new features of the charge dynamics in the nonadiabatic regime.
On the other hand, the coupling to the right lead closes before the level within the bias window depopulates. As a consequence, at the end of the pumping cycle there is a residual charge (∼ 0.1) which is stuck within the bias window. By further increasing the frequency up to ω = 1.75 another effect appears in Fig. 3b. Although there is a nonvanishing pumped current during the first cycle J r takes negative values at the end of the cycle, which means that in this interval the system absorbs charge rather than pumps it. Looking at the corresponding occupation number in Fig. 3d one infers why the system does not act like a pump over the entire pumping sequence. N (t) goes slightly above 1 then drops to 0.9; during this interval the system pushes electrons to the right lead and therefore J r (t) > 0. Then the occupation number increases, leading to negative values of J r (t) (the pumping period in Fig. 3b is 3.7). The physical picture behind this is the following: i) In the first halfperiod both levels are populated, though not completely; ii) During the first part of the pumping sequence the level within the bias window depopulates, generating therefore a positive current J r ; iii) When this happens the total occupation number goes below 1; also, since the chemical potential of the right lead is higher than the lower level of the dot, the system absorbs charge form the right lead and the occupation number increases again.
Finally, Fig. 3c shows the currents in a highly nonadiabatic regime ω = 3.5. The steplike features within each pumping cycle are washed out and on the first cycle the system does not pump any charge (the occupation number is simply increasing and stays below 1). Moreover, even when a stable pumping regime is achieved (for t > 5) it is not effective at all since very little charge is pumped.
The transient regime should be noticed as well in the period-averaged currents. In order to check this we give in Fig. 4a these currents for the three frequencies considered in Fig. 3. The following things are observed: i) In all three cases the d.c. components of the currents become eventually equal and their value does not depend anymore on the period index. Moreover, the averaged current is conserved. The passage to this 'stationary' regime is faster at low frequencies; ii) In the transient regime J l,k exceeds J r,k since there is a net charge ac-cumulation in the dot along each transient cycle (see the occupation numbers in Fig. 3d); iii) At large frequencies J r,k takes negative values in the transient regime (see the first cycle at ω = 3.5) because the system absorbs charge from the right lead. Fig. 4b shows that the average occupation number depends strongly on the period index in the transient regime and settles down to 1.5 in the long-time limit. Notice that this means that the level which contributes to the transport is half occupied.
The bias applied on the leads is an important parameter in turnstile operation because it controls the number of levels giving the main contribution to the current. As reported experimentally 6 the number of electrons pumped during one pumping cycle is given by these levels. Fig. 5a shows the occupation numbers of a three sites turnstile pump submitted to a different bias. We take a small frequency (ω = 0.3) and therefore the pumping cycle is long enough to allow the filling of the lowest levels (the dot has three eigenstates E ± = ±1.4 and E 0 = 0). For µ l = 3 and µ r = 1 only the highest level is located within the bias window and the pump transfers one electron. The occupation number at the end of the cycle is N = 2. We emphasize that the frequency is much smaller than the gap between E 0 and E − and therefore the lowest level cannot give an important contribution to the current via excited sidebands.
By decreasing the chemical potential of the right lead to µ r = 0 the middle level alligns to µ r and the turnstile pumps 1.5 electrons at W = 3. Note that due to the rather large coupling to the leads there is no charge quantization condition to be fullfilled. The dotted line shows the occupation number for the same bias W with µ l = 1.5 and µ l = −1.5. In this case we have a symmetric bias window (as marked in the figure) containing the middle level in the center. The other two levels are close to the chemical potentials of the leads. The transferred charge is Q p ∼ 2, suggesting that each level alligned to one chemical potential pumps only half electronic charge. This happens because the level close to µ r (µ l ) are difficult to depopulate (populate). Also, by tuning the width and the position of the bias window it is possible to transfer the same charge in different ways (i.e. by involving different levels of the pump). For example, seting µ l = 1.5 and µ r = 0 one can still transfer one electron at a bias window W = 1.5. Of course, the efficiency of the pump increases when the number of levels participating in transport increases. Fig. 5b shows the pumped currents associated to the occupation numbers in Fig. 5a. Although both cases W = 1.5 and W = 2 correspond to a pumped charge Q p = 1, the current is higher at W = 1.5. Note also that the currents have a peak structure when two levels participate in transport.
The discussion around Figs. 3 and 4 suggests that at a given frequency the transient regime would cover more pumping cycles if the number of levels located below the bias window increases. This is confirmed in Fig. 6 which shows the occupation number of a 3-site turnstile. We take µ l = 3 and µ r = 1 and in this case the BW contains only the highest level E + = 1.4. N (t) shows clearly that the other two levels are filled only after k = 5 pumping cycles. Notably, even in the transient regime the system pumps a small amount of charge (∼ 0.25) because of the highest level from the BW. This level cannot be completely filled in the transient cycles because most of the charge populates the lowest two levels. The behavior of the individual occupation numbers N i , i = 1, 2, 3 confirms the intuitive picture of the internal charge dynamics. It is easily seen that for any pumping cycle except the first, the occupation number of the 3rd site decreases even in the first halfperiod, namely during the charging process. Since in this time range the contact to the right lead is turned off the charge from the right contact can only flow back to the middle site. Indeed, N 2 increases as N 3 decreases.
As the dot charges from the left lead the inverse flow is noticed. In the pumping halfperiod the middle site occupation firstly increases because the charge accumulated in the first site passes to the right contact which is now open, and finally decreases. All these internal bouncing trajectories are taken accurately into account in our calculation because the method we used to solve for the Dyson equation gives the entire matrix of the Green function (not only the diagonal matrix elements entering the current formula).
We turn now to another important feature of timedependent transport, namely the contribution due to inelastic scattering processes in which the incident electrons gain or loose energy quanta from the driving fields. Within the Floquet scattering approach it was shown 9,25 that the outgoing electrons can have any energy E n = E + nhω in the so called sideband ladder, E being the energy of the incident electrons. In the present model the pumping potential is not periodic because of the condition V l (t) = V r (t) = 0 for t < 0, therefore the Floquet theorem cannot be rigorously applied. At best, one can use results of the Floquet theory in the long time limit where all relevant quantities will oscillate with the frequency of the driving signals (see Ref. 26 for a discussion). Nevertheless, inelastic tunneling processes are physically possible especially in the transient regime and should be noticed in the averaged currents. In particular one expects to see additional contributions to the current at frequencies matching the gaps µ l,r ± E i between the levels of the isolated system and the chemical potentials of the leads. In order to check these features we tune the bias window below the three levels of the 3-site pump by setting µ l = −2.5 and µ r = −3.5 such that E − − µ l ∼ 1. We take a small pumping amplitude v l = v r = 0.35 and look at the currents for several frequencies, as shown in Figs. 7a-d. For small frequencies (ω = 0.3 and ω = 0.5 -not shown) there is no pumped current and J l takes both positive and negative values, while N (t) increases and decreases accordingly during the charging halperiod. This suggests that the system repels incident electrons back to the left lead. This happens because on one hand there is no level below or within the bias window and, on the other hand, the frequency is too small to allow the population of the lowest level. The situation changes for ω = 1 (see Fig. 7a). A pumped current appears in all three cycles presented. We related this to the occupation number plotted in Fig. 7d which shows that there is a charge pumped into the right lead. When comparing Fig. 7a and Fig. 7d (the dotted line) one notes that in the second half of the charging cycle J l < 0 and N (t) decreases. This signals the energy relaxation process from the lowest level of the system to the left lead only, because the contact to te right lead is not yet open. Only the second (lower slope) decrease of N (t) is associated to pumping trough the right lead. At ω = 1.6 ( Fig. 7b) the pumped charge and current increase. J l is mostly positive because the fast oscillating signals prevent relaxation to the left lead. The pumped current still shows a peak structure. In Fig. 7c we show a highly non-adiabatic regime at ω = 2.75. In this case the second level E 0 = 0 will contribute as well. In order to avoid tunneling from the right lead to the level E − we have considered for this curve µ r = −5. The peak structure of the pumped current dissapears and a periodic regime establishes slowly after 9 cycles. As we have said in the introduction, the present approach takes into account all tunneling processes between the leads and the system and within the system. This nonperturbative treatment might be crucial for describing the relaxation processes mentioned above, especially in the transient regime where the resonances are not well defined.
Finally we investigate briefly the satellite peaks appearing in the averaged current when a gate potential is used to move the levels of the dot. These peaks were observed experimentally 3 and are associated with absorbtion and emission proces of energy quanta from the pumping fields. Theoretically they were obtained first in the Master equation framework. 27 In the Keldysh formal-ism a calculation of the stationary current in the WBL approximation was also presentent in Ref. 28. Here we want to check wether the satellite peaks appear also in the first first pumping cycles, namely in the transient regime. To this end we consider the dependence of the pumped current for a two site turnstile on a gate potential V g (see the Hamiltonian given in the introduction) which shifts the two levels of the isolated system. Since we are interested in observing satellite peaks associated to one level only we take the parameters such that v l,r < ω < δ2, where δ = E 1 −E 2 = 1 is the level spacing. Fig. 8a shows the averaged current associated to the 2nd pumping cycle One notices as once two satellite peaks located on each side of main resonant peaks (at V g = ±1 the levels of the isolated dot are shifted to zero, threfore they are located in the middle of the bias window). Also, the distance between the sattelite peaks and the associated main peak equals roughly the frequency ω = 0.65 and confirms the absorbtion/emission picture. There are however several aspects due to the transient regime in which these peaks appear. i) As shown in Fig. 8b the averaged current on the first pumping cycle does not display satellite peaks and is mostly negative. ii) Also, in Fig. 8a the first sattelite peak has a negative value. This can be understood by looking at Fig 8c which gives the 3D plot of the pumped current as a function of time and gate potential. Clearly, around V g = −1.65 there is a maximum positive current at the beginning of the pumping cycle. In this range the system effectively pumps charge to the right lead via photon assited tunneling involving the highest level of the dot. This process is rather weak because in the transient regime the occupation of the levels below the bias window is not complete. This is why after a short pumping regime the system absorbs charge from the lead and threfore J r (t) < 0. As the first level enters the bias window a main peak appears around V g = 0. Similar description can be made for the contribution of the second level. iii) For V g > 0.3 the average current is positive; the photon-assisted tunneling contribution to the pumping process is amplified and the current vanishes as the lowest level of the dot is pushed upwards.
Our calculations show that photon assisted tunneling takes place also in the transient regime.
IV. CONCLUSIONS
We have studied the turnstile pump regime of a few level noninteracting quantum dot within the non-equilibrium Green-Keldysh formalism for timedependent transport. The out-of-phase oscillating barriers coupling the system to the leads are described by time-dependent hopping terms. In the numerical calculations we have considered a train of trapezoidal pulses that mimick the configuration used in the experimental work of Kouwenhoven et al. 6 . Our approach includes explicitely the starting time of the pumping cycles and therefore captures the transient behavior of the time- dependent and period-averaged current. To our best knowledge, such a calculation is presented here for the first time. We identify basically two stages of transport.
The system experiences first a transient regime with poor pumping efficiency. This is due to the fact that the dot rather absorbs charge from the leads in order to populate the levels located below the bias window. The number of pumping cycles in which the transient features should be observed in future experiments with many level turnstiles increases if the bias window contains only the highest levels. In the second stage the occupation number of the dot and the currents oscillate with the pumping period. We show that at low frequency and strong coupling to the leads an integer or half-integer number of electrons are pumped, depending on the number and the location of the levels within the bias window. In the highfrequency case the pumped charge is rather small even if additional contributions appear due to scattering processes involving energy sidebands. We show that satellite peaks due to the photon-assisted tunneling appear also in the transient regime. The present analysis could be extended to two-dimensional systems in order to discuss magnetic field effects. Also, different types of potentials (e.g. harmonic or damped pulses) can be considered.
PACS numbers: 73.23.Hk, 85.35.Ds, 85.35.Be, 73.21.La
online) The pumping potentials applied on the left (V l -solid line) and right (Vr -dashed line) contacts. We show two pumping cycles k = 2. The parameter ∆ = 0.6, v l = vr = 0.75 and the frequency ω = 1.
FIG. 2 :
2(Color online) (a) The pumped current in the right lead (Jr) and the current entering the system from the left lead (J l ) for a 2-site turnstile. (b) The occupation number of the dot N (t) and the on-site occupations Ni(t), i = 1, 2. The pumping potentials are also given. During the second halfperiod of each pumping cycle the turnstile expells one electron to the right lead. The bias is fixed to W = 3.0, ω = 0.6, v l = vr = 0.75 and kT = 0.0001.
FIG. 3 :FIG. 4 :
34(Color online) The shape of the currents Jr and J l depends on the pumping frequency: (a) ω = 1.25, (b) ω = 1.75, (c) ω = 3.5. At large frequency Jr takes negative values in the first pumping cycles. (d) The occupation number N (t) corresponding to the three frequencies taken in (a), (b), (c). W = 3.0, v l = vr = 0.75, kT = 0.0001. (Color online) (a) The average currents J r,k and J l,k as a function of the period index k. There are three pairs of curves, corresponding to the frequencies ω = 0.6 (k = 4), ω = 1.75 (k = 5) and ω = 3.5 (k = 11). For each frequency the two averaged currents are drawn with the same type of line (color); J r (J l ) has a positive (negative) slope. (b) The average occupation number (see the discussion in the text). W = 3.0, v l = vr = 0.75, kT = 0.0001.
FIG. 5 :
5(Color online) (a) The occupation number at different values of the bias window and different allignment of the turnstile levels with respect to the bias window. (b). The pumped current Jr(t) corresponding to the occupation numbers in (a). For all curves the frequency is set to ω = 0.3. We show only the first pumping cycle.
FIG. 6 :
6(Color online) (a) The total occupation number N (t) and the on-site occupation numbers Ni (i, 1, 2, 3) for a 3-site turnstile submitted to a bias W = 2 and to a pumping signal of frequency ω = 1.57.
FIG. 7 :
7(Color online) For a 3 site turnstile with all the levels above the bias window one can notice sideband contributions to the two time-dependent currents at frequency (a) ω = 1.0, (b) ω = 1.6 and (c) ω = 2.75. (d) The occupation number at different values of the frequency.
FIG. 8 :
8(Color online) (a) The averaged current Jr,2 as a function of the gate potential for a two sites turnstile shows satellite peaks associated to photon-assisted tunneling during the second pumping cycle. (b) The averaged current corresponding to the first pumping cycle Jr,1 does not contain satellite peaks. (c) The current Jr as a function of time (only the second cycle is shown) and gate potential. The discussion is made in the text. Other parameters ω = 0.65, v l = vr = 0.35, µ l = 0.2, µr = −0.2, kT = 0.0001.
AcknowledgmentsThis work was supported in part by the Icelandic Science and Technoloy Reasearch Programme for Postgenomic Biomedicine, Nanoscience and Nanotechnology. V.M was also supported by CEEX Grant D11-45/2005. We acknowledge useful discussions with C. S. Tang.
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[] |
[
"NONLINEAR SCALAR MULTIPOINT BOUNDARY VALUE PROBLEMS AT RESONANCE",
"NONLINEAR SCALAR MULTIPOINT BOUNDARY VALUE PROBLEMS AT RESONANCE"
] |
[
"Daniel Maroncelli "
] |
[] |
[] |
In this work we provide conditions for the existence of solutions to nonlinear boundary value problems of the formfor i = 1, · · · , n. The existence of solutions will be proved under a mild growth condition on the nonlinearity, g, which must hold only on a bounded subset of {0, · · · , N } × R.
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[
"https://arxiv.org/pdf/1811.06466v1.pdf"
] | 119,723,144 |
1811.06466
|
55e2fc19d49df8f9fc13fb0bd176b95faa4381b0
|
NONLINEAR SCALAR MULTIPOINT BOUNDARY VALUE PROBLEMS AT RESONANCE
15 Nov 2018
Daniel Maroncelli
NONLINEAR SCALAR MULTIPOINT BOUNDARY VALUE PROBLEMS AT RESONANCE
15 Nov 2018arXiv:1811.06466v1 [math.DS] ( * * The final version of this manuscript has been accepted for publication in Journal of Difference Equations and Applications)Multipoint boundary value problemsResonanceLyapunov-Schmidt procedureBrouwer's degree
In this work we provide conditions for the existence of solutions to nonlinear boundary value problems of the formfor i = 1, · · · , n. The existence of solutions will be proved under a mild growth condition on the nonlinearity, g, which must hold only on a bounded subset of {0, · · · , N } × R.
Introduction
In this paper we provide criteria for the solvability of nonlinear scalar multipoint boundary value problems of the form (1) y(t + n) + a n−1 (t)y(t + n − 1) + · · · a 0 (t)y(t) = g(t, y(t + m − 1)) subject to (2) n j=1 b ij (0)y(j − 1) + n j=1 b ij (1)y(j) + · · · + n j=1 b ij (N )y(j + N − 1) = 0 for i = 1, · · · , n.
Throughout our discussion we will assume that g : R × R → R is continuous, m is fixed with 1 ≤ m ≤ n, N is an integer greater than 2, the coefficients b ij (·) and a 0 (·), · · · , a n−1 (·) are real-valued with a 0 (t) = 0 for all t, and the boundary conditions are independent.
We focus on the solvability of nonlinear boundary value problems at resonance; that is, problems where the solution space of the associated linear homogeneous problem, (6), subject to boundary conditions, (4), is nontrivial. We will assume throughout that the solution space of this linear homogeneous problem is 1-dimensional.
In a vast majority of the literature on resonant boundary value problems, see [1,6,16,17,20,22,24,25,29,30,31,33,34], it is assumed that the nonlinearities of the difference (differential) equation are bounded. Recently, there has been a large push to obtain existence results in cases where the nonlinearity of the difference (differential) equation is unbounded. There have been several results in this regard, most of which require g to satisfy a growth condition on intervals of the form (−∞, z 0 ] and [z 0 , ∞). For interested readers, we mention [3,4,19,21,23,26,27]. Our focus will be on the case where the nonlinearity is allowed to be unbounded, but must satisfy a mild growth condition on a bounded subset of {0, · · · , N } × R. Those readers interested in results obtained for the case of nonresonant difference equations may consult [10,11,13,12,14,37,38].
Our main result is Theorem 3.1, which establishes the existence of solutions to (1)- (2) under suitable interaction of the solution space of the linear homogeneous problem and the nolinearity g. We would like to remark that the result we obtain in Theorem 3.1 constitutes a significant generalization of the work found in [18,31]. In [18], the author discusses the existence of solutions to (1)- (2) in the special case of nonlinear Sturm-Liouville problems with standard two-point linear boundary conditions. In [31], the authors discuss the existence of solutions to (1)-(2) under the assumption of bounded nonlinearities that must also satisfy a limit condition at ±∞. This limit assumption is quite standard, often referred to as a Landesman-Lazer type condition. In section 4, we give a detailed comparison between Theorem 3.1 and the work from [18,31].
Our main tool in the analysis of Theorem 3.1 will be the application of an alternative method in combination with Brouwer's degree theory. The application of these ideas to discrete and continuous nonlinear boundary value problems is extensive. For those readers interested in fixed point methods, coincidence degree theory, the Lyapunov-Schmidt procedure or more general alternative methods, and their application to difference and differential equations, we suggest [2,6,7,8,9,17,24,27,28,29,30,32,33,34,35,36] and the references therein.
Preliminaries
The nonlinear boundary value problem (1)-(2) will be viewed as an operator problem. To help facilitate in the construction of this problem, we define
A(t) = 0 1 0 · · · 0 0 0 1 · · · 0 . . . . . . . . . . . . 0 0 0 · · · 1 −a 0 (t) −a 1 (t) −a 2 (t) · · · −a n−1 (t) , f : R × R n → R n by f t x 1 x 2 .
. .
x n−1 x n = 0 0 . . . 0 g(t, x m ) ,
and n × n matrices B k , k = 0, · · · , N , by
(B k ) ij = b ij (k).
The nonlinear boundary value problem (1)-(2) is now equivalent to the nonlinear system
(3) x(t + 1) = A(t)x(t) + f (t, x(t))
subject to boundary conditions
(4) N i=0 B i x(i) = 0.
The underlying function spaces for our operator problem are as follows:
X = φ : {0, 1, 2, · · · , N } → R n | N i=0 B i φ(i) = 0 , and Z = {φ : {0, 1, 2, · · · , N − 1} → R n } .
The topologies used on X and Z are that of the supremum norm. We use · to denote both norms and we will use | · | to denote the standard Euclidean norm.
We define operators as follows:
L : X → Z by (Lx)(t) = x(t + 1) − A(t)x(t), and F : X → Z by F (x)(t) = f (t, x(t)).
Solving the nonlinear boundary value problem (1)-(2) is now equivalent to solving
(5) Lx = F (x).
Remark 2.1. It will be important to know that the very natural assumption regarding the independence of the boundary conditions is equivalent to the augmented matrix [B 0 , · · · , B N ] having full row rank, and thus is also equivalent to
Ker(∩ N k=0 B T k ) = {0}
. See Definition 2.6 and [32].
Crucial to the use of any alternative method is the construction of projections onto the kernel and image of L; to aid in the construction of these projections, we obtain a complete description of these spaces. The following characterization of kernel and image of L can be found in [20].
Let
Φ(t) = I if t = 0 A(t − 1)A(t − 2) · · · A(0) if t = 1, 2, · · · .
We then have that Φ is the principal fundamental matrix solution to linear homogeneous problem (6) x(t + 1) = A(t)x(t).
For those readers interested in the general theory of difference equations, we suggest [5,15].
Proposition 2.
2. An element h ∈ Z is contained in the Im(L) if and only if
B 1 Φ(1)Φ −1 (1)h(0)+· · ·+B N Φ(N ) N −1 i=0 Φ −1 (i+1)h(i) ∈ Ker N i=0 B i Φ(i) T ⊥ .
The proof of Proposition 2.2 is trivial and can be found in [20]. It follows easily from the variation of parameters formula, (7) x
(t) = Φ(t)x(0) + Φ(t) t−1 i=0 Φ −1 (i + 1)h(i),
and an application of the boundary conditions (4). Since we are assuming that the solution space of the linear homogeneous problem, (6), subject to boundary conditions, (4), is 1-dimensional, it follows from Proposition 2.3 that we may pick a vector u ∈ R n which forms a basis for Ker
N i=0 B i Φ(i) .
We define S : {0, 1, · · · , N } → R n by
S(t) = Φ(t)u.
It follows that a function x ∈ Ker(L) if and only if x(·) = S(·)α for some α ∈ R.
Using the fact that Ker
N i=0 B i Φ(i) and Ker N i=0 B i Φ(i) T
have the same dimension, we may also pick a vector w ∈ R n which forms a basis for
Ker N i=0 B i Φ(i) T
. We introduce the following notation which simplifies our characterization of Im(L). We define Ψ T :
{0, 1, 2, · · · , N − 1} → R n by Ψ T (t) = N i=t+1 w T B i Φ(i)Φ −1 (t + 1).
We now have the following characterization of the Im(L).
Proposition 2.4. An element h ∈ Z is contained in the Im(L) if and only if N −1 i=0 Ψ T (i)h(i) = 0.
Having characterized the kernel and image of L, we are now in a position to construct the projections which will form the basis of the Lyapunov-Schmidt projection scheme. In this regard, we choose to follow [20,32]. The proofs that the following operators, P and I − Q, are projections onto the kernel and image of L, respectively, are simple consequences of our previous characterization of these spaces. Proofs may be found in [32].
Definition 2.5. Let V : R n → R n be the orthogonal projection onto Ker N i=0 B i Φ(i) . Define P : X → X by [P x](t) = Φ(t)V x(0).
Then P is a projection onto Ker(L).
Definition 2.6. Define Q : Z → Z by [Qh](t) = Ψ(t) N −1 j=0 |Ψ(j)| 2 −1 N −1 i=0 Ψ T (i)h(i).
Then I − Q is a projection onto Im(L).
Remark 2.7. That Q is well-defined is a consequence of Remark 2.1, see [32].
The following is the formulation of the alternative problem which we will use to analyze the nonlinear boundary value problem, (1), subject to boundary conditions, (2). It is often referred to as the Lyapunov-Schmidt projection scheme. This type of projection scheme has become quite standard in resonant boundary value problems, we include the proof simply for the convenience of the reader.
Proposition 2.8. Solving Lx = F (x) is equivalent to solving the following system x − P x = M p (I − Q)F (x) and N −1 i=0 [Ψ(i)] n g(i, [x(i)] m ) = 0 ,
where M p is L |Ker(P ) −1 and [e] k denotes the kth row of a vector e in R n .
Proof.
Lx = F (x) ⇐⇒ (I − Q)(Lx − F (x)) = 0 and Q(Lx − F (x)) = 0 ⇐⇒ Lx − (I − Q)F (x) = 0 and QF (x) = 0 ⇐⇒ M p Lx − M p (I − Q)F (x) = 0 and QF (x) = 0 ⇐⇒ (I − P )x − M p (I − Q)F (x) = 0 and N −1 i=0 Ψ T (i)f (i, x(i)) = 0 ⇐⇒ (I − P )x − M p (I − Q)F (x) = 0 and N −1 i=0 [Ψ(i)] n g(i, [x(i)] m ) = 0 .
Main Results
We now come to our main result. We start by introducing some notation that will be useful in what follows. We introduce the following sets:
O ++ = {t ∈ {0, 1, · · · , N − 1} | [Ψ(t)] n > 0, [S(t)] m > 0}, O +− = {t ∈ {0, 1, · · · , N − 1} | [Ψ(t)] n > 0, [S(t)] m < 0}, O −+ = {t ∈ {0, 1, · · · , N − 1} | [Ψ(t)] n < 0, [S(t)] m > 0}, O −− = {t ∈ {0, 1, · · · , N − 1} | [Ψ(t)] n > 0, [S(t)] m < 0}, O 0 = {t ∈ {0, 1, · · · , N − 1} | [Ψ(t)] n = 0, [S(t)] m = 0}, and O = O ++ ∪ O +− ∪ O −+ ∪ O −− .
We also define A :
= M p (I − Q) (Operator norm), s max := max t∈{0,··· ,N −1} |[S(t)] m |, s min := min O |[S(t)] m |, g r = sup t∈{0,··· ,N −1},x∈[−r,r] |g(t, x)|, and p : R × Im(I − P ) → Im(I − P ) by p(α, v) = M p (I − Q)F (αS(·) + v).
Theorem 3.1. Suppose the following conditions hold:
C1. O 0 is empty C2.
There exists positive real numbers c and d, with c < d, and functions
W 1 , U 1 , W 2 , U 2 , w 1 , u 1 , w 2 and u 2 such that if x ∈ [c, d], then W 1 (t) < g(t, x) for t ∈ O ++ if x ∈ [−d, −c], then g(t, x) < U 1 (t) for t ∈ O ++ if x ∈ [c, d], then g(t, x) < u 1 (t) for t ∈ O +− if x ∈ [−d, −c], then w 1 (t) < g(t, x) for t ∈ O +− if x ∈ [c, d], then g(t, x) < W 2 (t) for t ∈ O −+ if x ∈ [−d, −c], then U 2 (t) < g(t, x) for t ∈ O −+ if x ∈ [c, d], then u 2 (t) < g(t, x) for t ∈ O −− and if x ∈ [−d, −c], then g(t, x) < w 2 (t) for t ∈ O −− C3. d > cs max + A g d (s max + s min ) s min C4. J 2 ≤ 0 ≤ J 1 , where J 1 = N −1 i=0 [Ψ(i)] n K 1 (i), J 2 = N −1 i=0 [Ψ(i)] n K 2 (i),
and K 1 and K 2 are defined by
K 1 (t) = W 1 (t) t ∈ O ++ w 1 (t) t ∈ O +− W 2 (t) t ∈ O −+ w 2 (t) t ∈ O −− , and K 2 (t) = U 1 (t) t ∈ O ++ u 1 (t) t ∈ O +− U 2 (t) t ∈ O −+ u 2 (t) t ∈ O −− .
Then there exists a solution to the nonlinear boundary value problem (1)- (2).
Proof. Define H : R × Im(I − P ) → R × Im(I − P ) by (8) H(α, x) = N −1 i=0 [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m ) v − p(α, v) .
From Proposition 2.8, the zeros of H are precisely the solutions of (1)-(2). We will show the existence of a solution to the nonlinear boundary value problem by showing that the Brouwer degree of H, deg(H, Ω, 0), is nonzero for some appropriately chosen set Ω.
To this end, endow R × Im(I − P ) with the product topology and define
Ω = {(α, v) | |α| ≤ α * and v ≤ r * },
where α * = c + A g d s min and r * = A g d .
Define Q : [0, 1] × Ω → R × Im(I − P ) by Q(γ, (α, v)) = (1 − γ)α + γ N −1 i=0 [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m ) v − γp(α, v) .
It is evident that Q is a homotopy between the identity mapping and H. In what follows, we will show that Q(γ, (α, v)) is nonzero for each γ ∈ (0, 1) and every (α, In what follows, it will be useful to note that
α * s max + r * = c + A g d s min s max + A g d = cs max + A g d s max + A g d s min s min < d.(9)
We now turn our attention to showing that Q(γ, (α, v)) = 0 for each γ ∈ (0, 1) and every (α, v) ∈ ∂(Ω). We start by assuming (α, v) ∈ ∂(Ω), with |α| ≤ α * and v = r * .
Since for every i, |α[S(i)] m + [v(i)] m | ≤ α * s max + r * , we have, using (9), that
α[S(i)] m + [v(i)] m ∈ [−d, d]. It follows that p(α, v) = M p (I − Q)F (αS(·) + v) ≤ M p (I − Q) F (αS(·) + v) = A max {0,··· ,N −1} |f (i, αS(i) + v(i))| = A max {0,··· ,N −1} |g(i, α[S(i)] m + [v(i)] m )| ≤ A g d = r * .
Thus, p(α, v) ≤ r * = v and it becomes clear that Q(γ, (α, v)) = 0 for every γ in (0,1), since v − γp(α, v) = 0.
We finish the proof by looking at the case when (α, v) ∈ ∂(Ω) with |α| = α * and v ≤ r * . Combining the fact that p(α, v) ≤ r * with (9), we conclude that for each i and for every (
α, v) ∈ ∂(Ω), |α[S(i)] m + [p(α, v)(i)] m | ≤ d.
Further, if |α| = α * , then for all i ∈ O, we have
|α[S(i)] m + [p(α, v)(i)] m | ≥ α * s min − p(α, v) ≥ α * s min − A g d = c + A g d s min s min − A g d = c.
Thus, we have shown that when (α, v) ∈ ∂(Ω) with |α| = α * and v ≤ r * , then
for all i ∈ O, |α[S(i)] m + [p(α, v)(i)] m | ∈ [c, d].
In fact, we have shown that if α = α * and i ∈ O
++ ∪ O −+ , then α[S(i)] m + [p(α, v)(i)] m ∈ [c, d] and if i ∈ O +− ∪ O −− , then α[S(i)] m + [p(α, v)(i)] m ∈ [−d, −c]. Similarly, if α = −α * and i ∈ O ++ ∪ O −+ , then α[S(i)] m + [p(α, v)(i)] m ∈ [−d, −c] and if i ∈ O +− ∪ O −− , then α[S(i)] m + [p(α, v)(i)] m ∈ [c, d].
Using C2., we now conclude that when α = α * ,
W 1 (i) < g(i, α[S(i)] m + [p(α, v)(i)] m ) for i ∈ O ++ w 1 (i) < g(i, α[S(i)] m + [p(α, v)(i)] m ) for i ∈ O +− g(i, α[S(i)] m + [p(α, v)(i)] m ) < W 2 (i) for i ∈ O −+ and g(i, α[S(i)] m + [p(α, v)(i)] m ) < w 2 (i) for i ∈ O −− .
Thus, since O 0 is empty,
N −1 i=0 [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m ) = i∈O [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m ) > O [Ψ(i)] n K 1 (i) = N −1 i=0 [Ψ(i)] n K 1 (i) = J 1 ≥ 0.
Similarly, we may conclude that if α = −α * ,
N −1 i=0 [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m ) = i∈O [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m ) < O [Ψ(i)] n K 2 (i) = N −1 i=0 [Ψ(i)] n K 2 (i) = J 2 ≤ 0. It follows that α · N −1 i=0 [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m ) > 0. Since (1 − γ)α + γ N −1 i=0 [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m )
would be 0 for some γ ∈ (0, 1) if and only if
α · N −1 i=0 [Ψ(i)] n g(i, α[S(i)] m + [p(α, v)(i)] m ) < 0,
we have that Q(γ, (α, v)) is nonzero in these cases. We now conclude, by the homotopy invariance of the Brouwer degree, that deg(H, Ω, 0) = deg(I, Ω, 0) = 1.
The result now follows.
if x ∈ [c, d], then W 1 (t) > g(t, x) for t ∈ O ++ if x ∈ [−d, −c], then g(t, x) > U 1 (t) for t ∈ O ++ if x ∈ [c, d], then g(t, x) > u 1 (t) for t ∈ O +− if x ∈ [−d, −c], then w 1 (t) > g(t, x) for t ∈ O +− if x ∈ [c, d], then g(t, x) > W 2 (t) for t ∈ O −+ if x ∈ [−d, −c], then U 2 (t) > g(t, x) for t ∈ O −+ if x ∈ [c, d], then u 2 (t) > g(t, x) for t ∈ O −− and if x ∈ [−d, −c], then g(t, x) > w 2 (t) for t ∈ O −− ,
then provided J 1 ≤ 0 ≤ J 2 , (1)-(2) has a solution. The proof is essentially the same.
The following corollary isolates the special case in which [Ψ(i)] n and [S(i)] m have the same sign for all i = 0, · · · , N − 1. This case is of special interest since it occurs in all 'self-adjoint' boundary value problems, Sturm-Liouville boundary value problems being a special case, specific cases of second-order periodic difference equations being another. It also happens in several other cases, as we will see in our example in section 5. Proof. It suffices to show that conditions C2. and C4. of Theorem 3.1 hold. We will assume [Ψ(i)] n · [S(i)] m ≥ 0, that g(t, x) > 0 for every x ∈ [c, d] and each t = 0, · · · , N − 1, and that g(t, x) < 0 for every x ∈ [−d, −c] and each t = 0, · · · , N − 1.
Since [Ψ(i)] n · [S(i)] m ≥ 0, we have that O +− and O −+ are empty. It follows that condition C2. of Theorem 3.1 reduces to (NC2.) There exists positive numbers c and d, with c < d, and functions
W 1 , U 1 , w 2 and u 2 such that if x ∈ [c, d], then W 1 (t) < g(t, x) for t ∈ O ++ if x ∈ [−d, −c], then g(t, x) < U 1 (t) for t ∈ O ++ if x ∈ [c, d], then u 2 (t) < g(t, x) for t ∈ O −− and if x ∈ [−d, −c], then g(t, x) < w 2 (t) for t ∈ O −− .
However, using C3*., NC2. is clearly satisfied by taking W 1 = U 1 = u 2 = w 2 = 0. It then trivially follows that J 1 = J 2 = 0, so that condition C4. of Theorem 3.1 is satisfied. This completes the proof for this case. The other cases are similar.
The following corollary is an application of Theorem 3.1 to cases in which the nonlinearities satisfy a sublinear or 'small' linear growth condition. C3**. d > cs max + (K 1 (1 − β) + K 2 )(s max + s min ) s min − K 1 β(s max + s min ) , where K 1 = AM 1 , K 2 = AM 2 , and we are assuming s min − K 1 β(s max + s min ) > 0; that is, K 1 < s min β(s min + s max ) .
Then the nonlinear boundary value problem, (1)- (2), has at least one solution.
Proof. From (C2**.), we get g d ≤ M 1 d β + M 2 . Thus,
cs max + A g d (s max + s min ) s min ≤ cs max + A(M 1 d β + M 2 )(s max + s min ) s min = cs max + (K 1 d β + K 2 )(s max + s min ) s min .
Using (C3**.), we have
cs max + (K 1 (1 − β) + K 2 )(s max + s min ) s min − K 1 β(s max + s min ) < d, from which we conclude cs max + K 2 (s max + s min ) < ds min − dK 1 β(s max + s min ) − K 1 (1 − β)(s max + s min ) = ds min − K 1 (1 + β(d − 1))(s max + s min ) ≤ ds min − K 1 (1 + (d − 1)) β (s max + s min ) = ds min − K 1 d β (s max + s min ).
Rearranging, it follows that
cs max + A g d (s max + s min ) s min ≤ cs max + (K 1 d β + K 2 )(s max + s min ) s min < d.
Remark 3.5. We would like to point out that if g is sublinear on all of R; that is, there exist positive numbers M 1 , M 2 and a constant β, 0 ≤ β < 1, such that |g(t, x)| ≤ M 1 |x| β + M 2 for every x ∈ R and each t = 0, · · · , N − 1, and there is a
R > 0 such that if x > R, then W 1 (t) < g(t, x) for t ∈ O ++ if x < −R, then g(t, x) < U 1 (t) for t ∈ O ++ if x > R, then g(t, x) < u 1 (t) for t ∈ O +− if x < −R, then w 1 (t) < g(t, x) for t ∈ O +− if x > R, then g(t, x) < W 2 (t) for t ∈ O −+ if x < −R, then U 2 (t) < g(t, x) for t ∈ O −+ if x > R, then u 2 (t) < g(t, x) for t ∈ O −− and if x < −R, then g(t, x) < w 2 (t) for t ∈ O −− ,(10)
then C3. of Theorem 3.1 holds, since lim r→∞ Rs max + A g r (s max + s min ) s min r = 0 < 1. Thus, if conditions C1. and C4. of Theorem 3.1 are satisfied, then the nonlinear boundary value problem has a solution.
In fact, if g has 'small' linear growth; that is, |g(t, x)| ≤ M 1 |x| + M 2 for every x ∈ R and each t = 0, · · · , N − 1 with AM 1 s max + s min s min < 1, then provided (10) holds we have that C3. of Theorem 3.1 holds, since in this case lim r→∞ Rs max + A g r (s max + s min ) s min r ≤ AM 1 s max + s min s min < 1. Thus, again, if conditions C1. and C4. of Theorem 3.1 are satisfied, then the nonlinear boundary value problem has a solution.
Comparision to previous results
In this section we show how Theorem 3.1 improves upon existing results in the literature.
. General Multipoint. In [31] the authors look at the existence of solutions to (1)- (2). They obtain results by placing conditions on the nonlinearity, g, which are much more restrictive than Theorem 3.1. Their main result, written in terms of the notation of this paper, is the following:
H4. L 1 L 2 < 0, where L 1 = g(+∞) · O++∪O−+ [Ψ(i)] n + g(−∞) · O+−∪O−− [Ψ(i)] n and L 2 = g(−∞) · O++∪O−+ [Ψ(i)] n + g(+∞) · O+−∪O−− [Ψ(i)] n
Then there exists a solution to the nonlinear boundary value problem (1)-(2). Theorem 4.1 is a simple consequence of Theorem 3.1. To see this, suppose the conditions of Theorem 4.1 hold and assume L 2 < 0 < L 1 . We will abuse notation, slightly, and use g to denote the time dependent function defined on {0, · · · , N } × R by g(t, x) = g(x). Since g(±∞) exist, we must have that g is bounded. Let ε > 0 and define the functions W 1 , U 1 , W 2 , U 2 , w 1 , u 1 , w 2 and u 2 in Theorem 3.1 as follows: W 1 (t) = g(+∞) − ε, U 1 (t) = g(−∞) + ε, W 2 (t) = g(+∞) + ε, U 2 (t) = g(−∞) − ε, w 1 (t) = g(−∞) − ε, u 1 (t) = g(+∞) + ε, w 2 (t) = g(−∞) + ε, u 2 (t) = g(+∞) − ε. It is clear that for these functions there exists an R, depending on ε, such that (10) of Remark 3.5 holds.
Now, if we calculate J 1 = N −1 i=0 [Ψ(i)] n K 1 (i), we get O++ [Ψ(i)] n (g(+∞) − ε) + O+− [Ψ(i)] n (g(−∞) − ε) + O−+ [Ψ(t)] n (g(+∞) + ε) + O−− [Ψ(i)] n (g(−∞) + ε), or g(+∞) · O++∪O−+ [Ψ(i)] n + g(−∞) · O+−∪O−+ [Ψ(t)] n − O |[Ψ(i)] n |εdt.
However, this is equal to
L 1 − O |[Ψ(i)] n |ε. Similarly, J 2 = L 2 + O |[Ψ(i)] n |ε.
Since we are assuming L 2 < 0 < L 1 , it is easy to see that for small enough ε, J 2 < 0 < J 1 . The case where L 1 < 0 < L 2 follows from Remark 3.2 by a similar argument. The result is now a consequence of Remark 3.5.
Remark 4.2. The above discussion shows that Theorem 3.1 is a substantial improvement of the result found in [31]. It is a generalization in two significant ways. Firstly, Theorem 3.1 does not require the nonlinearity, g, to be bounded as is required in [31] where they impose that g(±∞) exist. Secondly, the assumptions of . Sturm-Liouville. In [18], the author proves the existence of solutions to nonlinear Sturm-Liouville problems of the form (11) ∆(p(t − 1)∆x(t − 1)) + q(t)x(t) + λx(t) = f (x(t)); t ∈ {a + 1, · · · , b + 1} subject to (12) a 11 x(a) + a 12 ∆x(a) = 0 and a 21 x(b + 1) + a 22 ∆x(b + 1) = 0, where throughout it is assumed that f : R → R, p : [0, 1] → R and q : [0, 1] → R are continuous, p(t) > 0 for all t ∈ [0, 1], a 2 + b 2 , c 2 + d 2 > 0, and λ is an eigenvalue of the associated linear Sturm-Liouville problem.
Their main result is the following:
Example
We now provide an example which shows the application of Theorem 3.1. Consider y(t + 2) + y(t + 1) + y(t) = g(y(t + 1)) subject to y(5) + y(8) + y(9) = 0 and y(2) + y(8) + y(9) = 0
Looking at equations (1) and (2), we see that n = m = 2. Writing this in system form, we have
x(t + 1) = Ax(t) + f (x(t)) subject to B 2 x(2) + B 5 x(5) + B 8 x(8) = 0, where x(t) =
y(t) y(t + 1) , Since A is constant, it follows that Φ(t) = A t . We then have that
A = 0 1 −1 −1 ,B 2 Φ(2) + B 5 Φ(5) + B 8 Φ(8) = −1 −2 −1 −2 .
If we choose 2 −1 as a basis for Ker(B 2 Φ(2) + B 5 Φ(5) + B 8 Φ(8)), then we get
S(t) = 2 −1 if t ≡ 0 mod 3 −1 −1 if t ≡ 1 mod 3 −1 2 if t ≡ 2 mod 3
.
We now take −1 1 as a basis for Ker (B 2 Φ(2) + B 5 Φ(5) + B 8 Φ(8)) T , which gives Ψ(t) = Notice that [Ψ(t)] 2 [S(t)] 2 ≥ 0 for all t = 0 · · · N − 1, so that Theorem 3.1 is applicable for an abundance of real-valued functions, g, provided g is such that conditions C3*. and C4*. of Corollary 3.3 hold for some positive real numbers c and d. We point out again, as in Remark 3.5, that if C3*. holds eventually, then C4*. is automatically satisfied.
Further, if the end behavior of g is not 'sign-changing', then (1)-(2) may still have a solution. It is of interest to note that this may happen for a g which satisfies lim x→∞ g(±x) = ∞ (or lim x→∞ g(±x) = −∞), and so is not of standard Landesman-Lazer form.
For a specific instance of this, fix c > 2 and let g be a continuous function with
g(x) = β ln(1 + |x|) if x ∈ [−d, −c] γ ln(1 + x) if x ∈ [c, d] ,
where d = e γ (1 + c) − 1, 0 < β < 1, and γ > ln(1 + c) ln(1 + c) − 1 . We will assume that g(x) < β ln(1 + c) for x ∈ [−c, 0] and g(x) < γ ln(1 + c) for x ∈ [0, c].
i Φ(i) and the solution space of the linear homogeneous problem, (6), subject to the boundary conditions ,(4), have the same dimension.Proof. Taking h = 0 in the variation of parameters formula,(7), and applying the boundary conditions, we have Lx = 0 if and only if ∃u ∈ R n such that x(·)
v) in ∂(Ω) = {(α, v) | |α| = α * and v ≤ r * or |α| ≤ α * and v = r * }, so that,by the invariance of the Brouwer degree under homotopy, deg(H, Ω, 0) = deg(I, Ω, 0) = 1.
Remark 3. 2 .
2If the inequalities of Theorem 3.1 are reversed; that is,
Corollary 3. 3 .
3Suppose the following conditions are satisfied:C1*. O 0 is empty. C2*. [Ψ(i)] n · [S(i)] m ≥ 0 (or [Ψ(i)] n · [S(i)] m ≤ 0) for all i = 0, · · · , N − 1. C3*.There exists positive real numbers c and d, with c < d, such that g(t, x) > 0 (or g(t, x) < 0) for every x ∈ [c, d] and each t = 0, · · · , N −1 and g(t, x) < 0 (or g(t, x) > 0) for every x ∈ [−d, −c] and each t = 0, · · · , N − 1.C4*. d > cs max + A g d (s max + s min ) s min Then the nonlinear boundary value problem (1)-(2) has a solution.
Corollary 3 . 4 .
34Suppose the following conditions are satisfied: C1**. Conditions C1., C2., and C4. of Theorem 3.1 hold. C2**. There exist positive constants M 1 and M 2 such that |g(t, x)| ≤ M 1 |x| β +M 2for every x ∈ [−d, d] and each t = 0, · · · , N − 1, where 0 < β ≤ 1.
Theorem 4 . 1 .
41Suppose (1) subject to boundary conditions (2) has a 1-dimensional solution space. If H1. g is independent of t H2. g(±∞) := lim x→±∞ g(x) exist H3. O 0 is empty
required only on a bounded interval. In Theorem 4.1 the existence of g(±∞) requires very specific behavior of g on intervals of the form [z 0 , ∞) and (−∞, −z 0 ], for z 0 large.
Theorem 4 . 3 .
43Suppose f : R → R satisfies |f (x)| ≤ M 1 |x| β + M 2 ,where M 1 and M 2 are nonnegative constants and β ∈ [0, 1). If there exist z * such that ∀z > z * , f (z) > 0 and ∀z < −z * , f (z) < 0, then there exists a solution to (11)-(12).
Theorem 4.3 is also a consequence of Theorem 3.1. This follows from the fact that in the case of the Sturm-Liouville problem, because of the self-adjointness associated with it, [Ψ(i)] 2 and [S(i)] 1 (Theorem 3.1), may be chosen to be equal. The result is now a consequence of Corollary 3.3 and Remark 3.5.
.
Observe, O ++ = {2}, O +− = ∅, O −+ = ∅, O −− = {3}, and O 0 = ∅.
Let f (x) := e x (1 + c) − 1 − (2c + 3A ln(1 + c)x 2 ), where A is as in the definition of Theorem 3.1 and choose x c such that if x > x c , then f (x) > 0. If γ > x c , then we have the following:(1) g(−d) = β ln(1 + d) < ln(1 + d) = γ + ln(1 + c) < γ ln(1 + c) = g(c) (2) cs max + A g d (s max + s min ) s min = 2c + 3Aγ ln(1 + d) < 2c + 3A ln(1 + c)γ 2 < e γ (1 + c) − 1 = d.We now verify that conditions C1.-C4. of Theorem 3.1 are satisfied for this choice of g and c, d. It has already been noted that O 0 is empty, so that C1. holds. In this specific example, C2. reduces to the existence of numbers w 1 , w 2 , u 1 , andu 2 such that if x ∈ [c, d],then w 1 < g(x) := g(2, x) if x ∈ [−d, −c], then g(2, x) := g(x) < u 1 if x ∈ [c, d], then u 2 < g(x) =: g(3, x) if x ∈ [−d, −c], then g(3, x) := g(x) < w 2 .
We take w 1 = g(c), w 2 = g(−d), u 1 = g(−d), and u 2 = g(c), so that C2. is clearly satisfied for these choices of w 1 , w 2 , u 1 , and u 2 . (2) shows that condition C3. holds. Finally, if J 1 and J 2 are as in Theorem 3.1, then we have that J 1 = w 1 − w 2 = g(c) − g(−d) > 0 and J 2 = u 1 − u 2 = g(−d) − g(c) < 0 (see(1)). Thus, C4. is satisfied and we conclude, using Theorem 3.1, that there exists a solution to the nonlinear boundary value problem (1)-(2).
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|
[] |
[
"APPROXIMATELY COUNTING BASES OF BICIRCULAR MATROIDS",
"APPROXIMATELY COUNTING BASES OF BICIRCULAR MATROIDS"
] |
[
"Heng Guo ",
"Mark Jerrum "
] |
[] |
[] |
We give a fully polynomial-time randomised approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.
|
10.1017/s0963548320000292
|
[
"https://arxiv.org/pdf/1808.09548v3.pdf"
] | 52,119,907 |
1808.09548
|
493a3dfbe6869f66071b98bec84347b7cf64ab2a
|
APPROXIMATELY COUNTING BASES OF BICIRCULAR MATROIDS
24 Nov 2018
Heng Guo
Mark Jerrum
APPROXIMATELY COUNTING BASES OF BICIRCULAR MATROIDS
24 Nov 2018
We give a fully polynomial-time randomised approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.
Introduction
In this note we introduce a new application of the "popping" paradigm that has been used to design efficient perfect samplers for a number of combinatorial structures. Existing examples are cycle popping [Wil96,PW98], sink popping [CPP02] and cluster popping [GP14,GJ18] that, respectively, produce uniformly distributed spanning trees, sink-free orientations in undirected graphs, and root-connected subgraphs in directed graphs (and, as a consequence, connected subgraphs of an undirected graph). In doing so we provide an example of a natural class of matroids for which the bases-counting problem is hard (#P-complete) to solve exactly, but which is polynomial time to solve approximately in the sense of Fully Polynomial-time Randomised Approximation Schemes (or FPRAS). For basic definitions connected with the complexity of counting problems refer to [MR95] or [Jer03].
Towards this end, we introduce "bicycle popping" as a means to sample, uniformly at random, bases of a bicircular matroid. 1 Bicircular matroids are associated with undirected graphs and will be defined in the next section. Note that the main result and its proof can be understood in graph-theoretic terms, and no knowledge of matroid theory is needed beyond the exchange axiom. Our perfect sampling approach can be implemented to run in O(n 2 ) time, where n is the number of vertices in the instance graph. (Refer to Section 4.) Using a standard reduction, such a sampler can be used to construct an efficient randomised algorithm, indeed an FPRAS, for estimating the number of bases within specified relative error (Theorem 6 and Theorem 10).
The computational complexity of counting bases of a matroid exactly is still only partially understood. According to the class of matroids under consideration, the exact counting problem may be polynomial time, #P-complete or unresolved. Counting bases of a graphic matroid (i.e., counting spanning trees of a graph) is a classical problem and is well solved by Kirchhoff's matrix-tree theorem. This method extends fairly directly to the wider class of regular matroids [Mau76]. The bases-counting problem for bicircular matroids, a restriction of the class of transversal matroids, was shown to be #P-complete by Giménez and Noy [GN06]. The status of the important case of binary matroids appears to be open [Sno12].
Jerrum [Jer06] showed that it is #P-hard to exactly count bases of certain sparse paving matroids. Combined with the approximation algorithm of Chávez Lomelí and Welsh [CW96], this result highlights a (presumably) exponential gap between exact and approximate counting. However, it could be said that this example is not particularly natural. Piff and Welsh [PW71] demonstrated that the number of paving matroids on a ground set of n elements is doubly exponential in n, so even representing the problem instance raises significant issues. Combined with the completeness result of Giménez and Noy [GN06], our FPRAS provides a more convincing and natural demonstration of the gap between exact and approximate counting for matroid bases.
The work described here was supported by the EPSRC research grant EP/N004221/1 "Algorithms that Count".
1 "Bicycle popping" has the advantage of being easy to remember, but it is important to note that the term is unconnected with the concept of bicycle space of a graph.
After posting our paper on arXiv, we were made aware of an independent work of Kassel and Kenyon [KK17], who have proposed essentially the same algorithm 2 for sampling from a weighted distribution on cycle-rooted spanning trees. Their interest in the algorithm is as a component in their proofs, for which correctness of the algorithm is obviously important and is proved in detail. The time-complexity of their algorithm is not analysed in detail, though Kassel and Kenyon offer some brief remarks about the run-time of the algorithm on a square grid. Kassel [Kas15] also observes the connection to sampling bases of a bicircular matroid, and notes that the corresponding counting problem is #P-complete.
Even more recently, Anari, Liu, Gharan and Vinzant [ALOV18] have posted on the arXiv a proof that the expansion of the so-called "basis exchange graph" for any matroid is at least 1. This result implies that a random walk on the bases exchange graph is rapidly mixing, and provides a Markov Chain Monte Carlo (MCMC) approach to sampling bases of any matroid. The only requirement is that there exists an efficient independence oracle to verify whether a given set is a basis. Since this requirement certainly holds for bicircular matroids, their work yields an alternative approach to sampling bases of a bicircular matroid. Their method is very different to ours and does not give a perfect sampler (though the deviation of the output distribution from uniformity decays exponentially fast in the run-time). Also, the analysis of the expansion factor of the basis-exchange graph is technically challenging, while the analysis of our popping algorithm is relatively elementary. Before the recent work of Anari et al., the basesexchange graph was known to be an expander only in special cases. Most notably, Feder and Mihail [FM92] showed that the class of so-called "balanced matroids", a strict superset of the class of regular matroids, has expansion factor at least 1. (See also [JSTV04] for improvements and simplifications.) Furthermore, all paving matroids admit an FPRAS for the number of their bases, as shown by Chávez Lomelí and Welsh [CW96], through the straightforward Monte-Carlo method.
Bicycle-popping
For a graph G = (V, E), let n = |V | and m = |E|. When m ≥ n and G is connected, we associate a bicircular matroid B(G) with G. The ground set is E, and a subset R ⊆ E is independent if every connected component of (V, R) has at most one cycle. Thus, the set of bases of B(G) is
B = {R | every connected component of (V, R) is unicyclic}.
In particular, if R ∈ B, then |R| = n. Denote by π B (·) or simply π(·) the uniform distribution over B. We refer the reader to [Mat77] for more details on bicircular matroids. Giménez and Noy [GN06] have shown that counting the number of bases for bicircular matroids is #Pcomplete. See also [GMN05] for extremal bounds on this number.
We now associate a random arc a v = (v, w) to each vertex v ∈ V , which is uniform over all neighbours w of v. Given an arbitrary assignment σ = (a v ) v∈V , consider the directed graph (V, σ) with exactly those |V | arcs. It is easy to see that each (weakly) connected component of this graph has the same number of arcs as vertices. Thus, there is exactly one (directed) cycle per connected component. Let U (σ) ⊆ E be the subset of edges of G obtained by dropping the direction of arcs in σ. Consider the distribution τ (·) on subsets of E induced by σ via the mapping U (σ). There are two reasons why τ (·) is not quite the same as π(·).
(1) It is possible to have 2-cycles in σ, in which case at least one connected component of U (σ) will be a tree rather than a unicyclic graph.
(2) Every cycle in σ of length greater than 2 may be reversed without changing U (σ).
Thus, in τ (·), each subgraph with k connected components arises in 2 k ways, skewing the distribution towards configurations with more connected components. For each edge e ∈ E, let B e denote the event that a 2-cycle is present at e, i.e., both orientations of e appear in σ. For each cycle C in G, we fix an arbitrary orientation and denote by B C the event that C is oriented this way in σ. If we further condition on none of B e or B C happening, the resulting distribution τ induced by U (σ) is exactly π(·).
Partial rejection sampling [GJL17] provides a useful framework to sample from a product distribution conditioned on a number of bad events not happening. In particular, we call a collection of bad events extremal if any two bad events are either probabilistically independent or disjoint (i.e., cannot both occur). It is straightforward to verify that the collection of bad events {B e | e ∈ E} ∪ {B C | C is a cycle in G} is extremal. (The reason is similar to the cyclepopping algorithm. See [GJL17, Section 4.2]. In fact, the bad events here are either identical or more restrictive than those for cycle-popping.) For an extremal instance, to draw from the desired distribution, we only need to randomly initialize all variables, and then repeatedly rerandomize variables responsible for occurring bad events. This is Algorithm 1, which we call "bicycle-popping".
Algorithm 1: Bicycle-popping Let S be a subset of arcs obtained by assigning each arc a v independently and uniformly among all neighbours of v; while a bad event B e or B C is present do Let Bad be the set of vertices involved in any occurring bad event;
Re-randomize {a v | v ∈ Bad} to get a new S; end return the undirected version of S We need to be a little bit careful about bad events (B C ), since there are potentially exponentially many cycles in G. We cannot afford to dictate the unfavourable orientation a priori, but rather need to figure it out as the algorithm executes. This is not difficult to get around, since we only need an arbitrary (but deterministic) orientation of each cycle. For example, we may arbitrarily order all vertices, and give a sign ± to each direction of an edge according to the ordering. The sign of an odd-length cycle is the product over all its edges, and the sign of an even-length cycle is the product over all but the least indexed edge. Then, we can simply declare all orientations with a + sign "bad". An alternative is to reject cycles randomly, which is considered in [KK17] and is described in Section 6.
Since the extremal condition is satisfied, applying [GJL17, Theorem 8] we get the correctness of Algorithm 1.
Proposition 1. Conditioned on terminating, the output of Algorithm 1 is exactly π(·).
We remark that bicircular popping, Algorithm 1, differs from cycle-popping [PW98] by associating random variables to all vertices, and differs from cluster-popping [GP14,GJ18] by associating random variables to vertices rather than edges.
Run-time analysis
An advantage of adopting the partial rejection sampling framework is that we have a closed form formula for the expected run-time of these algorithms on extremal instances.
In the general setting of partial rejection sampling, the target distribution to be sampled from is a product distribution over variables, conditioned on a set of "bad" events (A i ) i∈I not happening for some index set I. Let T i be the number of resamplings of event A i . Let q i be the probability such that exactly A i occurs, and q ∅ be the probability such that none of (A i ) i∈I occurs, both under the product distribution. Suppose q ∅ > 0 as otherwise the support of π(·) is empty. For extremal instances, [GJL17, Lemma 12] and the first part of the proof of [GJL17, Theorem 13] yield
E T i = q i q ∅ .(1)
Let T be the number of resampled variables. By linearity of expectation and (1), [GH18,Eqn. (2)].) We note that an upper bound similar to the right hand side of (2) was first shown by Kolipaka and Szegedy [KS11], in a much more general setting but counting only the number of resampled events.
E T = i∈I q i · |var(A i )| q ∅ . (2) (See also
Specialising to Algorithm 1, let q e and q C be the corresponding quantity for bad events B e and B C , respectively. Let Ω 0 be the set of assignments so that no bad event happens, and Ω e (or Ω C ) be the set of assignments of (a v ) v∈V so that exactly B e (or B C ) happens and none of the other bad events happen. Then |Ω 0 | = |B|. For a bad event B, let var(B) be the set of variables defining B, namely, var
(B e ) = {a u , a v } if e = (u, v) ∈ E, and var(B C ) = {a v | v ∈ C} if C is a cycle in G. Define Ω var E := {(σ, a v ) | ∃e ∈ E, σ ∈ Ω e , a v ∈ var(B e )}, and Ω var cycle := {(σ, a v ) | ∃a cycle C, σ ∈ Ω C , a v ∈ var(B C )}. Then e∈E qe·|var(A i )| q ∅ = Ω var E |Ω 0 | and C is a cycle q C ·|var(A i )| q ∅ = Ω var cycle |Ω 0 | . Proposition 2.
Let T be the number of resampled variables of Algorithm 1. Then
E T = Ω var E |Ω 0 | + Ω var cycle |Ω 0 | .
We bound these ratios using a combinatorial encoding idea. Namely, we want to design an injective mapping from Ω var E or Ω var cycle to Ω 0 . To make the mapping injective, we in fact have to record some extra information. We first deal with Ω var cycle . Lemma 3. For a connected graph G = (V, E) with m ≥ n where m = |E| and n = |V |, Ω var cycle ≤ n |Ω 0 |. Proof. We define a "repairing" mapping ϕ : Ω var cycle → Ω 0 × V , as follows. For σ ∈ Ω C , we define σ fix to be the same as σ except that the orientation of C is reversed. Clearly σ fix ∈ Ω 0 . Let
ϕ(σ, a v ) = (σ fix , v) if σ ∈ Ω C and v ∈ C.
We claim that ϕ is injective. To see this, given σ fix and v, we simply flip the orientations of the cycle containing v to recover σ. Since ϕ is injective, we have that Ω var cycle ≤ n |Ω 0 |. For Ω var E , the proof is slightly more involved. For σ ∈ Ω e , if we contract e, this component is a directed tree rooted at e.
Lemma 4. For a connected graph G = (V, E) with m ≥ n where m = |E| and n = |V |, Ω var E ≤ 2n(n − 1) |Ω 0 |. Proof. Let Ω E := e∈E Ω e . Then Ω var E = 2 |Ω E |.
Fix an arbitrary ordering of all vertices and edges. Our goal to define an injective "repairing" mapping ϕ : Ω E → Ω 0 × V × E. For σ ∈ Ω e , find the connected component of U (σ) containing the edge e = (v 1 , v 2 ), and let its vertex set be S. Depending on whether S = V , there are two cases.
(1) If S = V , then, since the graph G is connected, there must be at least one edge joining the component to the rest of the graph. Pick the first such edge (u, u ′ ) where u is in S and u ′ is not.
(2) Otherwise S = V ; then since the graph has at least n edges, there must be at least one edge not in U (σ). Let e ′ = (u, u ′ ) be the first such edge, and C be the cycle resulting from adding e ′ to U (σ). Suppose the correct orientation on C induces the orientation u → u ′ on e ′ .
Let u = u 1 , u 2 , . . . , u ℓ = v 1 be the unique path between u and v 1 in U (σ). (v 1 is chosen arbitrarily from the two endpoints of e.) Let σ fix be the assignment so that a u i points to u i−1 , where u 0 = u ′ , and all other variables are unchanged from σ. It is easy to verify that σ fix ∈ Ω 0 . Define ϕ(σ) = (σ fix , u, e) where e = (v 1 , v 2 ). We claim that ϕ is injective. We just need to recover σ given (σ fix , u, e). We first figure out whether S = V . Notice that u ′ can be recovered as u → u ′ is in σ fix . If S = V , then the edge (u, u ′ ) is a bridge under σ fix ; whereas if S = V , (u, u ′ ) is not.
In the first case, simply find the path between u and v 1 , and reverse the "repairing" to yield the original σ. In the second case, we remove (u, u ′ ) first, and then recover the unique path between u and v 1 . The rest is the same as the first case.
Note that |σ fix | = n, u → u ′ ∈ σ fix , and e ∈ U (σ fix ), but (v 1 , v 2 ) = (u, u ′ ). Thus, fixing σ fix , there are n choices for u, and (n − 1) choices for e = (v 1 , v 2 ). Since ϕ is injective, we have that
Ω var E = 2 |Ω E | ≤ 2n(n − 1) |Ω 0 |.
Combining Lemma 3, Lemma 4, and Proposition 2, we have the following theorem.
An implementation based on a loop-erasing random walk
In the execution of Algorithm 1, during each iteration, one needs to find all bad events, and a naive implementation may take up to O(n) time for this task, giving another factor on top of the bound in Theorem 5. Here we provide an implementation that has expected run-time O(n 2 ), similar to the loop-erasing random walk of Wilson [Wil96]. A formal description is given in Algorithm 2.
Algorithm 2: A random walk implementation of bicycle-popping
V u ← V ; S ← ∅; while V u = ∅ do v ← an arbitrary vertex in V u ;
Start a random walk from v, and erase any cycle C having length 2 or a wrong orientation, until some vertex in V \ V u is reached, or a good cycle C is formed; Remove all vertices of the walk from V u ; Add all (undirected) edges along the walk to S; end return S Observe that, in Algorithm 2, once a cycle is orientated correctly, none of its associated arcs will be resampled ever again, and the same holds for any arc attached to it. We will call such arcs "fixed". Starting from an arbitrary vertex v, we assign a random arc from v to u, and continue this for u. So far this is just the normal random walk with memory. The difference is that whenever a cycle appears, we check whether it has length > 2 and the correct orientation. If not, then we erase it, and continue the random walk. Otherwise, we keep all random arcs leading towards this cycle, and mark them as fixed. Thus, Algorithm 2 amounts to a loop-erasing random walk with a special erasing rule.
Once the first random walk stops with a correctly oriented cycle, we do the same for the next vertex that has not been fixed yet. Now the new walk has two possible terminating conditions. Namely it is fixed if it has reached some fixed vertex, or a correctly oriented cycle of length > 2 is formed. This process is repeated until all vertices are fixed.
Algorithm 1 specifies a particular order of resampling bad events, modulo the ordering of bad events within each iteration of the while-loop. However, bad events can be sampled in any order, without affecting correctness or the expected number of resampled variables. Although the proof of this key fact has appeared in the context of specific instances of partial rejection sampling, such as cycle-popping [PW98] and sink-popping [CPP02], we are not aware that the argument has been presented in generality, so we do so presently. As a consequence of this key fact, Algorithm 2, which is sequential, has the same resulting distribution and expected number of resampled variables as Algorithm 1, which is parallel. In particular, the expected run-time of Algorithm 2 has the same order as the number of resampled variables, which is at most O(n 2 ) by Theorem 5.
The correctness of Algorithm 2 is due to the aforementioned fact that the ordering of resamplings does not matter for extremal instances. We now formalise and verify this fact. Consider a generic partial rejection sampling algorithm that repeatedly locates an occurring bad event and resamples the variables on which it depends. A specific implementation will choose a particular order for resampling the bad events. We can represent the choices made as a path in a countably infinite, directed "game graph" Γ = (Σ, A). The vertex set Σ of Γ contains all multisets of bad events. We refer to these vertices as states. The arc set A is defined relative to a resampling table, as used in [GJL17], following Moser and Tardos. As the algorithm proceeds, the "frontier" in the table between used and fresh random variables advances; in the notation of [GJL17], the frontier at time t is specified by the indices (j i,t : 1 ≤ i ≤ n). At time t, the implementation will have sampled a certain multiset M ∈ Σ of bad events: an event B * that has been resampled k times will occur k times in M . Note that M determines the number of times each variable has been resampled, and hence the frontier of the table. So, even though we don't know the order in which those bad events were resampled, we do know the occurring bad events at time t. For each M ∈ Σ and each possible occurring bad event B * , we add an arc in Γ from M to M ′ = M + B * . A state with outdegree 0 is a terminating state. Given a fixed resampling table, an implementation of partial rejection sampling will generate a directed path in Γ starting at the state ∅. With probability 1 (over the choice of resampling table), this path will be finite, i.e., end in a terminating state.
We now apply a Lemma of Eriksson [Eri96], which is similar in spirit to Newman's Lemma, but which is both more elementary and better suited to our needs. Observe that if two bad events occur at time t then they can be resampled in either order without altering the result; this is a consequence of the fact that the events are on disjoint sets of variables. In the terminology of [Eri96], the game graph Γ has the polygon property. It follows from his Theorem 2.1 that Γ has the strong convergence property: if there exists a path starting at ∅ and terminating at M , then every path starting at ∅ will terminate at M in the same number of steps. Since a terminating path exists with probability 1, we see that both the output and the number of resampled variables is independent of the order in which the implementation decides to resample bad events. In other words, the correctness of Algorithm 2 follows from that of Algorithm 1, and the distribution of the number of resampled variables is identical in the two algorithms.
Approximating the number of bases
For completeness, we include a standard self-reduction to count the number of bases of a bicircular matroid, utilising Algorithm 1.
Theorem 6. There is an FPRAS for counting bases of a bicircular matroid, with time complexity O(n 3 m 2 ε −2 ).
Proof. Let 0 < ε < 1 be a parameter expressing the desired accuracy. Also let N (G) be the number of bases of B(G), the bicircular matroid associated with G.
The technique for reducing approximate counting to sampling is entirely standard [Jer03, Chap. 3], but we include the details here for completeness. Fix any sequence of graphs G = G m , G m−1 , . . . , G n+1 , G n , where each graph G i−1 is obtained from the previous one G i by removing a single edge e i , and G n is a disjoint union of unicyclic components. (Thus the edge set of G n is a basis of B(G).) Then, noting N (G n ) = 1,
(3) N (G) −1 = N (G m ) −1 = N (G m−1 ) N (G m ) × N (G m−2 ) N (G m−1 ) × · · · × N (G n+1 ) N (G n+2 ) × N (G n ) N (G n+1 ) .
Let X i be the random variable resulting from the following trial. Select, uniformly at random, a basis R from B(G i ) and set
X i = 1, if e i / ∈ R; 0, otherwise. Note that µ i = E X i = N (G i−1 )/N (G i ), so that N (G) −1 = E(X m X m−1 . . . X n+2 X n+1 ) = µ m µ m−1 . . . µ n+2 µ n+1 ,
where the m − n trials are assumed independent.
Now let X i be obtained by taking the mean of t independent copies of the random variable X i .
Since E X i = µ i , we have N (G) −1 = E Z, where Z = X m X m−2 · · · X n+1 . Also, Var X i = t −1 Var X i ,
so if t is large enough the variance of Z will be small, and Z −1 will be a good estimate for N (G). For this approach to yield a polynomial-time algorithm, we need that all the fractions appearing in the product (3) are not too small. In fact we will show that they are all bounded below by 1/2n, which is sufficient.
For the moment, assume this claim, i.e., that 1/2n ≤ µ i ≤ 1, for all n < i ≤ m. Note that
E X 2 i = E X i = µ i since X i is a 0,1-variable. Standard manipulations give E X 2 i = Var X i + (E X i ) 2 = t −1 Var X i + µ 2 i = t −1 (E X 2 i − µ 2 i ) + µ 2 i ≤ µ 2 i 1 + 1 tµ i , whence E Z 2 = E X 2 m . . . E X 2 n+1 ≤ µ 2 m . . . µ 2 n+1 1 + 2n t m = (E Z) 2 1 + 2n t m
Setting t = 40nmε −2 , we obtain E Z 2 = exp(ε 2 /20)(E Z) 2 ≤ (1 + ε 2 /16)(E Z) 2 , which implies Var Z ≤ (ε 2 /16)(E Z) 2 . Thus, by Chebyshev's inequality,
Pr |Z − N (G) −1 | ≤ 1 2 εN (G) −1 = Pr |Z − E Z| ≤ 1 2 ε E Z ≥ 3 4 . It follows that Pr |Z −1 − N (G)| ≤ εN (G) ≥ 3
4 . In other words, the algorithm that returns the estimate Z −1 satisfies the conditions for an FPRAS.
To complete the proof we just need to bound the ratio µ i = N (G i−1 )/N (G i ). Let R be a basis of B(G i ) that contains the edge e i , i.e., that is not a basis of B(G i−1 ). Let R 0 be the unique basis in B(G n ) and note that e i / ∈ R 0 . Since R 0 is also a basis of B(G i ), the exchange axiom for matroids asserts that there is an edge f ∈ R 0 \ R such that R + f − e i is a basis of B(G i ) and hence of B(G i−1 ). This exchange operation associates a basis in B(G i−1 ) with each basis in B(G i ) that is not a basis in B(G i−1 ); furthermore, every basis in B(G i−1 ) arises at most |R 0 | = n times in this way. It follows that N (G i ) ≤ (n + 1)N (G i−1 ) ≤ 2nN (G i−1 ), as required.
Overall we need O(nmε −2 ) samples for m − n estimators each. For each sample, we use Algorithm 2 which has expected run-time O(n 2 ) by Theorem 5, yielding the claimed time complexity.
Faster approximate counting
Similar to [KK17], let Ω be the set of configurations consisting of directed edges so that every vertex is the tail of exactly one arc. Consider the following Gibbs distribution:
ρ γ 2 ,γ (S) ∝ γ C 2 (S) 2 γ C(S) ,(4)
where S ∈ Ω, γ 2 , γ ≥ 0 are two parameters, and C(S) (or C 2 (S)) is the number of cycles of length greater than 2 (or 2-cycles) present in (V, S). We adopt the convention that 0 0 = 1.
Then Algorithm 1 and Algorithm 2 sample from the distribution µ 0,0.5 . We define also the corresponding partition function
Z γ 2 ,γ (G) = S∈Ω γ C 2 (S) 2 γ C(S) .(5)
Then Z 0,0.5 (G) = |B|. Another interesting special case is ρ 1,1 , with the corresponding partition function Z 1,1 (G) = v∈V deg(v). This is because ρ 1,1 corresponds to choosing a random neighbour of v for each v ∈ V . Note that each cycle longer than 2 has two potential orientations.
In order to sample from ρ γ 2 ,γ in (4) where γ 2 , γ ∈ [0, 1], we introduce the following variant of Algorithm 2. Again, this is very similar to the sampling algorithm of Kassel and Kenyon [KK17].
Algorithm 3: Sample ρ γ 2 ,γ in (4)
V u ← V ; S ← ∅; while V u = ∅ do v ← an arbitrary vertex in V u ;
Start a random walk from v, and erase any cycle C formed with probability 1 − γ 2 if the length is 2 or with probability 1 − γ otherwise, until some vertex in V \ V u is reached, or a cycle C is accepted; Remove all vertices of the walk from V u ; Add all arcs along the walk to S; end return S The correctness of Algorithm 3 also follows from [GJL17, Theorem 8] and the argument in Section 4. We introduce an auxiliary variable for each cycle, which is false with probability 1−γ 2 if the cycle has length 2, or with probability 1 − γ if the cycle is longer. A cycle is "bad" if and only if it is present and the auxiliary variable is false. Although there are exponentially many such auxiliary variables, we only reveal them when necessary. Every time a cycle is popped, the auxiliary variable is reset. By the same reason as for Algorithm 1, such an instance is extremal and the correctness follows.
Since the extremal condition holds, the run-time of Algorithm 3 can be analysed analogously to that of Algorithm 1 and Algorithm 2. Let T be the number of resampled variables of Algorithm 3. We apply (2). First consider the case of γ 2 = 0. To bound e∈E qe|e| q ∅ , we use the injective mapping in Lemma 4, and to bound C is a cycle q C |C| q ∅ , we use an injective mapping similar to the one in Lemma 3 by simply flipping the auxiliary variable. Observe that both mappings preserve all cycles other than the one repaired, and as a consequence, preserve all weights up to a factor 1−γ γ in the latter case. It implies that when γ 2 = 0, the run-time can be bounded as follows:
E T ≤ 2n(n − 1) + 1 − γ γ · n.(6)
Otherwise γ 2 > 0. To bound e∈E qe|e| q ∅ , again, we need to resort to the injective mapping in Lemma 4. However, now the "perfect" configurations and "one-flaw" configurations allow more than one 2-cycles. Let Ω (k) ⊂ Ω be the set of configurations with k 2-cycles, for any k ≤ n/2. Lemma 7. There is an injective mapping ψ k : Ω (k) → Ψ (k−1) for any k ≥ 1, where Ψ (k−1) := {(σ, v, e) | σ ∈ Ω (k−1) , v ∈ V, e ∈ U (σ)}. Moreover, ψ k preserves all cycles except one with length 2.
Proof. The mapping is similar to the one in Lemma 4. Given S ∈ Ω (k) , we choose an arbitrary 2-cycle, and apply the "fix" of the injective mapping in Lemma 4. It is straightforward to check that this operation will only destroy the chosen 2-cycle, and it is reversible given the auxiliary information.
A consequence of Lemma 7 is that e∈E qe|e| q ∅ ≤ 2n 2 , similar to Lemma 4. This is because for any configuration with exactly one "bad" 2-cycle, applying the mapping in Lemma 7 yields a configuration without the presence of the bad 2-cycle, and other cycle structures are all preserved. The overhead is to remember one vertex and one edge from the undirected version of the image.
To bound C is a cycle q C |C| q ∅ , consider again the injective mapping similar to the one in Lemma 3, by simply flipping the value of the auxiliary variable. It implies that C is a cycle
q C |C| q ∅ ≤ 1−γ γ · n.
Thus we have that for γ 2 > 0, the following also holds:
E T ≤ 2n 2 + 1 − γ γ · n.(7)
Combining the two cases (6) and (7), we have the following corollary. The Gibbs formulation (4) allows us to utilise faster annealing algorithms to reduce approximate counting to sampling. See [ŠVV09,Hub15], and the current best algorithm is due to Kolmogorov [Kol18].
In fact, we will need a slight generalisation from [GH18] as follows. Let Ω be a finite set, and the generalised Gibbs distribution ρ β (·) over Ω takes the following form:
ρ β (X) = 1 Z(β) exp(−βH(X)) · F (X),(8)
where β is the temperature, H(X) ≥ 0 is an integer function called the Hamiltonian, F : Ω → R + is a non-negative function, and, with a little abuse of notation, Z(β) = X∈Ω exp(−βH(X)) · F (X) is the normalising factor. We would like to turn the sampling algorithm into an approximation algorithm to Z(β). Typically, this involves calling the sampling oracle in a range of temperatures, which we denote [β min , β max ]. (This process is usually called simulated annealing.) Let Q := Z(β min ) Z(βmax) , q = log Q, and N = max X∈Ω H(X). The following result is due to Kolmogorov [Kol18,Theorem 8], as extended in [GH18, Lemma 8].
Proposition 9. Suppose we have a sampling oracle from the distribution ρ β for any β ∈ [β min , β max ]. There is an algorithm to approximate Q within 1 ± ε multiplicative errors using O(q log N/ε 2 ) oracle calls in average.
Theorem 10. There is an FPRAS for counting bases of a bicircular matroid, with time complexity O(n 3 (log n) 2 ε −2 ).
Proof. We will do a two stage annealing. Our starting point is Z 1,1 (G) = v∈V deg(v). We first apply the annealing algorithm in Proposition 9 between γ 2 = 1 and γ 2 = 0. We only treat γ 2 as the temperature in this stage but not γ. 3 Clearly H(S) = C 2 (S) ≤ m in this case, and log N = O(log n). The more complicated estimate is Q = Z 1,1 (G) Z 0,1 (G) . For any S ∈ Ω, we apply the mapping in Lemma 7 at most C 2 (S) ≤ n times to get S ′ where no 2-cycle is present. Given S ′ , and a sequence of vertex-edge pair generated by this repairing sequence, we may uniquely recover S since the mapping in Lemma 7 is injective. Moreover, the longer cycles in S are all preserved in S ′ . It implies that Q = Z 1,1 (G) Z 0,1 (G) ≤ (mn) n and q = log Q = O(n log n). Hence, the number of samples required in this step is O(n(log n) 2 ε −2 ).
In the second stage, we apply Proposition 9 between γ = 1 and γ = 0.5, while γ 2 = 0 is fixed. Then H(S) = C(S) ≤ n in this case, and log N = O(log n). Moreover Q = Z 0,1 (G) Z 0,0.5 (G) ≤ 2 max S C(S) ≤ 2 n . Thus q = log Q = O(n). The number of samples required in this step is O(n log nε −2 ).
Overall, the number of samples required is O(n(log n) 2 ε −2 ). Together with Corollary 8, this implies the claimed run-time.
Theorem 5 .
5Let G = (V, E) be a connected graph, n = |V |, m = |E|, and m ≥ n. The expected number of random variables sampled in Algorithm 1 on G is at most 2n 2 . The bound in Theorem 5 is tight. Consider a cycle of length n. Clearly |Ω 0 | = 1 and Ω var cycle = n as there is only one cycle containing n edges. Moreover, |Ω e | = n − 1 for there are n − 1 choices of the missing edge. Thus Ω var E = 2 |Ω E | = 2n(n − 1) and the upper bound is achieved.
Corollary 8 .
8The run-time of Algorithm 3, in expectation, is O(n 2 ) if γ 2 ∈ [0, 1] and γ ∈ [1/2, 1].
There are a couple of fine differences, such as not rejecting 2-cycle, and popping cycles randomly instead of deterministically. These differences do not change the nature of the algorithm.
This explains why we need F (X) in (8).
AcknowledgementsWe thank Adrien Kassel for bringing reference[KK17]to our attention. We also thank Kun He for pointing out an improvement in Lemma 4. Part of the work was done while HG was visiting the Institute of Theoretical Computer Science, Shanghai University of Finance and Economics, and he would like to thank their hospitality.
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United Kingdom. E-mail address: [email protected] (Mark Jerrum) School of Mathematical Sciences. Queen Mary; London, E1 4NS, United KingdomHeng Guo) School of Informatics, University of Edinburgh, Informatics Forum, Edinburgh, EH8 9AB ; University of London, Mile End Road(Heng Guo) School of Informatics, University of Edinburgh, Informatics Forum, Edinburgh, EH8 9AB, United Kingdom. E-mail address: [email protected] (Mark Jerrum) School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, United Kingdom. E-mail address: [email protected]
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"Carnero Rosell"
] |
It is well known that the probability distribution function (PDF) of galaxy density contrast is approximately lognormal; whether the PDF of mass fluctuations derived from weak lensing convergence (κ WL ) is lognormal is less well established. We derive PDFs of the galaxy and projected matter density distributions via the Counts in Cells (CiC) method. We use maps of galaxies and weak lensing convergence produced from the Dark Energy Survey (DES) Science Verification data over 139 deg 2 . We test whether the underlying density contrast is well described by a lognormal distribution for the galaxies, the convergence and their joint PDF. We confirm that the galaxy density contrast distribution is well modeled by a lognormal PDF convolved with Poisson noise at angular scales from 10 -40 (corresponding to physical scales of 3-10 Mpc). We note that as κ WL is a weighted sum of the mass fluctuations along the line of sight, its PDF is expected to be only approximately lognormal. We find that the κ WL distribution is well modeled by a lognormal PDF convolved with Gaussian shape noise at scales between 10 and 20 , with a best-fit χ 2 /DOF of 1.11 compared to 1.84 for a Gaussian model, corresponding to p-values 0.35 and 0.07 respectively, at a scale of 10 . Above 20 a simple Gaussian model is sufficient. The joint PDF is also reasonably fitted by a bivariate lognormal. As a consistency check we compare the variances derived from the lognormal modelling with those directly measured via CiC. Our methods are validated against maps from the MICE Grand Challenge N-body simulation.
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10.1093/mnras/stw2106
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[
"https://arxiv.org/pdf/1605.02036v1.pdf"
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1605.02036
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92d07fb06998c491a8e37307ffdfeb03167495a8
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Testing the lognormality of the galaxy and weak lensing convergence distributions from Dark Energy Survey maps
2016
L Clerkin
D Kirk
M Manera
O Lahav
F Abdalla
A Amara
D Bacon
C Chang
E Gaztañaga
A Hawken
B Jain
B Joachimi
V Vikram
T Abbott
S Allam
R Armstrong
A Benoit-Lévy
G M Bernstein
R A Bernstein
E Bertin
D Brooks
D L Burke
A
Testing the lognormality of the galaxy and weak lensing convergence distributions from Dark Energy Survey maps
Carnero Rosell
162016Preprint 6 August 2016 Compiled using MNRAS L A T E X style file v3.0 (Affiliations are listed at the end of the paper)
It is well known that the probability distribution function (PDF) of galaxy density contrast is approximately lognormal; whether the PDF of mass fluctuations derived from weak lensing convergence (κ WL ) is lognormal is less well established. We derive PDFs of the galaxy and projected matter density distributions via the Counts in Cells (CiC) method. We use maps of galaxies and weak lensing convergence produced from the Dark Energy Survey (DES) Science Verification data over 139 deg 2 . We test whether the underlying density contrast is well described by a lognormal distribution for the galaxies, the convergence and their joint PDF. We confirm that the galaxy density contrast distribution is well modeled by a lognormal PDF convolved with Poisson noise at angular scales from 10 -40 (corresponding to physical scales of 3-10 Mpc). We note that as κ WL is a weighted sum of the mass fluctuations along the line of sight, its PDF is expected to be only approximately lognormal. We find that the κ WL distribution is well modeled by a lognormal PDF convolved with Gaussian shape noise at scales between 10 and 20 , with a best-fit χ 2 /DOF of 1.11 compared to 1.84 for a Gaussian model, corresponding to p-values 0.35 and 0.07 respectively, at a scale of 10 . Above 20 a simple Gaussian model is sufficient. The joint PDF is also reasonably fitted by a bivariate lognormal. As a consistency check we compare the variances derived from the lognormal modelling with those directly measured via CiC. Our methods are validated against maps from the MICE Grand Challenge N-body simulation.
INTRODUCTION
It was first noted by Hubble that the distribution of galaxies in angular cells on the celestial sphere is well approximated by a lognormal (Hubble 1934). This has been confirmed observationally (Coles & Jones 1991, Wild et al. 2005) as well as in N-body simulations (Bernardeau & Kofman 1995, Bernardeau 1994, Kayo, Taruya & Suto 2001, which have shown that the underlying mass density field is expected to be lognormal.
Since the weak lensing convergence field along the line of sight is a weighted projection of the mass density contrast field, one might suspect that the lognormal distribution is a reasonable, if not exact, model of this too. This has been tested on numerical simulations and a lognormal PDF shown to be a reasonable model (Taruya, Hamana & Kayo 2002, Hilbert, Hartlap & Schneider 2011. Even better fits to the convergence PDF, particularly in the tails of the distribution, have been obtained by generalisations of a lognormal PDF (Das & Ostriker 2006, Takahashi et al. 2011, Joachimi, Taylor & Kiessling 2011. The Dark Energy Survey (DES) (Dark Energy Survey Collaboration 2005, 2016 presents an excellent opportunity to study both of these fields. DES was specifically conceived to produce cutting edge science from four different cosmological probes -large-scale structure, weak gravitational lensing, galaxy clusters and supernovae -using the same instrument. The full survey involves five years of observations, currently in progress. In this paper we focus on data produced during the pre-survey Science Verification (SV) series of observations. This early data from DES allowed for the construction of two types of density fields. One is from luminous matter, i.e. galaxies of various types, δg, which are biased tracers of the underlying dark matter field, δm. The other uses the weak lensing of galaxy shapes to construct a convergence, or κ map (Vikram et al. 2015;Chang et al. 2015) that is directly sensitive to the integrated dark matter field out to the lensed galaxies.
Both maps trace the underlying density distribution in the Universe. Galaxies are biased tracers of matter density, preferentially clustering in overdense regions. Galaxy density contrast can then be considered a biased local tracer of the density field.
Weak lensing convergence on the other hand responds directly to the underlying density field and is therefore unbiased. However gravitational deflection of light is a cumulative effect, sensitive to the integrated matter density along the line of sight from source galaxy to observer. The convergence field for a given galaxy source distribution therefore gives us information about the cumulative density field between observer and source, with the exact contribution of matter at different distances along the line of sight governed by the lensing efficiency function.
The purpose of the present study is to analyse the galaxy and mass maps from DES SV simultaneously, testing the two maps separately for log-normality, as well as analysing the joint distribution. To our knowledge this is the first time that the log-normality of the weak lensing convergence field alone has been tested using data rather than numerical simulation (Taruya et al. 2002), and the first time the joint distribution has been tested for log-normality.
The Counts in Cells (CiC) method (e.g. Hubble 1934;White 1979;Gaztanaga 1992;Szapudi 1997;Bernardeau et al. 2002) is a natural way to measure the individual and joint PDFs. The CiC technique splits up a particular data set into spatial cells, in two or three dimensions, and takes an aggregate of the available information inside each cell. Statistical variation between cells then provides information on the properties of that cosmological field.
DES observations are ideally suited to this sort of analysis. The fact that DES provides a joint galaxy survey and convergence map data set produced from the same observations makes it easier to ensure consistency between data and to control for systematics. The SV data we use were taken before the start of the full five year survey, covers 139 deg 2 to full survey depth and forms a test-bed for the kind of analyses planned on the final DES data. All of the analyses in this work are done first on galaxy and convergence maps from MICE simulations in order to test our methodology.
The outline of the paper is as follows. In section 2 we review the theory and formalism used. We describe the galaxy and weak lensing convergence maps from MICE simulations and DES used in section 3, and summarise our CiC method in section 4. In section 5 we validate our CiC method on MICE Grand Challenge Nbody simulations, checking that we see the expected lognormality in MICE δg and the noise-free convergence. In section 6 we present lognormal fits to the individual DES galaxy and convergence field distributions as well as their joint distribution. We check the validity of the log-normal model by measuring the variance of the fields and comparing this to the variance derived under the assumption of log-normality. We discuss the results in 7, and in the Appendices we give the formalism used to calculate moments from CiC, test the impact of systematic effects, and confirm that assumptions we make in our method do not significantly affect our results.
LOGNORMAL MODELLING OF COSMIC FIELDS
Lognormal distributions are very common in nature, from the sizes of clouds, pebbles on a beach, or crystals in icecream; the length of sentences or words in a conversation; to populations of bacteria (see Limpert, Stahel &Abbt 2001, Gaskell 2004 andreferences therein). Many of these examples involve multiplicative processes, of either merging or fragmentation. Any process that can be written as a product of terms will, if there are many terms, tend to a lognormal distribution. This is because if a process X can be written as a product of independent terms, then ln(X) will be a sum of independent terms, and via the central limit theorem these will be normally distributed. So ln(X) is normally distributed, or, X is lognormally distributed.
There are many examples of the hierarchical merging or fragmenting of structure leading to lognormal distributions, such as: the initial mass function of field stars, explained in terms of cloud fragmentation (Zinnecker 1984); X-ray flux variations, suggesting lognormal distribution of emitting regions (Gaskell 2004); luminosity functions of central galaxies, explained in terms of BCGs being formed by steps of mass adding/stripping (e.g. Taghizadeh-Popp, Heinis & Szalay 2012); and the angular momentum of disc galaxies (Marr 2015).
In this paper we test the lognormality of the distribution of matter in the Universe via the galaxy density contrast field, δg, and via the weak lensing convergence field, κWL. Each approach has particular observational and astrophysical noise associated with it, which we discuss and propose models for in this section.
Galaxies
In the standard picture of gravitational instability, the primordial density fluctuations that are the precursor of all structure in the universe are assumed to be a random Gaussian field. Once they enter the non-linear regime, with finite rms fluctuations, their PDF must deviate from a Gaussian to avoid non-zero probabilities being attached to regions with negative densities (Fry 1984). The exact form of the PDFs in this regime is not known but there are various phenomenological models that are fully specified statistically and satisfy the common sense requirement that the matter density, ρ 0 (e.g. Saslaw & Hamilton 1984, Suto, Itoh & Inagaki 1990, Lahav et al. 1993, Gaztanaga & Yokoyama 1993Suto 1993, Ueda & Yokoyama 1996.
One such model that is widely used is the lognormal. As well as being completely specified statistically and always having ρ 0, it becomes arbitrarily close to Gaussian statistics at early times and has the advantage that it can be handled analytically. The merits of this model in a cosmological context are discussed extensively in Coles & Jones (1991). They explain possible motivations for using the lognormal model as: empirical; kinematic; an application of the central limit theorem (as described above); and importantly, simplicity. It is one of the few random fields for which interesting quantities such as its moments can be calculated analytically.
It should be noted that despite these compelling motivations to use a lognormal in the statistical treatment of density perturbations, it does have shortcomings. In particular, it is not uniquely specified by its moments; many distributions can lead to the same set of moments. It must then be the case that information is lost in going from a lognormal field to its moments, an effect quantified in Carron (2011). However, it remains a popular and useful tool in analysis of the mass density contrast field.
Galaxies are biased tracers of the mass density contrast field. The 1D log-normal distribution of galaxy density contrast δg = (ρ− ρ)/ρ is given by:
f (δg)dδg = 1 w √ 2π exp −x 2 2w 2 dx(1)
where x = ln(1 + δg) + w 2 /2 and w 2 is the variance of the corresponding normal distribution f [ln (1 + δg)]. The offset w 2 /2 ensures that δg = 0. The width w is then the single free parameter of the 1D lognormal distribution. If the lognormal distribution correctly describes the data, the variance of the overdensities will be related to the variance, w, of the underlying Gaussian distribution by
δgδg = e w 2 − 1.(2)
Due to the discrete nature of galaxies, shot noise is present. Assuming Poisson sampling of galaxies, the shot noise in the measurement of the distribution of galaxy overdensities can be accounted for by convolving the log-normal model with a Poisson distribution. The probability distribution function of the galaxy counts N in a cell of given size is then given by:
P (N ) = ∞ −1N N (1 + δg) N N ! e −N (1+δg ) f (δg)dδg(3)
This Poisson sampled lognormal distribution has been shown to be a good fit to different galaxy populations in Coles & Jones (1991), Blanton (2000) and Wild et al. (2005). In this work the smallest number of DES galaxies in a cell considered is around 300, so the shot noise term is important.
Weak Lensing Convergence
Various expressions for the convergence PDF have been proposed (Munshi & Jain 2000;Valageas 2000;Kainulainen & Marra 2011).
The lognormal model has the advantage -as in the case of the matter density contrast -of mathematical convenience, while offering the chance to extract more information than assuming a purely Gaussian model for the convergence field. Following a study that showed that a lognormal transformation of the matter density contrast increases the signal to noise (Neyrinck et al. 2009), Seo et al. (2011) performed an analogous study of the weak lensing convergence. They found that such a transform, when applied to the positively offset convergence, decorrelated angular frequencies and increased the signal-to-noise in the transformed power spectrum. The convergence field along a line-of-sight can be expressed as a weighted projection of the mass density contrast field:
κ(θ) = χ hor 0 dχw(χ)δ[r(χ)θ, χ],(4)
where χ is the comoving distance, χ hor is the angular diameter distance to the horizon and δ[r(χ)θ, χ] is the underlying matter density contrast field. w(χ) is a geometrical weight function that depends on the relative separations of sources, lens and the observer (see e.g. Mellier 1999; Bartelmann & Schneider 2001 for reviews). It takes the form
w(χ) = 3 2 H0 c 2 χΩ0 a(χ) χ hor χ dχ n(χ ) χ − χ χ ,(5)
where n(χ) is the source galaxy distribution. The distribution of κ is not expected to be exactly lognormal, even if δ is, since κ is a weighted projection of the mass density contrast field along line of sight. However, simulations have shown (Taruya et al. 2002) that the convergence field is well approximated by a lognormal outside the regime of extremely high κ. Hence we choose in this work to model the noise free κ field distribution with a shifted lognormal where the shift κ0 = −κmin and is called the 'minimum convergence parameter ' (Hilbert et al. 2011). The lowest possible value of κ is given by −κ0 since the lognormal is only defined for a positive range. The mean is given by
P (κ) = exp − (ln(κ0 + κ) − µ) 2 2σ 2 ln(κ0 + κ) √ 2πσ for κ > −κ0, 0 otherwise,(6)µ = ln(κ0 + κ ) − σ 2 /2(7)
and the second moment
κκ = e (2µ+σ 2 ) (e σ 2 − 1).(8)
The value assigned to κ0 is a modelling choice that can be approached in several ways. The minimum measured value of κ could be used, but this is a noisy quantity and should not be used unless one has access to many realisation of κ. Or, treating κ analogously to δg, we could consider a theoretical minimum corresponding to the convergence we would see, for a given source distribution, if the mass distribution was a pure void along the entire line of sight. For the MICE source distribution used in this work this value is −0.050, and for the DES source distribution it is −0.053. However simulations show that there are no empty lines of sight in a ΛCDM universe (Taruya et al. 2002;Vale & White 2003;Hilbert et al. 2011). So we choose, where possible, to treat κ0 as a free parameter and fit it jointly with the lognormal width. As for galaxies, we need to modify the lognormal to account for noise. The DES κ map is constructed (see section 3.2) from measurements of shear, which is the change in the ellipticity of galaxies resulting from weak gravitational lensing. Since galaxies are intrinsically elliptical (i.e. in the absence of lensing), the observed shear is the sum of this intrinsic ellipticity and the shear caused by lensing. The variance of the intrinsic ellipticity, called shape noise, is the dominant source of noise in shear measurements, typically by a factor of more than an order of magnitude. An estimate of the shape noise in the DES κ map is provided by the 100 noise realisations described in section 3.2.
To analyse the shape of the noise distribution we construct PDFs via CiC (as described in section 4) on each of the 100 maps. The resulting distributions appear Gaussian, as shown in figure 1, where the thick blue curve is made up of 100 superimposed noise distributions with jackknife error bars, and the magenta line shows the mean bestfit Gaussian PDF. A Gaussian model provides an excellent fit, with average goodness of fit over the 100 maps χ 2 /dof = 0.95 ± 0.73.
We therefore propose that the 1D probability distribution for the weak lensing convergence field is then given by a convolution of a lognormal distribution with a noise contribution modelled as Gaussian:
f (κ) = 1 √ 2πσn ∞ −κ 0 exp − (κ − κ) 2 2σ 2 n P (κ )dκ(9)
where P (κ) is the noise free log-normal distribution given in equation 6, and σn is the Gaussian width of the shape noise. In this work, σn is determined from the 100 noise realisations.
Joint Galaxy and Weak Lensing Convergence Field
We can try to model the joint distribution of galaxy density contrast δg and weak lensing convergence κWL as a bivariate lognormal with PDF f (δg, κWL). Following the notation used in Coles & Jones (1991); Wild et al. (2005), this is given by
f (δg, κ) = |V | −1/2 2π exp − (g 2 δ +g 2 κ − 2rLNg δgκ ) 2(1 − r 2 LN ) ,(10)
where gx = ln(x)− ln(x) , with x = (1+δg) and x = (1+κ/κ0) for the galaxy and convergence fields respectively, andgx = gx/ωx where ωx is the variance of the underlying Gaussian field ln(x). The lognormal correlation coefficient rLN is given by
rLN = g δ gκ ω δ ωκ ≡ ω 2 δκ ω δ ωκ(11)
and |V | is the determinant of the covariance matrix
V = ω 2 δ ω 2 δκ ω 2 δκ ω 2 κ .(12)
Note that rLN and V are defined in log-density space, and so rLN is not the same as the (linear) Pearson correlation coefficient ρ. The conditional probability
f (δg|κ) = f (δg, κ) f (δg) (13) = w δ (2π|V |) 1/2 exp − (g δ − rLNgκ) 2 2(1 − r 2 LN )
.
Since f (δg, κ) = f (κ)f (δg|κ) we can combine equations 9 and 13 to give the joint probability distribution function f (δg, κ). Including the convolutions with Poisson shot noise and Gaussian shape noise then gives
P (N, κ) = ∞ −1 ∞ −κ 0 1 √ 2πσn exp − (κ − κ) 2 2σ 2 n f (κ) ×N N (1 + δg) N N ! e −N (1+δg ) f (δg|κ)dδgdκ (15)
THE DATA
This paper uses the DES Science Verfication (SV) galaxy and shape catalogues. The SV data were gathered between November 2012 and February 2013, shortly after DECam (Flaugher et al. 2015) commissioning, and before the beginning of the (five year) DES survey proper in August 2013. The operation of the camera, survey Figure 2. Redshift kernels of the observables considered in this paper: the galaxy redshift distribution of the DES Benchmark galaxy sample using the best-fit Skynet photo-z estimation (black line), and the lensing efficiency function of the sources used to make the DES κ map (red line). Also shown is the redshift distribution of the source galaxies (shaded region). Each is shown with an arbitrary normalisation to make comparison easier.
planning, data analysis and reduction were all tested in preparation for starting year one of DES itself. The SV goal was to reproduce the properties of the full five-year DES survey over a much smaller sky area. Five optical filters (grizY ) are used, with exposure times of 90 seconds for griz and 45 seconds for Y . The final median depth in our region of interest, per band, was g ∼ 24.0, r ∼ 23.9, i ∼ 23.0 and z ∼ 22.3.
In total the SV data covered ∼ 250 deg 2 at close to the nominal depth of the full DES survey. The observing footprint was divided into regions to maximise overlap with other surveys and with several small fields used for SNe searches.
In this paper we concentrate on a large contiguous region of ∼139 deg 2 called the SPT-E field due to its overlap with the South Pole Telescope CMB survey. This amount of contiguous data makes the SV SPT-E field a powerful data set in its own right, particularly for weak gravitational lensing where it rivals the full CFHTLenS (Erben et al. 2013) and is only slightly shallower.
DES Galaxy Sample
We use a particular subset of the DES SV galaxy catalogue known as the "Benchmark" sample (Crocce et al. 2015). First a catalogue of galaxies suitable for LSS analysis was constructed from the SV data and dubbed the "Gold" sample (Rykoff et al. 2016). Objects were included if detected in all five of the DES photometric bands. This covered ∼244 deg 2 , restricted to dec > −61 to avoid the Large Magellenic Cloud and R Doradus regions. In addition the Gold catalogue included masking of satellite trails and other artifacts, removal of regions where colors are severely affected by stray light and the application of additional stellar locus correction (Kelly et al. 2014).
From this Gold sample, the Benchmark sample was selected for cosmological analysis by imposing the additional conditions:
• 18.0 < i < 22.5 • 0 < g − r < 3, 0 < r − i < 2 and 0 < i − z < 3. • wavg spread model > 0.003 (star-galaxy separation) • 60 < ra < 95 and −62 < dec < −40 (SPT-E),
where i refers to SExtractor's MAG_AUTO quantity. The cuts on position restrict our analysis to the SPT-E region. The redshifts used in this paper come from the Skynet photo-z pipeline (Bonnett et al. 2015, Graff & Feroz 2013. The galaxy redshift distribution is shown in figure 2. The redshift range we use throughout this paper is 0.1 < z < 1.5, chosen as in this region the galaxy redshift distribution overlaps with the lensing efficiency function used to make the DES κW L map (see next section).
DES κ Map
Shear measurement on DES SV galaxy images was performed with two independent pipelines: IM3SHAPE 1 (Zuntz et al. 2013) and NG-MIX 2 (Sheldon 2014). Extensive testing of both codes was carried out by the DES collaboration (see Jarvis et al. 2015 for details) and both pipelines passed all requirement tests for measurement of cosmic shear with the SV data set. A number of cuts were applied to both catalogues to remove stars, spurious detections, poor measurements and other effects that could bias shear measurement; these are also described in Jarvis et al. (2015).
Shear measurements for a given galaxy are headless vectors and the cosmic shear field is therefore a spin-2 quantity. To allow us to perform our CiC analysis on a scalar quantity we work with maps of weak lensing convergence, κ, a spin-0 field. This κ-reconstruction was performed using the Kaiser-Squires method (Kaiser & Squires 1993), and the production and initial analysis of these κ maps is described in detail in Vikram et al. (2015).
The Kaiser-Squires reconstruction method uses the relation of the Fourier transform of the observed shear,γ, to that of the convergence,κ,
κ = D * γ ,(16)D = 2 1 − 2 2 + 2i 1 2 | | 2 ,(17)
where i are the Fourier counterparts of the angular coordinates, θi, i = 1, 2. The inverse Fourier transform of equation 16 gives the convergence for the observed field in real space. In the absence of noise, systematics and masking, the convergence will be a real (spin-0) quantity. In reality these effects produce a non-zero imaginary component. It is most convenient to express the real part of the convergence map as a map of curl free E-modes, and the imaginary part as divergence free B-modes. The κ maps have pixels of size 2 . For use in this analysis the original flat sky κ maps are transformed into HEALPix (Gorski et al. 2005) maps at resolution Nside=4096. This is done by dividing each pixel of the flat sky maps into 25 sub-pixels, and creating a HEALPix map by combining these sub-pixels. This procedure reduces inaccuracies in changing from one mapping system to another, and in tests gives the same angular power spectrum measurements as the flat sky map to well within the errors. The source galaxy selection used to construct the κ map used in this paper took galaxies with redshifts in the range 0.6 < z < 1.3. The resulting redshift efficiency function is shown in figure 2. The lensing efficiency function peaks at z ∼ 0.3, and our selection of galaxies at 0.1 < z < 0.5 overlaps significantly with the range of redshifts to which the κ map is sensitive.
In addition to the E-mode κ map we make use of a . Standard deviations of the κ E , κ B signal (magenta) and 100 realisations in which the shears have been randomised (blue) at a cell size of 20 . The random realisations of κ E give an estimate of the shape noise contribution to κ E ; this accounts for 80% of the κ E signal. The κ B signal is also a good estimate of the shape noise, with the standard deviation of the κ B signal agreeing with the rms standard deviation of κ E random realisations within 2%. These standard deviations are calculated via CiC and errors are from jackknife sampling.
of other products made in the course of the DES mass-mapping analysis. A B-mode map was constructed by rotating the measured galaxy ellipticities by 45 degrees. The physical process of weak gravitational lensing does not induce B-modes in the convergence field so the B-mode map is a test of systematic effects in our observations, shear measurement and κ-reconstruction; it should be consistent with zero within our reconstruction noise. We will refer to the B-mode reconstructed map as κB.
In addition to the E-and B-mode maps we also make use of a series of noise-only realisations, made by taking the galaxy shape catalogue and rotating the measured shape of each galaxy by some random angle. κ maps were then constructed from each randomised catalogue in the usual way. This has the effect of destroying all cosmological information in the resulting maps, while retaining the same noise properties as the data (because the distribution of galaxies on the sky and in redshift remains the same, as does the overall ellipticity distribution across the sample). 100 of these noise realisations were made and we use them to estimate the noise contribution in our measurement, as described in more detail in section 2.2. Fig. 3 shows the standard deviations of the κE, κB signal (magenta) and 100 noise realisations (blue) for a cell size of 20 . This shows that the shape noise (given by the random realisations of κE) accounts for 80% of the κE signal, underlining the importance of accounting for shape noise in our modelling (as described in section 3.2). The shape noise dominates the signal most at small scales, accounting for 89% of the signal at 10 and dropping to 64% at 40 . We can see that the κB signal is also a good estimate of the shape noise, with the standard deviation of the κB signal agreeing with the rms standard deviation of κE random realisations within 2%. These standard deviations are calculated via CiC (see section 6.3 for a prescription for calculating moments from CiC) and errors are from jackknife sampling (see section 4). . Distribution of MICE κ WL at an angular scale of 10 when DESlike shape noise is added. An estimate of the width of the shape noise distribution is obtained by fitting a Gaussian to the 100 random realisations of DES κ WL . A noise contribution drawn from a Gaussian of this width is added to MICE κ WL at the level of the cells used to construct the CiC distribution. The darker, narrow histogram is that of the shape noise free MICE κ WL ; the lighter histogram shows the distribution once the Gaussian shape noise is added; the black dashed line shows the observed distribution of DES κ WL . The Gaussian width of the DES shape noise estimate, σn is 0.0099 at this scale, which is 89% of the width of the resulting noisy MICE κ WL distribution.
MICE Simulations
We validate our measurement of CiC from DES SV data using a special set of mock catalogues produced from N-body simulations for the DES collaboration. These come from the Marenostrum Institut de Ciències de l'Espai Grand Challenges (MICE-GC hereafter) lightcone N-body simulation and associated halo catalogue.
These simulations have been used to produce mock galaxy catalogues for ∼200 million galaxies over 5000 deg 2 up to a redshift of z = 1.4. There are also shear estimates for each galaxy made by ray-tracing through the N-body simulations. Every galaxy has a κWL value assigned from the integrated dark matter field.
The simulations are made with 4096 3 particles of mass For use in this paper we have reduced the effective number densities in the mock galaxy and shear catalogues to reflect the statistics of the DES SV samples as well as normalising the redshift to reflect the distribution shown in Fig. 2. Each mock catalogue is projected onto a HEALPix map of N side = 8192, which is then degraded to match the resolution of our data maps where appropriate.
In order to be able to compare the distribution of DES κWL (which we know has a significant shape noise contribution) with simualtions, we create a second MICE κWL sample that has DESlike shape noise added. An estimate of the width of the shape noise distribution is obtained by fitting a Gaussian to the 100 random realisations of DES κWL. A noise contribution drawn from a Gaus-sian of this width is added to MICE κWL at the level of the cells used to construct the CiC distribution. Figure 4 shows the effect of adding shape noise to MICE κWL in this way at an angular scale of 10 . The darker, narrow histogram is that of the shape noise free MICE κWL; the lighter histogram shows the distribution once the Gaussian shape noise is added; and the black dashed line shows the distribution of DES κWL. At a smoothing scale of 10 arcmin the Gaussian width of the DES shape noise estimate is 0.0099, which is 89% of the width of the resulting noisy MICE κWL distribution; this falls to 63% at 40 .
METHOD
Constructing PDFs via Counts-in-Cells
The CiC approach is a relatively simple way to measure the distribution of galaxies in a survey, but it is a surprisingly powerful tool. A general CiC distribution for galaxies can be denoted by f (N, V ), the probability of finding N galaxies in a volume of space V . This can be a 3D volume or, as is the case in this paper, a 2D area on the sky where we count over a population projected along the line of sight. Repeating this procedure with cells of varying radii, r, gives us the distribution fr(N ), where the moments of fr(N ) are related to the volume integrals of the correlation functions of our underlying observable (Peebles 1980;Fry 1985;Saslaw 2000;Fry & Gaztanaga 1994).
We perform our CiC analysis on the galaxy density contrast and weak lensing covergence maps with HEALPix pixelisation of resolution Nside = 4096, which corresponds to an average pixel size of 0.9 . For galaxies, to construct the PDF we sum the galaxy counts, N , inside 2D circular cells of fixed radius r in the range 10-40 . At the median redshift, z = 0.3, of the sources considered this corresponds to physical scales of 3-10 Mpc. The smallest cells used are 10 times larger than the HEALPix pixels in order to minimise edge effects, and this also avoids any difference in counts across our survey area due to the changing geometry of the HEALPix pixels (see Appendix C for a discussion of this assumption). We chose to use randomly positioned circular cells rather than using the HEALPix pixels themselves as this allows us to repeat the analysis straightforwardly at any smoothing scale, rather than using only the fixed scales of HEALPix pixels. The criterion for accepting a cell is that 80% of its area should fall in unmasked regions (again see Appendix C for discussion of this choice). We want to use enough cells that all pixels in the map are covered at least once, and find that this is achieved when the total area of the cells is 20 times that of the survey. We use a coverage of 100 times the total area.
Histograms of the counts give us the distribution f (N ), and this procedure is repeated with cells of different radii to obtain the distribution fr(N ). Double counting of pixels is accounted for by jackknife errors on the height of each bin in the resulting histogram of counts. We divide the survey area into 152 approximately equal area (1 deg 2 ) jackknife patches. For a fixed set of randomly generated cells, and removing one patch at a time, we re-make the galaxy and convergence PDFs and re-calculate the statistics of interest in order to produce covariances.
We repeat our CiC analysis on the DES reconstructed κ maps. The 'count' in each cell is now the average of the weak lensing convergence κ in pixels contained in that cell.
In Appendix B we test the impact of spatially varying systematic effects on the DES δg and κWL CiC distributions.
It is straightforward to generalise our CiC method to more than one observable. We simply throw the same circles onto each map (using the same mask for each), allowing us to compare counts at the same position for different observables.
Fitting the PDFs
We fit the lognormal models described in section 2 to these distributions. For the MICE and DES galaxy density contrast distributions we fit a Poisson sampled lognormal using equation 3. For MICE κWL, which has no shape noise, we fit a plain lognormal model (equation 6). For the κWL distributions which include shape noise (i.e. DES κWL and the MICE κWL to which we add shape noise), we use equation 9.
The histogram bins in δg or κWL are correlated. This is demonstrated in figure 5, which shows the correlation matrix of bin heights of DES κWL at a smoothing scale of 10 . In fitting the lognormal model we take into account these correlations by minimising
χ 2 = ( f − d)C −1 ( f − d) .(18)
Here f is the data vector of the lognormal fit at the bin centres, d is the data vector of bin heights, and C is the covariance matrix. We remove weak eigenvectors of the covariance matrix via singular value decomposition.
VALIDATING METHODS ON MICE
In this section we verify the methods used to test the lognormality of DES δg and κWL fields. After checking that the MICE δg field is lognormal as we would expect, we see if this is true of the noiseless convergence field.
To enable easier comparison with the DES κWL results we also look at the distribution of the simulation κWL for the MICE sample with number of galaxies and n(z) matched to our DES sample, and with DES-like shape noise added. We then look at the joint distribution of δg and κWL, for the cases with and without shape noise.
As an additional check of the validity of the lognormal model, we compare the second moments of the distributions as calculated via CiC with those derived under the assumption of lognormality.
Testing Lognormality of MICE Density and Convergence Fields
One-dimensional PDFs and log-normal fits
We first construct a simple histogram of δg from the CiC to estimate the 1D PDF of δg. The histogram uses 50 bins and we calculate jackknife errors on the bin heights as described in the previous section. The result for cells of radius of 10, 15 and 30 is shown in the upper panel of Fig. 6. We fit a Poisson sampled lognormal distribution as described in eqn. 1 with w as the single free parameter. The best-fit lognormal, which minimises χ 2 , is shown as a solid black line and the best fit Gaussian (magenta) is shown for comparison. At a cell size of 10 (corresponding to about 3 Mpc at the median redshift z = 0.3) it is clear that the lognormal model fits the data better, reflecting the non-linear clustering at this scale. The counting of information inside a cell can be thought of as a form of smoothing where the cells form a top-hat filter with a fixed size. As the size of our cells increases we average information on increasingly large scales and lose sensitivity to the effects of non-linear clustering on small scales. The lognormal distribution is designed to capture some of the information present as a result of non-linear evolution, so we would expect it to become less pronounced as the effective smoothing scale increases. This is indeed the case: at a cell radius of 10 the lognormal model is highly favoured, with a χ 2 /DOF = 1.13, compared to 9.66 for the Gaussian. At a cell size 30 (corresponding to a physical scale of 8Mpc at the median redshift) the distribution has become much more Gaussian with best-fit χ 2 /DOF for the Gaussian model now 1.50. The lognormal model is still favoured at this scale, with best-fit χ 2 /DOF = 0.95.
The result for the MICE κWL PDF is shown in the middle panel of fig. 6. Since there is no shape noise in the simulation we fit a plain lognormal, shown by the black line. As discussed in section 2.2, in order to fit a lognormal model to κWL one must assign a value to κ0, the minimum convergence parameter in equation 6. At 10 we jointly fit κ0 and the lognormal width in equation 6, finding best-fit κ0 = 0.021. For larger scales we find that it is not possible to jointly constrain κ0 and the width of the lognormal as they are degenerate. We therefore use the theoretically derived κ0 = 0.050, described in section 2.2.
Since the convergence is the weighted sum of the mass fluctuations along the line of sight we expect it to be only approximately lognormal. At a smoothing scale of 10 the lognormal is a good fit, with χ 2 /DOF= 1.19, and it is significantly preferred to the Gaussian model, which has a best-fit χ 2 /DOF= 14.43. This lognormality of κWL at small scales is in line with Taruya et al. (2002) who found that a lognormal model was a good fit to simulated κWL at angular scales of 2 -4 . Increasing the cell radius above 10 removes the clear preference for the lognormal, and the lognormal and Gaussian models fit the data equally well at cell radii of 30 . The fixed, theoretically derived κ0 = 0.050 allows the lognormal model with a single free parameter to fit the distribution well at 15 , but at larger cell radii this model does very slightly worse than the Guassian model. This suggests that this value of k0 may not be a good estimate for the minimum κ in the CiC PDF for larger cells. This makes sense as this κ0 corresponds a pure void along the line of sight, which is a decreasingly likely observation as the cell radius increases.
The final row of figure 6 shows the distribution of κ using the sub-sample of MICE with DES-like galaxy density and n(z), and to which DES-like shape noise has been added, as described in section 3.3. The shape noise dominates the resulting distribution, particularly at smaller scales. The width of the distribution of shape Figure 6. UPPER ROW: measured 1D PDF of MICE galaxies at a smoothing scales of 10, 15 and 30 . The Poisson sampled lognormal fit (black) provides a better fit to the galaxy CiC distribution than the Gaussian (magenta) at a scale of 10 . The distribution becomes increasingly Gaussian at larger scales. MIDDLE ROW: same as above but for the MICE κ WL PDF. Again the lognormal provides a good fit at the smallest scale, with the κ WL distribution becoming more Gaussian at larger scales. BOTTOM ROW: Fits to κ WL using the sub-sample of MICE with DES-like galaxy density and n(z), and to which DES-like shape noise has been added. This shape noise makes the distribution of κ WL more Gaussian at all scales. All χ 2 are per degree of freedom. noise is 74% of the width of the noisy κ distribution at 40 , and at 10 it accounts for 89%. We model the noisy κ distribution with a lognormal convolved with Gaussian noise as described in section 2.2, using equation 9. Again we find that it is not possible to jointly constrain κ0 and the width of the lognormal at scales above 10 as they are degenerate. We therefore use the theoretically derived κ0 = 0.049. It can be seen from figure 6 that at all scales the shape noise makes the noisy κ distribution much more Gaussian.
Despite the low signal to noise, at 10 the lognormal convolved with Gaussian noise provides a better fit than the simple Gaussian, with χ 2 /DOF= 1.06 and 1.56 respectively. At scales larger than this the Gaussian model performs as well as the lognormal. As with the noise free convergence distribution, the theoretically derived κ0 seems to be a less suitable choice at larger scales as the Gaussian model provides a better fit for scales above 30 .
Joint galaxy-convergence distribution
In this sub-section we study the joint distribution of galaxy overdensities and weak lensing convergence and determine to what extent it can be described as a bivariate lognormal distribution. We look at joint distributions using both the full MICE sample, and the subsample with DES-like galaxy density and n(z) and the addition of DES-like shape noise. As in the 1D case, the full sample with higher galaxy density allows us to better capture any lognormal behaviour, and the DES-like sample allows us to compare the results for DES data given in the next section with simulations.
We can make a simple quantitative estimate of the relative correlation of δg and κWL by calculating the Pearson product-moment correlation coefficient, r, for the joint PDF, where
ρX,Y = Cov(X, Y ) σX σY = (X −X)(Y −Ȳ ) σX σY .(19)
We begin with the joint distribution of δg and κW L with no shape noise, which is shown in the upper panel of Fig. 7 for a smoothing scale of 15 . The blue contours in the top right section of this plot show the joint PDF, and the dashed magenta contours show the bivariate lognormal fit. Since there is no shape noise in this case the bivariate fit is given by equation 15 but omitting the Gaussian convolution. We expect the correlation coefficient ρ to be high (close to one) since the galaxies considered are responsible for the lensing. This is indeed what we see: the Pearson correlation coefficient is 0.81 at a smoothing scale 10 and 0.89 at 40 . We do not see full correlation because the relevant window functionsthe lensing efficiency function of the source sample and the galaxy redshift distribution of the galaxy sample -do not overlap precisely.
The lower panel of this figure shows the case where MICE κWL has had shape noise added. This noise reduces the correlation of the κWL with δg, smearing out the joint distribution (shown on the top right of the figure) versus the shape noise free case. The Pearson correlation coefficient is reduced to 0.45.
Comparison of Moments
We can use the second moments to check the validity of the lognormal modelling by comparing the moments derived directly from the CiC with those derived by fitting a lognormal model to the CiC PDF. The second moments of the MICE galaxy and convergence fields δ 2 g and κ 2 can be calculated from the CiC (as described in Appendix A). The moments derived under lognormal modelling are given by equations 2 and 8.
First we calculate the variance of the MICE galaxy PDF, shown in the first panel of Fig. 8. Blue data points show the ratio of the variance δ 2 g from fitting a lognormal to the CiC PDF and that calculated directly from the CiC. Errors on δ 2 g directly from CiC are produced by jackknife sampling; errors on the δ 2 g derived from the lognormal fit are from the 1σ width of the likelihood of the lognormal width.
The lognormal model gives a good estimate of the variance of the MICE galaxy density contrast distribution (with Poisson shot noise accounted for) at all scales. It gives a better estimate of the variance than a Gaussian model at all scales, and particularly at 10 . The lognormal model also gives a good estimate of the variance of the weak lensing convergence distribution at scales up to 20 . The poorer estimates at 30 and 40 are due to the fact that we fix κmin to the theory value at these scales.
These results suggest that within the ranges of scales discussed, the lognormal model can be used to estimate the two point statistics of both the galaxy density contrast and weak lensing convergence distributions to reasonable accuracy in these simulations. Contours for the simulation are given by the solid blue lines, with dashed magenta contours for the fit. Also shown are the 1D PDFs for 1 + δg and 1 +κ W L individually. PDFs are calculated via the CiC method with cells of radius 15 . As in the rest of this paper, galaxies are selected over the redshift range 0.1 < z < 0.5 and WL sources are restricted to the range 0.6 < z < 1.3. This joint distribution has a Pearson correlation coefficient of r = 0.83. LOWER PANEL: Same but with DES-like shape noise added to κ WL . The Pearson correlation coefficient drops to 0.45 with the addition of this shape noise. g from our fits to the 1D lognormal distribution to that calculated directly from the CiC PDF; black data points show the same but for the Gaussian fit. Data points are offset slightly in scale for clarity. LOWER PANEL: same but for shape noise free MICE weak lensing convergence.
TESTING LOGNORMALITY OF DES DENSITY AND CONVERGENCE FIELDS
Here we repeat the analysis of the previous section with DES galaxy and convergence maps, looking first distributions individually and then at their joint distribution.
6.1 One-dimensional PDFs and log-normal fits Fig. 9 shows 1D CiC PDFs for the DES galaxy density contrast (top row) and κWL (second row) fields for different cell radii. The lognormal fit is again shown in black, and for comparison a Gaussian fit is shown in magenta. For the δg PDF at 10 the lognormal model is clearly favoured, with χ 2 /DOF= 1.28 compared to 6.55 for the Gaussian model. This confirms the expected lognormal behaviour at non-linear scales, indicating that our CiC procedure is capturing non-linear clustering information beyond the Gaussian assumption at smaller radii. As in the simulations the δg PDFs clearly appear more Gaussian at larger cell radii, although the lognormal model still provides a better fit than the Gaussian even at 30 , with χ 2 /DOF of 0.97 and 1.82 respectively.
The best-fit values of the free parameters of the lognormal fits to the DES galaxy density contrast distribution, the χ 2 , the number of degrees of freedom (DOF) and the second moment of the bestfit lognormal PDF are given in table 1, for smoothing scales of 10 -40 .
The second row of Fig. 9 shows the DES κWL distribution. We find that it is possible to jointly constrain κ0, the minimum convergence parameter in equation 6, and the width of the lognormal at Table 1. Best-fit parameters and derived statistics from lognormal fits to CiC PDFs of DES galaxy density contrast, for varying cell radii. First first column gives the cell radius, and the second column is the width of the best fitting Poisson-sampled lognormal. The following columns are the minimum χ 2 for Gaussian and Poisson sampled lognormal fits, and the number of degrees of freedom. The final column is the second moment of the bestfitting lognormal PDF, derived from the lognormal width, with 1σ errors given by the likelihood of the lognormal width. Table 2. Same as Table 1 but for DES weak lensing convergence. The lognormal fit accounts for shape noise, so the statistics quoted are for the de-noised κ W L distribution. The additional information given vs. Table 1 , in the second column, is the best-fit minimum convergence parameter
r, arcmin κ 0 σ χ 2 G χ 2 LN DOF κκ × 10 −κ 0 = −κ min .
all smoothing scales. The best-fit values of κ0 as well as the lognormal width σ, best-fit χ 2 and the second moment of the best-fit lognormal PDF are given in table 2. The best-fit κ0 = 0.021 and σ = 0.235 at cell radius 10 are in good agreement with the results from the MICE simulation at this scale, which are 0.023 and 0.226 respectively. Note that for larger scales we fix κ0 at the theory value of 0.05 in the simulations, so would not expect close agreement of the best-fit lognormal width with that of the data at the these scales.
The κWL distribution appears quite Gaussian at all scales due to the Gaussian shape noise, the distribution of which has a width of 70-90% of the width of the κWL distribution. Despite this low signal to noise, as in the case of simulated κWL, we find that the lognormal model with Gaussian shape noise (black line) provides a better fit than the simple Gaussian model (magenta line) at small scales. At 10 the lognormal model has χ 2 = 1.11 and the Gaussian 1.84, corresponding to p-values of 0.35 for the lognormal model (i.e. within one σ) and 0.07 for the Gaussian model. At 15 the advantage of the lognormal model over the Gaussian is clear, with best-fit χ 2 /DOF of 1.01 and 2.13 respectively. At scales larger than this the Gaussian model provides a good fit with best-fit χ 2 /DOF of 1.09, 1.06 and 1.14 at 20, 30 and 40 . The lognormal model is over-fitting the data at these scales, with χ 2 /DOF of 0.46, 0.57 and 0.66 at the same scales, so the Gaussian model is sufficient in this regime.
Joint galaxy-convergence distribution
The joint distribution of DES galaxy density contrast and weak lensing convergence data at an angular scale of 15 is shown in the top right panel of fig. 10. The data are shown by the blue contours, and the bivariate fit is shown by the dashed magenta contours. The individual 1D PDFs for 1 + δg and 1 + κ/κ0 are also shown. Figure 9. UPPER ROW: measured 1D PDF of DES galaxies at a smoothing scales of 10, 15 and 30 . At 10 the Poisson sampled log-normal fit (black) provides a much better fit than the Gaussian (magenta), demonstrating the log-normality of the galaxy CiC distribution at this scale. At larger scales the distribution becomes more Gaussian. LOWER ROW: same but for κ WL . Here the lognormal model includes Gaussian shape noise, which provides a good fit at all scales. Error bars on the counts PDFs are jackknife errors. All χ 2 are per degree of freedom.
Before we account for shot noise in the galaxies and shape noise in the convergence, the galaxy counts and κWL have a Pearson correlation coefficient of 0.45. This is in line with what we see in the MICE simulations once DES-like shape noise is added (bottom row of Fig. 7).
Once we account for these sources of noise, the correlation coefficient is 0.82, again in line with the noise-free MICE simulations, where the Pearson correlation coefficient was 0.83 (top row of Fig. 7).
Comparison of Second Moments
In this section we check the validity of the lognormal model by comparing second moments derived from the log-normal assumption with those measured directly from the data.
The variance of the DES galaxy PDF is shown in the first panel of Fig. 11. Blue data points show the ratio of the variance δ 2 g from our fits to the 1D lognormal distribution to that calculated directly from the CiC PDF. Errors on κ 2 directly from CiC are produced by jackknife sampling; errors on κ 2 derived from the lognormal fit are from the 1σ width of the likelihood of the lognormal width. The second panels shows the same for DES κWL. The lognormal model with appropriate noise contribution gives an estimate of the variance that is consistent with that calculated directly from the CiC, for both galaxies and κWL, at all scales from 10 -40 .
For the galaxy density contrast distribution, the Gaussian model provides a less accurate estimate of the variance calculated directly from the CiC at all scales. For the convergence distribution the Gaussian model again gives variance estimates less accurate than the lognormal model at all scales.
For both galaxies and weak lensing convergence, the Gaussian and lognormal approaches underestimate the variance as compared to measuring it directly from the CiC. This is because in constructing the CiC PDF to which we fit the lognormal model, we bin the cell counts. We account for noise via singular value decomposition, and one of the things this removes is contributions to the fit from the outermost bins, which have very few cell counts. This makes the effective distribution narrower, with lower second moment, than if these noisy data points were included. In calculating the variance directly from the CiC (as described in Appendix A) this binning is not necessary and all cells, including those with the most extreme values of δg or κWL, are included in the calculation, resulting in a larger variance in δg or κWL. This effect is less stark in the MICE simulations where there are a greater number of galaxies than in the DES data, so fewer bins are discarded due to low counts of cells. This underestimation of the variance, however, is not significant within the errors.
DISCUSSION
We have tested the lognormality of the DES galaxy density contrast and weak lensing convergence PDFs at angular scales of 10 -40 (corresponding to physical scales of 3 -10 Mpc at median redshift (1.65 ± 0.14) × 10 −2 (9.84 ± 1.66) × 10 −6 (1.18 ± 0.06) × 10 −5 40
(1.38 ± 0.16) × 10 −2 (8.40 ± 1.30) × 10 −6 (6.75 ± 0.44) × 10 −6 Table 3. Second moments of DES galaxy density contrast and weak lensing convergence, as calculated by CiC, for different cell radii. Shot and shape noise have been accounted for, and these are the de-noised moments. The final column gives our estimate the shape noise of the weak lensing convergence. This is derived from the 100 realisations of the κ WL map with randomised shears, which we find to agree with the second moment of the κ WL B-mode within 2%. In the context of this work, estimating the CiC PDF is a way of quantifying the non-linear growth of mass and galaxy fluctuations, as well as the visual impression of comparing the κWL mass maps with the galaxy distribution on the same patch of the sky. It is also a test of systematics. Our main findings are as follows:
• In agreement with many earlier papers we find that the 1D DES galaxy PDF is well fitted by a lognormal model, taking into account Poisson shot noise, with best-fit χ 2 /DOF= 1.28 vs. 6.55 for a Gaussian model at a scale of 10 .
• In modelling the weak lensing convergence distribution it is important to account for shape noise since the width of this noise is a significant fraction (70-90%) of the width of the κWL signal. We find that the shape noise estimate derived from the 100 realisations of DES κWL in which the shears have been randomised agrees with Figure 11. Same as figure 8, but for DES galaxies (upper panel) and convergence (lower panel). that of the κB mode within 2% at all scales from 10 to 40 , and that the distribution of the shape noise can be well modeled by a Gaussian PDF. This allows us to model the κWL distribution with a lognormal convolved with Gaussian PDF. In future work it would be interesting to investigate the spatial correlation of this noise.
• The convergence field is not expected to be exactly lognormal even if the mass density contrast field is, as it is a weighted projection of the mass density field along the line of sight. We find however, in agreement with previous work on simulations, that the 1D κWL PDF is well fitted by a lognormal model, taking into account shape noise. This is the first such measurement on data. The best-fit χ 2 /DOF for the lognormal model is 1.11, compared to 1.84 for a Gaussian model, corresponding to p-values of 0.35 (i.e. within one σ) and 0.07 respectively. At scales above 15 the Gaussian model is a sufficient approximation.
• The bivariate (κWL, δg) PDF is also well fitted by a bivariate lognormal.
• De-noised second moments derived via the lognormal fit are consistent with variances derived directly from the data up to scales of 40 , for both the DES galaxy density contrast and weak lensing convergence distributions. This pilot study could be extended to much larger areas with weak lensing surveys such as the full DES (5000 deg 2 ) survey, LSST (20,000 deg 2 ) and Euclid (15,000 deg 2 ). In this work we have tested the lognormality of the κWL PDF; with the higher signal/noise that future surveys will provide it might be possible to deduce from the observed κWL PDF whether or not the underlying matter density field is lognormal -essentially inverting equation 4.
In this work we have used the CiC to probe lognormality, but there is a wealth of information contained within it that could be exploited in future work. The CiC contains the full PDF so as well as the variance, higher order moments such as skewness and kurtosis can also be extracted.
The method used in this work, of constructing PDFs via CiC and cross correlating them, could be used to extract information on galaxy bias and to derive cosmological parameters. It could also be interesting to repeat this analysis using manipulations of the shear field than other κWL that avoid the reconstruction noise due to the Kaiser Squires method.
Quantifying P (κWL) will be important for the emerging field of mass reconstruction using κWL, since it is required as a prior input for this process. We have demonstrated that a lognormal model is a better choice than a Gaussian model at scales of 10 -20 . As well as the improved ability to capture non-linear behaviour versus a Gaussian model, the lognormal model still allows fast production of, for example, simulated realisations of the convergence field for testing, and covariance matrices.
APPENDIX A: MEASUREMENT OF MOMENTS FROM COUNTS-IN-CELLS
In section 5 we use the second moment, as calculated via CiC, as a check that the lognormal model accurately recovers the characteristics of the galaxy and κWL distributions. In this section we show how these are calculated for the galaxy and weak lensing convergence distributions, including how noise is accounted for.
Moments of a distribution are easily obtained via the CiC technique, with the p th central moment of the distribution of the number of objects n at angular scale θ given by:
mp(θ) = 1 N (θ) nc(θ) i=1 (xi(θ) −x) p (A1)
where N (θ) is the number of cells of angular size θ used,x is the mean count of observable x in a cell, and xi is the count in cell i. For the distribution of galaxies xi is the number of galaxies in a cell, and for the convergence xi is the average κ within a cell. The connected central moments µp(θ) can be derived using the moment generating function (see 3.2.4 of Bernardeau et al. 2002 for a derivation and nice diagrammatic representation of the connected moments). The second connected moment is equal to the second central moment, µ2 = m2.
For the galaxy distribution, shot noise can be accounted for by assuming that galaxies form a Poisson sampling of the underlying matter density field:
P (λ) = ∞ n=0 λ n P (n).(A2)
Taylor expanding this around λ = 1 gives n(n − 1)...(n − p − 1) =n p kp where kp is the p th moment of the local density distribution. The 'de-noised' second moment is:
k2 = µ2 −n,(A3)
wheren is the mean number of galaxies in a cell. The area averaged correlations are then given bȳ
wp = kp np ,(A4)
For both the MICE and DES galaxy distributions, the second moment as calculated via CiC, with shot noise removed, is given by equations A1 -A4.
For κW L, since there is no need to model shot noise, k2 = µ2. The second moment for MICE κW L is then given by equation A4. In the case of DES κ we need to remove shape noise. The shape noise in the DES κW L map is estimated from the 100 noise realisations, as discussed in section 3.2. Following Van Waerbeke et al. 2013, we assume the de-noised second moment of DES κW L is then given bȳ
w2 =w 2,data −w2,noise(A5)
wherew2,noise is given by equations A1 -A4 (with k2 = µ2), andw2,noise is the mean of the second moments measured via CiC from each of the noise maps. The second moment, estimated jointly from two distributions becomes
m2(θ) = 1 N (θ) nc(θ) i=1 (n δ,i (θ) −n δ ) (κi(θ) −κ) (A6) =n δnκ (θ) δi(θ)κi(θ) (A7)
APPENDIX B: SYSTEMATIC EFFECTS
We investigate the potential impact on our results of spatially varying systematics. The systematics we consider are varying the amount of air mass dependent on the distance of the observed field from the zenith, exposure time, magnitude limit, atmospheric seeing, and sky brightness. The values of these properties were mapped across the DES-SV area as described in Leistedt et al. (2015). We compare PDFs of δg and κWL for the full samples used in this work versus when the areas worst-affected by these systematics are removed. We produce PDFs with the 20% of worst-affected pixels masked, for each systematic in turn. Figure B1 shows the resulting distributions. The top row shows DES galaxy number density at cell radii of 10, 15, 30 (blue data points) with jackknife errors. Coloured lines show PDFs with cuts for each systematic effect in turn. The lower panel shows the fractional difference between the full sample and those with systematics cuts, with errors shown by the grey shaded region. The same for DES κWL is shown in the bottom row.
Here we can see that the PDFs of the cut data are broadly consistent with that of the full data, given the jackknife errors. For DES galaxies, for each systematic effect at least 95% of the bin heights after the cuts are made fall within the jackknife errors of the original distribution, and all are within 1.5 sigma of the original distribution. For DES κWL, at least 93% of the new bin heights fall within the jackknife errors of the original distribution. All are Figure B1. UPPER ROW: PDF of DES galaxy number density at cell radii of 10, 15, 30 (blue data points) with jackknife errors. Coloured lines show PDFs with cuts for each systematic effect in turn. Lower panel shows the fractional difference between the full sample and those with systematics cuts, with errors shown by the grey shaded region. LOWER ROW: Same but for DES κ W L . within 1.9 sigma of the original data points. We can see that the effect of the systematics on the distribution increases with scale. Importantly PDFs of the cut κWL data at scales below 20 , which is where we detect lognormality of κWL, are completely consistent with the original distributions, i.e. all of the new bin heights fall within the errors of the original distribution. This simple test is reassuring and indicates that our lognormal fits to the DES δg and κWL distributions are not likely to be affected by these systematic effects.
APPENDIX C: TESTS OF SAMPLING METHODS
Our CiC analysis has made particular choices for cell size and distribution when accounting for the mask and creating the underlying HEALPix maps. In this Appendix we test each of these assumptions and demonstrate that the conclusions of our analysis are robust to our methodological choices.
HEALPix tessellations are made up of pixels with equal area, but not equal shape. Using circles that encompass several pixels will reduce the effect of the different pixel shapes, more so the larger the circles relative to the pixels. To check that the effect of varying shapes is effectively mitigated in this way we measure the area averaged 2-point correlationw2(θ) for DES galaxies at different HEALPix resolutions (512, 1024, 2048, and 4096). Figure C1 shows that when the cells size is close to the pixel size the correlation function is not smooth due to the effects of pixel shape. Once the cells are several times larger than the average pixel separation the correlation function becomes smooth, and the correlation func- Figure C1. Second moment of MICE galaxy density contrast distribution as a function of scale as calculated via CiC, using underlying HEALPix maps with different resolutions. The HEALPix maps have nside (see main text) 512, 1024, 2048, 4096 corresponding to pixels of sizes shown by the solid circles. At scales approaching the pixel size edge effects are visible. tions based on the different HEALPix resolutions converge. This confirms that the method of using circular cells several times larger than the pixel resolution does not suffer from the effects of pixel shape, and that the underlying pixel resolution is not important as long as the minimum cell size considered is sufficiently large. Figure C2. Left panel: Effect of the mask threshold (fraction of a cell that must be unmasked in order to be included in the analysis) on the resulting probability distribution of MICE galaxies. Middle Panel: Effect of this threshold on the second moment. Right panel: The fraction of the cells randomly thrown that are kept in the analysis, and, of those kept, the fraction of the data that comes from re-weighting these cells.
The other sampling assumption we test is the threshold at which we decide to discard a randomly positioned cell because too much of it is masked. If unmasked fraction of a randomly positioned cell is less than f , the counts are up-weighted to make it the equivalent of a whole cell. Here we explain the choice of f = 0.8 used in this work.
The first panel of figure C2 shows the PDF of MICE galaxy counts for which the f takes different values. For values of f > 0.5 there is not much difference in the histograms by eye. The middle panel shows the variances of these different distributions, with errors produced by jackknife sampling. We find that the effect on the variance of changing f is not significant within the errors.
If we chose a very high threshold, such as requiring 90% of a cell to be unmasked in order for it to be used, we would throw away a lot of cells landing near the edge of the survey and give greater statistical weight to areas away from the edges. If we set f too low, so that cells with a large fraction of their area masked are kept, we will end up re-weighting a lot the data nearer the edges. So we would like to strike a balance between these two effects. From the right panel of figure C2 we can see than f = 0.8 is the highest value that can be allowed before the number of cells we discard drops off significantly, and that at this value the fraction of data reweighted is not too high (around 10%). Hence f = 0.8 seems to be a sensible choice.
Unions Seventh Framework Programme (FP7/2007(FP7/ -2013 including ERC grant agreements 240672, 291329, and 306478. This paper has gone through internal review by the DES collaboration.
LC thanks the Perren Fund for a studentship. OL, DK and MM acknowledge support from a European Research Council Advanced Grant FP7/291329.
Figure 1 .
1Demonstration of the Gaussianity of the noise in DES weak lensing convergence, κ, at a smoothing scale of 20 . Probability density distributions of the 100 realisations of the κ map, in which shears were randomised, are shown in blue, with jackknife errors. A Gaussian PDF was fitted to each, with the mean of the best fitting PDFs shown in magenta. A Gaussian model is an excellent fit to the noise, with average goodness of fit χ 2 /DOF= 0.95 ± 0.73.
Figure 3
3Figure 3. Standard deviations of the κ E , κ B signal (magenta) and 100 realisations in which the shears have been randomised (blue) at a cell size of 20 . The random realisations of κ E give an estimate of the shape noise contribution to κ E ; this accounts for 80% of the κ E signal. The κ B signal is also a good estimate of the shape noise, with the standard deviation of the κ B signal agreeing with the rms standard deviation of κ E random realisations within 2%. These standard deviations are calculated via CiC and errors are from jackknife sampling.
Figure 4
4Figure 4. Distribution of MICE κ WL at an angular scale of 10 when DESlike shape noise is added. An estimate of the width of the shape noise distribution is obtained by fitting a Gaussian to the 100 random realisations of DES κ WL . A noise contribution drawn from a Gaussian of this width is added to MICE κ WL at the level of the cells used to construct the CiC distribution. The darker, narrow histogram is that of the shape noise free MICE κ WL ; the lighter histogram shows the distribution once the Gaussian shape noise is added; the black dashed line shows the observed distribution of DES κ WL . The Gaussian width of the DES shape noise estimate, σn is 0.0099 at this scale, which is 89% of the width of the resulting noisy MICE κ WL distribution.
2927M h −1 in a box of side 3072 h −1 Mpc. The MICE-GC has an assumed flat ΛCDM cosmology: Ωm = 0.25, Ω b = 0.044, ΩΛ = 0.75, σ8 = 0.8, h = 0.7, ns = 0.95. The MICE-GC DES mocks approximately reproduce the magnitude limits of the DES survey and are complete down to apparent magnitude i < 22.0 at z = 0.5.
Figure 5 .
5Correlation matrix of bin heights for a histogram of DES κ WL , at a smoothing scale of 10 . Derived from jackknife sampling of the DES κ WL map.
Figure 7 .
7UPPER PANEL: Joint CiC distribution of weak lensing convergence and galaxy density contrast for MICE simulation at a smoothing scale of 15 . The top right plot shows the bivariate lognormal fit to MICE simulations.
Figure 8 .
8UPPER PANEL: comparison of second moments of MICE galaxy density contrast as a function of smoothing scale, directly measured via CiC and from lognormal and Gaussian fits to the CiC PDF. Blue data points show the ratio of the variance δ 2
Figure 10 .
10Joint distribution of weak lensing convergence and galaxy density contrast for DES at smoothing scale 15 . Upper right panel: Fit of bivariate lognormal to DES SV data. Contours for the data are given by the solid blue lines, with dashed magenta contours for the fit. Also shown are the individual 1D PDFs for 1 + δg and 1 +κ W L . DES Benchmark galaxies are used, selecting the redshift range 0.1 < z < 0.5 and WL sources from the imshape catalogue are used over the range 0.6 < z < 1.3. All redshifts are best-fits from the Skynet pipeline. PDFs are calculated via the CiC method with cells of radius 15 . This joint distribution has a Pearson correlation coefficient of r = 0.45. z = 0.3).
number https://github.com/esheldon/ngmix1 The
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ACKNOWLEDGEMENTSThe authors would like to thank Ludovic Van Waerbeke for extremely useful exchanges on the formalism for the calculation of the convergence second moment.We are grateful for the extraordinary contributions of our CTIO colleagues and the DECam Construction, Commissioning and Science Verification teams in achieving the excellent instrument and telescope conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the DES Data Management group.Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Founda-
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. Astrophysics Group, Department of Physics and Astronomy. 1Astrophysics Group, Department of Physics and Astronomy, University College London, 132 Hampstead Road, London, NW1
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. PO Box. 94Department of Physics and Electronics, Rhodes UniversityDepartment of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa
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. P. O. Box. 500Fermi National Accelerator LaboratoryFermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA
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|
[
"https://github.com/esheldon/ngmix1"
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[
"Mixed Virtual Element approximation of linear acoustic wave equation",
"Mixed Virtual Element approximation of linear acoustic wave equation",
"Mixed Virtual Element approximation of linear acoustic wave equation",
"Mixed Virtual Element approximation of linear acoustic wave equation"
] |
[
"Franco Dassi ",
"Giuseppe Vacca ",
"\nDipartimento di Matematica e Applicazioni\nUniversità degli Studi di Milano Bicocca\nVia R. Cozzi, 5520125MilanoItaly\n",
"\nDipartimento di Matematica\nALESSIO FUMAGALLI MOX\nPolitecnico di Milano\nP.za L. da Vinci, 3220133MilanoItaly\n",
"\nDipartimento di Matematica\nDipartimento di Matematica\nILARIO MAZZIERI* MOX\nPolitecnico di Milano\nP.za L. da Vinci, 32, Università degli Studi di Bari, via E. Orabona, 420133, 70125BariMilanoItaly, Italy\n",
"Franco Dassi ",
"Giuseppe Vacca ",
"\nDipartimento di Matematica e Applicazioni\nUniversità degli Studi di Milano Bicocca\nVia R. Cozzi, 5520125MilanoItaly\n",
"\nDipartimento di Matematica\nALESSIO FUMAGALLI MOX\nPolitecnico di Milano\nP.za L. da Vinci, 3220133MilanoItaly\n",
"\nDipartimento di Matematica\nDipartimento di Matematica\nILARIO MAZZIERI* MOX\nPolitecnico di Milano\nP.za L. da Vinci, 32, Università degli Studi di Bari, via E. Orabona, 420133, 70125BariMilanoItaly, Italy\n"
] |
[
"Dipartimento di Matematica e Applicazioni\nUniversità degli Studi di Milano Bicocca\nVia R. Cozzi, 5520125MilanoItaly",
"Dipartimento di Matematica\nALESSIO FUMAGALLI MOX\nPolitecnico di Milano\nP.za L. da Vinci, 3220133MilanoItaly",
"Dipartimento di Matematica\nDipartimento di Matematica\nILARIO MAZZIERI* MOX\nPolitecnico di Milano\nP.za L. da Vinci, 32, Università degli Studi di Bari, via E. Orabona, 420133, 70125BariMilanoItaly, Italy",
"Dipartimento di Matematica e Applicazioni\nUniversità degli Studi di Milano Bicocca\nVia R. Cozzi, 5520125MilanoItaly",
"Dipartimento di Matematica\nALESSIO FUMAGALLI MOX\nPolitecnico di Milano\nP.za L. da Vinci, 3220133MilanoItaly",
"Dipartimento di Matematica\nDipartimento di Matematica\nILARIO MAZZIERI* MOX\nPolitecnico di Milano\nP.za L. da Vinci, 32, Università degli Studi di Bari, via E. Orabona, 420133, 70125BariMilanoItaly, Italy"
] |
[] |
We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In absence of external load, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical θ -method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering application of the method.
|
10.48550/arxiv.2209.11603
|
[
"https://export.arxiv.org/pdf/2209.11603v1.pdf"
] | 252,519,248 |
2209.11603
|
1bbd1bcad8ba005447158d72a69f4d96c6d02697
|
Mixed Virtual Element approximation of linear acoustic wave equation
Franco Dassi
Giuseppe Vacca
Dipartimento di Matematica e Applicazioni
Università degli Studi di Milano Bicocca
Via R. Cozzi, 5520125MilanoItaly
Dipartimento di Matematica
ALESSIO FUMAGALLI MOX
Politecnico di Milano
P.za L. da Vinci, 3220133MilanoItaly
Dipartimento di Matematica
Dipartimento di Matematica
ILARIO MAZZIERI* MOX
Politecnico di Milano
P.za L. da Vinci, 32, Università degli Studi di Bari, via E. Orabona, 420133, 70125BariMilanoItaly, Italy
Mixed Virtual Element approximation of linear acoustic wave equation
[Received on Date Month Year; revised on Date Month Year; accepted on Date Month Year]ANDMixed Virtual Elementsacoustics wave equationspolygonal meshesenergy conservation
We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In absence of external load, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical θ -method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering application of the method.
Introduction
The numerical simulation of acoustic, elastic or electromagnetic wave propagation finds application in many scientific disciplines, including aerospace, geophysics, civil engineering, telecommunications, and medicine for instance.
The present work considers a Mixed Virtual Element method on general polytopal grids for the discretization of the acoustic wave equation written as a first order system of hyperbolic partial differential equations.
In general, mixed methods consider the discretization of vector fields in some H(div)-conforming spaces while scalar fields in some L 2 spaces. Classes of mixed methods include the well known Raviart-Thomas (RT) [9,58,61], the Brezzi-Douglas-Marini (BDM) [26,30,31,56] finite element schemes and more recently the Mixed Virtual Element Methods (MVEM) [15,39]. The VEM, introduced firstly in [14], have been applied recently to various differential problems, including elasticity [10], Stokes [20,33], Navier-Stokes [21], Cahn-Hilliard [2], Darcy [39], Helmholtz [53], Maxwell [18] and wave [4,40,62] equations.
The major benefit of using VEM, instead of classical approaches, is the fact that it gives the opportunity to preserve at the discrete level some important properties valid at the continuous level. In particular, it is possible to design discrete spaces with global high regularity, which preserve the arXiv:2209.11603v1 [math.NA] 23 Sep 2022 polynomial divergence/curl, that are robust with respect to mesh-locking phenomena. Moreover, VEM can handle general polytopal meshes, that are particularly useful to account for small features in the model (such as cracks, holes and inclusions), and treat in an automatic way hanging nodes, movable meshes and adaptivity.
The very first analysis of RT finite element discretization applied for the spatial approximation of the acoustic wave equation is presented by Geveci in [46]. He showed that even if the RT finite elements conserve the energy of the system, when a time discretization is applied, the fully-discrete method produces an implicit time-marching scheme which is inefficient, since the mass matrix is nondiagonal. For this reason it is preferred to use mass lumping techniques, see e.g. [13,37] or symplectic schemes that conserve a positive-definite perturbed energy functional [52]. A slightly different formulation, in which two time derivatives appear on the vector variable and none on the other equation has been analysed in literature, see, for instance, [34,38,49,50,51].
Discontinuous Galerkin (dG) methods have been also considered for the solution of the wave equation in the mixed form, see for instance [11,34,35,36,55]. However, they have been mostly studied for the second order hyperbolic version: we mention [48,60] for the scalar case, [5,8,42,43,44,54,59] for the vectorial case and [3,6,7] where a dG approximation on polygonal grids is considered. Within the framework of polytopal methods we mention a recent work by [32] where the Hybrid High-Order (HHO) method is applied to the wave equation in either first and second order form and [19] where the Mimetic Finite Differences is applied in the context of Hamiltonian wave equations.
Here, for the first time, a MVEM is considered for the solution of linear wave acoustics written as a system of first order partial differential equations. The analysis is carried out by taking inspiration from the approaches proposed in [17,22] and [39]. Concerning the choice of the degrees of freedom, the proposed scheme can be seen as the extension on polytopal grids of RT finite elements. The integration in time of the semi-discrte problem is achieved by considering a θ -method scheme.
The paper is organised as follows: in Section 2 we review the mathematical model, its weak formulation, a stability result and the energy conservation of the system. In Section 3 after introducing the virtual element spaces with the associated set of degrees of freedom and defining the discrete bilinear forms, we present the semi-discrete virtual element formulation and establish the well-posedness of the semi-discrete problem, the stability bound for discrete solution in a discrete energy norm and the energy conservation. In Section 4 we analyse the theoretical properties of the proposed method: by considering the Fortin operator introduced in [39], we recall the interpolation estimates. Then, we prove optimal order of convergence for the proposed method. Moreover we estimate the error between the energy of the exact solution and the energy of the virtual element solution. In Section 5 we introduce the family of θ -method schemes for integrating in time the semidiscrete problem in Section 3.4 and discuss the property of energy conservation. In Section 6 we provide some experiments to give numerical evidence of the behaviour of the proposed scheme. Finally, Section 7 is devoted to conclusions and future perspectives.
Notations and Preliminaries
Throughout the paper we will follow the usual notation for Sobolev spaces and norms as in [1]. Let Ω ⊂ R 2 be the computational domain with Lipschitz continuous boundary ∂ Ω and external unit normal n n n, we denote with x x x = (x 1 , x 2 ) the independent variable. With a usual notation the symbols ∇ and curl denote the gradient and curl for scalar functions, while div denotes the divergence operator for vector fields. For an open bounded domain ω, the norm in the space L p (ω) is denoted by · L p (ω) , norm and seminorm in H s (ω) are denoted respectively by · s,ω and |·| s,ω , while (·, ·) ω and · 0,ω denote the L 2 -inner product and the L 2 -norm (the subscript ω may be omitted when ω is the whole computational domain Ω ).
We recall the following well known functional spaces which will be useful in the sequel
H(div, ω) := {v v v ∈ [L 2 (ω)] 2 : div v v v ∈ L 2 (ω)} , H(curl, ω) := {v v v ∈ [L 2 (ω)] 2 : curl v v v ∈ L 2 (ω)} ,
and introduce the following spaces
V V V := {v v v ∈ H(div, Ω ) s.t. v v v · n n n = 0 on Γ N } , Q := L 2 (Ω ) ,
where Γ N ⊂ ∂ Ω , equipped with natural inner products and induced norms.
Since we are dealing with a time dependent problem, we will also consider the following Bochner spaces. Let T > 0, for space-time functions v(x x x,t) defined on ω ×(0, T ), we denote with v t the derivative with respect to the time variable. Furthermore, using standard notations [57], for a Banach space V with norm · V , we introduce the space
L 2 (0, T ;V ) := v : (0, T ) → V s.t. v measurable, T 0 v(t) 2 V dt < +∞ .
In similar way, for n ≥ 0, we consider the space C n (0, T ;V ).
Mathematical Model
Let Ω ⊂ R 2 be the polygonal domain. The boundary ∂ Ω is divided in three parts, with mutually disjoint interiors, denoted by Γ D , Γ N and Γ R , corresponding to Dirichlet, Neumann and Robin boundary conditions, respectively; one or two of them may be empty. In a time interval (0, T ], for a piece-wise constant positive real valued function c (representing the characteristic velocity of the medium), and a scalar source f , the following problem is set in Ω .
Problem 1 (Model problem) Find (u u u, p) such that u u u t (x x x,t) − ∇p(x x x,t) = 0 0 0 c −2 p t (x x x,t) − div u u u(x x x,t) = f (x x x,t) in Ω × (0, T ],
supplied with the following boundary conditions
p(x x x,t) = g D (x x x,t) on Γ D × (0, T ], u u u(x x x,t) · n n n = g N (x x x,t) on Γ N × (0, T ], u u u(x x x,t) · n n n + α −1 c −1 p(x x x,t) = g R (x x x,t) on Γ R × (0, T ],
being α > 0 an impedance parameter, and g D , g N and g R given functions, and initial conditions
p(x x x, 0) = p 0 (x x x) in Ω , u u u(x x x, 0) = u u u 0 (x x x) in Ω .
This problem describes the space-time variation of particle velocity u u u and acoustic pressure p in a heterogeneous medium where waves propagate with characteristic velocity c = µ ρ , being ρ > 0 the density and µ > 0 the viscosity of the medium, respectively. Notice that the condition on Γ R × [0, T ] is known in the literature as impedance boundary condition and includes the low-order absorbing condition when α = 1 and g R = 0, [11]. In the following we suppose u u u 0 ∈ Q 2 , p 0 ∈ Q, f ∈ L 2 (0, T ; Q) and for the sake of presentation we set g D = g N = g R = 0 and α = 1. The general case, i.e., with non-homogeneous boundary conditions con be treated similarly. Next, we define the following bilienar forms m(·, ·) :
V V V ×V V V → R m(u u u, v v v) := (u u u, v v v) Ω ∀(u u u, v v v) ∈ V V V ×V V V , n(·, ·) : Q × Q → R n(p, q) := (c −2 p, q) Ω ∀(p, q) ∈ Q × Q, b(·, ·) : V V V × Q → R b(u u u, q) := (div u u u, q) Ω ∀(u u u, q) ∈ V V V × Q, r(·, ·) : V V V ×V V V → R r(u u u, v v v) := (cu u u · n n n, v v v · n n n) Γ R ∀(u u u, v v v) ∈ V V V ×V V V ,(2.2)
and the linear functional associated to given data is defined as
F(·) : Q → R F(q) := ( f , q) Ω ∀q ∈ Q.
Then the weak formulation of Problem 1 reads as follows
Problem 2 (Weak problem) Find the velocity u u u ∈ L 2 (0, T ;V V V ) ∩ C 0 (0, T ; Q 2 ), and the pressure p ∈ L 2 (0, T ; Q) ∩C 0 (0, T ; Q) s.t. m(u u u t , v v v) + b(v v v, p) + r(u u u, v v v) = 0 ∀v v v ∈ V V V , n(p t , q) − b(u u u, q) = F(q) ∀q ∈ Q,
with initial condition u u u(·, 0) = u u u 0 and p(·, 0) = p 0 in Ω .
Using standard arguments is possible to prove that Problem 2 is well posed, [11,27,51] and satisfies the following stability estimate, cf. also [45] sup 0≤t≤T
(u u u, p)(t) E ≤ T 0 c f (s) 0,Ω ds + (u u u 0 , p 0 ) E , where (v v v, q) 2 E := v v v 2 0,Ω + c −1 q 2 0,Ω ∀ (v v v, q) ∈ [L 2 (Ω )] 2 × Q . (2.3)
We finally observe that if f = 0 and Γ R = / 0 Problem 2 is energy conservative, i.e. the solution (u u u, p) of Problem 2 satisfies
(u u u, p)(t) E = (u u u 0 , p 0 ) E ∀t ∈ [0, T ] . (2.4)
Mixed Virtual Elements
In this Section we describe the virtual element discretization of Problem 2 on general polygonal meshes. In particular, in Section 3.1 we introduce the assumptions on the regularity of the polygonal meshes together with the definition of crucial projector operators that will be fundamental in the construction of the VE discretization. In Section 3.2 we describe the H(div)-conforming VE spaces, whereas in Section 3.3 we present the discrete forms. Finally, in Section 3.4 we introduce the semi-discrete VE discretization of Problem 2 and we provide the stability bound of the solution and the energy preserving property of the semi-discrete system.
Mesh assumptions and polynomial projections
From now on, we will denote by E a general polygon having E edges e. For each polygon E and each edge e of E we denote by |E|, h E the measure and diameter of E respectively, by h e we denote the length of e. Furthermore n n n e E (resp. n n n E ) denotes the unit outward normal vector to e (resp. to ∂ E).
Let {Ω h } h be a sequence of decompositions of Ω into general polygons E, where the granularity h is defined as h = sup E∈Ω h h E .
We suppose that {Ω h } h fulfills the following assumption: (A1) Mesh assumption. There exists a positive constant ρ such that for any
E ∈ {Ω h } h • Any E ∈ {Ω h } h is star-shaped with respect to a ball B E of radius ≥ ρ h E ; • Any edge e of any E ∈ {Ω h } h , h e ≥ ρ h E .
We remark that the hypotheses above, though not too restrictive in many practical cases, could possibly be further relaxed, combining the present analysis with the studies in [23,24,28].
Referring to Problem 2, we assume that for any h the decomposition Ω h matches with the subdivision of ∂ Ω into Γ D , Γ N , Γ R and with the definition of the piece-wise constant velocity c. We denote by Σ h the set of all the mesh edges and, for any E ∈ Ω h , we define Σ E h the set of the edges of E. The total number of vertexes, edges and elements in the decomposition Ω h are denoted by L V , L e and L P , respectively.
Using standard VE notations, for any mesh object ω ∈ Ω h ∪ Σ h and for any n ∈ N let us introduce the space P n (ω) to be the space of polynomials defined on ω of degree ≤ n, with the extended notation P −1 (ω) = {0}. For any n ∈ N and for any non-negative s ∈ R we consider the broken spaces
• P n (Ω h ) = {q ∈ L 2 (Ω ) s.t. q| E ∈ P n (E) for all E ∈ Ω h }, • H s (Ω h ) = {v ∈ L 2 (Ω ) s.t. v| E ∈ H s (E) for all E ∈ Ω h },H s (Ω h ) = ∑ E∈Ω h v 2 H s (E) , |v| 2 H s (Ω h ) = ∑ E∈Ω h |v| 2 H s (E) .
For any E ∈ Ω h , let us introduce the L L L 2 2 2 -projection Π 0,E n :
L 2 (E) → P n (E), given by E q n (v − Π 0,E n v) dE = 0 for all v ∈ L 2 (E) and q n ∈ P n (E),
with obvious extension for vector functions Π 0,E n :
[L 2 (E)] 2 → [P n (E)] 2 . The global counterpart Π 0 n : L 2 (Ω h ) → P n (Ω h ), is defined for all E ∈ Ω h by (Π 0 n v)| E = Π 0,E n v . (3.1)
In the following the symbol will denote a bound up to a generic positive constant, independent of the mesh size h and of the time step τ introduced in Section 5, but which may depend on Ω , on the "polynomial" order k and on the regularity constant appearing in the mesh Assumption (A1).
Virtual Element spaces
We start by presenting an overview of the H(div)-conforming Virtual Element space, cf. [15,16,39].
Let k ≥ 0 be the "polynomial" order of the method. We thus consider on each polygonal element E ∈ Ω h the virtual space
V V V k (E) = v v v ∈ H(div, E) ∩ H(curl, E) s.t. (i) div v v v ∈ P k (E) , (ii) curl v v v ∈ P k−1 (E) , (iii) (v v v · n n n e E )| e ∈ P k (e) ∀e ∈ Σ E h .
In the following, we summarize the main properties of the space V V V k (E). We refer to [15,16,39] for a detailed analysis.
(P1) Polynomial inclusion: P k (E) ⊆ V V V k (E); (P2) Degrees of freedom: the following linear operators D V constitute a set of DoFs for V V V k (E): for any w w w ∈ V V V k (E) we consider D V 1 the element moments of the divergence E (div w w w) p k dE ∀p k ∈ P k (E) \ R, D V 2 the element moments E w w w · (p k−1 x x x ⊥ ) dE ∀p k ∈ P k−1 , where x x x ⊥ := (y, −x) T ; D V 3 the edge moments e (w w w · n n n e E ) p k de ∀p k ∈ P k (e), ∀e ∈ Σ E h . Therefore the dimension of V V V k (E) is dim(V V V k (E)) = E (k + 1) + k 2 + 2k . (P3) Computable quantities: for any w w w ∈ V V V k (E) the DoFs D V allow to compute Π 0,E k w w w , div w w w , (w w w · n n n e E )| e ∀e ∈ Σ E h .
The global virtual element space V V V k is defined by gluing together all local spaces, that is: we require that for any internal edge e ∈ Σ E h ∩ Σ E h , shared by E and E , w w w| E · n n n e E + w w w| E · n n n e
E = 0 ∀w w w ∈ V V V k ,
that is in accordance with the DoFs definition D V 1. Therefore we have
V V V k := {v v v ∈ V V V s.t. v v v| E ∈ V V V k (E) ∀E ∈ Ω h }. (3.2)
The dimension of V V V k is thus given by
dim(V V V k ) = (k + 1)L e + (k 2 + 2k)L P .
The discrete pressure space Q k is given by the piecewise polynomial functions of degree k, i.e.
Q k := P k (Ω h ) . (3.3)
Virtual Element forms
The next step in the construction of our method is the definition of a discrete version of the continuous forms in (2.2). Following the usual procedure in the VE setting, we need to construct discrete forms that are computable through the DoFs. Notice that in the light of property (P3), for any
v v v h , w w w h ∈ V V V k and q h , p h ∈ Q k the quantities n(p h , q h ) , b(v v v h , q h ) , r(v v v h , w w w h ) , F(q h ) are computable.
Whereas for arbitrary functions in V V V k the form m(·, ·) is not computable since the discrete functions are not known in closed form. Employing property (P3) for any
v v v h , w w w h ∈ V V V k (E) we define the computable local discrete bilinear form: m E h (v v v h , w w w h ) := E (Π 0,E k v v v h ) · (Π 0,E k w w w h ) dE + h −2 E S E (v v v h , w w w h ) . (3.4)
The stabilizing term in (3.4) is given by
S E (v v v h , w w w h ) = S E (I − Π 0,E k )v v v h , (I − Π 0,E k )w w w h , where S E (·, ·) : V V V k (E) ×V V V k (E) → R is a computable symmetric discrete form.
In the present paper we consider the so-called dofi-dofi stabilization [14] defined as follows: let v v v h and w w w h denote the real valued vectors containing the values of the local degrees of freedom (properly scaled) associated to
v v v h , w w w h in the space V V V k (E) then S E (v v v h , w w w h ) = v v v h · w w w h .
Under mesh Assumption (A1) the form S E (·, ·) satisfies the following bounds (we refer to [39] for the
details) v v v h 2 0,E h 2 E S E (v v v h , v v v h ) v v v h 2 0,E , (3.5) for all v v v h ∈ V V V k (E) ∩ ker(Π 0,E k ). The global form m h (·, ·) : (V V V k ∪ [P k (Ω h )] 2 ) × (V V V k ∪ [P k (Ω h )] 2 ) → R can be derived adding the local contributions m h (v v v h , w w w h ) := ∑ E∈Ω h m E h (v v v h , w w w h ) ∀(v v v h , w w w h ) ∈ (V V V k ∪ [P k (Ω h )] 2 ) . (3.6)
Notice that, from (3.4) and (3.5), it follows that
m h (p p p k , v v v h ) = m(p p p k , v v v h ) ∀p p p k ∈ [P k (Ω h )] 2 , v v v h ∈ (V V V k ∪ [P k (Ω h )] 2 ) , (3.7) v v v h 2 0,Ω m h (v v v h , v v v h ) v v v h 2 0,Ω ∀v v v h ∈ (V V V k ∪ [P k (Ω h )] 2 ) . (3.8)
Virtual Element semi-discrete problem
Referring to the spaces (3.2) and (3.3), the forms (2.2) and the discrete bilinear form (3.6), we can state the following semi-discrete problem.
Problem 3 (VEM problem) Find u u u h ∈ L 2 (0, T ;V V V k ) ∩ C 0 (0, T ; Q 2 ), and p h ∈ L 2 (0, T ; Q k ) ∩ C 0 (0, T ; Q k ) s.t. m h (u u u ht , v v v h ) + b(v v v h , p h ) + r(u u u h , v v v h ) = 0 ∀v v v h ∈ V V V k , n(p ht , q h ) − b(u u u h , q h ) = F(q h ) ∀q h ∈ Q k ,
with initial condition u u u h (·, 0) = Π 0 k u u u 0 and p h (·, 0) = Π 0 k p 0 in Ω .
The well-posedness of Problem 3 follows from:
• discrete inf-sup condition [16]: there exists β > 0 s.t.
inf
q h ∈Q k sup v v v h ∈V V V k b(v v v h , q h ) v v v h V V V q h Q ≥ β .
• coercivity on the discrete kernel: the bilinear form m h (·, ·) satisfies
m h (v v v h , v v v h ) v v v h 2 V V V ∀v v v h ∈ K K K k , where K K K k := {v v v h ∈ V V V k s.t. b(v v v h , q h ) = 0 ∀ q h ∈ Q k }.
Let us introduce the discrete energy norm
(v v v h , q h ) 2 E h := m h (v v v h , v v v h ) + c −1 q h 2 0,Ω ∀ (v v v h , q h ) ∈ (V V V k ∪ [P k (Ω h )] 2 ) × Q k . (3.9)
Then, using analogous argument to that used for the continuous case, the discrete solution (u u u h , p h ) of Problem 3 satisfies the following stability estimate
sup 0≤t≤T (u u u h , p h )(t) E h ≤ T 0 c f (s) 0,Ω ds + (Π 0 k u u u 0 , Π 0 k p 0 ) E ,
where, from (3.7), we have used that
(p p p k , q h ) E h = (p p p k , q h ) E ∀(p p p k , q h ) ∈ [P k (Ω h )] 2 × Q k . Furthermore if f = 0 and Γ R = / 0, Problem 3 is energy conservative, i.e. the solution (u u u h , p h ) of Problem 3 satisfies (u u u h , p h )(t) E h = (Π 0 k u u u 0 , Π 0 k p 0 ) E ∀t ∈ [0, T ] ,(3.10)
that is the semi-discrete counterpart of (2.4).
Remark 1 Notice that definitions (2.3) and (3.9) and bounds (3.8) imply the following norm equivalence
(v v v h , q h ) E (v v v h , q h ) E h (v v v h , q h ) E ∀ (v v v h , q h ) ∈ V V V k × Q k . (3.11)
Remark 2 The proposed approach can be easily extended to more general situations such as the three dimensional case [15], and domains with curved boundary/interfaces [39,41]. The analysis could be developed with very similar arguments to the ones in the forthcoming sections.
Theoretical analysis
In this section, we present some theoretical results for the virtual element Problem 3. In Section 4.1 we review the interpolation estimates, whereas in Section 4.2 and Section 4.3 we provide the convergence analysis and the energy error estimate respectively.
Interpolation estimates
We now recall the optimal approximation properties for the space V V V k (see [39]). We define the linear Fortin operator Π F k : [H 1 (Ω )] 2 → V V V k in the following way. For any w w w ∈ [H 1 (Ω )] 2 and for all e ∈ Σ h and E ∈ Ω h , Π F k w w w is determined by
E div(w w w − Π F k w w w) p k dE = 0 ∀p k ∈ P k (E) \ R , E curl(w w w − Π F k w w w) p k−1 dE = 0 ∀p k−1 ∈ P k−1 (E) , e (w w w − Π F k w w w) · n n n e p k de = 0 ∀p k ∈ P k (e) .
The conditions above implies that the following diagram is commutative, i.e.,
[H 1 (Ω )] 2 div − −− → Q 0 −−→ 0 Π F k Π k 0 V V V k div − −− → Q k 0 −−→ 0
where 0 is the map that associates to every function the value 0. In particular, for any w w w ∈ [H 1 (Ω )] 2 it holds (see [39]) div(Π F k w w w) = Π 0 k div w w w , curl(Π F k w w w) = Π 0 k−1 curl w w w , ((Π F k w w w) · n n n e )| e = (Π 0,e k (w w w · n n n e ))| e ∀e ∈ Σ h .
V V V k ) Under Assumption (A1) for any v v v ∈ V V V ∩ [H s (Ω h )] 2 where 1 ≤ s ≤ k + 1, Π k F v v v satisfies the estimate v v v − Π k F v v v 0,Ω h s |v| s,Ω h .
We now review a classical approximation result for polynomials on star-shaped domains, see for instance [29].
Convergence analysis
In this section we provide the convergence property for the semi-discrete scheme. Then for all t ∈ (0, T ) the following error estimate holds:
(u u u−u u u h , p− p h )(t) E h k+1 |u u u 0 | k+1,Ω h + c |p 0 | k+1,Ω h +|u u u t | L 1 (0,t;H k+1 (Ω h )) + c |p t | L 1 (0,t;H k+1 (Ω h )) , where c := c −1 L ∞ (Ω ×(0,T )) .
Proof For all t ∈ (0, T ), let us introduce the following error quantities e e e I (t) The first term on the right-hand side of (4.2) can be bounded by using Proposition 4.1 and Lemma 4.1 getting
:= u u u(t) − Π F k u u u(t) , ρ I (t) := p(t) − Π 0 k p(t) , e e e h (t) := Π F k u u u(t) − u u u h (t) , ρ h (t) := Π 0 k p(t) − p h (t) ,(e e e I , ρ I )(t) E h (k+1) |u u u(t)| k+1,Ω h + c |p(t)| k+1,Ω h h k+1 |u u u 0 | k+1,Ω h + c |p 0 | k+1,Ω h + t 0 |u u u t (s)| k+1,Ω h + c |p t (s)| k+1,Ω h ds = h k+1 |u u u 0 | k+1,Ω h + c |p 0 | k+1,Ω h + |u u u t | L 1 (0,t;H k+1 (Ω h )) + c |p t | L 1 (0,t;H k+1 (Ω h )) .
(4.3)
To estimate the second term on the right-hand side of (4.2) we proceed as follows. We consider Problem 2 and Problem 3 and obtain First, we notice that the term η b vanishes, in fact The first term on the right-hand side of the previous equation can be bounded as follows
η b = b(Π F k u u u, p − p h ) − b(u u u h , p − p h ) + b(u u u h , Π 0 k p − p h ) − b(u u u, Π 0 k p − p h ) = b(Π F k u u u, p − p h ) + b(u u u, p h − Π 0 k p) + b(u u u h , Π 0 k p − p) = b(Π F k u u u, p − p h ) + b(u u u, p h − Π 0 k p) (div u u u h ∈ P k (Ω h ) & def. Π 0 k ) = b(Π F k u u u, p − p h ) + b(Π F k u u u, p h − Π 0 k p) (p h − Π 0 k p ∈ P k (Ω h ) & (4.1)) = b(Π F k u u u, p − Π 0 k p) = 0. (div(Π F h u u u) ∈ P k (Ω h ) & def.η u u u = m h (Π F k (u u u t ) − Π 0 k (u u u t )
, e e e h ) + m(Π 0 k (u u u t ) − u u u t , e e e h ) (by (3.7))
Π F k (u u u t ) − Π 0 k (u u u t ) 0,Ω + u u u t − Π 0 (u u u t ) 0
,Ω e e e h 0,Ω (by (3.8)) u u u t − Π F k (u u u t ) 0,Ω + u u u t − Π 0 (u u u t ) 0,Ω e e e h 0,Ω (by tri. ineq.) h k+1 |u u u t | H k+1 (Ω h ) m h (e e e h , e e e h ) 1/2 (4.9) where in the last inequality we used Proposition 4.1, Lemma 4.1 and equation (3.8). Whereas η p is estimated by
η u u u = n(Π 0 (p t ) − p t , ρ h ) ≤ c −1 (Π 0 (p t ) − p t ) 0,Ω c −1 ρ h 0,Ω c h k+1 |p t | H k+1 (Ω h ) c −1 ρ h 0,Ω .
(4.10)
Therefore recalling definition (3.9), from (4.8), (4.9) and (4.10) and the Cauchy-Schwarz inequality, we
obtain d dt (e e e h , ρ h )(t) E h h k+1 |u u u t (t)| H k+1 (Ω h ) + c |p t (t)| H k+1 (Ω h ) .
By integrating the previous bound on (0,t) we obtain
(e e e h , ρ h )(t) E h (e e e h , ρ h )(0) E h + h k+1 t 0 |u u u t (s)| H k+1 (Ω h ) ds + c t 0 |p t (s)| H k+1 (Ω h ) ds
(e e e h , ρ h )(0) E + h k+1 u u u t L 1 (0,t;H k+1 (Ω h )) + c p t L 1 (0,t;H k+1 (Ω h )) .
(4.11)
We finally bound the initial data error
(e e e h , ρ h )(0) E Π F k u u u 0 − Π 0 k u u u 0 0,Ω + c p 0 − Π 0 k p 0 0,Ω h k+1 |u u u 0 | H k+1 (Ω h ) + c |p 0 | H k+1 (Ω h ) .
(4.12)
The thesis now follows combining (4.11), (4.12) and (4.3) in (4.2).
Energy error
An important aspect in wave propagation problems is the energy conservation of the continuous and semi-discrete system (cf. (2.4) and (3.10) rispectively). In the following proposition we estimate the errors between the energy of the continuous and the semi-discrete energy of the system. Then for all t ∈ (0, T ), the following estimate holds:
0 ≤ (u u u, p)(t) 2 E − (u u u h , p h )(t) 2 E h h 2(k+1) |u u u 0 | 2 k+1,Ω h + c 2 |p 0 | 2 k+1,Ω h , where c := c −1 L ∞ (Ω ×(0,T )) .
Proof . From (2.4) and (3.10) we infer Direct application of Pythagorean Theorem yields
(u u u, p)(t) 2 E − (u u u h , p h )(t) 2 E h = (u u u 0 , p 0 ) 2 E − (Π 0 k u u u 0 , Π 0 k p 0 ) 2 E = u uζ 2 0,Ω = Π 0 k ζ 2 0,Ω + ζ − Π 0 k ζ 2 0,Ω , ∀ζ ∈ L 2 (Ω ) ,
therefore, recalling that c is piece-wise constant w.r.t. Ω h , we infer
(u u u, p)(t) 2 E − (u u u h , p h )(t) 2 E h = u u u 0 − Π 0 k u u u 0 2 0,Ω + c −1 (p 0 − Π 0 k p 0 ) 2 0,Ω . (4.13)
Then, from (4.13) and Lemma 4.1, the following bounds hold
(u u u, p)(t) 2 E − (u u u h , p h )(t) 2 E h ≥ 0 , (u u u, p)(t) 2 E − (u u u h , p h )(t) 2 E h h 2(k+1) |u u u 0 | 2 k+1,Ω h + c 2 |p 0 | 2 k+1,Ω h .
Time integration
In the light of energy conservation (2.4) and (3.10) and the energy error estimate in Proposition 4.3, it is crucial to understand how the energy is preserved or not under time-stepping schemes.
In the present section we formulate a fully discrete version of Problem 3 aiming at preserving the energy of the system. Therefore, we introduce a sequence of time steps t n = nτ, n = 0, . . . , N, with time step size τ. Next, we define v v v n h,τ := v v v h (·,t n ) (resp. q n h,τ := q h (·,t n )) as the approximation of the function v v v h (·,t) ∈ V V V k (resp. q h (·,t) ∈ Q k ) at time t n , n = 0, . . . , N. To integrate in time Problem 3, we take under consideration the family of θ -method schemes and analyse their energy conservation properties. The fully discrete systems consequently reads as follows:
Given (u u u 0 h,τ , p 0 h,τ ) = (Π 0 k u u u 0 , Π 0 k p 0 ), find (u u u n h,τ , p n h,τ ) for n = 0, . . . , N s.t. m h u u u n+1 h,τ − u u u n h,τ τ , v v v h + b(v v v h , θ p n+1 h,τ + (1 − θ )p n h,τ ) + r(θ u u u n+1 h,τ + (1 − θ )u u u n h,τ , v v v h ) = 0 ∀v v v h ∈ V V V k , n p n+1 h,τ − p n h,τ τ , q h − b(θ u u u n+1 h,τ + (1 − θ )u u u n h,τ , q h ) = θ F n+1 (q h ) + (1 − θ )F n (q h ) ∀q h ∈ Q k ,(5.1)
where F n (q h ) := ( f (t n ), q h ) Ω and θ ∈ [0, 1]. It is well known that all θ -methods are first order accurate in time, except for θ = 1/2, i.e., the Crank-Nicolson method, which is second order accurate.
To analyse the energy conservation of the proposed schemes we consider null forcing terms, i.e. f = 0, and Γ R = / 0. Thus, the above system reduces to
m h u u u n+1 h,τ − u u u n h,τ τ , v v v h + b(v v v h , θ p n+1 h,τ + (1 − θ )p n h,τ ) = 0 ∀v v v h ∈ V V V k , n p n+1 h,τ − p n h,τ τ , q h − b(θ u u u n+1 h,τ + (1 − θ )u u u n h,τ , q h ) = 0 ∀q h ∈ Q k . (5.2)
We start by considering in (5.2) v v v h = θ u u u n+1 h,τ + (1 − θ )u u u n h,τ ∈ V V V k , and q h = θ p n+1 h,τ + (1 − θ )p n h,τ ∈ Q k , and by summing together the two above equation we get
m h u u u n+1 h,τ − u u u n h,τ τ , θ u u u n+1 h,τ + (1 − θ )u u u n h,τ + n p n+1 h,τ − p n h,τ τ , θ p n+1 h,τ + (1 − θ )p n h,τ = 0, that is θ (u u u n+1 h,τ , p n+1 h,τ ) 2 E h + (1 − 2θ ) m h (u u u n+1 h,τ , u u u n h,τ ) + n(p n+1 h,τ , p n h,τ ) = (1 − θ ) (u u u n h,τ , p n h,τ ) 2 E h ,
which, rearranging the terms, is
(u u u n+1 h,τ , p n+1 h,τ ) 2 E h + (2θ − 1) (u u u n+1 h,τ − u u u n h,τ , p n+1 h,τ − p n h,τ ) 2 E h = (u u u n h,τ , p n h,τ ) 2 E h . (5.3)
Now, it is easy to see that for n = 0, . . . , N:
• if θ = 1/2, i.e., for the Crank-Nicolson method, the discrete energy is conserved, i.e.
(u u u n h,τ , p n h,
τ ) E h = (u u u 0 h,τ , p 0 h,τ ) E h = (Π 0 k u u u 0 , Π 0 k p 0 ) E ,
• if 1/2 < θ ≤ 1, that is 2θ − 1 > 0, the second term in (5.3) is positive and then we can obtain
(u u u n+1 h,τ , p n+1 h,τ ) 2 E h ≤ (u u u n h,τ , p n h,τ ) 2 E h ,
that means dissipation of energy, cf. [46], • if 0 ≤ θ < 1 2 , that is 2θ − 1 < 0, the second term in (5.3) is negative and then we have
(u u u n+1 h,τ , p n+1 h,τ ) 2 E h ≥ (u u u n h,τ , p n h,τ ) 2 E h ,(5.4)
that is the method produces a nondecreasing energy at each time step, in agreement with [47].
To obtain an explicit nearby energy-conservative scheme, one can consider, for example, the symplectic Euler scheme, that reduces to:
n p n+1 h,τ − p n h,τ τ , q h − b(u u u n h,τ , q h ) = F n (q h ) ∀q h ∈ Q k , m h u u u n+1 h,τ − u u u n h,τ τ , v v v h + b(v v v h , p n+1 h,τ ) + r(u u u n h,τ , v v v h ) = 0 ∀v v v h ∈ V V V k . (5.5)
We refer the reader to [52] for the analysis.
Numerical tests
In this section we provide some numerical examples to show the behaviour of the method and give numerical evidence of the theoretical results derived in the previous sections. We consider three test cases: the first in Subsection 6.1 presents the expected convergence rates for different approximation degree over several mesh families. The second test case, given in Subsection 6.2, is focused on the energy conservation properties of the scheme. Finally, in Subsection 6.3, we consider a wave propagation problem in a domain having with small curved inclusions. The latter is handled by using the extension of VEM to curved edges (cf. Remark 2).
Error decay
In this first test case, we verify the expected convergence rate of the method for different mesh families:
• tria a simplicial mesh;
• quad a Cartesian mesh; • hexa composed by distorched hexagons;
• voro a mesh made of Voronoi cells optimized via a Lloyd algorithm.
In order to compute the VEM errors between the exact solution (u u u, p) and the VEM solution (u u u h , p h ) at the final time T , we consider the computable L 2 -like error quantities, Let us set the mesh-size parameter h := 1
L P ∑ E∈Ω h h E .
For each family, we build a sequence of four meshes with decreasing mesh size h. In Figure 1 we depict one mesh as representative of each family. In accordance with Proposition 4.2, the trend of each error indicator is computed and compared to the expected convergence trend. We consider a problem on Ω = (0, 1) 2 with analytical solution given by u u u(x x x,t) = −2π cos(2πx) cos(2πy) sin(t) 2π sin(2πx) sin(2πy) sin(t) ,
p(x x x,t) = cos(2πy) sin(2πx) cos(t).
We set Γ D = ∂ Ω , Γ N = Γ R = / 0 and choose c = 1. Boundary and initial conditions as well as the forcing term f in Problem 1 are computed accordingly. Since we are considering the space discretization error, we set T = 1.e − 7 and τ = 1.e − 8 with θ = 1 in (5.1).
The computed errors are given in Figure 2 and 3. Such convergence lines are coherent with the estimates derived in the theoretical results, see Proposition 4.2. For high approximation degree and fine mesh size, we notice stagnation of the error of the hexa and voro families. This is an expected behaviour discussed in [25]. A possible solution is to introduce a different basis for the polynomial expansion that makes the local systems better conditioned. However, this is out of the scope of the current work and it may be a good starting point for future investigations.
O(h 4 ) O(h 3 ) O(h 2 ) e u 2 × 10 −2 5 × 10 −2 10 −1 h O(h 5 ) O(h 4 ) O(h 3 ) O(h 2 ) e p hexa 5 × 10 −2 10 −1 2.5 × 10 −1 h 10 −7 10 −5 10 −3 10 −1 error O(h 5 ) O(h 4 ) O(h 3 ) O(h 2 ) e u 5 × 10 −2 10 −1 2.5 × 10 −1 h O(h 5 ) O(h 4 ) O(h 3 ) O(h 2 ) e p quad k = 1 k = 2 k = 3 k = 4 FIG. 2.
Error decay for different type of meshes. Example in Subsection 6.1.
Conservation of energy
In this second example, we test the conservation of energy given, for the semi-discrete system, in (3.10). We consider the same mesh families and approximation degrees of the previous example and we choose four different time integration schemes: Explicit Euler, i.e. θ = 0 in (5.2), Implicit Euler, i.e. θ = 1 in (5.2) Crank-Nicolson, i.e. θ = 1 2 in (5.2), and Symplectic Euler (5.5). As observed in Section 3.4, the space discretization does locally conserve the energy so the possible lack of conservation will be due to the temporal scheme. The conservation of energy is depicted in Figure 4. In these graphs we compute the difference between the initial energy and the one at a specific time step and we report such values multiplied by a factor of 10 4 . We notice that for the Explicit Euler the energy tends to increase during time in agreement with (5.4). This trend becomes more evident for higher approximation degree. For the Implicit Euler the situation has an opposite behaviour. Indeed, the energy is now dissipated during time. Also for this scheme, higher approximation degrees tend to dissipate more energy. The semi-implicit Symplectic Euler scheme mitigates this effect: the energy is not conserved during time, but the absolute error does not grow and stay limited. Finally, Crank-Nicolson shows a perfect energy conservation property. To better appreciate this behaviour we report in Table 1 the values
E = (u u u T h,τ , p T h,τ ) E h − (u u u 0 h,τ , p 0 h,τ ) E h (u u u 0 h,τ , p 0 h,τ ) E h ,(6.
Multiple scattering in curved configurations
In this example, we show the qualitative behaviour of the solution computed by the proposed approach in a domain with multiple circular inclusions. We consider two different configurations that are inspired by [12]. Both domain are defined in Ω = (0, 1) 2 , but the first one has five holes whose centres are located at
x x x 1 = [a , c] , x x x 2 = [a , b] , x x x 3 = [a , a] , x x x 4 = [b , a] , x x x 5 = [c , a] ,
while the second one has three additional circles centred at
x x x 6 = [b , c] , x x x 7 = [c , c] , x x x 8 = [c , b] ,
where the values of the three parameters are given by a = 0.6052631578947355 , b = 0.5, and c = 0.3947368421052622 .
All circles have diameter 0.02. We refer to the first and the second case as fiveHoles and eightHoles, respectively. The grid is build starting from a structured quadrilateral mesh, where the cells intersecting with a circle are properly cut. Moreover, since we exploit the possibility to include the geometry information within VEM spaces [39], we do not need to over-refine edges to get an accurate representation of the circles themselves. As a consequence, we can reduce the computational effort without losing the accuracy of the solution, by neglecting spurious waves due to a piece-wise linear representation of the circles.
For both fiveHoles and eightHoles examples, we build two meshes, see Figure 5. The first one starting from a square divided in 38 × 38 uniform squares. Then, the second one is constructed by refining 2 times only the quadrilateral elements of the first one, see the detail in Figure 6.
Along the external boundaries of Ω , we impose absorbing boundary conditions, while on all the small circles perimeters we use homogeneous Neumann conditions. Then, we consider c = 1 and the following source term f (x x x,t) = 10. e −(t−1) 2 /σ 1 e − x x x−x x x c 2 /σ 2 , with σ 1 = 0.01 and σ 2 = 0.00125 and x x x c = [0.5, 0.5] . The initial values of both u u u and p are set to zero. We assume approximation degree equal to k = 2. As final time we take T = 10, we divide the interval [0, T ] into 50 equally spaced sub-intervals and we use the Crank-Nicolson time integration scheme.
In Figure 7 we show the computed pressure solution p h evaluated at quadrature points at different time steps for the fiveHoles case. On the top panels we show the coarse mesh while, on the bottom In Figure 8 we consider the eightHoles case. Also for this experiment we observe that the coarser mesh provides similar results of the ones obtained with the refined mesh. Moreover, since the holes are displaced in a symmetric way with respect to x x x c we expect the waves preserve a symmetric structure. This is confirmed by the snapshots represented in Figure 8.
Conclusions
In this work, we presented a general order virtual element numerical discretization scheme to approximate the wave equation written in mixed form. The latter results in a first order system of differential equations in both space and time dimension. We presented the a-priori stability analysis as well as the convergence property of the scheme in a suitable energy norm and we showed that the proposed virtual element scheme preserve the semi-discrete energy of the system in absence of external load. To integrate in time the semi-discrete problem we consider the family of the θ -method schemes and we discuss their energy conservation properties. We verified the theoretical results on different benchmark tests and we applied the proposed scheme on a domain with circular inclusions to show the capabilities of the method in term of accuracy and of flexibility in handling complex geometries. To conclude, the presented mixed virtual element method allows a robust and flexible numerical discretization that can be successfully applied to wave propagation problems. Future developments in this direction may include the study of multi-physics problems (written in a mixed form) such as vibro-acoustics (with elastic or poroelastic structure) interaction problems. step 13 step 26 step 42
Fortin operator we have the following interpolation estimate (see [39, Proposition 4.1]).
Proposition 4. 1 (
1Approximation property of
Lemma 4.1 (Bramble-Hilbert) Under Assumption (A1) and referring to (3.1), for all q ∈ H s (Ω h ) where 0 ≤ s ≤ k + 1, it holds q − Π 0 k q Ω ,0 h s |q| Ω h ,s .
Proposition 4. 2
2Under Assumption (A1), let (u u u, p) be the solution of Problem 2 and (u u u h , p h ) be the solution of Problem 3. Assume that u u u t , p t ∈ L 1 (0, T ; H k+1 (Ω h )) and u u u 0 , p 0 ∈ H k+1 (Ω h ) .
From triangle inequality and (3.11) it holds(u u u − u u u h , p − p h )(t) E(e e e I , ρ I )(t) E + (e e e h , ρ h )(t) E h .(4.2)
m(u u u
t , e e e h ) + b(e e e h , p) + r(u u u, e e e h ) + n(p t ,ρ h ) − b(u u u, ρ h ) = F(ρ h ), m h (u u u ht , e e e h ) + b(e e e h , p h ) + r(u u u h , e e e h ) + n(p ht , ρ h ) − b(u u u h , ρ h ) = F(ρ h ).Then subtracting the previous equations we get the following error equationm(u u u t , e e e h ) − m h (u u u ht , e e e h ) + n((p − p h ) t , ρ h ) + r(u u u − u u u h , e e e h ) + η b = 0 , (4.4) where η b := b(e e e h , p − p h ) + b(u u u h − u u u, ρ h ).
(e e e ht , e e e h ) + n(ρ h t , ρ h ) + r(e e e h , e e e h ) = (m h (Π F k (u u u t ), e e e h ) − m(u u u t , e e e h )) − n(ρ I t , ρ h ) − r(e e e I , e e e h ) . (4.6) Recalling that (e e e h · n n n e )| e ∈ P k (e) for all e ∈ Σ h , and employing (4.1) we have r(e e e I , e e e h ) = r(u u u − Π F k u u u, e e e h ) = 0 . (4.7) Therefore, since r(e e e h , e e e h ) ≥ 0, from (4.6) and (4.7), and recalling definition (3.e e e h , e e e h ) 2 E h ≤ (m h (Π F k (u u u t ), e e e h ) − m(u u u t , e e e h )) − n(ρ I t , ρ h ) =: η u u u + η p . (4.8)
Proposition 4. 3
3Under assuption (A1), let (u u u, p) be the solution of Problem 2 and (u u u h , p h ) be the solution of Problem 3 with f = 0 and Γ R = / 0. Assume that u u u 0 , p 0 ∈ H k+1 (Ω h ).
e u u u := u u u(T ) − Π 0 k u u u h (T ) 0,Ω and e p := p(T ) − Π 0 k p h (T ) 0,Ω .
FIG. 1 .
1Example of the adopted polygonal meshes. Example in Subsection 6.1.
The spatial domain is still the unit square, Γ D = [0, 1] × {0, 1} i.e. the top and the bottom boundaries, Γ N = {0, 1} × [0, 1] i.e. the left and right boundaries, and Γ R = / 0. The final time is T = 1 and τ = 1/200. We set to zero the source term, the characteristic velocity c = 1 and the initial velocity and pressure as u u u(x x x, 0) = sin(x) cos(y) , and p(x x x, 0) = cos(2πy) sin(2πx) .
FIG. 3 .
3Error decay for different type of meshes. Example in Subsection 6.1.
FIG. 5 .
5Representation of the computational grid for fiveHoles (left) for eightHoles (right). Example in Subsection 6.3. FIG. 6. Detail of one hole in the inital mesh (left) and in the refined one (right). Example in Subsection 6.3.
FIG. 7 .
7Solution obtained with the fiveHoles mesh, on the top with a coarse grid and on the bottom with a finer grid. Example in Subsection 6.3.
1 )
1being (u u u T h,τ , p T h,τ ) E h the energy of the system at final time T . The behaviours of these time discretisation schemes perfectly match the arguments in Section 5. .4038e-16 8.6677e-15 2.0568e-15 9.0382e-15 k = 2 1.2555e-14 2.7043e-14 2.2665e-15 2.4104e-15 k = 3 2.8832e-14 2.2205e-13 6.3020e-16 1.0358e-14 k = 4 2.9968e-15 1.2107e-13 2.2041e-16 1.6068e-14 TABLE 1 Energy errors E computed as in (6.1) for the Crank-Nicolson scheme in (5.2).E
tria
quad
hexa
voro
k = 1 7
AcknowledgmentsThe authors acknowledge the financial support of INdAM-GNCS through Project "Sviluppo ed analisi di Metodi agli Elementi Virtuali per processi accoppiati su geometrie complesse". Moreover, the authors would like to thank Anna Scotti for many fruitful discussions.ones, we consider the fine mesh. From a qualitative point of view we observe no much difference between them.
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[] |
[
"Free energy calculations along entropic pathways III. Nucleation of capillary bridges and bubbles",
"Free energy calculations along entropic pathways III. Nucleation of capillary bridges and bubbles"
] |
[
"Caroline Desgranges \nDepartment of Chemistry\nUniversity of North Dakota\n58202Grand ForksND\n",
"Jerome Delhommelle \nDepartment of Chemistry\nUniversity of North Dakota\n58202Grand ForksND\n"
] |
[
"Department of Chemistry\nUniversity of North Dakota\n58202Grand ForksND",
"Department of Chemistry\nUniversity of North Dakota\n58202Grand ForksND"
] |
[] |
Using molecular simulation, we analyze the capillary condensation and evaporation processes for Argon confined in a cylindrical nanopore. For this purpose, we define the entropy of the adsorbed fluid as a reaction coordinate and determine the free energy associated with both processes along entropic pathways. For capillary condensation, we identify a complex free energy profile resulting from the multi-stage nature of this phenomenon. We find capillary condensation to proceed through the nucleation of a liquid bridge across the nanopore, followed by its expansion throughout the pore to give rise to the stable phase of high density. In the case of capillary evaporation, the free energy profile along the entropy pathway also exhibits different regimes, corresponding to the initial destabilization of the layered structure of the fluid followed by the formation, and subsequent expansion, of a bubble across the nanopore.
|
10.1063/1.4982943
|
[
"https://arxiv.org/pdf/2108.10383v1.pdf"
] | 99,937,508 |
2108.10383
|
9a105c48cf13b0796ce9b6ddc753dc30743b89de
|
Free energy calculations along entropic pathways III. Nucleation of capillary bridges and bubbles
Caroline Desgranges
Department of Chemistry
University of North Dakota
58202Grand ForksND
Jerome Delhommelle
Department of Chemistry
University of North Dakota
58202Grand ForksND
Free energy calculations along entropic pathways III. Nucleation of capillary bridges and bubbles
(Dated: August 25, 2021)
Using molecular simulation, we analyze the capillary condensation and evaporation processes for Argon confined in a cylindrical nanopore. For this purpose, we define the entropy of the adsorbed fluid as a reaction coordinate and determine the free energy associated with both processes along entropic pathways. For capillary condensation, we identify a complex free energy profile resulting from the multi-stage nature of this phenomenon. We find capillary condensation to proceed through the nucleation of a liquid bridge across the nanopore, followed by its expansion throughout the pore to give rise to the stable phase of high density. In the case of capillary evaporation, the free energy profile along the entropy pathway also exhibits different regimes, corresponding to the initial destabilization of the layered structure of the fluid followed by the formation, and subsequent expansion, of a bubble across the nanopore.
I. INTRODUCTION
The stability of liquid bridges between solid surfaces, as well as the mechanism by which they form, is a central phenomenon in interface science and adhesion [1][2][3][4][5][6][7] . The formation of these bridges, as well as the nucleation of bubbles, actually occurs in a wide range of systems that span many length scales, from colloidal systems to nanosized capillaries [8][9][10] . This has been shown e.g. by atomic force microscopy and surface force apparatus experiments, which have revealed how liquid contacts can form and snap off, paving the way for applications in nanotribology and nanolithography [11][12][13][14][15][16][17][18][19][20] . In recent years, the use of molecular modeling and simulation to provide a molecular level understanding of this phenomenon has drawn considerable interest [21][22][23][24][25][26][27][28] . Density functional theory calculations [29][30][31][32] , as well as molecular dynamics [33][34][35][36][37] and Monte Carlo simulations [38][39][40][41][42][43][44][45][46][47][48][49][50] Condensation and evaporation processes are challenging processes to model and simulate. This is due to the large free energy barrier of nucleation that the system has to overcome to complete the phase transition [59][60][61][62][63][64][65][66] . Simulating the pathways underlying condensation and evaporation in nanoscale capillaries requires suitable simulation techniques that allow for the sampling of these rare events by driving the formation of the new phase. In this work, we focus on elucidating both the capillary condensation and evaporation pathways and on shedding light on the relation between the structure of the nanoconfined fluid and its entropy. For this purpose, we extend the recently developed µV T −S simulation method 67,68 to study these two phenomena. The µV T − S approach is implemented within the grandcanonical ensemble, in which the chemical potential µ, the temperature T and the volume V of the system are held fixed. As shown in the case of the nucleation of liquid droplets for single component systems 67 and binary mixtures 68 , this method provides a direct way to calculate the entropy of the system throughout the nucleation process and, therefore, to use it as a reaction coordinate for such activated processes. In this work, we carry out µV T − S simulations with the aim of (i) simulating the adsorption and desorption of the fluid in the grand-canonical statistical ensemble, that is especially suited to study adsorption phenomena, and (ii) using the entropy of the confined fluid as the reaction coordinate to elucidate the two pathways underlying nanoscale capillary condensation and evaporation. To achieve this, we impose a target value for the entropy of the system, through an umbrella sampling bias potential 69 . Then, by varying the value of the entropy of the system, we are able to sample the entire pathway connecting the two nanoconfined phases, starting from the metastable confined vapor and ending with the nanoconfined liquid in the case of capillary condensation. Doing so, we are able to identify the different intermediate structures, specifically bridges and bubbles, that form in the nanopores during the capillary condensation and evaporation processes.
The paper is organized as follows. We start by discussing how we extend the µV T − S approach to study the condensation and evaporation processes of Argon in MCM-41 silica mesoporous molecular sieves. We also present the potential models used for the adsorbate and its interaction with the cylindrical nanopore, and provide the technical details for the simulations. We then examine the capillary condensation process for Argon in the nanopore and determine the free energy barrier associated with the process. We carry out a detailed structural analysis to identify the formation of the liquid bridge along this pathway. We also discuss the capillary evaporation process that occurs during the desorption of the confined fluid and apply the same analyses as for the reverse pathway. We finally draw our main conclusions in the last section.
II. MODELS
We study the capillary condensation and evaporation of Argon in a cylindrical pore of 10 atomic diameters. The geometry is defined as follows. The cylindrical pore is aligned with the z-axis, and we define the length, or lateral dimension, as L z . Simulations are carried out at the boiling point for Argon (T = 87.3 K) within a nanopore found in MCM-41 silica mesoporous molecular sieves 54 . We model Argon with a Lennard-Jones potential and take the following parameters for the exclusion diameter σ = 3.4Å and for the potential well-depth /k B = 119.8 K. The interactions between the adsorbed Argon atoms and the nanopore are modeled with the following functional form [52][53][54][55][56][57] U sf (r, R) = π 2 ρ s sf σ 2 sf 63 32
R−r σ sf 1 + r R −10 F −9 2 , −9 2 ; 1; r R 2 − 3 R−r σ sf 1 + r R −4 F −3 2 , −3 2 ; 1; r R 2(1)
in which r is the radial coordinate of the Ar atom adsorbed in the pore, R is the pore radius (here 5σ), ρ s is the surface density of adsorption centers and F (α, β; γ; δ) is the hypergeomet-
III. SIMULATION METHOD
To span the entire pathway underlying the capillary condensation and evaporation processes, we carry out a series of µV T − S simulations and gradually vary the entropy of the system to drive the phase transition within the confined fluid. We briefly outline here the principles of the µV T − S method (more details may be found in previous work 67, 68 ).
S loading = U − µN T (2)
In this equation, S loading denotes the total entropy of the system and, as such, increases with the total number of atoms N adsorbed in the nanopore. We add that the pV term is omitted in Eq. 2 since it is negligible when compared to the other terms (pV actually accounts for less than 0.5 % of the lowest values sampled for T S loading ). This approximation is similar to the one made by Waghe, Rasaiah and Hummer in their calculation of the entropy of water adsorbed in carbon nanotubes 72 . To sample the transition pathway, we use S loading as the reaction coordinate and drive the nucleation event through an umbrella sampling approach 69 . For this purpose, we define the following bias energy:
U b = 1 2 k(S loading − S 0 ) 2(3)
in which S 0 is the target value for the total entropy. This bias energy is added to the total potential energy of the system, which appears in the Metropolis criteria used in the Monte with µ for the capillary condensation process being slightly above the chemical potential at coexistence and µ for the capillary evaporation process being slightly below. A total of 35 umbrella sampling windows are performed to connect the vapor and liquid phases for the confined fluid and to sample the entire condensation and evaporation pathways. For each window, we first carry out an equilibration run of 1 × 10 8 MC steps to allow the system to relax towards the target value for the entropy. We then perform a production run of 2 × 10 8 MC steps, during which averages are collected collect and structural analyses are run.
IV. RESULTS AND DISCUSSION
We start by discussing the results obtained for the capillary condensation process. To sample the entropic pathway for this process, we carry out a series of µV T − S simulations at µ = −10.48 . We gradually increase the value for the target entropy, as we go from one umbrella sampling window to another, and, as a result, increase S loading for the confined fluid. We show in Fig. 1 how the system responds to this increase in the entropy of the adsorbed fluid. The top panel of Fig. 1 consists of a plot of the number of Argon atoms N adsorbed in the nanopore as a function of the reduced entropy of the adsorbed fluid (noted as S * loading ). We observe the following trends. First, we find that N increases with S * loading , starting from about 900 Ar atoms adsorbed for S * loading = 3300, and reaches a value of about 1600 Ar atoms for S * loading = 6800. This plot shows that the successive µV T − S simulations, with increasing values for the total entropy of the system, allow to simulate the phase transition process from a metastable phase of low density to the stable phase of high density. The bottom panel of Fig. 1 shows the corresponding variation of the interaction energy within the confined fluid during the µV T − S simulations. Overall, we observe a decrease in the interaction energy as S * loading increases. This behavior is consistent with the increase in the amount of Ar adsorbed in the cylindrical nanopore observed when S * loading increases.
We plot in Fig. 2 the free energy profile obtained for the capillary condensation process.
The starting point for the µV T −S simulations is a metastable phase of low density adsorbed in the nanopore, with a reduced entropy of S * loading = 3300. As shown on the left of Fig. 2, the free energy profile exhibits a local minimum for this metastable phase. For convenience, we have chosen to assign the origin for the free energy to this local minimum. Several regimes can be identified in this plot of the free energy as a function of the entropy of the adsorbed phase. First, the free energy profile exhibits a steep increase, over entropies ranging from S * loading = 3300 to S * loading = 4800. This increase in free energy corresponds to the steady increase in N , and accordingly to the steady decrease of U * interaction , over the same range of entropies. Then, for S * loading = 4800 to S * loading = 6000, the free energy profile exhibits a second regime which can be characterized as an almost flat top, that very slowly decreases with S * loading . This second regime is associated with changes in both the slopes for the variations of N and U * interaction in Fig. 1. Then, for S * loading = 6000 to S * loading = 6850, we observe a third regime for the free energy profile and a steady decrease in free energy.
During this stage, the system completes the phase transition towards the stable phase of high density. As expected, we obtain a free energy for the stable phase that is below, by 11 k B T , the free energy of the metastable phase of low density. This establishes that the end point for the µV T − S simulations is indeed the stable phase. As for the previous two regimes in the free energy profile, we find that the decrease for the free energy, observed during the third regime, is connected to sharp changes in the variations of both N and U * interaction for the adsorbed phase. This indicates that significant structural changes take place within the fluid confined in the nanopore.
To elucidate the mechanism underlying capillary condensation, we carry out a series of structural analyses. We begin by determining the density profiles for the adsorbed fluid across the nanopore. The results obtained during several of the µV T − S simulations are shown in Fig. 3. At the start of the capillary condensation process, i.e. for the metastable phase of low density, the adsorbed fluid mostly consists of two layers close to the wall, as shown by the two peaks on the density profile for radii of about 4 σ (fluid layer closest to the wall) and 3 σ (second layer). As S * loading increases, Ar atoms start to populate the center of the nanopore, and a third peak for a radius around 2 σ develops (see the density profile in red for S * loading = 4290). We detect the formation of a fourth and then of a fifth peak when the system reaches the top of the free energy barrier (see density profile in green). Then, as S * loading further increases, the height of the peaks associated with the innermost fluid layers continue to increase, until the nanopore has completely filled and the system reaches the stable phase of high density.
We now more closely look at the radial density profiles of Fig. 3 to determine why, for intermediate values of S * loading , we find lower peaks found for the innermost fluid layers. This prompts the following question. Are these lower peaks associated with a uniform distribution of Ar atoms along the side of the tube (along the z direction)? Alternatively, do these lower peaks signal the formation of a liquid contact within the nanotube? To answer this question, we need to determine how the distribution of atoms evolves along the side of the cylindrical nanopore, i.e. along the z axis. For this purpose, we refine our structural analysis by computing the density profiles against both the radius and the z coordinate and show how the radial density varies along the nanopore. We show in Fig. 4(a) the results obtained for the metastable phase of low density. The plot for the density profile shows that the atoms adsorbed are very predominantly located close to the wall. More specifically, the fluid is organized into two layers close to the wall as shown by the two bright lines (i.e. regions of high density) on the density profile of Fig. 4(a). This is consistent with the profiles of Fig. 3 that showed the formation of two layers close to the wall. The additional information brought by Fig. 4(a) is the fact that Ar atoms in these two layers are distributed uniformly along z. We also show in Fig. 4(a) a snapshot of a configuration of the system for the metastable phase of low density and highlight in cyan the region where the void dominates (we define void any region of space for which the reduced density is below a threshold value).
We carry out the same analysis along the rest of the capillary condensation pathway. At the top of the free energy barrier, we obtain the density profile and snapshot shown in Fig. 4(b). Both show that, at this stage, the Ar atoms located in the inner part of the nanopore are not uniformly distributed along z and that a liquid-like region has developed for a value of z around 20 σ on the density profile. This can best be seen on the density profile with the onset, for z = 20, of 3 bright spots for r around 0, 1 and 2, that indicate the formation of 3 partial liquid layers. The snapshot of a configuration of the system at the top of the free energy barrier (see Fig. 4(b)) confirms that a liquid bridge has developed across the nanopore for this value of z = 20. As the capillary condensation further proceeds, the liquid bridge becomes wider and wider along the nanopore, resulting in a recess of the void region. Finally, as the system reaches the stable high density phase, we see that the structure of the adsorbed fluid exhibits a uniform structure with multiple layers (see Fig. 4(c)). The mechanism observed for capillary condensation is thus consistent with the scenario proposed by Everett and Haynes 58 for a macroscopic capillary and the results from gauge cell Monte Carlo simulations from Neimark et al. 54 , which both pointed to a liquid bridge-mediated capillary condensation process. We add that the free energy barriers are of the same order as those previously obtained for the capillary condensation process by Vishnyakov and Neimark 54 using the gauge cell method.
We now turn to the results obtained for the capillary evaporation process (µ = −10.54 ).
We show in Fig. 5 the free energy profile obtained as the system moves from a metastable phase of high density to the stable phase of low density. In line with our results for the capillary condensation process, we set the origin for the free energy axis to the free energy of the metastable phase. The main difference with the case of the condensation process, studied in the first part of the paper, is that we now carry out a series of µV T − S simulations with decreasing values for the total entropy of the system. Therefore, S * loading now varies in the opposite sense to the one followed for the condensation process. Looking at the free energy profile as a function of S * loading , we find three successive regimes for the variations of the free energy of the system. First, for S * loading varying from 7000 (metastable phase of high density) to 6500 (top of the free energy barrier), we observe a steep increase in free energy. Then, for S * loading between 6500 and 4800, the free energy profile exhibits a much more gradual behavior, as the free energy slowly decreases with the entropy of the system, in an almost linear fashion. The third regime is observed for lower values of S * loading (between 4800 and 2900). For this range of entropies, the free energy decreases more rapidly with entropy, until the system reaches the stable phase of low density. At this point, the free energy of the system is 55 k B T lower than the metastable state from which the evaporation pathway started. This confirms that the system has reached a thermodynamically stable phase at the end of the µV T − S simulations.
We now analyze the structure of the adsorbed fluid along the evaporation pathway. Fig. 6 shows the density profiles across the cylindrical nanopore for different values of the entropy.
The starting point for the µV T − S simulations displays the expected multi-peaked density profile. This density profile is consistent with the organization of the metastable phase of high density in multiple fluid layers within the nanopore (S * loading = 6980). Close to the top of the free energy barrier (S * loading = 4761), we observe a small, but noticeable, increase in the number of Ar atoms adsorbed in the nanopore. This indicates that the capillary evaporation process starts with the adsorption of a few extra Ar atoms, amounting to about 1−2% of the overall number of Ar atoms within the system. These extra Ar atoms destabilize the organization within the nanopore and trigger the desorption process, as shown by the decrease in the amplitude of the peaks in the density profile that follows (see e.g. the density profile obtained for S * loading = 5750). This leads to a decrease in the free energy as S * loading further decreases. Then, at the end of the second regime found in the free energy plot, we observe that the density profile exhibits two main peaks, the third inner most layer showing a reduced fluid density (S * loading = 4761). Finally, we recover the structure of the metastable phase of low density (see the density profile for at S * loading = 2887), for which the Ar atoms are very predominantly adsorbed close to the wall of the cylindrical nanopore.
We finally comment on the local structure within the fluid along the side of the nanopore.
We observe that once the organization of the high density phase has been destabilized, capillary evaporation proceeds through the formation of a bubble within the nanopore.
This can best be seen in Fig. 7 for S * loading = 5750. The density profile of Fig. 7 shows that a dark spot develops for values of z around 15 and that the innermost layers of the adsorbed fluid start to be depleted. Furthermore, this plot establishes that a non-uniform distribution of Ar atoms has formed along the side of the nanopore. In this case, the dark spots on the density profile on the left of Fig. 7, as well as the reduction in the height of the peaks for the innermost layers in Fig. 6, are due to the formation of a bubble, as shown through the low density region indicated on the snapshot on the right of Fig. 7.
V. CONCLUSIONS
In this work, we carry out µV T − S simulations to elucidate the capillary condensation and evaporation processes. Using the total entropy for the adsorbate, we are able to drive these processes along the entropic pathways underlying these processes. This allows us to shed light on the structural changes that occur within the confined fluid and give rise to the phase transition. Considering the example of Argon adsorbed in a smooth cylindrical nanopore typical of the MCM-41 silica adsorbent, we start by focusing on capillary condensation. Our simulations allow us to identify a complex free energy profile, corresponding to the successive stages in the condensation process. Our results show that capillary condensation from a metastable phase of low density starts with the nucleation of a liquid bridge within the nanopore. This liquid bridge then expands, becomes wider and wider, as the total entropy of the system increases, and finally takes over the system to yield the stable phase of high density. Applying this approach to the phenomenon of capillary evaporation similarly uncovers a multi-stage process. In this case, the layered structure of the metastable phase of high density is first destabilized by the adsorption of a few extra atoms. This is then followed by the nucleation of a bubble within the nanopore. Overall, the mechanism identified for capillary condensation is consistent with the scenario of Everett and Haynes 58 for a macroscopic system and with gauge cell Monte Carlo simulations by Neimark et al. 54 . The results obtained here further establish the key role played by the nucleation of liquid bridges in the capillary condensation process, and of the nucleation of bubbles during capillary evaporation, and shed light on the interplay between the entropy and the structure of the adsorbed fluid throughout these processes.
have started to shed light on the formation of solid-like and liquid-like junctions in nanopores and on the dynamics of cavitation between solid surfaces. In particular, Neimark and coworkers 51-57 have examined the capillary condensation process in cylindrical nanopores and have shown that the scenario proposed by Everett and Haynes 58 for a macroscopic capillary could also extend to nanosized capillaries. This suggested a very complex and intriguing pathway for the capillary condensation process, involving a series of structural changes in the adsorbed fluid and the formation of a liquid bridge across the pore section as an intermediate towards the high density adsorbed phase.
ric series. The parameters for the solid-fluid interactions were taken as ρ s sf = 2253 K/nm 2 and σ sf = 3.17Å. This functional form accurately models the interaction between adsorbate and the structureless cylindrical layer of adsorption centers on the pore wall as shown by Ravikovitch et al. 70 . Following Vishnyakov and Neimark 54 , we carry out simulations of capillary condensation and evaporation in nanopores with a long lateral dimension L z = 30σ to allow for the sampling of symmetry breaking configurations containing bubbles and liquid bridges. Periodic boundary conditions are applied along this lateral direction z. We also calculate explicitly the interactions between Argon atoms up to a distance of 5σ and neglect the fluid-fluid interactions beyond that cutoff distance. Finally, in the rest of this work, we use the conventional set of reduced units 71 , with respect to the Lennard-Jones parameters of the fluid.
µV T − S simulations are carried out in the grand-canonical ensemble (µ, V, T ), i.e. at constant chemical potential µ, temperature T and volume V . This statistical ensemble is especially well suited to study adsorption phenomena, since the number of atoms N adsorbed in the nanopore is allowed to vary in the (µ, V, T ) ensemble. Thus, this ensemble mimics what is observed experimentally during the adsorption/desorption processes. This ensemble has also the advantage of providing a direct way of estimating the entropy of the confined fluid. Taking the total Helmoltz free energy A = U − T S of the confined fluid to be equal to N µ, we obtain the following equation for the entropy of the adsorbed fluid S loading
Carlo simulations. In practice, for capillary condensation, we perform a series of µV T − S simulations with increasing values for the target entropy S 0 . This promotes the uptake of additional Ar atoms since the total entropy of the system is a function of N . Similarly, the pathway for the desorption process is sampled by carrying out successive µV T − S simulations, with decreasing values for the target entropies S 0 .Previous work 73 has shown that the conditions for liquid-vapor equilibrium in the pore are obtained for µ = −10.53 . To observe the capillary condensation/evaporation processes, the chemical potential must be close enough to the chemical potential at equilibrium. More specifically, to simulate the capillary condensation process, we carry out µV T −S simulations for a chemical potential of µ = −10.48 , while for the evaporation process, we use a chemical potential of µ = −10.54 . These two values for µ show that our findings are consistent with the results from Peterson and Gubbins 73 . As expected, these two values of µ bracket the estimate of Peterson and Gubbins for the chemical potential at the liquid-vapor equilibrium,
FIG. 1 :
1Capillary condensation (µ = −10.48 ): variation of the number of atoms adsorbed N (top) and of the interaction energy for the confined fluid U * potential (bottom) as a function of the total entropy of the adsorbed fluid S * loading . FIG. 2: Free energy profile for the capillary condensation at µ = −10.48 . The origin for the energy is set to 0 for the starting point (metastable phase of low density -see the local minimum on the left of the plot). After condensation, the system reaches a free energy minimum corresponding to the stable phase of high density (see the minimum on the right of the plot). FIG. 3: Capillary condensation (µ = −10.48): density profiles across the nanopore (r = 0 denotes the center of the pore) for increasing values of S * loading . The profile in black (S * loading = 3364) is for the metastable phase of low density, while the profile in cyan (S * loading = 6859) corresponds to the stable phase of high density.
FIG. 4 :
4Mechanism underlying capillary condensation. Radial density profile along the side (z) of the nanopore for increasing values of the entropy S * loading = 3364 in (a), S * loading = 4929 in (b) and S * loading = 6859 in (c), together with the corresponding snapshots. In the snapshots, Ar atoms are shown as orange spheres, while the regions of low density are highlighted in cyan. The bright spots on the left of the density profile around z = 25 indicate the formation of the liquid bridge in (b), as shown by the discontinuity of the cyan region in the corresponding snapshot. FIG. 5: Free energy profile for the capillary evaporation at µ = −10.54 . The origin for the energy is set to 0 for the starting point (metastable phase of high density). After evaporation, the system reaches a free energy minimum corresponding to the stable phase of low density.
FIG. 6 :
6Capillary evaporation (µ = −10.54 ): density profiles across the nanopore (r = 0 denotes the center of the pore) for decreasing values of S * loading . The profile in black (S * loading = 6980) is for the metastable phase of high density, while the profile in cyan (S * loading = 2887) corresponds to the stable phase of low density.
FIG. 7 :
7Bubble formation during capillary evaporation. Radial density profile along the side (z) of the nanopore for S * loading = 5750 and its corresponding snapshot. The darker spots on the left of the density profile around z = 15 indicate the formation of the bubble, which is highlighted in cyan in the snapshot.
Acknowledgements Partial funding for this research was provided by NSF through CAREER award DMR-1052808. Acknowledgement is made to the Donors of the American
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|
[] |
[
"Graph sampler : a simple tool for fully Bayesian analyses of DAG-models",
"Graph sampler : a simple tool for fully Bayesian analyses of DAG-models"
] |
[
"Sagnik Datta [email protected] ",
"Ghislaine Gayraud [email protected] ",
"Eric Leclerc [email protected] ",
"Frederic Y Bois ",
"Sagnik Datta ",
"Ghislaine Gayraud ",
"Eric Leclerc ",
"Frederic Y Bois ",
"\nInstitute of Industrial Science\nLIMMS/CNRS-IIS (UMI 2820\nSorbonne Universités\nUniversité de Technologie de Compiègne\nBMBI (S. Datta)\nLMAC (G. Gayraud) Bât. G.I\n60 319, 60 203Compiègne cedexCSFrance\n",
"\nThe University of Tokyo\n4-6-1 Komaba, Meguro-ku153-8505TokyoJapan\n",
"\nINERIS Parc ALATA\nBP2 60550Verneuil en HalatteFrance\n"
] |
[
"Institute of Industrial Science\nLIMMS/CNRS-IIS (UMI 2820\nSorbonne Universités\nUniversité de Technologie de Compiègne\nBMBI (S. Datta)\nLMAC (G. Gayraud) Bât. G.I\n60 319, 60 203Compiègne cedexCSFrance",
"The University of Tokyo\n4-6-1 Komaba, Meguro-ku153-8505TokyoJapan",
"INERIS Parc ALATA\nBP2 60550Verneuil en HalatteFrance"
] |
[] |
Bayesian networks (BNs) are widely used graphical models usable to draw statistical inference about Directed acyclic graphs (DAGs). We presented here Graph sampler a fast free C language software for structural inference on BNs. Graph sampler uses a fully Bayesian approach in which the marginal likelihood of the data and prior information about the network structure are considered. This new software can handle both the continuous as well discrete data and based on the data type two different models are formulated. The software also provides a wide variety of structure priors which can be informative or uninformative. We proposed a new and much faster jumping kernel strategy in the Metropolis-Hastings algorithm. The source C code distributed is very compact, fast, uses low memory and disk storage. We performed out several analyses based on different simulated data sets and synthetic as well as real networks to discuss the performance of Graph sampler.Keywords Bayesian networks · structure learning · posterior distribution · MCMC · Metropolis-Hasting algorithm Acknowledgment S.Datta is funded by a Ph.D. studentship for the French Ministry of Research.
|
10.1007/s00180-017-0719-1
|
[
"https://arxiv.org/pdf/1505.07228v2.pdf"
] | 62,090,733 |
1505.07228
|
5f22d4890631b0b4dd943db3b10235ab737442e1
|
Graph sampler : a simple tool for fully Bayesian analyses of DAG-models
4 Feb 2016
Sagnik Datta [email protected]
Ghislaine Gayraud [email protected]
Eric Leclerc [email protected]
Frederic Y Bois
Sagnik Datta
Ghislaine Gayraud
Eric Leclerc
Frederic Y Bois
Institute of Industrial Science
LIMMS/CNRS-IIS (UMI 2820
Sorbonne Universités
Université de Technologie de Compiègne
BMBI (S. Datta)
LMAC (G. Gayraud) Bât. G.I
60 319, 60 203Compiègne cedexCSFrance
The University of Tokyo
4-6-1 Komaba, Meguro-ku153-8505TokyoJapan
INERIS Parc ALATA
BP2 60550Verneuil en HalatteFrance
Graph sampler : a simple tool for fully Bayesian analyses of DAG-models
4 Feb 2016Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor) 2 Sagnik Datta et al.
Bayesian networks (BNs) are widely used graphical models usable to draw statistical inference about Directed acyclic graphs (DAGs). We presented here Graph sampler a fast free C language software for structural inference on BNs. Graph sampler uses a fully Bayesian approach in which the marginal likelihood of the data and prior information about the network structure are considered. This new software can handle both the continuous as well discrete data and based on the data type two different models are formulated. The software also provides a wide variety of structure priors which can be informative or uninformative. We proposed a new and much faster jumping kernel strategy in the Metropolis-Hastings algorithm. The source C code distributed is very compact, fast, uses low memory and disk storage. We performed out several analyses based on different simulated data sets and synthetic as well as real networks to discuss the performance of Graph sampler.Keywords Bayesian networks · structure learning · posterior distribution · MCMC · Metropolis-Hasting algorithm Acknowledgment S.Datta is funded by a Ph.D. studentship for the French Ministry of Research.
Introduction
Representing knowledge with uncertainty and automatic reasoning is often carried out using graphical models (Pearl, 1988;Lauritzen, 1996;Neapolitan, 1990). Judea Pearl and Richard E. Neapolitan were the first to summarize the properties of directed acyclic graphs (DAGs) and established them as a new field of study. In the recent years, formal statistical inference on systems of multiple interacting components is often done using DAGs (Heckerman et al, 1995) or Markov networks (Edwards, 2000). A Bayesian network (BN) or a belief network is a probabilistic model denoted by a graph G = (V, E) in which each node or vertex v ∈ V represents one of the random variables in set X = (X 1 , X 2 , ..., X N ), where N is the number of nodes and each edge e ∈ E express the dependence among the variables in X. A BN is always directed and acyclic and is therefore a DAG. Besides static BNs, there are also dynamic BNs (Husmeier, 2003;Friedman et al, 1998), which are actually generalizations of hidden Markov models. In that paradigm, all the random variables of the network are considered to be potentially related to each other over adjacent time points.
DAGs have been used for more than two decades in biomedicine and healthcare for handling uncertainty in disease diagnosis, selecting the optimal treatment and predicting treatment outcome (Andreassen et al, 1999). Other applications are found in social network analyses, high dimensional data analyses etc. In computational biology and bioinformatics, BNs have been proposed to model DAGs, for example, for modeling gene regulatory networks, protein structure and gene expression. Many of the available software (Korb and Nicholson, 2010) programs which do BN parameter estimation or structural inference run on commercial platforms. The others which are readily available and free are generally not well maintained and not updated regularly. However there are some widely used packages scripted in R that are free and properly maintained.
In this article we present an efficient C language software Graph sampler for Bayesian inference on the structure of BNs; this latter is based on the wellknown Metropolis-Hastings(MH) algorithm that allows to sample DAGs from the posterior distribution. Unlike the existing BN learning software Graph sampler propose a new jumping kernel in the MH algorithm making it more time efficient. The software is easy to use and works well with both discrete and continuous data. For each of these data types we formulated two different models. One of the key feature of this software is that it handles large network structure efficiently. A number of different priors are included in this software to reflect the prior knowledge on the graph structure. This software is quite flexible for using different combination of prior knowledge. We made several performance test on our software with intensive simulation studies regarding all possible choices of models and priors to venture the robustness of Graph sampler. Alongside we considered structmcmc, a versatile and easy to use R package (Mukherjee and Speed, 2008) for detailed discussion on the performance of Graph sampler. Like Graph sampler, this software also aims to learn structure on DAGs via a fully Bayesian approach.
The remainder of the paper is organized as follows: Section 2 briefly discusses the statistical inference for BNs with Graph sampler ; Section 3 is about the new jumping kernel, the installation and running of Graph sampler ; Section 4 discusses about some of the widely used software and their approach. This section also describes the methodology to check the efficiency of Graph sampler for discrete as well as continuous data set along with the difficulties in the flipping technique. In Section 5 we present our results along with their significance. In Section 6 we summarize all our results in the form of a discussion.
Statistical inference for BNs
Recent work on BNs by Mukherjee and Speed (Mukherjee and Speed, 2008), used Markov chain Monte Carlo (MCMC) simulations to infer on network structure from node values. They also considered priors encoding information relative to existence of edges, degree distribution and sparsity structure of the graph. Bois and Gayraud (2015) recently extended informative priors to account for motifs frequencies in order to generate realistic gene regulating graphs (Bois and Gayraud, 2015). For baseline information they proposed to use Bernoulli distributions to model prior knowledge on individual edges.
Variational Bayes (VB) methods have also been proposed as an alternative to full Bayesian inference. VB is an efficient way to deal with intractable integrals arising in a full Bayesian context and can be considered as an entension of the expectation-maximization (EM) algorithm (Beal and Ghahramani, 2003). VB can also help in model selection by providing a lower bound for the data marginal likelihood. However it provides only an approximate analytical solution for the posterior probability of the parameters and latent variables involved in a graphical model. Hence we focus here on a full Bayesian approach.
The following sections describe the prior densities and data likelihood available in Graph sampler.
Priors on graph structure
It is quite a challenging problem to make inference on graphical model structure even with moderate number of nodes, partly because of the huge number of possible graphs. For N nodes, the number of possible DAGs can be computed recursively as (Robinson, 1973):
a N = N k=1 (−1) k−1 N k 2 k(N −k) a N −k
where a 0 = 1 by convention. So for 2, 3, 4 and 5 nodes network there are 3, 25, 543 and 29281 possible DAGs respectively.
All of those possible graphs are usually not equally plausible a priori and thus certain features may be incorporated as more likely than the others, a priori. To make inference on graph structure, Bois and Gayraud (Bois and Gayraud, 2015) considered three different priors. In the most general case, a set of independent Bernoulli priors (P B ) is used to model the prior knowledge on individual edges. If p i,j is the probability of existence of a directed edge e i,j from node i to node j, and e i,j ∼ B(p i,j ) for all (i, j) ∈ {1, ..., N } × {1, ..., N }, then the global Bernoulli prior P B on the graph G is
P B (G) = N i =j=1 (p i,j ) ei,j (1 − p i,j ) 1−ei,j
The choice of the value for each hyperparameter p i,j depends on the prior evidence we have on the existence of the given edge from the scientific literature.
The degree deg(v) of a vertex v is defined as the number of edges involving v. We can also define the degree distribution (P D ) for G as a function P d = card{v ∈ V : deg(v) = d} for all the vertices having degree d. The prior on degree can be expressed as a power law given by
P d ∝ d −γ , with γ > 0
Thus we can define the prior degree distribution for the graph G as
P D (G) ∝ N i=1 N j=1 e i,j −γ , with N j=1 e i,j > 0
In addition to P B and P D , if we consider a Beta-Binomial prior (P M ) as it is done in Bois and Gayraud (2015) on the occurence of three-nodes motifs then the total prior on the graph G can be expressed in a product form as:
P T (G) ∝ P B (G) × P D (G) × P M (G)
Alternative to the Bernoulli prior for the presence of edges is the so called concordance prior (P C ) (see Mukherjee and Speed (2008)). The latter required the specification of a prior matrix E with elements E i,j = 1 representing a desired edge and -1 representing a non-desired edge. At each iteration the prior is calculated by counting the number of disagreements with the adjacency matrix A with elements A i,j = 1 or 0 representing the presence or absence of a directed edge starting from node i and ending in node j of the graph G. The form of the concordance prior is then
P C (G) ∝ exp(−ρ( N i,j=1 |A i,j − E i,j |))
where ρ is a positive valued hyperparameter. If P C is used, then the total prior is: or, equivalently, if P B is a flat prior (with p i,j = 0.5 for i = j) to avoid any conflict or double accounting with P C , the total prior is:
P T (G) ∝ P B (G) × P C (G) × P D (G) × P M (G)
(1)
Data likelihood and prior predictive distribution
Our main interest is to uncover the underlying structure of BNs. In any Bayesian network, a parent node has always an influence on its child nodes. Let us denote by x = (x 1 , ..., x N ) the data we have on N nodes, where x i is a n-dimensional vector; where n is the number of data points per node. Even though the model considered involves many parameters, the posterior distribution of each of these parameters are not of our primary interest. Integrating out the parameters in a Bayesian context leads to the prior predictive distribution (or the joint marginal likelihood of the data):
f (x|G) = N i=1 f (x i |P a (x i )),
where P a (x i ) is the set of parent values of x i in graph G (Heckerman et al, 1995) and f (·) is the prior predictive distribution of x i given its parenthood. For a global parent P a (x i ) = ∅ and thus f (x i |P a (x i )) reduces to f (x i ).
Continuous data
Gaussian regression model with a Normal-Gamma prior
The general expression for the linear Gausian regression for a given node x i is:
x i = M (x i )β + u(2)
where x i = (x 1,i , x 2,i , · · · , x n,i ) is a vector of n observations of the dependent variable x = ((x i,j ) 1≤i≤n;1≤j≤k ), k is the cardinality of P a (x i ), M (x i ) is a so-called design matrix of order (n × (k+1)) with the first column as 1's and other columns as P a (x i ), β is a real valued vector of regression parameters of length k, and u follows a Gaussian distribution N (0, λ −1 I n ) distribution with λ being a positive real valued precision and I n is the identity matrix of size n.
The likelihood for this regression model is therefore:
L(λ, β|x i , P a (x i )) = ( λ 2π ) n/2 exp(− λ 2 (x i − M (x i )β) t (x i − M (x i )β))
where C t denotes the transpose of the matrix C.
It is classical to choose conjugate form for the priors on the parameters (i.e. β and λ) involved in the regression model :
P (β, λ) = N k (β | β 0 , (n 0 λ) −1 )Ga(λ | α, ω) = (ω) α (n 0 ) k/2 (2π) k/2 Γ (α) (λ) k 2 +α−1 exp − λ 2 (β − β 0 ) t n 0 (β − β 0 ) − ωλ
where β 0 (real valued vector) and n 0 (matrix of dimension k×k) are hyperparameters related to β and α, ω, both being real valued positive numbers, are hyper-parameters of λ. In the above equation Γ (.) represents the Gamma function.
In that case, the prior predictive distribution is t ν (µ, Σ), the n-dimensional multivariate t-distribution with parameters µ, Σ and ν, whose density function is:
f (x i |P a (x i )) = Γ (ν + n)/2 Γ (ν/2)(νπ) n/2 | Σ | −1/2 1 + 1 ν (x i − µ) t Σ −1 (x i − µ) − ν+n 2(3)
where, µ = [µ 1 , . . . , µ n ] t is the location parameter, Σ is the scalar matrix of dimension (k × k) and ν is the degrees of freedom such that
µ = M (x i )β 0 Σ −1 = h(M (x i ))αω −1 h(M (x i )) = I n − M (x i )[M (x i ) t M (x i ) + n 0 ] −1 M (x i ) t ν = 2α
Gaussian regression model with a Zellner g-prior
With Graph sampler we can also use the Zellner g-prior (Mukherjee and Speed, 2008;Nott and Green, 2004) for the parameters β and (λ) −1 :
P (β|λ −1 ) = N k (0, gλ −1 [M (x i ) t M (x i )] −1 ), P (λ −1 ) ∝ λ
where g is a user defined positive scale factor. The prior predictive distribution of the data is given by:
f (x i |P a (x i )) ∝ (1 + g) −(k+1)/2 s −n/2(4)
where k is the cardinality of P a (x i ) and
s = x i x i − g 1 + g x t i M (x i )[M (x i ) t M (x i )] −1 M (x i ) t x i provided the term M (x i ) t M (x i ) is invertible. A sufficient condition for M (x i ) t M (x i )
to be invertible is that k n, that is, the number of parents should be less than the number of data points for per node. Thus Zellner g-prior fails to work when number of parents is greater than the number of data points per node.
Discrete data
For discrete data, Graph sampler offers the possibility to use a Multinomial model with a Dirichlet prior on its parameters (see (Heckerman et al, 1995)). For such a prior on the parameters we have a closed form of the prior predictive:
f (x|G) = n i=1 si j=1 Γ (D ij ) Γ (D ij + D ij ) · mi k=1 Γ (D ijk + D ij ) Γ (D ijk )(5)
where D ijk is the number of components of x i that takes the value k given that P a (x i ) has configuration j, D ijk are the Dirichlet hyperparameters, s i represents the possible number of configurations of the parents of x i , m i stands for the number of possible values of components of x i and D ij and D ij are given by:
D ij = mi k=1 D ijk and D ij = mi k=1 D ijk .
3 Graph sampler : sampling and installation
Efficient sampling: fast jumping kernel
Graph sampler can efficiently generate random samples for general directed graphs (Bois and Gayraud, 2015), but we focus here on the sampling of BNs from a posterior distribution conditionned by data (observed node values). We use an adjacency matrix representation for the graph and store only the eventual difference between adjacency matrix as it is a fast and efficient storage method. A Metropolis-Hasting sampler (Robert and Casella, 2004) is used to sample random graphs according to the prescribed posterior probability distribution. The algorithm of the simplified jumping kernel is as follows for the t − th iteration: We denote the current graph by G and its adjacency matrix by A G t . The proposal graph is denoted by G and its adjacency matrix by A G in our algorithm.
Algorithm
Step 1: Select
A G t i,j while scanning A G t Step 2: (a): Sample z i,j ∼ Bernoulli(p i,j ) where p i,j is the Bernoulli prior for edge e i,j (b): (i): Adding an edge if z i,j = 1, then A G i,j = A G t i,j if A G t i,j = 1 1 o.w. and provided G is still a DAG, o.w. go back to Step 1 (i): Deleting an edge if z i,j = 0, then A G i,j = A G t i,j if A G t i,j = 0 0 if A G t i,j = 1
Step 3: Calculate the acceptance ratio. We accept G with probability
δ = min(1, f (x, G )P T (G )P (A G t i,j |A G i,j ) f (x, G t )P T (G t )P (A G i,j |A G t i,j ) )
Note that due to Step 2, the acceptance ratio can be rewritten as follows:
δ = min(1, f (x|G )P T (G )P B (G t ) f (x|G t )P T (G t )P B (G ) ) where f (x|G) = N i=1 f (x i |P a (x i ))
is the prior predictive given by Eq (3, 4 and 5) and P T is the total prior on the graph structure. Clearly this simplifies since P B is a part of P T (see Eq (1)).
Step 4: Choose G t+1 as follows: G t+1 = G with probability δ G t with probability 1-δ The procedure is repeated until convergence in probability is attained. Gelman and Rubin's (GR)R criterion (Gelman and Rubin, 1992) is used on each element of the graph's adjacency matrix to check the convergence of several simulation chains. The advantage of using the GR criterion for the convergence is that, we do not have to save the whole chains from the start. We can check the convergence using the final posterior edge probability matrix obtained after the specified number of iterations neglecting the burning runs. For this criterion, we consider the three edge probability matrices and calculate the within-chain and the between-chain variance. Then the estimated variance of the parameter is calculated as a weighted sum of the within-chain and between-chain variance. Based on the potential scale reduction factor we infer on the convergence of the three chains. For BNs we need to ensure that the proposed graphs are DAGs. This is done with a fast topological sorting algorithm (similar to that of (Pearce and Kelly, 2006)) operating on a list index of the nodes.
Graph sampler installation
Graph sampler is an easily available free software that can be redistributed or modified under the terms of the GNU General Public License as published by the Free Software Foundation. It is an inference as well as simulation tool for DAGs and can simulate random graphs for general directed graphs as well as for DAGs. In the case of BNs, we infer about their probable structure through the joint use of priors and data about node values.
Graph sampler is written in ANSI-standard C language and can be compiled in any system having a ANSI C compliant compiler. The GNU gcc compiler (freeware) is highly recommended and the automated compilation script (called Makefile) can be successfully used if the standard 'make' command is available. In order to modify the input file parser, the 'lex' and ' yacc' are highly recommended. The full software along with the manual can be downloaded from:
https://sites.google.com/site/utcchairmmbsptp/software Once downloaded, the software should be decompressed using 'gunzip' and 'tar' commands. Other archiving tools can also be used. Graph sampler can be compiled using the 'make' command. On successful compilation of Graph sampler, it is ready for running. In order to run Graph sampler, an input file specifying the simulation parameters should be provided. In Unix the command-line syntax to run that executable is:
"graph sampler [input-file [output-prefix]] "
where the brackets indicate optional arguments. If no input file and/or output prefix are not specified, the program uses the defaults. The default input file is script.txt and the output files created depends on the parameters specified in the input file. Default output file names are best graph.out, graph samples.out, degree count.out, motifs count.out, edge p.out and results mcmc.bin.
A Graph sampler input file is a text (ASCII) file that obeys relatively simple syntax (see the manual). Values of all the predefined variables in the input file should be properly defined. Description and range of each variable is illustrated in the manual. In case of improper assignment of values, Graph sampler post error messages during runtime.
Efficiency analyses of Graph sampler
In order to evaluate the efficiency and accuracy of Graph sampler, we performed several experimental runs based on different network structure, size and data type. We even varied the underlying model based on the data type to have a clear idea about the efficiency of the software. There are other very well known packages in R like the deal and bnlearn. The package deal (Boettcher and Dethlefsen, 2003) is scripted in R language and uses BNs to analyse the data which can be discrete and/or continuous types; for the network parameters, suitable priors can be constructed and parameter estimation is possible using successive updating. This package is useful for structure learning of the network and uses the heuristic greedy search algorithm. The scoring function is based on maximising the Bayes factor. At each step of the greedy search algorithm, we either add, delete or reverse an edge and calculate the Bayes factor for all the possible graphs. The proposal graph with the maximum Bayes factor is selected to update the current graph. The package bnlearn (Scutari, 2010) is also scripted in R to do structural inference on BNs. This package is efficient to work with both discrete as well as continuous data. For the BN structure learning, various constraint-based algorithms like the Grow-shrink, Incremental association and Max-min parents and children algorithms are implemented. Hill-Climbing greedy search algorithm is the only score-based algorithm implemented in this package. The sampling algorithm consists of first recovering the skeleton of the desired graph and then evaluating the directionality of the edges. Both of these packages sample the best graph that maximizes the Bayes factor or the posterior odds from amongst all possible proposal graphs. This is quite different from the sampling scheme that we proposed. Our interest is to sample graphs based on the specified posterior distribution. Another efficient Matlab package for BN analysis is bnt (Murphy, 2007). This tool does inference on parameters as well as on network structure. This software is suitable for both decision networks and dynamic Bayesian networks. However this tool lacks the GUI. Inference is carried out using various algorithms like the Junction tree, variable elimination and Pearl's polytree. bnt has various options for parameter learning as well as for structure learning. It is very clearly documented, free and is very object-oriented. Besides these well known packages, we also have structmcmc (Mukherjee and Speed, 2008) which is an efficient software coded in R for BN structure learning. We selected structmcmc because of its similarities with Graph sampler regarding the model formulation and the Metropolis-Hasting algorithm. Unlike Graph sampler, the sampling technique in structmcmc also allows flipping of edges between two selected nodes. This package also propose to sample graphs based on the posterior distribution. Thus we selected structmcmc as a baseline software to discuss in details the advantages and limitations of Graph sampler. structmcmc proposes a Zellner g-prior for continuous datasets and a Multivariate Dirichlet prior for discrete datasets. The Normal Gamma model is not available to structmcmc users.
In all the cases we used a null matrix as the initial adjacency matrix. Results showed that they are robust with respect to the initial adjacency matrix. For the prior on edges, we used the concordance prior (Mukherjee and Speed, 2008) (P C ) with ρ = 1 since structmcmc does not provide a Bernoulli prior. The prior on the loop motifs P M was not used and therefore set to 1. We also used the degree prior (P D ) to check the increase in efficiency of Graph sampler. As an alternative to (P C ), we also used both informative as well as uninformative (P B ) prior as a structure prior to check the efficiency of the software. We followed the simulation procedure described in (Mukherjee and Speed, 2008) to generate discrete datasets. For the continuous case, we generate data as described in Equation (2).
To start with we considered a real life biological network specifically the EGFR system. For this actual network we simulated a discrete dataset. This real life network consisted of only 14 nodes. In order to check efficiency of Graph sampler for larger networks, we simulated networks of 5 to 120 nodes, with 100 data points for each node. Figure 1 represents the network with 120 nodes. It is clearly a descending tree network. We used three different seeds to run three chains for each software program. We saved the three chains separately and calculated Gelman'sR convergence diagnostic at each iteration.
The first iteration for whichR attained at most 1.05 for all edges of the graph was considered as the minimum number of iterations required for convergence. Graph sampler was compiled with gcc version 4.2.1 (Apple Inc. build 5666) while structmcmc was run with R 3.0.2 (R Core Team, 2013). We performed a time and convergence comparison between both the software. In order to check the performance and efficiency of both the software, we use R language script to plot the heat map and the accuracy curve. The heat map is used to summarize for all edges the edge posterior distribution through its means. We used the R language package lattice to plot the heat maps. The accuracy curve is given by accuracy = (true positive edges + true negative edges)/ total number of edges and is a function of the probability threshold above which an edge is declared present. We used SDMTools, to plot the accuracy curves.
Dealing with Discrete data
An actual biological network
To study the practical efficiency of Graph sampler for a true biological network, we considered the same network as studied by S. Mukherjee (Mukherjee and Speed, 2008). In their work they considered a biological network known as the epidermal growth factor receptor (EGFR) system. Figure 2 gives a pictorial representation of the EGFR system. This biological network involves 14 proteins each of which is a ligand, a receptor or a cytosolic protein. Data for this study was synthesized based on the network and following the model as used by S. Mukherjee (Mukherjee and Speed, 2008
Fig. 2: Graph of the EGFR system
To make our experimental runs coherent with structmcmc, we defined the prior matrix on the graph structure based on the concordance prior proposed in (Mukherjee and Speed, 2008). As the model proposed was Multinomial model, we defined a Dirichlet prior for the parameters involved. In case of Graph sampler we strengthen the prior on the structure of the network by considering a degree prior with γ = 3. Figure 3 represent our comparative study in the form of heat maps and accuracy curve. We observe that for a small network like the EGFR system there is no significant difference in accuracy between Graph sampler and structmcmc for low threshold values but for higher threshold values, structmcmc has an advantage. Observing the heat maps we find that even though Graph sampler produces lower edge probability matrix but does not give rise to false negative edges. On the other hand structmcmc generates high edge probability matrix but leads to a higher number of false negative edges that were not present in the prior network. Thus we can take this as a trade off where we have to be careful while working with both of these software and decide accordingly. If we focus on time efficiency of the two software, we observe that Graph sampler is almost 200 times faster than structmcmc. Thus taking both the criteria together, we observe that Graph sampler has its own advantages and limitations like structmcmc. An alternative other than the Dirichlet prior on the parameter could be the Zellner g-prior with g-prior equals 1. Even though the Zellner g-prior should be used for continuous data, we did use it for the discrete data to check its efficiency. For a binary data set as described above, we can easily fit a linear regression model and thus fall back to the continuous scenario making the use of Zellner prior valid. We still used the priors P C and P D . Figure 4 represents the heat map and the accuracy curve for the two software. Comparing the heat map of figure 3(B) and figure 4(A), we observe that there is an improvement in the posterior edge probability matrix obtained from Graph sampler. There is also a reduction in the number of false negative edges and this is true even for structmcmc. As far as accuracy is concerned, both the software are quite efficient and there was no significant difference in accuracy between the two. However considering the time scale, Graph sampler is again 100 times after than structmcmc and thus have a slight advantage. Zellner g-prior, here the x-axis represents the parents and y-axis the corresponding children. Accuracy curve for the two software (C)
Simulated networks
We studied the performance of Graph sampler for discrete datasets, using the Multinomial model. Figure 5 gives a graphical summary of our timing results. Because of memory problems we could not achieve convergence with structmcmc for networks of more than 60 nodes. Similarly for 120 nodes, Graph sampler did not converge with a billion iterations. One of the reason behind this could be that with larger network size we have to increase the dataset. Graph sampler was about 100 times faster than the structmcmc for the same number of iterations. Graph sampler running time was also less influenced by the network size (i.e. it increased by a factor 4.06 when going from 5 to 60 nodes with Graph sampler and that factor being 16.5 for structmcmc).
With 30 nodes, Graph sampler took about 2 × 10 7 iterations (275 seconds) to converge while for structmcmc it was 10 6 iterations (3848 seconds). Thus structmcmc required 10 to 100 times less iterations to reach convergence but was about 14 times slower than Graph sampler . We compared the edge probability matrices from both the software by the accuracy curve to check whether they converge to the true graph. Figure 6 panel (D) shows the accuracy of the software in retrieving actual edges as a function of the probability threshold above which an edge is declared present. It was observed that with small threshold values structmcmc had higher accuracy than Graph sampler. We observe that with the single use of concordance prior the accuracy of Graph sampler in retrieving edges correctly is low for small threshold values. However with a threshold value above 0.5, both the software had almost the same accuracy. Altogether, Graph sampler reaches convergence faster in time and has equal accuracy as structmcmc for threshold values above 0.5. The accuracy of Graph sampler can be improved with the introduction of the degree prior (P D ) with γ = 2. As our network is a descending tree network, the inclusion of the degree prior is very beneficial. Figure 7(A) represents the posterior probabilities of the edges and resembles more like the true graph. It was observed that with the prior P D , the accuracy of Graph sampler improved significantly (Figure 7(B)) and even with very low threshold values, the accuracy was almost equal to 0.9. Fig. 7: Heat map of the posterior edge probabilities obtained from Graph sampler with discrete data with concordance and degree priors (A) and the improvement in the accuracy curve due to the use of degree prior in Graph sampler (B).
Number of iterations
Time (seconds) 1e+03 1e+05 1e+07 1e+09
Dealing with Continuous data
For continuous data, structmcmc uses a Zellner g-prior on regression parameters. With Graph sampler either a normal-gamma prior or a Zellner g-prior can be used. The two models are compared here.
Normal-gamma model
For the continuous data set, convergence with Graph sampler was achieved with almost 10 times less iteration number compared to structmcmc. It was observed that with networks having 60 nodes, structmcmc faced problems in convergence. Figure 8 represents our study in a graphical way. Regarding the time taken for iterations, Graph sampler was almost 10 times faster than structmcmc. For networks with more than 20 nodes, time efficiency of structmcmc decreases sharply. The very narrow band width (A) reveals that for Graph sampler the increase in network size does not have much influence on time. Figure 8 represents our study in a graphical way.
We also studied the posterior edge probabilities obtained from each of the software to draw inference on their efficiency to retrieve the true graph structure. We plotted the posterior edge probabilities in the form of heat maps to understand sampling scheme prescribed in Graph sampler. Figure 9 represents the three heat maps for a network with 30 nodes. We observe that Graph sampler perform well in retrieving edges present higher up in a tree network. However as we go down the tree structure, the efficiency of Graph sampler decreases. As our model used simulated data, so there is an underlying correlation in the dataset. For this reason as we go down the network Graph sampler uses a normal gamma prior while structmcmc uses a Zellner g-prior structure Graph sampler samples many new edges. Interestingly the directionality of the network structure is maintained. On the other hand structmcmc is efficient as we move down the network structure. However structmcmc fumble with the directionality of the edges. We plotted the accuracy curve for the two software. Figure 9(D) represents that the accuracy of structmcmc is slightly higher than that of Graph sampler for smaller threshold values. However this difference is not much and for higher threshold values, both the software have almost equal accuracy.
Strengthening the prior model by considering the hierarchy of the structure and defining a degree prior
The degree prior (P D ) with γ = 2 was introduced to check the improvement in the accuracy of Graph sampler. Figure 10(A) represents the posterior probabilities of the edges and resembles more like the graph in Figure 9(B). It was observed that with the prior P D , there was no significant improvement in the accuracy of Graph sampler (Figure 10(B)).
An alternative: Zellner g-prior model
For continuous datasets, we can also use the Zellner g-prior as used in structmcmc. Unlike structmcmc, we run Graph sampler for various g-prior values. We started with a g-prior of 1 for all the networks considered in our study. We checked the time required and the convergence point. Later we performed similar runs for different g-prior with values 5, 10, 50 and 100. In each case we observed the convergence rate and the posterior edge probabilities. With a g-prior value of 1 or 5, convergence was achieved for all the networks. As we increased the value of g-prior to 10, runs with networks having 5 and 10 nodes converged with the maximum value ofR being 1.12. This was not the case for networks with 20, 30, 40 and 60 nodes as they converged (R = 1.05 approx). However increasing the g-prior value to 50 and 100, convergence was not achieved for any of the networks. Thus in such a situation, Graph sampler and structmcmc differs. For smaller network sizes, Graph sampler performed well with g-prior value equal to 1 or 5. With bigger networks sizes having 100 data points for each nodes, the g-prior value can range from 1 to 40. Graph sampler failed to converge when the g-prior value was equal to the number of data points. On the other hand, structmcmc performed well with higher g-prior values and was most efficient when the g-prior value was equal to the number of data points.
In order to understand the reason behind such a difference in convergence rate between the two software, we discuss the flipping technique used in Graph sampler along with its advantages and disadvantages.
Problem with flips
The primary advantage of the reduced jump kernel (only adding or deleting one edge at a time) used in Graph sapmpler is that it is faster than the jump kernel allowing flips. Since the choice of pairs of nodes is systematic, there is no need to check the neighbourhood cardinality (Husmeier, 2003) as done in structmcmc.
This jump kernel has some drawbacks also. Consider a network with 5 nodes where node 4 is a parent of node 5. In the MCMC simulations, at a particular step, we propose to add an edge from node 5 to node 4. As the log posterior for the proposed network is quite high (if 4 conditions 5, the two are correlated and 5 conditioning 4 has high probability), we accept such a proposal. According to our flipping technique, in order to retrieve the true edge (from 4 to 5), we need to first delete the edge from 5 to 4, leaving them independent. However a network with 4 and 5 independent has low log posterior probability and we rarely accept such a move. Figure 11 is a dot plot where the blue dots and the red circles appearing in pairs represents the difference in log probability for a network when passing from an edge (4 to 5) to an edge (5 to 4) respectively using the jump kernel specified in Graph sampler. For a pair A, a move from the red dot (log probability -675) to a blue dot (log probability -676), the log probability has to pass through -720 (4 and 5 independent) making such a move impossible. So in the most likely regions of G the flip is very unlikely. For the pair B, a move from the red dot (log probability of -698) to a state of independence (log probability of -685) is easy, but the next move to a blue dot (log probability of -700) is not easy. This is also an unlikely region where some flips are possible. The flip for the pair C is unlikely to occur as the log probability has to pass by -687 while going from -678 to -681. Flips would occur easily when they are close to the diagonal.
The difficulty with the jump kernel can be due to the large data set for which the posterior mass favours fewer graphs. Under such a situation, the standard MCMC scheme faces difficulty in moving between graphs, or finding the high-scoring graphs. In such a case parallel tempering is a proficient option to speed up the MCMC-based convergence of network inference. This tempering approach is generally referred to as Model Composition by Metropolis-Coupled Markov Chain Monte Carlo (MC 4 ) (Barker et al, 2010). The parallel tempering involved in this MC 4 (beyond the scope of this paper) approach allows proper mixing of the Markov chain and helps to escape the local maxima.
Sensitivity to the prior on graph structure
To check the sensitivity of the normal-gamma model with respect to informative and non informative priors, we considered a network with 40 nodes having 100 data points for each node and defined only the P B priors on the edges. We first considered an informative prior on edges with each desired edge having a prior probability of 0.9 and 0.1 for others except for autoloops for which probability was 0. We define a less informative prior with 0.8 and 0.2 and carry out our experimental run. We repeated the process and finally defined a flat non informative prior of 0.5 for all the edges.
For each run with different prior probabilities, convergence was obtained in Graph sampler and then retrieved the posterior edge probabilities as it conveyed the information regarding the sensitivity of Graph sampler in selecting the desired graph out of all the equivalent graphs.
With a strong informative prior of 0.9 and 0.1, Graph sampler converged and was able to retrieve all the desired edges with a high probability. The posterior probabilities of some undesired edges were also high. This is mainly observed as we move down the network due to the presence of partial correlation between the nodes higher up in the network and those at the bottom. As we use less informative priors, this behaviour becomes more prominent and the efficiency of Graph sampler decreases (Figure 12). With a flat prior of 0.5 for all the edges, the efficiency of Graph sampler is the least. Figure 1 and the heat map of Figure 12(A) show that the true network is a descending tree (the upper triangular matrix of the heat map has zero edge probability). For the informative prior, we observe that the normal-gamma model works well log probability of 4 independent of 5 log probability of 4<-->5 for the upper part of the tree network. However as we descend down the tree, the sensitivity of the model decreases as more children are involved. One way to increase the performance of the model can be by increasing the number of data points for each node present in the lower part of the network. We plotted the accuracy curve to depict the efficiency of Graph sampler for the various informative priors. Figure 12(D) represents the accuracy curve of Graph sampler. We observed that Graph sampler was very versatile with the type of prior information provided. With very strong prior the accuracy was almost equal to 0.9 for threshold values above 0.2. This stated that Graph sampler was efficient enough to retrieve the true edges with higher posterior probabilies and allot low probability (less than the threshold) to false edges. For noninformative priors Graph sampler had an accuracy of 0.65 for threshold below 0.3. The accuracy increased to 0.8 and above with the increase in threshold from 0.5 to 1.0.
We checked the sensitivity of Graph sampler for degree prior (P D ) with a noninformative Bernoulli prior on edges. We observed that there was not much improvement in the accuracy of Graph sampler in this case ( Figure 13).
Discussion
We observed that like other widely used software (bnlearn, deal ), Graph sampler works well with both discrete as well as continuous data. For continuous data, we considered a regression model and we fitted a multinomial model for the discrete data. These two model representations are commonly used when dealing with BNs. For continuous data, Graph sampler allow the user to use either the Normal-gamma prior or the Zellner g-prior for the parameters involved in the regression model. Our results showed that Graph sampler performs better with the Normal-gamma prior than the Zellner g-prior. Another drawback of using a Zellner g-prior is that while using this prior the number of parents should be less than the number of data points per node which can be unrealistic while dealing with real observations. For the multinomial model we defined a Dirichlet prior for the parameters. This software provides a number of different structure priors which are mutually independent and can be used in varied combination to improve the results. In particular we considered the concordance prior (P C ), the Bernoulli prior (P B ), the degree prior (P D ) and the motif prior (P M ). Even though we did not utilize the motif prior in this work, but one can use this prior according to the needs. Taking into consideration the data type and the structure prior we are able to perform full Bayesian analyses to sample graphs from the specified posterior distribution. We observed that while dealing with large networks the choice of priors and their combination is quite essential for retrieving the true edges because of the huge number of possible DAGs. It means that larger networks generally require strong informative priors on the structure of the network. Hence for specific network structures (like, tree structure), we have to think of incorporating a more appropriate structure prior. For a large tree structure we observed that the efficiency of Graph sampler in retrieving the true edges decreases as we move down the tree. An alternative way to solve this problem could be to use larger data sets for nodes present way down in the tree structure.
We find that Graph sampler is very efficient with respect to time. Both for continuous as well as discrete data, we observed that Graph sampler was atleast 10 times faster than structmcmc. One of the primary reason behind Graph sampler being time efficient is due to its fast jumping kernel. The systematic scanning of the edges and proposing to add or delete an edge at each iteration makes this kernel very efficient with respect to time. Moreover this jumping kernel requires less memory for storage. However as explained in Section 5, this jumping kernel has some limitations. The kernel faces difficulty with large data sets for which the posterior mass favours fewer graphs. The standard MCMC scheme faces problem with the local maxima and thus fails to search for the high-scoring graphs. A proficient option would be to use parallel tempering (beyond the scope of this paper) to speed up the MCMC based convergence. Secondly, Graph sampler is scripted in C language. C language being a compiled language is much more faster than R which is an interpreted language. Our results showed that for large networks i.e. networks with more than 100 nodes, Graph sampler was efficient in convergence. To resume, we observed that Graph sampler is a flexible software which is efficient with respect to time and convergence even for large networks.
We observed that Graph sampler could be a good software apart from the widely used R packages to perform Bayesian analyses of DAGs models. Besides structmcmc, bnlearn and deal which are scripted in R, there are several other very efficient BN software that are capable of performing Bayesian inference on parameter estimation and/or on network structure. Some of these software are free while others run on commercial platforms. Each software run on different platforms and follow specific sampling scheme. For interested readers, Korb and Nicholson (2010) reviewed some of the software available at the time that do inference on network structure.
Fig. 1 :
1Hierarchical representation of the 120 nodes network used to generate our simulated data. All the other smaller networks were subsets from this network 5 Results
Fig. 3 :
3Heat map of the real network with 14 nodes (A), edge posterior heat maps of Graph sampler (B) structmcmc (C) with discrete data. The x-axis represents the parent nodes and the y-axis the corresponding children. Accuracy curve of the two software (D).
Fig. 4 :
4edge posterior heat maps of Graph sampler (A), structmcmc (B) with
Fig. 5 :
5Time and convergence comparison of Graph sampler (A) and structmcmc (B) performance for various network size (N) with discrete data. The x-axis represents the number of iterations performed and the y-axis the time taken on an Apple mac book 2.53 GHz Intel Core 2 Duo processor. The black lines with circles give the minimum number of iteration required to reach convergence.
Figure 6 panels
6(A -C) represent the posterior edge probabilities in the form of heat maps.
Fig. 6 :
6Heat map of the true network with 30 nodes (A), edge posterior heat maps of Graph sampler (B) structmcmc (C) with discrete data. The x-axis represents the parent nodes and the y-axis the corresponding children. Accuracy curve of the two software (D).
Fig. 8 :
8Time and convergence comparison of Graph sampler (A) with the structmcmc(B) for varying network size with continuous data set. The
Fig. 9 :
9Heat map of the true network with 30 nodes (A), edge posterior heat maps of Graph sampler (B) structmcmc (C) with continuous data, the x-axis represents the parents and y-axis the corresponding children. Accuracy curve for the two software (D).
Fig. 10 :
10Heat map of the posterior edge probabilities obtained from Graph sampler with Normal-gamma model with concordance and degree priors (A). Accuracy curve with the use of degree prior in Graph sampler (B).
Fig. 11 :
11The flips for a network with 5 nodes along with the log posterior probability after each flip. The blue dots represent the presence of the edge from 4 to 5 and the red dots for the edge from 5 to 4 in a 5 node network. The vertical and the horizontal lines in the graph are kept for easy reference of the marked pairs with both the axis.
Fig. 12 :
12Heat map of the true (desired) network with 40 nodes (A), heat map of posterior edge probabilities obtained from Graph sampler with an informative prior of 0.9 for desired edges (B) and with a noninformative prior of 0.5 (C) with a Normal gamma likelihood; the x-axis represents the parents and y-axis the corresponding children. Accuracy curve for the various informative priors (D)Fig. 13: Heat map of the posterior edge probabilities obtained from Graph sampler with Normal-gamma model with a noninformative Bernoulli prior and degree priors (A). Accuracy curve due to the use of degree prior in Graph sampler (B).
P T (G) ∝ P C (G) × P D (G) × P M (G)
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Mc4: A tempering algorithm for largesample network inference. D Barker, S Hill, S Mukherjee, Pattern Recognition in Bioinformatics. 6282Barker D, Hill S, Mukherjee S (2010) Mc4: A tempering algorithm for large- sample network inference. Pattern Recognition in Bioinformatics 6282:431- 442
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|
[] |
[
"Anti-mirror effect: A perfect lens brings a brighter feature",
"Anti-mirror effect: A perfect lens brings a brighter feature"
] |
[
"Yadong Xu \nSchool of Physical Science and Technology\nSoochow University\n215006SuzhouJiangsuChina\n",
"Shengwang Du \nDepartment of Physics\nThe Hong Kong University of Science and Technology\nClear Water BayKowloon, Hong KongChina\n",
"Lei Gao \nSchool of Physical Science and Technology\nSoochow University\n215006SuzhouJiangsuChina\n",
"Huanyang Chen \nSchool of Physical Science and Technology\nSoochow University\n215006SuzhouJiangsuChina\n"
] |
[
"School of Physical Science and Technology\nSoochow University\n215006SuzhouJiangsuChina",
"Department of Physics\nThe Hong Kong University of Science and Technology\nClear Water BayKowloon, Hong KongChina",
"School of Physical Science and Technology\nSoochow University\n215006SuzhouJiangsuChina",
"School of Physical Science and Technology\nSoochow University\n215006SuzhouJiangsuChina"
] |
[] |
In this letter, we show that a perfect lens can be employed to make multiple objects appear like only one in the far field, leading to a new concept of illusion optics. Numerical simulations are performed to verify the functionalities for both passive and active objects. The conceptual device can be utilized to enhance the illumination brightness for both incoherent and coherent systems.A perfect lens 1 made of a negative refractive index material can form ideal images beyond the diffraction limit. Recently, combined with transformation optics 2, 3, 4 , it has been realized that many "magic" illusion effects can be obtained with extended perfect lens geometries 5, 6, 7 . In this work, we show that such a perfect lens can also act as an "anti-mirror" that makes multiple objects appear like only one in the far field. Our numerical simulation verifies that this "overlapped illusion optics" effect works for both passive and active objects (or sources). When applied to incoherent and coherent illumination systems, such as solid-state lighting, this technique can lead to dramatic enhancement in the illumination brightness and spatial-mode quality, as well as the heat-dissipation efficiencies.
|
10.1088/1367-2630/13/2/023010
|
[
"https://arxiv.org/pdf/1011.0542v1.pdf"
] | 119,214,809 |
1011.0542
|
b0100a0fa2b662a9b574a028ac75fed49ae234b7
|
Anti-mirror effect: A perfect lens brings a brighter feature
Yadong Xu
School of Physical Science and Technology
Soochow University
215006SuzhouJiangsuChina
Shengwang Du
Department of Physics
The Hong Kong University of Science and Technology
Clear Water BayKowloon, Hong KongChina
Lei Gao
School of Physical Science and Technology
Soochow University
215006SuzhouJiangsuChina
Huanyang Chen
School of Physical Science and Technology
Soochow University
215006SuzhouJiangsuChina
Anti-mirror effect: A perfect lens brings a brighter feature
In this letter, we show that a perfect lens can be employed to make multiple objects appear like only one in the far field, leading to a new concept of illusion optics. Numerical simulations are performed to verify the functionalities for both passive and active objects. The conceptual device can be utilized to enhance the illumination brightness for both incoherent and coherent systems.A perfect lens 1 made of a negative refractive index material can form ideal images beyond the diffraction limit. Recently, combined with transformation optics 2, 3, 4 , it has been realized that many "magic" illusion effects can be obtained with extended perfect lens geometries 5, 6, 7 . In this work, we show that such a perfect lens can also act as an "anti-mirror" that makes multiple objects appear like only one in the far field. Our numerical simulation verifies that this "overlapped illusion optics" effect works for both passive and active objects (or sources). When applied to incoherent and coherent illumination systems, such as solid-state lighting, this technique can lead to dramatic enhancement in the illumination brightness and spatial-mode quality, as well as the heat-dissipation efficiencies.
It is well known that, as an object is placed in front of a plane mirror, a virtual image is formed on the other side. This image looks identical (except the opposite handedness) to the object viewed by an observer in front of the mirror. In other words, the mirror transforms a single object into two separate objects, as illustrated in Fig. 1a. One may ask an interesting question: Is this mirror effect invertible? Or, is there any way to make two objects look like one, which we shall call the "anti-mirror" effect? Our answer is yes! To illustrate the basic idea for simplicity, we consider here a two-dimensional (2D) case with transverse electric (TE) polarized waves. Two identical cylindrical perfect electric conductors (PECs) are placed on both sides of a perfect lens. The distance between them is 2d, where d is the thickness of the perfect lens (see the schematic plot in Fig. 1b). Such a system displays the "anti-mirror" effect because the two PECs look like one to observers on both sides of the lens. More interestingly, the effect is also valid for active sources, as we will show later. Figure 1 Mirror and "anti-mirror" effect. a, an object (green circle) in front of a plane mirror is equivalent to two identical objects (green circles). b, two identical PECs (green circles) on both sides of a perfect lens (red region) is equivalent to one PEC (green circle). Illusion optics 7 and transformation optics 2, 3, 4 tell us that a perfect lens can be viewed as a transformation medium from a simple one-dimensional (1D) folded coordinate transformation 8 . The folded coordinate transformations can also bring a perfect lens with finite size. Here we use the folded coordinate transformations in Ref. [9] to illustrate the basic ideas. Figure 2a and 2b is the extension of Fig. 1b. In Fig. 2a, an elliptic cylindrical PEC is embedded in the restoring medium (blue regions) while an circular cylindrical PEC is located in another side of the perfect lens (red region) so that the positions and shapes of the two PECs (green regions) follow the image-forming principle of transformation optics 7 . As a result, the whole system in Fig. 2a looks like a bare circular cylindrical PEC (see in Fig. 2b) for the far-field observers. Such a phenomenon can even become more interesting. As we have known from illusion optics 7 , an illusion device with an elliptic cylindrical PEC (green circle in Fig. 2c) embedded in the restoring medium (blue regions in Fig. 2c) looks like a bare circular cylindrical PEC (dashed green circle in Fig. 2c) 10 . Hence, we can replace the real cylindrical PEC in Fig. 2a with such an illusion device. Figure 2d gives a schematic plot showing that two objects look like one PEC (dashed green circle in Fig. 2d) to the far-field observers. This effect has not been found in nature before. In fact, we can understand it easily. The two illusion devices are close to each other, and their virtual illusion spaces 7 share a common region. Inside the shared region, the same PEC image is formed simultaneously by both illusion devices. Our numerical simulations show that this "overlapped illusion optics" (multiple objects look like one) works for both PECs and active sources. Figure 2 Overlapped illusion optics. a, the "anti-mirror" effect with finite size. b, a bare cylindrical PEC. c, an illusion device to form a virtual PEC image outside it. d, two illusion devices to form the same virtual PEC image, which shall be termed as the "overlapped illusion optics". The PECs are denoted by green color. The virtual PEC images are denoted by dash green circle. The perfect lenses are denoted by red color. The restoring media are denoted by blue color.
To demonstrate the above effect, we perform full-wave simulations using the COMSOL Multiphysics finite-element-based electromagnetics solver. We set the unit to be a wavelength. All the circular cylindrical PECs (both real and virtual) are located at the origin, whose radii are 0.4. The material parameters are related to the geometric shapes of the illusion devices. We use similar shapes to that in Ref. [9] The TE waves is incident upward along the y-axis. Figure 3a shows the scattering patterns of the devices in Fig. 2a. The same rule applies in the rest parts of Fig. 3 and The above overlapped illusion optics may provide solutions to modern solid-state illumination systems. Light-emitting diodes (LEDs) have been considered as the next generation lighting source because of their low operating voltage, small size, high energy-conversion efficiency and long lifetime. However, it is still a big challenge and a high cost to produce commercial single-LED bulbs to meet resident illumination requirement due to the heat dissipation and other manufacturing difficulties. To increase the illuminance level, a common solution is to package many LEDs inside a lamp. As a result, it is extremely difficult to generate spatial illumination uniformity for residence use. With the overlapped illusion optics proposed in this paper, this problem can be solved by overlapping the illusion images from all LEDs located physically at different positions -from an observer, it looks just like a single-LED source! Such a solid-state lighting device not only provides high illuminance level with spatial uniformity, but also dissipates heat efficiently.
Here we construct a model to simulate the proposed LED bulb. For comparison, in c, an improved double-LED bulb: double line current sources in the "anti-mirror" system. The intensity of the electric field are plotted. high-power and high radiance coherent illumination sources with preserved spatial quality. In Fig. 5a, we simulate a single coherent source with a line current 1
1 I A = at 1 (0, 0.5) r = −
. In Fig. 5b, the spatial quality degrades when two coherent sources are aligned in parallel because of their interference. Figure 5c shows the simulation result of our anti-mirror effect (the same configuration as the LED bulb in Fig. 4c where the two LEDs are replaced with two coherent sources here). When the two sources are operated in the same optical frequency and phase, the light amplitude increases by a factor of 2 and thus the total power by a factor of 4! Such coherent system can be achieved using feed-back control with heterodyning detection. The increase of the output of the energy is not surprising because the overlapped illusion optics affects also the optical fields inside the two sources and increases the output conversion efficiency. This technique may have potential applications in beam-combining technique 12,13 for developing high-power laser sources with preserved beam quality. Such effects cannot be obtained from the traditional beam-combining techniques. In summary, we have demonstrated the anti-mirror effect of the perfect lens. Transformation optics extends such an effect to make multiple objects look like one in the far field. Based on this concept, we proposed and numerically verified the overlapped illusion optics. When applied to incoherent illumination systems, we designed a many-LED bulb with brighter feature and much better spatial uniformity than a conventional one. Such a method may also be potentially applied in the beam-combining technique to generate high power coherent laser beams from multiple laser diodes (LDs) with preserved beam and spatial mode qualities. Therefore, the proposed anti-mirror effect and overlapped illusion optics may have wide applications.
− . The distances between the centers of the circular cylindrical PECs and the perfect lens interfaces are set to be 0.5, from which we can obtain the detailed shapes and positions of the elliptic cylindrical PECs.
Fig. 2 .
2The identical far-field patterns in each part ofFig. 3conform the above finding.
Figure 3
3Simulation results. a, b, c, and d are the scattering patterns of the devices in Figs. 2a, 2b, 2c, and 2d.
Fig
by two incoherent sources) sit in parallel, as shown inFig. 4b, the far-field light intensity is a sum of those of the two incoherent sources, with unavoidable spatial fluctuations. Here we set one source to be at 1 phase. To eliminate the intensity fluctuations at far field, we follow the proposed overlapped illusion optics and replace the two PECs inFig. 2a(or Fig. 3a) with two line current sources. For aesthetic reasons, we rotate the device by 90 deg around the origin. The two line current sources are at the same positions of those in Fig. 4b. Region 'I' is anisotropic materials − .Figure 4cshows the intensity distribution of the improved double-LED. Compared toFig. 4b, ithas the same level of light brightness but the device behaves like a single-LED bulb with perfect spatial quality. The result can be simply extended to many (>2)-LED bulbs.
Figure 4
4An improved incoherent illumination system with overlapped illusion optics. a, a single LED bulb: single line current source in vacuum. b, a double-LED bulb: double line current sources in vacuum.
Figure 5
5Coherent sources with overlapped illusion optics (to motivate a new laser beam combining technique). a, a single line current source in vacuum. b, two coherent line current sources in vacuum. c, a combined double-coherent source: two coherent line current sources in the "anti-mirror" system.
Negative Refraction Makes a Perfect Lens. J B Pendry, Phys. Rev. Lett. 85Pendry, J. B. Negative Refraction Makes a Perfect Lens. Phys. Rev. Lett. 85, 3966-3969 (2000).
. U Leonhardt, Optical Conformal Mapping. Science. 312Leonhardt, U. Optical Conformal Mapping. Science 312, 1777-1780 (2006).
. J B Pendry, D Schurig, D R Smith, Controlling Electromagnetic Fields. Science. 312Pendry, J. B., Schurig, D. & Smith, D. R. Controlling Electromagnetic Fields. Science 312, 1780-1782 (2006).
Transformation optics and metamaterials. H Y Chen, C T Chan, P Sheng, Nature Mater. 9Chen, H. Y., Chan, C. T. & Sheng, P. Transformation optics and metamaterials. Nature Mater. 9, 387-396 (2010).
Superscatterer: Enhancement of scattering with complementary media. T Yang, Opt. Express. 16Yang, T. et al. Superscatterer: Enhancement of scattering with complementary media. Opt. Express 16, 18545-18550 (2008).
Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell. Y Lai, Phys. Rev. Lett. 10293901Lai, Y. et al. Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell. Phys. Rev. Lett. 102, 093901 (2009).
Illusion optics: The optical transformation of an object into another object. Y Lai, Phys. Rev. Lett. 102253902Lai, Y. et al. Illusion optics: The optical transformation of an object into another object. Phys. Rev. Lett. 102, 253902 (2009).
General relativity in electrical engineering. U Leonhardt, T G Philbin, New J. Phys. 8247Leonhardt, U. & Philbin, T. G. General relativity in electrical engineering. New J. Phys. 8, 247 (2006).
A simple route to a tunable electromagnetic gateway. H Y Chen, New J. Phys. 1183012Chen, H. Y. et al. A simple route to a tunable electromagnetic gateway. New J. Phys. 11, 083012 (2009).
Wave and ray analysis of a type of cloak exhibiting magnified and shifted scattering effect. Y Luo, Pr. Electromag. Res. S. 95Luo, Y. et al. Wave and ray analysis of a type of cloak exhibiting magnified and shifted scattering effect. Pr. Electromag. Res. S. 95, 167-178 (2009).
Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect. F Zolla, Opt. Lett. 321069Zolla, F. et al. Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect. Opt. Lett. 32, 1069 (2007).
Laser diode combining for free space optical communication. G S Mecherle, Proc. SPIE. SPIE616281Mecherle, G. S. Laser diode combining for free space optical communication. Proc. SPIE 616, 281 (1986).
Laser beam combining for high-power, high-radiance sources. T Y Fan, IEEE J. Sel. Top. Quantum Electron. 11Fan, T. Y. Laser beam combining for high-power, high-radiance sources. IEEE J. Sel. Top. Quantum Electron. 11, 567-577 (2005).
|
[] |
[
"NON-SEMIGROUP GRADINGS OF ASSOCIATIVE ALGEBRAS",
"NON-SEMIGROUP GRADINGS OF ASSOCIATIVE ALGEBRAS"
] |
[
"Pasha Zusmanovich "
] |
[] |
[] |
It is known that there are Lie algebras with non-semigroup gradings, i.e. such that the binary operation on the grading set is not associative. We provide a similar example in the class of associative algebras. . arXiv:1609.03924. † "Magma" means a set with an (everywhere defined) binary operation on it, without any additional conditions. In the older literature, the term "groupoid" was used instead, but since then the latter term was taken by category theorists.
|
10.1016/j.laa.2017.01.038
|
[
"https://arxiv.org/pdf/1609.03924v2.pdf"
] | 119,332,321 |
1609.03924
|
3beee9e3e12e9bf31f6faf8bc18a8b5f34a44cf4
|
NON-SEMIGROUP GRADINGS OF ASSOCIATIVE ALGEBRAS
May 2018
Pasha Zusmanovich
NON-SEMIGROUP GRADINGS OF ASSOCIATIVE ALGEBRAS
May 2018
It is known that there are Lie algebras with non-semigroup gradings, i.e. such that the binary operation on the grading set is not associative. We provide a similar example in the class of associative algebras. . arXiv:1609.03924. † "Magma" means a set with an (everywhere defined) binary operation on it, without any additional conditions. In the older literature, the term "groupoid" was used instead, but since then the latter term was taken by category theorists.
INTRODUCTION
Let A be a (generally, not necessarily associative) algebra, and A = g∈Γ A g its grading over a set (Γ, * ), i.e. * : Γ × Γ → Γ is a partial binary operation defined for each pair (g, h) such that A g A h = 0, in which case A g A h ⊆ A g * h . How the identities satisfied by the algebra A are related to identities satisfied by the grading set Γ? Since the operation * on Γ is partial, in the latter case it makes sense to speak about the (im)possibility to complete * in such a way that it will satisfy that or another identity, or, in more strict terms, about the (im)possibility of embedding of (Γ, * ) into an appropriate magma † (G, ·) such that g * h = g · h whenever A g A h = 0.
It is immediate that commutativity or anticommutativity of A implies that Γ can be embedded into a commutative magma. Elementary manipulations involving homogeneous components A g 's of graded Lie and associative algebras may suggest that both Jacobi identity and associativity of the algebra A are strongly connected with the associativity of the grading set Γ. In the Lie case, it was believed for a while (and even claimed in an influential paper [PZ] as Theorem 1(a)) that each grading of a Lie algebra is a semigroup grading, i.e. the grading set (Γ, * ) can be embedded into a semigroup. This is indeed so for all gradings of Lie and associative algebras appearing naturally (root space decompositions with respect to a Cartan subalgebra, gradings arising from various group or Hopf algebra actions on the algebra, Z-gradings providing connection between Lie and Jordan algebras, semigroup algebras and their twisted variants, grading by Pauli matrices motivated by physics, etc.). However, in [E1] and [E2] examples of non-semigroup gradings of Lie algebras were given. The aim of this note is to provide an example of a non-semigroup grading of an associative algebra. This is done in §1 following approach of [Z, §3], where it was shown how non-semigroup gradings of Lie algebras can be constructed using δ -derivations. §2 contains some further questions.
AN EXAMPLE OF A NON-SEMIGROUP GRADING
In the associative case, instead of δ -derivations we may consider a slightly more general notion of (δ , γ)-derivations, i.e. linear maps D : A → A on an algebra A such that
D(xy) = δ D(x)y + γxD(y)
for any x, y ∈ A, and some fixed elements of the ground field δ , γ. In the Lie case, due to anticommutativity, any such condition implies that either δ = γ, i.e. D is a δ -derivation, or that D is an element of "generalized centroid", i.e. D(xy) = (δ + γ)D(x)y = (δ + γ)xD(y)
for any x, y ∈ A, the latter condition being too restrictive to be interesting. (The same dichotomy holds for commutative algebras).
Lemma 1. Let A be a finite-dimensional algebra over an algebraically closed field K, and A
= λ ∈K A λ is the root space decomposition with respect to an (δ , γ)-derivation of A. Then A λ A µ ⊆ A δ λ +γ µ for any λ , µ ∈ K.
Note that the algebra A here and below is not assumed to be associative, or Lie, or to satisfy any other distinguished identity.
Proof. It is trivial to check that if x and y are eigenvectors of an (δ , γ)-derivation of A, corresponding to eigenvalues λ and µ respectively, then the product xy is an eigenvector corresponding to δ λ + γµ (or zero, if δ λ + γµ is not an eigenvalue). Then proceed by induction on the sum of multiplicities of the respective eigenvalues, exactly the same way as in, for example, [J,Chapter III,§2].
The following is a slightly modified "nonassociative" analogue of the Lie-algebraic statement [Z,Proposition 3.1].
Proposition. Let A be a finite-dimensional algebra over an algebraically closed field, and D an (δ , γ)derivation of A. Suppose that there are roots λ , µ, η, θ , ξ (not necessarily distinct) in the root space decomposition of A with respect to D such that
0 = A λ A η ⊆ A θ , A θ A µ = 0, (1) 0 = A η A µ ⊆ A ξ , A λ A ξ = 0,(2)
and (δ 2 − δ )λ = (γ 2 − γ)µ. Then the said root space decomposition is a non-semigroup grading of A.
Note that the conditions (1) and (2)
are somewhat weaker than (A λ A η )A µ = 0 and A λ (A η A µ ) = 0, respectively.
Proof. The conditions (1) and (2) ensure that both expressions (λ * η) * µ and λ * (η * µ) are defined. If the root space decomposition of A with respect to D is a semigroup grading, then these two expressions are equal: (λ * η) * µ = λ * (η * µ). By Lemma 1, this equality is equivalent to (δ 2 − δ )λ = (γ 2 − γ)µ, a contradiction.
Corollary. The conclusion of Proposition holds in each of the following cases:
(i) δ = γ = 0, 1, and λ = µ; (ii) δ = γ, δ + γ = 1, and λ = µ = 0.
Proof. Obvious. Now we will provide an example of a family of associative algebras having δ -derivations as in heading (i) of the Corollary, and hence admitting a non-semigroup grading. Let V be a vector space over a field K, and f L , f R , g L , g R : V → V be four linear maps. Consider the vector space direct sum
Ke ⊕ Ka ⊕V ⊕V ′ ,
where Ke and Ka are one-dimensional vector spaces spanned by elements e and a respectively, and V ′ is a second copy of V , identified with V via a nondegenerate linear map v → v ′ . Define the multiplication on this direct sum as follows:
e 2 = e, a 2 = 0, av = f L (v) ′ , va = f R (v) ′ , av ′ = g L (v), v ′ a = g R (v),
where v ∈ V , and the rest of the products between basic elements are zero. The associativity of the so defined algebra, let us denote it as A( f L , f R , g L , g R ), is equivalent to the following conditions:
f L • g L = g L • f L = 0 f R • g R = g R • f R = 0 g R • f L = g L • f R f R • g L = f L • g R .
Lemma 2. Suppose that each of the maps f L , f R , g L , g R is nonzero, and (δ , γ) = (0, 0). Then each (δ , γ)-derivation D of the algebra A( f L , f R , g L , g R ) is of the following form:
D(e) = 0 if δ + γ = 1 β e if δ + γ = 1 D(a) = αa + v a + w ′ a D(v) = ϕ(v) + ψ(v) ′ , v ∈ V D(v ′ ) = ϕ(v) + ψ(v) ′
where α, β ∈ K, v a , w a ∈ V , ϕ, ϕ, ψ, ψ : V → V are linear maps, and the following conditions are satisfied:
(δ f R + γ f L )(v a ) = 0 (δ g R + γg L )(w a ) = 0 and ϕ • f L = γg L • ψ ψ • f L = δ α f L + γ f L • ϕ ϕ • f R = δ g R • ψ ψ • f R = γα f R + δ f R • ϕ ϕ • g L = δ αg L + γg L • ψ ψ • g L = γ f L • ϕ ϕ • g R = γαg R + δ g R • ψ ψ • g R = δ f R • ϕ.
Proof. Direct calculations.
The non-vanishing conditions of Lemma 2 are merely technical ones, to avoid consideration of numerous degenerate tedious cases.
We may specialize this setup in many different ways to get an example of an algebra having a (δ , γ)derivation satisfying the condition of Proposition or its Corollary, and hence admitting a non-semigroup grading. One of the easiest ways is to set f L = f R = g L = g R = f , where f • f = 0 (say, V is 2dimensional, and f has the matrix 0 1 0 0 in the canonical basis), δ = γ = −1, α = β = 0, v a = w a = 0,
and ψ = ϕ = 0, ϕ = id V , ψ = − id V . Then D from Lemma 2 is a (−1)-derivation (or, antiderivation) of the algebra A( f , f , f , f ). The eigenvalues of D are 0, 1, −1, with eigenspaces A 0 = Ke⊕Ka, A 1 = V , and A −1 = V ′ . Then by heading (i) of Corollary, the root space decomposition A( f , f , f , f ) = A 0 ⊕ A 1 ⊕ A −1
is a non-semigroup grading. This fact can be also verified directly: as (Note that this is the same non-associative grading set as in the Lie-algebraic example in [E2]).
A 2 0 = Ke ⊂ A 0 , A 0 A 1 = A 1 A 0 = (Im f ) ′ ⊂ A −1 , and A 0 A −1 = A −1 A 0 = Im f ⊂ A 1 ,
The algebra A( f , f , f , f ) is, obviously, commutative, with a commutative grading. By modifying this example to make the maps f L , f R , g L , g R different, it is possible to get various examples of associative non-commutative algebras with a non-semigroup grading, commutative or not. The relevant calculations are trivial, but somewhat cumbersome, and are left to the interested reader.
FURTHER QUESTIONS
If L = g∈Γ L g is a Lie algebra graded by an abelian group Γ, then its universal enveloping algebra U (L) is a Γ-graded associative algebra, with the graded components U (L) g linearly spanned by monomials of the form x 1 . . . x k , where x i ∈ L g i and ∑ k i=1 g i = g (see, e.g., [SF,Theorem 4.3]). The algebra U (L) is infinite-dimensional, what, perhaps, is not that interesting in our context. In the positive characteristic it is possible, however, to define the same grading on the finite-dimensional restricted universal enveloping algebra of a graded restricted Lie algebra. However, the facts that multiplication in Γ is defined everywhere, and is associative, are crucial in this construction, and it is unclear how to extend or modify it to grading by an arbitrary set Γ. Question 1. Is it possible to construct a grading of the (restricted) universal enveloping algebra, given (arbitrary, not necessarily semigroup) grading of the underlying Lie algebra?
A positive answer to this question will produce a plethora of non-semigroup gradings of finitedimensional associative algebras in positive characteristic, different from those exhibited in §1: take any of the examples from [E1] or [E2] over a field of characteristic p > 0, pass, if necessary, to the p-envelope, and consider the restricted universal enveloping algebra.
Question 2. What is the minimal dimension of an associative algebra admitting a non-semigroup grading?
It is, probably, possible to prove, following the approach of [E2,Theorem in §1], and classification of low-dimensional associative algebras, that any grading of an associative algebra of dimension ≤ 3 is a semigroup grading. Since the underlying algebra is not necessarily commutative, there are apriori much more possibilities for a noncommutative partial operation on a 2-and 3-element grading set. The relevant calculations should be straightforward, but definitely cumbersome.
We also failed to find examples of non-semigroup gradings of associative algebras of dimension 4 and 5. The minimal dimension of an algebra with non-semigroup grading following the scheme of §1 is 6.
By analogy with the question about gradings of simple Lie algebras from [E1], one may ask Question 3. Is it true that any grading of a full matrix algebra is a semigroup grading?
Note that this question cannot be approached by constructing an appropriate (δ , γ)-derivation as in §1: it is easy to see that any (δ , γ)-derivation of a full matrix algebra is either an (inner) derivation, or a scalar multiple of the identity map (see, e.g., [S, Theorem 1] for a slightly more general statement).
Finally, note that, in principle, the same approach as in §1 may be used to construct examples of nonsemigroup gradings in varieties of algebras satisfying other identities of degree 3 (like Leibniz, Zinbiel, left-symmetric, Lie-admissible, Alia algebras, etc.). Another interesting topic would be to explore the question from the point of view of operadic Koszul duality: for example, does the presence/absence of non-semigroup gradings of algebras over a binary quadratic operad P entails the same for algebras over the operad Koszul dual to P?
we have the following (partial) operation on the grading set:0 * 0 = 0, 0 * 1 = 1 * 0 = −1, 0 * (−1) = (−1) * 0 = 1,what contradicts associativity: 1 = 0 * (−1) = (0 * 0) * (−1) = 0 * (0 * (−1)) = 0 * 1 = −1.
ACKNOWLEDGEMENTSThanks are due to Miroslav Korbelář for asking questions which prompted me to write this note. This work was supported by the Statutory City of Ostrava (grant 0924/2016/SaŠ), and the Ministry of Education and Science of the Republic of Kazakhstan (grant 0828/GF4).
A Lie grading which is not a semigroup grading. A Elduque, arXiv:math/0512618Lin. Algebra Appl. 418A. Elduque, A Lie grading which is not a semigroup grading, Lin. Algebra Appl. 418 (2006), N1, 312-314; arXiv:math/0512618.
More non-semigroup Lie gradings. arXiv:0809.4547Lin. Algebra Appl. 431, More non-semigroup Lie gradings, Lin. Algebra Appl. 431 (2009), N9, 1603-1606; arXiv:0809.4547.
Lie Algebras. N Jacobson, Interscience Publ. reprinted by DoverN. Jacobson, Lie Algebras, Interscience Publ., 1962; reprinted by Dover, 1979.
. J Patera, H Zassenhaus, On Lie gradings. I, Lin. Algebra Appl. 112J. Patera and H. Zassenhaus, On Lie gradings. I, Lin. Algebra Appl. 112 (1989), 87-159.
Ternary derivations of separable associative and Jordan algebras. A I Shestakov, Sibirsk. Mat. Zh. 53in RussianA.I. Shestakov, Ternary derivations of separable associative and Jordan algebras, Sibirsk. Mat. Zh. 53 (2012), N5, 1178-1195 (in Russian);
. Siber. Math. J. 53English translationSiber. Math. J. 53 (2012), N5, 943-956 (English translation).
H Strade, R Farnsteiner, Modular Lie Algebras and Their Representations. Marcel DekkerH. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations, Marcel Dekker, 1988.
On δ -derivations of Lie algebras and superalgebras. P Zusmanovich, arXiv:0907.2034J. Algebra. 324P. Zusmanovich, On δ -derivations of Lie algebras and superalgebras, J. Algebra 324 (2010), N12, 3470-3486; Erra- tum: 410 (2014), 545-546; arXiv:0907.2034.
. Department Of Mathematics, Ostrava Ostrava, Czech Republic E-, mail address: [email protected] OF MATHEMATICS, UNIVERSITY OF OSTRAVA, OSTRAVA, CZECH REPUBLIC E-mail address: [email protected]
|
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[
"Online Learning for Loss Functions with Memory and Applications to Statistical Arbitrage",
"Online Learning for Loss Functions with Memory and Applications to Statistical Arbitrage",
"Online Learning for Loss Functions with Memory and Applications to Statistical Arbitrage",
"Online Learning for Loss Functions with Memory and Applications to Statistical Arbitrage"
] |
[
"Oren Anava [email protected] \nTechnion, Haifa, Haifa, HaifaIsrael, Israel, Israel\n",
"Elad Hazan [email protected] \nTechnion, Haifa, Haifa, HaifaIsrael, Israel, Israel\n",
"Shie Mannor \nTechnion, Haifa, Haifa, HaifaIsrael, Israel, Israel\n",
"Oren Anava [email protected] \nTechnion, Haifa, Haifa, HaifaIsrael, Israel, Israel\n",
"Elad Hazan [email protected] \nTechnion, Haifa, Haifa, HaifaIsrael, Israel, Israel\n",
"Shie Mannor \nTechnion, Haifa, Haifa, HaifaIsrael, Israel, Israel\n"
] |
[
"Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel",
"Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel",
"Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel",
"Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel",
"Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel",
"Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel"
] |
[] |
In many online learning scenarios the loss functions are not memoryless, but rather depend on history. Our first contribution is a complete characterization of sufficient and necessary conditions for learning with memory, accompanied with a novel algorithm for this framework that attains the optimal O( √ T )-regret. This improves previous online learning algorithms that guaranteed O(T 2/3 ) regret and required more stringent conditions. As an application of the new technique, we address the classical problem in finance of constructing mean reverting portfolios. We design an efficient online learning algorithm for this problem, and provide guarantees for its performance. We complement our theoretical findings with an empirical study that verifies our theoretical results on financial data.
| null |
[
"https://arxiv.org/pdf/1302.6937v2.pdf"
] | 18,433,040 |
1302.6937
|
ef48164520bfe75bf611effe22c745011edce827
|
Online Learning for Loss Functions with Memory and Applications to Statistical Arbitrage
27 Feb 2013
Oren Anava [email protected]
Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel
Elad Hazan [email protected]
Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel
Shie Mannor
Technion, Haifa, Haifa, HaifaIsrael, Israel, Israel
Online Learning for Loss Functions with Memory and Applications to Statistical Arbitrage
27 Feb 2013
In many online learning scenarios the loss functions are not memoryless, but rather depend on history. Our first contribution is a complete characterization of sufficient and necessary conditions for learning with memory, accompanied with a novel algorithm for this framework that attains the optimal O( √ T )-regret. This improves previous online learning algorithms that guaranteed O(T 2/3 ) regret and required more stringent conditions. As an application of the new technique, we address the classical problem in finance of constructing mean reverting portfolios. We design an efficient online learning algorithm for this problem, and provide guarantees for its performance. We complement our theoretical findings with an empirical study that verifies our theoretical results on financial data.
Introduction
In numerous online learning scenarios the environment is not completely oblivious to the decision maker, and the decision maker's historical actions affect her current state and future rewards. We are particularly concerned in scenarios in which this historic effect is relatively short-term and simple, in contrast to stateaction models in which better tailored reinforcement learning models have been devised [YMS09].
Thus, our focus is on online learning in which actions have short-term effects on future losses. This model was initially considered in the information theory community [MOSW06], with an eye on applications in compression, coding and portfolio selection. In this work, after describing our contributions to this framework, we apply this model to finding statistical arbitrage opportunities in financial market data.
We start by studying the framework of online learning with memory. Our first result is a novel algorithm for this framework that attains optimal regret bounds on the order of O( √ T ), where T is the number of prediction iterations. The algorithm is also computationally efficient, and we show that its assumptions are not only sufficient, but in fact necessary for any efficient algorithm for learning with memory. These bounds improve [MOSW06], who attains a regret bound of O(T 2/3 ). Thus, our results show that in fact the optimal regret bounds for this framework are of the same order as standard memoryless learning, and the overhead for the more complicated model is in the memory effect only.
Next, we proceed to study our motivating problem -constructing mean revering portfolios. In the literature, "statistical arbitrage" refers to statistical mispricing of one or more assets based on the expected value of these assets. One of the most common trading strategies, known as "pairs trading", seeks to create a mean reverting portfolio by combining two different assets (typically using both short and long sales). Then, by buying the combined portfolio below its mean and selling it above, one can have an expected positive profit with low risk.
This strategy consists of three main steps: choosing the two underlying assets, then finding appropriate distribution of weights over them, and finally applying trading algorithms (which determine the buying and selling points) to maximize profit. In this work we focus on the first two steps with the following extensions: we allow portfolios that consist of more than two assets, and we consider an online scheme in which the distribution of weights can be updated (up to some extent). I.e., given a set of n different assets we wish to isolate a subset of k assets that has a large amount of mean reversion, and then determine a certain weight distribution over this subset.
The problem of modifying the weights of a portfolio in order to maximize mean reversion is a learning problem with memory: the weights of previous iterations determine the price of the mixed asset, and thus the overall performance in terms of mean reversion. We cast this problem formally as a learning with memory problem, and utilize our new technique to solve it online. This yields the first sub-linear regret algorithm for this problem. Besides the theoretical sublinear regret bound, we test the resulting algorithm and demonstrate its effectiveness on financial data.
Related work
Statistical arbitrage and in particular pairs trading strategies initially took place in the mid 80's [EG87]. Since then, a great deal of work has been done on the problem of assembling mean reverting portfolios, mostly using cointegration techniques (see [Vid11] for more comprehensive information). In order to quantify the amount of mean reversion in various portfolios, different proxies are often suggested such as zero-crossing and predictability [Sch11,D'A11]. In this work, we consider a new proxy for mean reversion which is aimed at maximizing fluctuation, as well as keeping the mean close to zero. Furthermore, whereas classical cointegration techniques require training period before applying a trading strategy (see for instance [AL10,AM12]), the online approach does not require that, in addition to providing a performance guarantee against the best mean reverting portfolio in hindsight.
Online learning with memory
Online learning is a game-theoretic optimization framework, where iteratively an online player chooses a decision x t , and suffers loss f t (x t ). The loss function f t is chosen by an all-powerful adversary with full knowledge of our learning algorithm (see for instance [CBL06]).
Here we consider the following extension. At iteration t, an online player chooses a decision x t ∈ K, where K is called the decision set. Then, an adversary chooses a loss function f t : K m → R, and the online player suffers loss of f t (x t−m+1 , . . . , x t ). Notice that the loss at iteration t depends on the previous m − 1 decisions of the player, as well as on his current decision.
Our goal in this framework is to minimize the sum of losses over predefined number of iterations T . A reasonable benchmark is to try to be not much worse than the total loss suffered by the best decision in hindsight. More precisely, we define the regret as
R T = T t=m f t (x t−m+1 , . . . , x t ) − min x T t=m f t (x, . . . , x),(1)
and wish to obtain efficient algorithms, whose regret grows sub-linearly in T , corresponding to an average per-round regret going to zero as T increases. 1 We henceforth make the following assumptions. The first two assumptions are standard and necessary for any regret minimization algorithm to apply, even without memory. Assumption three is the only new assumption we make, and as we explain -it is a necessary assumption if one considers efficient algorithms.
1. The diameter of K is bounded, i.e., there exists D > 0, such that sup x,y∈K x − y ≤ D , where · refers to the ℓ 2 norm.
2. There exists G > 0, such that
sup (x t−m+1 ,...,xt),t ∇f t (x t−m+1 , . . . , x t ) ≤ G.
It follows that f t is Lipchitz continuous with a Lipchitz constant G.
3. Define g t (x) = f t (x, . . . , x). Then, we assume that g t (x) is convex in x, for all t. This assumption is essentially necessary for an efficient algorithm, since achieving sublinear regret bound for {f t } T t=1 implies also that T t=1 g t (x) can be minimized efficiently.
Algorithm and analysis
By assumption 3 the functions {g t } T t=1 are convex, and hence we can apply the Online Gradient Descent (OGD) algorithm of [Zin03] with some modifications.
Algorithm 1 OGD with Memory 1: Input: Learning rate η. 2: Choose x 1 ∈ K arbitrarily. 3: for t = 1 to T do 4:
Play x t and suffer loss f t (x t−m+1 , . . . , x t ). 5: Set x t+1 ← Π K x t − η∇g t (x t ) 6: end for
Here, Π K is the Euclidean projection onto K, i.e.
Π K (y) = arg min x∈K y − x .
For Algorithm 1 we can prove the following bound.
Theorem 2.1. Let G and D be as defined in Section 2, and set η = D G √ mT . Then, Algorithm 1 achieves the following regret bound for {f t } T t=1 :
R T = T t=m f t (x t−m+1 , . . . , x t ) − min x T t=m f t (x, . . . , x) ≤ 2 · GD √ mT .
(2)
Proof. By applying algorithm 1 for the loss functions {g t } T t=1 we have that
T t=m g t (x t ) − min x T t=m g t (x) ≤ D 2 2η + ηG 2 T 2 ,
using the analysis of [Zin03], and from the definitions of f t and g t it follows that
T t=m f t (x t , . . . , x t ) − min x T t=m f t (x, . . . , x) ≤ D 2 2η + ηG 2 T 2 .(3)
On the other hand, since f t is Lipshitz continuous for the Lipshitz constant G we have:
|f t (x t , . . . , x t ) − f t (x t−m+1 , . . . , x t )| 2 ≤ (G · (x t , . . . , x t ) − (x t−m+1 , . . . , x t ) ) 2 = G 2 · m−1 j=1 x t − x t−j 2 ≤ G 2 · m−1 j=1 j l=1 x t−l+1 − x t−l 2 ≤ G 2 · m−1 j=1 j l=1 η∇g t−l (x t−l ) 2 ≤ G 2 · m−1 j=1 j l=1 η 2 G 2 = G 4 · m−1 j=1 j l=1 η 2 ≤ m 2 η 2 G 4 , which implies that |f t (x t , . . . , x t ) − f t (x t−m+1 , . . . , x t )| ≤ mηG 2 .
Summing the above for t = m, . . . , T we get:
T t=m f t (x t , . . . , x t ) − T t=m f t (x t−m+1 , . . . , x t ) ≤ T t=m mηG 2 = mηG 2 T.(4)
Now, by combining Equations (3) and (4) we get the following inequality:
T t=m f t (x t−m+1 , . . . , x t ) − min x T t=m f t (x, . . . , x) ≤ D 2 2η + ηG 2 T 2 + mηG 2 T. Finally, substituting η = D G √ mT yields R T = T t=m f t (x t−m+1 , . . . , x t ) − min x T t=m f t (x, . . . , x) ≤ 2 · GD √ mT ,
which completes the proof.
Application to finance
In this section we use the technique just developed to find and exploit statistical arbitrage opportunities. Roughly speaking, the goal is to synthetically create a mean reverting portfolio, exploiting correlation between similar assets. That is, we are seeking a strategy that maintains weights upon predefined set of assets, such that the combined portfolio is mean reverting.
As a first step we define a criterion for measuring mean reversion, that is empirically well behaving. Unfortunately, this criterion is not convex (as are most of other previously considered criteria), and we define a semi-definite relaxation to cope with the problem. Another difficulty comes from the very nature of the problem: weights of one iteration affect future performance, thus memory comes unavoidably into the picture.
We proceed to formally define the new mean reversion criterion, its semi-definite relaxation, and the use of our new memory-learning algorithm in this model.
Problem definition
Tendency to return to the mean is not a quantifiable criterion for mean reversion, and the literature addresses several proxies to capture the notion of mean reversion, e.g., in [Sch11, D'A11]. In this work, we present a new criterion for mean reversion effectiveness: low squared mean and high variance. More precisely, we denote by y t ∈ R n the prices of n assets at time t, and by x t ∈ R n a distribution of weights over these assets (we allow short selling, thus x t can contain negative entries).
Since short selling is allowed, the norm of x t can sum up to an arbitrary number, determined by the loan flexibility of the back. Thus we assume without loss of generality that x t 2 = 1, and define the following loss function:
f t (x t−m−1 , . . . , x t ) = m−1 i=0 x T t−i y t−i 2 − λ · m−1 i=0 x T t−i y t−i 2
for some λ > 0. Notice that minimizing this loss function iteratively yields a process {x T t y t } T t=1 such that its mean is close to 0, while its variance is maximized. Variance maximization is crucial here, since high variance processes tend to create larger amount of statistical arbitrage opportunities.
We use the regret criterion to measure our performance against the best distribution of weights in hindsight, and wish to obtain online algorithm that generates a series {x t } T t=1 such that
T t=m f t (x t−m−1 , . . . , x t ) − min x T t=m f t (x, . . . , x) = o(T ).
The previous distributions of our weights affect the mean reversion amount of our portfolio, and hence f t is a loss function with memory. As in Section 2, we define g t (x) = f t (x, . . . , x), and show that by obtaining regret bound for {g t } T t=1 we also guarantee a regret bound for {f t } T t=1 . Notice that g t is of the form
g t (x) = x T A t x − x T B t x (5) for A t = m−1 i=0 m−1 j=0 y t−i y t−j , and B t = λ · m−1 i=0 y t−i y T t−i ,
The function g t is not convex in general, and hence we cannot apply the technique detailed in Section 2 straightforwardly. Instead, we define
h t (X) = X • A t − X • B t ,(6)
where
X • A = n i=1 n j=1 X(i, j) · A(i, j),
and X is a PSD matrix with T r(X) = 1.
Now, the problem of minimizing T t=m h t (X) is a PSD relaxation to the problem of minimizing T t=m g t (x), and for the optimal solution
x ⋆ = arg min x T t=m g t (x),
it holds in particular that
min X T t=m h t (X) ≤ T t=m h t (x ⋆ x ⋆⊤ ) = T t=m g t (x ⋆ ).
Notice that h t is linear in X for all t, and hence we can apply regret minimization techniques on the loss functions {h t } T t=1 .
Parameter setting
Throughout this section we use the following parameters and notations:
1. The decision set K is defined as:
K = {X|X is PSD and T r (X) = 1}.
From this definition we can bound the diameter of K by sup X,Y ∈K X − Y F ≤ D = √ 2 , when · F refers to the Frobenius norm.
2. There exists G > 0, such that
sup (x t−m+1 ,...,xt),t ∇f t (x t−m+1 , . . . , x t ) ≤ G.
It follows that f t is Lipshitz continuous for the Lipshitz constant G, and also that sup
Xt,t ∇h t (X t ) F ≤ G.
Clearly, the value of G depends on the prices of assets we are considering, and its computation is done accordingly.
Algorithm and analysis
We turn now to present our online algorithm.
Algorithm 2 Online Statistical Arbitrage (OSA) 1: Input: Learning rate η, X 0 = 1 n I n×n . 2: Randomize x 0 ∼ X 0 . 3: for t = 1 to (T − 1) do 4:
Set X t ← Π K (X t−1 − η∇h t (X t )) 5: Set x t = x t−1 w.p. 1 − 1 m √ T , 6:
Otherwise, randomize x t ∼ X t .
7:
Play x t and suffer loss f t (x t−m+1 , . . . , x t ).
8: end for
Here, Π K refers to the following projection onto K:
Π K (X) = arg min Y ∈K X − Y F .
Also, x t ∼ X t refers to the eigenvector decomposition of the matrix X t . I.e, we represent X t = n i=1 λ i v i v ⊤ i , where each v i is a unit vector and n i=1 λ i = 1, when λ i ≥ 0. Then, we randomize the eigenvector x t = v i with probability λ i . Technically, this decomposition is possible due to the fact that X t is positive semidefinite with T r(X t ) = 1 for all t.
For Algorithm 2 we can prove the following bound.
R T = T t=m E [f t (x t−m+1 , . . . , x t )] − min x T t=m f t (x, . . . , x) ≤ 3 · √ mGDT 3/4 .
Proof. By applying Algorithm 2 for the loss functions {h t } T t=1 we get a series of matrices {X t } T t=1 , such that
T t=m h t (X t ) − min X T t=m h t (X) ≤ D 2 2η + G 2 2 T t=m η,
using the analysis of [Zin03]. From the definitions of g t and h t it exists that
min X T t=m h t (X) ≤ min x T t=m g t (x), and hence T t=m h t (X t ) − min x T t=m g t (x) ≤ D 2 2η + G 2 2 T t=m η.
Now, from Lemma 3.2 we know that
T t=m h t (X t ) − E [g t (x t )] ≤ √ mGDT 3/4 , which yields T t=m E [g t (x t )] − min x T t=m g t (x) ≤ D 2 2η + G 2 2 T t=m η + √ mGDT 3/4 .
From the definition of g t it follows that
T t=m E [f t (x t , . . . , x t )] − min x T t=m f t (x, . . . , x) ≤ D 2 2η + G 2 2 T t=1 η + √ mGDT 3/4 .(7)
Next, we bound the distance between x t and x t−1 in expectation for all t. Unlike presented in Section 2, we cannot rely on the closeness of x t and x t−1 that follows from the step size of the online update. However, we can use the fact that x t = x t−1 with probability 1 m √ T , and therefore
E x t − x t−1 2 ≤ 0 · 1 − 1 m √ T + D 2 · 1 m √ T = D 2 m √ T .
Now, similarly to Section 2 we rely on the fact that f t is Lipchitz continuous with a Lipchitz constant G, and thus
|E [f t (x t , . . . , x t )] − E [f t (x t−m+1 , . . . , x t )]| 2 ≤ E [|f t (x t , . . . , x t ) − f t (x t−m+1 , . . . , x t )|] 2 ≤ (G · E [ (x t , . . . , x t ) − (x t−m+1 , . . . , x t ) ]) 2 ≤ G 2 · m−1 j=1 E x t − x t−j 2 ≤ G 2 · m−1 j=1 j l=1 E x t−l+1 − x t−l 2 ≤ G 2 · m−1 j=1 j l=1 D 2 m √ T ≤ mG 2 D 2 √ T ,
and it follows that
|E [f t (x t , . . . , x t )] − E [f t (x t−m+1 , . . . , x t )] | ≤ √ mGD T 1/4 . Summing the above for all t yields T t=m E [f t (x t , . . . , x t )] − T t=m E [f t (x t−m+1 , . . . , x t )] ≤ √ mGDT 3/4 ,(8)
and by combining (7) and (8) we get that
T t=m E [f t (x t , . . . , x t )] − min x T t=m f t (x, . . . , x) ≤ D 2 2η + G 2 2 T t=1 η + 2 · √ mGDT 3/4 . Finally, substituting η = D √ mGT 3/4 yields R T = T t=m E [f t (x t−m+1 , . . . , x t )] − min x T t=m f t (x, . . . , x) ≤ 3 · √ mGDT 3/4 ,
which completes the proof.
We now turn to prove Lemma 3.2. (5) and (6). Then, Algorithm 2 generates online
Lemma 3.2. Let g t and h t be as denoted in Equations and
sequences {X t } T t=1 and {x t } T t=1 such that T t=m h t (X t ) − E [g t (x t )] ≤ √ mGDT 3/4 .
Proof. Notice that g t (x t ) = h t x t x ⊤ t from the definitions of g t and h t . We show by induction that
X t − E x t x ⊤ t F ≤ √ mD T 1/4 ,
and from the Lipchitz property of {h t } T t=1 this also implies that
h t (X t ) − E h t x t x ⊤ t ≤ √ mGD T 1/4 . Thus, for t = 0 we have that X 0 − E x 0 x ⊤ 0 F = 0 ≤ √ mD T 1/4 , since x 0 ∼ X 0 . For t = 1 we have that X 1 − E x 1 x ⊤ 1 F = 0 · 1 m √ T + X 1 − E x 0 x ⊤ 0 F · 1 − 1 m √ T = X 1 − X 0 + X 0 − E x 0 x ⊤ 0 F · 1 − 1 m √ T ≤ X 1 − X 0 F · 1 − 1 m √ T + X 0 − E x 0 x ⊤ 0 F · 1 − 1 m √ T ≤ (ηG) · 1 − 1 m √ T = D √ mT 3/4 − D m 3/2 T 5/4 ≤ √ mD T 1/4 .
Next, we assume that
X t − E x t x ⊤ t F ≤ √ mD T 1/4 and prove that X t+1 − E x t+1 x ⊤ t+1 F ≤ √ mD T 1/4 . Thus, X t+1 − E x t+1 x ⊤ t+1 F = 0 · 1 m √ T + X t+1 − E x t x ⊤ t F · 1 − 1 m √ T = X t+1 − X t + X t − E x t x ⊤ t F · 1 − 1 m √ T ≤ X t+1 − X t F · 1 − 1 m √ T + X t − E x t x ⊤ t F · 1 − 1 m √ T ≤ ηG + √ mD T 1/4 · 1 − 1 m √ T = D √ mT 3/4 + √ mD T 1/4 · 1 − 1 m √ T = √ mD T 1/4 − D m 3/2 T 5/4 ≤ √ mD T 1/4 ,
which completes the induction. Now, from the Lipchitz property of {h t } T t=1 this implies that
h t (X t ) − E h t x t x ⊤ t ≤ √ mGD T 1/4 ,
and summing the above for all t yields
T t=m h t (X t ) − E h t (x t x ⊤ t ) ≤ √ mGDT 3/4 .
Experiments
The following experiments demonstrate the effectiveness of the proposed algorithms, under two different settings. In the first setting, we consider the Dow Jones Index (30 stocks) and apply our criterion to isolate subsets of stocks that have a maximal amount of mean reversion. In the second setting, we consider pairs of sector related stocks, and apply the proposed algorithm to construct mean reverting portfolios ("pairs trading").
Comparison method
We compare the amount of mean reversion of the different portfolios using the Portmanteau test from [LB78]. This test is aimed at determining whether a process is close to be pure noise process, and hence can be effectively applied in our case as a measure for mean reversion. More accurately, we define ∆ t = x T t y t − x T t−1 y t−1 to be the daily change of our portfolio, and consequently
Q m {∆ t } T t=1 = T (T + 2) L k=1 ρ (k) 2 T − k ,
to be the Portmanteau statistic, where
ρ (k) = T t=k+1 ∆ t ∆ t−k T t=1 ∆ 2 t
is the sample autocorrelation at lag k, and L is the number of autocorrelation lags (chosen to be 20 by default). Under the null hypothesis, the asymptotic distribution of Q m is chi-square with L degrees of freedom, and therefore we can use the p-value as our measure for mean reversion.
Additionally, we compare the revenue obtained by applying the following trading strategy to each of the portfolios: buy whenever it reaches a certain lower threshold, and sell whenever it reaches an upper threshold (we assume no transaction cost). We arbitrarily use (−1) and (+1) as lower and upper thresholds in our experiments, but similar results can be shown for any other choice. It is highly likely that dynamic trading strategies such as those presented in [GGR99,JY06] would yield higher revenue. However, the design of a trading strategy is completely orthogonal to our work, and our goal here is simply to compare various portfolios with unified trading strategy.
Data set
For the first setting, we consider time series of daily closing rates of all 30 stocks in the Dow Jones. For the second setting, we consider time series of daily closing rates of 8 pairs of stocks. The selection of the pairs relies on their sectoral belonging (Financials, Energy, Telecommunication services, etc.). In both settings, we use data between the dates 01/01/2008 and 01/02/2013, which is taken from Yahoo! Finance.
Isolating mean reverting portfolio
In this setting, we test our criterion on the Dow Jones Index (30 stocks) and isolate subsets of stocks that have large amount of mean reversion. This can be done by setting a certain threshold, and including only those stocks with corresponding weights above this threshold in our portfolio. We compare the performance of our criterion for various values of λ (recall that higher value of λ corresponds to more fluctuating portfolios). p-value Index λ = 6 λ = 3 λ = 0 Dow Jones 6.03 · 10 −5 6.47 · 10 −5 6.89 · 10 −5 This setting is aimed at testing the effect of the λ parameter in the proposed criterion. In Figure 1 one can clearly see that the proposed algorithm generates distributions that create mean reverting portfolios, regardless of the value of λ. The significance of the Portmanteau test can be clearly seen in Table 1 for all three values. The supplementary material contains detailed information regarding the proposed portfolios for each of the values of λ.
Pairs trading
In this setting, we compare the performance of the proposed algorithm (OSA) to the trading strategy of distributing the weight proportionally to the price of the stocks (this strategy is referred to as "Benchmark"). I.e., assume that the average prices (over certain period of time) of stocks A and B are $10 and $20. Then, we would sell two shares of A against each share of B we buy, or vise versa. We also compare the performance to the offline optimal distribution of weights (this strategy is referred to as "Off-opt"), which follows from our criterion:
x ⋆ = arg min x T t=m m−1 i=0 x T y t−i 2 − λ · m−1 i=0 x T y t−i 2 .
In all runs, we use the parameters λ = 2 and m = 5, which were chosen arbitrarily.
In Tables 2 and 3 one can clearly see the advantage of the proposed algorithm (OSA) over the offline benchmarks, in the compared parameters -revenue and closeness to pure noise. In Figure 2(b) we demonstrate the performance of the proposed algorithm visually, by applying it on the pair AT&T and Verizon (Telecommunication services).
p-value
Conclusion
Motivated by financial applications, we have considered the setting of online learning with memory, and gave efficient and asymptotically-optimal regret algorithms for this general setting. Application to constructing mean-reverting instruments is explored theoretically and empirically.
The following research directions remain: First, whereas the proposed algorithm for the online learning with memory framework is optimal in the number of iterations T , the optimal dependence on the memory parameter m remains unknown. We conjecture that the regret bound of Θ( √ mT ) is tight. Second, it would be interesting to explore the optimal trading strategy in conjunction to a mean reverting portfolio.
Theorem 3. 1 .
1Set η = D √ mGT 3/4 . Then, Algorithm 2 achieves the following regret bound for {f t } T t=1 :
Figure 1 :
1Three mean reverting portfolios, based on the Dow Jones index, each assembled by executing the proposed algorithm with certain λ parameter.
of mean reverting portfolio.
Figure 2 :
2Experimental results for T and VZ.
Table 1 :
1p-values for the Portmanteau test of three mean reverting portfolios based on the Dow Jones index (in all cases m = 10).
MSFT&INTC 5.13 · 10 −4 5.1 · 10 −4 2 · 10 −4Pair
Benchmark
Off-opt
OSA
KO & PEP
0.2906
0.2786
0.1746
T & VZ
0
0
0
MMM & DD
0.5694
0.5619
0.5896
PFE & MRK
0.1984
0.1973
0.3247
JNJ & PG
0.2484
0.2471
0.1687
XOM & CVX
0.2374
0.2671
0.4083
HD & WMT
0.5934
0.5679
0.5376
Table 2 :
2p-values for the Portmanteau test (smaller is better). The results are averaged over 50 runs for stability.
Table 3 :
3Revenues in the pairs trading setting. The results are averaged over 50 runs for stability.
The iterations in which t < m are ignored since we assume that the loss per iteration is bounded by a constant, this adds at most a constant to the final regret bound.
Marco Avellaneda, Jeong-Hyun Lee, Statistical arbitrage in the US equities market. Quantitative Finance. 10Marco Avellaneda and Jeong-Hyun Lee. Statistical arbitrage in the US equities market. Quan- titative Finance, 10:761-782, 2010.
Optimal portfolio selection in nonlinear arbitrage spreads. Hamad Alsayed, Frank Mcgroarty, European Journal of Finance. Hamad Alsayed and Frank McGroarty. Optimal portfolio selection in nonlinear arbitrage spreads. European Journal of Finance, January 2012.
Prediction, learning, and games. N Cesa-Bianchi, G Lugosi, Cambridge University PressD'A11N. Cesa-Bianchi and G. Lugosi. Prediction, learning, and games. Cambridge University Press, 2006. [D'A11]
Identifying small mean-reverting portfolios. Alexandre D' Aspremont, Quant. Finance. 113Alexandre D'Aspremont. Identifying small mean-reverting portfolios. Quant. Finance, 11(3):351-364, 2011.
Co-Integration and Error Correction: Representation, Estimation, and Testing. F Robert, C W J Engle, Granger, Econometrica. 552Robert F. Engle and C. W. J. Granger. Co-Integration and Error Correction: Representation, Estimation, and Testing. Econometrica, 55(2):251-276, 1987.
Pairs trading: performance of a relative-value arbitrage. Evan Gatev, William N Goetzmann, K Geert Rouwenhorst, Review of Financial Studies. Evan Gatev, William N. Goetzmann, and K. Geert Rouwenhorst. Pairs trading: performance of a relative-value arbitrage. Review of Financial Studies, pages 797-827, 1999.
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On a measure of lack of fit in time series models. G M Ljung, G E P Box, Biometrika. 65G.M. Ljung and G.E.P. Box. On a measure of lack of fit in time series models. Biometrika, 65:297-303, 1978.
On sequential strategies for loss functions with memory. N Merhav, E Ordentlich, G Seroussi, M J Weinberger, IEEE Trans. Inf. Theor. 487N. Merhav, E. Ordentlich, G. Seroussi, and M. J. Weinberger. On sequential strategies for loss functions with memory. IEEE Trans. Inf. Theor., 48(7):1947-1958, September 2006.
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Online convex programming and generalized infinitesimal gradient ascent. M Zinkevich, ICML. M. Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In ICML, pages 928-936, 2003.
|
[] |
[
"Characterization of Reflection Positivity: Majoranas and Spins",
"Characterization of Reflection Positivity: Majoranas and Spins"
] |
[
"Arthur Jaffe [email protected] \nHarvard University\n02138CambridgeMAUSA\n",
"Bas Janssens [email protected] \nUniversiteit Utrecht\n3584 CDUtrechtThe Netherlands\n"
] |
[
"Harvard University\n02138CambridgeMAUSA",
"Universiteit Utrecht\n3584 CDUtrechtThe Netherlands"
] |
[] |
We study linear functionals on a Clifford algebra (algebra of Majoranas) equipped with a reflection automorphism. For Hamiltonians that are functions of Majoranas or of spins, we find necessary and sufficient conditions on the coupling constants for reflection positivity to hold. One can easily check these conditions in concrete models. We illustrate this by discussing a number of spin systems with nearest-neighbor and long-range interactions.
|
10.1007/s00220-015-2545-z
|
[
"https://arxiv.org/pdf/1506.04197v2.pdf"
] | 117,414,229 |
1506.04197
|
69919ce82981d2ec80e3d25fa97096ada444ccba
|
Characterization of Reflection Positivity: Majoranas and Spins
26 Oct 2015
Arthur Jaffe [email protected]
Harvard University
02138CambridgeMAUSA
Bas Janssens [email protected]
Universiteit Utrecht
3584 CDUtrechtThe Netherlands
Characterization of Reflection Positivity: Majoranas and Spins
26 Oct 2015
We study linear functionals on a Clifford algebra (algebra of Majoranas) equipped with a reflection automorphism. For Hamiltonians that are functions of Majoranas or of spins, we find necessary and sufficient conditions on the coupling constants for reflection positivity to hold. One can easily check these conditions in concrete models. We illustrate this by discussing a number of spin systems with nearest-neighbor and long-range interactions.
Introduction
We consider a finite-dimensional Z 2 -graded * -algebra A = A even ⊕ A odd . The algebra A is a graded (super) tensor product of two algebras A ± , related by an anti-linear automorphism Θ : A → A, satisfying Θ(A ∓ ) = A ± and Θ 2 = I. In this sense, A is the double of A + . Such automorphisms often arise from geometric reflections on an underlying manifold, so we refer to Θ as the reflection automorphism. The main results summarized in Theorem 1.1 do not refer to an underlying geometry-while in the examples of §7 this becomes relevant.
In this context, we are interested in even linear functionals ω : A → C that are both reflection invariant and reflection positive with respect to the reflection Θ. A functional is called even if ω(A odd ) = 0, and just like in the ungraded case, it is called reflection invariant if ω(Θ(A)) = ω(A) for all A ∈ A. The notion of reflection positivity has to be adapted to the Z 2 -grading; we call ω reflection positive on A + if 0 ω(Θ(A)A) , for A ∈ A even + , (1.1)
0 ζ ω(Θ(A)A) , for A ∈ A odd + ,
where ζ = ±i is fixed once and for all. We introduce the twisted product • : [DM99]. We elaborate on this relation to super-Hilbert spaces in §2.
A − × A + → A with A • B = AB , if A or B ∈ A
We consider in detail the case that ω = ω H is a Boltzmann functional. By this we mean that there is an element H ∈ A called the Hamiltonian, such that In statistical physics, H ∈ A is self-adjoint. In this case Z H > 0, and ρ H is a state, meaning that ρ H is positive and normalized. Furthermore, it has the KMS property with respect to the automorphisms of A induced by e itH . But in lattice approximations to fermionic quantum fields, the action plays the role of H and often is not hermitian. In any case we do not assume that H is hermitian.
Here we specialize to two types of algebras A. In the first part of the paper, §1- §4, A will be an algebra of Majoranas, whereas in the second part §5- §7, A will generally be an algebra of spins.
An algebra of Majoranas is a * -algebra generated by self-adjoint operators c i . They are labeled by indices i running over a finite set Λ, and satisfy the Clifford relations c i c j + c j c i = 2δ ij I , i, j ∈ Λ .
(1.5)
The Z 2 grading of A is defined as +1 on the even and −1 on the odd monomials in the c i . Even elements of A are often called globally gauge invariant.
The reflection automorphism Θ of the Majorana algebra A comes from a fixed point free reflection ϑ : Λ → Λ. If Λ is the disjoint union of Λ + and Λ − with ϑ(Λ ± ) = Λ ∓ , then the algebras A ± are generated by the Majoranas c i with i ∈ Λ ± . In many applications, Λ will be a finite lattice in Euclidean space, and ϑ the reflection in a hyperplane which does not intersect Λ.
We give necessary and sufficient conditions such that the functionals ω H and ρ H are reflection positive on A + . Every Hamiltonian H ∈ A is defined by a coupling-constant matrix J as
H = − J i1,...,i k ; i ′ 1 ,...,i ′ k ′ Θ(c i1 · · · c i k ) • (c i ′ 1 · · · c i ′ k ′ ) ,(1.6)
where k and k ′ range over N, and i 1 , . . . , i k and i ′ 1 , . . . , i ′ k ′ range over Λ + . In fact one restricts the set over which one sums, in order to make the expansion unique, as explained in §1.3- §1.5. The conditions on reflection positivity are expressed in terms of the submatrix J 0 of J for which k = 0 and k ′ = 0. If ϑ comes from a reflection in Euclidean space, J 0 describes the couplings across the reflection plane. We use this terminology even if a geometric interpretation is lacking.
The central result in this paper, which also holds with ρ βH replaced by ω βH , is the following:
Theorem 1.1 Let H be reflection invariant and globally gauge invariant. Then ρ βH is reflection positive for all 0 < β, if and only if 0 J 0 .
In the second part of the paper, we focus on spin algebras A spin , generated by the Pauli matrices σ 1 j , σ 2 j , σ 3 j associated to each lattice site j ∈ Λ. In §5 we study Hamiltonians of the form H spin = − J a1,...,a k i1,...,i k σ a1 i1 · · · σ a k i k .
(1.7)
By expressing the spins σ a j as even polynomials in the Majoranas, we translate Theorem 1.1 to the spin context. This yields necessary and sufficient conditions on reflection positivity in terms of the coupling constants J a1,...,a k i1,...,i k . Again, the condition involves only the couplings across the reflection plane.
In §6 we analyze different reflections Θ and Θ ′ = αΘα −1 , both of which interchange the same A ± . If they are related by a reflection-invariant gauge automorphism α, then our characterization of reflection positivity applies to Θ ′ as well as Θ.
In §7 we illustrate the main results by showing that a number of spin Hamiltonians with nearest neighbor as well as long-range interactions fit naturally into our general framework. The central point of these examples is that our characterization of reflection positivity in Theorem 1.1 can be applied easily to realistic physical systems.
Reflection positivity of functionals has a long history in physics, as well as mathematics. On the one hand, reflection positivity gives the relation between classical systems and quantum theory. Furthermore reflection positivity is central in proving the existence of phase transitions/multiple equilibrium states in a number of classical and quantum systems. Concrete examples include, among many others, classical and quantum Heisenberg antiferromagnets, hardcore nearest-neighbor and Coulomb lattice gases, and φ 4 2 quantum fields. Some earlier work can be found in [OS73,OS75,GJS75,FSS76,DLS78,FILS78,KL81, FOS83,Lie94,MN96,NÓ14,NÓ15].
The present work was inspired by [JP15a,JP15b], and generalizes that work: here we obtain reflection positivity for couplings that are not necessarily diagonal (including long-range interactions), for observables that are not necessarily even, and with hypotheses that are necessary as well as sufficient.
1.1. Reflections. Here we study a finite set Λ which is an index set for the generators c i of our algebra. We assume that Λ is invariant under an involution ϑ : Λ → Λ that we call a reflection. We assume that ϑ exchanges two subsets Λ ± whose union is Λ, and that ϑ has no fixed points.
In specific models, Λ is often a finite subset of a manifold M, and ϑ is the restriction to Λ ⊂ M of a reflection We give a number of examples of this situation, where M is the Euclidean space R d , a torus T d , or a Riemann surface.
If M = R d , the reflection ϑ R d : R d → R d is given in suitable coordinates by ϑ(x 0 , x 1 , . . . , x d−1 ) = (−x 0 , x 1 , . . . , x d−1 ) .
The half-spaces R d ± = {x ∈ R d : ±x 0 0} have as a common boundary the reflection plane R d 0 = {x ∈ R d : x 0 = 0}. Then Λ + ⊂ R d + is a finite set of points on one side of the reflection plane R d 0 , the set Λ − is its reflection, and Λ = Λ + ⊔ Λ − . Note that Λ contains no points in the reflection hyperplane.
An important example is the d-dimensional simple cubic lattice One often has periodic boundary conditions, in which case M is the torus T d instead of R d . The invariant hypersurface M 0 is then the union of two (d − 1)tori.
Λ cubic = {−L − 1 2 , −L + 1 2 , . . . , L − 1 2 , L + 1 2 } d ,
Examples where M is a Riemann surface of arbitrary genus arise from considering the conformal inversion ϑ of a Schottky double of an open Riemann surface T , with Λ + a finite set of points in T .
In §1- §5, including the main Theorems 3.4, 4.2, 4.4, and 5.2, we only need Λ to be an abstract set with a fixed point free involution ϑ. In the discussion of examples in §7, we require additional structure for Λ, involving its geometric significance as a subset of a manifold M, as explained later.
1.2. Majoranas. One defines an algebra of Majoranas on the lattice Λ as the *algebra A with self-adjoint generators c i = c * i that satisfy the Clifford relations (1.5). For any subset Γ ⊂ Λ, let A(Γ ) denote the algebra generated by the c j 's with j ∈ Γ . In particular, A = A(Λ), and we define A ± := A(Λ ± ).
We call the automorphism α : A → A that implements the Z 2 grading a global gauge automorphism. On the generators, it satisfies
c j → α(c j ) = −c j .
(1.8)
The algebra A decomposes into the spaces A even and A odd of elements that are even and odd for the Z 2 -grading,
A = A even ⊕ A odd .
In the same vein, A(Γ ) = A(Γ ) even ⊕ A(Γ ) odd . An element A ∈ A that is either even or odd is called homogeneous. Since A ∈ A is even if α(A) = A and odd if α(A) = −A, the even elements are also called globally gauge invariant.
Define the degree |A| of A as |A| = 0 for A ∈ A even , and |A| = 1 for A ∈ A odd . The algebra A(Γ ) commutes with A even (Γ ′ ) when Γ ∩ Γ ′ = ∅. More generally, if A ∈ A(Γ ) and B ∈ A(Γ ′ ) are both eigenvectors of α, then
AB = (−1) |A||B| BA , when Γ ∩ Γ ′ = ∅ .
One says that A(Γ ) and A(Γ ′ ) supercommute if Γ and Γ ′ are disjoint.
Reflections and Invariant
Bases. The reflection ϑ : Λ → Λ defines an antilinear *-automorphism Θ : A → A given by
Θ(c i1 · · · c i k ) := c ϑ(i1) · · · c ϑ(i k ) .
(1.9)
Note that Θ exchanges A + with A − , namely Θ(A ± ) = A ∓ , and satisfies Θ 2 = Id. We construct bases of A that are adapted to this reflection. For Γ ⊆ Λ, let S Γ denote the set of sequences I = (i 1 , . . . , i k ) of distinct lattice points i 1 , . . . , i k ∈ Γ . For the important choices Γ = Λ, Γ = Λ + and Γ = Λ − , we denote S Γ by S, S + , and S − , respectively. For I ∈ S Γ , define the monomial C I := c i1 · · · c i k , and define C I := I for I = ∅. Each C I is an eigenvector of the gauge automorphism α, and we denote its degree by
|I| := |C I | .
(1.10)
Then |I| = 0 if k is even, and |I| = 1 if k is odd. Also
C * I = (−1) 1 2 k(k−1) C I .
(1.11)
The algebra A(Γ ) is spanned by the operators C I with I ∈ S Γ , but they are linearly dependent. In fact, C I = ±C I ′ if the sets {i 1 , . . . , i k } and {i ′ 1 , . . . , i ′ k ′ } are the same. A choice P + ⊆ S + such that every set {i 1 , . . . , i k } of distinct lattice points corresponds to precisely one tuple (i 1 , . . . , i k ) in P + yields a basis
B + = {C I ; I ∈ P + } of A + . This, in turn, yields a basis B − = Θ(B + ) of A − .
1.4. The Twist. From the two bases B + and B − , we construct a basis B of A, that is adapted to the reflection Θ. For this, fix a square root of minus one,
ζ = ± √ −1 , (1.12)
and define a basis for A by
B := {ζ |I||I ′ | Θ(C I )C I ′ ; C I , C I ′ ∈ B + } .
Although the main results on reflection positivity will hold for both twists ζ = √ −1 and ζ = − √ −1, the class of allowed Hamiltonians will not be the same. Note that, in a sense, the basis elements in B are the geometric mean of the operators Θ(C I )C I ′ and C I ′ Θ(C I ), which differ by a factor (−1) |I||I ′ | . The identity I = C ∅ = Θ(C ∅ ) = Θ(C ∅ )C ∅ is a basis element in all three bases B + , B − and B. Every A ∈ A has an expansion
A = I,I ′ a II ′ ζ |I||I ′ | Θ(C I )C I ′ ,
(1.13) which is unique if the I, I ′ are restricted to be in P + .
1.5. Twisted Product. In order to streamline notation, introduce the following (non-associative) twisted product
• : A × A → A .
(1.14)
Definition 1.2 Let A ∈ A be of the form A = A − A + with A ± ∈ A ± , and similarly B = B − B + with B ± ∈ A ± . If A ± and B ± are homogeneous, then A • B is defined by A • B := ζ |A−||B+|−|A+||B−| AB .
This extends bilinearly to the product • : A × A → A.
Note that the formula A • B := ζ |A−||B+|−|A+||B−| AB also holds for A = A + A − and B = B + B − . One finds
X I1I ′ 1 • X I2I ′ 2 = ζ |I1| |I ′ 2 |−|I ′ 1 | |I2| X I1I ′ 1 X I2I ′ 2 (1.15)
for twisted products of elements of the form X II ′ = Θ(C I )C I ′ or X II ′ = C I ′ Θ(C I ).
In terms of the twisted product, the basis B can be written
B = {Θ(C I ) • C I ′ : I, I ′ ∈ P + } .
(1.16)
Correspondingly we can rewrite the expansion (1.13) of a general element A ∈ A in basis elements as A =
I,I ′ ∈P+ a II ′ Θ(C I ) • C I ′ .
(1.17)
The twisted product has a number of useful properties. For example A + and A − commute with respect to the twisted product.
Proposition 1.3 If A + ∈ A + and B − ∈ A − , then A + • B − = B − • A + .
Proof. It suffices to prove this for homogeneous elements, in which case the result follows from
A + • B − = ζ −|A+||B−| A + B − , B − • A + = ζ |A+||B−| B − A + , and A + B − = (−1) |A+||B−| B − A + .
The twisted product respects the reflection:
Proposition 1.4 For all A, B ∈ A, one has Θ(A • B) = Θ(A) • Θ(B).since Θ(A ± ), Θ(B ± ) ∈ A ∓ .
It follows that the reflection permutes the basis B.
Corollary 1.5 The twisted product satisfies
Θ Θ(A) • B = Θ(B) • A , for A, B ∈ A + , or A, B ∈ A − .
(1.18)
In particular, the basis B is permuted by Θ, Proof. This follows from anti-linearity of Θ and Corollary 1.5.
Θ(Θ(C I ) • C I ′ ) = Θ(C I ′ ) • C I . (1.19) Proof. By Proposition 1.4, one has Θ(Θ(A) • B) = A • Θ(B),
Define k : Λ → N by k I = r for I = (i 1 , . . . , i r ). When dealing with adjoint operators, one frequently encounters the derived expressions (1.20)
Note that q I is 4-periodic in k, and s I is 8-periodic.
Proposition 1.7 Let I, I ′ ∈ P + . Then
(Θ(C I ) • C I ′ ) * = q I q I ′ Θ(C I ) • C I ′ .
(1.21)
Proof. As Θ is a * -automorphism,
(Θ(C I ) • C I ′ ) * = ζ −|I||I ′ | (Θ(C I )C I ′ ) * = ζ −|I||I ′ | C * I ′ Θ(C * I ) .
Inserting (1.11) gives
(Θ(C I ) • C I ′ ) * = ζ −|I||I ′ | q I q I ′ C I ′ Θ(C I ) = q I q I ′ Θ(C I ) • C I ′ .
In the last equality we use C I ′ Θ(C I ) = ζ 2|I||I ′ | Θ(C I )C I ′ and the definition of the circle product to give the desired relation (1.21).
Using this, one derives the following characterization of hermiticity.
Corollary 1.8 If A ∈ A has an expansion (1.17) with coefficients a II ′ , then A * has coefficients q I q I ′ a II ′ . The operator A is hermitian if and only if s I s I ′ a II ′ is real for all I, I ′ ∈ P + .
Proof. The first statement follows from
A * = I,I ′ ∈P+ a II ′ (Θ(C I ) • C I ′ ) * = I,I ′ ∈P+ a II ′ q I q I ′ Θ(C I ) • C I ′ . (1.22) Therefore, A is hermitian if and only if a II ′ = a II ′ q I q I ′ . Since s 2 I = s −2 I = q I , this is equivalent to s I s I ′ a II ′ = s I s I ′ a II ′ .
1.6. The Tracial State. Define the functional Tr : A → C by
Tr(A) = a ∅∅ , (1.23)
where a II ′ are the coefficients in (1.17).
Proposition 1.9 Let I 0 , I 1 and I ′ 0 , I ′ 1 be elements of P + . Then
Tr Θ(C I0 ) • C I ′ 0 * Θ(C I1 ) • C I ′ 1 = δ I0I1 δ I ′ 0 I ′ 1 . (1.24) Also Tr Θ(C I0 ) • C I ′ 0 Θ(C I1 ) • C I ′ 1 = q I0 q I ′ 0 δ I0I1 δ I ′ 0 I ′ 1 . (1.25)
Proof. The identity (1.25) is equivalent to (1.24) as a consequence of (1.21). The left hand side of (1.24) vanishes unless I 0 = I 1 and
I ′ 0 = I ′ 1 , in which case (1.21) along with C * I1 C I1 = C * I ′ 1 C I ′ 1 = I yields (Θ(C I0 ) • C I ′ 0 ) * · Θ(C I1 ) • C I ′ 1 = ζ −|I0||I ′ 0 |+|I1||I ′ 1 | C * I ′ 0 Θ(C * I0 )Θ(C I1 )C I ′ 1 = C * I ′ 1 Θ(C * I1 C I1 )C I ′ 1 = I .q I q I ′ a II ′ b II ′ . (1.30)
Hence the state Tr is cyclic.
(d) As Θ is antilinear and the basis elements satisfy (1.19), it follows that Tr satisfies (1.28).
(e) To demonstrate factorization, consider A − = I∈P+ a I∅ Θ(C I ) and
B + = K ′ ∈P+ b ∅K ′ C K ′ . In this case, identity (1.30) takes the form Tr(A − B + ) = a ∅∅ b ∅∅ = Tr(A − ) Tr(B + )
. So the factorization property follows.
Corollary 1.11 If H ∈ A is reflection invariant, Θ(H) = H, then the partition sum Z H = T(e −H ) is real. Proof. Since Θ is an automorphism, it follows from Θ(H) = H that Θ(e −H ) = e −
Reflection Positive Functionals
In this section, we characterize reflection invariance and reflection positivity of linear functionals in terms of their density matrix.
Reflection Invariance.
Let ω : A → C be a linear functional on A. From Proposition 1.10.b, we infer that every functional can be written
ω(A) = Tr(AR) (2.1) for a unique density matrix R ∈ A.
If ω is a state, then R is a positive operator with trace 1. Consider the sesquilinear form · , · R,Θ on A given as
A, B R,Θ := ω(Θ(A) • B) = Tr((Θ(A) • B)R) . (2.2)
If we expand R in terms of matrix elements r II ′ as
R = I,I ′ ∈P+ r II ′ Θ(C I ) • C I ′ , (2.3)
then (1.11) and Proposition 1.9 ensure that
r II ′ = C * I , C * I ′ R,Θ , where C I , C I ′ ∈ B + . (2.4) Definition 2.1 (Reflection Invariance) The linear functional ω is reflection invariant on A if ω(Θ(A)) = ω(A) for all A ∈ A.
Proposition 2.2 (Reflection-Invariant Functionals) The following conditions are equivalent:
(a) The functional ω(A) = Tr(AR) is reflection invariant on A. (b) The operator R is reflection invariant, Θ(R) = R. (c) The matrix r II ′ is hermitian, r I ′ I = r II ′ . (d) The sesquilinear form · , · R,Θ is hermitian on A + , A, B R,Θ = B, A R,Θ , for all A, B ∈ A + .
Proof. Thus
ω(A) = ω(Θ(A)), and ω is reflection invariant. (a)⇒(d): If ω is reflection invariant, then ω(Θ(B) • A) = ω(Θ(Θ(B) • A)) = ω(Θ(A) • B) ,
where the second equality follows from Proposition 1.5.
(d)⇒(b): Since B, A R,Θ = Tr((Θ(B) • A)R)
, reflection invariance of the trace and Proposition 1.5 yield
B, A R,Θ = Tr(Θ(Θ(B) • A)Θ(R)) = Tr((Θ(A) • B)Θ(R)) , for all A, B ∈ A + . Since A, B R,Θ = Tr((Θ(A)•B)R), we infer from A, B R,Θ = B, A R,Θ that Tr((Θ(A) • B)R) = Tr((Θ(A) • B)Θ(R)) . Since A is spanned by elements of the form Θ(A) • B with A, B ∈ A + , nonde- generacy of the trace implies Θ(R) = R.
We conclude that (a)⇔(b)⇔(d). The equivalence (b)⇔(c) was already proven in Proposition 1.6. 2.2. Reflection Positivity. In this section, we characterize reflection positive functionals in terms of their density matrix.
A linear functional ω : A → C is called even if ω(A odd ) = {0}. Note that if R isDefinition 2.4 The linear functional ω in (2.1) is reflection positive on A + with respect to Θ, if the form · , · R,Θ in (2.2) is positive, semidefinite on A + .
The reflection positive Hilbert space H is defined as the completion with respect to · , · R,Θ of the quotient of A + by the null space. If ω is even, H will be a super Hilbert space in the following sense.
Definition 2.5 ([DM99], §4.4) A super Hilbert space is a Z 2 -graded vector space H = H even ⊕ H odd with a form ( · , · ) : H × H → C that is -linear in the second argument, -graded symmetric, (w, v) = (−1) |v||w| (v, w)
for v, w ∈ H homogeneous -even, (v, w) = 0 for v ∈ H even and w ∈ H odd -positive, in the sense that
0 < (v, v) for 0 = v ∈ H even (2.5) 0 < ζ(v, v) for 0 = v ∈ H odd .
Furthermore, the total space H is required to be complete for the scalar product defined by v, w :
= (v, w) for v, w ∈ H even , v, w := ζ(v, w) for v, w ∈ H odd , and v, w := 0 for v ∈ H even , w ∈ H odd .
Proposition 2.6 If ω is even, reflection invariant, and reflection positive, then the completion H with respect to · , · R,Θ of A + modulo the null space is a super Hilbert space with the form
(A, B) := ω(Θ(A)B). Proof. The form A, B R,Θ = ω(Θ(A) • B)
is hermitian by Proposition 2.2, and positive semidefinite by reflection positivity of ω. As ω is even, Proposi-
tion 2.3 yields H even ⊥ H odd . Since A, B R,Θ = (A, B) for A, B ∈ A even + and A, B R,Θ = ζ(A, B) for A, B ∈ A odd
+ , graded symmetry of ( · , · ) follows from hermiticity of · , · R,Θ , and positivity of ( · , · ) (equation (2.5)) follows from the fact that · , · R,Θ is positive semidefinite.
As H even ⊥ H odd , the value of A • B for A ∈ A even and B ∈ A odd is quite immaterial for even functionals; the relevant property of the twisted product is that A • B = AB for A, B ∈ A even + , and A • B = ζAB for A, B ∈ A odd + . Our choice for Definition 1.2 was merely motivated by the wish to treat A + and A − on equal footing.
Θ(A), Θ(B) R,Θ = B, A R,Θ , for A, B ∈ A + .
(2.6)
Proof. For A, B ∈ A + , we infer from Proposition 1.3 and Corollary 1.5 that
ω(Θ(Θ(A)) • Θ(B)) = ω(A • Θ(B)) = ω(Θ(B) • A) .
The first term equals Θ(A), Θ(B) R,Θ and the last one B, A R,Θ . Proof. Expand A, B ∈ A + as A = I∈P+ a I C I and B = I∈P+ b I C I . Using (1.11) and (2.4) we obtain
A, B R,Θ = Tr((Θ(A) • B)R) = I 0 ,I ′ 0 ∈P + I 1 ,I ′ 1 ∈P + a I0 b I ′ 0 , r I1I ′ 1 Tr Θ(C I0 ) • C I ′ 0 (Θ(C I1 ) • C I ′ 1 ) = I,I ′ a I q I b I ′ q I ′ r II ′ .
It follows that A, A R,Θ 0 for all A ∈ A + if and only if the matrix r II ′ is is positive semidefinite.
Sufficient Conditions for Reflection Positivity
In statistical physics, Gibbs states are defined in terms of a Hamiltonian H, which in turn is given by a matrix J of coupling constants. In this section, we provide a sufficient condition on J for the associated Gibbs state to be reflection positive. This will be further refined to a necessary and sufficient condition in Section 4.
Density Matrices and Hamiltonians.
For a (not necessarily hermitian) Hamiltonian H ∈ A, consider the unnormalized density matrix R = e −H . We now focus on the Hamiltonian H rather than R, and define the Boltzmann functional
ω H : A → C by ω H (A) = Tr(A e −H )
. In §4.2 we show reflection positivity of the Boltzmann functional ω H for a large class of reflection symmetric, globally gauge invariant Hamiltonians H, namely all those for which the matrix of coupling constants is positive semidefinite. For such Hamiltonians Z H 1.
We use this result to prove reflection positivity for an even wider class of Hamiltonians, namely those for which the matrix of coupling constants across the reflection plane is positive semidefinite.
Neither result will require H to be hermitian, but if this happens to be the case, Z H is automatically nonzero, and ρ H is the Gibbs state with respect to the Hamiltonian H.
3.2.
Hamiltonians. The class of Hamiltonians for which these reflection positivity results hold, is defined in terms of the matrix of coupling constants,
J = (J II ′ ) , where I, I ′ ∈ P + .
(3.5)
By definition, these are the coefficients J II ′ ∈ C of the Hamiltonian H in its expansion with respect to the basis B,
H = − I,I ′ ∈P+ J II ′ Θ(C I ) • C I ′ .
(3.6)
The We give some preliminary results before proving the theorem.
Lemma 3.5 Let I 1 , . . . , I k , I ′ 1 , . . . I ′ k ∈ S + and |I j | = |I ′ j | for j 1. Then for all I 0 ,
I ′ 0 ∈ S + , Tr(C I0 · · · C I k ) Tr(C I ′ 0 · · · C I ′ k ) (3.7) is nonzero only if |I 0 | = |I ′ 0 |.
Proof. For every lattice point i ∈ Λ, let k i (I) be 1 if i occurs in I = (i 1 , . . . , i s ), and 0 otherwise. Then s = k I = i∈Λ k i (I). If Tr(C I0 · · · C I k ) is nonzero, then k j=0 k i (I j ) is even, as every i ∈ Λ must occur an even number of times. Therefore,
i∈Λ k j=0 k i (I j ) = k j=0 i∈Λ k i (I j ) = k j=0 k Ij is even. Since |I| = k I mod 2, the sum k j=0 |I j | is even. Similarly, one finds that k j=0 |I ′ j | is even if Tr(C I ′ 0 · · · C I ′ k ) is nonzero. Since |I j | = |I ′ j | for j 1 by assumption, we infer that |I 0 | = |I ′ 0 | if 3.7 is nonzero.
Lemma 3.6 Under the hypotheses of Lemma 3.5,
Tr (Θ(C I0 ) • C I ′ 0 ) · · · (Θ(C I k ) • C I ′ k ) = Tr(C I0 · · · C I k ) Tr(C I ′ 0 · · · C I ′ k ) . (3.8)
Proof. Use the definition of • to write
Tr((Θ(C I0 ) • C I ′ 0 ) · · · (Θ(C I k ) • C I ′ k )) = ζ k j=0 |Ij ||I ′ j | Tr Θ(C I0 )C I ′ 0 · · · Θ(C I k )C I ′ k ,(3.9)
and bring the terms of the form Θ(C Ij ) to the left. In doing so, one has to exchange Θ(C Ij ) with C I ′ j ′ for each j ′ < j, yielding a factor
(−1) j j ′ =0 |I ′ j ′ ||Ij | = ζ 2 j j ′ =0 |I ′ j ′ ||Ij | .
The right hand side in equation 3.9 can thus be written
ζ k j=0 |Ij ||I ′ j |+2 0 j ′ <j k |Ij ||I ′ j ′ | Tr(Θ(C I0 · · · C I k )C I ′ 0 · · · C I ′ k ) ,(3.10)
where we used that Θ(C I0 ) · · · Θ(C I k ) equals Θ(C I0 · · · C I k ).
Using the factorization of the trace, Tr(X − X + ) = Tr(X − ) Tr(X + ) for X ± ∈ A ± , and reflection invariance, Tr(Θ(X)) = Tr(X), given in Proposition 1.10.d and e, (3.10) becomes
ζ k j=0 |Ij ||I ′ j |+2 0 j ′ <j k |Ij ||I ′ j ′ | Tr(C I0 · · · C I k ) Tr(C I ′ 0 · · · C I ′ k ) .
(3.11)
Using Lemma 3.5, we rewrite the phase in (3.10)
ζ k j=0 |Ij ||I ′ j |+2 0 j ′ <j k |Ij ||I ′ j ′ | = ζ k j=0 |Ij| 2 = 1 . (3.12)
The last equality holds as k j=0 |I j | must be even, so its square is 0 mod 4, and the phase vanishes. Combining (3.12) with (3.11), the proof is complete.
A, B 0 H,Θ = ∞ k=0 1 k! I0,...I k I ′ 0 ,...,I ′ k a I0 b I ′ 0 J I1,I ′ 1 · · · J I k ,I ′ k × Tr(C I0 · · · C I k ) Tr(C I ′ 0 · · · C I ′ k ) . (3.13)
From the power series for e −H with H given by (3.6), one obtains the expansion
A, B 0 H,Θ = ∞ k=0 1 k! I0,...I k ∈P+ I ′ 0 ,...,I ′ k ∈P+ a I0 b I ′ 0 J I1I ′ 1 · · · J I k I ′ k × Tr((Θ(C I0 ) • C I ′ 0 ) · · · (Θ(C I k ) • C I ′ k )) .
(3.14)
The terms with I 0 and I ′ 0 arise from A and B, while the remaining I j , I ′ j come from powers of H. By Proposition 3.2, global gauge invariance of H ensures that |I j | = |I ′ j | for all j 1. From Lemma 3.6, we conclude that
Tr((Θ(C I0 ) • C I ′ 0 ) · · · (Θ(C I k ) • C I ′ k )) = Tr(C I0 · · · C I k ) Tr(C I ′ 0 · · · C I ′ k ) .
(3.15) So by Lemma 3.5, |I 0 | = |I ′ 0 | unless (3.15) vanishes. Using this and the expansion 3.14, one obtains 3.13.
Let χ k , ψ k denote vectors with components
χ k I1,...,I k = I0∈P+ a I0 Tr(C I0 · · · C I k ) , and ψ k I1,...,I k = I0∈P+ b I0 Tr(C I0 · · · C I k ) ,
labelled by P k + . Let J ⊗k I1,...,I k ;I ′ 1 ,...,I ′ k := J I1I ′ 1 · · · J I k I ′ k be the k th tensor power of the matrix J II ′ . Since J II ′ is a positive semidefinite matrix, J ⊗k is also positive semidefinite. Then
A, B 0 H,Θ = ∞ k=0 1 k! χ k , J ⊗k ψ k ,(3.16)
with the inner product
χ k , ψ k := I1,...I k ∈P+ χ k I1...I k ψ k I1...I k .
Setting B = A one has ψ k = χ k . Since each term in the sum (3.16) is nonnegative, the theorem follows. with q I = (−1) k I (k I −1) as defined in equation (1.20). Since the matrix J II ′ is positive semidefinite, the Boltzmann functional ω H is reflection positive by Theorem 3.8. The matrix r II ′ of coefficients of R = e −βH is positive semidefinite, as a consequence of Theorem 2.8. It follows that the Hadamard product matrix K, with matrix elements K II ′ = J II ′ r II ′ , is also positive semidefinite. From 3.18, we infer that
dZ βH dβ = I,I ′ ∈P+ q I K II ′ q I ′ = q, Kq ℓ 2 0 . (3.19)
Thus Z βH is a non-decreasing function of β.
Necessary and Sufficient Conditions
In Theorem 3.4 we have given sufficient conditions for reflection positivity of the Boltzmann functional ω H ; it is reflection positive if J 0, where J is the matrix (3.5) of couplings by which H is defined. Now we establish a stronger result, providing necessary and sufficient conditions in terms of the submatrix J 0 of J that contains only the couplings between Majoranas on different sides of the reflection plane. If J is positive semidefinite, then J 0 is positive semidefinite, but the converse does not hold.
In Section 4.2 we prove that ω H is reflection positive if and only if J 0 0. Using this, we prove the analogous statement for the Gibbs functional ρ H in §4.3.
Coupling Constants Across the Reflection Plane.
Let H be reflection invariant, so that the coupling-constant matrix J is hermitian. Order the basis elements in B + so C ∅ = I is the first one, and consider the decomposition of J, The interaction across the reflection plane is Proof. (a) Since H is reflection invariant, we infer from Proposition 3.2 that J is hermitian. Writing J as in (4.1), recall that reflection positivity of ω H is independent of the value of E. So for simplicity we can add a constant to H so that E = 0. Now we approximate J by J ε defined as the matrix
J = J ∅∅ J ∅I ′ J I∅ J II ′ = E V * V J 0 .−H 0 = I,I ′ ∈P+−{∅} J 0 II ′ Θ(C I ) • C I ′ .J ε := 0 V * V J 0 + ε 0 0 0 V V * , (4.8)
where 0 ε is a small parameter. Here V V * denotes the matrix with elements (V V * ) II ′ = V I V I ′ with I, I ′ ∈ P − ∅. Clearly J ǫ → J as ε → 0, so that H ε → H as ε → 0. Hence ω Hǫ → ω H as ε → 0. Assume that the functional ω Hε satisfies reflection positivity on A + for every ε > 0. Then the convergence explained above means that for A ∈ A + , the expectations ω Hε (Θ(A) • A) 0 converge to ω H (Θ(A) • A) 0 as ε → 0. We infer that ω H is reflection positive. Now we show that ω Hε does satisfy reflection positivity for every ε > 0. In order to see this, we make a second modification to J, by adding the constant ε −1 to H ǫ . Thus we obtain a new matrix of couplings J ε defined as
J ε = J ε + 1 ε 1 0 0 0 = 0 0 0 J 0 + ε −1 V * V εV V * . (4.9)
The couplings J ε correspond to a Hamiltonian H ε , that differs from H ε only by an additive constant. So ω Hε satisfies reflection positivity if and only if ω Hε does. Furthermore we can appeal to Theorem 3.4, so it is sufficient to show that the matrix J ε is positive semidefinite for every ε > 0.
We claim that J ε is positive semidefinite, since each of the two matrices on the right of (4.9) are positive semidefinite, as is the sum of two positive semidefinite matrices. The first matrix on the right is positive semidefinite by the assumption that J 0 is positive semidefinite. The second matrix is also positive definite, since it can be written as
ε −1 V * V εV V * = ε −1/2 0 ε 1/2 V 0 ε −1/2 ε 1/2 V * 0 0 .
This concludes the proof that ω H is reflection positive on A + . (b). Suppose that ω βH is reflection positive on A + for β ∈ [0, ε). Choose A = I∈P+ a I C I with a ∅ = 0, so A is in the null space of the form A, A 0 0,Θ , as Tr(Θ(A) • A) = |a ∅ | 2 = 0. Reflection positivity then ensures that the first derivative cannot be negative, Since this holds for all f ∈ ℓ 2 (P + ) with f ∅ = 0, this assures that the matrix J 0 is positive semidefinite.
0 d dβ A, A 0 βH,Θ β=0 = − Tr((Θ(A) • A)H) ,
Remark 4.3
This theorem is somewhat similar in flavor to Schoenberg's Theorem [Sch38a,Sch38b], which states that e −βK is a positive definite kernel for all β ∈ [0, ε) if and only if K is conditionally negative definite. In particular, it is striking that just as in Schoenberg's Theorem, J is only required to be positive definite on a subspace of codimension 1. (a) Suppose that Z H = 0, and that the matrix J 0 of coupling constants across the reflection plane is positive semidefinite. Then ρ H is reflection positive, and Z H > 0. (b) If there exists an ε > 0 such that ρ βH is reflection positive for all β ∈ [0, ε), then the matrix J 0 of coupling constants across the reflection plane is positive semidefinite.
Reflection Positive Gibbs
Remark 4.5 If H ∈ A even is self-adjoint, then the condition Z βH = 0 is automatically satisfied for all β 0.
Proof. (a) If J 0 is positive semidefinite, then ω H is reflection positive by Theorem 4.2. Reflection positivity of the Gibbs functional ρ H then follows by Remark 4.1.
(b) The partition function Z βH = Tr(e −βH ) is analytic in β, and real by Corollary 1.11. Since Z 0 = 1, the expression
ρ βH (X) = Z −1 βH Tr(Xe −βH )
is well defined and analytic in a neighborhood U of β = 0. The inequality
0 ρ βH (Θ(A) • A) for β ∈ U thus yields 0 Z βH ρ βH (Θ(A) • A) = ω βH (Θ(A) • A) .
Since this holds for all A ∈ A + and β ∈ U , the Boltzmann functional ω βH is reflection positive for all β ∈ U , and J 0 is positive semidefinite by Theorem 4.2.
Reflection Positivity for Spin Systems
From the corresponding result for Majoranas, we now derive necessary and sufficient conditions for reflection positivity in the context of spin systems. As in the case of Majoranas, these will be formulated in terms of the matrix of coupling constants across the reflection plane.
Spin Algebras.
In spin models, the algebra of observables for a lattice site j ∈ Λ is M 2 (C), spanned by I and the Pauli spin matrices σ 1 j , σ 2 j , σ 3 j . The operators σ a j and σ b j ′ commute for j = j ′ , and otherwise satisfy the familiar relations σ a j σ b j = δ ab I + i c ǫ abc σ c j . In this context, the full algebra of observables is
A spin = j∈Λ M 2 j (C) ,
and the algebra of observables on the ± side of the reflection plane is
A spin ± = j∈Λ± M 2 j (C) .
Define the operators Σ (I,A) as the product of spins
Σ (I,A) = σ a1 i1 . . . σ a k i k .
They are labelled by sets of the form a 1 ), . . . , (i k , a k )} ,
(I, A) := {(i 1 ,(5.1)
where i s is a lattice point in Λ, a s is a spin label in {1, 2, 3}, and i s = i t for s = t. Together with the identity Σ ∅ := I, the operators Σ (I,A) constitute an orthonormal basis of A with respect to the bilinear trace pairing,
Tr spin (Σ (I,A) Σ (I ′ ,A ′ ) ) = δ AA ′ δ II ′ . (5.2)
Definition 5.1 (Standard Reflection) The standard reflection Θ on A spin is defined by Θ(σ a j ) = −σ a ϑ(j) , for j ∈ Λ and a ∈ {1, 2, 3}.
The standard reflection satisfies Θ(Σ (I,A) ) = (−1) k I Σ ϑ(I,A) . 5.3. Mapping Spins to Majoranas. Spin models map to Majorana models by a well-known transformation. For a single site, this is similar to the infinitesimal rotation written in terms of Dirac matrices. The tensor product construction, projected to a chiral subspace, is known in the condensed matter literature as the Kitaev transformation. This map X → X from the algebra A spin of spins to the algebra A of Majoranas is constructed as follows.
Choose four Majoranas at site j denoted c α j , for α = 1, 2, 3, 4. (The superscripts denote labels, not powers.) The Majoranas satisfy the Clifford relations {c α j , c β j ′ } = 2δ αβ δ jj ′ I and c α * j = c α j . They generate the Majorana algebra A indexed by Λ = Λ × {1, 2, 3, 4}.
The product γ 5 j = c 1 j c 2 j c 3 j c 4 j is both self adjoint and unitary, so P 5 j = 1 2 (I + γ 5 j ) is the projection corresponding to the +1 eigenvalue. The projections P 5 j mutually commute, and also commute with all even elements of A. Their product P 5 := j P 5 j is called the chiral projection. It can be written as a product P 5 = P 5 − P 5 + (5.8) of the two commuting projections P 5 ± = j∈Λ± P 5 j in A ± . The map from spins to Majoranas is given by σ a j := ic a j c 4 j (5.9) on single spins σ a j , and extends to a linear map A spin → A by
Σ (I,A) := σ a1 j1 · · · σ a k j k .
The resulting linear map X → X is a homomorphism when restricted to P 5 , in the sense that for all X, Y ∈ A spin , one has XY P 5 = X Y P 5 .
J AA ′ I I ′ i k+k ′ Θ c a1 ϑ(i1) c 4 ϑ(i1) . . . c a k ϑ(i k ) c 4 ϑ(i k ) c a ′ 1 i ′ 1 c 4 i ′ 1 . . . c a ′ k i ′ k ′ c 4 i ′ k ′ (5.13)
in the Majorana algebra A. Equation (5.13) can thus be written
H = − I, I ′ J M I I ′ Θ(C I ) • C I ′ ,
where J M is the matrix of Majorana coupling constants. It equals
J M I I ′ = i k I +k I ′ J AA ′ ϑ(I)I ′ (5.14)
for the indices I = ((i 1 , a 1 ), (i 1 , 4), . . . , (i k , a k ), (i k , 4)) , (5.15) Here, we used Tr spin (X) = Tr M ( XP 5 ), equation (5.10), and the fact that P 5 + and Θ(P 5 + ) = P 5 − commute with the other factors, with P 5 = P 5 − P 5 + . (b): This is analogous to the proof of Theorem 4.2.b. Choose X ∈ A spin + such that Tr spin (Θ(X)X) = 0. Expand X as
I ′ = ((i ′ 1 , a ′ 1 ), (i ′ 1 , 4), . . . , (i ′ k , a ′ k ′ ), (i ′ k ′ ,X = (I ′ ,A ′ ) x A ′ I ′ Σ (I ′ ,A ′ ) , with the coefficient b ∅ of Σ ∅ = I(−1) k I x A I J AA ′ ϑ(I)I ′ x A ′ I ′ .
Since x ∅ = 0, only the coupling constants J 0 AA ′ ϑ(I)I ′ across the reflection plane contribute. Substituting y A
I := i k I x A I yields 0 (I,A) (I ′ ,A ′ ) y A I i k I +k I ′ J 0 AA ′ ϑ(I)I ′ y A ′ I ′ ,
so that i k I +k I ′ J 0 AA ′ ϑ(I)I ′ is positive semidefinite, as required.
Automorphisms that Yield New Reflections
In Sections 4 and 5, we have given a characterization of reflection positivity with respect to a standard reflection Θ. In this section, we show how these results extend to other reflections Θ ′ = α −1 Θα, where α is an automorphism. The special case where α is a gauge transformation, can be very useful in applications.
6.1. Relation to Other Reflections. We formulate this in the more general context of a Z 2 -graded algebra A which is the super tensor product of two isomorphic subalgebras A + and A − . This means that A is A + ⊗ A − as a vector space, with multiplication defined by
(A ⊗ B)(A ′ ⊗ B ′ ) = (−1) |A ′ ||B| AA ′ ⊗ BB ′
on homogeneous elements. The twisted product A • B is then defined as in Definition 1.2. It reduces to the ordinary product on algebras that are purely even, such as the spin algebra A spin . A reflection Θ : A → A is an antilinear automorphism such that Θ(A ± ) = A ∓ and Θ 2 = I. Two different reflections Θ and Θ ′ are related by the linear automorphism β := ΘΘ ′ , which maps A ± to A ± , and satisfies Θβ = β −1 Θ. Conversely, if Θ is a reflection and β satisfies β(A ± ) = A ± and Θβ = β −1 Θ, then Θ ′ := Θβ is also a reflection.
Recall that a linear functional ω :
A → C is reflection positive on A + with respect to Θ ′ , if 0 ω(Θ ′ (A) • A) for all A ∈ A + .
If Θ ′ is related to Θ by a square β = α 2 , then reflection positivity with respect to Θ and Θ ′ are related as follows.
Proposition 6.1 Let α be a linear automorphism of A such that α(A ± ) = A ± and Θα = α −1 Θ. Let Θ ′ := α −1 Θα .
Then the pullback α −1 * ω(A) := ω(α −1 (A)) is reflection positive with respect to Θ on A + , if and only if ω is reflection positive with respect to Θ ′ on A + .
Proof. Since α is a linear automorphism, α(
A − • A + ) = α(A − ) • α(A + ) for A ± ∈ A ± . For A ∈ A + , one has α −1 * ω(Θ(A) • A) = ω(α −1 Θ(A) • α −1 (A)) = ω(Θ ′ (α −1 (A)) • α −1 (A))) .
Thus the first term is positive for all A ∈ A + , if and only if the last term is positive.
We apply this to the algebras of Majoranas and spins, with the Gibbs functional ρ H (A) = Z −1 H Tr(Ae −H ). Proof. The first statement follows as Θ(α(H)) = α(H) is equivalent to α −1 Θα(H) = H. For the second statement, note that the normalized trace is unique on the algebras of Majoranas and spins. Thus α * Tr = Tr for every automorphism α, and one has
α −1 * ρ H (A) = Z −1 H Tr(α −1 (A)e −H ) = Z −1 H Tr(α(α −1 (A)e −H )) = Z −1 H Tr(Ae −α(H) ) = ρ α(H) (A)
. Note that in the above, we do not require Θ, Θ ′ or α to respect the involution * on the algebra A. If A is either the spin algebra or the algebra of Majoranas, then the canonical reflection Θ preserves the involution. In this case, Θ ′ = α −1 Θα will preserve the involution if and only if α 2 does so.
6.2. Gauge Automorphisms. In the context of a (super) tensor product A of
Z 2 -graded * -algebras A j A = j∈Λ A j ,
we define the gauge automorphism α τ , parameterized by a collection {τ j } j∈Λ of automorphisms of A j , as α τ := ⊗ j∈Λ τ j .
If the A j can be canonically identified which each other, and all τ j are the same, then α τ is called a global gauge transformation.
Suppose that A has a reflection Θ such that Θ(A j ) is isomorphic to A ϑ(j) . Then the gauge automorphism α τ is called reflection invariant if τ j = Θτ −1 ϑ(j) Θ for all j ∈ Λ. Every reflection invariant gauge automorphism satisfies α τ (A ± ) = A ± and α τ Θ = Θα −1 τ .
6.2.1. Majorana Algebras with 1 generator. In the case of the Majorana algebra generated by c j with j ∈ Λ, A j is the two-dimensional algebra generated by I and c j , and the only two automorphisms are τ j (c j ) = ±c j . There is a unique nontrivial global gauge automorphism c j → −c j .
6.2.2. Majorana Algebras with 4 generators. In the case of the algebra generated by Majoranas c α j with j ∈ Λ and α ∈ {1, 2, 3, 4}, the algebra A i is the Clifford algebra Cl(4, C) generated by the c α i with i fixed. The automorphisms τ j can be taken to be conjugation by an invertible element g j ∈ Cl(4, C) × , that is, τ j (A) = g j Ag −1 j . The spin group Spin(4) is the group of even elements g ∈ Cl(4, C) × such that gc α g −1 = R α β c β for some R ∈ SO(4, R).
Spin Algebras.
In the next section the most relevant case will be the spin algebra A spin , where A i is the purely even algebra M 2 (C). If τ i is conjugation by a matrix g i ∈ SL(2, C), we denote the gauge automorphism corresponding to the collection {g j } j∈Λ by α g . The requirement g ϑ(j) = Θg −1 j Θ translates to g ϑ(j) = g * j . It is an automorphism of * -algebras if and only if g i ∈ SU(2, C) for every i ∈ Λ + .
Examples of Spin Models
We apply the characterization of reflection positivity in Theorem 5.2 to a number of spin systems: the Ising model, the quantum rotator, and the anti-ferromagnetic Heisenberg model. Nearest neighbor couplings are treated in §7.1, and long range interactions in §7.2.
Many of these examples are well-understood, and we include those mainly to show that they have a natural interpretation within our general framework. Some relevant references are [DLS76,FILS78,DLS78,FL78,Bis09].
In this section, the lattice Λ has a geometric interpretation. It is a finite, fixed point free subset of a manifold M with involution ϑ M , as explained in §1.1. An important example is M = R d with ϑ : R d → R d the orthogonal reflection in a hyperplane Π. Periodic boundary conditions can be handled by taking M = T d the d-dimensional torus.
7.1. Nearest Neighbor Couplings. The nearest neighbor Heisenberg model is given in terms of the Pauli matrices σ a j on a lattice j ∈ Λ by the Hamiltonian
−H = 3 a=1 jj ′ J a jj ′ σ a j σ a j ′ + 3 a=1 j h a j σ a j . (7.1)
Here the sum is over the nearest neighbor pairs jj ′ , and J a jj ′ = J a j ′ j . As H is Hermitian, the partition sum Z H = Tr(e −βH ) is nonzero.
In order to define nearest neighbor models, we assume that the lattice Λ ⊆ M has the property that "bonds are perpendicular to the reflection hyperplane". This means that two lattice points j ∈ Λ + and j ′ ∈ Λ − can only be nearest neighbors if j ′ = ϑ(j). (For example, this is the case in Fig. 1 and Fig. 2.) Let J 0 AA ′ ϑ(I)I ′ denote the matrix of couplings across the reflection plane, defined in (5.4), (5.5). It is given by Here j, j ′ ∈ Λ + and a ∈ {1, 2, 3}. Note that J a ϑ(j)j ′ is only nonzero if j = j ′ , as sites j ′ ∈ Λ + and ϑ(j) ∈ Λ − on different sides of the reflection plane can only be neighbors if j = j ′ .
7.1.1. Anti-Ferromagnetic Heisenberg Models. In order to show reflection positivity for the anti-ferromagnetic Heisenberg model, we restrict the coupling constants in (7.1) as follows:
The full matrix of coupling constants is ϑ-symmetric, J a jj ′ = J a ϑ(j)ϑ(j ′ ) . The external field is antisymmetric, h a ϑ(j) = −h a j , and couplings across the reflection plane are anti-ferromagnetic, J ϑ(j)j 0.
Proposition 7.1 (Anti-ferromagnetic Heisenberg Model) For the above restrictions on the coupling constants in the Hamiltonian H of (7.1), the Gibbs state ρ βH is reflection positive with respect to the standard reflection Θ(σ a j ) = −σ a ϑ(j) .
Proof. Under the standard reflection Θ, the first term on the right side of (7.1) is invariant if J a jj ′ = J a ϑ(j)ϑ(j ′ ) , while the second term is invariant if the external field satisfies h a ϑ(j) = −h a j . By Theorem 5.2, the Gibbs state ρ βH is reflection positive for all β 0, if and only if the matrix i k I +k I ′ J 0 AA ′ ϑ(I)I ′ is positive semidefinite. As k I = k I ′ = 1, this matrix is diagonal with entries −J a ϑ(j)j , labelled by the j ∈ Λ + for which ϑ(j) ∈ Λ − . This matrix is positive definite if and only if J a ϑ(j)j 0.
This includes the usual anti-ferromagnetic Heisenberg model, with constant couplings J 1 ij = J 2 ij = J 3 ij = J 0, and vanishing external field h a ij = 0. The quantum rotator model is the special case J 3 ij = 0, and the Ising model is the special case J 2 ij = J 3 ij = 0. By the above proposition, they are reflection positive in the anti-ferromagnetic case of negative coupling constants with vanishing external field h a j .
7.1.2. Ferromagnetic Quantum Rotator Model. The next example illustrates the gauge transformation method introduced in §6. In order to show reflection positivity for the ferromagnetic quantum rotator model, we restrict the coupling constants in (7.1) as follows: We require J 3 jj ′ = 0 and 0 J a j ′ j for a = 1, 2. (In fact, the proof only uses that the bonds j ′ = ϑ(j) across the reflection plane are ferromagnetic.) We assume that the couplings are symmetric around the reflection plane, J a jj ′ = J a ϑ(j)ϑ(j ′ ) 0 for a = 1, 2. Finally, we require that the first two components of the external field are reflection symmetric, h a ϑ(j) = h a j for a = 1, 2, and that the third component is antisymmetric, h 3 ϑ(j) = −h 3 j . For long-range interactions, the matrix of coupling constants across the reflection plane will not be diagonal, as was the case for nearest neighbor models. As f is reflection positive, this matrix is positive semidefinite if and only if J a 0 for a = 1, 2, 3.
Long-Range
Ferromagnetic Rotator Model.
The long-range rotator model is given by the Hamiltonian (7.5) with J 3 = 0.
In the anti-ferromagnetic case J 1,2 0, Proposition 7.3 shows that it is reflection positive with respect to the standard reflection Θ, satisfying Θ(σ a j ) = −σ a ϑ(j) for a = 1, 2, 3. As in the nearest neighbor case, the ferromagnetic model 0 J 1,2 is reflection positive for a different reflection Θ ′ , satisfying (7.2).
Proposition 7.4 (Long-Range Quantum Rotator) The Gibbs state ρ βH for the Hamiltonian (7.5) is reflection positive with respect to the anti-linear reflection Θ ′ for all β 0, if and only if 0 J a for a = 1, 2.
even ζ AB , if both A, B ∈ A odd . (1.2) In terms of this twisted product, the reflection positivity equation (1.1) becomes simply 0 ω(Θ(A) • A) , for A ∈ A + . (1.3) This definition of a reflection-positive form is natural in the context of super algebras. The completions of A even + and A odd + with respect to the form (1.3) are then the orthogonal, even and odd parts of a super-Hilbert space H, see Deligne and Morgan
ω
H (A) = Tr(A e −H ) , where Tr is a tracial state on A. If the partition sum Z H := Tr(e −H ) is nonzero, define the Gibbs functional ρ H as the normalized Boltzmann functional, ρ H (A) = Z −1 H Tr(A e −H ) . (1.4)
ϑ M : M → M. In the examples of interest, M is a disjoint union M = M + ⊔ M 0 ⊔ M − , where ϑ M interchanges M + and M − , and leaves the hypersurface M 0 invariant. The set Λ + is then a finite set of points in M + , and Λ − is its reflection.
with the reflection plane illustrated by the dashed line inFigure 1.
Fig. 1 .
1Reflection in a cubic lattice.Another example, with M = R 2 , is the honeycomb lattice inFigure 2.
Fig. 2 .
2Reflection in the 2-dimensional honeycomb lattice.
Proof. It suffices to check this for A = A − A + and B = B − B + as in Definition 1.2. By antilinearity of Θ, one then finds Θ(A • B) = Θ(ζ |A−||B+|−|A+||B−| AB) = ζ −|A−||B+|+|A+||B−| Θ(A)Θ(B) for the left side of the equation. For the right side, one finds the same expression Θ(A) • Θ(B) = Θ(A − )Θ(A + ) • Θ(B − )Θ(B + ) = ζ |A+||B−|−|A−||B+| Θ(A)Θ(B) ,
I (k I −1) and s I := ζ 1 2 k I (k I −1) .
Proposition 1.10 (The Normalized Trace) The functional Tr is a tracial, factorizing, reflection-invariant state. Namely (a) It is normalized, Tr(I) = 1. (b) It is positive definite, Tr(A * A) 0 for all A ∈ A, with equality only for A = 0. (c) It is cyclic, Tr(AB) = Tr(BA) for all A, B ∈ A . (1.27) (d) It satisfies Tr(Θ(A)) = Tr(A) for all A ∈ A . (1.28) (e) It factorizes, Tr(A − A + ) = Tr(A − ) Tr(A + ) , for A ± ∈ A ± . (1.29) Proof. (a) As I = Θ(C ∅ ) • C ∅ , one has Tr(I) = 1. (b) From (1.24) and the expansion (1.17), one finds Tr(A * A) = I,I ′ ∈P+ |a II ′ | 2 0 . Furthermore Tr(A * A) = 0 only if all the a II ′ = 0. As the Θ(C I ) • C I ′ are a basis, the vanishing of a II ′ ensures that A = 0. Hence Tr is positive definite. (c) From equation (1.25), one obtains Tr(AB) = Tr(BA) = I,I ′ ∈P+
H . Using Proposition 1.10.d, one then finds Z H = Tr(e −H ) = Tr(Θ(e −H )) = Tr(e −H ) = Z H , so that Z H is real.
(b)⇒(a): By Proposition 1.10.d, the trace is reflection invariant, Tr(Θ(X)) = Tr(X). If Θ(R) = R, one finds Tr(AR) = Tr(Θ(AR)) = Tr(Θ(A)Θ(R)) = Tr(Θ(A)R) .
even, then also ω(A) = Tr(RA) is even. Proposition 2.3 If ω is even, then A Proof. For A ∈ A even + and B ∈ A odd + , one has A, B R,Θ = ω(Θ(A) • B). This equals zero, as Θ(A) • B ∈ A odd .
Proposition 2. 7
7The functional ω in (2.1) is reflection positive on A + , if and only if it is reflection positive on A − . In fact
Theorem 2. 8 (
8Basic Reflection Positivity) The linear functional ω in (2.1) is reflection positive on A + , if and only if the matrix r II ′ defined in (2.3) is positive semidefinite.
partition function Z H := Tr(e −H ) is nonzero, then define the Gibbs functional ρ H : A → C as the normalization of ω H , ρ H (A) := ω H (A) Z H = Tr(Ae −H ) Tr(e −H ) . (3.2) Using equation 2.2, the (unnormalized) Boltzmann functional ω H yields the sesquilinear form A, B 0 H,Θ := Tr((Θ(A) • B) e −H ) (3.3) on A + . Similarly, the (normalized) Gibbs functional ρ H yields the form A, B H,Θ := Tr((Θ(A) • B) e −H ) Tr(e −H ) . (3.4) Remark 3.1 The functional ω H in (3.1) is reflection positive on A + if A, B 0 H,Θ in (3.3) is positive semidefinite on A + , and the functional ρ H defined in (3.2) is reflection positive on A + if the form A, B H,Θ in (3.4) is positive semidefinite on A + . Note that ω H and ρ H are even if H is globally gauge invariant.
following proposition expresses some relevant properties of H in terms of the matrix J. Recall that H is called reflection invariant if Θ(H) = H, and globally gauge invariant if α(H) = H, where α is the global gauge automorphism of (1.8).Proposition 3.2 The Hamiltonian H in (3.6) is RI: reflection-invariant if and only if J is hermitian, J I ′ I = J II ′ . GI: globally gauge-invariant if and only if J II ′ = 0 for |I| = |I ′ |. H: hermitian if and only if ζ1 2 k I (k I −1)+ 12 k I ′ (k I ′ −1) J II ′ is real.Proof. The first statement is Proposition 1.6. For the second statement, note that the global gauge transformation α leaves the basis element Θ(C I ) • C I ′ fixed if |I| = |I ′ |, and otherwise multiplies it by −1. Linear independence of the basis B ensures that each term in the expansion of H must be gauge invariant. The third statement is a consequence of Proposition 1.7.Proposition 3.3 If H is reflection invariant, then the sesquilinear form A, B 0 H,Θ on A + given by (3.3) is hermitian, and Z H = Tr(e −H ) is real: Θ(H) = H ⇒ A, B 0 H,Θ = B, A 0 H,Θ , and Z H = Z H . If both H is reflection invariant and Z H = 0, then the form A, B H,Θ is defined in (3.4) and is hermitian. Proof. The operator R of §2 equals e −H here. So Θ(R) = e −Θ(H) , and if H is reflection invariant, then so is R. By the implication (b)⇒(d) of Proposition 2.2, the form · , · R,Θ is hermitian. Also (b)⇒(a) ensures that Z H = Tr(e −H ) = Tr(Θ(e −H )) = Z H is real. Hence if Z H = 0, the form A, B H,Θ is also hermitian. 3.3. Reflection Positivity: Preliminary Results. We now prove reflection positivity of the Boltzmann functional ω H for Hamiltonians H that arise from a positive semidefinite matrix J of coupling constants. Theorem 3.4 (Reflection Positivity of ω H , Part I) Let H ∈ A be reflection symmetric and globally gauge invariant. If the matrix J of coupling constants for H, defined in equation (3.6), is positive semidefinite, then ω H is reflection positive on A + .
Proof (Proof of Theorem 3.4). Expand A, B ∈ A + as A = I∈P+ a I C I and B = I∈P+ b I C I , with C I ∈ B + . We claim that the sesquilinear form A, B 0 H,Θ = Tr(Θ(A) • B e −H ) can then be written in the form
Corollary 3. 7
7If A ∈ A + has the expansion (1.17), then under the conditions of Theorem 3.The right side of 3.17 is the k = 0 term in 3.13. This yields a lower bound, as all the other terms are nonnegative by the proof of Theorem 3.4. Proposition 3.8 Suppose that the matrix J of coupling constants for H, defined in (3.6), is positive semidefinite. Then Z βH is a non-decreasing function of 0 β with Z 0 = 1. In particular, 1 Z βH for all 0 β. Proof. Let R = e −βH and consider Z βH = Tr(e −βH ) = Tr(R) for β 0. Note that Z 0 = 1 by Proposition 1.10.a. Using Proposition 1.9 to evaluate the trace, one obtains dZ βH dβ = − Tr(He −βH ) = − Tr(HR) = I,I ′ ∈P+ q I J II ′ r II ′ q I ′ , (3.18)
J
= J ∅∅ yields the additive constant −E in H. Reflection invariance of H ensures that E is real. In fact E is not of physical relevance. It does not affect whether the functional ω H is reflection positive. Furthermore it does not even enter the normalized Gibbs functional. The energy shift H → H − E multiplies both ω H and Z H by e E , so it does not affect their quotient ρ H . The column vector V I = J I∅ has indices labelled by I ∈ P + − {∅}, as does its hermitian adjoint V * . The hermitian matrix J 0 = (J 0 I,I ′ ) , with indices I, I ′ ∈ P + − {∅} (4.2) is called the matrix of coupling constants across the reflection plane. The matrix decomposition (4.1) corresponds to the four terms in the decomposition H = H − + H 0 + H + − E , I∅ Θ(C I ) = I∈P+−{∅} V I Θ(C I ) ∈ A − (4.4) is the sum of the interactions on one side of the reflection plane, namely on sites in Λ − . The reflection H + of H − is the interaction within Λ + , −H + = Θ(−H − ) = I∈P+−{∅} V I C I ∈ A + . (4.5)
of Reflection Positivity. We give necessary and sufficient conditions on the Hamiltonian H ∈ A for the Boltzmann functional ω H (A) = Tr(Ae −H ) to be reflection positive on A + .Remark 4.1 Reflection positivity of ω H means that the hermitian form on A + defined by A, B 0 H,Θ = Tr(Θ(A) • B · e −H ) is positive semidefinite; 0 A, A 0 H,Θ for A ∈ A + . In particular, Z H = Tr(e −H ) = I, H = 0, reflection positivity of the Boltzmann functional ω H therefore implies reflection positivity of the (physically relevant) Gibbs functional ρ H = Z −1 H ω H . Theorem 4.2 (Reflection Positivity of ω H , Part II) Let H ∈ A be reflection symmetric and globally gauge invariant. Let J 0 be the matrix of coupling constants across the reflection plane, defined in (4.1)-(4.2). Then: (a) If J 0 is positive semidefinite, the functional ω H is reflection positive on A + . (b) Conversely, if there exists an ε > 0 such that ω βH is reflection positive on A + for all β ∈ [0, ε), then the matrix J 0 is positive semidefinite.
reflection positivity would be violated for small β. One can evaluate (4.10) in a fashion similar to the computation of (3.18), but with Θ(A) • A replacing R.Expanding Θ(A) • A as I,I ′ a I a I ′ Θ(C I ) • C I ′ , and using Proposition 1.9 to evaluate the trace, one obtains0 − Tr(Θ(A) • AH) = I,I ′ ∈P+−∅ (q I a I ) J 0 II ′ (q I ′ a I ′ ),(4.11) with q I = (−1) k I (k I −1) as in (1.20). As a ∅ = 0, the sum restricts to P + −∅, and only J 0 contributes. From equation 4.11, one then obtains 0 f, J 0 f . (4.12)
Functionals. Using Theorem 4.2, we obtain the following necessary and sufficient conditions on H for the Gibbs functional ρ H (A) = Z −1 H Tr(Ae −H ) , to be reflection positive on A + . In this expression, Z H = Tr(e −H ) denotes the partition sum.
Theorem 4. 4 (
4Reflection Positivity of Gibbs Functionals) Let H ∈ A be a reflection symmetric, globally gauge invariant Hamiltonian.
J
Hamiltonians. Any Hamiltonian H spin ∈ A spin , not necessarily Hermitian, takes the formH spin = − k j1,...,j k a1,...,a k J a1 j1 . . . a k j k σ a1 j1 . . . σ a k j k .(5.4) Partition j 1 . . . j k into the sets ϑ(I) ⊆ Λ − and I ′ ⊆ Λ + , where both I and I ′ are subsets of Λ + . Using (5.3) and setting J AA ′ ϑ(I)I ′ = J a1 j1 . . . a k j k , AA ′ ϑ(I)I ′ Σ (ϑ(I),A) Σ (I ′ ,A ′ ) k I J AA ′ ϑ(I)I ′ Θ(Σ (I,A) )Σ (I ′ ,A ′ ) . (5.7)
.
Reflection Positivity for Spin Hamiltonians. Recall that for a (not necessarily Hermitian) Hamiltonian H ∈ H spin , the Boltzmann functional ω H (X) = Tr spin (Xe −H ) is called refection positive on A + if 0 ω H (Θ(X)X) = Tr spin (Θ(X)X e −H ) . (5.11) If the partition sum Z H = Tr spin (e −H ) is nonzero, then the Gibbs functional is defined by ρ H (X) := Z −1 H ω H (X). Reflection positivity of ρ H is equivalent to 0 ρ H (Θ(X)X) = Z −1 H Tr spin (Θ(X)X e −H ) . (5.12) From Theorem 4.2 for Majoranas, one derives the following characterization of reflection positivity for spin systems. It is given in terms of the matrix J 0 AA ′ II ′ of coupling constants across the reflection plane. This is the submatrix of the matrix J AA ′ II ′ of coupling constants (5.5) with I = ∅ and I ′ = ∅. Theorem 5.2 (Reflection Positivity for Spins) Let H ∈ A spin be a (not necessarily Hermitian) reflection invariant Hamiltonian. (a) If the matrix i k I +k I ′ J 0 AA ′ ϑ(I)I ′ is positive semidefinite, then the Boltzmann functional ω H is reflection positive. If Z H = 0, then Z H > 0, and the Gibbs state ρ H is reflection positive. (b) If there exists an ε > 0 such that either ω βH or ρ βH is reflection positive on A spin + for all β ∈ [0, ε), then the matrix i k I +k I ′ J 0 AA ′ ϑ(I)I ′ is positive semidefinite.Remark 5.3 The requirement that H is reflection invariant is equivalent to Hermiticity of the matrix i k I +k ′ I J AA ′ ϑ(I)I ′ . Furthermore, the requirement that Z H = 0 is automatically fulfilled if H is Hermitian.Proof. It suffices to prove (a) and (b) for the Boltzmann functional ω H . Statement (a) for the Gibbs functional ρ H then follows from Remark 4.1. Following word by word the proof of Theorem 4.4.b, one obtains statement (b) for ρ H from statement (b) for ω H .
4)) and zero elsewhere. With respect to an appropriate choice of basis, the matrix i k I +k I ′ J AA ′ ϑ(I)I ′ is the only nonzero block in J M I I ′ . Therefore, the latter is positive semidefinite if and only if the former is. The same holds for the matrices J M0I I ′ and i k I +k I ′ J 0 AA ′ ϑ(I)I ′ ofcouplings across the reflection plane. The Majorana Hamiltonian H is globally gauge invariant since each spin involves two Majoranas, and it is reflection invariant as J is Hermitian. Since J M0 I I ′ is positive semidefinite, Theorem 4.4 yields reflection positivity of H. This implies reflection positivity of H, since Tr spin (Θ(X)Xe −H ) = Tr M (Θ( X) Xe − H P 5 ) = Tr M (Θ( XP 5 + )( XP 5 + )e − H ) 0 .
k I x A I Σ ϑ(I,A) .Since ρ βH (Θ(X)X) = Tr spin (Θ(X)Xe −βH ) is nonnegative and zero for β expansion (5.6) and the orthogonality relations (5.2) of Σ (I,A) with respect to the trace pairing, one thus obtains 0 (I,A) (I ′ ,A ′ )
Corollary 6 . 2
62The Hamiltonian H ′ := α(H) is invariant under the reflection Θ if and only if H is invariant under Θ ′ := α −1 Θα. The Gibbs functional ρ H is reflection positive with respect to Θ ′ on A + , if and only if ρ H ′ is reflection positive with respect to Θ on A + .
indices (I, A) = {(j, a)} and (I ′ , A ′ ) = {(j ′ , a)} of equation (5.1), and zero in all other components.
,x ′ ∈Λ : x =x ′ } J a σ a x ′ σ a x f (x − x ′ ) . (7.5)Here f can be any reflection invariant, reflection positive function on R d , or on its compactificationT m × R d−m in m d directions. For such functions the matrix f (ϑ(x)−x ′ ) for x, x ′ ∈ Λ + ispositive semidefinite. Here there is extensive analysis, and some relevant papers are [OS73,OS74,LM75,GJ79,FL10]. An important example is f (x) = x −s on R d , which is reflection positive for s max{0, d − 2} by [NÓ14, Proposition 6.1]. Reflection positive functions on the compactification can be obtained from reflection positive functions on R d under suitable conditions on the rapidity of their decay, see for example [JJM14, Proposition 15].
7.2.1. Anti-Ferromagnetic Heisenberg Model. For J a 0 (the anti-ferromagnetic case), we can use the standard reflection Θ(σ a j ) = −σ a j . Proposition 7.3 (Long-Range Heisenberg Model) The Gibbs functional ρ βH for the Hamiltonian (7.5) is reflection positive with respect to Θ for all β 0, if and only if J a 0 for a = 1, 2, 3. Proof. The Hamiltonian (7.5) is hermitian and Θ-invariant, so by Theorem 5.2, it is reflection positive for all β 0 if and only if the matrix i k I +k I ′ J 0 AA ′ ϑ(I)I ′ is positive semidefinite. The matrix of coupling constants across the reflection plane has entries J 0 AA ′ ϑ(I)I ′ = J a f (ϑ(x) − x ′ ) for the indices (I, A) = {(x, a)} and (I ′ , A ′ ) = {(x ′ , a)} of equation (5.1), and all other entries are zero. Since k I = k I ′ = 1, one finds i k I +k I ′ J 0 AA ′ ϑ(I)I ′ = −J a f (ϑ(x) − x ′ ) .
if and only if the matrix a II ′ is hermitian, namely a I ′ I = a II ′ .which equals
Θ(B) • A by Proposition 1.3.
Proposition 1.6 Let A ∈ A have the expansion (1.17). Then A is reflection
invariant, namely Θ(A) = A,
AcknowledgementsThe authors thank the MFO (Oberwolfach) for hospitality at a conference where they first met, and had conversations leading to this research. A.J. was supported in part by a grant from the Templeton Religion Trust. B.J. was supported by the NWO grant 613.001.214 "Generalised Lie algebra sheaves." He thanks A.J. for hospitality at Harvard University and in Basel during the writing of the paper.Proposition 7.2 (Ferromagnetic Quantum Rotator) With the above restrictions on the coupling constants in the Hamiltonian H of (7.1), the Gibbs state ρ βH is reflection positive with respect to the anti-linear reflection Θ ′ that satisfies Θ ′ (σ 1 j ) = σ 1 ϑ(j) , Θ ′ (σ 2 j ) = σ 2 ϑ(j) , and Θ ′ (σ 3 j ) = −σ 3 ϑ(j) .Proof. We use the gauge transformation α g of §6.2.3, with g j = e i π 4 σ 3 j for j ∈ Λ + and g j = e −i π 4 σ 3 j for j ∈ Λ − . This yields the clockwise rotation over π/2 around the third axis,and the counterclockwise rotationAfter the gauge transformation, the Hamiltonian H of (7.1) becomes H ′ = α g (H), which decomposes aswith the sum over nearest neighbors j, j ′ ∈ Λ + . Similarly,with the sum over nearest neighbors j, j ′ ∈ Λ − . Finally,where j ∈ Λ + has j ′ ∈ Λ − as a nearest neighbor. The Hamiltonian H ′ is invariant under the standard reflection defined byThe matrix of coupling constants across the reflection plane is positive semidefinite if 0 J 1 ϑ(j)j and 0 J 2 ϑ(j)j . From Theorem 5.2, we see that under these conditions, the Gibbs state ρ βH ′ for the Hamiltonian H ′ is reflection positive with respect to Θ.Applying Corollary 6.2, we infer that the Gibbs state ρ βH for the original Hamiltonian H = α −1 (H ′ ) is reflection positive for the gauge transformed reflection automorphism Θ ′ = α −1 Θα = Θα 2 , given in equation (7.2).Proof. By Corollary 6.2, ρ βH is reflection positive for Θ ′ = α −1 Θα, if and only if ρ βH ′ is reflection positive for Θ. Here H ′ = α(H), and we choose α = α g to be the gauge transformation of equations (7.3) and (7.4).The gauge transformed Hamiltonian H ′ has the form H ′ = H ′ + + H ′ 0 + H ′ − , where the term H ′ 0 containing the couplings across the reflection plane isIt follows that the matrix of couplings across the reflection plane for the gauge transformed Hamiltonian H ′ is Since k I = k I ′ = 1, the matrix i k I +k I ′ J ′0 AA ′ ϑ(I)I ′ is positive semidefinite in the ferromagnetic case 0 J a .In order to apply Theorem 5.2 to H ′ , we still need to check that H ′ is reflection invariant under the standard reflection Θ. By Corollary 6.2, this is equivalent to reflection invariance of the original Hamiltonian H under Θ ′ . This is readily seen to be the case by using the explicit equation (7.2) for Θ ′ .As α g is a * -automorphism, H ′ is hermitian, so Z βH ′ 0. One then infers from Theorem 5.2, that ρ βH ′ is reflection positive with respect to Θ. As mentioned in the start of the proof, Corollary 6.2 then yields that ρ βH is reflection positive for Θ ′ .Remark 7.5 An external field 3 a=1 j h a j σ a j can be added to (7.5) under the same conditions as in the nearest neighbor case. For the anti-ferromagnetic Heisenberg model, h a ϑ(j) = −h a j for a = 1, 2, 3. For the ferromagnetic quantum rotator, h a ϑ(j) = h a j for a = 1, 2, and h a ϑ(j) = −h a j for a = 3.
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"Spectroscopic charaterization of FHLC stars from the Hamburg/ESO survey and a newly found HdC star",
"Spectroscopic charaterization of FHLC stars from the Hamburg/ESO survey and a newly found HdC star"
] |
[
"Aruna Goswami \nIndian Institute of Astrophysics\n560 034Bangalore\n"
] |
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"Indian Institute of Astrophysics\n560 034Bangalore"
] |
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"Conference Title ASI Conference Series"
] |
The sample of candidate faint high latitude carbon (FHLC) stars chosen from the Hamburg/ESO survey is a potential source to search for objects of rare types. From medium resolution spectral analyses of about 250 objects from this sample, the object HE 1015−2050, was found to be a hydrogen-deficient carbon (HdC) star. Apart from U Aquarii, HE 1015-2050 is the only example, till now, of a Galactic cool HdC star that is characterized by strong spectral features of light s-process element Sr, and weak features of heavy s-process elements such as Ba. This object, with its enhanced carbon and hydrogen-deficiency, together with anomalous sprocess spectral features, poses a challenge as far as the understanding of its formation mechanism is concerned. We discuss possible mechanisms for its formation in the framework of existing scenarios of HdC star formation.
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"https://arxiv.org/pdf/1204.1806v1.pdf"
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1204.1806
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b04537afe4c88846086ce8f2767904778f4575b7
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Spectroscopic charaterization of FHLC stars from the Hamburg/ESO survey and a newly found HdC star
2011
Aruna Goswami
Indian Institute of Astrophysics
560 034Bangalore
Spectroscopic charaterization of FHLC stars from the Hamburg/ESO survey and a newly found HdC star
Conference Title ASI Conference Series
002011Received 10th May 2014arXiv:1204.1806v1 [astro-ph.SR] 9 Apr 2012stars: carbon -stars: Late-type -stars: HdC -stars: spectral characteristics
The sample of candidate faint high latitude carbon (FHLC) stars chosen from the Hamburg/ESO survey is a potential source to search for objects of rare types. From medium resolution spectral analyses of about 250 objects from this sample, the object HE 1015−2050, was found to be a hydrogen-deficient carbon (HdC) star. Apart from U Aquarii, HE 1015-2050 is the only example, till now, of a Galactic cool HdC star that is characterized by strong spectral features of light s-process element Sr, and weak features of heavy s-process elements such as Ba. This object, with its enhanced carbon and hydrogen-deficiency, together with anomalous sprocess spectral features, poses a challenge as far as the understanding of its formation mechanism is concerned. We discuss possible mechanisms for its formation in the framework of existing scenarios of HdC star formation.
Introduction
The Hamburg/ESO survey was initiated in the year 1990, for the Southern Hemisphere to complement the Hamburg/Quasar survey (HQS) that covers the Northern sky except the Galactic plane (Wisotzki et al. 1996, Reimers & Wisotzki 1997, Wisotzki et al. 2000. This survey was based on digitized objective-prism photographs taken with ESO 1m Schmidt telescope. An area of 9500 deg 2 in the southern sky was covered with an average limiting magnitude B∼ 17.5 mag, on the prism plate. Christlieb et al. (2001) have used the Hamburg/ESO survey (HES) to augment the number of known FHLC stars. The HES spectra cover a wavelength range of 3200 -5200 Å and the seeing limited spectral resolution is typically 15 Å at H γ . An automated procedure based on the detection of C 2 and CN molecular bands of the spectra was used to identify the carbon stars. This procedure resulted 403 candidate faint high latitude carbon stars from a set of 329 plates that cover an area of 6400 deg 2 (87% of the survey area) to the magnitude limit V ∼ 16.5. The surface density of FHLC stars ∼ 0.072 ± 0.05 deg −2 , is about 2-4 times higher than those obtained from previous objective prism and CCD surveys at high Galactic latitude (Sanduleak & Pesch 1988;MacAlpine & Lewis 1978, Green et al. 1994. However, it is to be noted that the majority of the carbon stars known today mostly come from the Sloan Digitized Sky Surveys (SDSS).
Although, at high Galactic latitude the surface density of FHLC stars is low, different classes of carbon stars populate the region: a) the normal AGB stars, carbonenriched by dredge-up during the post main-sequence phase, which are found among the N-type carbon stars, b) FHLC stars showing significant proper motions and having luminosities of main-sequence dwarfs, called dwarf carbon stars (dcs) and c) the CH giants, similar to the metal-poor carbon stars found in Globular clusters and some dwarf spheroidal galaxies. Among these, warm carbon stars of C-R type are also likely to be present. The sample of stars listed by Cristlieb et al. being high latitude objects contains a mixture of these objects. Using medium resolution spectral analysis we have classified these objects based on their spectral chatacteristics. The spectra were obtained using the Himalayan Faint Object Spectrograph Camera (HFOSC) attached to the Himalayan Chandra Telescope (HCT) at the Indian Astronomical Observatory (IAO), Hanle during 2005 -2010. The grism and the camera combination used for observations provided a spectral resolution of ∼ 1330(λ/δλ); the observed bandpass is about 3800 -6800 Å . The spectra of a few objects were also acquired using the OMR spectrograph at the cassegrain focus of the 2-3-m Vainu Bappu Telescope (VBT) at Kavalur. With a 600 line mm −1 grating, we get a dispersion of 2.6 Å pixel −1 . The wavelength range covered is 4000 -6100 Å and the resolution ∼ 1000.
The membership of a star in a particular class is established from a comparison with the spectral atlas of carbon stars of Barnbaum et al. (1996). Further details on the detection procedure and the spectral characteristics of these objects are available in Goswami (2005) and Goswami et al. (2007Goswami et al. ( , 2010a. In our sample of 403 stars, HE 1015−2050 is found to exhibit spectral charateristics of HdC stars. Its photometric parameters, along with those of the comparison HdC star of RCB type U Aqr are given in Table 1. HdC stars are a rare class of objects; only five non-variable HdC and fifty five RCB type stars are known so far in our Galaxy. The origin of these objects is still debated and they are poorly understood due to a lack of statistically significant sample. Each addition to this rare group of objects is therefore important.
Spectral characteristics of HdC stars
The spectra were classified using the following spectral characteristics: a) strength of band-head of the CH band around 4300 Å , b) prominence of secondary P-branch head near 4342 Å , c) strength/weakness of the Ca I feature at 4226 Å , d) isotopic band strength of C 2 and CN, in particular the Swan bands of 12 C 13 C and 13 C 13 C near 4700 Å , e) strength of other C 2 bands in the 6000−6200 Å region, f) 13 CN band near 6360 Å and other CN bands across the wavelength range, g) strength of s-process elements such as Ba II features at 4554 and 6496 Å .
The Hydrogen-deficient supergiants comprise of three sub-classes: a) Extreme helium stars (EHes) of spectral types F and G, b) R Coronae Borealis (RCB) stars of spectral types F and G and c) hydrogen-deficient carbon (HdC) stars that are much cooler than EHes and RCBs. Similarities in chemical compositions indicate an evolutionary link between EHes and RCBs. Whether there is an evolutionary link between these stars and HdCs is not known. The abundance analysis of HdC stars is difficult because their spectra are dominated by molecular bands.
HdC stars are spectroscopically similar to the RCB stars. However, infra-red excesses and deep light minima that are characteristics of RCBs are absent in HdCs, instead they show small-amplitude light variations. Hydrogen deficiency and weaker CN bands relative to C 2 molecular bands are the two primary spectral characteristics of RCB stars. The strong C 2 molecular bands are seen in the spectra of cool RCB stars. They are weakly visible in warm RCB stars. As Balmer lines are weak in carbon stars, the strength/weakness of CH band in C-rich stars provides a measure of hydrogen deficiency. We have used hydrogen deficiency and the relative strength of C 2 bands in the 6000 -6200 Å region and the CN bands near 6206 Å and 6350 Å as important classification criteria for HdC stars.
The spectrum of HE 1015-2050 is characterized by strong C 2 molecular bands. However, G-band (CH around 4310 Å ) is only marginally detected indicating that HE 1015-2050 is a hydrogen-deficient carbon star. HE 1015−2050 bears a remarkable similarity with U Aqr, a cool RCB star. Similar to U Aqr, it exhibits an anomalously strong feature of Sr II at 4077 Å . Y II line at 3950 Å is also clearly detected in both spectra and no significant enhancement of Ba II features at 4554 Å and 6496 Å is seen. Fe I feature at 4045 Å is clearly detected. The feature due to Na I D also appears to be strong. The H α feature is not detected. The most striking feature in HE 1015−2050 and U Aqr is the Sr II at 4215 Å , this feature is blended with the nearby strong blue-degraded (0,1) CN 4216 band head in HD 182040 and ES Aql (Fig. 1).
RCB stars are also characterized by their location in the J-H and H-K planes with respect to cool carbon stars. The Two Micron All Sky Survey (2MASS) measurements (Skrutskie et al. 2006) place HE 1015−2050 on the J-H versus H-K plane along with the cool LMC RCB stars supporting our classification. Fig. 1, except for the wavelength region 5900-6700 Å . The CN bands which appear with almost equal strengths in the spectrum of the CN star V460 Cyg is almost absent (or barely detectable) in the spectra of HE 1015-2050 and the HdC star U Aqr of RCB type. H α feature is distinctly seen in the spectra of the CH and C-R star HD 209621 and Hd 156074 respectively. This feature is not detectable in the spectra of HE 1015-2050 and HdC stars. Non detection of H α , and marginal detection of G-band of CH (Fig. 1) hints at hydrogen-poor nature of the object (Goswami et al. 2010b).
Discussion and Conclusions
The strong Sr II features observed in HE 1015−2050 are indications of enhanced sprocess abundances. In late type stars such as CH giants and carbon-enhanced metalpoor objects, the observed enhanced abundances of s-process elements are generally explained on the basis of a binary picture in which the primary companion low-mass object while evolving through the Asymptotic Giant Branch (AGB) stage transfers the s-enhanced material to the secondary companion (McClure 1983(McClure , 1984McClure & Woodsworth 1990). In case of HE 1015−2050 such an explanation is presently not applicable as its binary status is not yet known. Although none of the RCB and HdC stars known so far, are known to be binary, long-term radial velocity monitoring of HE 1015−2050 would be useful to know its binarity.
The stellar atmosphere of HE 1015−2050 has an estimated effective temperature (T e f f ) of 5263 K, derived using semi-empirical temperature calibration relations from Alonso et al. (1996). This temperature estimate is very similar to those of cool Galactic RCB stars such as S Aps, WX CrA, and U Aqr (T e f f ∼ 5000 K, Lawson et al. 1990). Although the resolution of our spectrum is not adequate to derive quantitative estimates one could expect this object to have similar s-elements abundance as that of U Aqr, as they exhibit very similar spectra. Vanture et al. (1999) have determined abundances of Rb, Sr, Y, and Zr in U Aqr that are found to be greatly enhanced but Ba did not show any significant enhancement. Estimates of Vanture ([Y/Fe] = +3.3, [Zr/Fe] = +3.0, and [Ba/Fe] = +2.1) are in general agreement with those of Malaney (1985) but larger than the estimates of Bond (1979).
The abundances of light s-process elements in solar material is attributed to 'weak s-process' described by a single-neutron irradiation (Beer & Macklin 1989). The main component of s-process occurs through partial mixing of protons into the radiative 12 C layer during thermal pulses that initiate the chain of reactions 12 C(p, γ) 13 N(β) 13 C(α, n) 16 O, in a narrow mass region of the He intershell called the 13 C pocket. The reaction 13 C(α, n) 16 O acts as the source of neutrons (Iben & Renzini 1982, Lattanzio 1987 in this process. The 'weak s-process' is assumed to occur in massive stars in He or C-burning phase. Although, no observational support exists for this scenario, 22 Ne(α, n) 25 Mg reaction is believed to be the source of neutrons. This reaction has limited efficiency as most of the neutrons liberated are absorbed by light nuclei, and a few remain available for Fe-seed nuclei to capture. This process allows production of light s-process nuclei with mass numbers 65 < A < 90 (Prantzos et al. 1990) and Sr with mass number A = 87 falls within this range. Weak s-process could therefore be a likely mechanism responsible for the observed Sr in U Aqr and HE 1015−2050.
Main-sequence objects with strong Sr are quite common but they also show equally strong Ba. However, in the globular cluster ω Cen Stanford et al. (2006) have noted one object 2015448 with anomalously strong Sr and weak Ba. The effective temperature, surface gravity and the metallicity ([Fe/H]) of this object are respectively 5820 K, 4.2 and −0.7. The Sr and Ba abundances with respect to Fe are respectively [Sr/Fe] = +1.6 and [Ba/Fe] = 0.6 (Stanford et al. 2006). A similar mechanism responsible for 2015448 formation may also hold good for HE 1015−2050 and needs further investigation. We note, however, a primary difference between 2015448 and HE 1015−2050; while the former shows carbon depletion ([C/Fe] = −0.5), HE 1015−2050 shows enhancement of carbon.
Low mass hydrogen-deficient stars are associated with late stage of stellar evolution and are believed to be in a short-lived evolutionary phase. The characteristic light decline of five or more magnitudes shown by RCB stars within a few days from the onset of minimum (followed by slow recovery to maximum light) is suggested to be due to directed mass-ejections, which is believed to be primarily a signature of surface activity rather than chemical peculiarity. Hydrogen-deficient stars that do not exhibit such irregular fadings may indicate absence of such mass-ejection episodes; but, whether that is characteristic of a particular evolutionary stage is not yet established. Detailed spectroscopic as well as photometric studies could provide insight into these aspects. In addition, extended photometric observations of HE 1015−2050 would be useful to detect short as well as long term photometric variations observed in RCBs.
Figure 1 .
1A comparison between the spectrum of HE 1015-2050 with the spectra of V460 Cyg (C-N star), U Aqr, ES Aql (cool HdC stars of RCB type), HD 182040 (a non-variable HdC star), HD 156074 (C-R star), and HD 209621 (CH star) in the wavelength region 3850-4950 Å . G-band of CH distinctly seen in the CH and C-R star's spectra are barely detectable in the spectra of HE 1015-2050 and other HdC stars spectra. The large enhancement of Sr II at 4077 Å in the spectrum of U Aqr is easily seen to appear with almost equal strength in the spectrum of HE 1015-2050. Y II line at 3950 Å is prominent in the spectrum of HE 1015-2050, this line of Y II is also considerably strengthened in U Aqr. These two features of Sr II and Y II are not observed in the spectra of HD 182040 and ES Aql. The spectrum of HE 1015-2050 compares closest to the spectrum of the HdC star U Aqr of RCB type(Goswami et al. 2010b).
Figure 2 .
2Same as
Table 1 :
1Photometric parameters of HE 1015−2050 and the comparison star U AqrStar No.
l
b
B
V
B-V a
J
H
K
HE 1015-2050 261.31
29.08
16.9 16.3
0.67
14.977 14.778 14.504
HE 2200-1652
39.15
-49.81 12.1 11.1
0.99
9.562
9.283
8.961
( U Aqr)
a From Christlieb et al. (2001)
AcknowlegmentAG gratefully acknowleges the LOC for local hospitality during the workshop and thanks Sunetra Giridhar for her comments and suggestions on the manuscript.
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|
[] |
[
"ULTRA-LOW FREQUENCY GRAVITATIONAL RADIATION from MASSIVE BLACK HOLE BINARIES",
"ULTRA-LOW FREQUENCY GRAVITATIONAL RADIATION from MASSIVE BLACK HOLE BINARIES"
] |
[
"Mohan Rajagopal \nStanford University\n\n",
"Roger W Romani \nStanford University\n\n"
] |
[
"Stanford University\n",
"Stanford University\n"
] |
[] |
For massive black hole binaries produced in galactic mergers, we examine the possibility of inspiral induced by interaction with field stars. We model the evolution of such binaries for a range of galaxy core and binary parameters, using numerical results from the literature to compute the binary's energy and angular momentum loss rates due to stellar encounters and including the effect of back-action on the field stars. We find that only a small fraction of binary systems can merge within a Hubble time via unassisted stellar dynamics. External perturbations may, however, cause efficient inspiral. Averaging over a population of central black holes and galaxy mergers, we compute the expected background of gravitational radiation with periods P w ∼ 1 − 10y. Comparison with sensitivities from millisecond pulsar timing suggests that the strongest sources may be detectable with modest improvements to present experiments.
|
10.1086/175813
|
[
"https://arxiv.org/pdf/astro-ph/9412038v3.pdf"
] | 16,044,638 |
astro-ph/9412038
|
603a9a8890af3b8c8f10b96fd84a0354cd4a0bd4
|
ULTRA-LOW FREQUENCY GRAVITATIONAL RADIATION from MASSIVE BLACK HOLE BINARIES
7 Feb 1995
Mohan Rajagopal
Stanford University
Roger W Romani
Stanford University
ULTRA-LOW FREQUENCY GRAVITATIONAL RADIATION from MASSIVE BLACK HOLE BINARIES
7 Feb 1995Subject headings: black holes -galaxies:interactions -pulsars:timing -radiation mechanisms:gravitational
For massive black hole binaries produced in galactic mergers, we examine the possibility of inspiral induced by interaction with field stars. We model the evolution of such binaries for a range of galaxy core and binary parameters, using numerical results from the literature to compute the binary's energy and angular momentum loss rates due to stellar encounters and including the effect of back-action on the field stars. We find that only a small fraction of binary systems can merge within a Hubble time via unassisted stellar dynamics. External perturbations may, however, cause efficient inspiral. Averaging over a population of central black holes and galaxy mergers, we compute the expected background of gravitational radiation with periods P w ∼ 1 − 10y. Comparison with sensitivities from millisecond pulsar timing suggests that the strongest sources may be detectable with modest improvements to present experiments.
Introduction
The possibility that many galaxies contain a central black hole of 10 7 M ⊙ or more, coupled with the ideas that elliptical galaxies probably formed from mergers (c.f. Barnes & Hernquist, 1992) and that all galaxies may have formed from multiple mergers of subunits, has led naturally to consideration of the fate of two or more massive black holes in a merger remnant. Simulations suggest that the cores of victim galaxies merge via violent relaxation in a few crossing times to form the remnant core (Barnes & Hernquist); with some possible exceptions (Governato et al. 1994), black holes in the victims then end up in the core of an elliptical or a spiral bulge. The holes sink towards the center of the stellar distribution on the dynamical friction time scale of ∼ 10 6 years until they become bound to one another (Begelman, Blandford & Rees 1980, hereafter BBR), forming a massive black hole binary (BHB). If the binary loses enough energy and angular momentum to the field stars, which is by no means certain, it will enter a regime where gravitational radiation alone can bring about inspiral and coalescence within the Hubble time.
During BHB inspiral, the gravity waves generated sweep through a range of frequencies. In the final merger of two 10 9 M ⊙ black holes, a 'chirp' of gravitational radiation with periods as small as P GR ∼ 10 4 s will be emitted. It has been suggested that this radiation might be detected through the Doppler tracking of spacecraft and by using space-based interferometers (Thorne & Braginsky 1976, Thorne 1992, Fukushige et al. 1992a. Unfortunately, the binary spends only few wave periods in the final chirp, so if inspiral is rare such events will be difficult to observe. In contrast, at binary frequencies ∼ 0.01 − 1 µ-Hertz, the BHB persists for many orbits. As first pointed out by Detweiler (1979), the strain amplitudes of the ultra-low frequency (ULF) gravity waves generated by sufficiently massive BHBs may be detected out to cosmological distances as perturbations in the pulse arrival times of quiet pulsars (Romani & Taylor 1983). With a new generation of timeable millisecond pulsars (Kaspi, Taylor & Ryba 1994, hereafter KTR;Camillo, Foster & Wolszczan 1994) allowing remarkable improvements in sensitivity, it seems appropriate to assess the probability of massive BHB detection.
In this paper we examine the fate of BHBs in a population of merger remnants, estimating the frequency of events and using these results to outline the required pulsar timing sensitivity needed to effect a detection or constrain the merger hypothesis. First we consider inspiral as the average result of repeated interactions with the surrounding stars (Mikkola & Valtonen, 1992, hereafter MV;Hills, 1993). The binary loses energy, shrinks and moves faster (hardens), with moderate increase in eccentricity. Numerical simulation (Makino et al. 1993, hereafter MFOE) seems to confirm this evolution in general, but warns of the importance of back-action of the BHB on the surrounding core, neglected in previous analytical investigations. Since the characteristics of the host core are crucial to the success of the inspiral, in section 2 we develop a numerical integration of the evolution of binding energy and angular momentum (hence eccentricity) during a merger event, which can be applied to a range of black hole masses and core properties. This approximate trajectory includes back-action on the core via loss cone depletion (BBR) and heating. In Section 2.3, we estimate f SI , the fraction of mergers achieving a successful inspiral due to stellar scattering, by computing our evolutionary model for a range of core parameters and BHB properties. Additional mechanisms that may drive inspiral are also mentioned in Section 2.3. In section 3 we collect estimates of the number density of massive black hole cores and galaxy merger rates to estimate the number of merging sources visible as a function of gravity wave period and strain amplitude. In section 4 we compare these rates to present and anticipated pulsar timing sensitivities and conclude by commenting on the prospects for further constraining inspiral events.
Simulation of Inspiral
Binary Evolution Formulae
It was suggested by Fukushige et al. (1992b) that the evolution of a BHB can be computed from the classic Chandrasekhar formula (Chandrasekhar, 1943) for dynamical friction (DF) on a single object in a homogeneous background. Applying an approximate formula to both holes, they find a large growth in the eccentricity; this was our motivation to trace energy and angular momentum loss of a BHB. We use the results of MV, who track the net effect of an ensemble of three-body interactions between an equal-mass binary and an intruder from a background of light objects. They cast their results in the form (ȧ/a 2 ) = (R a /v ∞ )πGρ, where a is the binary semi-major axis, ρ the density of background stars, and v ∞ the pre-encounter intruder speed. They find the numerical factor R a to be well fit by R a = −6.5/[1 + 2.44(v ∞ /W ) 2 ], independent of binary eccentricity. Here W is the r.m.s. relative speed of the binary elements, i.e. the relative speed of a circular binary of equal energy.
We assume the galactic core to have a Plummer model velocity distribution, given by
f (v) = 16 21 √ 3π 2 σ 3 1 − v 2 12σ 2 7/2 .
(1)
Since the one-dimensional dispersion σ depends on position in a Plummer model, we use the central value σ 0 . Using the fitting function, we calculate (as a function of σ 0 ) σ 0 R a /v ∞ σ 0 , the average over Plummer distributed v ∞ . This giveṡ
E = GM 1 M 2 πρ 2σ 0 σ 0 R a v ∞ σ 0(2)
for the evolution of the binary energy. In Fig. 1, we compare (2) to the energy loss rate of a circular binary due to DF, applying the Chandrasekhar formula (c.f. Binney & Tremaine, 1987) independently to each hole. In the W < σ 0 limit, the MV formula agrees by construction with the analytic calculation of Gould (1991); it thus differs from the DF result in the form of the Coulomb logarithm Λ (see Gould for a discussion). For the DF calculations of Fig. 1, we take the maximum impact parameter b max contained in Λ to be a function of the binary period, as discussed in the next section. In the W ∼ > σ 0 regime which proves most important in our calculation, we see in the figure that DF does not approximate well the more physically grounded and asymptotically sound three-body results.
For the eccentricity evolution, MV give results in the form dX d ln a = X n+1/2 − X where X ≡ 1 − e 2 , and n = 8 v∞ W
For each X and W we again average over v ∞ with distribution (1), and convert (3) to a time derivative, givingẊ
= −G 2 M 1 M 2 πρ 2Eσ 0 σ 0 R a v ∞ σ 0 X n+1/2 − X X,W/σ 0(4)
In Fig. 1 we also compare the amount of eccentricity growth of the binary as a function of energy for DF and MV. Again, DF does not seem to be a suitable approximation. Since MV treat only the equal-mass binary, we impose a correction factor to force agreement with Gould's analytic mass ratio scaling for M 1 = M 2 . We use M 1 ≥ M 2 throughout. Mikkola and Valtonen (1992). The left scale applies to the solid lines givingĖ, and the one value in this range from Hills (1983); the right scale is for the eccentricity growth rate at eccentricity of 0.9 (dotted lines). To calculate b max and Λ for the DF formulae we use M 1 = M 2 = 1.0M 9 , r c = 200pc and σ = 300kms −1 .
In following the orbit evolution, we also include the energy and angular momentum losses to gravitational radiation (GR): (Peters 1964)
dE dt = − 32 5 G 4 M 2 1 M 2 2 (M 1 + M 2 ) c 5 a 5 (1 − e 2 ) 7/2 1 + 73 24 e 2 + 37 96 e 4 ,(5)dL dt = − 32 5 G 7/2 M 2 1 M 2 2 (M 1 + M 2 ) 1/2 c 5 a 7/2 (1 − e 2 ) 2 1 + 7 8 e 2 .(6)
Once these terms dominate, the BHB will circularize. When the time scale for binary evolution by GR alone is less than 10 10 years, we call this the GR regime.
Properties of the Core
We use a Plummer model core of stars with mean mass m s , core radius r c and isotropic central velocity dispersion (1). The core stars have total potential energy W and kinetic energy K, satisfying the virial relation 2K + W = 0. With
W (N s , r c ) = − 3π 32 G(N s m s ) 2 r c − GN s m s (M 1 + M 2 ) r c ,(7)
we use K(N s , σ 0 ) = (9π/32)N s m s σ 2 0 and the virial relation to find
N s = 6σ 2 0 r c Gm s − 32 3π M 1 + M 2 m s .(8)
The core mass is M c = N s m s .
Only a subset of the core stars can effectively exchange energy with the binary. According to Heggie (1975), these are the stars whose interaction time with the binary is less than its orbital time scale. This criterion defines a maximum impact parameter b max (a function of the binary period) which we also used in the DF calculation for Fig. 1. At any distance from the center of the core, the condition on impact parameter selects stars whose velocities are within a cone aimed towards or away from the center, the so-called loss cone (Frank & Rees 1976). For our core, the fraction of stars with impact parameter at most b max is found by direct integration to be
N avail N s = b 2 max b 2 max + r 2 c .(9)
Since scattered stars enter a solid angle at a rate proportional to the solid angle size, stars can enter this loss cone no faster than the rate N avail /t relax . Thus t relax is also the loss cone repopulation time, (Binney & Tremaine 1987).
t lc = t relax = N s 8 ln rcσ 2 0 Gms r c σ 0 ≃ 3.7 × 10 13 yr N s 10 10 r c 200pc 300kms −1 σ 0(10)
Once encounters with the binary deplete the loss cone (BBR), the binary evolution rate will be limited by the rate at which stars enter the loss cone, and the amount of energy each star can take from the binary. In the W > σ 0 limit, MV find the change in binary energy with each stellar encounter to be δE ∼ 2Em s /(M 1 + M 2 ). This is in good agreement with the results at several mass ratios of Roos (1988) and Hills (1983). Writing this as δE = (M 2 /M 1 )m s v 2 2c , where the circular velocity v 2c of the lighter binary component is roughly the closest approach velocity of a star, we pass to the W ≤ σ 0 regime by using σ 0 for v 2c . The energy the loss cone can take away without repopulation is thus E avail = N avail δE , and from (10), energy enters the loss cone at a rate (dE/dt) in = E avail /t relax . If each star which interacts with the binary is scattered into a random orbit, a fraction N avail /N s of stars will be returned to the loss cone. For a loss cone population in steady state, then, the limiting binary evolution rate is
dE bin dt max = N s N s − N avail dE dt in = b 2 max + r 2 c r 2 c δE N avail t relax(11)
This rate is generally much smaller than that due to unrestrained dynamical friction. We approximate the situation by using the average core density in (2) and (4) until E avail drops below the binding energy of the binary. From this point on the binary will affect the loss cone stellar density significantly, and we impose the steady-state evolution rate. Invariably this 'loss cone catastrophe' results in a binary evolution time scale E bin /Ė bin much longer than the Hubble time. Similar but less specific treatments have been presented by BBR and Roos (1981).
We also treat eviction of stars from the core, found to be significant in the N-body work of Makino et al. (1993, hereafter MFOE). We assume all stars which interact with the binary after v 2c is larger than the Plummer central escape velocity v esc = √ 12σ 0 receive enough recoil velocity to be kicked out of the core. We calculate the resultant change in the core energies assuming rapid revirialization, readjusting r c and σ 0 . The binary also heats the core before becoming hard, but this effect is small.
At the beginning of the simulation, the binary's semi-major axis is the binding radius r b , defined such that the mass of stars within r b is less than M 1 . Its center of mass is at the core center, and it has some initial eccentricity e i . We assume the core is not rotating after the merger, since N-body merger simulations show very small angular momentum in the remnant bulge (Barnes 1992), even when progenitor bulges rotate (Hernquist 1993). A simulation is terminated unsuccessfully if E bin /Ė bin exceeds 10 10 years, or successfuly if the GR regime is reached. To estimate the parameter space available for successful inspiral, we find log(1 − e 2 i ) for the marginally successful trajectory to one part in a thousand by using ten bisection steps on the initial value. Some sample trajectories are shown in Figure 2. If we ignore back-action and loss cone depletion the results are identical to those of MV.
Our simulations confirm the basic behavior described by MFOE. For a range of initial conditions, interaction with background stars can cause some increase in the binary eccentricity. If we choose core parameters and black hole masses to match those of MFOE, we find loss cone depletion to occur at binary radii similar to those at which they terminate their simulation, where they too note a slowdown in binary evolution which they attribute to a possible loss cone effect.
Success Rate and Alternative Merger Mechanisms
For all pairs of binary member masses drawn from the set log M/M ⊙ = {10,9.5,...,7.0,6.5} we calculated the average initial critical eccentricity required for successful inspiral in 80 galactic Fig. 2.-Trajectories for two BHB/cluster combinations and several e i . Slight eccentricity growth and termination at the loss cone limit are visible, as is recircularization in the GR regime. Binding radius r b , and the radius r h at which v 2c > σ 0 are indicated. cores, drawn from a smoothed distribution based on the data set of Lauer (1985). We chose the distribution by noting that the binned data show N (M/10 9 M ⊙ ) = exp(−M 9 /22) (presumably the tail of a Schechter function); matching this and the N (r c ) of the data set gives our core distribution in the r c − σ 0 plane. In drawing from the distribution we required that the stellar mass of the host core be at least twice the total mass of the holes.
To estimate the fraction of successful inspirals f SI for any given binary mass, we assume stochastically distributed initial eccentricities (P (e) = 2e), so the inspiral success probability is P (e > e crit ) = 1 − e 2 crit . This choice of probability may be somewhat too low since N-body simulations of collisions of bulge-disc-halo systems (Barnes 1992) show radial final infall of the bulges; high initial binary eccentricity seems a likely consequence. Some critical eccentricities for BHB (log M 1 M ⊙ , log M 2 M ⊙ ) which will later prove most relevant for pulsar detection are: (9.5, 9) → .985, (9, 9) → .987, and (9, 8) → .990. These yield f SI of 2.9%, 2.6% and 2.0% respectively. Over the whole range of binary masses, the f SI are ∼ 1% − 3%. From the dotted curves in Fig. 1 we see that the dynamical friction formulae yield much more eccentricity growth, hence earlier arrival in the GR regime as per equations (5) and (6), and larger success fraction. We ran calculations with all the above features but using DF, finding f SI ranging from 1% to 50%. In the case of DF, it is possible to account for rotation of the stellar core, which was found to be an important impediment to eccentricity growth in the simulations of MFOE. By giving all angular momentum lost by the binary to the core, and applying the DF formulae to the motion of the binary relative to the rotating background, we find much less eccentricity growth, resulting in f SI between 1% and 16%. If even the modest eccentricity growth in the MV picture, as seen in Figure 2, is suppressed by spinup of the core, the MV f SI could be driven slightly lower.
Clearly stellar action on an isolated BHB in a galaxy core is of limited efficacy. However, the absence of many double active galactic nuclei (AGN's) and of orbital variations in the broad line region, as well as short binary periods inferred from from AGN jet precession (Roos 1988) suggest that inspiral beyond the loss cone limit does occur in many cases. Other means of binary inspiral have been proposed: gas may be ejected during nuclear activity, taking with it energy and angular momentum acquired from the binary; or it may be accreted by the larger hole, causing orbital contraction as M 1 r 2 remains constant to conserve angular momentum (BBR). Roos (1988) has also pointed out that if merger events happen several times during the lifetime of a galaxy and if, in particular, mergers are related to the onset of accretion and nuclear activity in AGNs, then external perturbations due to incoming galactic masses should accelerate stellar scattering inspiral. Though a binary which fails to merge is left evolving on the relaxation (and loss cone filling) time scale, Roos estimates that by the time a new merging galaxy is within a few core radii of the remnant pair, the loss cone can be refilled quickly and repeatedly, driving the original binary to GR inspiral. His calculation of the flux of stars entering the loss cone in this tidal repopulation scheme suggests that the galactic orbit decays no faster than that of the binary. If the incoming galaxy also has a black hole, it seems likely to form a binary with the post-coalescence hole.
Accordingly, in the remainder of this paper we follow two hypotheses: that inspiral occurs with the low probabilities allowed before loss cone depletion by MV's stellar encounter dissipation, and that repopulation ensures that inspiral always occurs. The latter picture assumes a galaxy formation model involving repeated mergers; these two cases certainly bracket the actual situation.
Merger Rates and Populations of Nuclear Black Holes
We wish to compute the rate of mergers observed from some redshift z. In an Einsteinde Sitter universe (Λ = 0, Ω = 1) with the metric ds 2 = dt 2 − a(t) 2 [dχ 2 + χ 2 dΩ 2 ], let F m (z) be the number of merger events at redshift z in the history of today's bright (L * ) galaxies, per dimensionless comoving volume per unit redshift. With this definition, F m (z)dz = N n gal a 3 0 , where n gal is the current number density of such galaxies and a 0 the current scale factor; N can range from ∼ 1/3, if only single mergers occur to form elliptical galaxies, to ∼ 10 in scenarios where all galaxies are formed from repeated merging of building blocks. At redshift z we observe a dimensionless co-moving area 4πχ 2 (z), with χ(z) = 2c a 0 H 0 1 − 1 √ 1+z . During a period T z at redshift z, the comoving volume observed is V χ = 4πχ 2 (c/a)T z . Thus the number of mergers per unit observers' time is
ν(z)dz = V χ F m (z)dz T z (a 0 /a) = 16π c H 0 2 c a 3 0 1 − 1 √ 1 + z 2 F m (z)dz.(12)
Note that a 0 vanishes in any physical rate because of the normalization of F m (z).
The form of F m (z) is still quite uncertain. Burkey et al. (1994) find that the population of close galaxy pairs which seem certain to merge varies as ∼ (1 + z) 3.5±.5 , in the interval 0 ≤ z ≤ 0.6. Colín et al. (1994) find that a galaxy density ∼ (1 + z) 3.8 best fits the total galaxy count data in a model which accounts for photometric evolution. The 3.5 power law gives F m (z) ∼ (1 + z) 2.5 . We normalize F m (z) to the merger rate implied by the pair counts in Burkey et al.. Applying this rate in the interval 0 ≤ z ≤ 1 implies 40galaxies will have suffered a merging event; alternatively if this rate continues back to the epoch of high quasar activity, z ∼ 3, then each bright galaxy will have experienced roughly 5 merger events. We find:
ν(z) dz = 7.6 × 10 −2 yr −1 h −2 50 n gal 10 −3 Mpc −3 (1 + z) 2.5 1 − 1 √ 1 + z 2 dz,(13)
where H 0 = 50h 50 kms −1 /Mpc.
As an alternative less dominated by early merging, we take merging rate per comoving volume constant in time, so that F m (z) ∼ (1 + z) −5/2 . If we assume that merging began at some z m and continues to the present, resulting in N mergers per bright galaxy of number density n gal , we find ν(z)dz = 0.55 yr −1 N h 2 50 n gal 10 −3 Mpc −3
3 2 (1 + z m ) 3/2 (1 + z m ) 3/2 − 1 [1 − (1 + z) −1/2 ] 2 (1 + z) 5/2 dz(14)
If accretion onto massive central black holes is the source of AGN luminosity, the population of remnant holes can be estimated from models of AGN evolution (Cavaliere and Padovani, 1988;Small and Blandford, 1992). Recent HST detections of kinematic evidence for massive compact objects in nearby galaxy cores support this scenario. We adopt here Small and Blandford's more conservative model IA, with a flat local Seyfert luminosity function. This gives the black hole number density spectrum
Comparing the above to a Schecter luminosity function of bright galaxies N (L)dL = 2.5 × 10 −3 h 3 50 (L/L * ) −1.1 e −(L/L * ) d(L/L * )Mpc −3 (de Lapparent et al. 1989), we found that if all massive black holes are in bright galaxies, the integral of equation (15) implies that 21% of L > L * galaxies contain a black hole with M > 10 6.5 M ⊙ today. Conservatively, we hold the co-moving number density of holes fixed: if there are N m subunits at redshift z that will become a bright galaxy, then the number of BHs per subunit is 0.21/N m . We note that the high inspiral rates (∼ 1/3 − 10yr −1 ) estimated by Fukushige, et al. (1992a) are based on excessively optimistic assumption of an M > 10 8 M ⊙ black hole in each of ∼ 10 galaxy subcomponents forming an elliptical. In their picture there are 10 gravity wave chirps emitted for each bright elliptical seen today. As we shall see, such event rates are not consistent with bounds from pulsar timing.
Gravitational Wave Amplitude of BHB Inspiral
For a circular binary of orbital period P b , reduced mass µ and total mass M , Thorne (1987) gives a characteristic strain amplitude averaged over direction and polarization of h c = 8(2/15) 1/2 µ(2πM/P b ) 2/3 /r
for the emitted gravitational wave from a circular (e = 0) binary. In units convenient to the BHB problem this is
h c = 1.8 × 10 −15 M 1 M 2 (M 1 + M 2 ) −1/3 r −1 Gpc [g 1/2 (e, n)/n]P −2/3 w(17)
where masses are in units of 10 9 M ⊙, distances are in kiloparsecs, g(e, n) gives the fraction of the wave power in the n th harmonic as a combination of Bessel functions (Peters & Mathews 1963) and the observed gravitational wave period in years is P w = P b (1 + z)/n. Only the n = 2 harmonic is non-zero for a circular orbit.
Since the gravity wave flux scales as h 2 ω −2 , falling off with the luminosity distance as d −2 L , one can write the gravity wave amplitude from a source at z emitting waves that have period P w at redshift 0 in an Einstein-deSitter universe:
h −15 (z) = 0.29M 1 M 2 (M 1 + M 2 ) −1/3 h 50 [g 1/2 (e, n)/n]P −2/3 w (1 + z) 2/3 [1 − (1 + z) −1/2 ](18)
in dimensionless units of 10 −15 . The characteristic lifetime for the source at this period is
τ GR = 1.3 × 10 4 M −1 1 M −1 2 (M 1 + M 2 ) 1/3 f (e) −1 [n P w /2(1 + z)] 8/3 y(19)
where f (e) = (1 + 73 24 e 2 + 37 96 e 4 )/(1 − e 2 ) 7/2 , as in Eq. (5). We wish to estimate, for a set of assumptions about the population, the expected number of BHBs detectable at a given strain sensitivity h −15 . Inverting equation (18) gives z(h; M 1 , M 2 , P w , e). We can combine this with the merger frequency rate, two independent draws from the black hole mass function (15), the successful inspiral fraction for these hole masses (averaged over cores that might contain such holes), and the lifetime of the resulting binary at the indicated orbital period to get the number of visible gravity wave sources in a given amplitude range:
dN (h, M 1 , M 2 , P w , e) dM 1 dM 2 = ν(z)dz N R (M 1 )N R (M 2 ) N 2 L * f SI (M 1 , M 2 ) τ GR (M 1 , M 2 , P w , e, z) dh dz .(20)
Integration of this equation over M 1 , M 2 and a range of h gives the source population estimates shown in Figure 3a for waves of period P w = 10y. In practice, we find that even for inspiral driven by stellar encounters, the characteristic eccentricity of the binary decreases to ∼ < 0.3 by the time it reaches the range observable with pulsar timing, so restriction to circular orbits is reasonable for estimating fluxes. Depending on assumptions, we see that gravity waves should be detectable once a strain sensitivity of ∼ 10 −15 is reached. The properties of the detected BHBs can also be estimated by weighted sums over this population; for the case of f SI = 1, and mergers proceeding as in equation (13), average characteristics of the detected binaries are shown in Figure 3b. It is also worth noting that with these parameters, we expect ∼ 1 chirp per 10 4 yr with an amplitude greater than 10 −15 at periods P w ∼ 10 4 s, substantially lower than the rates quoted by Fukushige et al. (1992a). Right -weighted observed source properties for f SI = 1 mergers following rate (13). The binary mass ratio is q, and the GR inspiral time is τ .
Conclusions and Observational Prospects
Millisecond pulsar timing has been used to place strong bounds on the energy density in a stochastic background of ultra-low frequency gravitational waves (KTR and references therein). For example, these studies have placed an energy density limit Ω g < 2 × 10 −7 h −2 50 and have helped to rule out various exotic cosmologies, such as structure formation seeded by cosmic string loops. With a number of high-quality pulsars now being timed, it is worth considering whether astrophysical sources, such as BHBs, are within reach. In Figure 4 we show a simulated gravity wave spectrum computed from the amplitude number distributions (as in Figure 3a) for a range of wave periods. Two models are shown, the first with inspiral mediated by tidal repopulation (f SI = 1), the second for evolution limited by loss cone depletion. Amplitude distributions were computed at each frequency; then the power was integrated over the low h sources and Monte Carlo sampled from the rare bright sources. The corresponding amplitude is given in Figure 4, which thus shows the expected BHB ultra-low frequency gravity wave background. If one makes N obs ∼ 20 arrival time measurements per year over a period T obs ∼ 10y with an accuracy of δt ∼ 1δ −6 µs, then one can place a limit on the strain amplitude of a passing gravity wave with period P w of roughly h ∼ δt P w (N obs T obs ) −1/2 ∼ 2 × 10 −15 δ −6 (N 20 T 10 ) −1/2 P −1 y .
For this estimate to hold we must have T obs ∼ > P w , so that fitting pulsar parameters does not significantly absorb any gravity wave signal (Blandford et al. 1984). This also assumes that the pulsar timing residuals are distributed as white noise. According to this estimate, intensive long-term timing programs can reach the sensitivity needed to detect the brightest BHB gravity wave sources at periods near 10y.
Present timing results (KTR) show that PSR1937+21 which has been monitored for over eight years shows significant unmodeled timing noise, presumably due to rotational variations intrinsic to the pulsar. PSR1855+09, on the other hand, shows random variations at its ∼ 0.8µs arrival time accuracy with over seven years of timing; these variations are consistent with perturbations arising from instability of the best terrestrial clocks. PSR1855+09 residuals are constraining gravity wave sources at an amplitude of h ≈ 5 × 10 −15 (Fig. 4). Thus with the best present data, we estimate a chance ∼ 0.001 of detecting a merging BHB. However, with a factor of ∼ 5 increase in sensitivity (and a slightly increased experiment duration) we can anticipate detecting the brightest BHB gravity wave sources, for reasonable population assumptions. Whether this sensitivity increase can be effected is uncertain. In the last few years, timing programs have been initiated on several new pulsars (e.g. PSR J1713+0747, PSR J0437-4715) that provide sub-µs timing residuals. However, to take advantage of this precision it will be necessary to obtain improved atomic clock standards or time one pulsar against another. Finally, to reach interesting detection sensitivities we require the arrival time residuals to integrate down as white noise over the ∼ 10y periods. Clearly, intrinsic instabilities in PSR1937+21 prevent this; fortunately timing noise appears to correlate with period derivative and several of the new pulsars have period derivatives 10 -20 times smaller than that of PSR 1937+21. It should be noted that the required single order of magnitude improvement represents a much better prospect than most other gravitational radiation search techniques! In summary, we have developed a model of the inspiral of two massive black holes in a galaxy core driven by stellar encounters. We recover some of the behavior described by earlier workers, though the large eccentricity needed to circumvent the loss cone catastrophe in our calculations must be present ab initio. Our simplified sum allows computation of this process in a range of cluster cores, and we see that simple stellar encounter dissipation is only effective in 1%-3% of all BHB-producing merger events. Nonetheless, it seems likely that other processes will ensure that inspiral occurs in the majority of cases. Turning to the rate of galaxy mergers, and the fraction of merging cores containing high mass black holes, we estimate the event frequency and typical gravity wave amplitude expected from a cosmological population of merging cores with central BHBs. Simulating the gravity wave spectrum produced by this population shows that, while no detection of ULF GR sources is expected to date, moderate improvements in present sensitivities will make detection of waves from binary sources with periods of ∼ 10yr possible via timing observations of millisecond pulsars. Detection of correlated gravity wave signals in the arrival
Fig. 1 .
1-Comparison of binary evolution formulae given by dynamical friction and by
Fig. 3 .
3-Left -Source intensity distributions for several merger models. The filled dots give the distribution for tidally repopulated merging and open circles show numbers for evolution slowed by loss cone depletion, with the merger rate (13). Crosses show the population from mergers constant in t.
Fig. 4 .
4-Simulated ULF gravity wave backgrounds from f SI = 1 mergers and loss cone limited stellar encounter (MV) mergers. Bright nearby sources rise above the background. Approximate sensitivities from anticipated pulsar timing experiments are shown as dashed lines (see text).
Alfred P. Sloan Fellow
constitute an exciting confirmation of this astrophysical class of gravity wave sources. However, even upper limits a factor ∼ 5 lower than present bounds can constrain the population and merging behavior of massive BHBs throughout the universe. It is a pleasure to acknowledge useful discussions with Andrew Gould and Douglas Richstone, and helpful comments by the referee. RWR was supported in part by an Alfred P. Sloan fellowship, and MR. J.E. 393484National Science and Engineering Research Council of Canada. REFERENCES BarnesApJof several pulsars would, of course, constitute an exciting confirmation of this astrophysical class of gravity wave sources. However, even upper limits a factor ∼ 5 lower than present bounds can constrain the population and merging behavior of massive BHBs throughout the universe. It is a pleasure to acknowledge useful discussions with Andrew Gould and Douglas Richstone, and helpful comments by the referee. RWR was supported in part by an Alfred P. Sloan fellowship, and MR in part by a fellowship from the National Science and Engineering Research Council of Canada. REFERENCES Barnes, J.E. 1992, ApJ, 393, 484
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[
"A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency",
"A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency"
] |
[
"Iztok Bajc \nFakulteta za matematiko in fiziko\nUniverza v Ljubljani\nJadranska 19SI-1000LjubljanaSlovenija\n",
"Frédéric Hecht \nFakulteta za matematiko in fiziko\nUniverza v Ljubljani\nJadranska 19SI-1000LjubljanaSlovenija\n\nSorbonne Universités\nUMR 7598\nUPMC Univ Paris 06\nLaboratoire Jacques-Louis Lions\nF-75005ParisFrance\n\nUMR 7598\nCNRS\nLaboratoire Jacques-Louis Lions\nF-75005ParisFrance\n\nInštitut Jožef Stefan\nJamova 39SI-1000Ljubljana, Slovenija\n",
"\nDomaine de Voluceau\nINRIA-Paris-Rocquencourt\nEPC *\nBP105, 78153Le Ches-nay Cedex\n"
] |
[
"Fakulteta za matematiko in fiziko\nUniverza v Ljubljani\nJadranska 19SI-1000LjubljanaSlovenija",
"Fakulteta za matematiko in fiziko\nUniverza v Ljubljani\nJadranska 19SI-1000LjubljanaSlovenija",
"Sorbonne Universités\nUMR 7598\nUPMC Univ Paris 06\nLaboratoire Jacques-Louis Lions\nF-75005ParisFrance",
"UMR 7598\nCNRS\nLaboratoire Jacques-Louis Lions\nF-75005ParisFrance",
"Inštitut Jožef Stefan\nJamova 39SI-1000Ljubljana, Slovenija",
"Domaine de Voluceau\nINRIA-Paris-Rocquencourt\nEPC *\nBP105, 78153Le Ches-nay Cedex"
] |
[] |
This paper presents a 3D mesh adaptivity strategy on unstructured tetrahedral meshes by a posteriori error estimates based on metrics, studied on the case of a nonlinear finite element minimization scheme for the Landau-de Gennes free energy functional of nematic liquid crystals. Newton's iteration for tensor fields is employed with steepest descent method possibly stepping in.Aspects relating the driving of mesh adaptivity within the nonlinear scheme are considered. The algorithmic performance is found to depend on at least two factors: when to trigger each single mesh adaptation, and the precision of the correlated remeshing. Each factor is represented by a parameter, with its values possibly varying for every new mesh adaptation. We empirically show that the time of the overall algorithm convergence can vary considerably when different sequences of parameters are used, thus posing a question about optimality.The extensive testings and debugging done within this work on the simulation of systems of nematic colloids substantially contributed to the upgrade of an open source finite element-oriented programming language to its 3D meshing possibilities, as also to an outer 3D remeshing module.
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10.1016/j.jcp.2016.02.072
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[
"https://arxiv.org/pdf/1505.07046v1.pdf"
] | 26,981,014 |
1505.07046
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8ffd772eb665de8ce9a31f2fe1c0b809b4f7fb36
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A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency
26 May 2015
Iztok Bajc
Fakulteta za matematiko in fiziko
Univerza v Ljubljani
Jadranska 19SI-1000LjubljanaSlovenija
Frédéric Hecht
Fakulteta za matematiko in fiziko
Univerza v Ljubljani
Jadranska 19SI-1000LjubljanaSlovenija
Sorbonne Universités
UMR 7598
UPMC Univ Paris 06
Laboratoire Jacques-Louis Lions
F-75005ParisFrance
UMR 7598
CNRS
Laboratoire Jacques-Louis Lions
F-75005ParisFrance
Inštitut Jožef Stefan
Jamova 39SI-1000Ljubljana, Slovenija
Domaine de Voluceau
INRIA-Paris-Rocquencourt
EPC *
BP105, 78153Le Ches-nay Cedex
A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency
26 May 201513D mesh adaptivitymetricsfinite elementsFreeFem++nematic liquid crystalsPDEnonlinear analysis
This paper presents a 3D mesh adaptivity strategy on unstructured tetrahedral meshes by a posteriori error estimates based on metrics, studied on the case of a nonlinear finite element minimization scheme for the Landau-de Gennes free energy functional of nematic liquid crystals. Newton's iteration for tensor fields is employed with steepest descent method possibly stepping in.Aspects relating the driving of mesh adaptivity within the nonlinear scheme are considered. The algorithmic performance is found to depend on at least two factors: when to trigger each single mesh adaptation, and the precision of the correlated remeshing. Each factor is represented by a parameter, with its values possibly varying for every new mesh adaptation. We empirically show that the time of the overall algorithm convergence can vary considerably when different sequences of parameters are used, thus posing a question about optimality.The extensive testings and debugging done within this work on the simulation of systems of nematic colloids substantially contributed to the upgrade of an open source finite element-oriented programming language to its 3D meshing possibilities, as also to an outer 3D remeshing module.
Introduction
Effective minimization of functionals is an important topic in a variety of scientific tasks, in which the increasingly powerful computational capabilities of the last decades had allowed shifting from 2D systems to larger 3D ones. Confined nematic colloids with defects in directional ordering fields [1] are an example of such computational systems. In fact, the Landau-de Gennes free energy functional [2] for liquid crystals is a representative nonlinear functional from theoretical physics, along with similar ones, as for ex. the Gross-Pitajevski functional for Bose-Einstein condensates [3], or the Ginzburg-Landau for superconductivity [4]. All of them are phenomenologically describing critical phenomena in condensed matter systems with possible appearence of topological defects.
Advances in the computational science are relatively soon at hand for more classical computational fields, e.g., fluid dynamics [5], but usually not so readily used for theoretical physics porpouses, at least in three dimensions. Inter alia, the present paper tries to contribute also in this sense. The Landau-de Gennes free energy functional [2] is very well known in the realm of liquid crystals science. Plenty of physical systems have already been simulated by its minimization (for example [1,6,7]), with a good qualitative agreement of such calculations with physical experiments, thus empirically validating such approach. Also the mathematical task of wellposedness (existence and regularity) of the minimizers for particular forms of the Landau-de Gennes functional has been successfully analyzed [8]. Finite elements were used in [9,10], but without a truly systemic mesh adaptivity approach. The latter was employed in [11], with an empirical mesh estimator, upgrading the one used in a refining method [12] on a special symmetric case.
A lot of 3D simulations of nematic liquid crystals (NLC) employed the finite difference method (FD). In particular a set of codes, developed from methodologies introduced in previous NLC hydrodynamics works ([13] in 2D, and [14] in 3D) has proved to be robust, and been successfully used leading to some important theoretical results in the NLC field (see [15] for an essential shorter résumé).
Finite element methods [16] have the property that can considerably decrease the number of degrees of freedom by use of unstructured tetrahedral meshes. The latter can discretize the computational domain very flexibly, with (possibly larger) variations of magnitude of the mesh tetrahedra. Moreover, complicated surfaces can be modeled quite precisely with triangular surface meshes, with usually well defined boundary conditions, and a broad set of theoretically well-founded error analyses. On the other side, working with triangular and tetrahedral meshes implies an increased level of complexity for their generation and manipulation, which for three-dimensional domains is a field still reaching a complete operational maturity which could be available to a broader public. The present paper aims at contributing mostly to this point, in particular to aspects concerning the driving of a nonlinear scheme along with mesh adaptivity.
Concentrating on test examples of colloidal particles in confined nematic matrix (shortly, confined nematic colloids), which present a challenging, almost singular behaviour regarding mesh resolution requirements, the hereby presented scheme makes use of the mesh adaptivity tool of metric mappings, or shorter just metrics. These are representing a posteriori error estimates based on the Hessian of the solution(s), and are a still evolving [17] subfield of mesh adaptivity.
The overall scheme here used is programmed in FreeFem++, a complete and free (open source) C/C++ idiomatic programming language [18] with powerful commands and data types dedicated to the finite element method and its use for solution schemes of (systems of) partial differential equations and functional minimization. The overall work for the present paper contributed to the development and smoothing of some of the meshing-related parts used within it (with testing, debugging, and interacting with the modules' authors).
Summarizing, this paper will present a 3D mesh adaptivity strategy, based on isotropic metrics, with a finite element algorithm, implementated in FreeFem++ on one processor, on the case of the minimization of the Landaude Gennes free energy functional, modeling systems of confined nematic colloids. The driving of mesh adaptivity coupled with the nonlinear scheme will be found to be non-trivially dependent on parameters regarding tetrahedral meshes and metrics defined on them. The algorithmic behaviour with regard to two parameters will be particularly analyzed. Numerical experiments will show that the driving efficiency of the coupling of the nonlinear scheme with mesh adaptivity depends at least on what are the stopping criteria at which a new mesh adaptation is triggered, and to what precision each new mesh is rebuild. For the aims of this paper, sequences of different parameter values have been choosed into fixed arrays, but for the future better solutions could be enivisaged. The results and the correlations found along with the presented ideas and suggestions are supposed to be meaningful for wider classes of nonlinear systems and physics typologies.
2 Physical description of nematic liquid crystals
Nematic director and order parameters
Nematic liquid crystals are an oily material that can flow like a liquid, but also exhibit physical features (optical, for example), that are typical of crystals. They are a mesophase, i.e., more ordered than liquids, yet less ordered than crystals. These properties are mostly due to the elongated, rod-like form of their molecules, that in an appropriate temperature range (or under an applied electric/magnetic field) locally align into a preferential axis, called the director and denoted by a vector n. The degree of this alignment is described by another physical quantity, the scalar order parameter S. Both quantities are usually nonhomogeneous in space, thus formally represented by a vector and a scalar field (n(r) and S(r)), which can vary at each point of the nematic material. Only its direction being important, the nematic director is defined as a unit vector (field), |n| = 1. Being the sense in which the nematic molecules are pointing (statistically) the same, also the equivalence n ←→ −n must hold. (Sometimes the set of possible vectors n in a certain point r of the nematic is described mathematically with the equivalence class S 2 /Z 2 , which figuratively means approximately a hemisphere of the Euclidean 2dimensional sphere S 2 in R 3 , altough more precisely it is the real projective plane RP 2 .)
The possible values of the scalar order parameter S range between -1/2 and 1. As the negative values appear in situations not included here, we can concentrate our attention to the interval [0, 1]. Here, termodynamically speaking, S = 1 describes the ideal nematic phase, in which all the molecules are (would be) perfectly aligned, while the other extreme, S = 0, describes the high-temperature isotropic phase, in which the kinetic energy of the molecules is so large that they are completely disordered, like in a usual isotropic liquid. For classical nematic materials intermediate bulk values of S are the rule 6 , representing intermediate degrees of order.
The above description is not always enough to guarantee neither a truly correct physical picture, nor computational stability. Confined nematic systems with the inclusion of colloidal particles usually get frustrated, leading to appearence of topological defects. The latter are regions (usually line-like), where the scalar order parameter drops to a lower value. Some descriptions, as for ex. the above one with only director and scalar order parameter (in total only four scalar quantities), can lead to singularities. Thus, a second order tensor quantity must be introduced, that is, the tensor order parameter Q, related to the previous description by
Q = S 2 (3n ⊗ n − I) + P 2 (e (1) ⊗ e (2) − e (2) ⊗ e (1) ).(1)
Here, the greatest eigen value of Q is the scalar order parameter S, and its correspondent eigen vector is the director n. The other two orthonormal eigen vectors are e (1) , e (2) , and P the biaxiality parameter. When the latter may be negligible in some contexts, as, e.g., in setting boundary conditions, the second term can be dropped, leading to an uniaxial approximation model
Q(r) = S(r) 2 (3n(r) ⊗ n(r) − I).(2)
In both expressions I is the 3×3 identity matrix and ⊗ the tensor product. As the hereby notation stresses, Q(r) is a tensor field, and thus its components Q ij (r) are scalar fields. The tensor order parameter field is symmetric, Q ij = Q j i , and traceless, T r(Q) = 0, so it can be written as
Q = Q 11 Q 12 Q 13 Q 22 Q 23 −Q 11 − Q 22 ,(3)
where the non appearing lower off-diagonal components are meant to be equal to the corresponding symmetric upper ones. As it can be seen, only five components of the tensor are needed to represent the whole tensor field.
Landau-de Gennes model
From the Landau theory of phase transitions [19,20] it is known that appropriate thermodynamical systems with an order parameter can be described in a suitable temperature range of a phase transition by the Landau phenomenological expansion of their free energy, under the condition that it takes into account the symmetries of the system (i.e., the expansion must be invariant to them). Well known applications of this theory are found for example in magnetic systems, in the temperature range of the transition between the paramagnetic and ferromagnetic phase, or in superconductivity [4] by the Ginzburg-Landau equations [21], etc. In our case, the Landau-de Gennes model for liquid crystals will be employed, in which the Landau-de Gennes free energy functional will have the form
F (Q) = Ω [f e (∇Q) + f b (Q)] dV + Γp f s (Q)dA.(4)
Here, the first integral comprises the volume contributions (elastic density f e and bulk f b ) to the total nematic free energy in the interior of the (bounded) domain Ω enclosing the space filled with nematic (thus without colloidal particles, which are outside; Ω is thus a domain with holes).
The elastic energy density f e can in general be constructed with three constants. Here we use a simplyfied but qualitatively still accurate version, employing the one-constant approximation
f e (∇Q) = 1 2 L |∇Q| 2 = 1 2 L∇Q ij · ∇Q ij(5)
with L being the nematic elastic constant. The thermodynamic ( or bulk) energy has the form
f b (Q) = 1 2 A T r(Q 2 ) + 1 3 B T r(Q 3 ) + 1 4 C (T r(Q 2 )) 2 = 1 2 A Q ij Q ji + 1 3 B Q ij Q jk Q ki + 1 4 C (Q ij Q ji ) 2 ,(6)
with A, B, C being the material bulk constants. Here, as everywhere else in the text, Einstein double index summation is considered. The domain boundary ∂Ω is splitted in two disjoint subsets, ∂Ω = Γ p ∪Γ c . The first of the two, Γ p , consists of the colloidal particles surfaces 7 on which are defined the surface integrals, with penalty free energy density f s , having the form of the so-called Rapini-Papoular anchoring energy
f s (Q) = 1 2 W (Q ij − Q 0 ij )(Q ij − Q 0 ij ),(7)
where the constant W is the anchoring energy, and Q 0 ij the components of the reference tensor order parameter field on the surface of the particles.
The second one, Γ c , represents the computational cell walls, on which Dirichlet boundary conditions are employed, and which will be described in Section 4.
Free energy functional minimization -nematic structure calculation
The nematic liquid crystal systems here considered are at constant temperature and constant volume 8 . When such a system is physically let to evolve, its entropy grows driving the free energy potential to a minimum. If the latter is global, the equilibrium is stable, while if just a local minimum has been reached, the structure is considered to be metastable 9 . The Landau-de Gennes model is a static theory neglecting fluctuations. Its free energy functional minima describe the equilibrium configurations of a nematic system. Mathematically (computationally) this minimization can be achieved with finite elements in at least two ways.
The first one uses the elementary and well known necessary condition [23] for a differentiable functional F to have a minimum for Q = Q * , that is when its first variation vanishes, δF (Q * ) = 0. This in our case directly corresponds to the Euler-Lagrange equations in weak form. By solving this operator equation, i.e., by finding a numerical solution that sets it approximately to zero (here done by Newton's method [24,25]), one can achieve a minimum of the system. In the second way a direct minimization of the functional is performed, employing the steepest descent method [26]. In the present paper we use a hybrid technique, which starts in the first way, but possibly employs also the second one.
Newton's method/Steepest descent
The Newton's method (or Newton-Raphson method ) is a second order approximate iterative method for numerically solving (nonlinear) operator equations. Being the equation in our case δF (Q) = F (Q)δQ = 0, the iterative equation achieves the form
F (Q (i) )v (i) φ = −F (Q (i) )φ,(8)
where Q (i) is the current solution at step i (the last computed), and F (Q (i) ) and F (Q (i) ) are correspondingly the first and second variation of the func-
tional F in Q (i) , φ are the test functions (written in place of δQ), and v (i) is the equation solution (i.e. the move, or increment field v (i) = Q (i+1) − Q (i)
at the i-th step between two successive iterations). Possible situations exist, in which Newton's method can fail. Its iteration sequence can diverge, for ex. at bifurcation points, i.e., when the second variation of the free energy functional is singular, det(F (Q)) = 0, or trap itself into a (quasi)periodic orbit when in a neighbourhood of a saddle point.
To overcome such problems, it seemed convenient to introduce also a more stable method into which the algorithm could possibly switch in these cases, i.e., in the neighbourhood of such problematic nematic configurations.
One of the oldest gradient algorithms for functional minimization is the steepest descent method. Developed by Cauchy more than one century and half ago (altough for functions of several variables), it is a robust iterative algorithm, suitable for such situations. From an iterate Q (i) it proceeds by first calculating the (negative) gradient h (i) = −∇F (Q (i) ), then choosing a suitable parameter λ, and finally computing the next iterate by The obtained trajectory somehow resembles the natural path of a droplet of water descending a hill under the force of gravity. The main tasks to be accomplished during the steepest descent method are the calculation of the gradient, and the choice of the parameter λ, which was here done by the exact line search technique. As the Newton's algorithm, it must start in an initial configuration Q (0) .
Unstructured tetrahedral meshes and mesh adaptivity
When numerically solving (systems of) partial differential equations a mesh is intrinsically related to the solution computed on it, and it can be said of good quality, if leading to a good solution [27]. A computed solution is usually assumed to be such, if approximating the real solution with a low error. A general wish, aim, regarding meshes is to have the number of mesh vertices low as well. Assuming these definitions and goals, a mesh can be considered optimal [27], if leading to a solution computed within a prescribed error with the minimal number of degrees of freedom. Quantitatively, this can be obtained (and it is here done so, following [28,29,30,31]) by applying the equipartition of the total error to all the mesh elements. After setting a relative interpolation error threshold, the aim of the remeshing is to rebuild the tetrahedral mesh in such a way, that the interpolation error would be everywhere, i.e. on each element, below it.
In general this can be best achieved by building mesh elements by varying their size, and varying also their shape (i.e. edge lengths and angles between them) and orientation. The driving idea is that the size of the tetrahedra must get smaller in the regions of the computational domain, where the solution is spatially varying. The more it varies, the smaller the elements must be in order to catch the solution shape correctly enough -below the prescribed error. The motivation to locally vary also tetrahedra's shape and orientation is that if the solution locally doesn't change much along a direction, than the tetrahedra in that direction can be more elongated. This implies the use of a much lower number of elements. Examples exist [27], where the number of degrees of freedom used (with P1 elements), has been decreased for ten times, compared to the same computations done with the isotropic approach.
A contribution of the present paper to similar ones regarding liquid crystals and exploiting similar features or methods, e.g. [12,11,10], is the use of a systemic mesh adaptivity approach on three-dimensional unstructured tetrahedral meshes based on isotropic metrics.
Mesh generation
The basic ideas of unstructured mesh generation are quite similar irrespective of the dimensionality of the space in which the mesh is built. In 1D, segments of different length are generated, while in 2D/3D triangles/tetrahedra of different size, shape and orientation. Nevertheless, the 3D case is technically much more difficult to implement than the previous two [27].
Altough the implementation of mesh generation can be accomplished with an ample palette of approaches, its basic underlying idea is substantially the same. Regardless of the fact if we are using an advancing front technique, or a Delaunay approach, a their combination, or something else [27], we start with a closed surface mesh in 3D representing the domain boundary. Then, using a criterion dependent of the technique used, we add vertices in its interior until the whole domain is tetrahedralized. Such a tetrahedralization must simultaneously conform to both the domain geometry and the solution. Thus, when choosing the position where a vertex should be added, the information about both of them must be taken into account.
Metrics
This can be achieved by employing metric mappings, or simpler, metrics, with which it is possible to produce, via appropriate algorithms mentioned above, unstructured meshes with tetrahedra of the locally desired size and orientation, possibly with large scale variations within the same domain Ω.
The main idea is that the usual (classical) Euclidean length in space
d(r, r ) = ||r − r || 2 = < r − r , r − r >(9)
is changed by redefining the usual (Euclidean) scalar product < · , · > in R 3 , appearing in (9), with a new one, < · , · > M , defined as
< r, r > M =< r, M r >,
with M for being a constant symmetric positive definite matrix. By leaving it vary over the computational domain Ω, we obtain a 3×3 tensor field M(r), called the metric tensor field, or simply metric. With the domain Ω endowed with such a (Riemannian) structure, the theoretical distance l M (r 1 , r 2 ) between two points r 1 , r 2 ∈ Ω now equals to
l M (r 1 , r 2 ) = 1 0 < γ (t), M(γ(t))γ (t) > dt,(10)
where γ is the shortest possible path (the geodesic) between the two points. For practical purposes, the average lengthl M (r 12 ) of an edge between two vertices r 1 , r 2 , described by r 12 = r 2 − r 1 , can be computed as
l M (r 12 ) = 1 0 < r 12 , M(r 1 + tr 12 )r 12 > dt.(11)
As detailedly explained for ex. in [27,28,29,30,31], to have everything working correctly, M(r) must be symmetric and positive definite. This gives rise to anisotropic metrics.
In the hereby calculations isotropic metrics were used, which means that the diagonalization of the tensor M in the local coordinate frame has all of its three eigenvalues being equal. The diagonal metric tensor is then equivalent to a spatial distribution of the tetrahedral sides, i.e. a scalar field h(r).
Computations
For the computational examples we set and tested a code with calculations for the simplest case of confined colloidal nematic system, a colloidal particle immersed in confined nematic (i.e., the monomer ). The code has then been run for five different sizes of the system (all with the same length proportions), for three different types of convergence sequences (see explanation later), and for three different values of the computational parameter hmax (see definition later on).
Experimental setting in physical laboratory
In the concrete experimental set-up in the physical laboratory (see for ex. [1]) the particles of spherical shape have a diameter of order of magnitude of a couple or some microns, and are usually made of silica, glass, or metal. They are immersed in a nematic liquid (here 5CB), contained inbetween two glass plates, distant some microns one from the other for a distance at least a couple of times the magnitude of the particle's diameter.
The particles have homeotropic anchoring, which means that their surface is chemically treated with surfactant molecules attached perpendicular on it. Instead, the surfaces of the plates are treated mechanically (rubbed), in order to have horizontal anchoring with direction parallel to the sides of the cell. The nematic tends to align with the anchoring: at the sides of the cell parallel to them, and on the surface of the particle perpendicularly to it. Figure 1: Cross-section of the final tetrahedral mesh for one colloidal particle in confined nematic, obtained after the whole mesh adaptivity process: front perspective (leftx), and side perspective (right). Two cross-sections of the Saturn ring topological defect can be noticed symmetrically on particle's sides, where the mesh is very refined. The sorrounding green square is the cutting plane; tetrahedra at its intersection point out in hedgehog-style. Unstructured meshes can develop slight variations in density, e.g. near the particle, up on the left, which are mostly due to meshing technical reasons.
Computational details
The code has been written and tested first for the case of a spherical colloidal particle of diameter 2R = 1µm, posed in the center of a cubic cell with d = 2µm, full of nematic with values of the material constants L, A, B, C for the 5CB type (for their values see for ex. [1,6]). The boundary conditions matched the experimental ones described above. As at some sides the real (experimental) cell is very large (virtually infinite), and full of nematic, an approximation was made in the computations, putting at that sides, i.e., the walls of the computational cell, boundary conditions matching the behaviour of nematic at a longer distance.
All the computations runned on one processor of a 64-bit desktop machine with Intel Core 2Quad CPU [email protected]×4 processor, with 7.7GiB of RAM and the 64-bit Linux Ubuntu 12.04.4 LTS operating system. The Figure 2: Central cross-section of the computed nematic field around one colloidal particle correspondent to the final adapted mesh of Fig.1. As before, two crosssections of the Saturn ring topological defect can be noticed symmetrically on the sides of the particle. The visualization doesn't follow the standard LC literature for nematic fields. Theoretically, the length of each director "vector" (line) should be always equal to one. Here, each length is proportional to the volume of the tetrahedron on which it lies; consequenlty, the lines around the defect seem almost points.
FreeFem++ version used was 3.30. The main code in the remeshing process, mmg3d5ljll, for the moment not part of the standard FreeFem++ distribution yet, was cordially supplied by its authors Charles Dapogny, Cécile Dobrzynski, and Pascal Frey, and called as an external module. All runs were reniced at their beginning to a nice value of -10, i.e., to a higher priority.
Main scheme
After being launched, the overall algorithm works in the following way (see Alg. 1, written in pseudo-code, below).
First the initialization of the system is made. The initial mesh Th is built, and the starting guess for Qh set. Then, after the computation of the initial nematic structure into Qh, with Newton iteration (and possibly also steepest descent), the main loop is entered. Here, at each iteration k the current mesh is adapted into a new mesh, and a new nematic structure is computed on it. This is looped for totally NbOfAdapt times, which is a positive integer fixed by the length of the arrays of parameters tolAdapt and errm. Finally, the last computed mesh and nematic configuration on it are returned.
Initialization: initial mesh and starting guess
The surface mesh describing and enclosing the computational spatial domain was designed within the FreeFem++ built-in functionalities, and then tetrahedrized with TetGen [32] as one of its inner modules. More complicated surface meshes can be generated by Gmsh [33], or other (free) mesh generators, and then imported into FreeFem++.
A correct starting guess in this elementary case of a monomer was very simple, i.e., just the constant nematic configuration n = (0, 0, 1). The initial tetrahedral mesh was set fine enough in the neighborhood of the particle, where stronger variations of the nematic field and defects appear, and then linearly coarsened while approaching the cell walls, where the nematic conformation changes no more.
Algorithm 2
Newton iteration loop
This is the core, or in any case one of the innest parts of the overall algorithm (the other one is the mesh adaptivity loop). At each step i of the loop, the Newton equation (8) is solved. First the finite element stiffness matrix and load vector are obtained by discretizing the variational equation (8) on the fixed tetrahedral mesh T k , using P1 finite element basis functions. Being the sparse linear system symmetric positive definite, it can thus be solved by the conjugate gradient method (here with an ε < 0.5 · 10 −7 relative error bound, and a rough preconditioner, dividing each line of the sparse matrix with its largest element). This proved to be a good choice in this case, being direct factorization methods, as also GMRES, impracticable, due to the large sizes of the systems. The incremental solution v (i) of the sparse linear system is then added to the current solution Q (i) , obtaining Q (i+1) , in which the Newton step is again recomputed until the relative change of the functional value (of the system's free energy) is lower than the tolerance tol k , or the maximal number of iterations is reached. In the latter case the algorithm switches into the steepest descent method mood.
Alternatively, the normalized L 2 -norm of the move (increment) v (i) could be used as another (or concurrent) criterion. In our case was anyway being constantly monitored.
Mesh adaptivity loop
Also each mesh adaptation itself is computed iteratively. First a new tetrahedral mesh variable Thx is declared, which is then adapted several times during the loop. Its starting "value" is Th, i.e., the last computed mesh before entering into the mesh adaptivity procedure. Also a set of scalar fields scFields is declared, with regard to which the metrics will be computed.
Once the loop is started, at each new iteration a new finite element space Vhx, based on the current mesh Thx, is declared (which with FreeFem++ is done most easily and straightforwardly with just one short code line). An isotropic metric M is then declared as a scalar field from this FE space, and computed with a call of mshmet. One of the most important parameters of the latter call is errm k, representing the largest possible relative error of the solution on each element, at the k-th iteration (here ranging within a couple of percents, more precisely starting from 0.02 and ending with 0.01). The other parameter scFields represent the scalar functions with regard to which the metric is computed. Initially these were only the five components Q ij of the tensor order parameter field, and the scalar order parameter S (within the code written as Qh and S). After some experimentations it has been noticed and felt that also the inclusion of the (five) first variations of the free energy δF δQ ij could make sense, and so they have been added to the list (as DF).
With this metric M a new mesh is computed into Thx by mmg3d5. The latter takes care that the mesh contains only tetrahedra with side lengths inbetween the (argument) parameters hmin and hmax, and also that the ratio between the side lengths of any two neighbouring tetrahedra does not exceed the prescribed mesh graduality parameter value hgrad, (here fixed to 2.00 throughout all the calculations).
The loop ends when a condition characterizing some kind of convergence of the mesh (and/or the metric) is fullfilled. The condition must measure how much two subsequent meshes are close to each other. What we used here, was very simple, and most probably far from optimal, i.e., we used that the difference of the numbers of the vertices between two subsequent meshes in the loop does not exceed a certain number (here fixed to 300). Alternatively, another condition defined with the norm of the difference between two subsequent metrics could perhaps also be used, and would probably be more recommendable.
In any case, the loop was set to stop at NAdaptIter iterations (here fixed to 20).
Numerical results
The present overall scheme is supposed to serve as a case for a wider class of nonlinear calculations. Many other, similar nonlinear problems, perhaps arising from different types of physics, are awaiting to be solved, or the methods for their solution waiting to be improved. A topic in this context that seemed quite important to our perception, and not so much treated until now, is how to drive such a nonlinear algorithm in presence of mesh adaptivity. The latter didn't appear a so clear task about which it could be straightforwardly possible to make definite statements. But with systematic examination by running many computations, some patterns could be noticed indicating a kind of behaviour.
With the aid of collaborators who further developed the key codes mmg3d5 and mshmet, i.e. their authors, we first made smoothly work the trinom composed by the two and FreeFem++, the latter being the central programming language/software used, strongly FEM-oriented, in which our main code was written. This meant smoothing out their functioning as single entities, as well as their interfacing/communication with FreeFem++.
Plenty of preliminary tests were performed, in total several hundreds, may be thousand, each lasting from several hours to some days, on several cases of nematic colloidal systems. Apart from the monomer one, a lot of trials have been made also for the dimer, or for assemblies of several colloidal particles, i.e., for the so-called colloidal crystals 10 , which could be two-or three-dimensional, as for ex. 2 × 2, or 2×3, or 2 × 2 × 2, etc.
First it was recognized that the computations' behaviour of nonlinear finite elements based algorithms including mesh adaptivity is in general very parameter-dependent. Changing the value of only one parameter can quite boldly modify the behaviour of entire sets of calculations. This proved in the case of hmax, the parameter representing the maximally allowed length of tetrahedral edges, as it will be possible to notice further ahead, by comparing the computational results/measurements in the tables from Fig. 4.
After these very extensive preliminary tests, and after the above mentioned computational trinom was set and working, we performed three sets of computations on the simplest of nematic colloidal cases, the monomer, for three values of hmax. The latter indicated that the overall loop seems to be Mesh S7 S9 S12 adaptation tolAdapt errm tolAdapt errm tolAdapt errm 0. 0.5e-4 0.020 0.5e-4 0.020 0.5e-4 0.020 1.
0.5e-3 0.020 0.5e-3 0.020 0.5e-3 0.020 2.
0.5e-3 0.015 0.5e-3 0.015 0.5e-3 0.015 3.
0.5e-4 0.015 1.0e-4 0.020 1.0e-4 0.020 4.
0.5e-5 0.015 1.0e-4 0.015 1.0e-4 0.015 5.
0.5e-5 0.010 0.5e-4 0.015 0.5e-4 0.020 6.
1.0e-6 0.015 1.0e-5 0.015 0.5e-4 0.015 7.
1.0e-6 0.010 0.5e-5 0.015 1.0e-5 0.015 8.
0.5e-5 0.010 1.0e-5 0.015 9.
1.0e-6 0.010 0.5e-5 0.015 10.
0.5e-5 0.010 11.
1.0e-6 0.015 12.
1.0e-6 0.010 Table 1: Three sequences (arrays) used in calculations.
driven mostly by two factors. The first one is when (at what conditions) each new mesh adaptation is triggered. This is determined by the threshold values of the free energy relative variations, and by how are they distributed throughout the nonlinear computation.
The second factor influencing the algorithm's behaviour resulted to be how the mesh adaptivity is done, i.e., how the new mesh is rebuilt from the previous one at each mesh adaptation. This most strongly depends on the value of the solution error parameter, i.e. errm k, appearing as argument in mshmet. In fact, the call of the latter constructs the metric with which mmg3d5 then rebuilds the new mesh.
On empirical basis of the hereby presented computations, we argue that a more general algorithm regulating both factors (and possible others, which weren't explicitly detected yet) should be a loop, or possibly several nested ones, with appropriate stopping conditions. We guess this could guarantee the larger flexibility needed for more general purposes. In fact we recognized (had the confirmation), as said before, that with finite elements based nonlinear algorithms with mesh adaptivity is in general not so easy to predict exactly how a nonlinear computation will behave, thus neither how much it kcell/seq. will last before converging. Thus, to proceed by steps, we confined ourselves to arrange the threshold values in a fixed array we called tolAdapt, its constant length a priori determining how many times the mesh will be adapted, and its entries specifying at what thresholds. We set three such arrays, or sequences (see Table 1), calling them S7, S9, S12 with their integer suffix being their length, and used each of them in a set of computations with varying size of the system, determined by its coefficient kcell.
Summarizing, what mostly drives the overall nonlinear algorithm is when the mesh adaptivity is triggered, and how the respective new mesh is done. That is, at what free energy thresholds, and within what errors. An empirical proof of the fact, that it is not the same what strategy is brought into play, can be inferred from the tables in Fig. 4 , showing that computations with the sequence S9 were in almost all cases faster of those computed with the other two, S7 and S12, or in the worst case comparable -just slightly slower.
Regarding what properties the sequences of free energy threshold and mesh error values must have, it soon appeared quite evident that the values of the tolerances must be decreasing. In fact, at the start of a single simulation run, the initially computed nematic conformations are usually still quite far from the final (equilibrium) solution, i.e., the final nematic structure, and so the mesh adaptations must be more frequent. Here, at each adapted mesh there's still no real need for convergence to a higher accuracy. So the threshold values at the beginning of any sequence can be larger of those in the proceeding.
Similarly, also the sequence of values of the solution error parameter errm must tend to decrease, altough not necessarily completely monotonically. In reality, they must decrease at each (constant) threshold value.
Moreover, the present case suggested that the sequence of thresholds should be of intermediate length, as for ex. S9, i.e., neither too long, like S12, nor too short, like S7.
Therefore, since typologies of physical systems and their sizes vary in general, and the parameter sequences should in general vary with them too, in both length and values composition, it seemingly should make sense that the use of sequences in fixed arrays could, as mentioned earlier, be changed in the future by the use of one or more loops, possibly nested, satisfying suitable stopping conditions, that could drive the mesh adaptivity process optimally, or nearly so.
Conclusions and suggestions for the future
In this work a numerical method for functional minimization 11 of tensor fields on bounded, simply connected domains of Euclidean space R 3 has been developed on the Landau-de Gennes case describing confined nematic colloidal systems. Altough similar codes already exist, this finite elements based algorithm for the first time employs a systemic mesh adaptivity approach in 3D, with use of metrics (isotropic case).
Anyhow, coupling the mesh adaptation process with the nonlinear scheme shows a strong parametric dependence. For the time being we solved it by a priori setting sequences of the driving parameters into fixed arrays. Computations made for three different sequences, which were also of different length, empirically demonstrated the parametric dependence and gave some insight into the process behaviour.
For a more general solution of this mesh adaptivity-driving task, that would be appropriate for a more ample class of nematic colloidal systems and other kind of physics problems, possibly dynamical, we imagine and would like to advocate the introduction of a special auxiliary algorithm, using for ex. several nested while-loops, which would be flexible enough for such purposes.
When this and perhaps some other, more technical, questions will be optimized, the here presented methodology could be assumed to be ready for more intensive calculations, aimed at systematic research in theoretical pyhsics.
Extensions of the presented methodology could be envisaged also in directions of solving PDEs on more general manifolds [35], and/or the possible introduction of geometric integration [36] for dynamical problems.
Appendix
First and second variation of F
To implement equation (8) into our code following [25], the first and second variation of the free energy functional F need to be calculated. We compute them analytically, the first one for both Newton iteration and steepest descent, and the second one just for Newton.
Before that, we must first of all solve the task of preserving the traceless and simmetricity conditions of Q. This could have perhaps been done by the introduction of a special symmetric and traceless tensorial basis [37], which by construction preserves both conditions, as done for ex. in [9]. Alternatively, we have opted to just apply the substitutions Q 33 = −Q 11 −Q 22 and Q j i = Q ij into the free energy expressions, after which the forms of the free energy densities f e , f b and f s depend only on the five components Q 11 , Q 22 , Q 12 , Q 13 , Q 23 . All the following calculations have then been derived by taking into account only these components. So the elastic free energy density becomes
f e (∇Q) = L (∇Q ij · ∇Q ij + ∇Q 11 · ∇Q 22 ) ,(12)
and the surface energy density
f s (Q) = W (Q ij − Q 0 ij )(Q ij − Q 0 ij ) + (Q 11 − Q 0 11 )(Q 22 − Q 0 22 ) .(13)
After some symbolic computer calculations, omitted here for brevity, also f b has been transformed by substitutions into a polynomial of 4th degree in the actual five components Q ij . Without digging too deeply into formalism, we will just assume that the previous notation for the tensor field Q will from now on mean the five-tuple Q = (Q 11 , Q 22 , Q 12 , Q 13 , Q 23 ), and similarly for all the other tensor field quantities, as δQ, ϕ, and v. For a formally exhaustive and more abstract treatment in a Sobolev space setting, the reader is referred to [8].
The first variation of the Landau-de Gennes free energy F (4) is
δF (Q) = F (Q)φ = Ω ∂f e ∂∇Q ij · ∇φ ij + ∂f b ∂Q ij φ ij dV + Γp ∂f s ∂Q ij φ ij dA,(14)
where instead of δQ ij we already introduced the notation φ ij for the test functions, having well in mind that the pairs of indexes ij have only the five couples of values defined above. Variating again leads us to the second variation
δ 2 F (Q) = F (Q)φv = Ω ∂ ∂∇Q kl ∂f e ∂∇Q ij · ∇φ ij · ∇v kl + ∂ ∂Q kl ∂f b ∂Q ij φ ij v kl dV + Γp ∂ ∂Q kl ∂f s ∂Q ij φ ij v kl dA .(15)
The terms of the first variation for the elastic part are now easily obtained as ∂f e ∂∇Q ij · ∇φ ij = 2L ∇Q ij · ∇φ ij + 1 2 (∇Q 22 · ∇φ 11 + ∇Q 11 · ∇φ 22 ) , as also those for the second variation,
∂ ∂∇Q kl ∂f e ∂∇Q ij · ∇φ ij · ∇v kl = 2L ∇v ij · ∇φ ij + 1 2 (∇v 22 · ∇φ 11 + ∇v 11 · ∇φ 22 ) ,
where perhaps worth to be noted is the appearence of mixed terms. Similarly, the surface terms for the first variation are
∂f s ∂Q ij φ ij = 2W (Q ij − Q 0 ij )φ ij + 1 2 (Q 22 − Q 0 22 )φ 11 + (Q 11 − Q 0 11 )φ 22 ,
while those for the second read
∂ ∂Q kl ∂f s ∂Q ij φ ij v kl = 2W v ij φ ij + 1 2 (v 22 φ 11 + v 11 φ 22 ) ,
where again similar mixed terms appear. The concrete calculations for both variations of the concrete f b (Q) has been done with the help of the symbolic software Mathematica.
Steepest descent 7.2.1 Gradient calculation
The gradient, that we usually denote by h (here with h = −∇F (Q), following the notation of Polak [26]), is an element of the Hilbert space, in which we are seeking the solution Q * . For its calculation we use the Riesz theorem from basic functional analysis, which states that for each linear continuous functional G, mapping from a Hilbert space H into R, there exists exactly one element h ∈ H, such that the functional values G(Q) is equal to the scalar product < Q, h > for each element Q from H. In our case the functional G is the differential of the free energy functional F in a configuration Q, i.e., DF (Q). Denoting now the gradient by h, and expanding it as h = N i=1 h i φ i , i.e., by the basis functions of the Hilbert space H to which it belongs, we obtain
< Q, h >= N j=1 h j < Q, φ j > .(16)
As we want this to hold for every Q, we set Q = φ i , for each i, getting
< φ i , h >= N j=1 h j < φ i , φ j >, i = 1, . . . , N.(17)
Being < φ i , h > equal to the i-th gradient coefficient h i , and < φ i , φ j > to the ij-th element of the Gram matrix, the calculation of the Hilbert space gradient is accomplished by first computing the Gram matrix K, thus all the possible scalar products between the basis functions φ i , that is K ij = < φ i , φ j > (here we note that the Gram matrix is sparse). Besides, the negative of the differential (i.e. first variation) of F is evaluated in the momentary configuration Q, obtaining the right-hand side d = {−DF (Q)(φ i )} N i=1 of the linear system Kh = d. By solving it 12 , we obtain the gradient h = {−∇F (Q)(φ i )} N i=1 .
Scalar product choice
To implement this procedure, a choice of the scalar product must be made. Following the structure of the Landau-de Gennes free energy, we define, similarly as in [3], the scalar product as < Q, P >:= Ω 1 2 L ∇Q ij · ∇P j i + 1 2 A Q ij P j i dV + ∂Ω 1 2 W Q ij P j i dA, (18) where Einstein summation is here for now employed over all the indexes i, j = 1, 2, 3. The constant term Q 0 under the surface integral has been dropped to preserve the definition scalar product property of < Q, Q > vanishing only for Q = 0, and the constants left to mantain appropriate proportions between the addends. After applying the traceless condition Q 33 = −Q 11 − Q 22 , and symmetricity Q ij = Q j i , we obtained < Q, P > = Ω L (∇Q ij · ∇P ij + 1 2 (∇Q 11 · ∇P 22 + ∇Q 22 · ∇P 11 )) + A (Q ij P ij + 1 2 (Q 11 P 22 + Q 22 P 11 )) dV + ∂Ω W (Q ij P ij + 1 2 (Q 11 P 22 + Q 22 P 11 )) dA (19) 12 As before, we use conjugate gradients with relative tolerance = 0.5 × 10 −7 .
where, among the Einstein summation through only five index pairs, additional mixed terms in the index pairs 11 and 22 appear. For exactly traceless (and symmetric) tensor fields this is a scalar product. But for tensor fields which are numerically non-exactly traceless, it is no longer such, lacking again the property that < Q, Q > has to vanish only when Q = 0. Thus, after some experiments the mixed terms has been dropped, finally leaving
< Q, P > = Ω (L ∇Q ij · ∇P ij + |A| Q ij P ij ) dV + ∂Ω W Q ij P ij dA,(20)
where the absolute value brackets has been added to the constant A, otherwise the product could sometimes be negative, and thus obviously contraddicting the nonnegativity condition of the scalar product.
Exact line search
At each steepest descent iteration the calculated gradient gives only the direction of the maximal descent, but lets unsolved how much one should move in this direction. Thus, the iteration step must include also the choice of a proper coefficient λ ≥ 0. This can be crucial for the convergence itself, as for the time dependence of the iteration. The optimal choice for λ is the solution of the minimization problem λ * = arg min λ {F (Q + λh)},
which is called exact line search. This can seldom be too expensive, thus leading to a preference for approximative methods as for example the Armijo method [26]. But in the present case it leads to a not too complex or expensive situation. For the Landau-de Gennes functional the problem (21) means
F (Q + λh) = Ω [f e (∇Q + λ∇h) + f b (Q + λh)] dV + Γp f s (Q + λh)dA,(22)
which can be quite easily expanded and collected with regard to powers of λ, here done once with Mathematica, and then transcribed into the FreeFem++ code. After obtaining the coefficient terms by integration (here done within the code), one in fact gets a polynomial of fourth order in dependence of λ: p(λ) = a 0 + a 1 λ + a 2 λ 2 + a 3 λ 3 + a 4 λ 4 .
Extremal values are found when p (λ) = 0, that is, when p (λ) = a 1 + 2a 2 λ + 3a 3 λ 2 + 4a 4 λ 3 = 0, the three zeros of which are found with a GSL numerical procedure [34]. The minimal root between them is taken as the optimal λ * , after a verification of the positiveness of p in it as the minimum condition. During concrete computations λ * usually ranged around values between 0.01 and 0.3, while the other two roots were almost always pairs of complex conjugated zeros, thus not feasible candidates.
Figure 3 :
3Zoomed enlargments of the mesh (left) and the corresponding nematic director field (right) from Figs. 1 and 2 (both left) around the (Saturn ring) topological defect. In general, the volume (tetrahedral) mesh is more refined where the Hessian of the solution, or other appropriate functions, is larger (e.g., around the defect), and/or where the domain geometry varies (e.g., near the sphere's surface). // INITIALIZATION: Initialize(Sh, f, Th, Qh) { Sh= Construct_Surface_Mesh();// Constructs main surface mesh. Th= Tetgen(Sh, fine_density);// Fine tetrahedrization. f= Set_Initial_Mesh_Density(Sh);// Sets initial mesh density. Th= Tetgen(Sh, f); // Tetrahedrizes with density f. Qh= Set_Starting_Guess(Th); // Starting guess is set. return Th, Qh; }
Figure 4 :
4Computation times for values of hmax = 25, 50, 75. In all three cases the sequence S9 behaves better than the other two (almost everywhere).
Thx= Th; // Declares and initializes new mesh variable. scFields= {Qh, S, DF}; // Scalar fields for metric calculus.Algorithm 3
// MESH ADAPTATION:
Adapt_Mesh(Th, Qh; hmax, errm_k)
// Other possible parameters:
{
// hmin, hgrad (here fixed).
mesh3 for (j=1; j<=NAdaptIter; ++j) {
fespace Vhx(Thx, P13d);
// Declares new FE space.
Vhx M= mshmet(Thx, scFields, hmin, hmax, errm_k);//Metric.
Thx= mmg3d5ljll(Thx, M, hmin, hmax, hgrad); // Remeshing.
if (meshes close enough) break;
// Loop-exit condition.
}
return Th=Thx, Qh;
}
Table 2 :
2Numbers of vertices used in calculations for hmax = 25. Numbers
for hmax = 50 and 75 were similar, mostly slightly decreasing for a couple of
percents with increasing values of hmax.
Like, e.g., S ≈ 0.53 for pentylcyanobiphenyl (5CB), a well-known nematic material, extensively used in physical experiments, with nematic phase at room temperature range, the properties of which (values of physical constants) have also been used in the hereby simulations.
Here only spherical, but in general much more complicated shapes are possible.8 Which also justifies the choice of the free energy F as the appropriate thermodynamic potential.9 In both cases the nematic system is still fluctuating[22], employing a statistical equilibrium.
In these cases particular attention had to be brought to the setting of the starting guess.
As the minimization basically consists in resolving nonlinear systems of PDEs, the scheme can be used for them as well, thus regarded as more general.
Acknowledgments. Miha Ravnik, Daniel Svenšek, and George Mejak (University of Ljubljana) are acknowledged for useful discussions, and Pierre-Henri Tournier (UPMC, Paris) for help with debugging of the FreeFem++ related internal and external modules.
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|
[] |
[
"Shielding the vulnerable in an epidemic: a numerical approach",
"Shielding the vulnerable in an epidemic: a numerical approach"
] |
[
"Guus Balkema \nUniversity of Amsterdam\n\n"
] |
[
"University of Amsterdam\n"
] |
[] |
The death toll for Covid-19 may be reduced by dividing the population into two classes, the vulnerable and the fit, with different lockdown regimes. Instead of one reproduction number there now are four parameters. These make it possible to quantify the effect of the social distancing measures. There is a simple stochastic model for epidemics in a two type population. Apart from the size of the population of the vulnerable and the fit, and the initial number of infected in the two classes, only the four reproduction parameters are needed to run the two type Reed-Frost model. The program is simple and fast. On a pc it takes less than five minutes to do a hundred thousand simulations of the epidemic for a population of the size of the US. Epidemics are non-linear processes. Results may be counterintuitive. The average number of vulnerable persons infected by an infectious fit person is a crucial parameter of the epidemic in the two type population. Intuitively this parameter should be small. However simulations show that even if this parameter is small the death toll may be higher than without shielding. Under certain conditions increasing the value of the parameter may reduce the death toll. The article addresses these blind spots in our intuition.
| null |
[
"https://arxiv.org/pdf/2010.00959v1.pdf"
] | 222,125,115 |
2010.00959
|
67246fcc294d4976bbf913b750ac13bcec967082
|
Shielding the vulnerable in an epidemic: a numerical approach
October 5, 2020
Guus Balkema
University of Amsterdam
Shielding the vulnerable in an epidemic: a numerical approach
October 5, 2020
The death toll for Covid-19 may be reduced by dividing the population into two classes, the vulnerable and the fit, with different lockdown regimes. Instead of one reproduction number there now are four parameters. These make it possible to quantify the effect of the social distancing measures. There is a simple stochastic model for epidemics in a two type population. Apart from the size of the population of the vulnerable and the fit, and the initial number of infected in the two classes, only the four reproduction parameters are needed to run the two type Reed-Frost model. The program is simple and fast. On a pc it takes less than five minutes to do a hundred thousand simulations of the epidemic for a population of the size of the US. Epidemics are non-linear processes. Results may be counterintuitive. The average number of vulnerable persons infected by an infectious fit person is a crucial parameter of the epidemic in the two type population. Intuitively this parameter should be small. However simulations show that even if this parameter is small the death toll may be higher than without shielding. Under certain conditions increasing the value of the parameter may reduce the death toll. The article addresses these blind spots in our intuition.
Introduction
The reproduction number for Covid-19 lies between 2 and 3 if we take no action to stop it spreading. With proper measures such as social distancing it may be reduced to below one. It is known that infections are more lethal for the elderly (and for persons who suffer from obesity, diabetes, high blood pressure) than for healthy young persons 1 . These two facts suggest that a social distancing policy which takes into account the difference in risk for the vulnerable and for the fit might be effective in reducing the overall mortality.
So consider a population consisting of a million vulnerable persons and two million fit persons. Assume that the mortality is ten times as high for the vulnerable as for the fit.
To be concrete, assume an Infection Fatality Rate (IFR) of 0.01 for the vulnerable and 0.001 for the fit. This yields an overall IFR of 0.004 = (0.01 + 2 * 0.001)/3.
Assume a two type model. A vulnerable person infects r v = 0.7 persons on average, a fit person r f = 1.3. Ten vulnerable persons and twenty fit, all infectious, on average will yield 7 + 26 = 33 new infections (corresponding to the overall reproduction number r 0 = 1.1). Start with a hundred infections among the vulnerable and two hundred among the fit. The population is compartmentalized. If the compartments are watertight, there is no contact between the vulnerable and the fit. The epidemic will die out among the vulnerable since r v = 0.7 < 1, but it will be more severe among the fit than in the corresponding homogeneous model with reproduction number r 0 = 1.1 since r f = 1.3 > 1.1. Computations show that the total number of deaths is lower than in a homogeneous population.
In a more realistic model the expected number r f,v of vulnerable persons infected by an infectious fit person and the number r v,f of fit persons infected by an infectious vulnerable person are positive. So suppose ten infectious fit persons infect on average eleven fit persons and two vulnerable persons, and ten infectious vulnerable persons infect five vulnerable persons and two fit persons. This "standard model" will play a prominent role in the discussion below. The infection rate between the two groups is low. Such a society may be said to shield the vulnerable. How effective is this shield? Simulations show that in the heterogeneous society there will be more deaths than in a homogeneous population
INTRODUCTION
3 with reproduction number r 0 = 1.1. In the homogeneous population the death toll has a mean value of 2124; in the two type model the mean is ten per cent higher, 2347.
The result is not implausible. There is a considerable increase in the total number of infections among the fit. The positive cross infection r f,v = 0.2 from the fit to the vulnerable has the effect that the vulnerable are pulled along in this more severe epidemic, yielding a higher overall mortality. Compartmentalization increases the death toll. It is not clear how the transitions r f,v and r v,f are related. Intuitively for nursing homes one might interpret r f,v as measuring infections caused among the elderly by visits by the family, and r v,f as infections among the nurses and staff caused by illness among the elderly. Section 4 will look at this issue more closely.
The main purpose of the paper is to exhibit possible adverse effects of shielding in a heterogeneous population. This aim has been achieved already above by mentioning the results of the computations for the standard model. For the given values r v = 0.7 and r f = 1.3 above, and r v,f = 0.2, a large value of r f,v will reduce the total mortality. Based on that result the government might consider launching a campaign: "Tonight don't meet at the pub; visit your granny instead."
The benefits of social distancing measures which differentiate between the fit and the vulnerable are incontestable. This paper shows that policies have to be chosen with care to avoid adverse affects. Here we should mention a different beneficial effect of variations in the reproductive number across a population. Heterogeneity may reduce the herd immunity threshold 2,3 .
For an introduction to the mathematical background see 4,5 . The exposition below is self-contained. It relies on simulations. Readers with some experience in R are invited to use their skill to explore the effects of variations in the reproduction matrix.
THE MODEL AND ITS PROGRAM
4
Section 2 introduces the binomial Reed-Frost model. Section 3 presents the results.
These are discussed in Section 4. Section 5 contains our conclusion. The Appendix contains two sections: a non technical explanation of the decrease in mortality associated with large values of r f,v and a discussion of the role of eigenvalues and eigenvectors in the two-type Reed-Frost model.
The model and its program
In a homogeneous population of size n 0 with initially i 0 infections the probability of noninfection for any susceptible member of the population is q = q i 0 0 where q 0 = 1 − p 0 and p 0 = r 0 /n 0 is the probability of infection in a homogeneous population of size n 0 with reproduction number r 0 . The total number of new infections i among the n = n 0 − i 0 susceptibles is binomial-(n, p) with p = 1 − q. This yields a recursion starting with n = n 0 , i = i 0 . In R the recursion consists of three commands: n<-n-i; p<-1-q0^i; i<-rbinom(1,n,p);
The sequence of commands runs while i is positive. The total number infected is j = n 0 − n. This is the binomial or Reed-Frost model. In her exposition of this model in 1952 seconds on a ten year old iMac OS 10.11.6 with a 3.06 GHz Intel Core processor and 1067
MHz DDR3 memory modules in two out of four memory slots. The program determines the mean and sd of j, the total number infected. If the initial number of infections is small the epidemic may die out, but for i 0 = 300 initial infections each of the 10 5 simulations gives rise to a full blown epidemic.
THE MODEL AND ITS PROGRAM
5
The program for the multitype binomial model is similar. Start with a population n 0 = (10 6 , 2 * 10 6 ) of a million vulnerable and two million fit persons and i 0 = (100, 200) initial infections, a hundred among the vulnerable and two hundred among the fit. The probability that the infection will die out may be neglected. The reproduction number r 0 now becomes a matrix R which is akin to the transition matrix in a Markov chain where the kth row contains the probabilities p k,m of a transition from state k to m. We first look at a specific case, the standard model discussed in the introduction:
R = r v,v r v,f r f,v r f,f = 0.5 0.2 0.2 1.1 .
(2.1)
The first row states that ten infectious vulnerable persons will infect on average five vulnerable and two fit persons. The second row states that similarly ten infectious fit persons will infect on average two vulnerable and eleven fit persons.
For a vulnerable person the probability of not being infected is
q v = (1 − r v,v /n v ) iv * (1 − r f,v /n v ) i f . AQ = 1 − R[1, 1]/n 0 [1] 1 − R[1, 2]/n 0 [2] 1 − R[2, 1]/n 0 [1] 1 − R[2, 2]/n 0 [2] . (2.2)
The program for a simulation of the epidemic in the multitype model then is:
n<-n0; i<-i0; while(max(i)>0){ n<-n-i; q1<-Q[1,1]^i[1]*Q[2,1]^i[2]; q2<-Q[1,2]^i[1]*Q[2,2]^i[2]; i[1]<-rbinom(1,n[1],1-q1); i[2]<-rbinom(1,n[2],1-q2)} j<-n0-n;
The R program for the two type binomial Reed − Frost model.
RESULTS
3 Results
With r v = 0.7 and r f = 1.3 the reproduction matrix has the form The plot for r v = 0.8 and r f = 1.1 with r 0 = 1, Figure 3, left, paints a darker picture.
R = R(a, c) = 0.7 − a a c 1.3 − c R(0.2, 0.2) = 0.5 0.2 0.2 1.1 .
For a = 0.2 (green curve) and c = 0.1 the overall death toll is twice as high as in the homogeneous case. Mortality among the fit is one third of the total rather than one sixth as in the homogeneous case. The reason for this behaviour? In the heterogeneous case there will be a full blown epidemic among the fit because r f = 1.05 > 1. The constant c = r f,v > 0 will pull the vulnerable into the epidemic. The overall reproduction number here has the critical value r 0 = 1. In the homogeneous model epidemics will be short and die out.
The right side of Figure 3 shows the subcritical case, r 0 = 0.95. The adverse effects of shielding the vulnerable here are already apparent for c = 0. The mortality curves are decreasing. The excess death toll is small, less than ten. It is of little interest since it reflects the size of the contiguous initial infection rather than of the population.
RESULTS
9
In all three figures the situation brightens for c in the upper half of the interval [0, 0.8].
The total mortality decreases dramatically as c → 0.8. The low mortality for R(0.2, 0.8) on first sight is a mystery. A large value r f,v = 0.8 indicates many infections from fit to vulnerable. Why should the mortality almost vanish? Restrict attention to the fit. First assume a = r v,f = 0. The top row of R then is (0.7, 0), but if we turn to the fit we see that the infections will die out since 0.5 < 1. There also are infections from the fit to the vulnerable, but these do not concern the fit since we have assumed that the vulnerable do not infect the fit, a = r v,f = 0. The infections among the vulnerable will also die out (since r v,v = 0.7 < 1), apart from the import from the fit, but the import will die down as the epidemic among the fit dies out. If a = r v,f is positive this will not alter the situation as long as a is small. Section 6 contains a more detailed analysis.
Discussion
One cannot argue with the result of a computation. One can argue about the interpretation. We restrict the discussion to two topics.
1) How realistic are our assumptions on the entries of the reproduction matrix?
In the basic model the reproduction matrix is
R = R(a, c) = r v,v r v,f r f,v r f,f = 0.7 − a a c 1.3 − c .
A vulnerable infectious person infects on average 0.7 persons of whom a are fit; a fit infectious person infects on average 1.3 persons of whom c are vulnerable. Infection is due to contact. Social contacts of the healthy and young are more varied and more intense than for the old or sick. This difference has increased as the vulnerable have become more aware of their vulnerability. Human beings are social animals, but older people are perhaps better able to endure solitude and live with their thoughts and memories than the young. A factor 1.3/0.7 ≈ 2 may be excessive. In the example for the critical case, r 0 = 1, the factor is less, 1.05/0.9. The effect is similar.
In mathematics it is good practice to vary one variable at a time. The effect depends on the variables which are kept constant. If one entry of the reproduction matrix goes down and the other three are constant the epidemic will be less severe. In a partition the lockdown for the vulnerable becomes stricter while at the same time it is relaxed for the fit. This approach makes it natural to assume r f and r v to be constant. We then A healthy young person will cough with more force than a feeble old person. This may make the fit more infectious than the vulnerable. Thus there are indications that adults infect children but children hardly infect adults 7,8,9 . (There also is contrary evidence 10 .)
There are more reasons for a lack of symmetry. If the vulnerable are tested at regular intervals and visits are only allowed when the test result is negative this will not affect r f,v , but it will reduce the value of r v,f and hence increase the parameter t in Figure 4.
A representative list of pairs, infector and infectee, together with age and medical condition, might help to determine the role of the values of t > 0.5 in figure 4.
2) Do the results apply to real life?
The proportion of vulnerable to fit is 1:2. This is realistic for the Netherlands and In the more realistic model the infection curves in Figure 6 will have different shapes, but their relation to the dotted lines associated with the eigenvectors will be the same; the positions of the points describing the total number infected will change, but the shape of the curve which they form might well again be a question mark. This is speculation. Whether the suppositions hold can only be determined by doing the necessary simulations and observations.
Conclusion
The results presented in this article are indicative rather than descriptive. They suggest that our intuition fails us in understanding how the parameters of the reproduction matrix affect the outcome of an epidemic in a population divided into two classes, the vulnerable and the fit. It only takes a little effort to run the Reed-Frost model on a pc and simulate two type epidemics. Speed makes this model a viable alternative to our intuition. It may be a better guide to reality.
References
[1] Levin, A.T., Cochran, K.B. and Walsh, S.P. Assessing the age-specificity of infection fatality rates for COVID-19: Meta-analysis, and public policy implications. medRxiv, www.nber.org/papers/w27597 (2020).
[2] Britton, T., Ball, F. and Trapman, P. A mathematical model reveals the influence of Consider a billowing cloud of viruses in the region of the fit, a cloud which doubles in size every few days. In the region of the vulnerable the cloud of viruses shrinks and will fade away. Now assume the excess of viruses produced in the region of the fit every day is diverted to a region which is less hospitable and where the viruses will die out. The cloud in the region of the fit no longer grows. If one increases the part which is diverted by a fraction the cloud of viruses in the region of the fit will shrink at an exponential rate. This will be the case even if some of the viruses manage to find their way back to the region of the fit.
Can one arrange things such that in the region of the vulnerable the cloud will fade away in spite of the influx from the region of the fit? One may argue that the influx is only temporary since the cloud in the region of the fit will fade away at an exponential rate, and so will the fraction which is diverted to the region of the susceptible. The situation becomes less clear if a fraction of the cloud above the region of the susceptible manages to return to the region of the fit. In order to handle the situation where there is traffic between the two regions in both directions we have to be more specific. It helps to look at steady states. We give two examples. In both cases the top row of the matrix R is
i k+1 = i k * R * D k+1 D k+1 = diag( n k+1 / n 0 ) n k+1 = n k − i k . (7.8)
The new value i k+1 of the vector i is written as a linear transformation R k+1 of the old
value i k = (i k [v], i k [f ])
, where R k+1 = RD k+1 is a modulation of the reproduction matrix.
The key to linear dynamical systems x k+1 = x k R is the left eigenvector associated with the largest eigenvalue. In the multitype epidemic the maximal eigenvalue of the reproduction matrix R determines the severity of the epidemic; the corresponding left eigenvector determines the proportion of vulnerable to fit among the infected.
These words should not be taken literally. The relation between the eigenvalues and eigenvectors of the reproduction matrix R and the course of the epidemic is not perfect as one sees on comparing the curves on the left side and the right side in Figure 2. The coordinates associated with the left eigenvectors of R make R diagonal but destroy the diagonality of the modulator D k+1 in the recursion (7.8). decreases from slightly more than a hundred thousand to slightly more than a thousand. The death toll for R(0.2, c) in the Reed-Frost model and three approximations
Shielding the vulnerable may be counterproductive. That is what this paper is meant to show. For three values of r 0 , 1.1, 1.0 and 0.95, we choose values of r v and r f which satisfy (r v + 2r f )/3 = r 0 . We then plot the mortality as the transition rate r f,v from fit to vulnerable varies between 0 and 0.8. This is done for various values of r v,f . Figures 2, left side, and 3 show the effect on the mortality.
Abbey 6 writes: "Epidemics can be calculated stepwise from this model, with the aid of random numbers and a table of the cumulative binomial distribution (National Bureau of Standards, 1949)" and then explains how to use the seven digit tables of random numbers to create realizations of binomial variables and by repetition realizations of the epidemic, closing with the advice: "For values beyond the range of the binomial tables, the Poisson or normal distributions may be used as approximations to the binomial." Seventy years later for population size n 0 = 3 * 10 6 a batch of a hundred thousand simulations takes 72
similar expression hold for a fit person. The counterpart of the constant q 0 = 1 − r 0 /n 0 in the homogeneous model is the matrix:
Figure 1 :
1successive pairs i, starting with i 0 = (100, 200), and plot the infections i k [v] and i k [f ] to obtain logistic curves as in Figure 1. Two simulations of the epidemic with the reproduction matrix R in (2.1). The green curve describes the epidemic for the vulnerable, the red curve for the fit.For a batch of a hundred thousand simulations of the epidemic we compute the mean µ of j = n 0 − n, the total number of infections, the covariance matrix, the sd σ of the components j v and j f and the correlation ρ: µ = (180387, 543119) σ = (1362, 3988) ρ = 0.88. The mean number of deaths is 1803.9 + 543.1 = 2347.0. The sd of the mean is 0.054.Compare this to the situation of complete mixing.
Figure 2 :Figure 3 :
23two parameters. Recall that the parameter a = r v,f denotes the number of fit persons infected by an infectious vulnerable person; the parameter c = r f,v the number of vulnerables infected by an infectious fit person. The parameter c varies over [0, 0.8] for six values of a, a = 0, 0.1, . . . , 0.5 in Figure 2, left. This figure and Figure 3 show plots of the number of deaths for the whole population (solid curves) and for the vulnerable (dotted curves). The solid and dotted horizontal lines indicate the number of deaths in the corresponding homogeneous models with r 0 = 1.1 (Figure 2, left), r 0 = 1 (Figure 3, left), and r 0 = 0.95 (Figure 3,right). For r 0 = 1.1 the homogeneous model yields a mortality of 2123.9 of which the majority, 5/6, is vulnerable. The mortality for the homogeneous model with r 0 = 1 is 168.7 and for r 0 = 0.95 it is 23.4. This paper focuses on the supercritical case r 0 0.0 (black), 0.1 (red), 0.2 (green), 0.3 (blue), 0.4 (brown), 0.5 (purple) 0.0 (black), 0.1 (red), 0.2 (green), 0.3 (blue), 0.4 (brown), 0.5 (purple) The two type model with reproduction matrix R(a, c) in (3.5) for 0 ≤ c ≤ 0.8 and a = 0, 0.1, . . . , 0.5. On the left the mortality; on the right the maximal eigenvalue. The deaths are obtained from the infections as binomial variables: For the left side of Figure 2 the mean is computed over a hundred thousand simulations for the matrices R(a, c), with a in 0 : 5/10 and c in 0 : 40/50. black), 0.1 (red), 0.2 (green), 0.3 (blue), 0.4 (brown), 0.5 (purple) black), 0.1 (red), 0.2 (green), 0.3 (blue), 0.4 (brown), 0.5 (purple) The mortality for R a,c with 0 ≤ c ≤ 0.8 and a in 0 : 5/10. On the left r v = 0.9, r f = 1.05 and r 0 = 1. On the right r v = 0.85, r f = 1.0 and r 0 = 0.95.
Figure 4 :
4On the left j a [v](t) (dotted) and j a [f ](t) (dashed), the number of infections among the vulnerable and the fit, and 100d a (t) (solid), the number of deaths, for R(a, (0.3 + 0.8a) * t). On the right the differences with the standard graph, a = 0.2. The six mortality curves in the left part of Figure 2 all have the same shape and the same maximum. (So too for the six curves in the subfigures in Figure 3.) This suggests that the six curves can be derived from a common curve by suitable affine transformations of the horizontal coordinate. Introduce a new variable, t, and write c = (0.3+0.8a) * t. The 4 DISCUSSION 10 left side of Figure 4 plots j a [v](t) (dotted) and j a [f ](t) (dashed), the number of infections among the vulnerable and the fit, and a multiple of the death toll, 100d a (t) (solid) for the two type binomial model with reproduction matrix R a (t) = R(a, c) 0 ≤ t ≤ 1.5, a = 0, 0.1, . . . , 0.5; c = (0.3 + 0.8a) * t (3.6)in the same colours as theFigures 2 and 3. The curves almost coincide. They all seem to vanish for t → 1.5. The role of the simple affine transformation 3 + 8a is not clear.
results of the epidemic for various values of r f,v and r v,f . The homogeneous population with reproduction number r 0 = (r v + 2r f )/3 is the benchmark. This value of r 0 is somewhat arbitrary. It is not the reproduction number of the two type model.That depends on the proportion between the vulnerable and the fit among the infected.
Figure 6
6shows that this proportion varies in the course of the epidemic. The standard value r v,f = a = 0.2 is on the low side, but the graphs of the mortality for a = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5 all have the same shape. The assumption that one is free to vary c = r f,v is more problematic. The relation between the cross infection constants a = r v,f and c = r f,v is obscure. Contact is symmetric. Person A is within two meters of person B if and only if person B is within two meters of person A. If one replaces the infection rates r v,f and r f,v by contact rates c v,f and c f,v then symmetry of contact implies the conservation law: n f c f,v = n v c v,f . Contact is symmetric, but its effect on infection not.
perhaps for Italy and Japan. In countries with a younger population a proportion of 1:3 might be more appropriate. The population size is immaterial. The number of initial cases (100, 200) reflects the proportion of the vulnerable to the fit and is chosen large in order that the simulations all show full blown epidemics when the reproduction number is 1.05 or higher. The IFR of 0.004 is on the low side. It does not affect the results of the paper. The factor 10 between the IFR of 0.01 for the vulnerable and 0.001 for the fit is on the high side. The reader is invited to investigate the effect of increasing the IFR for 12 the fit to 0.004. What happens if one replaces the Reed-Frost model by a more realistic model? The assumption in the homogeneous model that everyone has the same probability of being infected is not realistic. In the two type model there is a reproduction matrix with four entries. If the more realistic model allows one to specify such a matrix, then -as in the Reed-Frost model -the maximal eigenvalue and the corresponding left eigenvector will determine the severity of the epidemic and the proportion of vulnerable to the fit among the infected.
non-technical explanation of the decrease in mortality The conclusion that the mortality may be reduced by increasing the transmission of Covid-19 from the fit to the vulnerable is hard to swallow. For a mathematician the eigenvalue argument may be convincing. For computing the eigenvalues and eigenvectors high school mathematics suffices. One has to solve a quadratic equation and a system of two linear equations. These calculations will not convince the general reader. Here is a more intuitive argument:
( 0 .Figure 5 :
055, 0.2). The bottom row is (c, 1.3 − c). Realizations of the epidemic i k (v) (green) and i k (f ) (red) for r v,f = 0.2 and r f,v = 0.5 with a mean of 214.8 + 21.8 deaths on the left and r f,v = 0.8 with a mean of 23.1 + 1.3 deaths on the right. Example 1 First assume c = 0.5. Consider a situation where a thousand fit are infectious and a thousand vulnerable. The thousand vulnerable infect 500 vulnerable and 200 fit; the thousand fit infect c * 1000 = 500 vulnerable and (1.3 − c) * 1000 = 800 fit. The result is a thousand new infections among the vulnerable and a thousand among the fit. At first sight nothing changes. But since there is a growing number of immune persons the new number of infectious persons decreases at an ever accelerating rate and the epidemic will die out. ♦ Example 2 Take c = 0.8. Let there be two thousand infectious vulnerable and a thousand fit. A calculation as above shows that these infect one thousand eight hundred vulnerable and nine hundred fit. The proportion 2 : 1 between the vulnerable and the fit is preserved. In this case the epidemic dies out at an exponential rate. ♦ The pairs (1000, 1000) and (2000, 1000) are called eigenvectors of the corresponding matrices R(0.2, 0.5) and R(0.2, 0.8) and the factors 1.0 and 0.9 linking the number of new infections to the old are called the eigenvalues. Figure 5 shows a realisation of each of 7 EIGENVALUES AND EIGENVECTORS OF THE REPRODUCTION MATRIX 16 these two type binomial epidemics. The proportions (1 : 1) and (2 : 1) are clearly visible in the number of infections of the vulnerable and the fit. 7 Eigenvalues and eigenvectors of the reproduction matrix Replace the binomial variables in (2.3) by their expectations to obtain the deterministic mean process with (i[1], i[2]) = (n[1] * (1 − q 1 ), n[2] * (1 − q 2 )). (7.7) The sequence of successive infections i k = (i k [v], i k [f ]) form the infection curve. The entries 1 − Q[i, j] = R i,j /n 0 [j] in (2.2) are of the order of 10 −6 . This suggests the approximation (1 − (1 − p) i ) ≈ ip and hence the recursion
Figure 6
6attempts to give an impression of the changes in the epidemic as the parameter c in the matrix R(0.2, c) in (3.5) varies from 0 to 1 over the twelve values listed above the plot. For each c the infection curve starts at i 0 = (100, 200) and finally moves off 7 EIGENVALUES AND EIGENVECTORS OF THE REPRODUCTION MATRIX 17 0.00, 0.01, 0.02, 0.04, 0.10, 0.20, 0.25, 0.30, 0.40, 0.50, 0.60, 1.00
Figure 6 :
6Infection curves for R(0.2, c) for various values of c, and the total number of infections, (j c [v], j c [f ]).to (0, 0) after attaining values which may be in excess of ten thousand, and the dotj c = (j c [v], j c [f ])depicts the total number of infections. The logarithmic scale on the two axes allows us to compress all this information into one figure. The logarithmic scale transforms rays y = cx in the positive quadrant into lines log 10 y = log 10 x+log 10 c parallel to the diagonal. Each dotted line corresponds to the ray through the left eigenvector e c associated with the maximal eigenvalue of R(0.2, c). Note that the quotient i k [v]/i k [f ] starts at 1/2 for k = 0, quickly approaches the quotient e c [v]/e c [f ] of the eigenvector, and then slowly backs off towards 1/2. As a result the quotient j c [v]/j c [f ] lies between e c [v]/e c [f ] and 1/2. Only the light blue infection curve (for c = 0.3) lies on the line associated with the eigenvector and so does the light blue point depicting the total number of infections. That is because the vector (1, 2) is a left eigenvector of the matrix R(0.2, c) for c = 0.3. The coloured points form a question mark. As c moves away from zero the total number of infections among the vulnerable, j c [v], increases rapidly, while the number among the fit decreases. The logarithmic scale makes it impossible to see what happens to the sum j c [v] + j c [f ]. However beyond the orange point (c = 0.25) both j c [v] and j c [f ] decrease as c increases. The number of infections among the vulnerable, j c [v]
A
policy directed at shielding the vulnerable should try to maximize the value of c = r f,v for the given boundary conditions r f = 1.3, r v = 0.7, r v,f = 0.2. Lines parallel to the diagonal through the coloured points drift to the South East as c increases. The proportion of deaths for the vulnerable and the fit, d c [v]/d c [f ] = 10j c [v]/j c [f ], increases steadily from 0.002 to 22 as c moves from 0 to 1. This may be expressed in the oracular rule: "Sacrifice the vulnerable to save the vulnerable." For the homogeneous Reed-Frost model there exists 4 a simple and intuitive good approximation τ = τ (r 0 ) to the fraction j/n 0 of infected when the reproduction number r 0 exceeds 1. It solves the equation 1 − τ = e −r 0 τ . Such a simple expression for the number infected is not available for multitype models. One may use τ = τ (ρ), where ρ is the largest eigenvalue of the reproduction matrix R, to estimate the total number of infections and use the corresponding left eigenvector to divide these among the vulnerable 7 EIGENVALUES AND EIGENVECTORS OF THE REPRODUCTION MATRIX 19 and the fit:(j v , j f ) = i 0 + τ (ρ) * (n 0 [v] − i 0 [v] + n 0 [f ] − i 0 [f ]) * e/(e[1] + e[2]).(7.9) This ad hoc estimate is not very accurate.Our benchmark is the mean value of (j v , j f ) for a hundred thousand simulations of the stochastic Reed-Frost model (RF). The table below lists estimates of the death toll for the two type model with reproduction matrix R(0.2, c) for various values of c. It gives anindication of the accuracy of the three models E, L, D defined by (7.9), (7.8) and(7.7)compared to the stochastic model RF in (2.3).
AcknowledgementsThe author thanks Alex van den Brandhof for his help in making the article accessible to the general reader.
. Science. 3692to SARS=CoV-2. Science 369, 846-849 (2020).
Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold. medRxiv. M G M Gomes, doi.org/10.1101/2020.04.27.20081893Gomes, M.G.M. et al. Individual variation in susceptibility or ex- posure to SARS-CoV-2 lowers the herd immunity threshold. medRxiv, doi.org/10.1101/2020.04.27.20081893 (2020).
Stochastic epidemic models: a survey. T Britton, Mathematical Biosciences. 225Britton, T. Stochastic epidemic models: a survey. Mathematical Biosciences 225, 24-35 (2010).
Susceptibility sets and the final outcome of collective Reed-Frost epidemics. F Ball, Methodol. Comput. Appl. Probab. 21Ball, F. Susceptibility sets and the final outcome of collective Reed-Frost epidemics. Methodol. Comput. Appl. Probab. 21, 401-421 (2019).
An examination of the Reed-Frost theory of epidemics. H Abbey, Human Biology. 24Abbey, H. An examination of the Reed-Frost theory of epidemics. Human Biology 24, 201-233 (1952).
De rol van kinderen in de transmissie van SARS-CoV-2. W Hoek, Van Der, Ned. Tijdschr. Geneeskd. 164Hoek, W. van der, et al. De rol van kinderen in de transmissie van SARS-CoV-2. Ned. Tijdschr. Geneeskd. 164:D5140, 1-7 (2020).
Transmission of SARS-CoV-2 by children. J Merckx, J A Labrecque, J S Kaufman, Dtsch Artzebl Int. 117Merckx, J., Labrecque, J.A. and Kaufman, J.S. Transmission of SARS-CoV-2 by children. Dtsch Artzebl Int 117, 33-34 (2020).
Role of children in household transmission of COVID-19. Archives of Disease in Childhood dx. J Kim, doi.org/10.1136/archdischild-2020-319910Kim, J. et al. Role of children in household transmission of COVID-19. Archives of Disease in Childhood dx.doi.org/10.1136/archdischild-2020-319910, 1-3 (2020).
Transmission dynamics of COVID-19 outbreaks associated with child care facilities -Slat Lake City. A S Lopez, MMWR Morb. Mortal. Wkly Rep. 69Lopez, A.S. et al. Transmission dynamics of COVID-19 outbreaks associated with child care facilities -Slat Lake City, Utah, April -July 2020. MMWR Morb. Mortal. Wkly Rep. 69, 1309-1323 (2020).
|
[] |
[
"Quantum tunneling of thermal protons through pristine graphene Quantum tunneling of thermal protons through pristine graphene",
"Quantum tunneling of thermal protons through pristine graphene Quantum tunneling of thermal protons through pristine graphene"
] |
[
"Igor Poltavsky ",
"Limin Zheng ",
"Majid Mortazavi ",
"Alexandre Tkatchenko ",
"Igor Poltavsky \nPhysics and Materials Science Research Unit\nUniversity of Luxembourg\nLuxembourg City L-1511Luxembourg\n",
"Limin Zheng \nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-614195BerlinGermany\n",
"Majid Mortazavi \nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-614195BerlinGermany\n",
"Alexandre Tkatchenko \nPhysics and Materials Science Research Unit\nUniversity of Luxembourg\nLuxembourg City L-1511Luxembourg\n"
] |
[
"Physics and Materials Science Research Unit\nUniversity of Luxembourg\nLuxembourg City L-1511Luxembourg",
"Fritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-614195BerlinGermany",
"Fritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-614195BerlinGermany",
"Physics and Materials Science Research Unit\nUniversity of Luxembourg\nLuxembourg City L-1511Luxembourg"
] |
[
"Citation: The Journal of Chemical Physics"
] |
Engineering of atomically thin membranes for hydrogen isotope separation is an actual challenge which has a broad range of applications. Recent experiments [M. Lozada-Hidalgo et al., Science 351, 68 (2016)] unambiguously demonstrate an order-of-magnitude difference in permeabilities of graphene-based membranes to protons and deuterons at ambient conditions, making such materials promising for novel separation technologies. Here we demonstrate that the permeability mechanism in such systems changes from quantum tunneling for protons to quasi-classical transport for heavier isotopes. Quantum nuclear effects exhibit large temperature and mass dependence, modifying the Arrhenius activation energy and Arrhenius prefactor for protons by more than 0.5 eV and by seven orders of magnitude correspondingly. Our findings not only shed light on the separation process for hydrogen isotope ions passing through pristine graphene but also offer new insights for controlling ion transport mechanisms in nanostructured separation membranes by manipulating the shape of the barrier and transport process conditions. Published by AIP Publishing.
|
10.1063/1.5024317
|
[
"http://orbilu.uni.lu/bitstream/10993/35796/1/Quantum%20tunneling%20of%20thermal%20protons%20through%20pristine%20graphene.pdf"
] | 46,931,612 |
1605.06341
|
023ff9607b3abf71c8cd5bbd299e6fb4d1e1c771
|
Quantum tunneling of thermal protons through pristine graphene Quantum tunneling of thermal protons through pristine graphene
2018. 2018
Igor Poltavsky
Limin Zheng
Majid Mortazavi
Alexandre Tkatchenko
Igor Poltavsky
Physics and Materials Science Research Unit
University of Luxembourg
Luxembourg City L-1511Luxembourg
Limin Zheng
Fritz-Haber-Institut
Max-Planck-Gesellschaft
Faradayweg 4-614195BerlinGermany
Majid Mortazavi
Fritz-Haber-Institut
Max-Planck-Gesellschaft
Faradayweg 4-614195BerlinGermany
Alexandre Tkatchenko
Physics and Materials Science Research Unit
University of Luxembourg
Luxembourg City L-1511Luxembourg
Quantum tunneling of thermal protons through pristine graphene Quantum tunneling of thermal protons through pristine graphene
Citation: The Journal of Chemical Physics
1482047072018. 201810.1063/1.5024317(Received 31 January 2018; accepted 9 April 2018; published online 31 May 2018)View Table of Contents: http://aip.scitation.org/toc/jcp/148/20 Published by the American Institute of Physics
Engineering of atomically thin membranes for hydrogen isotope separation is an actual challenge which has a broad range of applications. Recent experiments [M. Lozada-Hidalgo et al., Science 351, 68 (2016)] unambiguously demonstrate an order-of-magnitude difference in permeabilities of graphene-based membranes to protons and deuterons at ambient conditions, making such materials promising for novel separation technologies. Here we demonstrate that the permeability mechanism in such systems changes from quantum tunneling for protons to quasi-classical transport for heavier isotopes. Quantum nuclear effects exhibit large temperature and mass dependence, modifying the Arrhenius activation energy and Arrhenius prefactor for protons by more than 0.5 eV and by seven orders of magnitude correspondingly. Our findings not only shed light on the separation process for hydrogen isotope ions passing through pristine graphene but also offer new insights for controlling ion transport mechanisms in nanostructured separation membranes by manipulating the shape of the barrier and transport process conditions. Published by AIP Publishing.
I. INTRODUCTION
Atomically thin two-dimensional materials are increasingly being explored as a possible platform for developing novel separation technologies such as water desalination or proton-exchange membranes in fuel cells. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] Particular attention is given to the utilization of layered 2D materials for selective sieving of molecules, atoms, and ions. Recent experiments conducted by Geim's group have demonstrated that thermal protons (and deuterons) show appreciable conductance through a pristine graphene sheet (PGS) at ambient conditions. 3,4 Remarkably, the experimentally deduced proton transport barrier of 0.78 eV is at least 0.65 eV lower than those predicted by electronic structure calculations 17,18 for pristine graphene. Moreover, the difference in PGS areal conductivity for protons and deuterons cannot be explained by such calculations because the nuclear mass does not appear in the electronic Hamiltonian. 19 In Ref. 20 it was shown that the hydrogenation of graphene, neglected in previous firstprinciples calculations, can lead to a substantial decrease in the barrier to less than 1 eV, which partially explains the experimental observations. However, the explanation of the substantial isotope effect was restricted up to now to the consideration of zero-point energy (ZPE) contributions to the height of the barrier 20 and the energy of the initial state, 4 while the nuclear quantum effects (NQE) beyond ZPE have been neglected.
In general, nuclear quantum effects strongly depend on the anharmonicity of realistic interatomic potentials and can qualitatively change the transport mechanism leading to various striking effects. Examples are a substantial deviation of the temperature dependence of the transmission rate from the classical Arrhenius-like behavior or even temperature-independent membrane permeability. 21 In contrast to ZPE correction, where only the potential-energy profile near the classical transition state is relevant, the NQE beyond ZPE are dependent upon the entire shape of the potentialenergy surface (PES). Hence, to describe ionic transport, one must consider the ion-graphene interaction on ab initio level of accuracy as well as account for the thermal and quantum fluctuations of all atoms in the system. Existing methods for computing reaction rates 22 are not yet practically applicable for such complex multidimensional quantum-mechanical problems. The development of new complementary approaches is required to unambiguously reveal the mechanism of proton/deuteron transport through PGS as well as other complex transport phenomena.
In this work, we develop a hierarchical quantummechanical methodology for directly computing membrane permeabilities. Using this framework, we demonstrate the qualitative difference in the transport mechanisms for protons and heavier hydrogen isotopes. First, we consider an optimal one-dimensional transport pathway when ions follow a linear trajectory perpendicular to the PGS through the center of a carbon ring [see Fig. 1(a)]. This pathway corresponds to the minimal potential energy barrier and yields the largest contribution to graphene permeability. We find that protons tunnel through PGS, while heavier isotopes follow quasi-classical transport pathways. As a result, the average energy of protons contributing to the transmission process is considerably lower than the height of the potential energy barrier. This leads to a large reduction (more than 0.5 eV) of the Arrhenius activation energy for protons 3 as compared to the results of electronic structure calculations with classical nuclei. The difference in transport mechanisms is also responsible for the large proton-deuteron separation ratio. 4 In the second step, we account for the transport pathways beyond a single linear trajectory by performing free energy thermodynamic integration (FETI) using full densityfunctional theory (DFT) Born-Oppenheimer (BO) potentialenergy surface (PES) with fixed membrane geometry [see Fig. 1(b)]. We have found that the presence of chemisorption sites, which can bind ions, leads to negligibly small average transmission coefficients, making bare PGS impenetrable for hydrogen ions. 16 At the same time, blocking the chemisorption of H ions, for instance, by assuming pre-hydrogenation of graphene, leads to more than 10 20 increase in transmission coefficients. Thus, the pre-hydrogenation of graphene or covering it with some material (such as Nafion 3,4 ) plays a vital role in the transport process of H ions, defining not only the energy of an initial transition state 3 but also affecting the permeability of graphene 20 -an effect which is attributed to specific experimental conditions.
In the final step, our full-dimensional ab initio imaginarytime Feynman-Kac path integral molecular dynamics (AI-PIMD) simulations [23][24][25][26] demonstrate that the motion of carbon atoms in graphene is important for the transport process, concomitantly decreasing the proton activation energy [ Fig. 1(c)].
All in all, our hierarchical quantum model (see Fig. 2) provides novel interpretations for recent key experimental observations, 3,4 including (i) the quantum tunneling mechanism of proton transport, (ii) strong temperature dependence of the prefactor in the Arrhenius equation, (iii) the impact of chemisorption sites and covering material, and (iv) the role of mobile carbon atoms in hydrogen ion transport through PGS.
II. ARRHENIUS EQUATION AND QUANTUM PROCESSES
Transport experiments are often interpreted with the aid of the Arrhenius equation, which connects the rate constant (k) to the inverse absolute temperature ( β),
k = A exp(− βΛ).(1)
Here A is the pre-exponential factor and Λ is the so-called Arrhenius activation energy-the minimal energy required for reactants to transform into products. However, this equation is valid only for classical processes. Whenever quantum effects play a considerable role, a single energy cannot describe the corresponding reaction. In the case of transport, trajectories with different energies will contribute to the observed rates. Moreover, these contributions strongly depend upon temperature. This will result in a curvature in the Arrhenius plots and non-trivial temperature dependence of both A and Λ. 21 Below we will show that the thermal transport of protons through PGS at ambient conditions is strongly affected by tunneling, whereas heavier hydrogen ion isotopes follow nearly classical trajectories. Therefore, the pre-exponential factor A FIG. 2. The hierarchy of methods employed in this work. Density functional theory (DFT) and classical ab initio molecular dynamics (AI-MD) treat nuclei classically and cannot explain the experimental observations. The Wentzel-Kramers-Brillouin (WKB) model and free energy thermodynamic integration (FETI) include NQE and describe the isotope effect but still overestimate the activation energy. Ab initio imaginary-time path integrals molecular dynamic (AI-PIMD) simulations allow us to explain both the activation energy reduction and isotope effect by consistent inclusion of NQE, PGS geometry relaxation, and thermal fluctuations of carbon atoms. and the activation energy Λ in the Arrhenius equation (1) are temperature dependent for protons and different from those for deuterons or tritons. The value of Λ obtained by using Eq. (1) for protons is not the minimal energy required for the transport process to occur. To emphasize this fact, we will use hereafter the term "Arrhenius activation energy" instead of activation energy.
The following terminology is used below in this article: (i) Arrhenius activation energy-the slope of a logarithm of a transport rate versus the inverse temperature. Its value coincides with a commonly used activation energy only in the case of classical processes. (ii) Quasi-classical transport process-the mean energy of the particles participating in such a transport process is close to the height of the barrier. (iii) Quantum transport process-the energy distribution of the particles participating in such a transport process is considerably affected by tunneling and substantially extends into the classically forbidden low-energy region.
III. TRANSPORT ALONG AN OPTIMAL PATHWAY
The possible permeability of PGS is an open controversial question. In some experiments 16 protons are found to transfer only through rare, naturally occurring atomic defects. At the same time, experiments conducted by Geim's group 3,4 clearly demonstrate appreciable thermal proton transport through a PGS at ambient conditions. An essential common part of both sets of experiments is the graphene layer. Thus, to reveal the potential permeability of pristine graphene-based membranes, the conditions when both experimentally observed situations may happen, and the transmission mechanisms, we focus on the thermal flux of hydrogen isotope ions through PGS in vacuum. We compute the direct observable-membrane permeability-which allows us to extract Arrhenius activation energies without invoking any additional assumptions about the value of pre-exponential coefficients. 3,4,27 As a first step, we study the presumably most favorable transport pathway for a hydrogen isotope ion that follows a linear trajectory perpendicular to the PGS through the center of a carbon ring [see Fig. 1(a)]. This scenario corresponds to the lowest possible potential energy barrier U(z) and yields an upper bound to the permeability of PGS with fixed geometry. Henceforth in this work, we obtain U(z) from DFT calculations by employing the non-empirical exchangecorrelation functional of Perdew-Burke-Ernzerhof 28 (PBE) and the Tkatchenko-Scheffler method 29 to account for van der Waals interactions as implemented in the FHI-aims code. 30 An optimized free-standing graphene geometry is used for all ion-graphene distances. The unit cell consists of 32 carbon atoms, and periodic boundary conditions are applied. Further details of the calculations are presented in Sec. S1 of the supplementary material. Note that the particular choice of the PBE functional does not lead to any specific functional dependent conclusions and allows us to employ both analytical and numerical methods. For instance, the employment of the computationally more expensive PBE0 functional leads to the increase of the height of the barrier by 0.1 eV compared to PBE and modifies the isotope separation ratio by only 10%. The obtained barrier height is also in good agreement with the barrier computed with optB88-vdW potential in Ref. 20 for the transition from the physisorbed state.
The transport process in this approximation is simplified to an effective one-dimensional (1D) problem of a transport along a reaction coordinate, which is defined by the distance between the ion and the graphene plane. An average transmission probability k can be found as a ratio of passed and incident ion fluxes.
The flux of particles through the barrier can be found by employing a 1D transition-state model proposed in Ref. 21,
j = ∞ 0 v(p)f (p)T (p)dp ,(2)
where v(p) = p/m, m is the particle mass, p is its momentum, T (p) is the transmission coefficient of the barrier, and f (p) is the momentum probability distribution of the incoming flux of particles. Here we consider a thermal flux of particles, whose momentum distribution obeys the Maxwell-Boltzmann statistics (ions move freely from an infinite distance to the graphene plane interacting only with the potential barrier of the membrane),
f (p) = β 2πm exp − βp 2 2m .(3)
The influence of the momentum distribution of ions in the incident flux on the transparency of the barrier and the isotope separation ratio will be discussed below.
The transmission coefficient T of a single 1D barrier when the energy E of a particle is smaller than the maximum height of the barrier U max can be obtained using the Wentzel-Kramers-Brillouin (WKB) approximation, 31
T (E) = exp − 2 √ 2m z 2 (E) z 1 (E) U(z) − E dz .(4)
Here z i (E) are the distances where U(z i ) = E. Since we are interested in temperatures which are much smaller than U max , we can neglect the over-barrier reflection, setting T (E) = 1 for E > U max . Combining Eqs.
(2)-(4), we obtain
j q pass ( β) = j c pass ( β) + β 2πm 0 ∞ T (U(z))e −βU(z) ∂U ∂z dz ,(5)
where j q pass and j c pass are the fluxes of particles passing the barrier computed within classical and quantum-mechanical approaches, respectively,
j c pass ( β) = e −βU max 2πm β .(6)
Finally, for the average transmission probability k, we have further decrease in temperature, k becomes weakly temperature dependent indicating the increasing role of tunneling transport mechanism, which is characterized by a complete independence of the transmission coefficient upon temperature. A similar nonclassical behavior for deuterons is observed for considerably smaller temperatures (below 250 K) due to the larger mass of this isotope. In the experimentally relevant temperature range 270-330 K 3 all three curves in the inset of Fig. 3 exhibit nearly linear behavior. For deuterons and tritons, this is caused by a quasi-classical nature of the transport process. In contrast, for protons, the linear shape of the Arrhenius plot is a result of a large compensation of the temperature dependent Λ and A in Eq. (1) and a small width of the temperature window employed in the experiment. At higher and lower temperatures, the linear extrapolation obtained from the temperature range 270-330 K (red solid line in Fig. 3) considerably deviates from the actual transmission coefficient (red symbols in Fig. 3) for protons.
k α = 2πm β j α pass , α = c, q.(7)
The analytical expression for the transmission coefficient (see Sec. S1 of the supplementary material) allows us to compute separately the Arrhenius activation energy and the prefactor as a function of temperature literally repeating the experimental procedure. 3 The results for 270-330 K are shown in Fig. 4 (wider temperature range of 200-800 K is presented in Sec. S2 of the supplementary material). One can see that within the considered temperature range the prefactor, A, for protons changes by seven orders of magnitude, while the Arrhenius activation energy increases by more than 0.4 eV. In contrast, for heavier isotopes, both A and Λ are much less temperature dependent. This indicates a qualitative difference in the leading mechanisms of proton and deuteron/triton transport. From Fig. 4 it also follows that accounting for prefactors is crucial to obtain the correct value of the isotope effect. The difference in the Arrhenius activation energies for protons and deuterons at 300 K would lead to a separation ratio of exp{ β(Λ proton Λ deuteron )} ∼ 5 × 10 6 . To clarify the nature of transport mechanisms for different ions, we plot the contribution of ions to the transport process [f ( )T ( ) from Eq. (2)] as a function of their initial energy . The results for thermal flow through PGS at room temperature are shown in Fig. 5. Clearly, the energy windows for permeation of protons and deuterons/tritons are quite different. The main contribution to the heavy hydrogen isotope transport process is due to ions whose energies are comparable to the height of the barrier E DFT . This hints on the quasi-classical nature of the transport process for deuterons and tritons. In this case, the reduction of the activation energy is mainly caused by ZPE effects. 32 activation energy shows that both isotopes obey a similar transport mechanism.
For protons, we observe a qualitatively different behavior. The energies of transmitting particles are considerably lower than the height of the potential energy barrier which corresponds to the so-called tunneling regime. The obtained value of Arrhenius activation energy of 0.94 eV is caused by the quantum tunneling of protons through the graphene layer. This conclusion is also supported by the fact that the crossover temperature between shallow and deep tunneling regimes 33 for protons within the considered one-dimensional potential profile is 320 K. The fact that the crossover temperature falls within the range of the temperatures of practical interest makes the description of proton transport through PGS an especially challenging problem. All the existing methods for computing transmission rates demonstrate the largest errors within this regime. In Secs. IV and V, we will employ two more independent approaches to verify the conclusions obtained within our 1D model.
An important consequence of the quantum tunneling nature of proton transport is the critical influence of the shape of the potential energy barrier on the permeability of the system. The transmission coefficient can be varied by modifying the shape of the barrier without changing its height, 34 that is, without introducing defects in graphene. This explains the increase in permeability observed in experiments 3 upon decorating the PGS by nano-particles. A similar behavior should be obtained in the presence of electric fields. Evidently, the field which drives the ion flow through the graphene layer distorts the symmetry of the barrier suppressing the backscattering process.
IV. THREE-DIMENSIONAL TRANSPORT PROBLEM
The 1D transport model described in Sec. III has provided insights into the ion transport mechanisms, but it does not account for several important factors. First and foremost, the degrees of freedom transverse to the reaction coordinate, which may play a nontrivial role, have been neglected until now. 32 To address this issue, we compute the transport rate constant, k, by performing a free energy thermodynamic integration (FETI) using a three-dimensional (3D) Born-Oppenheimer DFT PES [for details of the calculations see Fig. 1(b) and Secs. S3 and S4 of the supplementary material],
k = e −β∆F .(8)
Here ∆F is the free energy change during the transport process,
∆F = − 0 ∞ f z d z,(9)
where f is the average force acting on the ion displaced from the graphene plane and z is the displacement vector pointing out of the graphene plane. The origin of the coordinates, z = 0, is assumed to be inside PGS. The averaging . . . z has been done by fixing the z coordinate of the centroid of the ion and performing imaginary-time path integral simulations using the precomputed PES. In contrast to the model described in Sec. III, the other two coordinates of the ion are not constrained, and the trajectories of ions can cross the carbon ring at any point. Similar to Sec. III, we determine k as a transport probability rather than the transport frequency, which explains the absence of a prefactor in Eq. (8). Note that in this section the geometry of the graphene layer is also fixed and does not depend upon the position of the ion. The role of the carbon atoms' motion and PGS geometry optimization will be considered in Sec. VI. By performing FETI we account for all possible transmission trajectories, also taking into consideration the chemisorption of ions on carbon atoms at a distance of 1.1 Å from the graphene plane. The chemisorption sites demonstrate large binding energy of 1.7 eV and have large cross-sectional area perpendicular to the graphene plane. The ions, trapped inside such sites, cannot participate in the transport process. As a result, PGS will behave as an impenetrable membrane for protons at the initial stage of the transport process. This is fully supported by the results of our simulations presented in Table I, where it can be seen that chemisorption leads to negligibly small average transmission coefficients.
However, effective blocking (saturation) of chemisorption sites in a given experiment may lead to qualitatively different observations. Such blocking can arise due to several factors, including the hydrogenation of graphene 20 or the presence of covering materials (Nafion, etc.) inhibiting chemical bond formation. To mimic the blocking of chemisorption sites and to avoid unnecessary speculations about the particular blocking mechanisms and the nature of the initial transport state, we introduce the following approximation. We assume that for any ion-graphene distance z [see Figs. 1(a) and 1(b)] the minimum of the interaction energy U min (z) lies on a straight line passing through the center of a carbon ring. Hence, whenever at a given point (x, y, z) the value of the interaction energy is lower than U min (z), we set U(x, y, z) = U min (z), otherwise we leave it unchanged. This allows us to retain the repulsion of an ion from carbon atoms at small ion-graphene distances ≤0.8 Å and to avoid the attraction to chemisorption sites at larger z. As a result, we observe an increase of average transmission coefficients by more than 25 orders of magnitude (see Table I) and the values becoming comparable to the predictions of the 1D model. A small reduction of the transmission coefficient is caused by less favorable pathways than the linear trajectory passing through the centers of carbon hexagons. Remarkably, the isotope effect does not change qualitatively within the 3D model as compared to the 1D case and both calculations yield the same results for the Arrhenius activation energy (within 20 meV). Namely, FETI yields Λ = 0.92 eV for protons and 1.34 eV for deuterons. TABLE I. Proton and deuteron reaction rate constants and their ratio at 300 K. WKB denotes the 1D transport model described above, while FETI implies the free energy thermodynamic integration approach with and without blocking the chemisorption sites. To understand why the transmission coefficients obtained within WKB and FETI methods are in such a good agreement, we plot the probability of locating a proton and a deuteron contributing to the transition state as a function of the distance from the graphene surface (see Fig. 6). The inclusion of NQE leads to a substantial delocalization of protons with the probability maximum located around 0.5 Å away from the graphene layer. This suggests that the in-plane ionic quantum fluctuations, which can considerably affect ionic transport in nanoporous materials, 32 are of minor importance for proton tunneling through graphene. In contrast, the deuteron wavefunction is much less delocalized in the direction perpendicular to the graphene plane. Deuterons come closer to the graphene plane than protons during the transport process. The effective repulsion caused by in-plane ionic quantum fluctuations becomes essential. This leads to a more significant difference in the deuteron reaction rate constant predicted within the one-dimensional WKB model and the threedimensional FETI approach comparing to the case of proton transport. Figure 6 also demonstrates that protons and deuterons, initially located at a distance larger than ∼0.8 Å from the graphene plane, do not contribute appreciably to the transport process. This provides further evidence in favor of the conclusion that the hydrogen ions trapped at chemisorption sites at a distance of ∼1.1 Å from the graphene layer cannot penetrate through a PGS.
V. CALCULATING THE TRANSPORT BARRIER WITH MOLECULAR DYNAMICS
To further asses the role of thermal and quantum fluctuations of the carbon atoms constituting the PGS in proton/deuteron transport process, we have performed classical and quantum ab initio (AI) molecular dynamics (MD) simulations. The barrier Λ is computed within the centroid-density quantum transition state theory (cd-QTST) as
Λ = E reacting complexes − E reactants .(10)
Here E reacting complexes is the average energy of a reacting complex and E reactants is the average energy of reactants, i.e., free ion and free-standing PGS. We define the reacting complex by constraining the vertical positions of the hydrogen isotope and two carbon atoms in the surrounding graphene hexagon. Other degrees of freedom were allowed to fluctuate freely. Trajectory snapshots of AI-MD and AI-PIMD simulations for deuteron and proton in the transition state are shown in Fig. 7 (Multimedia view). These snapshots clearly demonstrate a qualitative difference in the delocalization of protons and deuterons in the transition state which leads to different transport mechanisms. Details of the simulations are presented in Sec. S5 of the supplementary material.
In full agreement with previous calculations, 17,18 where only the electronic subsystem has been considered on a quantum-mechanical level, our AI-MD equilibrium proton transport barrier is 1.6 eV. The value of the classical AI-MD barrier is by 0.2 eV larger than that predicted by DFT calculations. This well-known effect is a result of ionmembrane repulsion inside the graphene plane caused by atomic motions.
In our imaginary-time Feynman-Kac path integral molecular dynamics (PIMD) simulations, we define the transmission state by fixing the out-of-plane coordinate of centroids for the hydrogen ion and two carbon atoms in the surrounding graphene hexagon. All the other simulation details are similar to classical AI-MD simulations. To compute the average energies entering Eq. (10), we utilize a recently developed perturbed path integral (PPI) approach. 35 This method is a combination of conventional ab initio imaginary-time path integral molecular dynamics simulations with a posteriori corrections for thermodynamic observables using perturbation theory. The PPI method enables a convergence of ∼10 meV for the total energy of the considered system with respect to the number of beads at ambient conditions using AI-PIMD trajectories with only 10 beads. Such convergence is an order of magnitude more accurate compared to the conventional second-order PIMD method and is required for a quantitative description of the tunneling process.
By allowing graphene atoms to move and fluctuate freely, we obtain the values of Λ for protons and deuterons of 0.4 and 1.4 eV correspondingly. Comparing the FETI activation energy for protons and the current result, one can see that strong interaction between protons and graphene at characteristic tunneling distances of ∼0.5 Å makes the separation of graphene and ions into two subsystems rather inaccurate. Importantly, the reduction of Λ for protons is not a result of graphene geometry relaxation ignored in FETI calculations. The same relaxation is present in AI-PIMD simulations for deuterons where, due to quasi-classical nature of the transport process, the difference between Λ obtained within these two methods is only 60 meV.
We remark that the value of Λ for protons computed by employing Eq. (10) is not exactly the Arrhenius activation energy. Indeed, cd-QTST may be inaccurate due to strong quantum fluctuations of protons and the anisotropy of the barrier caused by the motion of carbon atoms. 21 Unfortunately, the fact that the crossover temperature is within the temperature range of interest makes the thermal protons transport through PGS a challenging problem for any existing method for computing reaction rates. 22 Another complication preventing direct ab initio calculations of reaction rates by employing more sophisticated methods than cd-QTST is the undefined mechanism of blocking of the chemisorption sites. Even the hydrogenation can happen in many different ways. 20 Nevertheless, more than the twofold reduction of the Arrhenius activation energy obtained within cd-QTST compared to quantum calculations with fixed carbon atoms or classical AI-MD simulations unambiguously indicates the strong mutual influence of ion and carbon quantum fluctuations. These fluctuations should considerably affect the transmission rates and can further decrease the activation energy. Table II summarizes the results for the Arrhenius activation energy Λ obtained within different methods employed in this paper.
VI. DISCUSSION
Accounting for the quantum nature of an ion within WKB and FETI approaches leads to more than 0.5 eV decrease of the Arrhenius activation energy. Note that the results of our one-and three-dimensional calculations are in very good mutual agreement when the chemisorption sites are assumed to be blocked. Importantly, the shape of the BO PES for ion-graphene distances smaller than 0.8 Å, which defines the transmission state (see Fig. 6), is determined by ion-graphene repulsion and cannot be substantially affected by a particular mechanism for blocking of the chemisorption sites. Hence, the obtained qualitative difference in proton and deuteron transport processes and the large decrease of the Arrhenius activation energy for protons is independent of particular experimental conditions (the presence of water and Nafion and hydrogenation of graphene). Obviously, in the experiment, the interaction of ions with the surrounding atoms and molecules leads to the change of the energy of the initial state. The ions can be held tightly by either Nafion or another water molecule, which could potentially stabilize the initial state and, therefore, yield a higher overall barrier. On the other hand, the experimental observation of the proton transport at small bias fields suggests the presence of the mechanisms to overcome the binding of the ions to their carriers, while the nature of such mechanisms remains unclear. Hence, in principle, the energy of the initial state can be larger or smaller compared to the energy of a free proton. This may explain the overestimation of the activation energy obtained within our FETI method (0.92 eV), where the initial state is considered to be a vacuum, as compared to the experimental value of 0.78 eV. Another reason may be the change of the height of the barrier, for instance, caused by the hydrogenation of the graphene surface. From Fig. 5, it follows that one can modify the proton/deuteron separation ratio by changing the energy distribution of ions in the incident flux. For instance, the protons with energies below ∼0.4 eV are reflected by the membrane. Thus, different behavior of the proton transport may be observed for bias voltages below and above ∼0.4 eV. For instance, for voltage biases above ≥0.4 eV or due to the acquiring of an additional energy by the ion during the dissociation of an initial state, possible proton transport pathways with energies below the bias threshold will be excluded from the transmission process. This will reduce the proton-deuteron separation ratio. In contrast, cutting out the ions with energies above ∼1 eV will make the membrane impenetrable for deuterons without noticeable changes in the transmission rate for protons. This can be used to increase the proton/deuteron separation ratio to, theoretically, any desired value. In practice, the efficiency of such a mechanism will be a limiting factor.
VII. SUMMARY
Our first-principles quantum-mechanical approach has revealed the pronounced quantum nature of thermal proton transport through pristine graphene. We predict a substantial difference of 0.5 eV in the Arrhenius activation energies for protons and deuterons and rationalize the experimentally measured isotope effect, which is strongly influenced by the mass-dependent prefactor in the Arrhenius equation. We found that in order to observe non-negligible ion transport, chemisorption on the graphene surface has to be blocked. The motion of carbon atoms in graphene is also an important factor, yielding considerable reduction in the Arrhenius activation energy for protons, compared to a fixed membrane approximation. Further theoretical developments are needed to obtain accurate ionic transmission coefficients from full-dimensional ab initio calculations with an explicit mechanism of blocking chemisorption, an account for thermal and quantum fluctuations of membrane atoms, the presence of electric fields in the system, and ion-ion interaction. The incorporation of experimental conditions in simulations should reveal more subtle but potentially very important features of the transport process and control mechanisms for practical applications of hydrogen isotope ion tunneling.
SUPPLEMENTARY MATERIAL
See supplementary material for all the technical details of calculations employed in this work. Video files S1.mp4,
FIG. 1 .
1Proton and deuteron transport through pristine graphene. Color bars show the value of the Arrhenius activation energy obtained with different approaches that are schematically explained in sub-figures (a)-(c). (a) The one-dimensional potential-energy barrier used for the transport model based on the Wentzel-Kramers-Brillouin (WKB) approximation. (b) Examples of three-dimensional PES at different ion-graphene distances (0.0, 0.2, 0.4, 0.6 Å) used for the free energy thermodynamic integration (FETI) approach. (c) System geometry used in AI-PIMD simulations within centroid-density quantum transition state theory (cd-QTST) method.
Figure 3
3shows the Arrhenius plot for protons, deuterons, and tritons in the temperature range 200-800 K. One can see a pronounced deviation of the inverse temperature dependence of the transmission coefficient for a proton from the classical linear behavior for temperatures below 350 K. With a FIG. 3. Transmission coefficient as a function of inverse temperature for proton, deuteron, and triton transport through a pristine graphene layer for the temperature range 200-800 K. Symbols are the results of the one-dimensional tunneling model (see the text), while solid lines of the same color are the best linear fitting in the experimentally relevant temperature range 270-330 K (see the inset).
Quantum prefactors yield a separation ratio (A proton /A deuteron )× exp{ β(Λ proton Λ deuteron )} ∼ 16 in excellent agreement with experiment. 4 FIG.4. Activation energies (a) and prefactors (b) extracted from the classical form of Arrhenius equation when it is applied for proton, deuteron, and triton tunneling through a pristine graphene layer along the optimal transport pathway.
Indeed, the obtained value of the Arrhenius activation energy for deuterons of 1.36 eV is in a good agreement with the results of ZPE calculations in Ref. 27. The value of the activation energy for tritons is found to be of 1.39 eV. The small difference in the FIG. 5. Relative contribution to the tunneling process through PGS, in arbitrary units (a.u.), from ions with different initial energies (eV) at 300 K.
FETIFIG. 6 .
6FETI (blocked chem.) WKB Proton 3.1 × 10 50 1.9 × 10 24 7.1 × 10 23 Deuteron 2.0 × 10 52 7.8 × 10 26 4.3 × 10 24 Probability distributions in arbitrary units (a.u.)to find a proton/deuteron contributing to the transition state at a given distance from the graphene plane. The simulations have been preformed at the temperature of 300 K.
FIG. 7 .
7Trajectory snapshots of (a) AI-MD and AI-PIMD simulations for (b) deuteron and (c) proton in the transition state. Multimedia views: https://doi.org/10.1063/1.5024317.1; https://doi.org/10.1063/1.5024317.2; https://doi.org/10.1063/1.5024317.3
TABLE II .
IIProton and deuteron Arrhenius activation energies (in eV) obtained within different approaches used in this article. The FETI results correspond to PES where the chemisorption sites are blocked. The DFT barrier values correspond to the transport from a physisorbed state.DFT
AI-MD
AI-PIMD
WKB
FETI
Exp.
Proton
1.43
1.6
0.4
0.94
0.92
0.78
Deuteron
1.43
1.6
1.4
1.36
1.34
. . .
trajectories for deuterons and protons in the transition state (ion centroid is fixed in the plane of the graphene sheet), respectively. The PES.zip archive contains the PES code employed in Sec. IV. ACKNOWLEDGMENTS I.P. and A.T. acknowledge financial support from the Luxembourg National. S3 S2.Mp4, Ai-Md Mp4 Visualize, Ai-Pimd , Research within the FNR-CORE program (Grant No. FNR-11360857S2.mp4, and S3.mp4 visualize AI-MD and AI-PIMD trajec- tories for deuterons and protons in the transition state (ion centroid is fixed in the plane of the graphene sheet), respec- tively. The PES.zip archive contains the PES code employed in Sec. IV. ACKNOWLEDGMENTS I.P. and A.T. acknowledge financial support from the Lux- embourg National Research within the FNR-CORE program (Grant No. FNR-11360857).
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|
[] |
[
"Joint Optimization of Rate, Distortion, and Decoding Energy for HEVC Intraframe Coding",
"Joint Optimization of Rate, Distortion, and Decoding Energy for HEVC Intraframe Coding"
] |
[
"Christian Herglotz [email protected] \nMultimedia Communications and Signal Processing Friedrich-Alexander University Erlangen-Nürnberg\nCauerstr. 791058ErlangenGermany\n",
"André Kaup [email protected] \nMultimedia Communications and Signal Processing Friedrich-Alexander University Erlangen-Nürnberg\nCauerstr. 791058ErlangenGermany\n"
] |
[
"Multimedia Communications and Signal Processing Friedrich-Alexander University Erlangen-Nürnberg\nCauerstr. 791058ErlangenGermany",
"Multimedia Communications and Signal Processing Friedrich-Alexander University Erlangen-Nürnberg\nCauerstr. 791058ErlangenGermany"
] |
[] |
This paper presents a novel algorithm that aims at minimizing the required decoding energy by exploiting a general energy model for HEVC-decoder solutions. We incorporate the energy model into the HEVC encoder such that it is capable of constructing a bit stream whose decoding process consumes less energy than the decoding process of a conventional bit stream. To achieve this, we propose to extend the traditional Rate-Distortion-Optimization scheme to a Decoding-Energy-Rate-Distortion approach. To obtain fast encoding decisions in the optimization process, we derive a fixed relation between the quantization parameter and the Lagrange multiplier for energy optimization. Our experiments show that this concept is applicable for intraframe-coded videos and that for local playback as well as online streaming scenarios, up to 15% of the decoding energy can be saved at the expense of a bitrate increase of approximately the same magnitude.
|
10.1109/icip.2016.7532416
|
[
"https://arxiv.org/pdf/2203.01765v1.pdf"
] | 17,555,553 |
2203.01765
|
fa3371fa2da67ae57d21653c623be317e45ac785
|
Joint Optimization of Rate, Distortion, and Decoding Energy for HEVC Intraframe Coding
3 Mar 2022
Christian Herglotz [email protected]
Multimedia Communications and Signal Processing Friedrich-Alexander University Erlangen-Nürnberg
Cauerstr. 791058ErlangenGermany
André Kaup [email protected]
Multimedia Communications and Signal Processing Friedrich-Alexander University Erlangen-Nürnberg
Cauerstr. 791058ErlangenGermany
Joint Optimization of Rate, Distortion, and Decoding Energy for HEVC Intraframe Coding
3 Mar 2022HEVCenergydecodermodelrate-distortion- optimization
This paper presents a novel algorithm that aims at minimizing the required decoding energy by exploiting a general energy model for HEVC-decoder solutions. We incorporate the energy model into the HEVC encoder such that it is capable of constructing a bit stream whose decoding process consumes less energy than the decoding process of a conventional bit stream. To achieve this, we propose to extend the traditional Rate-Distortion-Optimization scheme to a Decoding-Energy-Rate-Distortion approach. To obtain fast encoding decisions in the optimization process, we derive a fixed relation between the quantization parameter and the Lagrange multiplier for energy optimization. Our experiments show that this concept is applicable for intraframe-coded videos and that for local playback as well as online streaming scenarios, up to 15% of the decoding energy can be saved at the expense of a bitrate increase of approximately the same magnitude.
I. INTRODUCTION
Nowadays, portable devices such as smartphones or tablet PCs have become an indispensable gadget for millions of users all over the world. A major drawback of these devices is that the battery capacity in conjunction with the power consumption of the system limits the operating time of the device. Hence, research aiming at reducing the power consumption of a process is a worthwhile task. This paper addresses the power consumption of the video decoding process. Analyzing the power consumption of a portable device it was shown that during video playback, the decoding process consumes a major part of the battery power [1]. To reduce the power consumption, great effort is pursued to optimize decoder implementations, not only on the software side [2], but also on the hardware side [3]- [5]. Using these techniques, energy can be saved in the range of several orders of magnitude. In this paper we proceed in a different direction and tackle the following question: If we have given a fixed (and maybe optimized) decoder solution, is it possible to further reduce the decoding energy using knowledge about its energetic properties? In this paper, we show that for intraframe coded videos this is indeed possible.
To address this problem, we propose to construct bit streams that consume less energy during the decoding process than conventional bit streams. To this end, we provide information about the energetic properties of the decoder to the encoder such that it can exploit this knowledge to decide for energy saving coding modes. This paper investigates how much decoding energy can be saved when such knowledge is available.
As in this context the decision criterion of the encoder is changed, the resulting bit streams may have a lower ratedistortion performance than before. That means that we trade decoding energy against bitrate. At first glance this may seem counter-intuitive as it is usually assumed that a larger bit stream requires more energy during decoding. In contrast we argue that depending on the encoding decision, the decoding process may show a different complexity. E.g., we can encode a so-called transform-skip flag. If the flag is false, an inverse, computationally complex transform is performed requiring a certain amount of processing energy. If the flag is true, the transform is skipped and thus no additional processing energy is required. We can see that, although both decisions are expressed by only one bit, the required processing energy on the decoder side may differ significantly.
There were similar works in the past where power or processing complexity, which can be interpreted as an approximation to processing power, was investigated. E.g., Zhihai He et al. [6] incorporated the encoding power into the Rate-Distortion calculation to save energy on the encoder side. Using the encoder's processing complexity, this idea has been deeply investigated, e.g., by Xiang Li et al. [7]. In another work, Yuwen He et al. [8] proposed a simple decoding complexity model for the modules deblocking filter and motion compensation and showed that operating times can be extended using power-aware optimization criteria. A more complex model for the decoder was presented by Ma et al. [9] where on the decoder side, dynamic voltage and frequency scaling is applied when the modeled processing complexity is small enough. A different approach using a highlevel power model for a hardware-accelerated decoder is used by Xin Li et al. [10]. In contrast, our approach uses energy values obtained from real measurements for a high number of different intraframe-coding tools such that more detailed and more accurate energy estimates are obtained. Furthermore, our work aims at optimizing the decoding energy at a constant objective visual quality.
The paper is organized as follows: Section II revisits the energy model that we use to obtain energy-aware encoding decisions. Afterwards, Section III explains how this model is incorporated into the encoder and according to which criteria the encoding decisions are taken. Then, Section IV introduces our test setup and gives evaluation results for the proposed algorithm in a local playback and a streaming scenario. Section V concludes the paper.
II. DECODER ENERGY MODEL
The energy model that we use to describe the energy consumption of the decoder was presented in [11] and is applicable for various different decoding systems [12]. It includes the processing energy of the CPU as well as the energy required by the random-access-memory (RAM). We adopt the accurate model to estimate the decoding energy for intra coded sequences that is calculated bŷ E dec = E 0 + e slice · n slice + ∀size ∀mode e mode,size · n mode,size + ∀size ∀comp e comp,size · n comp,size + e coeff · n coeff + e g1 · n g1 + e val · ∀c =0 log 2 |c| + e CSBF · n CSBF + e noMPM · n noMPM − e TSF · n TSF .
(1)
In this model, the values of the variables e ("specific energies") represent the energy required for executing different functions during the decoding process. These functions can be executed multiple times. For example, the coefficient e coeff can be interpreted as the energy required to decode a single nonzero residual coefficient, e g1 is the additional decoding energy if the coefficient's value is greater than one. Likewise, the variables e mode,size describe the intra-prediction process on a certain block size that can range from 32 × 32 to 4 × 4. CSBF corresponds to the coded subblock flags, MPM to the most probable modes, and TSF to the transform skip flags. e comp,size , where comp represents the color components Y, U and V, represents the energy required for transformation of the residual coefficients. Further information about these specific energies can be found online [13].
Counting how often these functions are executed during the decoding process of a single bit stream the so-called feature numbers n can be determined. Multiplying these numbers n with the specific energies e, we obtain an estimation for the complete required decoding energy. In [11] it was shown that estimation errors of less than 3% with respect to the measured true energy consumption can be achieved.
There are two important reasons for choosing this model: First, if specific energies are given, the decoding energy can be estimated without having to execute the decoding process or having the decoder at hand. The information about the bit stream features, which is inherently available in the encoder, is sufficient. Second, most of the specific energies can be assigned to encoding decisions. An example shall visualize this property: Consider the encoder needs to decide if residual coefficients for the luma component on a certain block size shall be coded or not. In the first option, a certain number n coeff of quantized coefficients will be coded where some of them are greater than one (n g1 ). Furthermore, the inverse transformation of this block has to be performed once (n comp=luma,size = 1). Multiplying the numbers with the corresponding specific energies and adding up the products we obtain an estimation on how much energy the decoder needs to process these tasks. For the second option, none of these tasks needs to be executed such that no additional energy is required.
With the help of this approach, the encoder gets a third encoding criterion next to rate and distortion: the decoding energy that we exploit for Decoding-Energy-Rate-Distortion-Optimization (DERDO) in the next section.
III. DECODING-ENERGY-RATE-DISTORTION
OPTIMIZATION In order to obtain savings in decoding energy we decided to modify the standard Rate-Distortion-Optimization (RDO) approach as presented in [14]. Therefore, we include the estimated decoding energy into the standard equation and obtain
min J DERD = D + λ R R + λ EÊ ,(2)
where we adopt the parameters distortion D, rate R, and the corresponding Lagrange multiplier λ R from the classic approach. In addition, we consider the estimated decoding energyÊ with a corresponding Lagrange multiplier λ E that will be derived in the next subsection. J DERD is the cost function to be minimized. During the encoding process, the decoding energyÊ is estimated at runtime using (1). Furthermore, we tested another approach neglecting the rate by calculating
min J DED = D + λ EÊ .(3)
The results to this approach will show the maximum potential energy savings. As the rate is not considered at all it can increase significantly. This approach is also used to obtain the λ E -QP relation as shown in the next subsection.
A. Lambda-QP Relation A major challenge in constructing a helpful optimization formula is finding a relation between the Lagrange multiplier λ E and the quantization parameter (QP). For optimal coding results, the QP should be independent from the choice of λ E such that an exhaustive search across different QP values would be necessary. To save encoding time, in standard RDO, it was shown that for a fixed λ R , a single QP is chosen in most cases [15], [16], such that a fixed equation relating λ R to the QP is proposed as
λ R = c · 2 QP −12 3 ,(4)
where c = 0.57 has been determined experimentally and is used for the HM encoder implementation [17]. We adopt this approach for DERDO and experimentally determine the relation between the QP and λ E . As decoding energy model, we take specific energies for the HM implementation that will be introduced in Section IV.
To get a first coarse approximation for λ E , we compared the decoding energy to the complete number of bits for several bit streams and found that the energy is usually lower by approximately six orders of magnitude (e.g., the sequence BasketballPass was encoded using 2.04 MBits and requires 1.78J for decoding). Hence, we multiplied the λ-values chosen in the reference software by 5·10 6 and tested eleven values for λ E in the range of [5.65 · 10 5 , 5.84 · 10 9 ]. To obtain the optimal QP, it was optimized on CTU-level in the corresponding range of [5±5, 45±5]. The resulting relative occurences are depicted in Figure 1.
We can observe that distinct peaks occur especially for lower values of λ E . For higher QPs the peaks shrink and even disappear. However, for simplification, we decided to stick to the traditional approach and found that .84 · 10 9 1.84 · 10 9 5.79 · 10 8 1.82 · 10 8 5.75 · 10 7 1.81 · 10 7 5.70 · 10 6 1.80 · 10 6 5.65 · 10 5
λ E = c E · 2 QP −12 3(5)
Relative occurence [%]
QP value Table I with c E = 0.57 · 10 7 is well suited to represent the relation. Figure 2 shows the experimental relation between the dominant QPs and the fixed λ E as well as the curve described by (5). We can see that for the more common intermediate QPs, the experimental result is well approximated.
IV. EVALUATION
We prove the applicability of our approach by testing the HM-13.0 decoder solution [17] and the optimized decoder in the FFmpeg framework [2] on a Pandaboard [18]. The Pandaboard is a development platform that is equipped with an OMAP-4430 System-on-Chip that is typical for portable devices like smartphones. To suppress impacts from background processes or peripheral hardware, we performed the decoding on runlevel 1 with disabled LEDs and disabled monitoring tasks. The measurement setup used to determine the processing energy of the pure decoding process is the same as in [19].
To estimate the decoding energy in the encoder, we used the accurate model with energy parameter values for the HMdecoder that can be found online [13]. For encoding, we adapted the HM-14.0 encoder by incorporating the specific energies e into the respective decision stages. To obtain the real resulting energy savings, the decoding energy for all resulting bit streams was newly measured. Hence, the results given in this section describe the real energy savings and not the savings estimated by the model which may be inaccurate.
As input sequences, we chose a subset of the HEVC test set as shown in Table I. The restricted number of frames was chosen to keep processing and measuring time at a reasonable level, where further tests indicated that coding more frames does not change the coding efficiency significantly. All sequences were encoded three times for optimal HM-decoding where all frames were coded as intra frames: First, we used the standard RDO and second, the proposed DERDO approach as shown in (2). In the third approach (DEDO), the rate was neglected as shown in (3). We used the following three properties to compare the performance of both algorithms: First the YUV-PSNR that is calculated by
PSNR YUV = 1 8 (6 · PSNR Y + PSNR U + PSNR V ) ,(6)
second the bitrate R in terms of bit stream file size, and third the measured complete decoding energy E in Joule [J]. We evaluate the coding efficiency for two use cases: Local playback that focuses on the pure processing energy (Sec. IV-A) and an online streaming application where transmission energies are considered (Sec. IV-B).
A. Local Playback
The results given in this section describe the energy savings in a local playback scenario. Figures 3(a) and 3(b) compare the RD-performance and the energy saving performance of the proposed approaches for the BlowingBubbles sequence using the HM-decoder. In Figure 3(a) we can see that as expected, using the new minimization functions results in losses in RD-performance. For the complete QP range, the RD-curves from DEDO and DERDO lie below the curve from RDO. To visualize how many more bits have to be spent to achieve the same visual quality, the right diagram gives the relative amount of extra bits needed for the proposed approach. We can see that especially for high QPs (low image qualities) more bits are required.
In contrast, regarding Fig. 3(b) we can see that a significant amount of decoding energy can be saved. The right diagram indicates possible savings of up to 15%. The curves of the other sequences show a similar behavior, though we observed that they can be shifted towards higher or lower rates and energies depending on their content. Furthermore, we can see that in terms of energy savings, the DEDO approach only performs slightly better than the DERDO approach. As in contrast DERDO shows a significantly lower bitrate increase, this approach seems to be more appropriate for practical use.
To summarize the results, we calculated the Bjøntegaard-Delta bitrate (BDBR) as proposed in [20] for all sequences and both decoders. Furthermore, to give mean values for the energy savings, we use the same approach but replace the bitrate by the consumed energy and calculated a BD-decoding energy Table II.
We can see that most energy can be saved for high resolution sequences. Furthermore, optimizing the bit stream for HM-decoding is also beneficial for a different decoding solution (FFmpeg), though savings are slightly lower (about 2% in average). Interestingly, in some cases (sequence Ki and vid) savings are higher for the FFmpeg solution, which can be explained by inaccuracies in the energy model and the estimated specific energies. Summing up, when using the DERDO approach, accepting bitrate increases of 6% to 24%, decoding energy can be saved in the range of 5% to 17%.
B. Online Streaming
In this subsection, in order to show how streaming of the video bit stream affects the overall energy consumption, we consider the transmission energy in a WiFi-streaming scenario. Therefore, we take the energy model proposed in [21]. In this model, a per-bit transmission energy E b in nJ per bit can be estimated depending on the throughput T h in megabit per second asÊ
b = a · T h −1 + b,(7)
where we take the values for parameter a = 305.3 and b = 13.1 for a universal transmission model (independent from protocol and packet size, values valid for a Google Nexus smartphone). Analog to our approach estimating the processing energy, we do not consider the idle transmission energy required for sustaining the network connection. To estimate the energy required in the streaming scenario, we calculate the throughput T h of each test sequence (which is the product of the bit stream file size R with the frame rate divided by the number of frames), derive the corresponding per-bit energyÊ b (23 − 8000nJ per bit), and add the estimated transmission energy (Ê tr = R ·Ê b ) to the measured decoding energy. The resulting energy values are used to calculate the Bjøntegaard-Delta values shown in Table III. We can see that energy savings are about 0.3 − 3% lower, where especially low resolution content is affected. For high resolution sequences, due to a much lower per-bit energy resulting from higher throughput, the transmission energy plays a minor role.
V. CONCLUSION In this paper we showed that it is possible to encode decoding-energy saving bit streams when the energetic properties of the decoding system are known to the encoder. Decoding energy can be saved in the range of 5% to 17% at a constant objective visual quality when accepting a compression efficiency loss of 6% to 24% in terms of rate-distortion performance. Online streaming only slightly affects the energy savings. Further work will extend this concept to interframe coding, test it on other decoding systems, and consider the transmission energy directly in the optimization process.
Fig. 1 .
1Relative frequency of occurences of the QPs for the λ E -values shown in the legend. The tested input sequences are listed in
Fig. 2 .
2Experimental relation between QP and λ E . The markers correspond to the maxima shown inFigure 1, the curve is the proposed approximation.
Fig. 3 .
3Rate-Distortion (a) and Decoding-energy-distortion (b) performance of the standard RDO (blue lines) and the proposed DEDO (green lines) and DERDO (red lines) approach for the BlowingBubbles sequence decoded with the HM-software. The left diagrams in each subfigure relate distortion with rate and energy, respectively, where the markers depict the QPs ranging from 15 to 50 in steps of 5. The right diagrams in each subfigure show the relative bitrate increases and energy savings in percentage in comparison to the standard RDO.
TABLE I .
IPROPERTIES OF INPUT SEQUENCES. THE SEQUENCES ARE TAKEN FROM THE HEVC TEST SET AND WERE ENCODED BEGINNING WITH THE FIRST FRAME WITH QPS RANGING FROM 15 TO 45 IN STEPS OF 5.Sequence
Class
Resolution
No. frames
PeopleOnStreet (PoS)
A
2560 × 1600
5
Traffic (Tr)
A
2560 × 1600
5
Kimono (Ki)
B
1920 × 1080
16
RaceHorses (RHC)
C
832 × 480
50
BasketballPass (BP)
D
416 × 240
50
BlowingBubbles (BB)
D
416 × 240
50
BQSquare (BQ)
D
416 × 240
50
RaceHorses (RHD)
D
416 × 240
50
vidyo3 (vid)
E
1280 × 720
50
SlideEditing (SE)
F
1280 × 720
50
TABLE II .
IIAVERAGE BITRATE INCREASES (BDBR) AND DECODING
ENERGY SAVINGS (BDDE) AS CALCULATED BY THE
BJØNTEGAARD-DELTA APPROACH (QPS 15, 25, 35, AND 45) FOR THE HM
AND THE FFMPEG DECODER.
DEDO
DERDO
Seq.
BDBR
BDDE
BDBR
BDDE
HM
FFmpeg
HM
FFmpeg
PoS
20.6%
17.4%
11.5%
16.4%
16.7%
10.3%
Tr
18.9%
16.9%
14.9%
15.3%
16.7%
14.2%
Ki
25.6%
10.6%
21.4%
21.3%
10.1%
19.2%
RHC
15.9%
15.0%
12.5%
12.5%
14.5%
11.0%
BP
22.8%
13.5%
12.8%
16.9%
12.9%
11.7%
BB
16.3%
15.8%
11.9%
12.4%
15.1%
10.8%
BQ
13.9%
9.9%
7.6%
9.9%
8.9%
5.9%
RHD
17.6%
14.8%
10.3%
13.0%
13.8%
9.4%
vid
28.7%
14.9%
17.4%
23.1%
14.2%
16.2%
SE
10.4%
6.3%
5.2%
6.7%
5.7%
5.5%
(BDDE). The resulting values for the HM decoder and the
FFmpeg decoder are summarized in
TABLE III .
IIIAVERAGE BITRATE INCREASES AND DECODING ENERGY SAVINGS AS CALCULATED BY THE BJØNTEGAARD-DELTA APPROACH (QPS 15, 25, 35, AND 45) IN AN ONLINE STREAMING APPLICATION (THE BDBR-VALUES ARE THE SAME AS IN TABLE II).DEDO
DERDO
Seq.
BDBR
BDDE
BDBR
BDDE
HM
FFmpeg
HM
FFmpeg
PoS
20.6%
17.2%
11.1%
16.4%
15.6%
10.0%
Tr
18.9%
16.5%
14.5%
15.3%
15.6%
13.8%
Ki
25.6%
9.2%
19.9%
21.3%
7.9%
17.9%
RHC
15.9%
14.0%
10.1%
12.5%
12.9%
8.9%
BP
22.8%
11.4%
8.0%
16.9%
10.2%
7.3%
BB
16.3%
14.2%
8.3%
12.4%
12.9%
7.6%
BQ
13.9%
9.4%
5.5%
9.9%
7.8%
4.3%
RHD
17.6%
12.2%
6.1%
13.0%
10.8%
5.6%
vid
28.7%
13.7%
16.1%
23.1%
12.0%
15.0%
SE
10.4%
6.4%
4.6%
6.7%
5.5%
4.9%
ACKNOWLEDGMENT This work was financed by the Research Training Group 1773 "Heterogeneous Image Systems", funded by the German Research Foundation (DFG).
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|
[] |
[
"A realizable data-driven approach to delay bypass transition with control theory",
"A realizable data-driven approach to delay bypass transition with control theory"
] |
[
"Pierluigi Morra \nLinné FLOW Centre\nKTH Royal Institute of Technology\nSE-10044StockholmSweden\n",
"Kenzo Sasaki \nInstituto Tecnológico de Aeronáutica\nSão José dos CamposBrazil\n",
"Ardeshir Hanifi \nLinné FLOW Centre\nKTH Royal Institute of Technology\nSE-10044StockholmSweden\n",
"André V G Cavalieri \nInstituto Tecnológico de Aeronáutica\nSão José dos CamposBrazil\n",
"Dan S Henningson \nLinné FLOW Centre\nKTH Royal Institute of Technology\nSE-10044StockholmSweden\n"
] |
[
"Linné FLOW Centre\nKTH Royal Institute of Technology\nSE-10044StockholmSweden",
"Instituto Tecnológico de Aeronáutica\nSão José dos CamposBrazil",
"Linné FLOW Centre\nKTH Royal Institute of Technology\nSE-10044StockholmSweden",
"Instituto Tecnológico de Aeronáutica\nSão José dos CamposBrazil",
"Linné FLOW Centre\nKTH Royal Institute of Technology\nSE-10044StockholmSweden"
] |
[] |
The current work presents a realizable method to control streaky disturbances in boundary layer flows and delay transition to turbulence by means of active flow control. Numerical simulations of the nonlinear transitional regime in a Blasius boundary layer are performed where streaks are excited in the boundary layer by means of a high level of free-stream turbulence. The occurring disturbances are measured by means of localized wall-shear-stress sensors and damped out using near-wall actuators, which resemble ring plasma actuators. Each actuator is powered by a time-varying signal whose amplitude is computed by processing signals from the sensors. The processed signal is the result of two control laws: the Linear Quadratic Gaussian regulator (LQG) and the Inverse Feed-Forward Control technique (IFFC). The use of the first control method, LQG, requires a state-space representation of the system dynamics, so the flow is described by means of a linear time-invariant operator that captures only the most relevant information of the dynamics and results in a reduced order model (ROM). The ROM is computed by means of the eigensystem realization algorithm (ERA), which is based on the impulse responses of the real system. Collecting such impulse responses may be unfeasible when considering free-stream turbulence because of the high dimensionality of the input forcing needed for a precise description of such a phenomenon. Here, a new method to identify the relevant system dynamics and generate the needed impulse responses is proposed, based on additional shear-stress measurements in an upstream location. Transfer functions between such measurements and other downstream sensors are obtained and allow the derivation of the ERA system, in a data-driven approach that would be realizable in experiments. Finally, the effectiveness of the technique in delaying bypass transition is shown. The work (i) presents a systematic and straightforward way to deal with high dimensional disturbances in order to build ROMs for a feasible control technique, and (ii) shows that even when considering practical constraints such as the type and size of actuators and sensors, it is possible to achieve at least as large delay of bypass transition as that obtained in more idealized cases found in literature.
|
10.1017/jfm.2019.793
|
[
"https://arxiv.org/pdf/1902.05049v1.pdf"
] | 119,483,981 |
1902.05049
|
ee897fed0f5db2287c8f74318176c6624c84eecc
|
A realizable data-driven approach to delay bypass transition with control theory
Pierluigi Morra
Linné FLOW Centre
KTH Royal Institute of Technology
SE-10044StockholmSweden
Kenzo Sasaki
Instituto Tecnológico de Aeronáutica
São José dos CamposBrazil
Ardeshir Hanifi
Linné FLOW Centre
KTH Royal Institute of Technology
SE-10044StockholmSweden
André V G Cavalieri
Instituto Tecnológico de Aeronáutica
São José dos CamposBrazil
Dan S Henningson
Linné FLOW Centre
KTH Royal Institute of Technology
SE-10044StockholmSweden
A realizable data-driven approach to delay bypass transition with control theory
Under consideration for publication in J. Fluid Mech. 1 (Received ?; revised ?; accepted ?. -To be entered by editorial office)
The current work presents a realizable method to control streaky disturbances in boundary layer flows and delay transition to turbulence by means of active flow control. Numerical simulations of the nonlinear transitional regime in a Blasius boundary layer are performed where streaks are excited in the boundary layer by means of a high level of free-stream turbulence. The occurring disturbances are measured by means of localized wall-shear-stress sensors and damped out using near-wall actuators, which resemble ring plasma actuators. Each actuator is powered by a time-varying signal whose amplitude is computed by processing signals from the sensors. The processed signal is the result of two control laws: the Linear Quadratic Gaussian regulator (LQG) and the Inverse Feed-Forward Control technique (IFFC). The use of the first control method, LQG, requires a state-space representation of the system dynamics, so the flow is described by means of a linear time-invariant operator that captures only the most relevant information of the dynamics and results in a reduced order model (ROM). The ROM is computed by means of the eigensystem realization algorithm (ERA), which is based on the impulse responses of the real system. Collecting such impulse responses may be unfeasible when considering free-stream turbulence because of the high dimensionality of the input forcing needed for a precise description of such a phenomenon. Here, a new method to identify the relevant system dynamics and generate the needed impulse responses is proposed, based on additional shear-stress measurements in an upstream location. Transfer functions between such measurements and other downstream sensors are obtained and allow the derivation of the ERA system, in a data-driven approach that would be realizable in experiments. Finally, the effectiveness of the technique in delaying bypass transition is shown. The work (i) presents a systematic and straightforward way to deal with high dimensional disturbances in order to build ROMs for a feasible control technique, and (ii) shows that even when considering practical constraints such as the type and size of actuators and sensors, it is possible to achieve at least as large delay of bypass transition as that obtained in more idealized cases found in literature.
(Received ?; revised ?; accepted ?. -To be entered by editorial office)
The current work presents a realizable method to control streaky disturbances in boundary layer flows and delay transition to turbulence by means of active flow control. Numerical simulations of the nonlinear transitional regime in a Blasius boundary layer are performed where streaks are excited in the boundary layer by means of a high level of free-stream turbulence. The occurring disturbances are measured by means of localized wall-shear-stress sensors and damped out using near-wall actuators, which resemble ring plasma actuators. Each actuator is powered by a time-varying signal whose amplitude is computed by processing signals from the sensors. The processed signal is the result of two control laws: the Linear Quadratic Gaussian regulator (LQG) and the Inverse Feed-Forward Control technique (IFFC). The use of the first control method, LQG, requires a state-space representation of the system dynamics, so the flow is described by means of a linear time-invariant operator that captures only the most relevant information of the dynamics and results in a reduced order model (ROM). The ROM is computed by means of the eigensystem realization algorithm (ERA), which is based on the impulse responses of the real system. Collecting such impulse responses may be unfeasible when considering free-stream turbulence because of the high dimensionality of the input forcing needed for a precise description of such a phenomenon. Here, a new method to identify the relevant system dynamics and generate the needed impulse responses is proposed, based on additional shear-stress measurements in an upstream location. Transfer functions between such measurements and other downstream sensors are obtained and allow the derivation of the ERA system, in a data-driven approach that would be realizable in experiments. Finally, the effectiveness of the technique in delaying bypass transition is shown. The work (i) presents a systematic and straightforward way to deal with high dimensional disturbances in order to build ROMs for a feasible control technique, and (ii) shows that even when considering practical constraints such as the type and size of actuators and sensors, it is possible to achieve at least as large delay of bypass transition as that obtained in more idealized cases found in literature.
Introduction
The laminar flow state is characterized by a lower friction drag than the turbulent one, which implies less energy consumption for many applications, such as transportation means like trains and aircrafts. Therefore, control of laminar-turbulent transition is of great interest in many technical areas. The transition scenario depends on a number of parameters, and an overall picture of these different scenarios can be found in Schmid & Henningson (2001). Transition to turbulence in boundary layer flows where free-stream turbulence has an intensity higher than ≈ 1% occurs rapidly and bypasses the classical scenario triggered by Tollmien-Schlichting (TS) waves, as showed by Arnal & Juillen (1978). When free-stream turbulence is present, a set of low-frequency vortices (Hultgren & Gustavsson (1981); Hunt & Durbin (1999); Zaki & Saha (2009)) enters the boundary layer and causes the appearance of elongated streaky structures of alternating high and low streamwise velocity. This was firstly observed in the experimental studies of Klebanoff (1971). The amplitude of such velocity fluctuations grows linearly along the streamwise direction (Andersson et al. 1999;Luchini 2000) and is accompanied by growing secondary fluctuations of the streaky structures on the planes perpendicular to the streamwise direction. When the amplitude of such secondary cross-flow fluctuations is sufficiently high turbulent spots appear (Brandt & Henningson 2002), which grow and merge further downstream and ultimately lead to a fully turbulent flow. This process was observed both in experiments (Matsubara & Alfredsson 2001) and simulations (Brandt et al. 2004). Thus, the boundary layer can be divided into three zones: (i) an upstream zone where there is high level of receptivity and free-stream turbulence triggers disturbances in the boundary layer, (ii) a middle zone where streaky disturbances grow due to the linear lift-up mechanism, and (iii) a downstream zone where the flow nucleates turbulent spots which grow and merge as they propagate downstream until the boundary layer becomes fully turbulent.
The boundary layer flow in the middle zone can often be described with sufficient accuracy by the linearized Navier-Stokes equations. The possibility to work with a linear system greatly facilitates the application of flow control techniques. The a priori knowledge of the linear behavior of the TS-waves was exploited in the experiments of Thomas (1983) and in the simulations of Laurien & Kleiser (1989) to counteract TS-waves and delay transition. Similarly, for bypass transition, Jacobson & Reynolds (1998) exploited the linearity of the dynamical system to show the possibility to damp streaky structures. In those works the a priori knowledge of the system dynamics was used to create ad hoc counter disturbances. Such ad hoc practice lacks in generality and may require tedious testing, therefore it is appealing to apply the optimal control theory to flow control problems. The control-theory community has produced many reliable and elegant techniques to tackle linear systems. Among the first successful applications of the optimal control theory in fluid mechanics are the works of Joshi et al. (1997), Bewley & Liu (1998), Högberg & Bewley (2000) and Högberg et al. (2003), where optimal control methods are applied to linearized systems and used in fully non-linear channel flows. More recently, Monokrousos et al. (2008) showed the successful application of the Linear Quadratic Gaussian regulator (LQG) for control of streaks triggered by the free-stream turbulence.
In optimal control techniques the final goal is finding the function that takes measurements as input and gives actuation signals as output while minimizing an objective function. Particularly, in classical optimal control methods the optimal solution for linear time-invariant systems is given by solving an algebraic Riccati equation, which consists of a matrix equation whose dimensions are roughly as those of the original linear system to control. If the original linear system has large dimensions, as is the case in fluid mechanics, the solution of the algebraic Riccati equation may be extremely computationally demanding. A possible solution is reducing the order of the optimal control problem by keeping only the information useful for the control. This is the idea behind reduced order models (ROMs). In fact, measurements usually contain only a portion of the total information present in the system and actuators can usually excite only certain structures. In control theory, such limitations posed by sensors and actuators define two properties of the system: its observability and its controllability, respectively. The control problem alone needs only the portion of the system that is observable and controllable. The practice of model reduction in flow control was treated in Bagheri et al. (2009), Semeraro et al. (2011), Poussot-Vassal & Sipp (2015 and Yao & Jaiman (2017). The approach was shown to be successful in the sense that the solution to the control problem was nearly unaffected by the use of a ROM. A classic technique for achieving a ROM is the Eigensystem Realization Algorithm (ERA) (Juang & Pappa 1985;Semeraro et al. 2013). ERA is based on a set of impulse responses from each input (actuators and disturbances) to each output (measurements). Ma et al. (2011) showed that the ROM achieved by the ERA is equivalent to that achieved by balanced truncation. This means that the ROM resulting from the ERA is a projection of the original system onto the set of modes given by the intersection of the set of the most observable flow structures and the set of the most controllable flow structures. Qualitatively, an observable structure is one that generates non-zero outputs whereas a controllable structure is one that can be excited by the inputs. The term "most" is obviously case dependent. For controllable structures it represents the number of flow structures used to recreate with an acceptable small error the flow field obtained by an impulse response. The same reasoning holds for the observable flow structures but with respect to the adjoint system (see Bagheri et al. (2009) for a more detailed discussion of controllability and observability in fluid mechanics systems). Examples of ERA applications in fluid mechanics are found in Semeraro et al. (2013), for control of a three-dimensional non-linear TS-wave packet, and Sasaki et al. (2018a), for control of three-dimensional TS-waves arising from stochastic disturbances.
A consistent modeling of the inputs implies correctly modeling the space spanned by the disturbances, which may require the use of a basis with as many degrees of freedom as the dimensions of the desired space. Thus, in case the space spanned by a disturbance or an actuator has large dimensions it may become unfeasible to collect all the impulse responses. A similar issue may also happen when the number of outputs is very high. Another possibility to avoid demanding computations for solving the control problem is dropping the use of model-based methods as discussed by Fabbiane et al. (2014), who made use of a learning algorithm that needs only minimal modeling.
The present work addresses the delay of bypass transition in a physically realizable framework. We use a finite number of localized near-wall actuators that resemble ring plasma actuators (Kim & Choi 2016;Kim et al. 2017;Shahriari et al. 2018) and localized wall-shear-stress sensors. We also assess behavior of two model-based optimal control methods: the LQG regulator and the Inversion Feed Forward Control (IFFC) (Sasaki et al. 2018a). Moreover, we make use of the ERA to generate the ROM, and present a technique that allows to account for the free-stream turbulence without resorting to the same high dimensional basis used for the description of the disturbance in the fully non-linear simulations. This permits us to collect a smaller set of impulse responses, which makes the system associated with the control problem much smaller and less computationally demanding. This technique is based on measurements only, which makes it feasible in experiments as well.
The paper is structured as follows. In § 2 the equations used to describe the full or reduced system dynamics are introduced; in § 3 the control techniques of interest are briefly described; in § 4 the details of the framework for the non-linear simulations are outlined; in § 5 the identification techniques and the used identified models are presented; in § 6 the behavior of the designed controller in the non-linear Navier-Stokes is assessed. A summary of the main conclusions is given in § 7.
Governing equations
Dynamical system
The Navier-Stokes equations can be expressed in terms of the perturbation quantities as
∂q ∂t = −(q · ∇)q B − (q B · ∇)q − (q · ∇)q − ∇p + Re −1 ∇ 2 q + f , (2.1a) ∇ · q = 0, (2.1b) q = q 0 at t = 0, (2.1c)
where q = q (x, t) is the perturbation velocity vector, q B = q B (x) the unperturbed velocity vector, p = p (x, t) the perturbation pressure, f = f (x, t) a body force vector, Re the Reynods number, t the time, x = (x 1 , x 2 , x 3 ) T the space variable, and ∇ = (∂ x1 , ∂ x2 , ∂ x3 ) the gradient operator.
Here, the unperturbed velocity vector q B is the solution of an evolving zero-pressuregradient boundary layer. The velocity perturbation q satisfies no-slip conditions at the wall x 2 = 0 and Neumann conditions at free-stream x 2 = L x2 . Periodicity is assumed along the spanwise direction x 3 and enforced along the streamwise direction x 1 by means of an artificial forcing f BC = λ(x 1 )q , which is placed in a fringe region at the outlet. λ(x) is a non-negative function which is non-zero only within the fringe region. f BC forces all perturbations to zero and modifies q B to be periodic (see Nordström et al. (1999) for details about f BC ).
The flow control problem consists in finding the correct external action that modifies the fluid dynamics to achieve a specific goal. In our case, such external action can take the form of a boundary condition or a body force and can be expressed as a function of time and space. It follows that the problem can be split into finding the correct spatial distribution of such an action and its time modulation. In the present work it is assumed that the spatial distribution and the time modulation of the external action are decoupled. The spatial distribution is prescribed, so the flow control problem reduces to the computation of its time-varying amplitude. From now on this time-varying scalar is referred to as input. Using a finite number of actuators N u , the external action used for control reads
f u = Nu k=1 u(t) k b(x) k , (2.2)
where b(x) k is the spatial shape of the k-th body force, and u(t) k the corresponding time variation. The latter represents the control input. Free-stream turbulence is modeled as a forcing in the fringe region. f BC is modified to force q to be equal to a prescribed perturbation that mimics the presence of free-stream turbulence. The prescribed perturbation is of the form
q F ST = α β ω Φ(α, β, ω)q (x 2 , α, β, ω)e i(αx1+βx3−ωt) ,(2.3)
withq an eigensolution to the Orr-Sommerfeld Squire eigenvalue problem for a parallel flow in a semi-bounded domain, α the streamwise wavenumber, β the spanwise wavenumber, and ω the angular frequency. Free-stream disturbances are thus expanded as a sum of eigenfunctions of the linearized parallel-flow problem (see Brandt et al. (2004) for more details). A linearized version of the Navier-Stokes equations about q B can be obtained by dropping the non-linear term (q · ∇)q from equation (2.1a).
Reduced-order dynamical system
The linear dynamical system used for the application of control theory techniques is a ROM and readsq
(t) = Aq(t) + Bu(t) + M d d(t),(2.4)
where q = q(t) is the N ×1 state vector (which generally is not exactly the same quantity represented by q ),q is its time derivative, A the N × N matrix that defines the system dynamics, B the N × N u matrix that characterizes the control inputs, u = u(t) a N u × 1 column vector containing all the input amplitudes u(t) k , M d the N × N d matrix that characterizes the disturbance inputs, and d = d(t) a N d × 1 column vector containing all the input amplitudes d(t) k . N is the degree of freedom of the ROM, N u the number of control inputs, and N d the number of disturbance inputs.
We also assume to have access to two finite sets of measurements: y(t), N y ×1, and z(t), N z × 1, where N y and N z represent the respective number of measurements available. It holds
y(t) = C y q(t), z(t) = C z q(t), (2.5)
where the N y × N matrix C y and the N z × N matrix C z characterize the measurements in the ROM. From now on y(t) and z(t) are referred to as outputs. Equations (2.4) and (2.5) form a ROM state-space representation of the system. A different description of the system can be given by means of transfer functions (TF). TFs are built by performing the Laplace transform on the state-space representation and in general describe the system as function of the angular frequency ω only. Here, sensors and actuators are placed on straight lines along the spanwise direction (figure 1), with N u = N y = N z . Since the flow is periodic in the spanwsie direction, TFs, inputs and outputs can be expressed as functions of the spanwise wavenumber β as well. The description by means of TFs readŝ
y(ω, β k ) =Ĝ uy (ω, β k )û(ω, β k ) +Ĝ dy (ω, β k )d(ω, β k ), z(ω, β k ) =Ĝ uz (ω, β k )û(ω, β k ) +Ĝ dz (ω, β k )d(ω, β k ),
(2.6) whereĜ =Ĝ(ω, β k ) are the TFs,ŷ =ŷ(ω, β k ) andẑ =ẑ(ω, β k ) the outputs in the frequency domain,û =û(ω, β k ) andd =d(ω, β k ) the inputs in the frequency domain, and k is used to stress the fact that the number of outputs is finite, so there is a finite amount of available wavenumbers. From now on all the variables denoted by a hat symbol are function of (ω, β k ), and the explicit writing (ω, β k ) is dropped.
The description of the system by means of TFs can be translated in the physical domain, leading to
y(t) k = t 0 Nu m=1 G uy km (t − τ )u(τ ) m dτ + t 0 N d m=1 G dy km (t − τ )d(τ ) m dτ, z(t) k = t 0 Nu m=1 G uz km (t − τ )u(τ ) m dτ + t 0 N d m=1 G dz km (t − τ )d(τ ) m dτ, (2.7)
with k the output index, and m the input index.
Control techniques
The present configuration of outputs and inputs together with the convective nature of the flow make all the control techniques described in this section be in a feed-forward configuration (Belson et al. 2013).
The control techniques used in the present work are all based on the assumption that the input u(t) k is a function of the upstream outputs y(t) m ,
u(t) k = t 0 Ny m=1 G yu km (t − τ )y(τ ) m dτ,(3.1)
with k = 1, 2, . . . , N u . Since q B is independent of the spanwise direction x 3 , the instantaneous linearized system dynamics is homogeneous along x 3 . The latter and the fact that the outputs are all given by the same type of sensor allows to drop the usage of the index k in (3.1) to have G yu m . Then, (3.1) can be rewritten as
u(t) k = t 0 Ny m=1 G yu m (t − τ )y(τ ) m+k−1 dτ, (3.2)
where for m + k − 1 > N y spanwise periodicity implies the use of m + k − 1 − N y .
Linear Quadratic Gaussian regulator
The technique is based on a linear model, aims at minimizing a quadratic cost function, and assumes the presence of Gaussian white noise disturbances. The addition of Gaussian white noise on the output y(t) in the state-space representation readsq
(t) = Aq(t) + Bu(t) + M d d(t), y(t) = C y q(t) + n(t), z(t) = C z q(t), (3.3)
where n(t), N y ×1, is the time modulation of the noise. d(t) is also treated as white noise.
There is no addition of noise on the output z(t) because it represents a reference output to minimize, whose measurement is not available in reality. The noise on the output y(t) corresponds to noise in a real available measurement.
The covariance matrices associated with d(t) and n(t) are V d , N d × N d , and V n , N y × N y , respectively, and are both diagonal and constant because of the assumption that d(t) and n(t) are white noise disturbances; in particular, it can be written
V d = v d I, V n = v n I, (3.4)
with v d > 0 and v n > 0 real scalars and I the identity matrix. The technique consists in finding G yu m by minimizing a prescribed H 2 -norm of interest. The disturbances d(t) and n(t) are treated as random variables, so the objective function of interest is defined as the expected value of an H 2 -norm. Here, the objective function contains both the reference output z(t) and the input for the control u(t), which is added to avoid an infinite amplitude of the input signal, penalizing excessive control action. The objective function reads
J = E lim T →∞ 1 T T 0 z(t) T Qz(t) + u(t) T Ru(t) dt , (3.5)
where the N × N matrices Q and R are design weights. The operator E[•] represents the expected value. From now on the matrices V d , V n , Q and R are referred to as weight matrices or design weights. In the LQG it is assumed that u(t) is a linear function of the states, but it is also assumed that not all the states are known at each time instant, so a second system for state estimation is introduced. The estimation system makes use of the known outputs to reconstruct the states at each instant of time, and is designed to minimize the estimation error. Thus, in addition to the minimization of the objective function (3.5) to compute the input that controls the system, the estimation introduces a second minimization problem. Generally these two minimization problems are coupled, but in the LQG they are independent and solved separately. They consists in the Linear quadratic regulator, which solves the control problem by assuming full-state information, and the Kalman filter, which solves the estimation problem by assuming stochastic disturbances on the outputs. The solution of the LQG is the combination of the two independent solutions. More details about the LQG are given in Appendix A, whereas a thorough description can be found in Lewis & Syrmos (1995).
Inversion Feed Forward Control
Inversion feed forward control (IFFC) is a technique developed in the frequency domain, and is based on a system described by TFs (2.6). The technique is exactly the same one used in Sasaki et al. (2018a), but in that work it is referred to as Wave Cancellation. The authors decided to change the nomenclature to adopt the name used in the control community.
The contribution of the disturbanced in the second equation of (2.6) may also be expressed asĜ dzd =Ĝ yzŷ +p, (3.6) whereĜ yz is a TF to design in order to maximize the extraction of information from y, whilep is the residual part of the information inẑ which is not retrieved byĜ yzŷ . The loss of information, i.e.p = 0, may be unavoidable and can be seen by resorting to the state-space representation. The matrix C y that characterizes the output y(t) does not necessarily span the same space spanned by the matrix that describes the system dynamics A, so the outputs y(t), being in a subspace, cannot reconstruct the whole state space. It follows that the only portion of the signalẑ(ω, β k ) which can be obtained from the outputs y(t) isz
=Ĝ uzû +Ĝ yzŷ , (3.7)
wherez is an estimate ofẑ. The objective of the control problem is the annihilation of the outputz. Then, a straightforward strategy to solve the problem is imposingz = 0, which is the basic idea behind IFFC. Assumingû =Kŷ in (3.7) giveŝ
K = (Ĝ uz ) −1Ĝyz , (3.8)
whereK solves the control problem in the frequency-wavenumber domain. The result in (3.8) is ill-conditioned in the zeros ofĜ uz , which may lead to spurious high amplitudes of the input. Moreover, model uncertainties are not considered. Such limitations are addressed in Devasia (2002), where the TFs, the inputs and the outputs are functions of the angular frequency ω only. Here, the same approach is used with some modification to account for a system description as function of ω and β k , following Sasaki et al. (2018a). The technique makes use of two weights,R andQ, which here are taken as constant, and solves the control problem by minimizing the following prescribed objective function
J = ∞ −∞ k û HRû +ẑ HQẑ ∆β k dω, (3.9)
where the superscript H indicates the complex conjugate transpose. The presence of the objective function turns the nature of the problem into an H 2 optimal control problem, whose solution is given byK
= (Ĝ uz ) HQĜyẑ R + (Ĝ uz ) HQĜuz .
(3.10)
The inverse Fourier transform ofK gives G yu m as in equation (3.2); only the causal part of G yu m is used for control, since actuation must be decided based solely on present or past information from the sensors.. This technique was already applied by Sasaki et al. (2018a,b) and showed to be successful in the control of Kelvin-Helmholtz and Tollmien-Schlichting waves, where the equivalence between LQG and IFFC for the damping of TS waves was also shown.
Plant
The domain of interest is a box as shown in figure 1, where the white symbols represent the outputs and the black symbols the inputs. For flow simulations the pseudo-spectral code SIMSON (Chevalier et al. 2007) is used. Here, the reference length is taken to be the displacement thickness of the boundary layer at the inlet δ * 0 and the reference velocity is the free-stream velocity U ∞ . In all of the present simulations the Reynolds number is Re = U ∞ δ * 0 /ν = 300. All the results that involve transition to turbulence are performed by means of LES simulations on a box of dimensions (L x1 , L x2 , L x3 ) = (4000, 60, 50) with (N x1 , N x2 , N x3 ) = (1024, 121, 108) points for the discretization. The effect of the large-eddy simulations (LES) filter (see Schlatter et al. (2004), Schlatter et al. (2006a), Schlatter et al. (2006b) for details) in the area where the flow dynamics is linear is negligible (Monokrousos et al. 2008). All the results that do not need to include the fully turbulent regime are performed by direct-numerical simulations (DNS) on a box of dimensions (L x1 , L x2 , L x3 ) = (1000, 60, 50) with (N x1 , N x2 , N x3 ) = (1152, 121, 108) points for the discretization. The points along the wall-parallel directions are equi-spaced, whereas along the wall-normal there are Gauss-Lobatto points.
The free-stream turbulence is modeled by superposition of 200 random modes from the continuous Orr-Sommerfeld-Squire spectrum. The integral length scale and the turbulent intensity used for all presented results are respectively L = 7.5δ * 0 and T u = 3.0%, considering the free-stream turbulence spectrum in Brandt et al. (2004).
As shown in figure 1, the input and output devices are placed along straight lines. The first set of outputs is placed at x 1,y = 250, which is downstream of the zone with high receptivity. The second set of outputs is placed at x 1,z = 400, since after that position the non-linearities start to be non-negligible. Input signals, corresponding to actuators, are generated at x 1,u = 325 to have the same ∆x 1 between input and outputs, such that the traveling time of a disturbance from the first set of outputs to the inputs is roughly the same to the traveling time from the inputs to the second set of outputs. The chosen location for the devices is also optimal in terms of identification accuracy for control design, as shown in Appendix B. The number of devices along the spanwise direction is the same for each set and it is equal to N u = N y = N z = 36. Such a choice is motivated by analyzing the wavenumber spectrum of the average disturbance energy. According to Shannon information theorem the sampling wavenumber needs to be at least twice the wavenumber of interest. In this case measuring the highest non-negligible spanwise fluctuation would require at least 18 devices. In order to have a better measurement of the spanwise fluctuations 36 devices are used. The devices are equi-spaced along the spanwise direction. The shape of the input actuator is given by
b(x) = {0, b 2 (x), 0} T , (4.1) with b 2 (x) = exp − (x 1 − x 1,0 ) 2 σ 2 x1 − (x 2 ) 2 σ 2 x2 − (x 3 − x 3,0 ) 2 σ 2 x3 , (4.2)
where σ x1 = 3, σ x2 = 5, and σ x3 = 1.5. The actuator shape resembles that of ring plasma actuators (see Shahriari et al. (2018); Kim & Choi (2016); Kim et al. (2017)), generating a body force in the wall-normal direction. This is efficient to excite or cancel streaks due to the lift-up effect. A detailed analysis on the effect of the actuator shape is presented in Sasaki et al. (2019), which is the parallel work to the present one. The outputs are computed as
1 S S ∂q 1 ∂x 2 x2=0 dS, (4.3)
with S the area on the wall where the measure is taken. This is an averaged measure of the shear stress associated to the perturbation part of the streamwise velocity component on the wall.
Reduced-order modelling and control design
Both control methods introduced in § 3 are model-based. The IFFC technique requires the knowledge of two TFs,Ĝ zu andĜ yz .Ĝ zu is by definition the Fourier transform of the output signal resulting from an impulse-response simulation of the linearized Navier-Stokes, whereasĜ yz needs to be modeled. The LQG technique, instead, requires the knowledge of the matrices A, B, C y , C z and M d , which characterize the ROM and need to be modeled.
The techniques used for this modeling are introduced in the remainder of this section, and are all based on input-output data, which is usually available in experiments. In input-output data part of the information about the system dynamics is lost. However, its usage is a reasonable design choice, since the control techniques work only with the observable and controllable structures, whose time evolution is described by input-output signals. In fact, by definition, the information lost in input-output data is the one associated to the unobservable and uncontrollable structures.
The present configuration of outputs and inputs together with the convective nature of the flow allow to estimate the downstream outputs z(t) from the upstream outputs y(t). This fact is exploited in the following part of this section.
Empirical TFs
The estimation of downstream outputsẑ by means of upstream outputsŷ can be performed by designing a TFĜ yz . Here,Ĝ yz is computed by means of an identification technique using the information extracted from the output data. The TF obtained in this way is referred to as empirical TF. This method was introduced in Sasaki et al.
(2017) for the estimation of a turbulent jet and applied in Sasaki et al. (2018b) for the closed-loop control of a two-dimensional shear layer. Here, the approach is extended to a flow with spanwise periodicity, i.e. outputs are function of β k as well. It was shown in Bendat & Piersol (2011) that the optimal frequency response, in the least square sense, is defined from the auto-and cross-spectra of the input and output signalŝ
G yz =Ŝ yẑ S yy ,(5.1)
whereŜ yy andŜ yz are respectively the auto-and cross-spectra of the input and output signals. BothŜ yy andŜ yz are computed as the expected values ofŷ Hŷ andŷ Hẑ , which are obtained via the process of ensemble averaging (Bendat & Piersol 2011). Equation (5.1), sometimes referred to as anĤ 1 estimator (Rocklin et al. 1985), minimizes the error due to noise in the output. One desirable property of an H 1 estimator is that the prediction error is linearly uncorrelated to the available output signal (Rocklin et al. 1985;Bendat & Piersol 2011). Any remaining errors correlated to the available signal are either due to the presence of noise in the measurements or to spectral leakage, which is unavoidable because the signal is not exactly periodic in time. Spectral leakage can be minimized by using long time series, by windowing the signal for the ensemble averaging, or via the calculation of an improved frequency response, as outlined in the following section.
Improved frequency response
The method considered here is referred to as improved frequency-response and consists of improving the accuracy of a TF by means of an iterative algorithm. This allows to obtain a more accurate linear approximation of a system and is particularly interesting when the impulse responses of the disturbances are not available or unfeasible to collect, as is the case for the free-stream turbulence or experimental implementations. The method is designed to minimize noise, spectral leakage and capture some nonlinearity (Schoukens et al. 1998).
The algorithm is initialized with a first-guess TFĜ yz 0 , which may be, for instance, the result obtained from equation (5.1). Then, the estimation error, which is the difference between the signal obtained by usingĜ yz 0 and the available outputŷ, is computed. The error reads
e(t) k = z(t) k − t 0 m G yz 0,m (t − τ )y(t) m dτ. (5.2)
with G yz 0,m (t) the inverse Fourier transform ofĜ yz 0 . Then, the TF between the error and the available outputŷ is computed asĜ yz e =Ŝ ye /Ŝ yy , which is used to update the initial TF asĜ yz 1 =Ĝ yz 0 +Ĝ yz e . Iterations are performed until the error TF is minimized.
Eigensystem realization algorithm using TFs
In § 3 it was shown that in order to design the LQG regulator it is necessary to solve two algebraic Riccati equations, and the computational power required for their solution grows quickly with the dimensions of the matrix A, which is the linear time-invariant operator used to describe the linearized system dynamics. Clearly, for fluid mechanics systems, which in general present numerous degrees of freedom, the usage of a ROM is preferable (Kim & Bewley 2007).
For the realization of the ROM we use the ERA (Juang & Pappa 1985). The ERA is based on output signals resulting from impulse responses. It is necessary to have access to an amount of impulse responses equal to the number of total inputs of the systems. The signals are written in a Hankel matrix whose dimensions depend on the total number of inputs and outputs and on the length of the saved time series, i.e. N t (N y + N z ) × N t (N u + N d ), N t being the number of time samples needed to have a good representation of the impulse response. In case the disturbance is free-stream turbulence the number of degrees of freedom used for the implementation of d(t) in the fully non-linear Navier-Stokes solver is of the order of hundreds. Then, since the Hankel matrix is decomposed by the Singular Value Decomposition, it is clear that collecting such a high number of impulse responses results in a heavy computational problem. Besides, in a practical application it is not possible to collect the impulse responses from the free-stream turbulence disturbance.
Therefore, in order to reduce the computational power required and to have a method that can be applied in experiments as well, a different approach is proposed. A new set N y × 1 of outputs y d (t), which measure the same quantity as y(t) and z(t), is introduced upstream of y(t), and the outputs generated by a non-linear Navier-Stokes simulation with free-stream turbulence and without control action are stored. An impulse response coincides with a TF by definition, so the following TFs can be computed as in equation
(5.1),Ĝ y d y =Ŝ y d ŷ S y d y d ,Ĝ y d z =Ŝ y d ẑ S y d y d ,(5.3)
and their inverse Fourier transforms can be used as a set of impulse responses to mimic the presence of free-stream turbulence upstream every control device. These estimated impulse responses are used in the ERA to model the impulse responses coming from M d d(t) in equation (3.3), which represents the disturbance in the system. The number of impulse responses from the actuators, which are characterized by Bu(t), is N u . Nevertheless, only one of these impulse responses is collected because the other ones can be computed by exploiting the homogeneity of q B and the periodicity of the flow along the spanwise direction.
Once the whole set of impulse responses is available it is possible to build the mentioned Hankel matrix, apply the ERA, and retrieve the ROM needed for the design of the LQG. To the best of the authors' knowledge, this is the first time this approach is used in fluid mechanics.
Identified reduced order model
Given the position of the sensors, data-driven TFs can be computed by exploiting the methods outlined in § 5.1 and § 5.3 and improved as described in § 5.2. Ensemble averaging is performed on time series with a sampling frequency of 1/0.3, over a sampling time of T = 25000, with a number 2500 of samples per each segment, an overlap of 80%, a total number of 166 segments, and by means of a triangular windowing function. The improvement of the empirical TF, with available outputs y(t) k at x 1 = 250 and estimated outputsz(t) k at x 1 = 400, is summarized in figure 2 and table 1. Figure 2 shows that the empirical TF estimates an outputz(t) k that is smoother than the original z(t) k taken from the DNS: the contours of the empirical TF do not show the sharp peaks of the DNS output (the darker areas). This implies that the empirical TF lacks accuracy in estimating the higher frequencies present in the original signal. The reason is likely spectral leakage together with the lower amplitude that the higher frequencies have with respect to the lower frequencies. This difference is decreased in the improved TF because the error, as in equation (5.2), is computed in the physical domain, where it is possible to isolate the higher frequencies bypassing the issue of relative amplitude and spectral leakage. In fact, the improved TF estimates better the higher frequencies, as in figure 2. Errors in amplitude estimation are also related to the windowing procedure of the time signal in the ensemble averaging. Even though the window is chosen to minimize such errors, its usage inevitably alters the computed amplitudes. Table 1 shows the normalized correlation value at zero delay, the mean-square (MS) and the variance (VAR) for the empirical TF and for the improved TF.
The TF that estimates the outputz(t) k given the available output y(t) k , i.e.Ĝ yz , is used in the IFFC control technique. The identified TFs used to mimic the effect of freestream turbulence, characterized by M d d(t) in the ROM, assume as available output a set of sensors y d (t) k at x 1 = 175 and estimate the outputs at x 1 = 250, 400, i.e. y(t) and z(t). These TFs correspond to equation (5.1).
The ROM resulting from the ERA consists of N = 387 degrees of freedom, which is considerably less than the degrees of freedom of the full system. The solution of the algebraic Riccati equations, which is the most computationally demanding step in the control design, with N = 387 can be computed within the order of minutes nowadays (on a laptop). The value N = 387 is found by imposing the error between the impulse response from the ROM and the original impulse response to be small enough. Since the ERA performs the Singular Value Decomposition of an Hankel matrix and the singular values are ordered such that σ i > σ i+1 , the ratio σ N /σ max ≤ 5 · 10 −4 is used to determine N . Figure 3 compares the impulse response from d(t) in the ROM resulted from the ERA against the estimated TF used as original impulse response in the ERA. The TFs are centered in zero and present a peak which is related to the group velocity of the structures. There clearly is good agreement between the ROM and the original data.
Control performance: transition delay
Here, the results from the non-linear simulations are presented.
Control method Control problem Estimation problem IFFC Q = 1, R = 2 · 10 4 none LQG Q = 1, R = 50 v d = 1, vn = 5 · 10 −4
Transition delay
The G yu m (t) used in the non-linear N-S simulations are those resulting in the best performance in the control design. The weight matrices of the J functionals of equations (3.5) and (3.9) are summarized in table 2 for IFFC and LQG, respectively. The choice of weights for control design is performed through input-output simulations based on time signals stored from uncontrolled non-linear N-S simulations. The input-output simulations consists in a linear superposition of signals, which makes them computationally inexpensive, such that their usage for control design becomes convenient to determine appropriate weights (details can be found in Appendix C).
In order to assess the performance of the controller, the following quantity is introduced
E = J M controlled J M uncontrolled , J M = 1 T T 0 z(t) T z(t) dt,(6.1)
which corresponds to the average behavior of the output to minimize. T is the total time of the simulation. Table 3 shows the performance of each control technique resulting from the inputoutput and non-linear N-S simulations. The comparison of the control techniques in the input-output simulation is consistent with the results of the fully non-linear Navier-Stokes: LQG outperforms IFFC. Moreover, both cases present a smaller reduction of the objective function in the results from the N-S simulations than the input-output ones, which may be attributed to non-linearities. In figure 4 the kernels from the IFFC and the LQG methods, respectively, are shown. These correspond to G yu m (t) as in § 3. It appears that the LQG weights a lot more the recent history of the signal than the IFFC does, which may be seen as the main reason for outperforming the IFFC result as further discussed next in § 6.2. In figure 5, the spanwise RMS values of G yu m (t)
RMS[u] x3 = 1 N u Nu k=1 u(t) 2 k 1/2 = 1 N u u(t) T u(t) 1/2 , (6.2)
are plotted. We observe that: (i) the magnitude of the averages and the fluctuation around the average values are higher for the signal generated by G yu m (t) from the LQG, and (ii) the actuation signal from the IFFC is smoother. The first difference can explain why the performance of the LQG drops more than that of the IFFC when moving from the inputoutput to the Navier-Stokes simulations. In fact, the actuation signal multiplies a fixed spatial support, and an increase in the magnitude of the actuation signal corresponds to a more intense forcing that leads to stronger non-linearities. The second difference comes from the shape of the G yu m (t) in figure 4. The G yu m (t) from the IFFC is less localized around t = 0 than the one from the LQG, which means that the latter only captures low-frequency dynamics of y(t) k signals. This can also be seen by comparing figures 5 and 6.
The effect of the actuation signal on the spanwise RMS values of the streamwise velocity component, shown in figure 7. It is clear that LQG outperforms slighlty IFFC also in terms of reduction of the disturbance amplitude throughout the boundary layer. The disturbance energy drops by a factor of ≈ 40% after the control action. As shown in figure 8, despite the growth of disturbance amplitude beyond x 1 = 400 (Re x ≈ 1.5 × 10 5 ) where the output z(t) k for the objective function is measured, a clear transition delay is achieved. A different measure of transition delay is the skin friction coefficient C f , which explicitly appears in the computation of the drag and is the measure of interest for many applications. The behavior of the C f based on the RMS values of the streamwise velocity component,
q 1,rms = < (q 1 − < q 1 > t ) 2 > t,x3 ,(6.C f (x 1 ) = ∂ ∂x 2 < q 2 1 > t,x3 x2=0 , (6.4)
is shown in figure 9. There, the threshold curves that represents the skin friction of a laminar and a fully turbulent flat-plate boundary layer are also presented. It clearly appears again that LQG performs slightly better than IFFC and that in the best case the transition delay is around ∆Re x = 1.5 × 10 5 , which is at least as good as most of the current results in literature where more idealized cases are studied, as in Monokrousos et al. (2008). There a transition delay around ∆Re x = 1.2 × 10 5 is achieved in the best possible scenario for a case with turbulence intensity T u = 3.0% and integral length scale L = 5.0δ * 0 . Nevertheless, in Monokrousos et al. (2008) actuation is performed by control of each point on a band of the flat plate, while measurements consists of wall shear-stress Bottom: controlled case with LQG. Black and white colors: c f < 2.5 × 10 −3 . Empty contours: c f ≥ 2.5 × 10 −3 . Same time t = t * for both simulations; the starting seed for the random free-stream turbulence generation is the same.
in both streamwise and spanwise directions and wall pressure fluctuations over a band of the flat-plate located downstream of the actuation. A visualization of the instantaneous behavior of the skin friction coefficient over the flat plate,
c f (x 1 , x 3 , t) = ∂ ∂x 2 q 1 (x, t) x2=0 (6.5)
is shown in figure 10. Although figure 10 is a snapshot at an arbitrary t = t * , it clearly appears that the controlled flow is smoother than the uncontrolled one for Re x < 5 · 10 5 , it does not present the wiggles associated to secondary instability, and presents strong chaotic structures further downstream than the uncontrolled case, consistently with figure 10. These are all evidences of transition delay. Figure 11. Full non-causal control TF G yu k (t) from the IFFC control method. Left: central line; the dot represents the peak value. Right: full TF.
Role of control methods, sensors and actuators for control performance
The reason behind the LQG outperforming the IFFC is now discussed from a physical point of view.
A limiting characteristic for control performance is the relative position of sensors and actuators, which was chosen to minimize prediction errors and exploit the linear behavior of the flow field. The input-to-output and the output-to-output time delays are respectively τ uz = 219.6 and τ IF F C yz = 216; these delays correspond to the peak of the transfer function, and approximate the average time for a structure induced by free-stream turbulence or the actuator, respectively, to arrive at the z(t) sensors. Therefore, the actuation is not sufficiently fast to cancel the streaks detected by the y(t) sensors, which is reflected by IFFC resulting in a non-causal G yu , as shown in figure 11. The result suggests that the performance of the controller may improve if it were possible to increase the difference between the time delays, which can be achieved either by changing the relative position of the devices or by changing the type of sensors or actuators. The limitations in the control performance are therefore not caused by the control methods, but by the structure of the plant. However, it appears that LQG can slightly compensate for this causality issue without any modification to the plant.
There exists a specific set of weights for which LQG outperforms IFFC, but there also exists a different set of weights for which LQG results in the same solution given by the IFFC. This is possible thanks to the presence of the estimation problem in the LQG, which introduces two more degrees of freedom in the control design. It follows that the reason behind the better performance of the LQG is the possibility of optimizing the estimation inside the control method, which is not included in the IFFC. Moreover, it appears that in the LQG keeping fixed the weight ratio R LQG /Q LQG associated to the best solution and increasing the value of the ratio v n /v d leads to worse performance (Appendix C). Thus, since the weights of the estimation problem define the dynamics of the estimator (Appendix A), the best solution comes from the estimator with the fastest response. In the present ROM the estimator that corresponds to τ LQG yz = τ IF F C yz = 216, as in the G yz used in IFFC, has a slower response than the one corresponding to the best solution found with LQG, where τ LQG yz = 213. Thus, it appears that LQG achieves better performance for a case where the difference between the time delays, τ uz − τ yz , is higher than it is in IFFC, as suggested by the non-causal result found from the IFFC, and that better performance is achieved by an estimator with a fast response. The latter occurs because the weights of the estimation problem define the dynamics of its error: an estimator with a fast response has a fast decaying error. This implies that after the same ∆t the faster estimator is more accurate.
The connection between the shorter time delay τ LQG yz = 213 mentioned earlier and the presence of a fast responding estimator may be explained by analyzing the capability of the wall shear-stress measurements to capture the dynamics of the streaks. An alternative measure of the streaky perturbations is their maximum streamwise velocity. Thus, a new set of outputs at (x 1 , x 2 ) = (250, 2.25) with the same spanwise positions of the other considered outputs is collected, as in figure 12b. These outputs are compared to those that measure the shear on the wall at (x 1 , x 2 ) = (250, 0), which are used to compute the input u(t) k . Figure 12a shows the time-space correlation between the two sets of measurements. It appears that the output resulting from the measurements of the streamwise velocity perturbation and the output resulting from the measurement of the shear on the wall are highly correlated in the positive time half plane. With the considered convention for the correlation, this implies that fluctuations at the higher wall-normal position precede those on the wall, and thus the instantaneous measure of the shear on the wall cannot predict the instantaneous or future maximum velocity fluctuation of the streak. In other terms, there is reverse causality between the measurements at x 2 = 0 and the measurements at x 2 = 2.25. This effect can be associated to the tilting of the streaky structures shown in figure 12b: an advecting streak first passes at higher wall normal positions (exemplified by the considered probe), and only later leaves a wall shear-stress signature. However, the wall shear-stress measurements are not completely unable to estimate the streaks velocity; they can effectively predict the velocity of the tilted structure at wall-normal positions closer to the wall. There the velocity is lower, the convection velocity of the streaks is reduced, and thus the time delay τ yz , which describes their traveling time, is larger. A more accurate estimation, which corresponds to an estimator with a faster response, can effectively predict the velocity further from the wall. There the velocity is higher, so closer to the real traveling speed of the streaks, and the resulting time delay τ yz smaller. This explains why to the best LQG solution corresponds a fast estimator and its connection to a shorter time delay τ LQG yz = 213. Moreover, the fact that the LQG results in a G yu which puts a lot of weight to the recent history of the output signal, around two orders of magnitude higher than the one from the IFFC (figure 4), is also explained by the presence of a fast estimator. In fact, to higher values of v n /v d , with R LQG /Q LQG fixed to the value of the best solution, correspond a shape of G yu which approaches the one of the best IFFC solution, so the difference between the best LQG and IFFC results must come from the presence of the fast estimator. The last statement is consistent with the present discussion, as it implies that the performance of the controller improves when it mainly makes use of the portion of the output that lies in a small neighborhood of t = T , with t ∈ (−∞, T ] the history time and T the running time. This neighborhood contains the meaningful information because of the mentioned reverse causality between the wall shear-stress measurements and the dynamics of the streaks, as shown by the non-positive time half plane in figure 12a.
Finally, it can be concluded that the performance of the best LQG result is better than those of the best IFFC thanks to the possibility of increasing the estimation accuracy through the estimation weights, which allows to slightly compensate for the reverse causality between the wall shear-stress measurements and the dynamics of the streaks.
Moreover, it appears that the limitation caused by the structure of the plant, including the location of sensors and actuators and their shapes, is more critical than the choice of control technique, and thus is the key design challenge (as further discussed in the parallel work Sasaki et al. (2019)).
Conclusions
The delay of bypass transition in a realistic scenario by means of active flow control, control theory and system identification is presented. Numerical simulations of the nonlinear transitional regime in a Blasius boundary layer are performed, where streaks are excited in the boundary layer by means of a high level of free-stream turbulence. A modelbased method for the delay of bypass-transition realizable in experiments is introduced. It makes use of a ROM representation of the system and is based on the signals from a finite number of localized sensors and actuators placed on the wall, which mimic real shear-stress sensors and ring plasma actuators, respectively. A technique for the characterization of disturbances with a large number of degrees of freedom for model-based approaches is presented, which allows to obtain reasonably low-dimensional ROMs by isolating the dynamics of interest via system identification of the effects of the disturbance on the system. The method is reliable, easy to implement, and based on measurement data, in a data-driven approach that would be realizable in experiments. The presented technique is applied to generate the ROM via ERA for solving the flow control problem by means of LQG, which to the best of the authors' knowledge has never been done in a flow control application.
The performance of the LQG is compared to that of the IFFC optimal control technique, which does not need the explicit characterization of the disturbance on the system, thus, simplifying the flow control problem. LQG is seen to perform slightly better than the simpler IFFC method once appropriate weights in the cost function are selected. The performance of the control techniques are compared in linear input-output and nonlinear Navier-Stokes simulations, showing that resorting to a linear ROM for control design is reasonable also in presence of the high-amplitude disturbances considered here. The effectiveness of the technique in delaying bypass transition is shown. Using LQG a transition delay of ∆Re x ≈ 1.4 × 10 5 for a case with turbulent intensity T u = 3.0% and integral length scale L = 7.5δ * 0 is achieved. This highlights the capability of the presented methods to achieve at least as large delay of bypass transition as that obtained in more idealized cases found in literature (Monokrousos et al. 2008).
Finally, the differences in the results obtained with IFFC and LQG are analyzed and related to the structure of the plant, so the limitations caused by the relative positions of sensors and actuators and by the shape of the sensor are outlined. In particular, a reverse causality issue arising from using wall streamwise-shear-stress sensors to predict the dynamics of the streaks is shown. The way in which this causality issue limits the control performance is described, and an explanation on the way in which the LQG can compensate for such issue is provided.
The authors would like to acknowledge the VINNOVA Projects PreLaFlowDes and SWE-DEMO and the Swedish-Brazilian Research and Innovation Centre CISB for funding. Moreover, part of this work was performed during an exchange programme at KTH, for which Kenzo Sasaki received a scholarship from Capes, project number 88881.132008/ 2016-01. Kenzo Sasaki work is also funded by a scholarship from FAPESP, grant number 2016/25187-4. André V. G. Cavalieri was supported by a CNPq grant 310523/2017-6. The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC, HPC2N and PDC.
Appendix A. Linear Quadratic Gaussian regulator
The LQG technique is designed to solve the control problem on a dynamical system subject to stochastic white noise disturbances. Here, the dynamical system is a ROM and readsq
(t) = Aq(t) + Bu(t) + M d d(t), y(t) = C y q(t) + n(t), z(t) = C z q(t). (A 1)
Since the LQG does not assume the full-state to be known, an estimation of the original dynamical system based on the known outputs is introduced,
q(t) = Aq(t) + Bu(t) − L(y(t) −ỹ(t)), y(t) = C yq (t), z(t) = C zq (t), (A 2)
whereq(t) andỹ(t) are estimates of q(t) and y(t), and L is an N × N y matrix to be designed. The estimated system accounts for the stochastic disturbances through the available outputs y(t). Subtracting (A 2) from (A 1) and substituting y(t) = C y q(t) and y(t) = C yq (t) givesė
(t) = (A + LC y )e(t) + M d d(t) + Ln(t), (A 3)
with e(t) = q(t) −q(t) the estimation error. Eq. (A 3) shows that the error dynamics is based on the matrix L and is driven by the stochastic disturbances. Thus, the matrix L should stabilize the error dynamics and dampen the amplitude of the stochastic disturbance n(t).
Since y(t) is an available measure, the estimated system is deterministic. Its solution is used to compute the actuation input u(t) = Kq(t), with K a matrix to be designed to solve the control problem. Substituting u(t) = Kq(t) in (A 2) giveṡ q(t) = (A + BK)q(t) − L(y(t) −ỹ(t)),
y(t) = C yq (t), z(t) = C zq (t). (A 4)
The LQG technique consists in computing K and L to solve the control and the estimation problem, respectively. These two problems are usually coupled in optimal control, but in the LQG technique they are not and result in the minimization of two different H 2norms (Skogestad & Postlethwaite 2005). The matrix K results from the linear quadratic regulator problem. It minimizes the objective function
J = lim T →∞ 1 T T 0 (z(t) T Qz(t) + u(t) T Ru(t)) dt, (A 5)
and results in solving the following algebraic Riccati equation
A T P u + P u A − P u BR −1 B T P u + C T z QC z = 0, (A 6)
where P u is a positive semi-definite N × N matrix which is the unknown of the equation. The relationship between K and P u reads
K = −R −1 B T P u . (A 7)
The matrix L results from the Kalman filter. It minimizes the expected value of the covariance matrix of the error at steady state,
J = lim t→∞ Tr(P e (t)), P e = E e(t)e(t) T , (A 8)
with Tr(•) the trace operator, and results in solving the following algebraic Riccati equation
P e A T + AP e − P e C T y V −1 n C y P e + M d V d M T d = 0, (A 9)
where P e is a positive semi-definite N ×N matrix and is the unknown of the equation, and V d and V n are the covariance matrices of d(t) and n(t), respectively. The relationship between L and P e reads L = −P e C T y V −1 n . (A 10)
Once both K and L are computed the state-space system based onq gives the input signal based on the history of the available output y(t),
u(t) = − t 0 Ke (A+BK+LC y )(t−τ ) Ly(τ ) dτ. (A 11)
Appendix B. Prediction for different streamwise positions
The streamwise position of sensors and actuators on the flat plate was chosen by also taking into account the estimation error. The behavior of the estimation error as function of the relative streamwise position between the available output and the output to estimate and as function of the absolute position of the available output is inspected. Given the set of N y = 36 estimated outputsỹ est (t) and the set of true outputs y DN S (t), the error is defined as ∆y (t) = y est (t) − y DN S (t). The available outputs are placed at a streamwise position x 1 = x up 1 and the estimated outputs y est (t) are placed at a streamwise position x 1 = x down 1 . The true outputs y DN S (t) are on the same place of the available outputs. It also holds that x down 1 > x up 1 , so the position of the outputs to be estimated never coincides to that of the available outputs. The available outputs are used to predict the downstream outputs in the future. Figure 13 shows the MS[∆y (t)]. Estimation is performed by means of empirical TFs because of their low computational cost. In figure 13 it is evident that the current positioning of sensors and actuators, 250 ≤ x 1 ≤ 400, is adequate. For x up 1 = 150, 250, the MS[∆y (t)] initially decreases due to decay of free-stream turbulence intensity along the streamwise direction. Left: Coherence γyz between the outputs y(t) k , at x1 = 250, and z(t) k , at x1 = 400. Right: normalized power spectral density of the output z(t) k :Ŝzz/ max(Ŝzz).
is adequate, the coherence coefficient between the measurements at x 1 = 250 and x 1 = 400 is calculated. The coherence γ yz is defined as
γ 2 yz = |Ŝ yz | 2 S yyŜzz , (B 1)
and measures the linearity between two different streamwise positions for each frequency. The definition holds for any pair of outputs. Its value varies between zero and one and indicates a complete random behavior (zero) and an exactly linear behavior (one) between the two signals.
It is desirable to have the highest values of coherence in (ω, β k ) regions where the signals are most energetic. This may be evaluated by computing the power-spectral density (PSD). The coherence coefficient between signals at x 1 = 250 and 400 and its normalized PSD is presented in figure 14. The results indicate an almost linear relation between signals at these two streamwise positions for part of the (ω, β k ) space that is of interest. It is noticeable that the most of the energy is strongly localized at very low frequencies, ω ≈ 0, with a spanwise wavenumber β ≈ 0.4, which corresponds to streaky motions with slow streamwise variation. This is the reason for the observed good accuracy of the estimation and justifies the choices of placements for sensors and actuators.
Appendix C. Choice of weights
The LQG has two objective functions, one for the control problem and one for the estimation problem, whereas the IFFC has only one objective function, the one for the control problem. The objective function for the control problem requires the definition of the weight matrix on the output z(t) (orẑ), Q (orQ), and the penalization matrix on the input u(t) (orû), R (orR), while the estimation problem requires the matrices that describe the covariance of the stochastic disturbance d, V d , and the noise n(t), V n . The control problem deals with finding the function that given an output provides an input to minimize an objective function, while the estimation problem deals with finding the function that allows to minimize the error in the estimation. The weights introduce more degrees of freedom in the design problem, and are usually left as free parameters. In fact, there is not a universally acclaimed method to compute those weights and close the control design problem, such that they are usually chosen iteratively (Skogestad & Postlethwaite 2005). Here, a brute force method is applied: a grid of arbitrarily chosen weights is used and a set of G yu m (t) is computed by means of the two control techniques as in § 3. The computed G yu m (t) are tested in input-output simulations based on the linear superposition of the input and output time series only. The input-output simulations make use of the second in equation (2.7) to compute the effect of G yu m (t) on the reference output to annihilate, z(t), and on equation (3.2) for the relationship between u(t) and y(t). The input-output simulation consists in computing for each time step, u(t) k = t 0 Ny m=1 G yu m (t − τ )y(τ ) m+k−1 dτ,
z(t) k = t 0 Nu m=1 G uz m (t − τ )u(τ ) m+k−1 dτ + z d (t) k , (C 1)
with y(t) k and z d (t) k , at x 1 = 250, 400 respectively, with the time series of outputs saved from the nonlinear uncontrolled Navier-Stokes simulations. The method is thus a simpler simulation of the control effect considering only a linear superposition of the open-loop output z d (t) and what would result from control action (via the transfer function G uz m ). The k index was dropped in G uz km (t) from equation (3.2) because the actuators have all the same spatial support and the linearized system dynamics is instantaneously homogeneous along the spanwise direction x 3 . G uz m (t) is found from an impulse-response simulation of the linearized Navier-Stokes equations.
This method avoids the use of computationally demanding Navier-Stokes simulations and is reliable to identify the best G yu m (t) and the associated weights. It also proves to be consistent with the results of the nonlinear N-S simulations. The time required to perform the input-output simulations is of the order of seconds on the average laptop.
Here, the covariance matrices are constants, as in equation (3.4), and the weight matrices for the control problem are chosen to be constants, Q = Q IF F C , Q = Q LQG I,R = R IF F C , R = R LQG I.
(C 2)
In figure 15 the results of the input-output simulations based on the IFFC or LQG methods are shown. From the results based on the IFFC technique it clearly appears that there exists a combination of weights (Q, R) where E is constant. This occurs because the J functional in equation (3.9) can be written as R, which is a constant in this case, times another functional with only one weight in the form Q/R. The constant R becomes irrelevant in the minimization problem, thus the minimization can be performed with respect to the functional with the weight Q/R. The weights can be related as The results show (figure 15) c 1 = 10 −4 . Moreover, by using equation (C 3), the objective function of the IFFC control technique (equation (3.9)) can be written as
Q = c 1 R, (C 3J = R ∞ −∞ k c 1û
Hû +ẑ Hẑ ∆β k dω, (C 5) which shows how under the assumption of constant weights the optimal solution depends only on the ratio of the weights. Writing the cost function as in equation (C 5) combines the physical meaning of the weights in one single parameter and constraints the solution of the minimization procedure to the isolines J(Q, R) = constant.
Since the LQG results in two independent optimization problems, the control problem and the estimation problem (Appendix A), both the objective functions can be expressed in a similar fashion as in equation (C 5). It follows that the performance E of the LQG can be expressed as function of two weight matrices only: one weight matrix from the control problem and one weight matrix from the estimation problem. This result is shown in figure 15 as function of R LQG /Q LQG and v n /v d (as in equation (3.4)). Since the variables are associated to two independent optimization problems, there is no general reason for the existence of a set of weights for which the performance parameter E is constant and has a minimum. In fact, figure 15 shows that E has a minimum for a specific combination of (R LQG /Q LQG , v n /v d ).
The minimum value achieved by the LQG is below the one achieved by the IFFC, E min LQG < E min IF F C . This occurs because the IFFC technique does not include the estimation problem. In figure 16 the estimation function of the LQG corresponding to E min LQG is compared to the G yz m , which is used in the IFFC and computed as in § 5.2. It can be seen that the two curves are slightly different. This occurs because the design parameters of the estimation problem can be tuned to seek for the overall optimal solution, which does not appear to be given by the IFFC, even though it was shown that G yz m is a good estimation function. In other terms, since the LQG has more design parameters than the IFFC, it can span a space of solutions of higher dimensions than the IFFC. The latter statement also implies that there exist a combination of parameters for the estimation problem in the LQG for which E LQG = E min IF F C holds, and that the result achieved by the IFFC can be seen as a suboptimal solution of the LQG.
The input-output simulations allowed to identify the weights that give the best performance if the system were to be completely linear.
Figure 1 .
1Plant. Computational box (cut along x1): frame of reference x1x2x3, sensors y and z (black circles), actuators u (white circles), and controller G yu . Contour plots: perturbation part of the streamwise velocity, q 1 , snapshot of an uncontrolled case at time t = t * ; the isolines of the contour plots are all for the same set of values. Boundary layer, δ99, shown on the left wall of the box.
Figure 2 .
2Comparison between the true z(t) k output from DNS data at x1 = 400 and the estimated outputz(t) k . The available output y avail (t) k is at x1 = 250. Top-Left: single output k = 9. Top-Right: DNS data z(t) k . Bottom-Left: Estimated dataz(t) k , empirical TF. Bottom-Right: Estimated dataz(t) k , improved TF.
Figure 3 .
3Original identified impulse responses used for the ERA (solid black lines) vs ROM impulse response (dashed red lines). Impulse response from d(t) to y(t) and z(t). The original identified impulse responses are built by the improved frequency response technique. Left: central output. Right: complete set of TF.
Figure 4 .
4Control TF G yu k (t). Left: IFFC technique. Right: LQG technique. Notice the different scales for the gains between the two subfigures.
Figure 5 .
5Spanwise RMS values of the actuation signals u(t) k at x1 = 325. Solid line: LQG. Dotted line: IFFC.
Figure 6 .
6Spanwise RMS values of the output y(t) k at x1 = 250.
Figure 7 .
7Left: q 1,rms /q uncontrolled 1,rms averaged along the spanwise direction at Rex = (1.51, 1.96, 2.40, 2.86) × 10 5 ; red dash-dotted line: uncontrolled case; black solid line: LQG. Right: q1,rms averaged along the spanwise direction at Rex = 1.5 × 10 5 ; red dash-dotted line: uncontrolled case; black dotted line: IFFC; black solid line: LQG.
Figure 8 .
8Maximum along the wall-normal direction of the q1,rms averaged along the spanwise direction. Red dash-dotted line: uncontrolled case. Black dotted line: IFFC. Black solid line: LQG.
Figure 9 .
9Average skin friction coefficient C f . Red dash-dotted line: uncontrolled case. Black dotted line: IFFC control method. Black solid line: LQG control method.
Figure 10 .
10Instantaneous skin friction coefficient c f (Rex, x3, t = t * ). Top: uncontrolled case.
Figure 12 .
12Left, a) : cross-correlation between the streamwise velocity fluctuation at the peak of the urms profile and the shear of the streamwise fluctuation at the wall. Right, b) : contours are the fluctuation of the streamwise velocity component; red and solid lines: positive values; blue and dashed lines: negative values; dashed-dotted line: δ99/δ * 0 , boundary layer.
For x down 1 >Figure 13 .
113350, the MS[∆y (t)] increases in all cases because from that position the nonlinear interactions become important. The value of MS[∆y (t)] grows faster with increasing x down 1 as the flow nonlinearity increases. In order to further confirm the fact that the chosen positioning of sensors and actuators Performance of the estimation of downstream outputs from available upstream outputs. x up 1 is represents the available output position; x down 1 represent the position of the estimated output. The outputs are the shear on the wall.
Figure 14 .
14Figure 14. Left: Coherence γyz between the outputs y(t) k , at x1 = 250, and z(t) k , at x1 = 400. Right: normalized power spectral density of the output z(t) k :Ŝzz/ max(Ŝzz).
Figure 15 .
15Performance parameter, log 10 (E), as in equation(6.1), as function of the weight matrices. Left: IFFC control technique. Right: LQG control technique.
) with c 1 a constant. By using equation (C 3), the solution to the control problem based on the IFFC (equation (3.10)) can be written aŝ
Figure 16 .
16Estimation TF. Estimatingz(t) from the available y(t). Solid black line: state observer. Dashed red line: empirical TF. Left: central TF. Right: complete TF.
Table 2 .
2Design weights of the G yu m (t) used in the fully non-linear Navier-Stokes simulations.Control method Input-output sim. Non-linear sim.
IFFC
37.31 %
40.43%
LQG
16.30 %
34.34%
Table 3. Performance, E, for the two control methods. Input-output vs non-linear N.-S.
simulations.
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|
[] |
[
"Floquet scattering matrix approach to the phase noise of a single-electron source in the adiabatic regime",
"Floquet scattering matrix approach to the phase noise of a single-electron source in the adiabatic regime"
] |
[
"Michael Moskalets "
] |
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We give the basic elements of the Floquet scattering matrix approach [1] to the dynamic quantum transport in mesoscopic and nanoscopic conductors. We use the scattering formalism to discuss the noise power spectrum of a single electron source working in the adiabatic regime and emitting particles into a chiral electron waveguide. The noise power is found to be quadratic at low frequencies and exponentially suppressed at high frequencies.Keywords Floquet scattering matrix · Single-electron source · Finite-frequency noise PACS 73.23.-b · 73.50.Td · 73.22.DjIntroductionRecent progress achieved with high-speed single-electron sources[2,3,4,5,6]have opened fascinating perspectives for the development of quantum electronics. On one side, it advances essentially the field of electrical quantum metrology[7,8,9,10]by offering a real possibility to close the quantum metrological triangle[11,12,13,14]. On the other side, the on-demand injection of a single-particle state makes it possible to develop an electronic analogue [15] of quantum optics having a great potential for quantum information processing[16]. For example, the tunable two-particle exchange was already proposed [17] and measured[18].The difference of a single electron source (SES) from a biased metallic contact, which is more conventional for electronics, is that SES emits electrons well separated in space
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10.1007/s10825-013-0481-8
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"https://arxiv.org/pdf/1303.5854v1.pdf"
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1303.5854
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Floquet scattering matrix approach to the phase noise of a single-electron source in the adiabatic regime
Michael Moskalets
Floquet scattering matrix approach to the phase noise of a single-electron source in the adiabatic regime
Received: date / Accepted: dateJournal of Computational Electronics manuscript No. (will be inserted by the editor)
We give the basic elements of the Floquet scattering matrix approach [1] to the dynamic quantum transport in mesoscopic and nanoscopic conductors. We use the scattering formalism to discuss the noise power spectrum of a single electron source working in the adiabatic regime and emitting particles into a chiral electron waveguide. The noise power is found to be quadratic at low frequencies and exponentially suppressed at high frequencies.Keywords Floquet scattering matrix · Single-electron source · Finite-frequency noise PACS 73.23.-b · 73.50.Td · 73.22.DjIntroductionRecent progress achieved with high-speed single-electron sources[2,3,4,5,6]have opened fascinating perspectives for the development of quantum electronics. On one side, it advances essentially the field of electrical quantum metrology[7,8,9,10]by offering a real possibility to close the quantum metrological triangle[11,12,13,14]. On the other side, the on-demand injection of a single-particle state makes it possible to develop an electronic analogue [15] of quantum optics having a great potential for quantum information processing[16]. For example, the tunable two-particle exchange was already proposed [17] and measured[18].The difference of a single electron source (SES) from a biased metallic contact, which is more conventional for electronics, is that SES emits electrons well separated in space
and time. It allows to manipulate in a controllable manner single particles and to engineer few-particle states necessary for storing and processing information. With this respect the crucial question is whether the emitted state is a genuine single-particle state or not. To answer this question the high-frequency noise of currents associated with a singleelectron emission was measured [19]. It was demonstrated that there exists a fundamental lower limit to the noise due to a quantum uncertainty in the time of emission. [19,20] The source, regularly emitting single particles, nevertheless generates fluctuating currents. This limiting noise is referred to as the phase noise. If, in addition, the emission is irregular or involves more than a single particle per time, the noise gets enhanced over this limit. The numerical calculations based on the Floquet scattering theory show an excellent agreement with experimental data. [21] In this contribution our aim is to develop an analytical theory of the phase noise of a single-electron emitter working in the adiabatic regime. For this purpose we use the Floquet scattering theory of a time-dependent quantum transport. [22] In the adiabatic regime the source is driven by an AC voltage, which varies slowly compared to the internal dynamics of an electron source. Therefore, the source stays in a close to equilibrium state while emitting electrons. In contrast, in the non-adiabatic regime of Refs. [19] the source is driven by a pulsed potential and, therefore, it emits electrons during a highly non-equilibrium, transient process. Despite of this difference we find the noise spectrum to be qualitatively similar: It shows a universal quadratic behavior at low frequencies that reflects a probabilistic tunneling dynamics of an electron leaving the source. At high frequencies the phase noise is suppressed. Using analytical calculations we show that in the adiabatic emission regime this suppression admits both a time-based and an energy-based interpretation. From the former point of view, the phase noise gets suppressed as a result of a current spectrum suppres-sion at high frequencies. From the latter point of view, the phase noise suppression is due to decrease in the number of electrons able to emit high-frequency energy quantum.
The paper is organized as follows. The basic elements of the Floquet scattering matrix formalism are presented in Sec. 2. In Sec. 3 the scattering amplitude of a single-particle emitter, the current generated, and the current correlation function are derived. In Sec. 4 the adiabatic working regime is discussed. The excess noise spectrum is calculated and analyzed in Sec. 5. We conclude in Sec. 6.
Floquet scattering matrix formalism
Here we closely follow the notations introduced in Ref. [1]. Let us consider a mesoscopic sample (a scatterer), connected to several, M r , metallic contacts via one-dimensional leads, Fig. 1. The metallic contacts play the role of electron reservoirs [23], which we label with greek letters, α = 1, . . . , M r . We consider a sample as mesoscopic if electrons propagating through it preserve phase coherence. Electrons in each reservoir α are assumed to be in equilibrium. An electron from any reservoir α can come to the sample. Then it will be either transmitted to another reservoir β or reflected back to the same reservoir. If the reservoirs have different chemical potentials, µ α = µ β , and/or temperatures, T α = T β , then electrons scattered between different reservoirs become nonequilibrium at the destination. This results in appearance of currents through the sample driven by the chemical potential (and/or temperature) difference. Another possibility to drive the system out of equilibrium is to act directly onto the sample and to change its properties in time. [24] For instance, one can vary a magnetic flux penetrating a scatterer, see Fig. 1, or an electrostatic potential induced by the nearby gate. After scattering off a dynamic sample, an electron changes its energy and becomes non-equilibrium even if it is scattered back to the reservoir it came from.
Current operator and current correlation function
Here we outline shorty how to calculate a current flowing through the system. We use a charge current as an example. However the same method is appropriate to address any kind of a flux, energy, spin, etc. First, we introduce second quantization operatorsâ † α (E),â α (E) which create, annihilate an electron with energy E incoming to the scatterer from the reservoir α. As we already mentioned, the incoming electrons are in equilibrium and, therefore, the quantumstatistical average of the product of creation and annihilation operators is given by the Fermi distribution function f α (E) of the corresponding reservoir,
â † α (E)a α (E ) = f α (E)δ (E − E ) .
(1) Fig. 1 (Color online) Mesoscopic scatterer connected to three contacts, M r = 3. An electron originated from the first contact (operatorâ 1 ) is shown to be transmitted to the second and the third contacts (operatorŝ b 2 andb 3 , respectively) or reflected back to the first contact (operatorb 1 ). The scatterer shown includes impurities (dark circles) and is threaded by the magnetic flux Φ.
Here δ (E − E ) is the Dirac delta function and the Fermi distribution function is,
f α (E) = 1 1 + e E−µα k B Tα ,(2)
where k B is the Boltzmann constant, µ α is the chemical potential, and T α is the temperature of the reservoir α. We stress the reservoirs are assumed to be macroscopic and their states are not changed by the coupling to a small mesoscopic system. Second, we introduce operatorsb † α (E),b α (E) for electrons scattered off the sample into the lead α. Since an electron scattered to the lead α can come from any other lead, the operator b α can be related to all the operators a β , β = 1, . . . , M r . The coefficients in such a relation are the elements S αβ of the scattering matrix S. If the scatterer is stationary the scattering matrix depends on only one energy argument, S(E). In contrast the scattering matrix for a dynamic sample depends on two energy arguments, S(E 2 , E 1 ). Since the energy of incoming, E 1 , and scattered, E 2 , electron can be different.
Here we are interested in an important case when the scatterer is driven periodically in time with frequency Ω . [25,26] In this case the energy, which an electron can pick up during scattering, is quantized in units ofhΩ . We call the corresponding scattering matrix as the Floquet scattering matrix, S F . Then the relation between b− and a−operators reads, [22] b
α (E) = M r ∑ β =1 ∞ ∑ n=−∞ S F,αβ (E, E n )â β (E n ) .(3)
Here E n = E +nhΩ . Strictly speaking the lower bound in the sum over n is restricted by the requirement, E n > 0, in order that incoming electrons are propagating. However if the energy quantumhΩ is much smaller than the bandwidth, the Fermi energy, the lower bound can be put to minus infinity, what we assume in equation (3). The current conservation requires the Floquet scattering matrix to be unitary. [1,22] For a single orbital channel case of interest here, the unitarity condition reads, [27]
∞ ∑ p=−∞ S * F (E p , E m )S F (E p , E n ) = δ m,n ,(4)∞ ∑ p=−∞ S * F (E m , E p )S F (E n , E p ) = δ m,n ,
where δ m,n is the Kronecker symbol.
If the relevant energy scales (such as the applied voltage, the temperature difference, the energy quantumhΩ , etc.) characterizing the non-equilibrium state are all small compared to the Fermi energy µ, then the current operator in lead α reads, [28]
I α (t) = e h dEdE e i E−E h t b † α (E)b α (E ) −â † α (E)â α (E ) ,(5)
where e < 0 is the elementary charge and h is the Planck constant. The quantum statistical average of this operator over the equilibrium state of electrons in the reservoirs gives a time-dependent current, I α (t) = Î α (t) , flowing into the lead α. To perform such averaging we use Eqs. (3) and (1).
Here we are interested in fluctuations of this current, ∆Î α =Î α − Î α . We characterize such fluctuations with the help of a current correlations function. In the frequency domain it reads as follows, [29]
P αβ (ω 1 , ω 2 ) = 1 2 ∆Î α (ω 1 )∆Î β (ω 2 ) + ∆Î β (ω 2 )∆Î α (ω 1 ) ,(6)
wherê
I α (ω) = e ∞ 0 dE b † α (E)b α (E +hω) −â † α (E)a α (E +hω) ,(7)
is the Fourier transform of a current operatorÎ α (t), Eq. (5).
Single electron source
Below, to be specific, we concentrate on a single-electron source used in Ref. [19]. The cartoon illustrating the features of the experimental set-up essential for us is shown in Fig. 2. The source is made of a mesoscopic capacitor [30] in a twodimensional electron gas in the quantum Hall effect regime In the capacitor electrons move clockwise. The capacitor is driven by the periodic potential applied to the nearby metallic gate (not shown). The electrons (holes) emitted by the capacitor flow to the metallic contact 2 where the current I 2 is measured. [31]. In this regime electrons propagate along chiral states located close to the edges of a sample [32,33] The capacitor, a small cavity with a circular edge state, is side-attached to the linear edge state playing the role of an electron waveguide. The length of a circular state is small such that the electron spectrum is quantized. The periodic potential of the top gate surrounding the capacitor (not shown in Fig. 2) drives its energy levels up and down. When some quantum level of the cavity rises above the Fermi level in the waveguide, an electron is emitted. In the opposite process, when the level in the cavity sinks below the Fermi energy, an electron enters the cavity, a hole is emitted into the waveguide. Since after completion of a period no net charge is emitted, such a cavity is called a capacitor. The current and its fluctuations at the contact 2 are of interest here. To calculate them we need the b 2 (E) operator, which is (see, Eq. (3)),
b 2 (E) = ∞ ∑ n=−∞ S F,21 (E, E n )â 1 (E n ) .(8)
The scattering element S F,22 = 0 since an electron motion is unidirectional. The non-zero scattering element is
S F,21 (E, E n ) = e ik(E)L 2c S F (E, E n ) e ik(E n )L c1 .(9)
Here L 2c (L c1 ) is a distance between the cavity and the contact 2 (1); k(E) is a wave number of an electron with energy E in the waveguide; S F (E, E n ) is the Floquet scattering matrix of the cavity. The exponential factors in Eq. (9) describe a free propagation of electrons from the contact 1 to the cavity and from the cavity to the contact 2. They are irrelevant for subsequent calculations. In contrast, the scattering off the cavity is crucial. The element S F (E, E n ) describes a forward scattering of electrons in the chiral waveguide the cavity is coupled to. During such a scattering an electron energy is changed from E n = E + nhΩ to E due to interaction with a dynamic cavity. Now we show how the Floquet scattering matrix of the cavity is calculated. We stress it is a matrix in the energy space.
Scattering amplitude of the SES
The cavity is modeled [34,35] as a single-channel chiral state of the length L with a uniform time-periodic potential U(t) = U(t + T ). This state is coupled with the help of a quantum point contact (QPC) to a single-channel linear edge state playing the role of an electron waveguide. The QPC is characterized by the energy independent reflection and transmission coefficients. The problem of scattering of electrons in the linear edge state off this scatterer can be solved exactly and the Floquet scattering matrix of the cavity can be calculated. [36] To simplify calculations, it is convenient to represent the Floquet scattering matrix elements as the Fourier transform of some auxiliary scattering amplitude dependent on one energy and one time arguments. Depending on which energy, for incoming or outgoing electrons, is kept fixed, this amplitude is referred to as S in (t, E) or S out (E,t), respectively. [37] For the purposes of this paper it is convenient to work with the latter amplitude, which is defined as follows,
S F (E, E n ) = S out,−n (E) ≡ T 0 dt T e −inΩt S out (E,t) .(10)
For the model of interest here the amplitude S out (E,t) has the following representation in terms of particular amplitudes S (q) (t) : [36] S out (E,t) =
∞ ∑ q=0 S (q) (t) ,(11)S (0) = r , S (q>0) (t) =t 2 r q−1 e iqkL e −iΦ q (t) , Φ q (t) = ē h t+qτ t dt U(t ) .
Here k ≡ k(E) is a wave number of an electron with energy E in the cavity; r andt are reflection and transmission coefficients of the QPC connecting the cavity and an electron waveguide. The partial amplitude S (q) (t) is an amplitude for an electron to enter the cavity at time t and to leave the cavity after completing q full turns (of duration τ each). The energy of an electron leaving the cavity is E. The quantity Φ q (t) is a time-dependent phase acquired by an electron during such a propagation. The potential energy eU is assumed to be small compared to an electron energy E that allows to separate the time-dependent phase Φ q (t) and the orbital phase qkL. This general solution is not restricted to any particular amplitude and/or time-dependence of the driving potential U(t).
Current
We substitute Eq. (8) into Eq. (5) and calculate a time dependent current, I 2 (t) = Î 2 (t) , measured at the contact 2 (see Fig. 2),
I 2 (t) = ∞ ∑ =−∞ e −i Ωt e h ∞ 0 dE (12) × ∞ ∑ n=−∞ S * F (E n , E) S F (E n+ , E) { f (E) − f (E n )} .
Here we suppose that both contacts, α = 1, 2, have the same chemical potential, µ 1 = µ 2 ≡ µ, the same temperature, T 1 = T 2 ≡ T , and thus are characterized by the same Fermi distribution function,
f 1 (E) = f 2 (E) ≡ f (E).
Current correlation function
In the set-up shown in Fig. 2 the current auto-correlation function is measured at the contact 2. We use Eq. (6) and find, [38]
P 22 (ω 1 , ω 2 ) = ∞ ∑ =−∞ 2πδ (ω 1 + ω 2 − Ω ) P (ω) ,(13a)
where
P (ω 1 , ω 2 ) = e 2 h ∞ 0 dE δ l0 F (E, E +hω 1 ) (13b) + ∞ ∑ n,m,p=−∞ F (E +n , E m +hω 1 ) S F E +p , E +n S * F (E, E +n ) ×S F (E +hω 1 , E m +hω 1 ) S * F (E p +hω 1 , E m +hω 1 ) , and F(E 1 , E 2 ) = f (E 1 ) [1 − f (E 2 )] + f (E 2 ) [1 − f (E 1 )] 2 .(14)
The current correlation function is non-zero even in equilibrium, when the driving potential is switched off. This is due to the quantum noise. [39,40] The important feature of the present set-up is the possibility to completely subtract the equilibrium quantum noise and to extract the contribution due solely to the emitted particles. For this purpose the excess noise was measured. [19] The excess refers to the difference between the current correlation functions measured with the source on and off. We will indicate the excess noise with the superscript (ex).
Here we discuss only the = 0 component of the excess auto-correlation function, which was measured in Ref. [19]:
P (ex) 0 (ω) = e 2 h ∞ 0 dE ∞ ∑ m=−∞ F(E, E m +hω)(15)× |Π m (E, ω)| 2 − δ m,0 , where Π m (E, ω) = ∞ ∑ q=−∞ S F (E q , E) S * F (E q +hω, E m +hω) . (16)
An equation similar to Eq. (15) was found in Ref. [21].
Adiabatic emission
We suppose that the potential U(t) changes slowly compared to the average time an electron spends in the cavity, the dwell time,
τ D = τ T .(17)
Here T = |t| 2 is the transmission probability of the QPC connecting the cavity and an electron waveguide; τ is a time of a single turn around the cavity. In such a case we can keep U(t) constant while calculating the time-dependent phase,
Φ q (t) = ē h qτU(t) .(18)
Then we can easily sum up over q in Eq. (11) and find S out (E,t) = S(E,U(t)), where
S(E,U(t)) = −e i(φ (E,t)+θ r ) 1 − √ Re −iφ (E,t) 1 − √ Re iφ (E,t) .(19)
Here we write the reflection coefficient as r = √ R exp(iθ r ) and introduce the phase φ (E,t) acquired by an electron during one turn,
φ (E,t) = θ r + k µ L + 2π E − µ − eU(t) ∆ ,(20)
where ∆ = h/τ is the level spacing in the cavity and k µ = k(µ) is the wave number of an electron with the Fermi energy µ. In above equation we expand the orbital phase, kL = k µ L + 2π (E − µ) /∆ close to the Fermi energy µ. This expansion is exact for a linear electron dispersion, E(k). For a nonlinear dispersion it is a good approximation if the relevant energies are close to the Fermi energy, |E − µ| µ. Note within this approximation the time of one turn around the cavity τ is energy independent. Respectively, the level spacing ∆ is also energy independent, hence the spectrum of electrons in the cavity is equidistant.
The amplitude S(E,U), Eq. (19), is nothing but the scattering amplitude of a cavity with a fixed potential U. Since now the potential is time-dependent, the amplitude S(E,U(t)) is referred to as the frozen scattering amplitude, meaning a scattering amplitude with a potential frozen at time t. This amplitude is manifestly unitary, |S(E,U(t))| 2 = 1, as the current conservation requires.
Quantized emission regime
If the transmission of the QPC is small, T 1, the quantum levels in the cavity are well defined, i.e., their width δ is small compared to the level spacing ∆ . One can model such levels as the Breit-Wigner resonances. [41] With varying potential U(t) the position of the levels in the cavity are changed. We suppose the only one level in the cavity crosses the Fermi level. Close to the time of an electron emission (when the level in the cavity rises above the Fermi level in an electron waveguide) the scattering amplitude can be represented as follows (up to the irrelevant phase factor), [17]
S(E,t) = t − t − (E) + iΓ τ (E) t − t − (E) − iΓ τ (E) ,(21)
where t − (E) is a time when the middle of an energy level is equal to E; the parameter Γ τ (E) is a time interval, during which the level having a width 2δ crosses the energy E. For an equidistant spectrum with level spacing ∆ the level halfwidth is δ = T∆ /(4π) and Γ τ = δ /|e∂ t U (t − ) | with ∂ t indicating a time derivative. Close to the time of a hole emission, t + , the complex conjugate to Eq. (21) should be considered.
Current in the adiabatic regime
The adiabatic regime implies that the Floquet scattering matrix changes only a little with energy on the scale put by the frequency, i.e., on the scalehΩ . [22] To the leading order in Ω its matrix elements are the Fourier coefficients of the frozen scattering amplitude, see Eqs. (10) and (19). Thus we can use S * F (E n , E) ≈ S * n (E) and S F (E n+ , E) ≈ S n+ (E). Moreover, expanding the Fermi function difference, f (E) − f (E n ) ≈ (−∂ f /∂ E) nhΩ , we rewrite Eq. (12) as follows,
I 2 (t) = −ie 2π ∞ 0 dE − ∂ f ∂ E S(E,t) ∂ S * (E,t) ∂t .(22)
At low temperatures, k B T ≤hΩ , we can keep the scattering amplitude as energy independent, S(E,t) ≈ S(µ,t), while integrating over energy in Eq. (22). Using Eq. (21) we find a time-dependent current generated by a single-particle emitter in the adiabatic regime: [17]
I 2 (t) = e π Γ τ (t − t − ) 2 + Γ 2 τ − Γ τ (t − t + ) 2 + Γ 2 τ .(23)
Here all quantities are calculated at the Fermi energy: t ∓ = t ∓ (µ) and Γ τ = Γ τ (µ). In above equation the current I 2 (t) is given for a single period, 0 < t < T and should be periodically extended to other times. Note for the adiabatic regime to be held the following inequality should be satisfied, [42] Γ τ τ D ,
i.e., the current pulse duration should be much larger than the dwell time.
The current I 2 (t) consists of two pulses of a Lorentzian shape: The first, positive pulse corresponds to an emitted electron (it carries a charge e), while the second, negative pulse corresponds to an emitted hole (it carries a charge −e). Consequently the current spectrum is bounded. The discrete Fourier transform of the current reads,
I 2,n = e T e −|n|ΩΓ τ e inΩt − + e inΩt + .(25)
Here we took into account that Γ τ T . As we will see below the exponential decay of a current spectrum results in an exponential decay of an excess noise spectrum.
Excess noise in the adiabatic regime
As we will show, the excess noise in the adiabatic regime exists at relatively low (but still finite) frequencies Therefore, we can expand Π m (E, ω), Eq. (16), and keep only the leading terms,
|Π m (E, ω)| 2 ≈ δ m,0 1 + ω 2 Re ∂ 2 Π m (E, ω) ∂ ω 2 ω=0 (26) +ω 2 ∂ Π m (E, ω) ∂ ω ω=0 2 .
Note, due to unitarity of the scattering matrix, Eq. (4), the linear in ω term vanishes in above equation. As it follows quite generally from Eqs. (15) and (26), the low frequency noise is quadratic in ω, that is quite expected for a capacitor. [30] In the adiabatic regime we use S F (E q , E) ≈ S q (E) and S * F (E q +hω, E m +hω) ≈ S * q−m (E +hω) in equation (16). With these simplifications and expansion (26) we find from Eq. (15) the following,
P (ex) 0 (ω) = e 2h ω 2 2π ∞ 0 dE ∞ ∑ m=−∞ S ∂ S * ∂ E m 2 (27) × {F(E, E m +hω) − F(E, E +hω)} .
If the temperature is of the order of the frequency, k B T ∼ hω ∼hΩ , then in the adiabatic regime the scattering amplitude can be kept constant over the entire energy interval relevant for the integration in above equation. That allow us to integrate over energy. With the scattering amplitude given in Eq. (21) we calculate,
P (ex) 0 (ω) = 2e 2 π (ωτ D ) 2 (ΩΓ τ ) 2 ∞ ∑ m=−∞ e −2|m|ΩΓ τ(28)× (mΩ + ω) coth mΩ + ω 2k B T /h − ω coth ω 2k B T /h ,
where the dwell time τ D = h/(T∆ ). Note, for the particular case, when the Fermi level aligns with a quantum level in the cavity at U = 0 and the time-dependent potential is eU(t) = (∆ /2) cos (Ωt), we find the parameter ΩΓ τ = T/(2π).
At zero temperature we sum up over m in Eq. (28) and find,
P (ex) 0 (ω) = 2 e 2 T (ωτ D ) 2 e −2|ω|Γ τ .(29)
This dependence is shown in Fig. 3. At small frequencies, ωΓ τ 1, the noise power is quadratic in ω. Such a quadratic dependence, found also for the cavity driven by a pulsed potential, is attributed to the probabilistic nature of tunneling of an electron leaving the cavity. [19,20,21] The factor 2 in front of Eq. (29) refers to two particles, one electron and one hole, emitted during one period. At higher frequencies, ωΓ τ 1, the excess noise gets suppressed. As we already mentioned, this suppression is a consequence of boundedness of a current spectrum, see Eq. (25). A complementary, energy-based interpretation of the excess noise suppression is also possible. It relies on a supposition [43] that an emitted electron does contribute to an excess noise power at frequency ω if it is able to emit an energy quantumhω. Taking into account that the energy of particles emitted by a single-electron source fluctuates, [45] we relate the factor exp(−2ωΓ τ ) in Eq. (29) to the number of electrons emitted with energy larger thanhω. We denote this number as N (ω). Provided the energy probability distribution p(E) for emitted electrons is known, this number can be calculated as follows,
N (ω) = ∞ µ+hω dE p(E) .(30)
To calculate p(E) we note that in the quantized emission regime only a single particle can be emitted at a time. Therefore, the probability p(E) can be calculated using the distribution function f (E) = b † (E)b(E) for electrons scattered off the cavity. The difference between p(E) and f (E) is twofold. First, they are differently normalized. And, second, for E < µ the hole distribution function, f h = 1 − f has to be considered. Thus for E > µ we have,
p(E) = f (E) ∞ 0 dE f (E) .(31)
To calculate f (E) we use Eq. (8) and find for the adiabatic regime and at zero temperature,
f (E) = ∞ ∑ q=[(E−µ)/(hΩ )] S q (E) 2 .(32)
Here [. . . ] stands for the integer part. For the Breit-Wigner scattering amplitude, Eq. (21), the Fourier coefficients are, S q = −2ΩΓ τ e −|q|ΩΓ τ e iqΩt ∓ θ (±q). Using these coefficients we get f (E) = 2ΩΓ τ e −2ΩΓ τ (n+1) for nhΩ < E − µ < (n + 1)hΩ with integer n. And, correspondingly, we calculate the energy probability distribution,
p(E) = 2Γ τ h e − 2Γτ h |E−µ| ,(33)
and the average number of electrons contributing to the excess noise,
N (ω) = e −2Γ τ |ω| .(34)
This factor is exactly what enters Eq. (29).
Conclusion
We presented here the scattering matrix approach to the nonstationary transport in coherent nanostructures. The essential feature of a non-stationary transport is an energy exchange between the carriers and the external driving force, that is naturally taken into account by the Floquet scattering matrix amplitudes dependent on two energies. Another advantage of the scattering formalism is that the results are presented in the form that admits an intuitive and transparent interpretation. For instance, the collective dynamics of electrons in a waveguide coupled to a driven cavity is naturally described in terms of extra particles, electrons and holes, emitted by the cavity. We use the scattering formalism to analyze the intrinsic noise, the phase noise, of a single-particle emitter recently investigated experimentally. The existence of this noise is rooted in the fundamental quantum-mechanical principle, the Heisenberg uncertainty principle. In the adiabatic emission regime analyzed here this relation manifests itself the most bright and the entire phase noise spectrum can be easily understood. The quadratic low frequency asymptotics is governed by the tunneling dynamics of an electron leaving the emitter: Before emission an electron occupies a quantum level in the cavity. Due to coupling between the cavity and an electron waveguide the quantum level acquires some width. Therefore, the initial energy of an electron has some uncertainty 2δ , which ultimately results in the uncertainty of the emission time, τ D ∼h/(2δ ). The latter in turn results in a quadratic phase noise spectrum at ωτ D 1. [19,20,21] With increasing frequency another energy, the energy absorbed by an emitted electron (hole) from the external driving force [46], comes into play. This energy E defines a time-extend, 2Γ τ =h/E , of an emitted single-particle state. [17,47] The finite duration of a current pulse bounds the frequency spectrum of a current and, therefore, the one of the phase noise.
Fig. 2 (
2Color online) Single-electron source (SES). The mesoscopic capacitor, a circular edge state, is attached to the Hall bar. The arrows indicate the direction of movement of electrons in the edge states (the blue strips).
Fig. 3 (
3Color online) Excess noise in the adiabatic regime as a function of frequency, P ex,ad (ω) = Cχ(2ωΓ τ ), Eq. (29), with C = 2(e 2 /T ) [τ D /(2Γ τ )] 2 and χ(x) = x 2 e −x . The parameter Γ τ is the halfwidth of an emitted current pulse. τ D is the dwell time.
Acknowledgements I thank Mathias Albert and Markus Büttiker for helpful discussions and valuable comments on the manuscript. I thank the University of Geneva for warm hospitality, where part of this work was carried out.
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|
[] |
[
"EFFECTIVE RANDOMNESS FOR CONTINUOUS MEASURES",
"EFFECTIVE RANDOMNESS FOR CONTINUOUS MEASURES"
] |
[
"Jan Reimann ",
"Theodore A Slaman "
] |
[] |
[] |
We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n, all but countably many reals are n-random for such a measure, where n indicates the arithmetical complexity of the Martin-Löf tests allowed. The proof is based on a Borel determinacy argument and presupposes the existence of infinitely many iterates of the power set of the natural numbers. In the second part of the paper we present a metamathematical analysis showing that this assumption is indeed necessary. More precisely, there exists a computable function G such that, for any n, the statement "All but countably many reals are G(n)-random with respect to a continuous probability measure" cannot be proved in ZFC − n . Here ZFC − n stands for Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom is replaced by the existence of n-many iterates of the power set of the natural numbers. The proof of the latter fact rests on a very general obstruction to randomness, namely the presence of an internal definability structure.Nauk, 25(6(156)): 85-127, 1970.
|
10.1090/jams/980
|
[
"https://arxiv.org/pdf/1808.10102v1.pdf"
] | 15,981,799 |
1808.10102
|
5d22e4eda6b686b92a4afb7e7b7741131c811a84
|
EFFECTIVE RANDOMNESS FOR CONTINUOUS MEASURES
30 Aug 2018
Jan Reimann
Theodore A Slaman
EFFECTIVE RANDOMNESS FOR CONTINUOUS MEASURES
30 Aug 2018
We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n, all but countably many reals are n-random for such a measure, where n indicates the arithmetical complexity of the Martin-Löf tests allowed. The proof is based on a Borel determinacy argument and presupposes the existence of infinitely many iterates of the power set of the natural numbers. In the second part of the paper we present a metamathematical analysis showing that this assumption is indeed necessary. More precisely, there exists a computable function G such that, for any n, the statement "All but countably many reals are G(n)-random with respect to a continuous probability measure" cannot be proved in ZFC − n . Here ZFC − n stands for Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom is replaced by the existence of n-many iterates of the power set of the natural numbers. The proof of the latter fact rests on a very general obstruction to randomness, namely the presence of an internal definability structure.Nauk, 25(6(156)): 85-127, 1970.
Introduction
The goal of this paper is study under what circumstances an infinite binary sequence (real) is random with respect to some probability measure. We use the framework of Martin-Löf randomness to investigate this question. Given a measure µ, a Martin-Löf test is an effectively presented G δ µ-nullset in which the measure of the open sets converges effectively to zero. As there are only countably many such tests, only measure-zero many reals can be covered by a Martin-Löf test for µ. The reals that cannot be covered are called Martin-Löf random for µ. Obviously, if a real X is an atom of a measure µ, then X is random for µ. If we rule out this trivial way of being random, the task becomes harder: Given a real X, does there exist a Reimann was partially supported by NSF grants DMS-0801270 and DMS-1201263. Slaman was partially supported by NSF grants DMS-0501167 and DMS-1001551. probability measure on the space of all infinite binary sequences such that X is not an atom of µ but X is µ-random?
In [41], we were able to show that if a real X is not computable, then such a measure exists. It is not hard to see that if a real X is computable, then the only way that X is random with respect to a measure µ is for it to be an atom of µ. Hence having non-trivial random content (with respect to any measure at all) in the sense of Martin-Löf is equivalent to being non-computable. Besides Martin-Löf randomness, various other notions of algorithmic randomness have been thoroughly investigated, such as Schnorr randomness or Kurtz randomness. Two recent books on algorithmic randomness [9,36] provide a good overview over the various concepts. They all have in common that they use algorithmic features to separate non-randomness from randomness. Moreover, in terms of the arithmetic hierarchy, the complexities of the underlying test notions usually fall within two or three quantifiers of each other.
This suggests that in order to study the random content of a real from the point of view of algorithmic randomness in general, we should look at how this content behaves when making tests more powerful by giving them access to oracles (or equivalently, considering nullsets whose definitions are more complicated). For Martin-Löf tests, this means the test has to be effectively G δ only in some parameter Z. This enlarges the family of admissible nullsets and, correspondingly, shrinks the set of random reals. If the parameter Z is an instance ∅ (n) of the Turing jump, i.e., Z is real that can decide all Σ n statements about arithmetic, we speak of (n + 1)-randomness.
Our goal is to understand the nature of the set of reals that are not nrandom with respect to any continuous probability measure. In particular, we want to understand how this set behaves as n grows larger (and more reals will have this property).
The restriction to continuous measures makes sense for the following reasons. By a result of Haken [17,Theorem 5], if a real is n-random, n > 2, with respect to some (not necessarily continuous) probability measure and not an atom of the measure, it is (n−2)-random with respect to a continuous probability measure. Thus considering arbitrary probability measures would only shift the question of how random a real is by a couple of quantifiers. And the core problem of finding a measure that makes a real random without making the real an atom of the measure remains. While we ignore features of randomness for arbitrary measures at lower levels, we develop insights into randomness for continuous measures. At the level of 1-randomness, there is an interesting connection with computability theory: In [41], drawing on a result of Woodin [47], we showed that if a real X is not hyperarithmetic, then there exists a continuous probability measure for which X is 1-random.
Our first main result concerns the size of the set of reals that are not n-random with respect to any continuous measure. The case n = 1 follows of course from the result in [41] mentioned above.
Theorem 1. For any n ∈ ω, all but countably many reals are n-random with respect to some continuous probability measure.
The proof features a metamathematical argument. Let us denote by NCR n the set of all reals that are not n-random with respect to any continuous probability measure. We show that for each n, NCR n is contained in a countable model of a fragment of set theory. More precisely, this fragment is ZFC − n , where ZFC − n denotes the axioms of Zermelo-Fraenkel set theory with the Axiom of Choice, with the power set axiom replaced by a sentence that assures the existence of n iterates of the power set of the natural numbers.
One may wonder whether this metamathematical argument is really necessary to prove the countability of a set of reals, in particular, whether one needs the existence of infinitely many iterates of the power set of ω to prove Theorem 1, a result about sets of reals. It turns out that this is indeed the case. This is the subject of our second main result. Theorem 2. There exists a computable function G(n) such that for every n ∈ ω, the statement "There exist only countably many reals that are not G(n)random with respect to some continuous probability measure."
is not provable in ZFC − n .
This metamathematical property of NCR is reminiscent of Borel determinacy [32]. Even before Martin proved that every Borel game is determined, Friedman [11] had shown that any proof of Borel determinacy had to use uncountably many iterates of the power set of ω. Borel determinacy is a main ingredient in our proof of Theorem 1. Theorem 2 establishes that this use is, in a certain sense, inevitable.
Theorem 2 is proved via a fine structure analysis of the countable models used to show NCR n is countable. These models are certain levels L β of Gödel's constructible hierarchy. In these L β (or rather Jensen's version, the J-hierarchy) we exhibit sequences of non-random reals with Turing degrees cofinal among those of the model. These reals are master codes [3,22], reals that code initial segments of the J-hierarchy in a way that arithmetically reflects the strong stratification of L. The main feature of this proof is a very general principle that manifests itself in various forms: an internal stratified definability structure forms a strong obstruction to randomness. This principle works for both iterated Turing jumps as well as certain levels of the J-hierarchy.
Before we proceed, we make one more comment on the restriction to continuous measures. Note that Theorem 1 is a stronger statement for continuous measures than for arbitrary measures. Furthermore, by Haken's result [17], Theorem 2 holds for arbitrary measures if we replace G(n) by G(n) + 2.
The paper is organized as follows. In Section 2, we introduce effective randomness for arbitrary (continuous) probability measures. We also prove some fundamental facts on randomness. In particular, we will give various ways to obtain reals that are random for some continuous measure from standard Martin-Löf random reals (i.e., random with respect to Lebesgue measure). We also consider the definability strength of random reals. Section 3 features the proof that for any n, all but countably many reals are n-random with respect to some continuous measure (Theorem 1). Finally, Section 4 is devoted to the metamathematical analysis of Theorem 1. In particular, it contains a proof of Theorem 2.
We expect the reader to have basic knowledge in mathematical logic and computability theory, including some familiarity with forcing, the constructible universe, and the recursion theoretic hierarchies.
In this section we review effective randomness on Cantor space 2 ω for arbitrary probability measures. We then prove some preliminary facts about random reals.
The Cantor space 2 ω is the set of all infinite binary sequences, also called reals. The topology generated by the cylinder sets
σ = {x : x⌈ |σ| = σ},
where σ is a finite binary sequence, turns 2 ω into a compact Polish space. 2 <ω denotes the set of all finite binary sequences. If σ, τ ∈ 2 <ω , we use ⊆ to denote the usual prefix partial ordering. This extends in a natural way to 2 <ω ∪ 2 ω . Thus, x ∈ σ if and only if σ ⊂ x. Finally, given U ⊆ 2 <ω , we write U to denote the open set induced by U , i.e. U = σ∈U σ .
2.1. Turing functionals. While the concept of a Turing functional is standard, we will later define a forcing partial order based on it, and for this purpose we give a rather complete formal definition here. The definition follows [46], with the one difference that we require Turing functionals to be recursively enumerable.
A Turing functional Φ is a computably enumerable set of triples (m, k, σ) such that m is a natural number, k is either 0 or 1, and σ is a finite binary sequence. Further, for all m, for all k 1 and k 2 , and for all compatible σ 1 and σ 2 , if (m, k 1 , σ 1 ) ∈ Φ and (m, k 2 , σ 2 ) ∈ Φ, then k 1 = k 2 and σ 1 = σ 2 .
We will refer to a triple (m, k, σ) as a computation in Φ, and we will say it is a computation along X when every X is an extension of σ.
In the following, we will also assume that Turing functionals Φ are usemonotone, which means the following hold.
(1) For all (m 1 , k 1 , σ 1 ) and (m 2 , k 2 , σ 2 ) in Φ, if σ 1 is a proper initial segment of σ 2 , then m 1 is less than m 2 . (2) For all m 1 and m 2 , k 2 and σ 2 , if m 2 > m 1 and (m 2 , k 2 , σ 2 ) ∈ Φ, then there are k 1 and σ 1 such that σ 1 ⊆ σ 2 and (m 1 , k 1 , σ 1 ) ∈ Φ.
We write Φ σ (m) = k to indicate that there is a τ such that τ is an initial segment of σ, possibly equal to σ, and (m, k, τ ) ∈ Φ. In this case, we also write Φ σ (m) ↓, as opposed to Φ σ (m) ↑, indicating that for all k and all τ ⊆ σ, (m, k, τ ) ∈ Φ. If, moreover, (m, k, τ ) is enumerated into Φ by time s, we write Φ σ s (m) = k. If X ∈ 2 ω , we write Φ X (m) = k (and Φ X s (m) = k, respectively) to indicate that there is an l such that Φ X⌈ l (m) = k (and this is enumerated by time s, respectively). This way, for given X ∈ 2 ω , Φ X defines a partial function from ω to {0, 1} (identifying reals with sets of natural numbers). If this function is total, it defines a real Y , and in this case we write Φ(X) = Y and say that Y is Turing reducible to X via Φ, Y ≤ T X.
By use-monotonicity, if Φ σ (m) ↓, then Φ σ (n) ↓ for all n < m. If we let m be maximal such that Φ σ (m) ↓, Φ σ gives rise to a string τ of length m + 1,
τ = Φ σ (0) . . . Φ σ (m).
If Φ σ (n) ↑ for all n, we put τ = ∅. On the other hand, if m does not exist, then Φ σ gives rise to a real Y . We write Φ(σ) = τ or Φ(σ) = Y , respectively. This way a Turing functional induces a function from 2 <ω to 2 <ω ∪ 2 ω that is monotone, that is, σ ⊆ τ implies Φ(σ) ⊆ Φ(τ ). Note that Φ(σ) is not necessarily a computable function, but we can effectively approximate it by prefixes. More precisely, there exists a computable mapping
(σ, s) → Φ s (σ) so that Φ s (σ) ⊆ Φ s+1 (σ), Φ s (σ) ⊆ Φ s (σ ⌢ i) for i ∈ {0, 1}, and lim s Φ s (σ) = Φ(σ).
If, for a real X, lim n |Φ(X⌈ n )| = ∞, then Φ(X) = Y , where Y is the unique real that extends all Φ(X⌈ n ). In this way, Φ also induces a partial, continuous function from 2 ω to 2 ω . We will use the same symbol Φ for the Turing functional, the monotone function from 2 <ω to 2 <ω , and the partial, continuous function from 2 ω to 2 ω . It will be clear from the context which Φ is meant. Φ is called total if Φ(X) is a real for all X ∈ 2 ω . If Φ is total and Φ(X) = Y , then Y is called truth-table reducible to X, Y ≤ tt X.
Turing functionals can be relativized with respect to a parameter Z, by requiring that Φ is r.e. in Z. We call such functionals Turing Z-functionals. This way we can consider relativized Turing reductions. A real X is Turing reducible to a real Y relative to a real Z, written X ≤ T(Z) Y , if there exists a Turing Z-functional Φ such that Φ(X) = Y .
2.2. Probability measures. By the Carathéodory extension theorem, a Borel probability measure µ on 2 ω is completely specified by its values on clopen sets, i.e., on finite unions of basic open cylinders. In particular, µ ∅ ) = 1, and the additivity of µ implies that for all σ ∈ 2 <ω ,
(2.1) µ σ = µ σ ⌢ 0 + µ σ ⌢ 1 .
An additive premeasure is a function η : 2 <ω → R ≥0 with η(∅) = 1 and η(σ) = η(σ ⌢ 0) + η(σ ⌢ 1) for all σ ∈ 2 <ω . Any additive premeasure induces a Borel probability measure, and if we restrict a Borel probability measure µ to its values on cylinders, we obtain an additive premeasure whose Carathéodory extension is µ. We can therefore identify a Borel probability measure on 2 ω with the additive premeasure it induces. We will exclusively deal with Borel probability measures and in the following simply write measure to denote a Borel probability measure on 2 ω . The Lebesgue measure λ on 2 ω is obtained by distributing a unit mass uniformly along the paths of 2 ω , i.e., by setting λ σ = 2 −|σ| . A Dirac measure, on the other hand, is defined by putting a unit mass on a single real, i.e., for X ∈ 2 ω , let
δ X σ = 1 if σ ⊂ X, 0 otherwise.
If, for a measure µ and X ∈ 2 ω , µ({X}) > 0, then X is called an atom of µ.
Obviously, X is an atom of δ X . A measure that does not have any atoms is called continuous.
Representation of measures and Martin-Löf randomness.
To incorporate measures into an effective test for randomness we represent them as reals. This can be done in various ways (for example, identify them with the underlying premeasure and code that), but in order for the main arguments in [41] to work, the representation has to reflect some of the topological properties of the space of probability measures.
Let M(2 ω ) be the set of all Borel probability measures on 2 ω . With the weak-* topology, this becomes a compact Polish space (see [26,Theorem 17.23]). It is possible to choose a countable dense subset D ⊆ M(2 ω ) so that every measure in M(2 ω ) is the limit of an effectively converging Cauchy sequence of measures in D. Moreover, the structure of the measures in D is such that they give rise to a canonical continuous surjection ρ : 2 ω → M(2 ω ) with the additional property that for every R ∈ 2 ω , ρ −1 ({ρ(R)}) is Π 0 1 (R). For details on the construction of ρ, see [4,Section 2]. If ρ(R) = µ, then R is called a representation of µ. A measure may have several distinct representations with respect to ρ. If µ is given, R µ will always denote a representation of µ.
Working with representations, we can apply computability theoretic notions to measures. The following two observations appeared as Propositions 2.2 and 2.3, respectively, in [41]. Proposition 2.1. Let R ∈ 2 ω be a representation of a measure µ ∈ M(2 ω ). Then the relations µ σ < q and µ σ > q (σ ∈ 2 <ω , q ∈ Q) are r.e. in R.
It follows that the representation of a measure can effectively approximate effectively approximate its values on cylinders to arbitrary precision. Proposition 2.2. Let R ∈ 2 ω be a representation of a measure µ ∈ M(2 ω ). Then R computes a function g µ : 2 <ω × ω → Q such that for all σ ∈ 2 <ω , n ∈ ω,
|g µ (σ, n) − µ σ | ≤ 2 −n .
We say a real X is recursive in µ if X ≤ T R µ for every representation R µ of µ. On the other hand, we say a real computes a measure if its computes some representation of it. A measure does not necessarily have a representation of least Turing degree [4,Theorem 4.2].
We will later show that the question of whether a real is random with respect to a continuous measure can be reduced to considering only continuous dyadic measures. A measure µ is dyadic if every measure of a cylinder is of the form µ σ = m/2 n with m, n non-negative integers.
For dyadic measures, it makes sense to speak of exact computability: A dyadic measure µ is exactly computable if the function σ → µ σ is a computable mapping from 2 <ω to Q. Note that for exactly computable measures, the relation µ σ > α, α rational, is decidable, whereas in the general case for µ with a computable representation we only know it is Σ 0 1 . If we encode a dyadic measure µ by collecting the ternary expansions of its values on cylinders in a single real Y , we obtain a representation not in the sense of ρ, but that is minimal in the following sense: Any real that can compute an approximation function to µ in the sense of Proposition 2.2 can compute Y .
We can now give the definition of a general Martin-Löf test. The definition is a generalization of Martin-Löf n-tests and Martin-Löf n-randomness for Lebesgue measure. We relativize both with respect to a representation of the measure and an additional parameter. Definition 2.3. Suppose µ is a probability measure on 2 ω , and R is a representation of µ. Suppose further that Z ∈ 2 ω and n ≥ 1.
(1) An (R, Z, n)-test is a set W ⊆ ω × 2 <ω which is recursively enu- merable in (R ⊕ Z) (n−1) , the (n − 1)st Turing jump of R ⊕ Z such that σ∈Wn µ σ ≤ 2 −n , where W n = {σ : (n, σ) ∈ W } (2) A real X passes a test W if X ∈ n W n . If X does not pass a test
W , we also say X is covered by W (or (W n ), respectively).
(3) A real X is (R, Z, n)-random if it passes all (R, Z, n)-tests.
(4) A real X is Martin-Löf n-random for µ relative to Z, or simply (µ, Z, n)-random if there exists a representation R µ such that X is (R µ , Z, n)-random. In this case we say R µ witnesses the µ-randomness of X.
If the underlying measure is Lebesgue measure λ, we often drop reference to the measure and simply say X is (Z, n)-random. We also drop the index 1 in case of (µ, Z, 1)-randomness and simply speak of µ-randomness relative to Z or µ-Z-randomness. If Z = ∅, on the other hand, we speak of (µ, n)-or µ-n-randomness. Note also that if µ is Z-computable, say R µ ≤ T Z, then (R µ , Z, n)-randomness is the same as (R µ , Z (n−1) , 1)-randomness.
Remark 2.4. The original definition of n-randomness for Lebesgue measure λ given by Kurtz [29] uses tests based on Σ n classes. However, it is possible to approximate Σ 0 n classes from outside in measure by open sets. Kurtz [29] and Kautz [25] showed that such an approximation in measure can be done effectively for classes of the lightface finite Borel hierarchy, in the sense that a Σ 0 n class can be approximated in measure by a Σ 0,∅ (n−1) 1 class. Therefore, while the definitions based on Σ 0 n nullsets and on Σ 0,∅ (n−1) 1 nullsets do not give the same notion of test, they yield the same class of random reals (see [9, Section 6.8] for a complete presentation of this argument).
The proof that the two approaches yield the same notion of n-randomness relativizes. Moreover, the approximation in measure by open sets is possible for any Borel probability measure, as any finite Borel measure on a metric space is regular. Finally, using Proposition 2.1 one can show inductively that, given a representation R of µ, the relations µ(S) > q and µ(S) < q (for q rational) are uniformly Σ 0,R n for any Σ 0 n class S. The latter fact is a key ingredient in the equivalence proof. Therefore, (µ, Z, n)-randomness could alternatively be defined via Σ 0,R⊕Z n tests. We prefer the approach given in Definition 2.3, because open sets are usually easier to work with, and because most techniques relativize.
Levin [30] introduced the alternative concept of a uniform test for randomness, which is representation-independent (see also [13]). Day and Miller [4,Theorem 1.6] have shown that for any measure µ and for any real X, X is µ-random in the sense of Definition 2.3 if and only if X is µ-random for uniform tests.
Since, for fixed R µ , Z, and n, there are only countably many (R µ , Z, n)tests, it follows from countable additivity that the set of (µ, Z, n)-random reals for any µ and any Z has µ-measure 1. Hence there always exist (µ, Z, n)random reals for any measure µ, any real Z, and any n ≥ 1.
However, µ, Z, and n put some immediate restrictions on the relative definability of any (µ, Z, n)-random real.
Proposition 2.5. If X is (µ, Z, n)-random via a representation R µ , then X cannot be ∆ 0 n (R µ ⊕ Z). Proof. If X is ∆ 0 n (R µ ⊕ Z), then X ≤ T (R µ ⊕ Z) (n−1)
, and we can build a (µ, Z, n)-test covering X by using the cylinders given by its initial segments.
It is also immediate from the definition of randomness that any atom of a measure is random with respect to it. This is a trivial way for a real to be random. The proposition below (a straightforward relativization of a result by Levin [48], see also [41,Proposition 3.3]) shows that atoms of a measure are also computationally trivial (relative to the measure). Proposition 2.6 (Levin). If for a measure µ and a real X, µ{X} > 0, then X ≤ T R µ for any representation R µ of µ.
Since we are interested in randomness for continuous measures, the case of atomic randomness is excluded a priori.
2.4.
Image measures and transformation of randomness. Let f : 2 ω → 2 ω be a Borel measurable function. If µ is a measure on 2 ω ,the image measure µ f is defined by
µ f (A) = µ(f −1 (A)).
It can be shown that every probability measure is the image measure of Lebesgue measure λ for some f . For continuous measures, Oxtoby [37] proved that any continuous, positive measure on 2 ω can be transformed into Lebesgue measure on the set of irrationals in [0, 1] via a homeomorphism.
Here, a measure is positive if µ σ > 0 for σ ∈ 2 <ω . Levin [48,Theorem 4.3], and independently Kautz [25,Corollary IV.3.18] (see also [2]) proved an effective version of these results. For a computable measure µ on 2 ω there exists a Turing functional Φ defined on almost every real such that µ is the image measure of λ under Φ. If µ is, moreover, continuous and positive, then Φ has an inverse that transforms µ into λ.
A consequence of the Levin-Kautz theorem is that every non-recursive real that is random with respect to a computable probability measure is Turing equivalent to a λ-random real. We will show now that for continuous measures, this can be strengthened to truth-table equivalence.
Proposition 2.7. Let X be a real. For any Z ∈ 2 ω and any n ≥ 1, the following are equivalent.
(i) X is (µ, Z, n)-random for a continuous measure µ recursive in Z.
(ii) X is (ν, Z, n)-random for a continuous, positive, dyadic measure ν exactly computable in Z. (iii) There exists a Turing Z-functional Φ such that Φ is an order-preserving homeomorphism of 2 ω , and Φ(X) is (λ, Z, n)-random. (iv) X is truth-table equivalent relative to Z to a (λ, Z, n)-random real.
Here, the order on 2 ω is the lexicographical order given by
X < Y :⇔ X(N ) < Y (N ) where N = min{n : X(n) = Y (n)}.
Proof. We give a proof for Z = ∅ and n = 1. It is routine to check that the proof relativizes and generalizes to higher levels of randomness.
(i) ⇒ (ii): Let X be µ-random, where µ is a continuous, computable measure. We construct a continuous, positive, dyadic, and exactly computable measure ν such that X is random with respect to ν, too. The construction is similar to Schnorr's rationalization of martingales [44] (see also [9, Proposition 7.1.2]).
We define ν by recursion on the full binary tree 2 <ω . To initialize, let ν * ∅ = 2. Now assume ν * σ is defined such that, µ σ < ν * σ < µ σ + 2 −|σ|+1 .
A simple case distinction shows that
max{µ σ ⌢ 0 , ν * σ −µ σ ⌢ 1 −2 −|σ| } < min{µ σ ⌢ 0 +2 −|σ| , ν * σ −µ σ ⌢ 1 }.
As the dyadic rationals are dense in R, there exists a dyadic rational r in this interval, and by Proposition 2.2 we can find such an r effectively in σ. Put
ν * σ ⌢ 0 = r, ν * σ ⌢ 1 = ν * σ − r.
Then clearly
ν * σ ⌢ 0 + ν * σ ⌢ 1 = ν * σ ,
and by the choice of r,
µ σ ⌢ 0 < ν * σ ⌢ 0 < µ σ ⌢ 0 + 2 −(|σ|) , µ σ ⌢ 1 < ν * σ ⌢ 1 < µ σ ⌢ 1 + 2 −(|σ|) .
We normalize by letting ν = ν * /2. By construction of ν * , the measure ν is dyadic and exactly computable. It is also clear from the construction that for all σ, µ σ < 2ν σ . In particular, ν is positive. Finally, if (V n ) is a test for ν, by letting W n = V n+1 we obtain a µ-test that covers every real covered by (V n ). Hence if X is µ-random, then X is also ν-random.
(ii) ⇒ (iii): Suppose ν is an exactly computable, continuous, positive, dyadic measure. Since ν is continuous and 2 ω is compact, for every m there exists a least l m ∈ ω such that whenever |σ| ≥ l m , then ν σ ≤ 2 −m . Without loss of generality, we can assume that l m < l m+1 . As ν is exactly computable, the mapping m → l m is computable.
We define inductively a mapping ϕ : 2 <ω → 2 <ω that will induce the desired homeomorphism. In order to do so, we first define, for every τ ∈ 2 <ω , an auxiliary finite, non-empty set E τ ⊆ 2 <ω . It will hold that (a) all strings in E τ are of the same length, and this length depends only on the length of τ ; (b) if σ ⊆ τ , then every string in E τ is an extension of some string in E σ ; (c) if σ and τ are incomparable, then E σ and E τ are disjoint; moreover, if |σ| = |τ | and σ is lexicographically less than τ , then all strings in E σ are lexicographically less than any string in Eτ ; (d) for all n, |τ |=n E τ = 2 ω ; (e) for all τ , 0 < ν E τ ≤ 2 −|τ | (2 − 2 −|τ | ).
Put E ∅ = {∅}. Suppose now that E τ is defined for all strings τ of length at most n, and that for these sets Eτ , (a)-(e) are satisfied.
Given any τ of length n, let
F τ = {σ : |σ| = l 2(n+1) & σ extends some string in E τ }.
Find the least (with respect to the usual lexicographic ordering) σ ∈ F τ such that η≤σ η∈Fτ
ν η ≥ ν E τ /2.
Let E τ ⌢ 0 = {η ∈ F τ : η < σ} and put the remaining strings of F τ into E τ ⌢ 1 . This ensures that E τ ⌢ 0 and E τ ⌢ 1 satisfy (a), (b), (c), and (d). Moreover, by the choice of the length of strings in F σ and property (e) for E τ , both E τ ⌢ 0 and E τ ⌢ 1 are non-empty. As ν is positive, this implies ν E τ ⌢ i > 0 for each i ∈ {0, 1}. Moreover, for each i ∈ {0, 1}, we can use the induction hypothesis for Eσ and deduce that
ν E τ ⌢ i ≤ ν E τ /2 + 2 −2|τ |−1 ≤ 2 −|τ |−1 (2 − 2 −|τ | ) + 2 2(|τ |+1) = 2 −|τ |−1 (2 − 2 −|τ |−1 ),
which yields the bound in (d). Now we define the mapping ϕ: Put ϕ(∅) = ∅. Suppose now ϕ(σ) is defined for all τ of length less than or equal to n. Given τ of length n, map all strings in E τ ⌢ 0 to τ ⌢ 0, and all strings in E τ ⌢ 1 to τ ⌢ 1. To make ϕ defined on all strings, map any string that extends some string in E τ but is a true prefix of some string in F τ to τ .
It is clear from the construction that ϕ induces a total, order preserving mapping Φ : 2 ω → 2 ω by letting Φ(X) = lim n ϕ(X⌈ n ).
Φ is onto since for every σ, E σ is not empty. We claim that Φ is also one-one. Suppose Φ(X) = Φ(Y ). This implies that for all n, ϕ(X⌈ n ) = ϕ(Y ⌈ n ), that is, for all n, X⌈ n and Y ⌈ n belong to the same E σ . Since ν is positive, the diameter of the E σ goes to 0 along any path. Hence X = Y .
It remains to show that Φ(X) is Martin-Löf random. Suppose not, then there exists an λ-test (W n ) that covers Φ(X). Let
V n = σ∈W n+2 E σ .
Then (V n ) covers X. Furthermore, the (V n ) are uniformly enumerable since the mapping σ → E σ is computable by the construction of the E σ . Finally,
τ ∈Vn ν τ = σ∈W n+2 ν E σ ≤ σ∈W n+2
thus X is not ν-random, contradiction.
(iii) ⇒ (iv): This is immediate. (iv) ⇒ (i): This follows from Theorem 5.7 in [41] The result also suggests that if we are only interested in whether a real is random with respect to a continuous measure, representational issues do not really arise. We can restrict ourselves to dyadic measures, which have a minimal representation.
Remark 2.8. We will henceforth, unless explicitly noted, assume that any measure is a dyadic measure. We drop reference to the representation and write µ instead of R µ .
2.5.
Continuous randomness via Turing reductions. While Proposition 2.7 gives a necessary and sufficient criterion for reals being random for a continuous measure, we will later need further techniques to show that a given real is random with respect to a continuous measure. As many of our arguments will involve arithmetic definability, it will be helpful to know to what extend randomness for continuous measures can be "transfered" via Turing reductions instead of truth-table reductions. The key ingredients are a theorem by Demuth [6] and a result by Kurtz [29].
Demuth [6,Theorem 17] showed that every non-recursive real truth-table below a Martin-Löf random real measure is Turing equivalent to a Martin-Löf-random real. The proof relativizes (as can be seen from the presentation in [9, Theorems 6.12.9 and 8.6.1]) and yields the next proposition.
Recall that we only consider dyadic measures and hence drop reference to a representation. Nevertheless, the results in this section are not dependent on the existence of a minimal representation and can be reformulated accordingly.
Proposition 2.9 (Demuth). Suppose Y is (µ, Z, n)-random (n ≥ 1) and X is truth-table reducible to Y relative to (µ ⊕ Z) (k) for some k ≤ n − 1 (i.e., X ≤ tt((µ⊕Z) (k) ) Y ). Further suppose X is not recursive in (µ ⊕ Z) (k) . Then X is Turing equivalent relative to (µ ⊕ Z) (k) to a (λ, µ ⊕ Z, n)-random real.
Kurtz [29,Theorem 4.3] observed that 2-random reals are ∅ ′ -dominated. More precisely, there exists a ∅ ′ -computable function dominating every function computable from a 2-random real.
The proof is based on the following idea (see [36,Proposition 5.6.28]): Given a Turing functional Φ, ∅ ′ can decide, given rational q and n ∈ ω,
whether λ {Y : Φ(Y )(k) is defined for all k ≤ n} > q.
For each n, let q n be maximal of the form i·2 −n so that the above holds, and let t n be such that Φ converges on at least measure q n -many strings of length t n by time t n . Construct a function f ≤ T ∅ ′ such that f (n) dominates all function values Φ(Y ) computed with use t n and within t n steps. Then the set of all Y for which Φ(Y ) is total and not dominated by f has Lebesgue measure 0 and can be captured by a ∅ ′ -Martin-Löf-test. The argument relativizes to other measures and parameters, and we obtain the following.
Proposition 2.10 (Kurtz). Given a measure µ and a real Z, there ex-
ists a function f ≤ T (µ ⊕ Z) ′ such that for every (µ, Z, 2)-random X, if g ≤ T(Z⊕µ) X, then g is dominated by f .
Together with Proposition 2.9 this yields a sufficient criterion for continuous randomness.
Lemma 2.11. Suppose n ≥ 3 and Y is (µ, Z, n)-random. If X ≤ T(µ⊕Z) Y and X is not recursive in (µ ⊕ Z) ′ , then X is (ν, (µ ⊕ Z) ′′ , n − 2)-random for some continuous measure ν ≤ T (µ ⊕ Z) ′′ .
Proof. We assume Z = ∅ to keep notation simple. Suppose X ≤ T(µ) Y via a Turing reduction Φ. By Proposition 2.10, the use and the convergence time of Φ on Y are dominated by some function recursive in µ ′ . We can modify Φ to Φ such that Φ is a truth-table reduction relative to µ ′ and Φ(Y ) = X. By Proposition 2.9, X is Turing equivalent relative to µ ′ to a (λ, µ, n)random real L. Any (λ, µ, n)-random real is also (λ, µ ′ , n − 1)-random, and so we can apply Proposition 2.10 to X and L to conclude that they are truth-table equivalent relative to µ ′′ . This in turn means that X is truthtable equivalent relative to µ ′′ to a (λ, µ ′′ , n − 2)-random real, which by Proposition 2.7 implies that X is (ν, µ ′′ , n − 2)-random for a continuous measure recursive in µ ′′ .
2.6. The definability strength of randomness. Lemma 2.11 shows that sufficiently high randomness for continuous measures propagates downward under Turing reductions (losing some of the randomness strength, however). This result was partly based on the fact that, for n ≥ 2, n-random reals cannot compute fast-growing functions (beyond what is computable by ∅ ′ ). There is further evidence that the computational strength of n-random reals is rather limited.
For example, random reals are generalized low (relative to the measure). This is a generalization of a result due to Kautz [25, Theorem III.2.1].
Proposition 2.12 (Kautz). Let µ be a continuous measure, and suppose X is µ-(n + 1)-random, where n ≥ 1. Then
(X ⊕ µ) (n) ≡ T X ⊕ µ (n)
The generalization works for the same reasons that n-randomness can be defined equivalently in terms of Σ 0,µ n -tests or Σ 0,µ (n−1)
1
-tests: Borel probability measures are regular, and the relations µ(S) > q and µ(S) < q (for q rational) are uniformly Σ 0,µ n for any Σ 0 n class S. Furthermore, one can generalize a result of Downey, Nies, Weber, and Yu [8], who show that every weakly 2-random real forms a minimal pair with 0 ′ . This will be of central importance in Section 4. For our purposes, it suffices to consider randomness instead of weak randomness, which we do in the following lemma.
Lemma 2.13. Suppose µ is a continuous measure and Y is µ-n-random, n ≥ 2. If X ≤ T µ (n−1) and X ≤ T Y ⊕ µ, then X ≤ T µ.
The structure of the proof is as follows: Following Downey, Nies, Weber, and Yu, we first show that the upper cone by Φ is Π 0 2 (relative to µ (n−2) ). Next, we argue that the upper cone has cannot have measure zero since it contains a random real. Finally, one uses this fact to isolate X as a path in a µ-r.e. tree. The last step is a generalized version of the result that if the Turing upper cone of a real has positive Lebesgue measure, then the real must be computable [5,42]. Our presentation follows [36].
Proof of Lemma 2.13. Suppose X ≤ T Y ⊕ µ via a Turing functional Φ and Y ≤ T µ (n−1) . Note that Y is ∆ 0 2 relative to µ (n−2) . Let Y (n, s) be a µ (n−2) - recursive approximation of Y , i.e., lim s Y (n, s) = Y (n). Given i, s ∈ ω, put U i,s = {X : ∃t > s Φ X⊕µ t (i) = Y (i, t) }. The set U i,s is Σ 0,µ (n−2) 1 uniformly in i, s and hence P = i,s U i,s is Π 0,µ (n−2) 2 . Note that P is the upper cone of X under Φ, P = {A : Φ(A) = X}.
P cannot have µ-measure 0: If it had then, since Borel probability measures on 2 ω are regular, for the sequence of open sets (V k ) k∈ω given by
V k = i,s ≤k U i,s , we have µV k ց 0. Since each V k is Σ 0,µ (n−2) 1
, µ (n−1) can decide whether µV k ≤ 2 −l for given l. Hence, we can convert (V k ) into a (µ, n)-test. Since k V k = P and P contains Y , this contradicts the fact that Y is µ-n-random.
Hence pick r rational such that µP > r > 0, where r is rational. Define a tree T by letting
σ ∈ T :⇔ µ{τ : Φ(τ ⊕ µ) ⊇ σ} > r,
and closing under initial segments. T is r.e. in µ and X is an infinite path through T .
Since µ is a probability measure, any antichain in T contains at most ⌈1/r⌉ strings. Choose σ = X⌈ n such that no τ ⊇ σ incompatible with X is in T . Such σ exists for otherwise we could find an antichain of more than ⌈1/r⌉ strings branching off X. To compute X⌈ m from µ, it suffices to enumerate T above σ until a long enough extension shows up.
We will later need the following relativization of the previous lemma. The proof is similar.
Lemma 2.14. Suppose µ is a continuous measure and Z is µ-(k + n)- random, k ≥ 0, n ≥ 2. If Y ≤ T µ (k+n−1) and Y ≤ T Z ⊕µ (k) , then Y ≤ T µ (k) .
One interpretation of Lemmas 2.13 and 2.14 is that µ-random reals are not helpful in computing (defining) reals arithmetic in µ. For example, if a real is properly ∆ 0 n relative to a measure µ, then it cannot be ∆ 0 k relative to µ ⊕ X where k < n and X is µ-(n + 1)-random.
In Section 4, we will also need a result similar to the previous lemmas regarding initial segments of linear orders, namely, that random reals are not helpful in the recognizing well-founded initial segments. The following lemma may appear technical at this point, but its importance will become clear towards the end of Section 4, in the proof of Theorem 4.50.
Lemma 2.15. Let j ≥ 0. Suppose µ is a continuous measure and ≺ is a linear order on a subset of ω such that the relation ≺ and the field of ≺ are both recursive in µ (j) . Suppose further X is (j + 5)-random relative to µ, and I ⊆ ω is the longest well-founded initial segment of ≺. If I is recursive in (X ⊕ µ) (j) , then I is recursive in µ (j+4) .
Proof. Suppose I ≤ T (X ⊕ µ) (j) , X is (j + 5)-random relative to µ, but I T µ (j+4) . By Lemma 2.11 (where X, µ j , 5, I are substituted for Y, Z, n, X in the statement of the Lemma, respectively), there is a continuous measure
µ I ≤ T µ (j+2) such that I is (µ I , µ (j+2) , 3)-random.
For given a ∈ Field(≺), let I(a) be the set of all reals Z ⊆ ω such that Z is an initial segment of ≺, and all elements of Z are bounded by a. I(a) is a Π 0 1 (µ (j) ) class. Let T a be a tree recursive in µ (j) such that [T a ] = I(a). Given n ∈ ω, let T a ⌈ n = {σ ∈ T a : |σ| = n} be the n-th level of T a . We have I(a) = n T ⌈ n . Now, if a ∈ I, then I(a) is countable (since in this case each element of I(a) is an initial segment of the well-founded part of ≺ and there are at most countably many such initial segments). Since µ I is continuous, it follows that I(a) has µ I -measure zero.
If, on the other hand, a ∈ I, then I ∈ I(a). Since I is (µ I , µ (j+2) , 3)random and I(a) is Π 0 1 (µ (j) ), I(a) does not have µ I -measure zero: Otherwise we could recursively in µ (j+2) , compute a sequence (l n ) such that µ I T a ⌈ ln ≤ 2 −n . This would be a (µ I , µ (j+2) , 1)-test that covers I, but I is
(µ I , µ (j+2) , 3)-random.
We obtain the following characterization of I.
a ∈ I ⇔ ∀n ∃l (µ I T a ⌈ l ≤ 2 −n ) Since µ I ≤ T µ (j+2)
, the property on the right hand side is Π 0 2 (µ (j+2) ), hence I is recursive in µ (j+4) , contradicting our initial assumption.
We conclude this section by establishing that Turing jumps cannot be µn-random, n ≥ 2, for any measure µ. While strictly speaking the following two results are not needed later, they are prototypical for a type of argument that will be important in Section 4, where we construct long sequences of reals with an internal definability hierarchy that are not random with respect to any continuous measure.
Proposition 2.16. For any k ≥ 0, if X ≡ T ∅ (k)
, then X is not 2-random with respect to any continuous measure.
Proof. The case k = 0 is clear, so assume k > 0. Suppose X ≡ T ∅ (k) is µ-2-random for some µ. Then ∅ ′ ≤ T X and also ∅ ′ ≤ T µ ′ . It follows from Lemma 2.13 that ∅ ′ is recursive in µ. Applying the same argument inductively to ∅ (i) , i ≤ k, yields ∅ (i) ≤ T µ, in particular X ≡ T ∅ (k) ≤ T µ, which is impossible if X is µ-2-random.
It may be helpful to picture the preceding argument as a "stair trainer machine": Using the supposedly random X, each step, that is, Turing jump, "sinks down" to µ, and eventually, X ≤ T µ, yielding a contradiction.
The non-randomness property of the jumps extends to infinite jumps, too.
Proposition 2.17. If X ≡ T ∅ (ω)
, then X is not 3-random with respect to a continuous measure.
Proof. Assume for a contradiction that X is µ-3-random for continuous µ.
By the inductive argument of the previous proof, 0 (k) ≤ T µ for all k ∈ ω. By a result of Enderton and Putnam [10], if Y is a ≤ T -upper bound for
{0 (k) : k ∈ ω}, then 0 (ω) ≤ T Y ′′ . Therefore, X ≤ T µ ′′ , contradicting that X is µ-3-random.
The Countability Theorem
In this section we will prove Theorem 1, which we restate here for convenience.
Theorem 1. Let n ∈ ω. Then the set NCR n = {X ∈ 2 ω : X is not n-random for any continuous measure} is countable.
As mentioned in the introduction, the case n = 1 was proved in [41]. The basic outline for the proof for n > 1 is as follows. We first show that the set of reals that are n-random for some continuous measure contains an upper cone in the Turing degrees. The argument uses Martin's result [31] that every Turing invariant Borel set that is cofinal in the partial ordering of the Turing degrees contains an upper cone with respect to Turing reducibility. The base of the cone is given by the winning strategy in a certain Borel game G(B). The constructive nature of Martin's proof of Borel determinacy yields that winning strategy is contained in a countable level L β n+3 of the constructible hierarchy. We use a forcing notion due to Kumabe and Slaman (see [46]) to show that given X / ∈ L β n+3 , there exists a forcing extension L β n+3 [G] in which every real is Turing reducible to X (relative to the generic G). In particular, X is in the upper cone of random reals above the winning strategy for the game G(B) (in L β n+3 [G]). Finally, we argue that the winning strategy is absolute and thus makes X random with respect a continuous measure.
3.1.
A cone of continuously random reals. In this section we will prove the existence of an upper cone of random reals.
Definition 3.1. A set A ⊆ 2 ω is Turing-invariant if X ∈ A and Y ≡ T X implies that Y ∈ A. A upper Turing cone is a Turing invariant set A ⊆ 2 ω of the form {Y ∈ 2 ω : X ≤ T Y } for some X ∈ 2 ω .
Borel-Turing determinacy ( [31,32]). If A ⊆ 2 ω is a Turing invariant Borel set, then there exists a real Y such that either A or 2 ω \ A contains an upper Turing cone.
Lemma 3.2. For every n ≥ 1, there exists a real X such that for all Y ≥ T X, Y is n-random with respect to some continuous measure.
Proof. Suppose X ≡ T(Z) R where R is (n + 2)-random relative to Z. By Lemma 2.11 X is n-random with respect to some continuous measure.
Let B ⊆ 2 ω be the set {Y ∈ 2 ω : ∃Z ∃R (Y ≡ T Z ⊕ R & R is (Z, n + 2)-random)}
Clearly, B is Turing invariant. To see that B is Borel, note that the set can be defined in the form
∃e, d (e, d are indices of Turing functionals such that Φ d (Φ e (Y )) = Y
and one half of Φ e (Y ) is (n + 2)-random relative to the other half).
As observed above, every real in B is n-random for a continuous measure (note that Y ∈ B cannot be recursive in Z ′ since R is (Z, n + 2)-random). Furthermore, B is cofinal in the Turing degrees, i.e., if S is any real, then there exists an Y ≥ T S such that Y ∈ B, since we can always find a real R that is (n + 1)-random relative to S and put Y ≡ T S ⊕ R.
By Borel-Turing determinacy, B contains an upper Turing cone, because B is cofinal in the Turing degrees.
3.2.
From upper cone to co-countably many. The determinacy argument of the previous subsection yields the existence of an upper cone of n-random reals. Martin's proof of Borel-Turing determinacy yields that the base of the cone is given by a winning strategy for a certain Borel game.
Although Martin's proof of Borel Determinacy [32] is of a constructive nature, its meta-mathematical complexity is high, in the sense that it makes inductive use of the power set operation: The higher the level of a Borel set, the more iterates of the power set of ω one needs to construct a winning strategy, in form of trees whose nodes are trees whose nodes are trees etc.
Friedman [11] showed that this is in fact an intrinsic feature of Borel determinacy. As we will see in the next section, this supplements Theorem 1 with an interesting metamathematical twist.
Nevertheless, Martin's proof of Borel determinacy is constructive. Therefore, it is not hard to locate a winning strategy within the constructible hierarchy.
Definition 3.3. Given n ∈ ω, ZFC − n denotes the axiom of ZFC, where the power set axiom is replaced by the sentence "There exist n-many iterates of the power set of ω".
Hence, in ZFC − 0 , for instance, we have the existence of the set of all natural numbers (since the Axiom of Infinity holds and ω is absolute), and various other subsets of ω as given by applications of separation or replacement, but we lack the guaranteed existence of the set of all such subsets.
Models of ZFC − n will play an important role throughout this paper. In particular, we are interested in models inside the constructible universe. As usual, L will denote the constructible universe, the limit of the cumulative hierarchy of sets obtained by iterating the power set operation restricted to definable subsets. For any ordinal α, L α denotes the α-th level of the hierarchy. A key property of this hierarchy is that |L α | = |α|. For more background on L, see [21,Chapter 13] or [28,Chapters V,VI]. For an indepth account, see [7]. Definition 3.4. Given n ∈ ω, let β n be the least ordinal such that
L βn |= ZFC − n .
By the Löwenheim-Skolem theorem and the Gödel condensation lemma, L βn , and hence β n , is countable.
Lemma 3.5. If A ⊆ 2 ω is Σ 0
n , then the Borel game G(A) with winning set A has a winning strategy S in L βn .
The proof given by Martin [33] is inductive. A key concept is the unraveling of a game. Simply speaking, a tree T over some set B unravels G(A) if there exists a continuous mapping π :
[T ] → 2 ω such that π −1 (A) is clopen in [T ]
, and there is a continuous correspondence between strategies on T and strategies on 2 <ω .
Martin first shows that Π 0 1 games can be unraveled. The argument is completely constructive, hence can be carried out in L. The unraveling tree T is given by the legal moves of some auxiliary game whose moves correspond to strategies in the original game on 2 <ω , that is, reals. To be able to collect all these legal moves requires the existence of the power set of ω.
The inductive step then shows how to unravel a given Σ 0
n set A. Sup- pose A = A i , where each A i is Π 0 n−1 .
By induction hypothesis, each A i can be unraveled by some T i via some mapping π i . Martin proves that the unravelings T i can be combined into a single one, T ∞ , that unravels each A i via some π ∞ . Since each of the sets
π −1 ∞ (A i ) is clopen, their union π −1 ∞ (A i ) = π −1 ∞ (A)
is open, and can in turn be unraveled by some T . Again, the proof is constructive. The last step in the construction (unraveling π −1 ∞ (A)) passes to a tree of higher type -its nodes correspond to strategies over T ∞ . Hence one more iterate of the power set of ω is introduced.
Therefore, Σ 0 n determinacy is provable in ZFC − n , and Martin's proof constructs a winning strategy in L βn , relative to L βn . By Mostowski's absoluteness theorem (see [21,Theorem 25.4]), this is also a winning strategy in V (see also Subsection 3.4).
In order to complete the proof of Theorem 1, it we need to prove a Posner-Robinson style theorem for reals not contained in L βn and use an absoluteness argument. We state Lemma 3.6 only in the case that n is greater than zero. Under this restriction, we can avoid class forcing and reduce to standard facts about set forcing. The case n equals zero is not needed for our argument.
Lemma 3.6. Suppose that n is a natural number greater than zero and X is a real number not in L βn . Then there exists a model L βn [G] of ZFC − n such that every real in L βn [G] is Turing reducible to X ⊕ G.
3.3.
Kumabe-Slaman forcing. This subsection is devoted to proving Lemma 3.6. We construct G by means of a notion of forcing due to Kumabe and Slaman. The forcing was an essential ingredient in the proof of the definability of the Turing jump by Shore and Slaman [46]. It allows for extending the Posner-Robinson Theorem to iterated applications of the Turing jump. It is based on the following partial order. The construction of the generic G follows [46] rather closely. We have to ensure, however, that forcing has the desired set theoretic properties.
In the following, we use the conventions and vocabulary of Section 2.1.
Definition 3.7. Let P be the following partial order.
(1) The elements p of P are pairs (Φ p , Z p ) in which Φ p is a finite, usemonotone Turing functional and Z p is a finite collection of subsets of ω. As usual, we identify subsets of ω with elements of 2 ω .
(2) If p and q are elements of P, then p ≥ q if and only if (a) (i) Φ p ⊆ Φ q and (ii) for all (x q , y q , σ q ) ∈ Φ q \ Φ p and all (x p , y p , σ p ) ∈ Φ p , the length of σ q is greater than the length σ p , (b) Z p ⊆ Z q , (c) for every x, y, and X ∈ Z p , if Φ q (x, X) = y then Φ p (x, X) = y.
In short, a stronger condition than p can add computations to Φ p , provided that they are longer than any computation in Φ p and that they do not apply to any element of Z p .
Let P n denote the partial order P as defined in L βn . By standard arguments, we show that if G ⊆ P n is a generic filter in the sense of L βn , then L βn [G] is a model of ZF − n . By inspection of P n , any such G naturally gives rise to a functional Φ G = {Φ p : p ∈ G}. To prove Theorem 3.6, given X not in L βn , we will exhibit a particular G so that G is P n -generic over L βn and so that every element in L βn [G] is computable from G ⊕ X.
Definition 3.8 (Definition III.3.3, [28]). Let P * be a partially ordered set. Then, p, q ∈ P * are compatible iff then have a common extension. An antichain is a subset of P * whose elements are are pairwise incompatible. P * has the countable chain condition (ccc) iff, in P * every antichain is countable. Lemma 3.9. Let n be a natural number greater than zero.
(1) P n is an element of L βn .
(2) L βn |= P n has the ccc.
Proof. Since n ≥ 1, L βn satisfies the statement that there is at least one uncountable cardinal. By usual structure theory for initial segments of L, the power set of ω as defined in L βn is a set in L βn . Since P is defined directly from ω, P n is a set in L βn .
If p and q are incompatible elements of P n , then Φ p and Φ q must be different. Since there are only countably many possibilities for Φ p and Φ q , any antichain in P n must be countable in L βn . (1) G is a filter on P n iff (a) G is not empty.
(b) ∀p, q ∈ G ∃r ∈ G[p ≥ r and q ≥ r].
(c) ∀p, q ∈ G[if p ≥ q and q ∈ G, then p ∈ G].
(2) G is P n -generic over L βn iff for all D such that D ⊆ P n is dense and D is definable from parameters in L βn , G ∩ D is not empty. Here, D is dense iff for every p ∈ P n there is a q ∈ D such that p ≥ q.
Lemma 3.11. If G is P n -generic over L βn , then L βn [G] |= ZFC − n .
Proof. By Lemma IV.2.26 of [28], it follows that if G is P n generic over L βn , then L βn [G] satisfies the axioms of ZF − except for possibly Replacement. It remains only to observe that L βn [G] satisfies Replacement, Choice and that there are n-many uncountable cardinals. The verification that L βn [G] satisfies Replacement is the same as given in Theorem IV.2.27 of [28]. That L βn [G] satisfies Choice follows from Replacement and the usual proof that the order of constructibility is a L βn -definable well-order of L βn applies relative to G. Finally, that L βn [G] has the same uncountable cardinals as L βn does follows from P n 's having the ccc in L βn by the argument given in the proof of Theorem IV.3.4 of [28]. Although this theorem is stated for models of ZFC, its proof does not invoke the power set axiom. Given that L βn [G] has n many uncountable cardinals, the Gödel Condensation Lemma relative to G ensures that it has n-many iterates of the power set of ω.
Next, we show that every dense set in L βn can be met via an extension adding no computations along X. This is crucial for the construction in [46]. Lemma 3.12. Let D ∈ L βn be dense in P n and suppose X / ∈ L βn . For any p ∈ P n , there exists a q ≤ p such that q ∈ D and Φ q does not add any new computation along X.
Proof. Suppose p = (Φ p , Z p ) is in P n . We say a string τ is essential for (p, D) if, whenever q < p and q ∈ D, there exists a triple (x, y, σ) ∈ Φ q \ Φ p such that σ is compatible with τ . In other words, whenever one meets D by an extension of p, a computation relative to some string compatible with τ is added. Note that τ 's being essential for (p, D) is definable in L βn .
If τ is a binary sequence and τ 0 is an initial segment of τ , then any sequence σ which is compatible with τ is also compatible with τ 0 . Thus, being essential is closed under taking initial segments. So, T (p, D) = {τ : τ essential for (p, D) } is a binary tree in L βn .
Assume now for a contradiction that a q as postulated above does not exist. This means that for any r ≤ p, either r / ∈ D or Φ r adds a computation along X. It follows that every initial segment τ ⊂ X is essential for (p, D).
Thus T (p, D) is infinite. Since L βn satisfies Kőnig's Lemma (equivalently, compactness of the Cantor set), there exists a real Y ∈ 2 ω ∩ L βn such that Y is an infinite path through T (p, D). Now consider the condition p 1 = (Φ p , Z p ∪{Y }). As Φ q = Φ p and Y ∈ L βn , we trivially have p 1 ≤ p in P n . Since every initial segment of Y is essential for (p, D), any extension of p in D must add a computation along Y . Since no extension of p 1 can add a computation along Y and every extension of p 1 is an extension of p, no extension of p 1 is in D. This contradicts the density of D.
We can now finish the proof of Theorem 3.6. It is sufficient to construct a P n -filter G that is generic over L βn such that for every A : ω → 2 in L βn [G] there is a k such that for all m, Φ X G ((k, m)) = A(m). The construction of such a G follows [46]. We fix countings of the set of terms in the forcing language for functions from ω to 2 in L βn and of the dense subsets D of P n which are definable over L βn . We proceed by recursion to define G. At stage n, we will have determined for each i less than n, an integer k i and we will ensure that for all m, Φ X G ((k i , m)) will have the same value as the interpretation of the i-th term does at m in L βn . By Lemma 3.12, we can meet dense sets and decide values of terms without adding any new values to Φ X G . We can then extend Φ G so that Φ X G ((k i , m)) takes the values equal to those already decided for the relevant terms. Finally, we can determine a value for k n that is greater than any argument for any computation mentioned in the construction so far.
We conclude by observing that the set Φ G satisfies the two conclusions of Theorem 3.6. First,
L βn [Φ G ] is model of ZFC − n since Φ G ∈ L βn [G] and L βn [G] is a model of ZFC − n . Second, for each Z : ω → 2, if Z ∈ L βn [Φ G ] then it is the denotation in L[G]
of some term in the forcing language for P n , and so there is a k such that for all n, Z(n) = Φ X G ((k, n)).
3.4.
Completing the proof of Theorem 1. We now put the pieces together to show that every real outside of L β n+3 is n-random with respect to a continuous probability measure. As L β n+3 is countable, this will complete the proof of Theorem 1. Given X ∈ L β n+3 , choose G as in Lemma 3.6. Consider the game defined in the proof of Lemma 3.2. Its winning set is
B = {X ∈ 2 ω : ∃Z ∃R (X ≡ T Z ⊕ R & R is (Z, n + 2)-random)}, which is Σ 0 n+3 .
Now relativize to G and denote the relativized winning set by B G . Since ZFC − n+3 proves Σ 0 n+3 determinacy, L β n+3 [G] contains a winning strategy for the relativized game
G(B G ) played inside L β n+3 [G].
The property of being a winning strategy for a given Borel game is Π Π Π 1 1 . By Mostowski's absoluteness theorem (see [21,Theorem 25.4]), this means that a winning strategy in L β n+3 [G] is actually a winning strategy "in the real world", i.e., it wins on all plays, not just the ones in L β n+3 [G].
Hence, following the proof of Lemma 3.2, every real in the upper cone (relative to G) of the winning strategy is (µ, G, n)-random for some continuous measure µ. By Lemma 3.6, X is (relative to G) in every upper cone with base in L β n+3 [G], X (µ, G, n)-random for some continuous measure µ. Finally, note that every (µ, G, n)-random real is (µ, n)-random.
We have shown that every real not contained in L β n+3 is n-random for a continuous measure. As β n+3 is countable, this completes the proof of Theorem 1.
The Metamathematics of Randomness
In this section, we will show that the metamathematical ingredients used to prove the countability of NCR n are necessary. More precisely, we will prove the Theorem 2, which we restate here for convenience. Theorem 2. There exists a computable function G(n) such that for every n ∈ ω, ZFC − n "NCR G(n) is countable."
Before starting the proof, we outline its basic idea. For given n, we will show that in the model L βn of ZFC − n , NCR G(n) is not countable. To this end, we find a sequence (Y α ) of reals that satisfies (1) (Y α ) is cofinal in the Turing degrees of L βn , hence not countable in L βn , (2) no Y α is G(n)-random for a continuous measure in L βn .
As we have seen in Propositions 2.16 and 2.17, iterating the Turing produces an increasing sequence of non-random reals. It makes sense therefore to look for a set-theoretic analogue of the jump hierarchy. This analogue is given by the master codes of the constructible hierarchy. Just as instances of the jump code levels of arithmetically definable subsets of ω, master codes code levels of the constructible universe. The master codes for a higher level of L can be obtained from codes for lower levels by iterating definability. This will be crucial for our proof, since it allows for applying the "Stair Trainer"-argument of Propositions 2.16 and 2.17 in this setting.
In order to define this sequence, we have to present a few more facts on L. In particular, we are interested how at each step new sets are added to L. This is the heart of the fine structure theory of L due to Jensen [22]. We will give a brief review of the core concepts and results. Readers familiar with fine structure theory can skip ahead to Subsection 4.3.
4.1.
Fine structure and Jensen's J-hierarchy. Fine structure provides a level-by-level, quantifier-by-quantifier analysis of how new sets are generated in L. Jensen defines the new constructible hierarchy, the J-hierarchy (J α ) α∈Ord that has all the important properties of the L-hierarchy (in particular, L = α J α ). In addition to this, each level J α has closure properties (such as under pairing functions) that L α may be lacking. While it is not strictly necessary for this paper to work with J α (we could work with (L ωα ) α∈Ord ), the J-hierarchy is the established framework for fine structure analysis, and we will adopt its basic concepts and terminology.
The sets J α are obtained by closing under a scheme of rudimentary functions. In contrast to L α+1 , J α+1 contains sets of rank up to ω(α+1), not just subsets of J α , e.g. ordered pairs. The rudimentary functions are essentially a scheme of primitive set recursion [23].
For transitive X, rud(X) denotes the smallest set Y that contains X ∪{X} and is closed under rudimentary functions (rud closed). The inclusion of {X} when taking the rudimentary closure guarantees that new sets are introduced even if X is closed under rudimentary functions.
The J-hierarchy is introduced as a cumulative hierarchy induced by the rud-operation:
J 0 = ∅ J α+1 = rud(J α ) J λ = α<λ J α for λ limit.
A fine analysis of the rudimentary functions reveals that the rud-operation can be completed by iterating some or all of nine basic rudimentary functions. Proposition 4.1 (Jensen [22]). Every rudimentary function is a composition of the following nine functions:
F 0 (x, y) = {x, y}, F 1 (x, y) = x \ y, F 2 (x, y) = x × y, F 3 (x, y) = {(u, z, v) : z ∈ x ∧ (u, v) ∈ y}, F 4 (x, y) = {(u, v, z) : z ∈ x ∧ (u, v) ∈ y}, F 5 (x, y) = x, F 6 (x, y) = dom(x), F 7 (x, y) = ∈ ∩ (x × x), F 8 (x, y) = {{x(z)} : z ∈ y}.
The S-operator is defined as taking a one-step application of any of the basic functions, The S-hierarchy is defined as the cumulative hierarchy induced by the Soperator and refines the J-hierarchy.
(4.1) S(X) = [X ∪ {X}] ∪ 8 i=0 F i [X ∪ {X}] .S 0 = ∅, S α+1 = S(S α ), S λ = α<λ S α for λ limit.
We obviously have
J α = β<ωα S β = S ωα .
We list a few basic properties of the sets J α . For details and proofs, see [22] or [7].
• Each J α is transitive and and is a model of a sufficiently large fragment of set theory (more precisely, it is a model of KP-set theory without Σ 0 -collection).
• The hierarchy is cumulative, i.e., α ≤ β implies J α ⊆ J β .
• rank(J α+1 ) = rank(J α )+ω. Each successor step adds ω new ordinals. J α ∩ Ord = ωα, in particular, J 1 = V ω and J 1 ∩ Ord = ω. • (J α ) α∈Ord and (L α ) α∈Ord generate the same universe: L = α J α .
Moreover, L α ⊆ J α ⊆ L ωα , and J α = L α if and only if ωα = α. Finally, J α+1 ∩ P(J α ) = P DEF (J α ), that is, J α+1 contains precisely those subsets of J α that are first order definable over J α . • The Σ n -satisfaction relation over J α , |= Σn Jα , is Σ n -definable over J α , uniformly in α.
• The mapping β → J β (β < α) is Σ 1 -definable over any J α . • There is a Π 2 formula ϕ V=L such that for any transitive set M ,
M |= ϕ V=L ⇔ ∃α M = J α .
The J-hierarchy shares all important metamathematical features with the L-hierarchy. We cite the two most important facts. The L-versions of the two propositions together constitute the core of Gödel's proof that GCH and AC hold in L.
Proposition 4.2.
There exists a Σ 1 -definable well-ordering < J of L and for any α > 1, the restriction of < J to J α is uniformly Σ 1 -definable over J α .
Proposition 4.3 (The condensation lemma for J).
For any α, if X Σ 1 J α , then there is an ordinal β and an isomorphism π between X and J β . Both β and π are uniquely determined.
For proofs of these results for the J-hierarchy again refer to [22] or [7].
4.2.
Projecta and master codes. The definable well-ordering < J together with the definability of the satisfaction relation can be used to show that each J α has definable Skolem functions, essentially by selecting the < J -least witness that satisfies an existential formula. The definable Skolem functions can in turn be used to define a canonical indexing of J α [22, Lemma 2.10].
Proposition 4.4 (Jensen [22]). For each α, there exists a Σ 1 (J α )-definable surjection from ωα onto J α .
While a simple cardinality argument yields that |J α | = |ωα|, Jensen's result shows that an ωα-counting of J α already exists in J α+1 . The indexing is obtained by taking (essentially) the Skolem hull of ωα under the canonical Σ 1 -Skolem function. The resulting set X is a Σ 1 -elementary substructure of J α , hence by the condensation lemma is isomorphic to some J β . The isomorphism taking X to J β is the identity on all ordinals below ωα, and one can show that this in turn implies that the isomorphism must be the identity on X, i.e., X = J β .
Boolos and Putnam [3] first observed that if a new real is defined in L α+1 , i.e., if P(ω) ∩ (L α+1 \ L α ) = ∅, then the strong absoluteness properties of L can be used to get a definable ω-counting of L α (instead of just an α-counting as above). Because, if a new subset Z of ω is constructed in L α+1 \ L α , one can take the Skolem hull of ω instead of ωα. The resulting X ∼ = L β is still equal to L α , since the definition of the new real applies in the elementary substructure L β . If β < α, then this would contradict the fact that Z ∈ L α .
Proposition 4.5 (Boolos and Putnam [3]). If
P(ω) ∩ (L α+1 \ L α ) = ∅, then there exists a surjection f : ω → L α in L α+1 .
Of course, at some stages no new reals are constructed. Boolos and Putnam [3] showed that the first such stage is precisely the ordinal β 0 , i.e., the least ordinal β such that L β |= ZF − . By Gödel's work, on the other hand, we know that no new real is constructed after stage ω L 1 . Jensen [22] vastly extended these ideas into the framework of projecta and master codes, which form the core concepts of fine structure theory. Definition 4.6. For natural numbers n > 0 and ordinals α > 0, the Σ nprojectum ρ n α is equal to the least γ ≤ α such that P(ωγ)∩(Σ n (J α )\J α ) = ∅.
We put ρ 0 α = α. Hence 1 ≤ ρ n α ≤ α for all n. As ρ n α is non-increasing in n, we can also define ρ α = min n ρ n α and n α = min{k : ρ k α = ρ α }.
Jensen [22,Theorem 3.2] proved that the projectum ρ n α is equal to the least δ ≤ α such that there exists a function f that is Σ n (J α )-definable over J α such that f (D) = J α for some D ⊆ ωδ, establishing the analogy with the Boolos-Putnam result. From this it follows that if ρ n α < α, it must be a cardinal in J α , for all n.
Jensen gave another characterization of the projectum, which in fact he used as his original definition in [22]. Suppose M, ∈ is a set-theoretic structure. We can extend this structure by adding an additional relation A ⊂ M . If we do this, we would like the structure to satisfy some basic set theoretic closure properties. For instance, we would like our universe to satisfy the comprehension axiom with respect to the new relation, that is, whenever we pick an x ∈ M , the collection of elements in x that satisfy A should be in M . Such structures are called amenable. Jensen [22,Theorem 3.2] showed that
ρ n α = the largest ordinal γ ≤ α such that J γ , A is amenable for any A ⊆ J γ that is in Σ n (J α ).
This means the projectum ρ n α identifies the "stable" core of J α with respect to Σ n definability over J α .
Being amenable with rud-closed domain can also be characterized via relative rud-closedness. This will be important later. The existence of a definable surjection between (a subset of) ωρ n α and Σ n (J α ) allows for coding Σ n (J α ) into its projectum. One way this can be implemented is via so-called master codes. Definition 4.10. A Σ n master code for J α is a set A ⊆ J ρ n α that is Σ n (J α ), such that for any m ≥ 1,
Σ n+m (J α ) ∩ P(J ρ n α ) = Σ m ( J ρ n α , A ).
A Σ n master code does two things:
(1) It "accelerates" definitions of new subsets of J ρ n α by n quantifiers. (2) It replaces parameters from J α in the definition of these new sets by parameters from J ρ n α (and the use of A as an "oracle").
The existence of master codes follows rather easily from the existence of a Σ n (J α )-mapping from ωρ n α onto J α . However, for n > 1, this mapping is not uniform. Jensen exhibited a uniform, canonical way to define master codes, by iterating Σ 1 -definability.
Put
A 0 α = ∅, p 0 α = ∅.
Assuming that A n α is a Σ n master code, it is not hard to see that every set x ∈ J ρ n α is Σ 1 -definable over J ρ n α , A n α with parameters from J ρ n+1 α and one parameter from J ρ n α (used to define a surjection from ωρ n+1 α onto J ρ n α ). Hence we can put
p n+1 α = the < J -least p ∈ J ρ n α such that every u ∈ J ρ n α is Σ 1 definable over J ρ n α , A n α with parameters from J ρ n+1 α ∪ {p} .
The p n α are called the standard parameters. Using p n+1 α , we can code the Σ 1 elementary diagram of the structure J ρ n α , A n α into a set A n+1 α :
A n+1 α := {(i, x) : i ∈ ω ∧ x ∈ J ρ n+1 α ∧ J ρ n α , A n α |= ϕ (2) i (x, p n+1 α )}, where (ϕ (k)
i ) is a standard Gödel numbering of all Σ 1 formulas with k free variables. It is not hard to verify that A n+1 α is a Σ n+1 master code for J α . Furthermore, the structure J ρ n α , A n α is amenable for each α > 1, n ≥ 0. We will call the structure J ρ n α , A n α the standard Σ n J-structure for J α .
Definition 4.11. We denote the standard J-structure over J α at the 'ultimate' projectum n α by J ρα , A α := J ρ nα α , A nα α .
One consequence of the A n α being master codes is that we can obtain the sequence of projecta of an ordinal by iterating taking Σ 1 -projecta relative to J ρ n α , A n α . Given an amenable structure J α , A , the Σ n -projectum ρ n α,A of J α , A is defined to be the largest ordinal ρ ≤ α such that J ρ , B is amenable for any B ⊆ J ρ that is in Σ n ( J α , A ).
Proposition 4.12 (Jensen [22]). For α > 1, n ≥ 0,
ρ n+1 α = ρ 1 ρ n α ,A n α .
In particular, the standard Σ n+1 J-structure for J α = J α , ∅ is the standard Σ 1 J-structure for J ρ n α , A n α .
ω-Copies of J-structures.
We later want to apply the recursion theoretic techniques of Section 2 to countable J-structures. We therefore have to code them as subsets of ω. If the projectum ρ n α is equal to 1, all set-theoretic information about the J-structure J ρ n α , A n α is contained in the master code A n α , which is simply a real, and hence lends itself directly to recursion theoretic analysis. Starting with the work by Boolos and Putnam [3], this has been studied in a number of papers (e.g. [24], [19]).
In this subsection we give a recursion theoretic analysis of the internal workings of a countable presentation of a J-structure.
Definition 4.13. Let X ⊆ ω. The relational structure induced by X is F X , E X , where xE X y ⇔ x, y ∈ X and F X = Field(E X ) = {x : ∃y (xE X y or yE X x)}.
The idea is that a number x represents a code for the set whose codes are the numbers y with yE X x,
Set X (x) = {y : yE X x}. The relational structure F X , E X is extensional if ∀x, y ∈ F X [(∀z zE X x ⇔ zE X y) ⇒ x = y],
that is ∀x, y ∈ F X (x = y ⇒ Set X (x) = Set X (y)).
Mostowski's Collapsing Theorem states that if F X , E X is extensional and well-founded, it is isomorphic to a unique structure (M, ∈), where M is a transitive set. In this sense we can speak of a countable set theoretic structure coded by X. If ϕ(v 1 , . . . , v n ) is a formula in the language of the set theory, we can interpret it over F X , E X and write X |= ϕ[a 1 , . . . , a n ] for F X , E X |= ϕ[a 1 , . . . , a n ] with a i ∈ F X . J-structures have an additional set A, and we capture this on the coding side via pairs X, M , where M ⊆ F X . Semantically, A and M are seen as interpreting a predicate added to the language. This way we can consider the satisfaction relation X, M |= ϕ, where ϕ is a set-theoretic formula with an additional unary predicate.
We are particularly interested in relational structures that code countable standard J-structures. The following is a generalization of the definition due to Boolos and Putnam [3] Definition 4.14. An ω-copy of a countable, extensional, set-theoretic structure S, A , A ⊆ S, is a pair X, M of subsets of ω such that X codes the structure F X , E X in the sense of Definition 4.13, and such that there exists a bijection π : S → F X such that
(4.2) ∀x, y ∈ S [ x ∈ y ⇐⇒ π(x)E X π(y) ], and (4.3) M = {π(x) : x ∈ A}.
The definition thus means an ω-copy X, M of S, A is isomorphic to S, A when seen as structures over the language of set theory.
If A = ∅, then necessarily M = ∅, and in this case we say X is an ω-copy of S.
We will now consider ω-copies of standard J-structures.
If ρ n α = 1, we have J ρ n α = L ω = V ω , i.e., the hereditarily finite sets. In this case, we obtain an ω-copy by fixing a bijection between ω and V ω , c.f. [1]. We let x, y ∈ X, i.e., x E X y, if and only if π ω (x) ∈ π ω (y) and x ∈ M if and only if π ω (x) ∈ A n+1 α . Then X, M is an ω-copy of J ρ n+1 α , A n+1 α via π −1 ω .
Definition 4.15. When ρ n α = 1, the canonical copy of J ρ n α , A n α is the ω-copy defined above.
We will now show that from a canonical copy of J ρ n+1 α , A n+1 α , we can extract ω-copies of all J ρ i α , A i α , i ≤ n, in an effective and uniform way. By choice of p n+1 α , for every u ∈ J ρ n α , there exists a Σ 1 -formula ψ(v 0 , v 1 , v 2 ) and x ∈ J ρ n+1 α such that u is the only solution over J ρ n α , A n α to ψ(v 0 , x, p n+1 α ).
Definition 4.16. A pair (i, x), i ∈ ω, x ∈ J ρ n+1 α
is an n-code if there exists a u ∈ J ρ n α such that u is the unique solution to
J ρ n α , A n α |= ϕ (3) i (v 0 , x, p n+1 α ).
(Recall that (ϕ (k) i ) is a standard Gödel numbering of all Σ 1 formulas with k free variables.)
We can check the property of being an n-code using Σ 1 formulas: (i, x) is an n-code if and only if (4.4) J ρ n α , A n α |= ∃v 0 ψ i (v 0 , x, p n+1 α ) and
(4.5) J ρ n α , A n α ∃v 0 , v 1 (ψ i (v 0 , x, p n+1 α ) ∧ ψ i (v 1 , x, p n+1 α ) ∧ v 0 = v 1 )
. This means a standard code has the information necessary to sort out n-codes among its elements. Relative to a canonical copy of J ρ n+1 α , A n+1 α , it is decidable whether some number is the image of an n-code, due to the effective way we translate between finite sets and their codes.
If X, M is an arbitrary ω-copy of J ρ n+1 α , A n+1 α via π, and (i, x) ∈ J ρ n+1 α is an n-code, then we call π((i, x)) a π-n-code. Being able to decide relative to X, M whether a number is a π-n-code of a pair hinges on knowledge of the following two functions (i) the mapping n X : n → π(n) (n ∈ ω), (ii) the mapping h X : (π(x), π(y)) → π((x, y)).
These mappings may not be computable relative to X, M , but they are definable, as follows.
If α > 1, then ω ∈ J α , and an ω-copy of any such J α must contain a witness for ω. In this case, we can recover n X recursively in X ′ . Lemma 4.17. If X is an ω-copy of J α , α > 1, then n X is computable in X ′ .
Proof. We approximate n X (i) from below. Let z = π(ω). At stage 0, put n X,0 (i) = 0 for all i. At stage s, we test whether sE X z. If yes, we can determine how s relates to the previous elements of z discovered, that is, we can compute the finite linear order of the elements of z seen so far, say n 0 E X . . . E X n k . We put n X,s (i) = n i for i ≤ k. The assertion now follows from the Limit Lemma.
This argument applies more generally as follows.
Lemma 4.18. If X, M is an ω-copy of J ρ n α , A n α , and ρ n α = 1, then X, M ′ uniformly computes the canonical copy of J ρ n α , A n α .
Proof. Suppose X, M is an ω-copy of J ρ n α , A n α via π. We need to show that the mapping π ω • π −1 is recursive in X, M ′ . The map n X,M is recursive in X, M ′ and therefore X, M ′ can compute the isomorphism between M and the image of A n α in the canonical copy.
We can also recover the function h X arithmetically in X.
Lemma 4.19. If X is an ω-copy of J α , then the function h X is computable in X (2) .
Proof. We have
h X (π(x), π(y)) = b ⇔ ∃c, d ∀z (zE X c ⇔ z = π(x)) ∧ ∀z (zE X d ⇔ z = π(x) ∨ z = π(y)) ∧ ∀z (zE X b ⇔ z = c ∨ z = d)
Definition 4.20. Suppose X, M is an ω-copy via π of a rud closed structure J, A . We say X, M is effective if the functions n X and h X are recursive in X ⊕ M .
Lemma 4.21. If ρ n α = 1, the canonical copy of J ρ n α , A n α is effective.
Proof. The mapping π −1 ω satisfies the conditions required in Definition 4.20 naturally. where u, v ∈ J ρ n+1 α , is recursive in X ⊕ M . The mapping π((i, u)) → (i, π(u)),
where i ∈ ω, is also recursive in X ⊕ M .
Proof. The first mapping can be computed by inverting h X (which must be one-one), the second mapping by additionally inverting n X . Lemma 4.23. If X, M is an effective copy of J ρ n+1 α , A n+1 α via π, then it is decidable in X ⊕ M whether a number y ∈ ω is a π-n-code.
Proof. Suppose y ∈ M (if not, it cannot be a π-n-code). Then y = π((i, x)) for some (i, x) ∈ A n+1 α . Since the copy is effective, we have π((i, x)) = h X (π(i), π(x)), and by Lemma 4.22 we can find i and π(x) recursively in X ⊕ M .
Recall that (ϕ (2) i ) is a standard Gödel numbering of the Σ 1 formulas with two free variables. There exist recursive functions g 1 , g 2 such that ϕ (2) g 1 (i) and ϕ (2) g 2 (i) are Σ 1 formulas equivalent (over J ρ n α , A n α ) to the formulas in (4.4) and (4.5), respectively. Then (i, x) is a n-code if and only if (g 1 (i), x) ∈ A n+1 α and (g 2 (i), x) ∈ A n+1 α . By the effectiveness of the ω-copy, the latter two conditions are equivalent to h X (π(g 1 (i)), π(x)) ∈ M and h X (π(g 2 (i)), π(x)) ∈ M, which is recursive in X ⊕ M .
Two n-codes (i 0 , x 0 ) and (i 1 , x 1 ) represent the same set u ∈ J ρ n α if u is the unique solution to J ρ n α , A n α |= ϕ
(3) i 0 (v 0 , x 0 , p n+1 α ) and J ρ n α , A n α |= ϕ (3) i 1 (v 0 , x 1 , p n+1 α )
. A similar property can be defined for π-n-codes.
J ρ n α , A n α |= ∃v 0 , v 1 , v 2 ϕ (3) i 0 (v 0 , x 0 , p n+1 α ) ∧ ϕ (3) i 1 (v 1 , x 1 , p n+1 α ) ∧ (v 2 ∈ v 0 ∧ v 2 ∈ v 1 ) ∨ (v 2 ∈ v 0 ∧ v 2 ∈ v 1 ) .
Let g 3 (i 0 , i 1 ) be a Gödel number for the Σ 1 formula
ψ(x, p n+1 α ) ≡ ∃v 0 , v 1 , v 2 ϕ (3) i 0 (v 0 , (x) 0 , p n+1 α ) ∧ ϕ (3) i 1 (v 1 , (x) 1 , p n+1 α ) ∧ (v 2 ∈ v 0 ∧ v 2 ∈ v 1 ) ∨ (v 2 ∈ v 0 ∧ v 2 ∈ v 1 ) .
Then (i 0 , x 0 ) and (i 1 , x 1 ) represent different sets if and only if
(g 3 (i 0 , i 1 ), (x 0 , x 1 )) ∈ A n+1 α ,
which in turn holds if and only if h X (π(g 3 (i 0 , i 1 )), h X (π(x 0 ), π(x 1 ))) ∈ M.
Since g 3 is computable, it follows from Lemma 4.22 that it is decidable in X ⊕ M whether two numbers are π-n-codes and whether they represent the same set.
Lemma 4.25. If X, M is an effective copy of J ρ n+1 α , A n+1 α via π, n X , h X , it computes an ω-copy Y, N of J ρ n α , A n α . Furthermore, the computation is uniform, and h Y and n Y can be computed uniformly from X ⊕ M ⊕ h X ⊕ n X .
Proof. By Lemmas 4.23 and 4.24, the set U = {y ∈ ω : ∃ u ∈ J ρ n α (y is the < ω -least π-n-code for u)} is recursive in X ⊕ M .
Let σ be the mapping σ : u ∈ J ρ n α → the unique π-n-code of u in U , and put
Y = { σ(x), σ(y) : x ∈ y ∈ J ρ n α }, N = {σ(x) : x ∈ A n α }.
Then Y, N is clearly an ω-copy of J ρ n α , A n α . To show that it is recursive in X ⊕ M , we note that for u, w ∈ J ρ n α , if (i, x) is an n-code for u and (j, y) is an n-code for w,
(4.6) u ∈ w ⇔ J ρ n α , A n α |= ∃v 0 , v 1 (ϕ i (v 0 , x, p n+1 α ) ∧ ϕ j (v 1 , y, p n+1 α ) ∧ v 0 ∈ v 1 ).
Moreover,
(4.7) u ∈ A n α ⇔ J ρ n α , A n α |= ∃v 0 (ϕ i (v 0 , x, p n+1 α ) ∧ v 0 ∈ A n α ).
There are recursive functions g 4 , g 5 that output Gödel numbers for Σ 1 formulas equivalent to the ones in (4.6) and (4.7), respectively. Given two numbers a, b ∈ U , we can use Lemma 4.22 to find (i, a 0 ) and (j, b 0 ) such that a = h X (π(i), π(a 0 )), b = h X (π(j), π(b 0 )). Then
aE Y b ⇔ h X (π(g 4 (i, j)), h X (a 0 , b 0 )) ∈ M.
Likewise, a ∈ N ⇔ h X (π(g 5 (i)), π(a 0 )) ∈ M.
To see that the functions h Y and n Y are uniformly recursive in X⊕M ⊕h x ⊕n X note that we can (i) given i ∈ ω, effectively compute the Gödel number of a Σ 1 formula that is satisfied by u if and only if u is the natural number i, (ii) given n-codes (i, x), (j, y) for elements u, w in J ρ n α , compute a Gödel number for a Σ 1 formula whose only solution is (u, w). By iterating the procedure described above, we obtain the following. If a copy is not effective, we can use Lemma 4.19 to decode the predecessor J-structures. (2) computes ω-copies of
J ρ n α , A n α , J ρ n−1 α , A n−1 α , . . . , and J ρ 0 α , A 0 α = J α , ∅ = J α .
Once we have an ω-copy of J α , we can use it to compute ω-copies of all J-structures "below" it. We introduce the following notation.
Definition 4.28. Given a structure F X , E X induced by X ⊆ ω and z ∈ ω, we define the segment F X , E X ⌈ z given by z, as
F X ⌈ z = {x ∈ F X : xE X z} and E X ⌈ z = E X ⌈ F X ⌈z . In particular, F X , E X ⌈ z = ∅ if z / ∈ F X .
If there is no danger of confusing it with the usual initial segment notation for reals, we will abbreviate F X , E X ⌈ z by X⌈ z . Lemma 4.29. If X is an ω-copy of J α , then X computes an ω-copy of J ρ n β , A n β , for all n ∈ ω, β < α.
Proof. Both J ρ n β and A n β are elements of J α . Let π be the isomorphism between J α and X, and let x β , a n β ∈ F X be such that
x β = π(J ρ n β ), a n β = π(A n β ).
Then X⌈ x β , Set X (a n β ) is an ω-copy of J ρ n β , A n β , clearly recursive in X.
A similar argument yields an analogous fact for the S-operator.
Lemma 4.30. If X is an ω-copy of J α , then X computes an ω-copy of S (n) (J β ), for all n ∈ ω, β < α.
4.4.
Defining ω-copies. In the previous section we saw how to effectively extract information from ω-copies of J-structures. Next, we describe how ωcopies of new J-structures can be defined from ω-copies of given J-structures.
The J-hierarchy has two types of operations that we need to capture: Defining new sets using the S-operator, and taking projecta and defining standard codes. We will analyze both operations from an arithmetic perspective.
An arithmetic analogue of the S-operator. The S-operator is defined by application of a finite number of explicit functions. This makes it possible to devise an arithmetic analogue, we which denote by S, and which is the subject of the following lemma.
Lemma 4.31. There exists an arithmetic function S(X) = Y such that, if X is an ω-copy of a transitive set U , S(X) is an ω-copy of the transitive closure of S(U ). Further, X is coded into a reserved column of ω, that is,
x, y ∈ X ⇔ 2 x , 2 y ∈ S(X), and 3 represents the element {F X } in S(X).
Proof. The elements of S(U ) are obtained by single applications of the functions F 0 , . . . , F 8 . Thus each element of S(U ) is the denotation of a term consisting of one of the functions and finitely many elements of U ∪ {U }.
Recursively in X, we can define an ω-copy of these terms. Membership of the set denoted by one term in the set denoted by another term or equality between the sets denoted by terms is arithmetic in X, since these are defined by quantification over X. The same applies for elements of the transitive closure of the thus coded structure. The additional uniformity condition on the coding of X does not change the calculation.
We can subject the S-operator to an analysis similar to that of the jump operator by Enderton and Putnam [10].
Lemma 4.32. (i) If A ∈ 2 ω is an arithmetic singleton, so is S(A). Furthermore, the arithmetic complexity of the formula for which S(A) is the unique solution is the maximum of the complexity of the formula for A and the complexity of the formula defining S.
(ii) There is an arithmetic predicate Q(n, X, Y ) such that
Q(n, X, Y ) ⇔ Y = S (n) (X).
Proof. To prove (i), note that if A is the unique solution to P (X), then S(A) is the unique solution to P (X [2] ) and X = S(X [2] ),
where X [2] = {a : 2 a ∈ X}. Claim (ii) follows similarly using the fact that for each k, there is a universal Π 0 k predicate.
Lemma 4.33.
If Z is such that Z ≥ T S (n) (X) for all n, then n S (n) (X)
is uniformly arithmetically definable from Z.
Proof. Define the predicate Q(n, e) as Q(n, e) :⇔ Φ Z e is total and Q(n, X, Φ Z e ).
To decide whether a ∈ S (n) (X), find, arithmetically in Z, the least e such that Q(n, e) and compute Φ Z e (a).
Corollary 4.34. If X is an ω-copy of J α and Z ≥ T S (n) (X) for all n, then Z uniformly arithmetically defines an ω-copy of J α+1 .
Proof. We can use n S (n) (X) to define a copy of J α+1 by 'stacking' the elements of S (n+1) (X) coded with base 3 and higher at the next 'available' prime column. Essentially this means that instead of moving S (n) (X) into the column given by powers of 2, we leave it unchanged and add new elements for S (n+1) (X) starting at the smallest available prime column.
An arithmetic version of the standard code. To define an arithmetic copy the Σ n -standard code for J α , we can simply interpret the set theoretic definitions as formulas of arithmetic. More precisely, suppose P is a definable predicate over a J-structure J ρ n α , A n α , and X, M is an ω-copy via π. Since the structure F X , E X , M is isomorphic to J ρ n α , ∈, A n α , we can use the same formula that defines P over J ρ n α , A n α and obtain a definition of π[P ] arithmetic in X, M . The problem, however, is that a bounded quantifier in set theory will not necessarily correspond to a bounded quantifier in arithmetic. This means the transfer of complexities between the Lévy-hierarchy and the arithmetical hierarchy may not result in uniform bounds.
However, we will use only a fixed, finite number of set-theoretic definitions. Most importantly, we use the uniform definability of the satisfaction relation |= over transitive, rud closed structures.
4.5.
Recognizing J-structures. Our goal is to show that there exists a recursive function G such that, for each n, no element of the sequence of canonical copies of J-structures with projectum equal to one in L βn can be G(n)-random with respect to a continuous measure. In the proof of this result (Theorem 4.50), we need to consider the initial segment of ω-copies computable in (some fixed jump of) µ.
The problem is that we cannot arithmetically define the set of ω-copies of structures J α . We can, however, define a set of "pseudocopies", subsets of ω that behave in most respects like ω-copies of actual J α , but that may code structures that are not well-founded.
By comparing the structures coded by these pseudocopies, we can also linearly order a subset of the latter (up to isomorphism), depending on whether a coded structure embeds into another. This ordering will be developed in Section 4.6.
Definition 4.37. A set X ⊆ ω is a pseudocopy if the following hold.
(1) The relation E X is non-empty and extensional.
(2) The structure F X , E X is rud-closed.
(3) The structure F X , E X satisfies ϕ V =J . (4) X's version of ω is isomorphic to ω; that is, X codes an ω-model.
To formalize these properties in arithmetic, we can take, as before, any formula in the language of set theory and interpret it over the structure F X , E X . This way we can define relations over F X intended to represent the corresponding set-theoretic relation. Extensionality can be formalized by a Π 0 2 (X) formula:
∀x, y(∀z(zE X x ↔ zE X y) → x = y).
Being rud-closed is an arithmetic property relative to X.
By Mostowski's Collapsing Theorem, if X satisfies (1) and E X is wellfounded, then F X , E X is isomorphic to a transitive set structure S, ∈ , and by (2) S will be rud-closed.
Property (3) is clearly arithmetic since it is defined by a single formula.
Finally, for (4), we can define ω using the usual Σ 0 set theoretic formula (the least infinite ordinal). In transitive, rud-closed sets, ϕ ω (x) holds if and only if x = ω. Interpreting ϕ ω over F X , E X , we obtain an arithmetic in X property. We require a pseudocopy F X , E X to satisfy ∃x ϕ ω (x). This x will be unique and define ω with respect to F X , E X . Let us denote this unique number by ω X .
Given ω X , we can also recover the mapping i → n X (i) as in Lemma 4.17. As the definition of ω X is uniform, we obtain that i → n X (i) is uniformly arithmetic in X. (4) holds exactly when this map from ω to ω X is a surjection, which is again uniformly arithmetic relative to X. Lemma 4.38. There exists an arithmetic formula ϕ PC (X) such that if ϕ PC (X) holds for a real X, then X is a pseudocopy. Moreover, if F X , E X is well-founded, then it is an ω-copy of a countable J β , β > 1.
4.6. Comparing pseudocopies. If two pseudocopies X and Y define wellfounded structures, they are ω-copies of sets J α and J β , respectively. Since α < β implies J α ∈ J β , it follows that one structure must embed into the other as an initial segment.
We want to find an arithmetic formula that compares two pseudocopies in this respect. The problem is that the isomorphism relation between countable structures need not be arithmetic. In our case, however, we can make use of the special set-theoretic structure present in the pseudocopies, by comparing the subsets of the cardinals present.
The complexity of the arithmetic operations involved in these comparisons will depend on the number of cardinals present in a pseudocopy.
Let us introduce the following notation. Recall that β N denotes the least ordinal such that L β N |= ZF − N . For any ordinal α, let (4.8) P α = max{n :
P (n) (ω) exists in J α },
if this maximum exists. We first note that for all α < β N , P α ≤ N . This is because, if P α were greater than or equal to n + 1 and β were the (n + 1)st cardinal in L α , then L β would satisfy ZF − N , hence α > β N . Hence P α is defined and uniformly bounded by N for all α < β N .
Using the predicate ϕ ω , we can formalize the (non-)existence of power sets of ω for pseudocopies. For any k ∈ ω, there exists a formula defining the predicate y = P (k) (ω).
Definition 4.39.
A pseudocopy X is an n-pseudocopy if it satisfies the uniformly arithmetic in X predicate ∃y(y = P (n) (ω)) ∧ ∀z(z = P (n+1) (ω)).
We now use the fact that pseudocopies are ω-models. Using the power sets of ω in each pseudocopy, we can check whether two pseudocopies have the same reals, sets of reals, etc.
First, we can check whether every real in X has an analogue in Y :
∀u (X |= u ⊆ ω → ∃v(Y |= v ⊆ ω ∧ ∀i(n X (i)E X u ↔ n Y (i)E Y v))).
By extensionality, such a v, if it exists, is unique. We can therefore define the mapping f X,Y 0 (u) = v which maps the representation of a real in F X , E X to its representation in F Y , E Y . We can similarly check whether every real in Y has an analogue in X. This gives rise to a function f Y,X 0 . Let ϕ (0) comp (X, Y ) be the arithmetic formula asserting that X and Y code the same subsets of ω.
We can continue this comparison through the iterates of the power set of ω. This will yield arithmetic formulas ϕ Given two n-pseudocopies, the above formulas allow for an arithmetic definition of isomorphic pseudocopies. Lemma 4.40. For given n and for any two n-pseudocopies X, Y that code well-founded structures F X , E X and F Y , E Y , respectively, if ϕ (n) comp (X, Y ), then X and Y code the same J α .
Proof. Assume for a contradiction X and Y are not isomorphic. Since they are well-founded pseudocopies, there must exist countable α, β such that F X , E X ∼ = (J α , ∈) and F Y , E Y ∼ = (J β , ∈). Without loss of generality, α < β. Since F X , E X and F Y , E Y code the same subsets of P (n) (ω) no new subset of P (n) (ω) is constructed between α and β. But this implies P (n+1) (ω) exists at α + 1, which is an immediate contradiction if β = α + 1. If β > α + 1, since P (n+1) (ω) does not exist in J β , a new subset of P (n) (ω) must be constructed between α + 1 and β, contradiction.
We will consider the comparison between ill-founded structures later. comp (X, Y ), then there exists an arithmetically in (X, Y ) definable function which maps the structure coded by X isomorphically onto the structure coded by Y .
Proof. Under the given hypothesis, there exists an α such that both X and Y are isomorphic to J α . By Proposition 4.4, there is a definable map from ωα onto J α . Moreover, there is a definable map from subsets of the greatest cardinal in J α onto ωα. Finally, there is a definable bijection from the greatest cardinal, which is some ω Jα k , to subsets of P (k) (ω). This gives us a definable isomorphism: We map an element in X to its least pre-image in ωα. This then gets mapped to the subset of P (k) (ω) to which it corresponds in X. Then we use our comparison formula between subsets of P (k) (ω) to map it to its counterpart in Y . Finally, in Y we map the subset of P (k) (ω) obtained this way to an ordinal and consequently to an element of Y .
We can use the transfer function of Corollary 4.41 to translate also between copies of S (n) (J α ).
Corollary 4.42. For every N , there exists a number d N , which can be computed uniformly from N , such that the following holds. Suppose X is an ω-copy of some J α with P α ≤ N . Suppose further that Z is an ω-copy of S (n) (J α ), for some n ∈ ω. Then S (n) (X) is recursive in (X ⊕ Z) (d N ) .
Proof. As Z computes an ω-copy of J α , Corollary 4.41 implies that some jump of Z ⊕ X computes the isomorphism between the two ω-copies of J α . Now apply Lemma 4.32.
For fixed N ∈ ω, let PC N = {X : X is an n-pseudocopy for some n ≤ N }.
This is an arithmetic set of reals. Restricted to PC N , the following relation, denoted by ∼ N is arithmetically definable:
X ∼ N Y : ⇔ X, Y are 0-pseudocopies and ϕ (0) comp (X, Y ) ∨ X, Y are 1-pseudocopies and ϕ (1) comp (X, Y ) ∨ . . . Working inside PC N , we can also use ϕ comp to arithmetically define a preorder ≺ on pseudocopies. The idea is that X ≺ Y if X embeds its structure into Y . For this purpose, we have to identify the "internal" J-hierarchy of a pseudocopy.
By Proposition 4.2, for any β, the sequence of J α (α < β) is uniformly Σ 1 -definable over J β . Let ϕ be the defining formula.
For z ∈ F X , we define Lemma 4.46. For every natural number N there is an arithmetic predicate such that for every real Z, the predicate defines a set of reals PC * N (Z) ⊆ PC N (Z) with the following properties:
(1) For every X ∈ PC * N (Z) and
x ∈ F X , if J X (x), then J X x ∈ PC * N (Z), (2) N is a total preorder on PC * N (Z), i.e., PC * N (Z)/ ∼ N is linearly ordered, (3) If X ∈ PC N (Z) is well-founded, then X ∈ PC * N (Z).
Proof. For property (1), suppose X ∈ PC N (Z) and z ∈ J X . We check that J X z ∈ PC N (Z). J X z is clearly recursive in Z. Since X codes an ω-model, properties (1)-(4) of Definition 4.37, which are satisfied by J X z within X, are true of J X z . For property (2), we investigate N -incomparability. Suppose X, Y are N -incomparable. We can use the ∼ N -relation to see if two J-segments of X and Y align: Consider the predicate
J X (x) ∧ J Y (y) ∧ J X x ∼ N J Y y .
It yields an arithmetic partial function from F X to F Y . It is single-valued by Lemma 4.43. Denote the domain of this function by D X and the range by R Y . Both D X and R Y are linearly ordered.
We consider the set of ordinals in a pseudocopy:
Ord X = {z : z is an ordinal in X}, Ord Y = {z : z is an ordinal in Y }.
Ord X , Ord Y are closed downward under E X , (E Y , respectively), and linearly ordered by E X (E Y ). If one structure is ill-founded, it must exhibit an instance of ill-foundedness among its ordinals, since an infinite descending ∈-chain in X would yield an infinite descending chain in the J X -hierarchy, which would correspond to an infinite descending chain in the ordinals in X. Let
Ord(D X ) = {Ord X ∩J X z : z ∈ D X }, Ord(R Y ) = {Ord Y ∩J Y z : z ∈ R Y }.
Both Ord(D X ) and Ord(R Y ) are initial segments of Ord X and Ord Y respectively, because the ordinals of any element of the J-hierarchy are an initial segment of the ordinals. Now we apply the incomparability of X and Y to show that there must be at least one instance of ill-foundedness in Ord(D X ) or Ord(R Y ).
Case 1: Ord(D X ) is cofinal in Ord X , Ord(R Y ) is cofinal in Ord Y .
This means, by the definition of the function for which D X and R Y are domain and range, respectively, the complete internal Jhierarchies of X and Y , respectively, are pairwise isomorphic. Furthermore, these isomorphisms are compatible by Lemma 4.43. Their union hence exhibits an isomorphism between the structure coded by X and the structure coded by Y , which would imply X ∼ N Y .
Case 2: Ord(D X ) is cofinal in Ord X , Ord(R Y ) is bounded in Ord Y .
Since X N J y for any y, Y must omit z∈R Y J z and hence is ill-founded. The case when Ord(D X ) is bounded and Ord(R Y ) is cofinal is analogous.
Case 3: Both Ord(D X ), Ord(R Y ) are bounded in Ord X , Ord Y , respectively.
In this case, Ord(D X ) and Ord(R Y ) are cuts in Ord X and Ord Y , respectively. If these cuts were principal in both structures, it would contradict the definition of D X and R Y by adding a new element to each set. In the limit case, reason as in Case 1: the union of the J X x , x ∈ D x , as evaluated in X, maps to the union of the J Y y , y ∈ R Y , as evaluated in Y . In the successor case, given an isomorphism between J X x and J Y y , because X and Y code ω-models, there also exists an isomorphism between S(J X x ), as evaluated in X, and S(J Y y ), as evaluated in Y .
Therefore, at least one of the two cuts is not principal, thereby exhibiting an instance of non-wellfoundedness.
We thus obtain the desired linearization of N . However, its definition involves quantification over all pseudocopies in PC N , and is therefore, if unrestricted, not arithmetic. We obtain the arithmetic set PC * N (Z) by considering all incomparable pairs in PC N (Z) and discarding all elements of PC N (Z) that are shown to be ill-founded by the above analysis. Since all pairs are being considered, PC * N / ∼ = N is linearly ordered by ≺ N . To see that this also ensures property (3) of the lemma, note that any element removed from PC N (Z) in this process is ill-founded.
Taking limits of ω-copies. We can use the ordering preceq N to construct limits of ω-copies. This will be needed in the proof of Theorem 4.50.
Lemma 4.47. For every N , there exists a number d N , which can be computed uniformly from N , such that the following holds. Suppose X = {X i : i ∈ ω} is a family of well-founded pseudocopies from PC N , in other words, each each X i codes a countable J α i in which there are at most N uncountable cardinals. Let γ be the supremum of the α i . Then there exists an ω-copy of J γ recursive in X (d) .
Proof. Using the N -predicate, we can arithmetically define a function f : ω → ω such that for all i, j,
f (i) ≤ f (j) ⇔ X i N X j .
This gives us a directed system of copies. We define a copy of J γ as a copy of the union of this directed system. Let Y 0 = X f (0) . Initialize by putting
U 0 = { 2 x+1 , 2 y+1 : x, y ∈ Y 0 }.
Suppose now we have defined U 0 ⊆ U 1 ⊆ · · · ⊆ U l with the property that
F U i ⊆ i k=0 {p m i : m ∈ ω},
where p i is the i-th prime number. If X f (l+1) ∼ N X f (l) , put U l+1 = U l . Otherwise, pick z such that X f (l+1) ⌈ z is isomorphic to U l , and let π l+1 be the isomorphism between X f (l+1) ⌈ z and U l . Given x, y such that x E X f (l+1) y, define E U l+1 as follows:
• If both x E X f (l+1) z and y E X f (l+1) z, add π l+1 (x), π l+1 (y) to U l+1 .
• If x E X f (l+1) z but not y E X f (l+1) z, add π l+1 (x), p y+1 l+1 to U l+1 . • If neither x E X f (l+1) z nor y E X f (l+1) z, add p x+1 l+1 , p y+1 l+1 to U l+1 . Putting U = l∈ω U l
yields an ω-copy of J γ arithmetic in X.
4.7.
Canonical copies are not random for continuous measures. We now want to use the framework of ω-copies to show that for any α < β N , the canonical copy of a standard J-structure J ρ k α , A k α cannot be K-random for a continuous measure, with K sufficiently large.
The argument rests mostly on various applications of the stair trainer technique introduced in Propositions 2.16 and 2.17, adapted to the notions of codings of countable J-structures developed in the previous sections. For convenience, we briefly review the core concepts.
ω-copy: A coding of a countable set-theoretic structure S, A , A ⊆ S, as a subset of ω; see Definitions 4.13 and 4.14. Canonical copy: The copy of a J-structure J ρ n α , A n α with projectum ρ n α = 1 by means of a canonical bijection V ω ↔ ω; see Definition 4.15. Effective copy: An ω-copy of a J-structure J ρ n α , A n α from which the internal fine structure hierarchy can effectively be recovered; see Definition 4.20 and Corollary 4.26. A canonical copy is always effective. Pseudocopy: An ω-copy of an ω-model of a rud-closed set satisfying ϕ V =J .
The set of pseudocopies is arithmetically definable. A pseudocopy may be ill-founded. If it is well-founded, it codes some countable level J α of the J-hierarchy; see Definition 4.37 and Lemma 4.38. By comparing their internal J-hierarchies, a subset of pseudocopies can be linearly ordered (up to isomorphism). This linear ordering is arithmetic, too; see Lemma 4.46.
We also fix some notation for the rest of this section. Given N ∈ ω, we fix c ∈ ω to be a sufficiently large number. It will be greater than the complexity of all arithmetic definitions (N -pseudocopies, comparison of pseudocopies and S-operators, linearization) introduced in the previous sections. In particular, Corollary 4.41 yields that, if X and Y are wellfounded ω-copies of a J α where α < β N , then the isomorphism between the two coded structures is recursive in (X ⊕ Y ) (c) . It will also be greater than the numbers d N from Corollary 4.42 and Lemma 4.47. After proving a couple of auxiliary results (Lemmas 4.48 and 4.49), we will also assume c to be greater than the constants appearing in these lemmas.
We give a first application of the stair trainer technique (as used in the proofs of Propositions 2.16 and 2.17) in the context of ω-copies and pseudocopies.
Lemma 4.48. There exist numbers d, e ∈ ω such that the following holds. Suppose µ is a continuous measure and X is an ω-copy of J α recursive in µ (m) , for some m ∈ ω. Suppose further that R is (m + e)-random with respect to µ and computes an ω-copy of J α+1 . Then there exists an ω-copy of J α+1 recursive in µ (m+d) .
Proof. By Lemma 4.30, R computes an ω-copy of S (n) (J α ), for all n ∈ ω. By Corollary 4.42, S (n) (X) is recursive in (R ⊕ µ) (m+d N ) . Lemma 4.31 on the other hand implies that S(X) is recursive in µ (m+a) , for some fixed a. We may assume that R is (m + d N + a + 1)-random for µ. By Proposition 2.12 and Lemma 2.14, S(X) is recursive in µ (m+d N ) . We can inductively use this line of reasoning and obtain that for each n ∈ ω, S (n) (X) is recursive in µ (m+d N ) . Now apply Corollary 4.34.
The lemma shows that with the help of a sufficiently random real that computes an ω-copy of the next level of the J-hierarchy, µ can reach a copy of this level arithmetically, too. Combined with Lemma 4.47, this will be the key ingredient in proving that canonical copies of standard codes cannot be random with respect to a continuous measure.
The next lemma establishes a similar fact for the standard J-structures J ρ n δ , A n δ over a given J δ .
Lemma 4.49. There exist numbers d, e ∈ ω such that the following holds. Suppose µ is a continuous measure and X is an ω-copy of J δ . Further suppose X is recursive in µ (m) . Finally, suppose that R is (m + e)-random with respect to µ and n is such that R computes an ω-copy of J ρ n δ , A n δ . Then there exists an ω-copy X n δ , M n δ of J ρ n δ , A n δ recursive in µ (m+d) .
Proof. The proof is similar to that of Lemma 4.48, inductively using Proposition 2.12, Lemma 2.14, Corollary 4.36, and Corollary 4.41.
From now on, we assume that c is also greater than the respective constants d, e from Lemmas 4.47, 4.48 and 4.49. We define G(N ) = (N +1)(3c+6).
Theorem 4.50. Suppose N ≥ 0, α < β N , and for some m > 0, ρ m α = 1. Then the canonical copy of the standard J-structure J ρ m α , A m α is not G(N )random with respect to any continuous measure.
Proof. We fix R and m R so that, when R is interpreted as a pair of reals, R is the canonical copy of some standard J-structure J ρ m R α , A m R α , where ρ m R α = 1. We assume for the sake of a contradiction that R is G(N )-random with respect to a continuous measure µ.
To obtain a contradiction similar to the proofs of Propositions 2.16 and 2.17, we inductively define a hierarchy of indices of (pseudo)copies arithmetic in µ. Definition 4.51. For each k with 0 ≤ k ≤ N , we let S k = {e ∈ ω : Φ µ (k(3c+6)) e is total and Φ µ (k(3c+6)) e ∈ PC N (µ (k(3c+6)) ) and for all d < e, Φ µ (k(3c+6)) d = Φ µ (k(3c+6)) e }.
The relation ≺ N induces an ordering on the indices in each S k , which will be denoted by ≺ N , too. The linearly ordered subsets corresponding to PC * N are given as L k = {e ∈ S k : Φ µ (k(3c+6)) e ∈ PC * N (µ (k(3c+6)) )}. Finally, we let I k = {e ∈ S k : Φ µ (k(3c+6)) e is well-founded}.
By Lemma 4.46, the sets S k and L k are arithmetic in µ (k(3c+6)) , for all k ≤ N . In particular, by choice of c, L k is recursive in µ (k(3c+6)+c) .
The following lemma shows that I k is the longest well-founded initial segment of L k . Lemma 4.52. Given k ≤ N , let I be a well-founded initial segment of L k . Then, for every e ∈ I, Φ µ (k(3c+6)) e is a well-founded pseudocopy.
Proof of Lemma 4.52. Suppose Φ µ (k(3c+6)) e , e ∈ I, is an ill-founded pseudocopy. Then it has an ill-founded sequence of ordinals, and hence also an ill-founded internal J-sequence. Since I is an initial segment, the entire internal J-structure must be present in I (via corresponding µ (k(3c+6)) -indices); see Lemma 4.46. This contradicts the fact that I is well-founded.
We will need an additional properties of I k .
Lemma 4.53. If J β is represented in I k , then β < α.
Proof of Lemma 4.53. Suppose J α is represented in I k . By Lemma 4.49, there exists an ω-copy X, M of J ρ m R α , A m R α recursive in µ (k(3c+6)+c) . (Recall m R is such that ρ m R α = 1 and R = J 1 , A m R α .) Comparing the canonical encoding of J 1 with X, we obtain that R is recursive in µ (k(3c+6)+2c) , contradicting the randomness of R.
We continue the proof of Theorem 4.50 and apply Lemma 2.15 to I k . L k is recursive in µ (k(3c+6)+c) . Since R is a canonical copy, we can use it to test for any pseudocopy M with an index in L k whether M embeds into J α . This can be done recursively in (µ (k(3c+6)) ⊕ R) (c) by Lemma 4.46 and the choice of c. By Lemma 4.52, every pseudocopy with an index in I k will embed into J α . Consequently, I k is recursive in (µ ⊕ R) (k(3c+6)+c) . By choice of G, R is at least (k(3c + 6) + c + 5)-random for µ, so Lemma 2.15 implies that I k is recursive in µ (k(3c+6)+c+4) .
Next we define by recursion a sequence of ordinals γ 0 , . . . , γ K , where K is at most N + 1.
• Let γ 0 = ω and ξ 0 = 1.
• Given γ k , we check whether γ k is a cardinal in each of the structures represented in I k .
-If so, we let γ k+1 = sup{β : ∃e ∈ I k (β has cardinality at most γ k in the structure represented by e)}, ξ k+1 = sup{β : J β is represented in I k }, and we continue the recursion. -Otherwise, there exists a j ≤ k such that γ j is not a cardinal inside some structure J δ represented in I k . Since the recursion made it to step k, δ is greater than any β such that J β has a representation in I k−1 . We terminate the recursion and let K = k. • If we reach γ N +1 , we terminate the recursion. Proof of Lemma 4.54. We first prove that J 2 is represented in I 0 . The canonical copy of J 1 is recursive. By Lemma 4.48, there exists an ω-copy of J 2 recursive in µ (c) . The canonical copy of J 2 , J 1 , A 2 is recursive in R. By Lemma 4.49, there exists an ω-copy of J 1 , A 2 recursive in µ (2c) . The isomorphism between this copy and the canonical copy of J 1 is recursive in µ (3c) . Therefore, by Lemma 2.13, the canonical copy of J 1 , A 2 is recursive in µ. By Corollary 4.26, there exists an ω-copy of J 2 recursive in µ. Now assume k > 0. J ξ k has a representation recursive in µ ((k−1)(3c+6)+c) . If ξ k is the maximum of the β for which J β is represented in I k−1 , there is an ω-copy of J ξ k recursive in µ ((k−1)(3c+6)) . Otherwise, Lemma 4.47 implies that there exists an ω-copy of J ξ k recursive in µ ((k−1)(3c+6)+c) . We can apply Lemma 4.48 to obtain an ω-copy of J ξ k +1 recursive in µ ((k−1)(3c+6)+2c) , which implies that it is represented in I k .
The lemma implies that for each i < K, γ i appears as a cardinal in some structure represented in I i .
in turn yields a an ω-copy of J γ N+1 recursive in µ (i(3c+6)) . This contradicts that J γ N+1 is not represented in any I i , for i ≤ N . This is sufficient to complete the proof of Theorem 4.50. 4.8. Finishing the proof of Theorem 2. We restate Theorem 2. Let G(n) be the recursive function defined before the statement of Theorem 4.50 in Section 4.7.
Theorem 2. For every n ∈ ω, ZFC − n "NCR G(n) is countable."
Proof. For any n ≥ 0, the set X of canonical copies of standard J-structures J ρ k α , A k α with ρ k α = 1 is not countable in L βn . For suppose f : ω ։ X were a counting of X such that f ∈ L βn . We may assume f is given as a real. By the closure properties of L βn , f ′ ∈ L βn . Let γ < β n be the least ordinal such that f ∈ J γ+1 \ J γ , and let m be such that f ′ is Σ m (J γ ). It follows that ρ m γ = 1. f ′ is computable in the canonical copy X m α , M m α of J ρ m γ , A m γ . It follows that X, M is not in the range of f . Since by Theorem 4.50, the set of canonical copies of standard J-structures is a subset of NCR G(n) . Therefore, NCR G(n) is not countable in L βn .
Definition 3 . 10 .
310Let G be a subset of P n .
Definition 4. 7 .
7Given A ⊆ M , the structure M, A is called amenable, if M is an amenable set and ∀x ∈ M [x ∩ A ∈ M ].
Definition 4 . 8 (
48Jensen[22]). A function f is A-rud if it can be obtained as a combination of the basis functions F 1 , . . . , F 8 and the functionF A (x, y) = x ∩ A. A structure M, A is rud closed if f [M n ] ⊆ M for all A-rud functions f .
Proposition 4 . 9 (
49Jensen [22]). A structure M, A , A ⊆ M , is rud closed if and only if M is rud closed and M, A is amenable.
Lemma 4 . 22 .
422If X, M is an effective copy of J
Lemma 4 . 24 .
424If X, M is an effective copy of J ρ n+1 α , A n+1 α via π, then it is decidable in X ⊕ M whether two numbers are π-n-codes of the same set. Proof. (i 0 , x 0 ) and (i 1 , x 1 ) represent different sets if and only if
Corollary 4 . 26 .
426If X, M is an effective copy of J . . , and J ρ 0 α , A 0 α = J α , ∅ = J α .
Proposition 4. 35 (
35Jensen [22], Corollary 1.13). For n ≥ 1, the satisfaction relation |= Σn M,A is uniformly Σ n ( M, A ) for transitive, rud closed structures M, A ).
Corollary 4 . 36 .
436Suppose X, M is an ω-copy of J ρ n α , A n α . Then there exists an ω-copy of J ρ n+1 α , A n+1 α uniformly arithmetically definable in X, M .
X, Y ) with the following property:If X and Y are pseudocopies in which P (n) (ω) exists, then ϕ(n) comp (X, Y ) holds if and only if X and Y have (representations of) the same subsets of P (i) (ω), for all 0 ≤ i ≤ n.
Corollary 4 . 41 .
441For given n, if X and Y are well-founded n-pseudocopies and ϕ (n)
X, Y are N -pseudocopies and ϕ (N ) comp (X, Y )
Definition 4 . 45 .
445Given a real Z, let PC N (Z) be the set of pseudocopies computable in Z.
Lemma 4 . 54 .
454For every 0 ≤ k ≤ K, J ξ k +1 is represented in I k .
. Randomness for Continuous Measures
−|σ|+2 ≤ 2 −n .
Acknowledgments. We would like to thank Sherwood Hachtman, Carl Jockusch Jr., Alexander Kechris, Donald Martin, and W. Hugh Woodin for many helpful discussions and suggestions. We would also like to thank the anonymous referees for their very careful reading of the manuscript and for their much-needed suggestions on how to improve the paper.the J-structure inside X (the internal J-structure of X) and, given z ∈ J X , write J X z for X⌈ z . Lemma 4.43. Suppose X ∈ PC N and z ∈ J X . Then J X z has no non-trivial automorphism. Further, if J X z 1 is isomorphic to J X z 2 , then z 1 = z 2 . Proof. Let π be an automorphism of J X z . Since X codes an ω-model, J X z is an ω-model. π fixes the ω of J X z and also fixes every natural number of J X z . By induction π must fix every set that is obtained by a finite number of power set operations to ω. There is an internally definable injection from J X z to sets obtained by a finite number of power set operations to ω. Hence π must fix all of J X z . (See proof of Corollary 4.41.) The second claim follows by the same reasoning.If X is well-founded, J X must be linearly ordered by E X . So if it is not (an arithmetic property relative to X), we can exclude X as ill-founded right away. From now on suppose J X is always linearly ordered.). This is an arithmetic property of the pair (X, Y ). We let X N Y if X ≺ N Y or X ∼ N Y . N is reflexive and transitive. Hence N defines a partial order on PC N .If both X and Y are well-founded pseudocopies in PC N , we have either X ≺ N Y or X ∼ N Y or Y ≺ N X, that is, "true" pseudocopies (i.e., those that code a J α ) are linearly ordered by N (up to isomorphism). Hence comparability can only fail if (at least) one of the pseudocopies is not wellfounded.Provided with a countable subset of PC N , such as all the elements of PC N recursive in a real Z, we will want to arithmetically define a subset that is linearly ordered by N by excluding some ill-founded pseudocopies.
By Lemma 4.49, there exists an ω-copy of J 1 , A δ recursive in µ (K(3c+6)+c+4+c) = µ (K(3c+6)+2c+4) . By Corollary 4.27, the canonical copy X, M of J 1 , A δ is recursive in µ (K(3c+6)+2c+4+2) = µ (K(3c+6)+2c+6). Case 1a: ρ δ = 1. The canonical copy of J 1 is recursive. Since α > γ K. R computes X, M , by Lemma 4.29. By Lemma 2.13, µ computes X, M . Since X, M is an effective copy, Corollary 4.26 implies µ computes an ω-copy of J δ . But this means J δ is represented in I 0 , which contradicts the definition of γ 1 . Case 1b: ρ δ > 1. Note that by Lemma 4.47, there exists an ω-copy Y ofCase 1a: ρ δ = 1. The canonical copy of J 1 is recursive. By Lemma 4.49, there exists an ω-copy of J 1 , A δ recursive in µ (K(3c+6)+c+4+c) = µ (K(3c+6)+2c+4) . By Corollary 4.27, the canonical copy X, M of J 1 , A δ is recursive in µ (K(3c+6)+2c+4+2) = µ (K(3c+6)+2c+6) . Since α > γ K , R computes X, M , by Lemma 4.29. By Lemma 2.13, µ computes X, M . Since X, M is an effective copy, Corollary 4.26 implies µ computes an ω-copy of J δ . But this means J δ is represented in I 0 , which contradicts the definition of γ 1 . Case 1b: ρ δ > 1. Note that by Lemma 4.47, there exists an ω-copy Y of
Since J δ is represented in I K , there exists an ω-copy of J δ recursive in µ (K(3c+6)) . By Lemma 4.49, there exists an ω-copy X, M of J ρ δ , A δ recursive in µ (K(3c+6)+c) . By comparing the coding of X and Y (using at most c jumps), µ (K(3c+6)+c+c) = µ (K(3c+6)+2c) can compute the transfer of M (the coding of A δ in X) to Y . This gives us an ω-copy Y, L of J ρ δ. A δ recursive in µ (K(3c+6)+2c)Since J δ is represented in I K , there exists an ω-copy of J δ recursive in µ (K(3c+6)) . By Lemma 4.49, there exists an ω-copy X, M of J ρ δ , A δ recursive in µ (K(3c+6)+c) . By comparing the coding of X and Y (using at most c jumps), µ (K(3c+6)+c+c) = µ (K(3c+6)+2c) can compute the transfer of M (the coding of A δ in X) to Y . This gives us an ω-copy Y, L of J ρ δ , A δ recursive in µ (K(3c+6)+2c) .
Using at most c jumps, the join of R and µ ((i−1)(3c+6)+2c+4) can compare Y and Y R and map L R , the encoding of A δ in Y R , to Y . This way we obtain an ω-copy Y, L Y of J ρ δ , A δ recursive in (R ⊕ µ ((i−1)(3c+6)+2c+4) ) (c) ≡ T R ⊕ µ ((i−1)(3c+6)+3c+4) . By Lemma 2.14, Y, L Y is recursive in µ ((i−1)(3c+6)+3c+4) . By Corollary 4.27, µ ((i−1)(3c+6)+3c+4+2) = µ (i(3c+6)) computes an ω-copy of J δ . But this implies J δ is represented in S i . In particular. Y R Since Α > Γ K , R Computes ; J Ρ Δ, L , J δ is. by Lemma 4.29. represented in I i . This contradicts the fact that δ is greater than any β such that J β has a representation in I K−1 . Case 2: K = N + 1Since α > γ K , R computes, by Lemma 4.29, another ω-copy of J ρ δ , A δ , say Y R , L R . Using at most c jumps, the join of R and µ ((i−1)(3c+6)+2c+4) can compare Y and Y R and map L R , the encoding of A δ in Y R , to Y . This way we obtain an ω-copy Y, L Y of J ρ δ , A δ recursive in (R ⊕ µ ((i−1)(3c+6)+2c+4) ) (c) ≡ T R ⊕ µ ((i−1)(3c+6)+3c+4) . By Lemma 2.14, Y, L Y is recursive in µ ((i−1)(3c+6)+3c+4) . By Corollary 4.27, µ ((i−1)(3c+6)+3c+4+2) = µ (i(3c+6)) computes an ω-copy of J δ . But this implies J δ is represented in S i . In particular, J δ is represented in I i . This contra- dicts the fact that δ is greater than any β such that J β has a representation in I K−1 . Case 2: K = N + 1.
Since R cannot be represented in any S N , γ N +1 < β N . Hence J γ N+1 is projectible, say to γ i . There is an ω-copy of J γ N+1 recursive in µ (N (3c+6)+2c+4) . By the same argument as above, there is a copy of J γ i. The analysis is similar to Case 1. γ N +1 is defined. A γ N+1 recursive in µ ((i−1)(3c+6)+3c+4). which ReferencesThe analysis is similar to Case 1. γ N +1 is defined. Since R cannot be rep- resented in any S N , γ N +1 < β N . Hence J γ N+1 is projectible, say to γ i . There is an ω-copy of J γ N+1 recursive in µ (N (3c+6)+2c+4) . By the same argument as above, there is a copy of J γ i , A γ N+1 recursive in µ ((i−1)(3c+6)+3c+4) , which References
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[] |
[
"Finite temperature quantum embedding theories for correlated systems",
"Finite temperature quantum embedding theories for correlated systems"
] |
[
"Dominika Zgid \nDepartment of Chemistry\nUniversity of Michigan\n48109Ann ArborMichiganUSA\n",
"Emanuel Gull \nDepartment of Physics\nUniversity of Michigan\n48109Ann ArborMichiganUSA\n"
] |
[
"Department of Chemistry\nUniversity of Michigan\n48109Ann ArborMichiganUSA",
"Department of Physics\nUniversity of Michigan\n48109Ann ArborMichiganUSA"
] |
[] |
The cost of the exact solution of the many-electron problem is believed to be exponential in the number of degrees of freedom, necessitating approximations that are controlled and accurate but numerically tractable. In this paper, we show that one of these approximations, the self-energy embedding theory (SEET), is derivable from a universal functional and therefore implicitly satisfies conservation laws and thermodynamic consistency. We also show how other approximations, such as the dynamical mean field theory (DMFT) and its combinations with many-body perturbation theory, can be understood as a special case of SEET and discuss how the additional freedom present in SEET can be used to obtain systematic convergence of results. * [email protected]; Corresponding author 1 P. A. M. Dirac, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and
|
10.1088/1367-2630/aa5d34
|
[
"https://arxiv.org/pdf/1611.00395v2.pdf"
] | 118,649,078 |
1611.00395
|
37fdd1e85d44c42cc2fab5f6c7e182370e0c6bea
|
Finite temperature quantum embedding theories for correlated systems
Dominika Zgid
Department of Chemistry
University of Michigan
48109Ann ArborMichiganUSA
Emanuel Gull
Department of Physics
University of Michigan
48109Ann ArborMichiganUSA
Finite temperature quantum embedding theories for correlated systems
The cost of the exact solution of the many-electron problem is believed to be exponential in the number of degrees of freedom, necessitating approximations that are controlled and accurate but numerically tractable. In this paper, we show that one of these approximations, the self-energy embedding theory (SEET), is derivable from a universal functional and therefore implicitly satisfies conservation laws and thermodynamic consistency. We also show how other approximations, such as the dynamical mean field theory (DMFT) and its combinations with many-body perturbation theory, can be understood as a special case of SEET and discuss how the additional freedom present in SEET can be used to obtain systematic convergence of results. * [email protected]; Corresponding author 1 P. A. M. Dirac, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and
I. INTRODUCTION
The computational cost of the exact solution of the realistic extended many-electron problem is believed to be exponential in the number of degrees of freedom, necessitating the development of accurate approximate methods able to capture interacting electron physics. 1 While mature tools for obtaining ground state energetics for both molecular and solid state problems exist, 2,3 solid state experiments are often performed at finite temperature and yield as the measured result not energy differences but single-and two-particle response functions, requiring a description of finite temperature excitations.
Many-body perturbation theory 3 accurately describes these phenomena where interactions are weak. However, many systems of interest are believed to be outside the regime of validity of perturbative approximations. In these systems, a non-perturbative solution is desired for a subset of the correlated degrees of freedom embedded into a background of more weakly correlated, perturbatively treated states. Ideally such an embedding construct should be numerically tractable and defined in terms of one or more small parameters that allow its tuning from a crude but computationally cheap, approximate solution to the exact but exponentially expensive one.
Several such theories have been developed. They include the dynamical mean field theory (DMFT), 4,5 its combination with electronic structure methods, such as LDA+DMFT [6][7][8] and GW+DMFT 9,10 , the self-energy functional theory, 11 and most recently the self-energy embedding theory (SEET). [12][13][14] All of them require a compromise between accuracy and numerical tractability or time to solution.
In this paper, we show that SEET can be understood as a conserving functional approximation to an exact Luttinger-Ward functional. 15 This functional framework of SEET allows us to compare this theory to other functional approximations, and show in particular that DMFT, HF+DMFT, and GW+DMFT can be understood as a special case of SEET and to illustrate how the additional freedom given by SEET can be employed to systematically improve results. In particular, we focus on various aspects of electron 'screening' and downfolding and how they are treated in various approximations.
This paper proceeds as follows. In Sec. II, we introduce the system under study, the SEET definition, DMFT, and several combinations of DMFT with many-body perturbation theory. In Sec. III, we compare the different approaches based on their functionals. In Sec. IV, we focus in detail on various aspects of electron screening. We form conclusions in Sec. V.
II. SYSTEM AND FORMALISM
We consider a system described by a Hamiltonian with full two-body interaction v ijkl and one-body terms t ij in a finite orbital basis:
H = N ij t ij a † i a j + N ijkl v ijkl a † i a † j a l a k ,(1)
where the indices i, j, k, and l enumerate all N basis orbitals present in the system. In case of a periodic system, Eq. 1 may in particular contain one-body terms connecting any orbital in any unit cell to any other orbital in any other unit cell, and general two-body integrals v mixing interactions between any of the orbitals in any of the unit cells in the system. Physical properties including thermodynamic quantities (energies and entropies), frequency dependent singleparticle (Green's functions and self-energies) and twoparticle quantities (susceptibilities) can be described in a functional approach. [15][16][17][18] In this approach, a Φ-functional Φ[G] of the Green's function G, which contains all linked closed skeleton diagrams, 15 is used to express the grand potential as
Ω = Φ − Tr log G −1 − TrΣG,(2)
and it satisfies δΦ δG
= Σ[G],(3)
where the self-energy Σ is defined with respect to a noninteracting Green's function G 0 via the Dyson equation
G = G 0 + G 0 ΣG.(4)
arXiv:1611.00395v2 [cond-mat.str-el] 1 Mar 2017
The functional formalism is useful because approximations to Φ that can be formulated as a subset of the terms of the exact Φ functional can be shown to respect the conservation laws of electron number, energy, momentum, and angular momentum by construction. 16,19 In addition, Φ-derivability ensures that quantities obtained by thermodynamic or coupling constant integration from non-interacting limits are consistent. 16 Functional theory therefore provides a convenient framework for constructing perturbative 16,[20][21][22] and non-perturbative 4,11-14 diagrammatic approximations.
On the other hand, approximations based on a Φ functional do not guarantee self-consistency on the twoparticle level, so that vertex functions which appear in the calculation of the one-particle self-energy may not the same as those generated by functional differentiation in two-particle correlation functions, and crossing symmetries may be violated. [23][24][25] The construction of methods for model systems that respect these symmetries by construction is an active topic of research. 26,27 The approximations we discuss in the following sections are all expressed in the functional form, thus making them straightforward to discuss and compare their respective assumptions, limits, and strengths.
A. The Self-energy Embedding Theory
Self-energy Embedding Equations
The self-energy embedding theory (SEET) 12-14 starts from the assumption that all orbitals present in the system can be separated into M distinct orbital subsets A i , each containing N A i orbitals, and a remainder R with N R orbitals, such that N A i N , for each i, and
N = M i=1 N A i + N R .
We assume that the orbitals within each subset A i are more strongly correlated among each other than with other orbitals present in the system, so that their intra-subset correlations need to be obtained in a nonperturbative way. Conversely, inter-set correlations between orbitals belonging to two different sets A i and A j , i = j, and correlations belonging to the remainder R are assumed to be weaker, such that they can be simulated perturbatively. The choice of orbital subsets and subset size N A i is general and will be commented on in Sec. II A 2.
SEET first approximates the solution of the entire system using an affordable but potentially inaccurate Φderivable method (weak coupling methods are a natural choice), and then corrects this approximation in the strongly correlated subspaces by a non-perturbative result. This is achieved by approximating the exact Φfunctional as
Φ SEET = Φ tot weak + M i=1 [Φ A strong ] i − [Φ A weak ] i .(5)
Here, Φ tot weak denotes a solution of the entire system using a conserving low-order approximation, for instance selfconsistent second order perturbation theory (GF2) [28][29][30][31][32][33] or the GW method. 20 Φ A denotes all those terms in Φ where all four indices i, j, k, l of v ijkl are contained inside orbital subspace A. Φ A weak is the approximation to Φ A within the weak coupling method used for solving the entire system, and Φ A strong the approximation or exact solution of Φ A obtained using the higher order method capable of describing 'strong correlation'.
Since the self-energy is a functional derivative of the Φ SEET -functional, the total self-energy Σ contains diagrams from both the 'strong' and 'weak' coupling methods and can be written in a matrix form reflecting the system separation onto different correlated blocks
Σ SEET = [Σ A ] 1 Σ int . . . . . . . . . Σ int [Σ A ] 2 Σ
.
. . Σ int [Σ A ] M Σ int . . . . . . . . . Σ int Σ R (6)
These blocks are obtained upon differentiation of the Φ SEET functional according to Eq. 3 and have the following form
[Σ A ] i = Σ tot weak + ([Σ A strong ] i − [Σ A weak ] i ),(7)Σ R = Σ R weak ,(8)Σ int = Σ int weak .(9)
Eq. 6 describes a subspace self-energy consisting of a contribution from the strongly correlated subspace embedded into a weakly correlated self-energy generated by all orbitals outside the subspace. This embedding of the selfenergy leads to the name 'self-energy embedding theory'. SEET satisfies the following limits:
• If the interaction v ijkl is zero or the temperature is infinity, β = 0, the self-energy is zero and therefore the method becomes exact.
• If M = 1 and the only subspace A includes all orbitals present in the system, N A = N , so that no orbitals are left in the perturbatively treated subspace, N R = 0, then the entire system is solved using the strong correlation method and Φ SEET = Φ A strong . Consequently, if the strong correlation method provides the exact solution, the exact solution of Eq. 1 is recovered.
• In the limit of non-interacting subsystems, when the interactions between strongly correlated subspaces are zero, together with a condition N R = 0 and i N A i = N , SEET recovers the solution of the system with the strong correlation method since
Φ SEET = M i [Φ A strong ] i .
• If the correlated subspaces are not treated exactly but using the same 'weak correlation' method as the rest of the system, the weak correlation solution for the full system is recovered since Φ tot
weak = Φ tot weak + M i=1 [Φ A weak ] i − [Φ A weak ] i .
While consideration of the exact limits is essential, the important practical question is whether (and where) one can expect SEET to be accurate away from these exact limits. As is evident in Eq. 5, SEET becomes accurate where the diagrams considered at the lower level method require no higher order corrections. This is the case in the high temperature, high energy, and high doping regimes where the self-energy is perturbative. Additionally, SEET is accurate if all non-perturbative correlations are restricted to the correlated subspaces, and its accuracy will therefore strongly depend on the choice of the correlated subspaces. Consequently, choosing the correlated subspaces is an important step in any SEET calculation. In many techniques, the 'strongly correlated' and 'weakly correlated' orbital subsets are chosen a priori. An example is DMFT, where Φ is truncated to local degrees of freedom, 5 or LDA+DMFT methods 5,6,34 where certain 'local' orbitals (usually orbitals with d or f -like character) are considered to be 'correlated', while 'wider' p and s orbitals are considered to be non-interacting.
The choice of SEET subspaces
While the same ad hoc orbital choice can be used for self-energy embedding theory, SEET also offers a different approach to the selection of correlated orbitals and in particular makes an adaptive choice of correlated orbitals 'a posteriori' possible, without the need to localize or 'downfold' orbitals.
A simple criterion for identifying the degree of orbital correlation is given by the frequency-dependence of the self-energy: the larger the frequency dependent part, the more 'non-Hartree-Fock' like an orbital is, and therefore the more it needs to be treated at the 'strongly correlated' level. Since Hartree-Fock only yields orbital occupancies of 0 and 2 (at zero T ), then any partial occupancy of an orbital obtained from diagonalizing a one-body density matrix obtained using a perturbative approach (used in the first step of SEET) indicates some degree of correlation. The larger the deviation from 0 or 2 the more "strongly correlated" an orbital is and the more likely it requires a non-perturbative treatment.
Consequently, the SEET calculations in Ref. 12-14 added orbitals to the strongly correlated subspace A using a criterion based on diagonalization of the one-body density matrix: chosen were those N A orbitals with the largest deviation of the occupancy from 0 and 2. This requires a basis transform of the hybridization function, non-interacting Hamiltonian, and two-body integrals into the basis that diagonalizes the one-body density matrix. While basis transforms for the two-body integrals are generally expensive, the transformed integrals are only necessary inside the correlated subspace, making the transform affordable in practice, such that the orbital transformation step is not a computational bottleneck.
Two possible subset selection schemes are illustrated graphically in Fig. 1. The upper panel shows a separation of orbitals based on energy or occupation scheme, where mostly unoccupied and mostly filled orbitals are treated as weakly correlated subspaces that can be treated by a weak correlation method. Partially filled orbitals are chosen as strongly correlated that will be treated by a nonperturbative method. The lower panel shows an alternative separation based on distance, where orbitals localized around a center position are treated as strongly correlated, whereas orbitals at farther distance are treated as uncorrelated or weakly correlated.
Self-consistent solution of the SEET equations
The Φ SEET functional of Eq. 5 defines the SEET approximation. 35 It requires the specification of the M correlated orbital subspaces A i and the subspace R, in addition to the 'strong coupling' and the perturbative weak coupling diagrams. We now describe an algorithm that generates a self-consistent solution of the SEET equations.
First, the weak coupling method is used to selfconsistently obtain the self-energy Σ tot weak and functional Φ tot weak of the entire system from a given initial Green's function, e.g. the Hartree-Fock (HF) or density functional theory (DFT) approximation. The self-consistency of the weakly correlated method eliminates all mem-ory of the initial starting point in its convergence to a fixed point.Upon convergence of the weakly correlated method, we choose the correlated subspaces according to Sec. II A 2. We then compute [Σ A weak ] i and [Φ A weak ] i in every orbital subspace i, i.e. the weak correlation approximation obtained with vertex indices exclusively contained in the correlated orbital subsets A i .
In a next step, [Σ A strong ] i needs to be obtained in each subspace i. To simplify notation, we select one particular subspace A i = A and absorb all other subspaces A j , j = i, and the remaining weakly correlated orbitals in space R. Using the non-interacting Green's function 36 in a block form
G 0 = ω − t A −t int −t † int ω − t R −1(10)
and the Dyson equation G = G 0 + G 0 ΣG, we express the interacting Green's function as
G tot = (G −1 0 ) A − Σ A (G −1 0 ) int − Σ int (G −1 0 ) int − Σ int † (G −1 0 ) R − Σ R −1 ,(11)
where (G −1 0 ) A denotes the inverse of the non-interacting Green's function restricted to the orbital subset A. Evaluation of G tot in the subset A yields
(G tot ) A = (G −1 0 ) A − Σ A − ∆ −1 ,(12)
where ∆ is defined as
∆ = (G −1 0 ) int − Σ int † ×(13)(G −1 0 ) R − Σ R −1 (G −1 0 ) int − Σ int .
Eq. 5, Eq. 12 and Eq. 13 show that the 'strongly correlated' A-subspace problem can be entirely formulated in the strongly correlated subspace as a problem in which the original interactions v ijkl have been restricted to the subspace A, but for which the bare Green's functions have been modified from G 0 to new propagators G 0 which contain a contribution from a frequency-dependent 'hybridization function' ∆. These propagators are defined as
G −1 0 = (G −1 0 ) A − ∆.(14)
Problems of this type are known as quantum impurity problems. A quantum impurity solver will obtain an expression for a correlated (G imp ) A given ∆ (Eq. 13) and G 0 (Eq. 10) as well as a subset of interactions v ijkl ∈ A i in either spatial or energy basis. Using the impurity problem Dyson equation, the self-energy for a strongly correlated orbital subset is obtained as
[Σ A strong ] i = G −1 0 − ((G imp ) A ) −1 .(15)
Once this strongly correlated Σ A strong is known, the total self-energy, Σ A , in subspace i is evaluated as
[Σ A ] i = Σ tot weak + ([Σ A strong ] i − [Σ A weak ] i ).(16)
We note in particular that there are contributions to the A-subspace self-energy from vertices and propagators with some indices outside of subspace A. These contributions are contained within (Σ tot weak − [Σ A weak ] i ) and only treated at the perturbative level. We would like to stress that these contributions provide an effective adjustment caused by non-local interactions to the [Σ A strong ] i that was evaluated using a subset of local interactions v ijkl ∈ A i .
While quantum impurity models were originally formulated in the context of dilute impurities in a metal, 37 they form the basis of many non-perturbative embedding schemes including DMFT. 4,5 Impurity problems are numerically tractable, with accurate or numerically exact methods ranging from continuous-time quantum Monte Carlo 38-42 to exact diagonalization 43,44 , configurationinteraction 45 , and numerical renormalization group theory 46 methods. The requirements for SEET impurity problems, i.e. general ('non-diagonal') hybridization functions ∆, multiple impurity and bath orbitals, and general interactions v ijkl currently make methods based on the configuration interaction hierarchy 45,47 most suitable for this task, despite the necessity to approximate the continuous hybridization function ∆ by a set of discrete bath levels and bath couplings.
If multiple correlated spaces are present, separate impurity problems need to be solved in each subspace A i , and correlated self-energies [Σ A ] i obtained. These selfenergies are then used to update each [Σ A ] i block of the self-energy Σ SEET obtained with the weak coupling method according to Eq. 6, and the Green's function for the entire system is evaluated using the Dyson equation. Iteration of this procedure, alternating weak coupling steps to update Σ int , Σ R , with impurity solver steps to obtain [Σ A ] i produces a converged Φ SEET and Σ SEET of the form of Eq. 5. Appendix A and Refs. 12-14 have detailed step-by-step instructions on the construction of the iterative procedure.
III. RELATIONSHIP TO OTHER FUNCTIONAL BASED THEORIES
A. DMFT DMFT 4,48,49 is a Φ-derivable theory that can be cast as an approximation to the exact Φ functional 5 :
Φ DMFT = M j=1 [Φ I ] j(17)
where j denotes unit cells, and [Φ I ] j contains all those diagrams of Φ where the interaction vertices have all four indices inside unit cell j. All diagrams in Φ connecting different unit cells, either via interactions or via propagators, are discarded. As a consequence, Σ DMFT = δΦDMFT δG is purely local to every cell. In a translationally invariant system where all unit cells are equal, Σ DMFT is independent of I, and only one impurity problem exists. In analogy to Eq. 12, an impurity model with G imp = G I , Σ imp = Σ I can be defined and the self-consistent solution of the Dyson equation G = G 0 + G 0 Σ DMFT G and the solution of the impurity problem leads to the DMFT approximation of Eq. 1. 50 Eq. 17 shows that DMFT can be understood as a special case of SEET in which the orbital subspaces A i are chosen to be the orbitals local to a unit cell, the 'weak correlation' method is skipped so that Φ weak = 0, and the strong-correlation problem is computed by the DMFT impurity solver. Correspondingly, DMFT will provide a good approximation to the physics of a correlated system as long as the following two criteria are fulfilled: first, the interactions are predominantly local; and second, selfenergy contributions from non-local terms (interactions or propagators) are negligible.
B. HF+DMFT
Similarly, HF+DMFT can be cast into this framework. The chosen correlated orbital subspaces I j are local to each unit cell, and the exact Φ is approximated as
Φ DMFT+HF = Φ tot HF + M j=1 [Φ I ] j − [Φ I HF ] j ,(18)
where [Φ I HF ] j is the HF Φ-functional with vertex indices restricted to unit cell j. To obtain a self-consistent Φ DMFT+HF , the Hartree Fock equations are solved for the entire system and subsequently some or all local orbitals are chosen to the correlated subspace I j . The impurity problem is then solved in the local subspace along the lines of DMFT.
Note that all the non-local contributions to the selfenergy of the unit cells that are frequency independent are generated by Φ tot HF . Any higher order contributions to the self-energy that are frequency dependent have purely local vertices and there are no non-local frequency dependent self-energy terms in the Φ DMFT+HF functional. Additionally, in the non-empirically adjusted HF+DMFT all impurity interactions remain the bare Coulomb interactions v pqrs and are local to the unit cell orbital subspaces I j .
Consequently, any adjustment or renormalization of the frequency dependent [Σ A strong ] i term due to the nonlocal effects that is present in SEET(ED-in-GF2 or ED-in-GW) is absent in HF+DMFT. This is the reason why spectral features and energies produced at the HF+DMFT or LDA+DMFT level using a bare, unrenormalized local Coulomb interaction are not recovered correctly. For small molecular systems, the incorrect energies resulting from employing HF+DMFT with bare Coulomb interactions can be found in Refs. 14 and 51.
C. GW+DMFT GW+DMFT 8,9,52 is based on the premise that both non-local interactions and non-local correlations are important and cannot be discarded; however, the non-local interactions can be treated perturbatively without a significant loss of accuracy.
The starting point of the GW+DMFT procedure is the GW approximation 20,53 for which the Φ functional consist of an infinite series of 'bubble' polarization diagrams, P = GG, connected by bare interaction lines. This series of bubbles can be resumed into a frequency-dependent 'screened' interaction W = v + vP W , where v is the bare Coulomb interaction. The self-energy is approximated as Σ = −GW , so that in the GW approximation Φ[G] = − 1 2 GW G. As Almbladh et al. 17 showed, it is convenient to define a functional Ψ, which is a functional both of the Green's function G and of the screened interaction W , 17 as
Ψ[G, W ] = Φ − 1 2 (P W − log(1 + P W ))(19)
which satisfies
δΨ δW G = − 1 2 P,(20)δΨ δG W = Σ.(21)
Together with the Dyson equation that relates G to Σ, these expression form a closed set of equations that allow the self-consistent computation of Σ and W . We note that while these equations are Φ (and Ψ)-derivable, and should be solved in a self-consistent manner, the size and complexity of W as well as the difficulty in carrying out the self-consistency necessitates additional approximations [53][54][55][56] in the case of large realistic systems, which may not respect the conserving properties of Hedin's 'fully self-consistent' GW approximation. Notable cases where these equations have been solved self-consistently without any approximations are the electron gas, 57 atoms and small molecules, 17,58-60 and lattice model systems. 61 GW+DMFT then makes use of the fact that, given W in all orbitals, there is a natural way of defining an 'effective' W in a subset d of correlated orbitals: 62 splitting the polarization into a contribution P d from the 'correlated' orbitals and a contribution from all other orbitals, P = P d + P r , one can define a screened interaction W r which does not contain any d-to-r processes and reformulate W as
W = [1 − W r P d ] −1 W r ,(22)W r = [1 − vP r ] −1 v.(23)
This identity is general and independent of the GW approximation. It allows to formulate non-perturbative corrections containing contributions by orbitals exclusively in the correlated subspace d without double counting.
Choosing as a subset of orbitals the ones that are local to the unit cell (or, equivalently, a subset of those local to the unit cell), it follows that 10
Ψ GW+DMFT = Ψ tot GW + M j=1 [Ψ I (G I , W r )] j − [Ψ I GW (G I , W r )] j .(24)
This defines the GW+DMFT approximation to the exact Ψ functional. The approximation is noteworthy because it is, as it is written, a diagrammatically sound method for solving realistic correlated many-body problems that includes renormalized interactions and non-perturbative local correlations. In practice, numerous technical and theoretical limitations exist. A fully self-consistent solution of the GW problem is technically very challenging. The various approximations employed (quasiparticles, no full selfconsistency, etc) at the level of GW along with the difficulty of numerically solving multi-orbital impurity problems with general non-local time-dependent interactions means that the rigorous diagrammatic footing described above is severely approximated in practical implementations of the GW+DMFT method 63 .
D. Comparison of SEET, DMFT, and GW+DMFT
The methods outlined above have several important commonalities. First, they require the self-consistent solution of a Φ (or Ψ)-derivable diagrammatic system. This implies that (provided the equations are actually solved to self-consistency) the important conservation laws are automatically fulfilled. They also consist of two-step procedures: an 'outer loop' that entails the solution of a system using a 'cheap' method (e.g. GW, GF2, or HF), and an 'inner' loop that requires the solution of a quantum impurity problem using non-perturbative techniques. All methods become exact at infinite temperature, at zero interaction, and when the system decouples into separate impurity problems without any inter-impurity interactions.
However, there are several important distinctions between these methods. The first is the choice of correlated orbital space. In DMFT and its variants, correlated subspace orbitals are chosen a priori to be the local orbitals or a subset of the local orbitals. This was historically motivated by an exact limit of infinite coordination number, 48,64 where the self-energy can be shown to reduce to the local form. The locality approximation can be controlled by systematically extending the size of the unit cell in the real 65,66 or reciprocal space, 67,68 or by introducing diagrammatic expansions in the non-local contributions. [69][70][71][72][73] In contrast, SEET uses insight from a low-order solution of the system to adaptively define the correlated subspace, e.g. via consideration of the elements of the diagonalized one-body density matrix that are different from 0 or 2. The control parameter used to converge SEET to the exact limit is the size of the correlated subspaces N A i , which can be systematically increased.
A second major difference between HF+DMFT, GW+DMFT, and SEET(ED-in-GF2 or ED-in-GW) is the way in which non-local interactions are treated. DMFT neglects any contribution from non-local interactions to the self-energy, here particularly any contributions from non-local interactions to the local self-energy are neglected. HF+DMFT evaluates the frequencyindependent part of the non-local self-energy at the HF level, but any non-local frequency-dependent contribution to the self-energy is neglected, as both interactions and propagators in Φ DMFT are chosen to be local. Both GW+DMFT and SEET(ED-in-GF2 or ED-in-GW) include frequency-dependent non-local correlations to some extent. Assuming that a local (rather than an energy) basis is chosen for SEET, the lowest order diagram contained in SEET(ED-in-GF2) but not in GW+DMFT is illustrated in the left panel of Fig. 2. Here, different indices are assumed to be in different unit cells. Conversely, SEET(ED-in-GF2) in a local basis would not include the diagram illustrated in the middle panel of Fig. 2. DMFT could in principle be extended to include the second order exchange diagram, such that the diagram in the left panel is contained, while a formulation of SEET around GW, i.e. SEET(ED-in-GW), would include the middle panel of Fig. 2. None of these methods includes the diagram illustrated in the right panel of Fig. 2. As a commonly used basis for SEET is an energy basis, rather than a local basis, a detailed comparison in the practically relevant case is not straightforward.
A third major difference consists of the selection of a basis. As DMFT-type methods perform a local approximation, the choice of basis functions strongly influences the types of correlations that can be contained in DMFT. In contrast, the adaptive choice of SEET basis does not require a localization procedure.
Finally, the nature of the correlated impurity problem is rather different in SEET and GW+DMFT. GW+DMFT, due to its construction of a screened interaction, requires impurity solvers able to evaluate problems with fully general frequency-dependent interactions. While efficient Monte Carlo methods exist that solve impurity problems with frequency-dependent densitydensity interactions, 74,75 efficient impurity solvers able to treat general frequency-dependent four-fermion interactions do not yet exist. SEET, on the other hand, due to the use of the Φ functional, requires no frequencydependence in the interactions. However, the rotation to the natural orbital basis in which the density matrix is diagonal usually mixes all orbitals and interactions, necessitating a treatment of the full four-fermion interaction terms (rather than just density-density interactions) with 'off-diagonal' hybridization functions.
IV. NON-LOCAL INTERACTIONS, CORRELATIONS, AND SCREENING
Non-local interactions and non-local dynamical correlations (caused both by local and non-local interactions) alter the local low-energy physics. A combination of these effects is colloquially summed up under the term 'screening', despite very different physical and diagrammatic origins. As the methods discussed above treat 'screening' to a different extent, we briefly discuss various aspects of it.
First, the 'screened interaction' W describes a way of re-summing certain classes of diagrams. W then takes the role of the bare interaction v in Φ and removes diagrams with repeated insertion of polarization parts, at the cost of introducing a frequency dependence. 20 The need for formulating perturbation theories in powers of W is motivated by a divergence of the perturbation theory in v, when truncated at any order, in the infinite system size (momentum q → 0) limit of the electron gas. 76 In contrast, a perturbation theory in W removes this divergence and stays finite. 20 Within GW and GW+DMFT, as well as within SEET(ED-in-GW), terms are included at least to lowest order in W , and W is approximated by the lowest order P .
SEET(ED-in-GF2) is based on a GF2 starting point that is divergent for metallic systems in the thermodynamic limit, as it is formulated in terms of the bare v. However, any finite system will yield a convergent answer. Thus, for a finite system, in an energy basis, the identification of the correlated orbitals will add near-Fermi-surface states to the correlated subspace and converge as the subspace is enlarged.
A second, entirely different effect also commonly referred to as 'screening' that leads to lowering of local bare Coulomb interactions is generated by the effect of non-local interactions on the local self-energy. 77 If the total orbital space is divided into a correlated subspace and the remainder, the correlated subspace self-energy acquires contributions due to non-local interactions with vertices and propagators in the remainder. This effect is general and present both for the frequency independent and dependent contribution to the self-energy. It is best illustrated for the frequency independent Hartree-Fock contribution Σ ∞ that can be separated into the following contributions:
[Σ ∞ ] ij∈A = kl γ kl (v ikjl − 0.5v iklj )(25)= [Σ ∞ ] embedded ij∈A + [Σ ∞ ] embedding ij∈A , [Σ ∞ ] embedded ij∈A = kl∈A γ kl (v ikjl − 0.5v iklj ),(26)[Σ ∞ ] embedding ij∈A = kl∈R γ kl (v ikjl − 0.5v iklj )(27)+ k∈A l∈R γ kl (v ikjl − 0.5v iklj ).
Here the matrix elements [Σ ∞ ] ij∈A have an 'embedded' contribution coming only from orbitals belonging to the subset A and an 'embedding' contribution where the summation runs over other orbitals R that are not contained in the subset A.
A model with non-local interactions often appears to have a smaller local self-energy than the same model with only on-site interactions. 78,79 Similarly, a multi-orbital model where inter-orbital interactions are truncated to density-density interactions encounters its metal-toinsulator transition at a weaker interaction than one with the full interaction structure. [80][81][82] As the DMFT approximation neglects all inter-unit-cell interactions inside the correlated subspace, and as technical limitations of the impurity solvers require restriction to density-density terms, the effective DMFT interactions are additionally lowered to account for these corrections.
In SEET, this method-dependent 'screening' contribution that results in the lowering of the correlated orbital subspace self-energy is not caused by introducing effective interactions. Rather, the 'embedded' subspace selfenergy [Σ A ] embedded ij∈A = [Σ A strong ] is evaluated using the bare Coulomb interactions (transformed to the appropriate basis) and is 'screened' due to the presence of the 'embedding' self-energy, [
V. CONCLUSIONS
We have discussed several diagrammatic approximations capable of describing a full Coulomb Hamiltonian. These approximate methods can then be used in ab initio calculations of realistic materials or molecular problems. We have paid particular attention to the functional interpretation and have shown that the DMFT -type approximations, where the correlated subspace orbitals are chosen to be local to the unit cell, are a subclass of a wider class of self-energy embedding theories, which can deal with both local and non-local orbitals present in the correlated subspace.
We have also shown that relaxing the locality approximation of the self-energy leads to additional freedom in choosing 'correlated' orbitals, and introduces a systematic small parameter that can be controlled in practice. Choosing the correlated orbital subspace as a set of onebody density matrix eigenvectors corresponding to eigenvalues with partial occupancy (most different than 0 or 2) provides an adaptive selection procedure.
While all the methods outlined here have a rigorous theoretical foundation, practical implementations of realmaterials embedding calculations remain extremely difficult and the approximations needed to lower the computational cost typically break Φ-derivability. While some of these approximations have the potential to be removed with future increases of computational power, calculating frequency dependent renormalized interactions in GW+DMFT for impurity models remains challenging. We therefore believe that embedding methods that do not rely on explicitly renormalized interactions in the correlated subspace, such as SEET(ED-in-GW) and SEET(ED-in-GF2), offer a promising route to the simulation of realistic materials with systematically improvable accuracy.
FIG. 1 .
1Illustration of two choices of SEET subspaces. Top panel: Selection of orbital subspaces based on the energy/occupation scheme: partially occupied orbitals near the Fermi level (µ, ν) are included in the correlated subspace, any other contribution is excluded. Bottom panel: Selection of orbitals based on a localization criterion: sets of neighboring orbitals are chosen as the correlated subspace.
panel: Example of a low order selfenergy diagram contained in SEET(ED-in-GF2) but not in GW+DMFT. Middle panel: low order diagram contained in GW+DMFT and SEET(ED-in-GW) but not in SEET(EDin-GF2). Right panel: low-order diagram not contained in GW+DMFT, SEET(ED-in-GW), and SEET(ED-in-GF2). Dashed lines denote interactions, solid lines Green's functions.
Σ] embedding ij∈A = [Σ tot weak ] ij∈A −[Σ A weak ]. Note that the internal summations in [Σ tot weak ] ij∈A extend over the orbitals that are not present in the correlated subspace, thus accounting for all the effects of the nonlocal interactions on the total frequency dependent subspace self-energy, [Σ A ] = [Σ A ] embedded ij∈A + [Σ] embedding ij∈A .
ACKNOWLEDGMENTSDZ and EG were supported by the Simons Foundation via the Simons Collaboration on the Many-Electron Problem. We thank Hugo Strand for insightful comments and a careful reading of the manuscript.Appendix A: Iterative updates in SEETThe SEET equations are formulated as a set of selfconsistent equations that need to be solved iteratively. The iteration procedure consists of several parts: (i) the problem setup with construction of two-body interactions and one-body integrals (hoppings), (ii) the solution of the entire system using a low-level, usually weak-coupling method (GF2, GW), (iii) the construction of the correlated subspace(s) and impurity problem(s), (iv) the solution of the impurity problems using a high-level, usually non-perturbative, impurity solver method, and (v) the adjustment of the chemical potential to match the target particle number of the system.The detailed SEET algorithm can be summarized as follows.The low level method loop and orbital basis choice.IN1: Choose a basis set.IN2:In the chosen basis evaluate t, and v to represent the Hamiltonian of interest.IN3: Solve the Hartree-Fock or Density Functional Theory equations to obtain an initial bare Green's function G 0 .LL0: Starting from a given G 0 , perform self-energy evaluation for the total system with a low level method and obtain Σ tot weak , [Σ A weak ] i , Σ R weak , Σ int weak , and G tot weak . LL3: Choose the most suitable basis for the subsystem A i that is physically motivated. It can be a energy, occupation, or a local orbital basis.LL4: Transform t, Σ tot weak , [Σ A weak ] i , Σ R weak , Σ int weak , and G tot weak to the new basis. Only a subset of v where all orbital indeces are belonging to subset A i ( v ijkl ∈ A i ) needs to be transformed to the new basis.The embedding loop. EL1: Construct G tot from Eq. 11 with the self-energy given by Eq. 6. Note, that in the first iterationEL2: If the particle number is different from the desired particle number, adjust the chemical potential until the desired particle number is reached.EL3: Find hybridization ∆ according to Eq. 13.The high level solver part.HL1: Define a non-interacting impurity Green's function from Eq. 14.TakeLL5: Go back to LL0 and use the resulting G tot as that starting Green's function to perform a single update of all the low level self-energies and return to EL1.Note, that in a "single shot SEET procedure" the LL5 point is not executed and the iterative loop is terminated once ∆ or electronic energy evaluated at the EL4 point stops to change.
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|
[] |
[
"Determination of the structure of the X(3872) inpA collisions",
"Determination of the structure of the X(3872) inpA collisions"
] |
[
"A B Larionov \nInstitute for Advanced Studies (FIAS)\nD-60438Frankfurt, Frankfurt am MainGermany\n\nNational Research Centre \"Kurchatov Institute\"\n123182MoscowRussia\n",
"M Strikman \nPennsylvania State University\n16802University ParkPAUSA\n",
"M Bleicher \nInstitute for Advanced Studies (FIAS)\nD-60438Frankfurt, Frankfurt am MainGermany\n\nInstitut für Theoretische Physik\nJ.W. Goethe-Universität\nD-60438Frankfurt am MainGermany\n"
] |
[
"Institute for Advanced Studies (FIAS)\nD-60438Frankfurt, Frankfurt am MainGermany",
"National Research Centre \"Kurchatov Institute\"\n123182MoscowRussia",
"Pennsylvania State University\n16802University ParkPAUSA",
"Institute for Advanced Studies (FIAS)\nD-60438Frankfurt, Frankfurt am MainGermany",
"Institut für Theoretische Physik\nJ.W. Goethe-Universität\nD-60438Frankfurt am MainGermany"
] |
[] |
Currently, the structure of the X(3872) meson is unknown. Different competing models of the cc exotic state X(3872) exist, including the possibilities that this state is either a mesonic molecule with dominating D 0D * 0 + c.c. composition, a ccqq tetraquark, or a cc-gluon hybrid state. It is expected that the X(3872) state is rather strongly coupled to thepp channel and, therefore, can be produced inpp andpA collisions at PANDA. We propose to test the hypothetical molecular structure of X(3872) by studying the D orD * stripping reactions on a nuclear residue.
|
10.1016/j.physletb.2015.07.045
|
[
"https://arxiv.org/pdf/1502.03311v2.pdf"
] | 118,623,537 |
1502.03311
|
d978899abd8aa3a9cb3b24f9d36e5416a7e03035
|
Determination of the structure of the X(3872) inpA collisions
11 Feb 2015
A B Larionov
Institute for Advanced Studies (FIAS)
D-60438Frankfurt, Frankfurt am MainGermany
National Research Centre "Kurchatov Institute"
123182MoscowRussia
M Strikman
Pennsylvania State University
16802University ParkPAUSA
M Bleicher
Institute for Advanced Studies (FIAS)
D-60438Frankfurt, Frankfurt am MainGermany
Institut für Theoretische Physik
J.W. Goethe-Universität
D-60438Frankfurt am MainGermany
Determination of the structure of the X(3872) inpA collisions
11 Feb 2015X(3872)pA reactionscharmed meson production PACS: 2543+t1440Rt1440Lb2410Ht
Currently, the structure of the X(3872) meson is unknown. Different competing models of the cc exotic state X(3872) exist, including the possibilities that this state is either a mesonic molecule with dominating D 0D * 0 + c.c. composition, a ccqq tetraquark, or a cc-gluon hybrid state. It is expected that the X(3872) state is rather strongly coupled to thepp channel and, therefore, can be produced inpp andpA collisions at PANDA. We propose to test the hypothetical molecular structure of X(3872) by studying the D orD * stripping reactions on a nuclear residue.
Introduction
The discovery of exotic cc mesons at B-factories and at the Tevatron stimulated interest to explore the possible existence of tetraquark and molecular meson states. The famous X(3872) state has been originally found by BELLE [1] as a peak in π + π − J/ψ invariant mass spectrum from exclusive B ± → K ± π + π − J/ψ decays. Nowadays the existence of the X(3872) state and its quantum numbers J P C = 1 ++ are well established [2]. In particular, radiative decays X(3872) → J/ψγ, X(3872) → ψ ′ (2S)γ [3] point to the positive C-parity of the X(3872). Probably the most intriguing feature is that the mass of the X(3872) is within 1 MeV the sum of the D 0 and D * 0 meson masses. This prompted the popular conception of the X(3872) being a DD * +DD * molecule. Other exotic X,Y,Z states, such as the X(3940) [4], Y (4140) [5], X(4160) [6] (c.f. recent reviews [7,8] for a more complete list), may be interpreted as molecular states of D * D * or D * SD * S . To probe the molecular nature of the X(3872) structure has been difficult. So far, most theoretical calculations have been focused on the description of radiative and isospinviolating decays of the X(3872). For example, the X(3872) → J/ψγ decay can be well understood within the DD * + c.c. molecular hypothesis [9]. On the other hand, the measured large branching fraction B(X(3872) → ψ ′ (2S)γ)/B(X(3872) → J/ψγ) = 3.4 ± 1.4 [3] seems to disfavour the molecular structure and requires a significant pure cc admixture in the X(3872) [10]. The theoretical predictions for the decay rates are, however, quite sensitive to the model details even within various approaches like charmonium or DD * + c.c. molecular models.
In this letter we suggest to test the charm meson molecular hypothesis of the X(3872) structure inpA collisions at PANDA. Assuming that the X(3872) is coupled to the pp channel, we consider the stripping reaction of the D-meson on a nuclear target nucleon such that aD * is produced and vice versa. We show that the distribution of the produced charmed meson in the light cone momentum fraction α with z-axis alongp beam momentum,
α = 2(ω D * (k D * ) + k z D * ) Ep + m N + p lab ,(1)
will be sharply peaked at α ≃ 1 at small transverse momenta which allows to unambiguously identify the weakly coupled DD * + c.c. molecule. Here, k D * and ω D * (k D * ) = (k 2 D * + m 2 D * ) 1/2 are, respectively, the momentum and energy of the producedD * meson in the target nucleus rest frame. Similar studies of hadron-, lepton-, and nucleus-deuteron interactions at high energy have been proposed long ago to test the deuteron structure at short distances as in the spectator kinematics the n-or p-stripping cross sections are proportional to the square of the deuteron wave function. For the X(3872) this idea is depicted in Fig. 1 (details follow below).
X(3872)-proton cross section
For brevity, R stands for X(3872) and the bar, which can be seen over the D * or D, will be dropped in many cases below. The most important ingredients of our calculations are the total Rp cross section and the momentum differential Rp → D * (D)X cross section. In the molecular picture, the latter cross section is the D(D * )-meson stripping cross section. To calculate the total Rp cross section within the Glauber theory, we start from the graphs shown in Fig. 1 which assume the DD * composition of R. It is convenient to perform calculations in the DD * molecule center-of-mass (c.m.) frame with proton momentum p p directed along z-axis. The invariant forward scattering amplitudes of the first two processes are
iM (1) (0) = d 3 k m R ω D |ψ(k)| 2 iM pD (0) ,(2)iM (2) (0) = d 3 k m R ω D * |ψ(k)| 2 iM pD * (0) ,(3)
where m R = ω D + ω D * is the mass of the molecule and ω D (ω D * ) is the energy of D (D * )-meson. (The different assumptions on the momentum dependence of meson energies Figure 1: Processes contributing to the forward scattering amplitude of a proton on the DD * molecule. Wavy lines denote the pD and pD * elastic scattering amplitudes. Straight lines are labeled with particle's four-momenta. The blobs represent the wave function of the molecule. discussed in the next section have practically no effect on the Rp cross section.) The molecule wave function in momentum space is defined as
p p k D * k D k D (1) p p p p k D * k D p p k D * (2) p p k D * k D p p k ′ D * k ′ D (3) p p k D * k D p p k ′ D * k ′ D(4)ψ(k) = d 3 r (2π) 3/2 e −ikr ψ(r) ,(4)
where k is the D * momentum in the DD * c.m. frame, with the normalization condition d 3 k|ψ(k)| 2 = 1. For the calculation of the third and forth processes in Fig. 1 we apply the generalized eikonal approximation (GEA) [11,12] which assumes the nonrelativistic motion of D and D * inside the molecule. In this approximation, the propagator of the intermediate proton depends only on the z-component of momentum transfer q ≡ k D * − k ′ D * , while the pD and pD * elastic scattering amplitudes depend only on the momenta of incoming particles and on the transverse momentum transfer. Thus, we obtain
iM (3) (0) = d 3 kd 3 q (2π) 3 im R 2ω D ω D * ψ * (k − q) iM pD * (q t )iM pD (−q t ) 2p p (q z + iε) ψ(k) ,(5)iM (4) (0) = d 3 kd 3 q (2π) 3 im R 2ω D ω D * ψ * (k − q) iM pD * (q t )iM pD (−q t ) 2p p (−q z + iε) ψ(k) .(6)
Therefore,
iM (3) (0) + iM (4) (0) = d 3 kd 2 q t (2π) 2 m R 4ω D ω D * p p ψ * (k − q t )iM pD * (q t )iM pD (−q t )ψ(k) .(7)
The optical theorem for the proton-molecule forward scattering amplitude is
ImM(0) = 2p p m R σ tot pR .(8)
Substituting (4) (0) and using the parameterization of the elementary amplitudes in the usual form as
M(0) = M (1) (0)+M (2) (0)+M (3) (0)+MM pD ( * ) (q t ) = 2iI pD ( * ) (k D ( * ) )σ tot pD ( * ) e −B pD ( * ) q 2 t /2 ,(9)
with
I pD ( * ) (k D ( * ) ) = [(E p ω D ( * ) − p p k z D ( * ) ) 2 − (m p m D ( * )
) 2 ] 1/2 being the Moeller flux factor we obtain the following expression for the proton-molecule total cross section:
σ tot pR = d 3 k|ψ(k)| 2 [I pD (−k)σ tot pD + I pD * (k)σ tot pD * ] − 1 2 d 3 kψ(k)I pD (−k)σ tot pD I pD * (k)σ tot pD * × d 2 q t (2π) 2 ψ * (k − q t )e −(B pD * +B pD )q 2 t /2 ,(10)
where the normalized flux factors are defined as I pD ( * ) (k) ≡ I pD ( * ) (k)/p p ω D ( * ) . In the small binding energy limit the molecule wave function decreases rapidly with increasing momentum k and becomes negligibly small at k ≪ B −1/2 pD . In this case one can set B pD = B pD * = 0 and perform the Taylor expansion of the flux factors in k z in Eq. (10). Then, for the S-state molecule with accuracy up to the linear terms in k z /m D and assuming that m D ≃ m * D , σ tot pD * ≃ σ tot pD we obtain the formula
σ tot pR = σ tot pD * + σ tot pD − σ tot pD * σ tot pD 4π r −2 DD * ,(11)
in line with previous calculations of the proton-deuteron total cross section [13]. We choose the wave function of a DD * molecule as the asymptotic solution of the Schroedinger equation at large distances:
ψ(r) = κ 2π e −κr r ,(12)
where the range parameter κ = √ 2µE b depends on the reduced mass µ = m D m D * /(m D + m D * ) and on the binding energy E B of the molecule. The corresponding momentum space wave function is
ψ(k) = κ 1/2 /π κ 2 + k 2 .(13)
Let us now discuss the input parameters of our model. Since there is no experimental information on Dp and D * p interactions, we rely on simple estimates in the high-energy limit. For small-size qq configurations the color dipole model predicts the scaling of the total meson-nucleon cross section with the average square of the transverse distance between quark and antiquark in the meson, which is proportional to the square of the Bohr radius r B = 3/4µα s . Here, µ = m q mq/(m q + mq) is the reduced mass with m q and mq being the constituent quark and antiquark masses. The Bohr radii of pion, kaon, and D-meson and J/ψ are ordered as r Bπ > r BK > r BD > r BJ/ψ . Hence, we expect that the total meson-nucleon cross sections follow the same order. At a beam momentum of 3.5 GeV/c (1/2 of the momentum of R formed in thepp → R process on the proton at rest) the total π + p and K + p cross sections are about 28 mb and 17 mb, respectively [2]. The J/ψp cross section is expected to be much smaller, 3.5 − 6 mb (c.f. [14] and refs. therein.). We assume the total Dp cross section σ tot pD = 14 mb, i.e. slightly below the K + p total cross section. This choice is in reasonable agreement with effective field theory calculations [15].
It is well known that at incident energies of a few GeV, the amplitude of meson (nucleon) -nucleon elastic scattering is (to a good approximation) proportional to the product of the electric form factors of the colliding hadrons (see e.g. [16] and refs. therein). Thus, in the exponential approximation for the t-dependence of the form factors, the slope parameters B pM of the transverse momentum dependence of the meson-proton cross section at small t should be proportional to r 2 p + r 2 M , where r 2 p and r 2 M are the mean-squared charge radii of the proton and meson, respectively. Since r 2 M ∝ r 2 BM , the slope parameters should be also ordered as the Bohr radii. Empirical values at p lab = 3.65 GeV/c are B pπ + = 6.75 ± 0.12 GeV −2 and B pK + = 4.12 ± 0.12 GeV −2 as fitted at 0.05 ≤ −t ≤ 0.44 GeV 2 [17]. On the other hand, B pJ/ψ = 3 GeV −2 at the comparable beam momenta [16]. We will assume the value B pD = 4 GeV −2 , since the Bohr radii of kaon and D-meson differ by ∼ 30% only. For the pD * interaction we assume for simplicity σ tot pD * = σ tot pD and B pD * = B pD .
Our educated guess on the D-and D * -meson-nucleon cross sections and slope parameters should of course be checked experimentally. The empirical information on σ tot pD can be obtained by measuring the A-dependence of the transparency ratio of D-meson production inpA reactions at beam momenta beyond the charmonium resonance peaks, where the backgroundpp →DD channel dominates. The slope parameter B pD can be addressed by measuring the transverse momentum spread of D-meson production inpA reactions.
We will further assume that the X(3872) wave function contains 86% of D 0D * 0 + c.c. and 12% of the D + D * − + c.c. component as predicted by the local hidden gauge approach [9]. The binding energy of D 0D * 0 is likely less than 1 MeV [8] and can not be determined from existing data [2] accurately enough. We set
E D 0D * 0 b = 0.5 MeV and E D + D * − b = 8
MeV in numerical calculations. This corresponds to the range parameters κ D 0D * 0 = 0.16 fm −1 and κ D + D * − = 0.64 fm −1 . With these parameters the total pR cross section (10) is σ tot pR = 25 and 23 mb for D 0D * 0 and D + D * − components, respectively, at the molecule momentum of 7 GeV/c in the proton rest frame.
D(D * ) stripping cross section
In high energy hadron-deuteron reactions, the main contribution to the fast backward nucleon production (in the deuteron rest frame or equivalently -fast forward in the deuteron projectile case) is given by the inelastic interaction of the hadron with second nucleon of the deuteron [18]. For large nucleon momenta the spectrum is modified as compared to the impulse approximation due to the Glauber screening and antiscreening corrections [19] since the hadron may interact with both nucleons. In a similar way, in calculations of the Rp → D * X cross section, we take into account the impulse approximation diagram (IA, Fig. 2a) and the single-rescattering diagrams of the incoming proton ( Fig. 2b) and of the outgoing proton or of the most energetic forward going baryon emerging from the inelastic pD interaction (Fig. 2c). The expressions for the invariant matrix elements for the processes (a) and (b) in Fig. 2 are straightforward to obtain in the c.m. frame of the molecule state R:
X p p k ′ D * k D } (a) + p p (b) + p p k ′ D * k ′ D (c) X } k D * k D * k ′ D * k ′ D X } p ′ pM (a) = 2m R ω D * ω D 1/2 (2π) 3/2 M X;pD ψ(k) ,(14)M (b) = im 1/2 R 2p p (2ω D ω D * ) 1/2 d 3 rψ(r)Θ(−z) d 2 q t (2π) 2 e −i(k+qt)r M X;p ′ D ′ M pD * (q t ) ,(15)
where k ≡ k ′ D * . In the case of M (b) we applied the GEA by expressing the propagator of the intermediate proton in the eikonal form and using the coordinate representation with r = r D * − r D . The explicit form of the amplitude M (c) can be written only for specific outgoing states X. However, for the diffractive states including the leading proton, the expression for M (c) can be obtained from the expression for M (b) by replacing Θ(−z) → Θ(z), which reflects the change of the time order of the pD * and pD interactions. Thus, for the diffractive outgoing state X the expression for M (b) + M (c) is given by Eq. (15) with replacement Θ(−z) → 1 (neglecting small differences in momenta of incoming and outgoing proton in elementary amplitudes). We assume that the same replacement can be done for any final state X. By summing over all states X we then obtain the momentum differential D * production (i.e. D-stripping) cross section in the molecule rest frame:
dσ pR→D * X = d 3 k ′ D * (2π) 3 2ω D * 4p p m R spins and sorts of X |M (a) + M (b) + M (c) | 2 (2π) 4 ×δ (4) (p X + k ′ D * − p p − P) d 3 p X 1 (2π) 3 2E X 1 · · · d 3 p Xn (2π) 3 2E Xn ,(16)
where P is the four momentum of the molecule (P 2 = m 2 R ). With a help of the unitarity relation for the elementary amplitudes [20] the sum over spin states and sorts of X and the integration over phase space volume can be reduced to the products of the imaginary parts of elastic scattering amplitudes. This leads to the following expression for the momentum differential cross section in the molecule rest frame:
d 3 σ pR→D * X d 3 k = σ tot pD I pD (−k)|ψ(k)| 2 κ ,(17)κ = 1 − σ tot pD * I pD * (k) d 2 q t (2π) 2 ψ * (k + q t ) ψ * (k) e −(B pD +B pD * )q 2 t /2 + (σ tot pD * I pD * (k)) 2 4 d 2 q t d 2 q ′ t (2π) 4 ψ(k + q t )ψ * (k + q ′ t ) |ψ(k)| 2 ×e −[B pD * (q 2 t +q ′2 t )+B pD (q ′ t −qt) 2 ]/2 .(18)
The first term in the r.h.s. of Eq. (18) is the pure IA contribution. The second and third terms are, respectively, the screening and antiscreening corrections (see Eqs. (8a) and (8b) in [19]). The D * meson is assumed to be on its vacuum mass shell, ω D * (k) = m 2 D * + k 2 , while the energy of the D meson is calculated from energy conservation, ω D (−k) = m R − ω D * (k). (The condition ω D > 0 constrains the maximum momentum of the emitted D * , k < 3.3 GeV/c. Above this value our model looses its applicability.) In the case of the D-meson production one has to exchange I pD ↔ I pD * , σ tot pD ↔ σ tot pD * and B pD ↔ B pD * in Eqs. (17), (18). In this case the on-shell condition is applied to the D-meson, while the D * energy is determined by energy conservation. It is convenient to express the differential invariant D * production cross section (17) in terms of the relative fraction α of the light cone momentum of the DD * molecule carried by the D * :
ω D * d 3 σ pR→D * X d 3 k = α d 3 σ pR→D * X dαd 2 k t ≡ G p→D * R (α, k t ) ,(19)
where α = 2(ω D * (k) − k z )/m R . Figures 3 and 4 show the differential cross section of D * 0 and D * − production from X(3872) collisions at 7 GeV/c with proton at rest as a function of α for several values of transverse momentum k t . At k t = 0, the cross section has a sharp maximum at α ≃ 2m D * /m R ≃ 1.04 and is almost unaffected by the screening and antiscreening corrections. With increasing k t , the width of α-distribution increases while the screening and antiscreening corrections to the IA term become important. This is expected since the large-k t component of the molecule wave function corresponds to small transverse separation between D and D * . The corrections become large for α ≃ 1 and large transverse momenta as can be directly seen from the structure of the integrands in Eq. (18). Indeed, α ≃ 1 corresponds to k z ≃ 0 in the molecule rest frame. Then at finite transverse momentum transfer q t the ratio ψ * (k t + q t )/ψ * (k t ) is less than unity at k t = 0 and asymptotically tends to unity with growing k t . Due to the extremely narrow wave function of the D 0 D * 0 molecule in momentum space, the screening and antiscreening corrections are sharply peaked at α ≃ 1.1 and develop structures in the α-dependence of the cross section at large transverse momenta. In the case ofpA reactions these structures are slightly smeared out due to the nucleon Fermi motion (see Fig. 6 below).
D * and D production off nucleus
In antiproton-nucleus interactions, we focus on the D * (or D) meson production in the two-step processpp → R, RN → D * (D)X. Similar to the case of Rp interactions, we apply the Glauber theory to calculate the differential cross sections of the D * production in antiproton-nucleus interactions. We start from the multiple scattering diagram shown in Fig. 5 which can be evaluated within the GEA. We will assume that the nucleus can be described within the independent particle model disregarding the c.m. motion corrections (c.f. [21]). The incoming antiproton, intermediate molecular state R and outgoing D *meson are allowed to rescatter on nucleons elastically an arbitrary number of times. The D * production cross section is proportional to the product of the sum of the amplitudes of Fig. 5 and their conjugated. The R state is formed on a proton 1, while the D * is produced in the collision of R with a nucleon 2. The nucleons 1 and 2 are fixed in the direct and conjugated amplitudes while the sets of other nucleon scatterers are arbitrary. The leading order contribution is given by the product term without elastic rescatterings. Nuclear absorption corrections are accounted for by summing all possible product terms with non-overlapping sets of nucleon scatterers. This gives the following expression for the momentum differential cross section of D * production on a nucleus:
α d 3 σp A→D * X dαd 2 k t = v −1 p d 3 r 1 Pp ,surv (b 1 , −∞, z 1 ) d 2 p 1t d 2 Γ 1→R p d 2 p 1t G 2→D * R (α, k t − α 2 p 1t ) × ∞ z 1 dz 2 P R,surv (b 1 , z 1 , z 2 )ρ(b 1 , z 2 )P D * ,surv (b 1 , z 2 , ∞) ,(20)
where
d 2 Γ 1→R p d 2 p 1t = |M R;p1 | 2 vp (2π) 2 4p 2 lab E 1 n p (r 1 ; p 1t , ∆ 0 R )(21)
is the in-medium width ofp with respect to production of R with transverse momentum p 1t ; vp = p lab /Ep is the antiproton velocity; n p (r 1 ; p 1t , ∆ 0 R ) is the proton occupation number. The longitudinal momentum ∆ 0 R of the proton 1 is obtained from the condition of on-shell production of the state R in the processp 1 → R:
∆ 0 R = m 2 N + E 2 1 + 2EpE 1 − m 2 R 2p lab .(22)
The proton occupation number is taken as the depleted Fermi distribution supplemented by high-momentum tail due to short-range quasideuteron correlations [22,23,21]:
n p (r; p) = (1 − P 2 )Θ(p F − p) + π 2 P 2 ρ p |ψ d (p)| 2 Θ(p − p F ) ∞ p F dp ′ p ′2 |ψ d (p ′ )| 2 ,(23)
where p F (r) = (3π 2 ρ p (r)) 1/3 is the proton Fermi momentum, P 2 = 0.25 is the proton fraction above the Fermi surface, ρ p (r) is the proton density, and ψ d (p) is the deuteron wave function. In Eq. (20), the nuclear absorption is given by the survival probabilities of the antiproton, the molecule, and the D * :
Pp ,surv (b 1 , −∞, z 1 ) = exp −σ tot pN z 1 −∞ dzρ(b 1 , z) ,(24)P R,surv (b 1 , z 1 , z 2 ) = exp −σ tot RN z 2 z 1 dzρ(b 1 , z) ,(25)P D * ,surv (b 1 , z 2 , ∞) = exp −σ tot D * N ∞ z 2 dzρ(b 1 , z) ,(26)
where ρ = ρ p + ρ n is the nucleon density. We use the two-parameter Fermi distributions of protons and neutrons [14]. As usual in the Glauber theory, Eqs.(24)-(26) neglect the Fermi motion of nucleon scatterers. In a similar way, in writing Eq. (20) we neglected the Fermi motion of nucleon 2 since the elementary cross section (19) depends only weakly on the proton momentum (via the flux factors, screening-and antiscreening contributions) and is in leading order proportional to the square of the molecule wave function. However, the transverse Fermi motion of proton 1 is taken into account in Eq. (20) in the high-energy approximation (c.f. [19]). The latter implies that the light cone momentum fraction α can be expressed in the target nucleus rest frame according to Eq.(1) where the Fermi motion of the proton 1 is still neglected. (We have numerically checked that using the exact Lorentz transformation to the c.m. frame of R to evaluate the invariant cross section ω D *
d 3 σ pR→D * X d 3 k
instead of using the infinite momentum frame in Eq. (20) which conserves α and assumes Galilean transformation for k t produces indistinguishable results.) Thepp → R matrix element in Eq.(21) is a major uncertainty in our calculations. Its modulus squared can be formally expressed in terms of the partial decay width Γ R→pp as
|M R;p1 | 2 = 4π(2J R + 1)m 2 R Γ R→pp m 2 R − 4m 2 N ,(27)
where the overline means averaging over antiproton and proton helicities and summation over the helicity of R. There is no experimental data on the partial decay width Γ X(3872)→pp .
(The recent LHCb data onpp invariant mass spectra from B + → ppK + decays [24] do not allow to clearly identify X(3872) in the pp decay channel due to statistical limitations.) In the present calculations, we will use the value Γ X(3872)→pp ≃ 30 eV as suggested by theoretical estimates [25]. This value is about two times smaller than Γ χ c1 (1P )→pp . However, one should note that, in the molecular picture, the decay of the X(3872) to the pp state requires the production of only two qq pairs, and not three qq pairs as in the ordinary charmonium decay to the pp channel. Thus, the partial decay width of the X(3872) into the pp channel may be even larger than that of the χ c1 (1P ) state [25]. Formula (20) has a simple physical interpretation if we express the integral d 3 r 1 as d 2 b 1 dz 1 . The factor Pp ,surv (b 1 , −∞, z 1 ) is the probability that the incoming from z = −∞ antiproton with impact parameter b 1 will reach the point z = z 1 . The combination (dz 1 /vp)d 2 p 1t d 2 Γ 1→R p /d 2 p 1t is the probability that the molecule R will be formed within the transverse momentum element d 2 p 1t when thep is passing the longitudinal element dz 1 . The factor P R,surv (b 1 , z 1 , z 2 ) is the probability that the molecule will reach the point z = z 2 . The combination dz 2 (dα/α)d 2 k t G 2→D * R (α, k t − α 2 p 1t )ρ(b 1 , z 2 ) is the probability that a D * will be produced in the kinematical element dαd 2 k t when the molecule R is passing the longitudinal element dz 2 . Finally, the factor P D * ,surv (b 1 , z 2 , ∞) is the probability that the D * will escape from the nucleus. In the spirit of the eikonal approach, all particles propagate parallel to the beam direction. (For example, we assumed that the transverse momentum of the molecule, p 1t , does not influence its trajectory.) The integration over z 2 can be taken with the explicit forms of the survival probabilities Eqs. (25), (26). As a result Eq.(20) takes the following simple form:
α d 3 σp A→D * X dαd 2 k t = 1 vp(σ tot RN − σ tot D * N ) d 3 r 1 Pp ,surv (b 1 , −∞, z 1 ) d 2 p 1t d 2 Γ 1→R p d 2 p 1t ×G 2→D * R (α, k t − α 2 p 1t )[P D * ,surv (b 1 , z 1 , ∞) − P R,surv (b 1 , z 1 , ∞)] .(28)
In Fig. 6 we display the differential cross sections of charmed meson production in antiproton collisions with lead nucleus at 7 GeV/c. The D + cross section is peaked at α ≃ 2m D /m R = 0.96 and behaves similar to the D * − cross section as a function of α and k t . The width of α-dependence of the D * 0 cross section is much smaller and the peak value is much larger as compared to the D * − and D + cross sections. The α-dependence of D * and D production inpA collisions is dominated by the elementary cross section (c.f. Figs. 3,4). However, a closer look reveals significant differences between D * production on a nucleus and on a proton due to the Fermi motion. These are better visible in the ratio of the two cross sections depicted in Fig. 7. At k t = 0 the ratio has a minimum at α ≃ 2m D * /m R because in this case the contribution from target protons with finite transverse momentum p 1t is suppressed by the factor |ψ(k t − α 2 p 1t )| 2 /|ψ(k t )| 2 . However, with increasing k t this factor becomes larger than unity for comoving proton 1. This leads to the observed local maximum in the α-dependence for k t ≃ 0.1 − 0.4 GeV/c. At large k t or for large deviations of α from unity the ratio tends to the constant value.
Discussion and conclusions
We proposed the idea of D(D * ) stripping from the X(3872) state, to investigate if X(3872) has a molecular structure. However, a major background is given by the direct processpp → DD * +c.c. in the target nucleus, because the thresholds of X(3872) and DD * production inpp collisions are almost the same. Due to the two-body final state, at small k t thepp → DD * cross section on the proton at rest has two peaks at
α = 2(ω D * ± k DD * )/ √ s where k DD * = [(s + m 2 D * − m 2 D ) 2 /4s − m 2 D * ] 1/2 is the DD * c.m. momentum and ω D * = (m 2 D * + k 2 DD * ) 1/2 .
Since the cross sectionpp → DD * grows with √ s close to threshold, the backgroundpA →D * X cross section at small k t will be widely distributed in α around α = 1 due to Fermi motion. Thus, the sharp peaks of stripping reactions at α = 1.04 forD *and at α = 0.96 for D-production should be clearly visible on the smooth background. As we have seen, these peaks are not influenced by intramolecular screening and antiscreening effects. Moreover, we expect that the elastic rescattering of antiproton and produced particles on the nucleons will practically not change theD * and D spectra at small k t [21]. There is another possible source of narrow peaks in α-distributions ofD * and D. The BELLE collaboration [26] has found a significant near-threshold enhancement in the D * 0D0 invariant mass spectrum from B → D * 0D0 K decays. We note that this does not exclude the existence of the D * 0D0 bound state. (One similar example is Λ(1405) which lies about 30 MeV below K − p threshold and can be treated as a K − p quasibound state although it strongly influences the K − p → Σ ± π ∓ and K − p → Σ 0 π 0 cross sections at small beam momenta [27,28].) But it is also possible that X(3872) is a resonance coupled to the D * 0D0 + c.c. channel. If such a resonance state is produced in peripheralpA collisions, it will decay far away from the nucleus, since the width of X(3872) is less than 1 MeV. The resulting α-distributions of D * 0 andD 0 will be also sharply peaked near α ≃ 1 at small k t . However, in this case both decay products can be detected. This gives a clear experimental condition for rejecting such decay events. In contrast, the stripping events would contain only one meson, D * 0 orD 0 , in the same kinematical region.
In conclusion, we have demonstrated that the spectra ofD * and D in the light cone momentum fraction at small transverse momenta allow to test the hypothetical DD * molecular structure of the X(3872) produced inpA collisions at threshold. We propose to search the narrow peak inD * or D production at α ≃ 1 and small k t as an unambiguous signal of the DD * molecular state formation inpA collisions in PANDA experiment at FAIR.
Figure 2 :
2The amplitude for the process Rp → D * X where X ≡ {X 1 , . . . , X n } is an arbitrary final state in the pD interaction. See also caption toFig. 1.
Figure 3 :Figure 4 :Figure 5 :
345The invariant differential cross section of D * 0 production in X(3872)p collisions at p lab = 7 GeV/c. Thick solid line -full calculation according to Eqs.(17)-(19). Thin solid line -the calculation taking into account only IA and screening term of Eq.(18). Dashed line -the calculation with κ = 1 in Eq.(17), i.e. only with the IA term. The inset at k t = 0 shows the behaviour of the differential cross section for a smaller range of α. Same asFig. 3but for D * − production. The amplitude of the processpA → D * X(A − 2) * . The wave functions of the initial and final nuclei are denoted as ψ A and ψ A−2 , respectively. The mass numbers are shown as subscripts. Wavy lines represent elastic scattering amplitudes on nucleons. "X" stands for the arbitrary final state particles in the semi-inclusive process R2 → D * X. The summation is performed over all possible sets of nucleon scatterers {n 1 },{n 2 } and {n 3 } for thep, R and D * , respectively.
Figure 6 :
6The invariant differential cross sections of D * 0 , D 0 , D * − and D + production inp 208 Pb collisions at p lab = 7 GeV/c. The calculations are done using full cross section X(3872)p → D * (D)X in Eq.(28) including the IA term as well as screening and antiscreening corrections (see Eqs.(17)-(19)). For k t = 0, the cross sections of D * 0 and D 0 production are divided by a factor of 100.
Figure 7 :
7The ratio of D * production cross sections inp 208 Pb and X(3872)p collisions at p lab = 7 GeV/c for several values of transverse momentum k t as a function of light cone momentum fraction α. The ratio is normalized at unity for α = 0.5. Top panel -D * 0 . Bottom panel -D * − .
AcknowledgementsThis work was supported by HIC for FAIR within the framework of the LOEWE program.
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[
"Comments on Kerr effect and gyrotropic order in cuprates",
"Comments on Kerr effect and gyrotropic order in cuprates"
] |
[
"Sudip Chakravarty \nDepartment of Physics and Astronomy\nUniversity of California Los Angeles\n90095Los AngelesCaliforniaUSA\n"
] |
[
"Department of Physics and Astronomy\nUniversity of California Los Angeles\n90095Los AngelesCaliforniaUSA"
] |
[] |
I comment on two recent papers on Kerr effect as evidence of gyrotropic order in cuprates, and I suggest that the arguments may not be sound. The difficulty is that in practically all cases the wave vector k z perpendicular to the copper-oxygen plane is not a good quantum number. This appears to be problematic for arXiv:1212.2698, whereas in arXiv:1212.2274 the symmetry arguments may turn out to be robust, but the microscopic picture is wanting. Thus, the Kerr effect in cuprates remains a puzzle, as there is little doubt that the arguments presented against time reversal symmetry breaking appear to be rather strong in both of these papers on experimental grounds.
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10.1103/physrevb.89.087101
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[
"https://arxiv.org/pdf/1301.0942v1.pdf"
] | 118,682,485 |
1301.0942
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d7e034ca3c1bf83a7f9e935396ca16aeaea32292
|
Comments on Kerr effect and gyrotropic order in cuprates
5 Jan 2013
Sudip Chakravarty
Department of Physics and Astronomy
University of California Los Angeles
90095Los AngelesCaliforniaUSA
Comments on Kerr effect and gyrotropic order in cuprates
5 Jan 2013(Dated: December 11, 2013)
I comment on two recent papers on Kerr effect as evidence of gyrotropic order in cuprates, and I suggest that the arguments may not be sound. The difficulty is that in practically all cases the wave vector k z perpendicular to the copper-oxygen plane is not a good quantum number. This appears to be problematic for arXiv:1212.2698, whereas in arXiv:1212.2274 the symmetry arguments may turn out to be robust, but the microscopic picture is wanting. Thus, the Kerr effect in cuprates remains a puzzle, as there is little doubt that the arguments presented against time reversal symmetry breaking appear to be rather strong in both of these papers on experimental grounds.
(k), v(k) = 1 ∂ε(k) ∂k − e E × Ω(k),(1)
where E is the electric field. Following Ref. 4 they then arrive at the expression for the maximum value of the Kerr rotation angle θ K at normal incidence on the optic axis of a uniaxial crystal, which is
θ K ∼ αl z ℜ 1 (1 − iωτ z ) 2 [ǫ (ω) − 1] ,(2)
where ǫ (ω) is the dielectric function in the plane perpendicular to the optic axis and l z is the mean free path in the z-direction, and α is the fine structure constant. This equation presupposes an order of magnitude estimate of the Berry phase contribution, 6
k F z −k F z dk z v z (k z ) k F (kz) 0 d 2 k Ω z (k, k z ) ∼ 1.(3)
The entire derivation is based on k z being a good quantum number, which it is not for any of the under doped cuprates. Therefore, the efficacy of the formula for θ K is moot. The authors then go on to make an estimate for the Kerr data in La 2−x Ba x CuO 4 by setting l z ∼ 1 (that is, the c-axis lattice constant), recognizing themselves that the c-axis transport is incoherent. For incoherent transport, the semiclassical dynamics involving k z , whether or not one includes the Berry curvature, is not particularly meaningful.
A microscopic Hamiltonian which is explicitly coherent in interlayer tunneling in Ref. 2 appears to be problematic. Take, for example, their Eq. (9), where the gyrotropic tensor is 2 expressed as
γ(ω) ∝ t ⊥ E F 2 · · ·(4)
with no other dependences on the interlayer hopping matrix element t ⊥ (assumed to be momentum independent) and E F the Fermi energy. It is difficult to see how this equation will survive if incoherence in the direction perpendicular to the copper oxide plane is included.
In
of measurements have found polar Kerr effect in a large number of cuprate superconductors at an onset temperature in the pseudogap regime; the experiments are surveyed in a recent article. 1 Initially, the experiments were interpreted in terms of time reversal symmetry breaking. However, an alternate proposal argues in favor of spontaneous breaking of inversion and mirror reflection symmetries. 2,3 Both papers invoke gyrotropy of the medium 4 in the pseudogap state of the cuprates. The arguments against time reversal symmetry breaking have been elegantly summarized, so there is little reason to duplicate them here. Here I want to question a fundamental assumption made in Ref. 3, which is also relevant for Ref. 2. However, the symmetry based arguments and a phenomenological model discussed in Ref. 2 could be robust. The starting point of Ref. 3 is the equation describing the velocity, v(k), where it is augmented correctly by a so-called anomalous term 5 containing the Berry curvature Ω
any case, a well defined cholesteric pitch requires coherent c-axis tunneling for which experimental evidence is non existent in any generic underdoped cuprates, some of which were investigated in Ref. 1.A few brief comments regarding incoherent interlayer hopping are in order. At temperatures above the superconducting transition temperature even the best underdoped cuprate superconductors show an insulating upturn as the temperature is lowered with the c-axis resistivity of the order of Ohm-cm or greater. In fact c-axis incoherence was used to explain quantum oscillations in ρ c (H), 7 as the magnetic field, electric field and the current are all in the same direction, and therefore normally there should not be any oscillations of the two dimensional density of states; the picture presented there was that the electron executes many cyclotron orbits in the plane and infrequently incoherently hops betweenn the planes. Second order perturbation theory, assuming coherence, in terms of (t ⊥ /E F ) must fail because of the degeneracy of the electronic states of the stacked copper-oxide planes, which leads to vanishing energy denominators. An exact analysis of the density of states with a coherent energy dispersion in a simplified model was discussed by Stephen 8 as to how the two-dimensional quantum oscillations can be inherited in ρ c (H), but unfortunately this model is totally inconsistent with the incoherent c-axis transport in underdoped cuprates.In my opinion, the true explanation of these remarkable Kerr measurements and their implications remain unexplained.This work was supported by by NSF under Grant No. DMR-1004520. * [email protected] 1 Aharon Kapitulnik, Jing Xia, Elizabeth Schemm and Alexander Palevski, New J. Phys. 11, 055060 (2009).
. Pavan Hosur, A Kapitulnik, S A Kivelson, J Orenstein, S Raghu, arXiv:1212.2274condmat.str-elPavan Hosur, A. Kapitulnik, S. A. Kivelson, J. Orenstein, S. Raghu, arXiv:1212.2274 [cond- mat.str-el].
. J Orenstein, Joel E Moore, arXiv:1212.2698cond-mat.str-elJ. Orenstein, Joel E. Moore, arXiv:1212.2698 [cond-mat.str-el].
L Landau, E Lifshitz, L Pitaevskii, Electrodynamics of Continuous Media. New YorkPergamon PressL. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon Press, New York, 1984).
. R Karplus, J M Luttinger, Phys. Rev. B. 951154R. Karplus and J. M. Luttinger, Phys. Rev. B 95, 1154 (1954).
. Jonghyoun Eun, Sudip Chakravarty, Phys. Rev. B. 8494506see, especiallyJonghyoun Eun and Sudip Chakravarty, Phys. Rev. B 84, 094506 (2011); see, especially, Sec. V.
. M J Stephen, Phys. Rev. B. 455481see, especiallyM. J. Stephen, Phys. Rev. B 45, 5481 (1992); see, especially, Sec. III.
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[] |
[
"Cross-Modality Paired-Images Generation for RGB-Infrared Person Re-Identification",
"Cross-Modality Paired-Images Generation for RGB-Infrared Person Re-Identification"
] |
[
"Guan-An Wang \nInstitute of Automation\nChinese Academy of Sciences\nBeijingChina\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Tianzhu Zhang [email protected] \nUniversity of Science and Technology of China\nBeijingChina\n",
"Yang Yang [email protected] \nInstitute of Automation\nChinese Academy of Sciences\nBeijingChina\n",
"Jian Cheng [email protected] \nInstitute of Automation\nChinese Academy of Sciences\nBeijingChina\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n\nCenter for Excellence in Brain Science and Intelligence Technology\nBeijingChina\n",
"Jianlong Chang [email protected] \nInstitute of Automation\nChinese Academy of Sciences\nBeijingChina\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Xu Liang \nInstitute of Automation\nChinese Academy of Sciences\nBeijingChina\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Zengguang Hou [email protected] \nInstitute of Automation\nChinese Academy of Sciences\nBeijingChina\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n\nCenter for Excellence in Brain Science and Intelligence Technology\nBeijingChina\n"
] |
[
"Institute of Automation\nChinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Science and Technology of China\nBeijingChina",
"Institute of Automation\nChinese Academy of Sciences\nBeijingChina",
"Institute of Automation\nChinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"Center for Excellence in Brain Science and Intelligence Technology\nBeijingChina",
"Institute of Automation\nChinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"Institute of Automation\nChinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"Institute of Automation\nChinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"Center for Excellence in Brain Science and Intelligence Technology\nBeijingChina"
] |
[] |
RGB-Infrared (IR) person re-identification is very challenging due to the large cross-modality variations between RGB and IR images. The key solution is to learn aligned features to the bridge RGB and IR modalities. However, due to the lack of correspondence labels between every pair of RGB and IR images, most methods try to alleviate the variations with set-level alignment by reducing the distance between the entire RGB and IR sets. However, this set-level alignment may lead to misalignment of some instances, which limits the performance for RGB-IR Re-ID. Different from existing methods, in this paper, we propose to generate cross-modality paired-images and perform both global set-level and fine-grained instance-level alignments. Our proposed method enjoys several merits. First, our method can perform set-level alignment by disentangling modalityspecific and modality-invariant features. Compared with conventional methods, ours can explicitly remove the modalityspecific features and the modality variation can be better reduced. Second, given cross-modality unpaired-images of a person, our method can generate cross-modality paired images from exchanged images. With them, we can directly perform instance-level alignment by minimizing distances of every pair of images. Extensive experimental results on two standard benchmarks demonstrate that the proposed model favourably against state-of-the-art methods. Especially, on SYSU-MM01 dataset, our model can achieve a gain of 9.2% and 7.7% in terms of Rank-1 and mAP. Code is available at https://github.com/wangguanan/JSIA-ReID.
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10.1609/aaai.v34i07.6894
|
[
"https://ojs.aaai.org/index.php/AAAI/article/download/6894/6748"
] | 211,076,134 |
2002.04114
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d83df9482f061f1cf43fc8f0d73ed894caf6ba4e
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Cross-Modality Paired-Images Generation for RGB-Infrared Person Re-Identification
Guan-An Wang
Institute of Automation
Chinese Academy of Sciences
BeijingChina
University of Chinese Academy of Sciences
BeijingChina
Tianzhu Zhang [email protected]
University of Science and Technology of China
BeijingChina
Yang Yang [email protected]
Institute of Automation
Chinese Academy of Sciences
BeijingChina
Jian Cheng [email protected]
Institute of Automation
Chinese Academy of Sciences
BeijingChina
University of Chinese Academy of Sciences
BeijingChina
Center for Excellence in Brain Science and Intelligence Technology
BeijingChina
Jianlong Chang [email protected]
Institute of Automation
Chinese Academy of Sciences
BeijingChina
University of Chinese Academy of Sciences
BeijingChina
Xu Liang
Institute of Automation
Chinese Academy of Sciences
BeijingChina
University of Chinese Academy of Sciences
BeijingChina
Zengguang Hou [email protected]
Institute of Automation
Chinese Academy of Sciences
BeijingChina
University of Chinese Academy of Sciences
BeijingChina
Center for Excellence in Brain Science and Intelligence Technology
BeijingChina
Cross-Modality Paired-Images Generation for RGB-Infrared Person Re-Identification
RGB-Infrared (IR) person re-identification is very challenging due to the large cross-modality variations between RGB and IR images. The key solution is to learn aligned features to the bridge RGB and IR modalities. However, due to the lack of correspondence labels between every pair of RGB and IR images, most methods try to alleviate the variations with set-level alignment by reducing the distance between the entire RGB and IR sets. However, this set-level alignment may lead to misalignment of some instances, which limits the performance for RGB-IR Re-ID. Different from existing methods, in this paper, we propose to generate cross-modality paired-images and perform both global set-level and fine-grained instance-level alignments. Our proposed method enjoys several merits. First, our method can perform set-level alignment by disentangling modalityspecific and modality-invariant features. Compared with conventional methods, ours can explicitly remove the modalityspecific features and the modality variation can be better reduced. Second, given cross-modality unpaired-images of a person, our method can generate cross-modality paired images from exchanged images. With them, we can directly perform instance-level alignment by minimizing distances of every pair of images. Extensive experimental results on two standard benchmarks demonstrate that the proposed model favourably against state-of-the-art methods. Especially, on SYSU-MM01 dataset, our model can achieve a gain of 9.2% and 7.7% in terms of Rank-1 and mAP. Code is available at https://github.com/wangguanan/JSIA-ReID.
Introduction
Person Re-Identification (Re-ID) (Gong et al. 2014;Zheng, Yang, and Hauptmann 2016) is widely used in various applications such as video surveillance, security and smart city. Given a query image of a person, Re-ID aims to find images of the person across disjoint cameras. It's very challenging due to the large intra-class and small inter-class variations caused by different poses, illuminations, views, and occlusions. Most of existing Re-ID methods focus on visible cam- Figure 1: Illustration of set-level and instance-level alignment (please view in color). (a) There is a significant gap between the RGB and IR sets. (b) Existing methods perform set-level alignment by minimizing distances between the two sets, which may lead to misalignment of some instances. (c) Our method first generates cross-modality paired-images. (d) Then, instance-level alignment is performed by minimizing distances between each pair of images.
eras and RGB images, and formulate the person Re-ID as a single-modality (RGB-RGB) matching problem. However, the visible cameras are difficult in capturing valid appearance information under poor illumination environments (e.g. at night), which limits the applicability of person Re-ID in practical. Fortunately, most surveillance cameras can automatically switch from visible (RGB) to nearinfrared (IR) mode, which facilitates such cameras to work at night. Thus, it is necessary to study the RGB-IR Re-ID in real-world scenarios, which is a cross-modality matching problem. Compared with RGB-RGB single-modality matching, RGB-IR cross-modality matching is more difficult due to the large variation between the two modalities. As shown in Figure 2(b), RGB and IR images are intrinsically distinct and heterogeneous, and have different wavelength ranges. Here, RGB images have three channels containing color information of visible light, while IR images have one channel containing information of invisible light.
The key solution is to learn aligned features to bridge the two modalities. However, due to the lack of correspondence Figure 2: (a) In the edge-photo task, we can get crossmodality paired-images. By minimizing their distances in a feature space, we can easily reduce the cross-modality gap. (b) In RGB-IR Re-ID task, we have only unpaired-images. The appearance variation caused by the cross-modality gap makes the task more challenging. (c) Our method can well generate images paired with given ones, which help us to improve RGB-IR Re-ID. (d,e) Vanilla image translation models such as CycleGAN and StarGAN (Choi et al. 2018) fail to deal with this issue. labels between every pair of images in different modalities like in Figure 2(a), existing RGB-IR Re-ID methods (Wu et al. 2017;Ye et al. 2018a;2018b;Dai et al. 2018;Hao et al. 2019) try to reduce the marginal distribution divergence between RGB and IR modalities, while cannot deal with their joint distributions. That is to say, as shown in Figure 1(b), they only focus on the global set-level alignment between the entire RGB and IR sets while neglecting the fine-grained instance-level alignment between every two images. This may lead to misalignment of some instances when performing the global alignment (Chen et al. 2018). Although we can alleviate this issue by using label information, in Re-ID task, labels of training and test sets are unshared. Thus, simply fitting training labels may not perform very well for unseen test labels.
Different from the existing approaches, a heuristic method is to use cross-modality paired-images in Figure 2(a). With the paired images, we can directly reduce the instance-level gap by minimizing the distance between every pair of images in a feature space. However, as in Figure 2(b),all images are un-paired in RGB-IR Re-ID task. This is because the two kinds of images are captured at different times. RGB images are captured at daytime while IR ones at night. We can also translate images from one modality to another by using image translation models, such as CycleGAN and StarGAN (Choi et al. 2018). But these image translation models can only learn one-to-one mappings, while mapping from IR to RGB images are one-to-many. For example, gray in IR mode can be blue, yellow even red in RGB mode. Under this situation, CycleGAN and StarGAN often generate some noisy images and cannot be used in the following Re-ID task. As shown in Figure 2(d,e), the generated images by CycleGAN and StarGAN are unsatisfying.
To solve the above problems, in this paper, we propose a novel Joint Set-level and Instance-Level Alignment Re-ID (JSIA-ReID) which enjoys several merits. First, our method can perform set-level alignment by disentangling modalityspecific and modality-invariant features. Compared with encoding images with only one encoder, ours can explicitly remove the modality-specific features and significantly reduce the modality-gap. Second, given cross-modality unpairedimages of a person, our method can generate cross-modality paired-images. With them, we can directly perform instancelevel alignment by minimizing the distances between the two images in a feature space. The instance-level alignment can further reduce the modality-gap and avoid misalignment of instances.
Specifically, as shown in Figure 3, our proposed method consists of a generation module G to generate crossmodality paired-images and a feature alignment module F to learn both set-level and instance-level aligned features. The generation module G includes three encoders and two generators. The three encoders disentangle a RGB(IR) image to modality-invariant and RGB(IR) modalities-specific features. Then, the RGB(IR) decoder takes a modalityinvariant feature from an IR(RGB) image and a modalityspecific feature from an IR(RGB) image as input. By decoding from the across-feature, we can generate cross-modality paired-images as in Figure 2(c). In the feature alignment module F, we first utilize an encoder whose weights are shared with modality-invariant encoder. It can map images from different modalities into a shared feature space. Thus, set-level modality-gap can be significantly reduced. Then, we further import an encoder to refine the features to reduce the instance-level modality-gap by minimizing distance between feature maps of every pair of cross-modality images. Finally, by jointly training the generation module G and feature alignment module F with the re-id loss, we can learn both modality-aligned and identity-discriminative features.
The major contributions of this work can be summarized as follows. (1) We propose a novel method to generate cross-modality paired-images by disentangling features and decoding from exchanged features. To the best of our knowledge, it is the first work to generate cross-modality paired-images for the RGB-IR Re-ID task. (2) Our method can simultaneously and effectively reduce both set-level and instance-level modality-variation. (3) Extensive experimental results on two standard benchmarks demonstrate that the proposed model performs favourably against state-of-the-art methods.
Related Works
RGB-RGB Person Re-Identification. RGB-RGB person re-identification addresses the problem of matching pedestrian RGB images across disjoint visible cameras (Gong et al. 2014). Recently, many deep ReID methods (Zheng, Yang, and Hauptmann 2016;Hermans, Beyer, and Leibe 2017;Wang et al. 2019a) have been proposed. Zheng et al. (Zheng, Yang, and Hauptmann 2016) learn identity-discriminative features by fine-tuning a pre-trained CNN to minimize a classification loss. In (Hermans, Beyer, and Leibe 2017), Hermans et al. show that using a variant of the triplet loss outperforms most other published methods by a large margin. Most of exiting methods focus on the RGB-RGB Re-ID task, and cannot perform well for the RGB-IR Re-ID task, which limits the applicability in practical surveillance scenarios.
RGB-IR Person Re-Identification. RGB-IR Person reidentification attempts to match RGB and IR images of a person under disjoint cameras. Besides the difficulties of RGB-RGB Re-ID, RGB-IR Re-ID faces a new challenge due to cross-modality variation between RGB and IR images. In (Wu et al. 2017), Wu et al. collect a cross-modality RGB-IR dataset named SYSU RGB-IR Re-ID and explores three different network structures with zero-padding for automatically evolve domain-specific nodes in the network. Ye et al. utilize a dual-path network with a bi-directional dualconstrained top-ranking loss (Ye et al. 2018a) and modalityspecific and modality-shared metrics (Ye et al. 2018b). In (Dai et al. 2018), Dai et al. introduce a cross-modality generative adversarial network (cmGAN) to reduce the distribution divergence of RGB and IR features. Hao et al. (Hao et al. 2019) achieve visible thermal person re-identification via a hyper-sphere manifold embedding model. In (Wang et al. 2019b) and (Wang et al. 2019c), they reduce modalitygap in both image and feature domains. Most above methods mainly focus on global set-level alignment between the entire RGB and IR sets, which may lead to misalignment of some instances. Different from them, our proposed method performs both global set-level and fine-grained instancelevel alignment, and achieves better performance.
Person Re-Identification with GAN. Recently, many methods attempt to utilize GAN to generate training samples for improving Re-ID. Zheng et al. (Zheng, Zheng, and Yang 2017) use a GAN model to generate unlabeled images as data augmentation. Zhong et al. (Zhong et al. 2018) translate images to different camera styles with CycleGAN , and then use both real and generated images to reduce inter-camera variation. Ma et al. (Ma et al. 2018) use a cGAN to generate pedestrian images with different poses to learn features free of influences of pose variation. All those methods focus on single-modality RGB Re-ID and cannot deal with cross-modality RGB-IR Re-ID. Different from them, ours generate cross-modality paired-images and learn both set-level and instance-level aligned features.
Image Translation. Generative Adversarial Network (GAN) (Goodfellow et al. 2014) learns data distribution in a self-supervised way via the adversarial training, which has been widely used in image translation. Pix2Pix ) solves the image translation by utilizing a conditional generative adversarial network and a reconstruction loss supervised by paired data. CycleGAN ) and StarGAN (Choi et al. 2018) learn images translations with unpaired data using cycle-consistency loss. Those methods only learn one-to-one mapping among different modalities and cannot be used in RGB-IR Re-ID, where the mapping from IR to RGB is one-to-many. Different from them, our method first disentangles images to modality-invariant and modality-specific features, and then generates crossmodality paired-images by decoding from exchanged fea-tures.
The Proposed Method
Our method includes a generation module G to generate cross-modality paired-images and a feature alignment module F to learn both global set-level and fine-grained instance-level aligned features. Finally, by training the two modules with re-id loss, we can learn both modality-aligned and identity-discriminative features.
Cross-Modality Paired-Images Generation Module
As shown in Figure 2(b), in RGB-IR task, the training images from two modalities are unpaired, which makes it more difficult to reduce the gap between the RGB and IR modalities. To solve the problem, we propose to generate pairedimages by disentangling features and decoding from exchanged features. We suppose that images can be decomposed to modality-invariant and modality-specific features. Here, the former includes content information such as pose, gender, clothing category and carrying, etc. Oppositely, the latter has style information such as clothing/shoes colors, texture, etc. Thus, given unpaired-images, by disentangling and exchanging their style information, we can generate paired-images, where the two images have the same content information such as pose and view but with different style information such as clothing colors.
Features Disentanglement. We disentangle features with three encoders. The three encoders are the modalityinvariant encoder E i of learning content information from both modalities, the RGB modality-specific encoder E s rgb of learning RGB style information, and the IR modalityspecific encoder E s ir of learning IR style information. Given RGB images X rgb and IR images X ir , their modalityspecific features M s rgb and M s ir can be learned in Eq.
(2). Similarly, their modality-invariant features M i rgb and M i ir can be learned in Eq. (1).
M s rgb = E s rgb (X rgb ), M s ir = E s ir (X ir ) (1) M i rgb = E i (X rgb ), M i ir = E i (X ir )
(2) Paired-Images Generation. We generate paired-images using two decoders including a RGB decoder D rgb of generating RGB images and an IR decoder D ir of generating IR images. After getting the disentangled features in Eq.(1) and Eq.(2), we can generate paired-images by exchanging their style information. Specifically, to generate RGB images X ir2rgb paired with real IR images X ir , we can use the content features M i ir from the real IR images X ir and the style features M s rgb from the real RGB images X rgb . By doing so, the generated images will contain content information from the IR images and style information from the RGB image. Similarly, we can also generate fake IR images X rgb2ir paired with real RGB images X rgb . Note that to ensure that the generated images have the same identities with their original ones, we only exchange features intra-person. This processes can be formulated in Eq.(3). Figure 3: Our proposed framework consists of a cross-modality paired-images generation module G and a feature alignment module F. G first disentangle images to modality-specific and modality-invariant features, and then decode from the exchanged features. F first use the modality-invariant encoder to perform set-level alignment, then further perform instance-level alignment by minimizing distance of each pair images. Finally, by training the two modules with re-id loss, we can learn both modalityaligned and identity-discriminative features.
X ir2rgb = D ir (M i ir , M s rgb ), X rgb2ir = D rgb (M i rgb , M s ir )(3)
Reconstruction Loss. A simple supervision is to force the disentangled features to reconstruct their original images. Thus, we can formulate the reconstruction loss L recon as below, where || · || 1 is L1 distance.
L recon = ||X rgb − D rgb (E i (X rgb ), E s rgb (X rgb ))|| 1 + ||X ir − D ir (E i (X ir ), E s ir (X ir ))|| 1(4)
Cycle-Consistency Loss. The reconstruction loss L recon in Eq.(4) cannot supervise the cross-modality paired-images generation, and the generated images may not contain the expired content and style information. For example, when translating IR images X ir to its RGB version X ir2rgb via Eq(3), the translated images X ir2rgb may not keep the poses (content information) from X ir , or don't have the right clothing color (style information) with X rgb . This is not the case we want and will harm the feature learning module. Inspired by CycleGAN ), we introduce a cycle-consistency loss to guarantee that the generated images can be translated back to their original version. By doing so, the consistency loss further limits the space of the generated samples. The cycle-consistency loss can be formulated as below:
L cyc = ||X rgb − X rgb2ir2rgb || 1 + ||X ir − X ir2rgb2ir || 1 (5)
where X ir2rgb2ir and X rgb2ir2rgb are the cyclereconstructed images as in Eq.(6).
X ir2rgb2ir = D ir (E i rgb (X ir2rgb ), E s ir (X rgb2ir )) X rgb2ir2rgb = D rgb (E i ir (X rgb2ir ), E s rgb (X ir2rgb ))(6)
GAN loss. The reconstruction loss L recon and cycleconsistency loss L cyc lead to blurry images. To make the generated images more realistic, we apply the adversarial loss (Goodfellow et al. 2014) on both modalities, which have been proved to be effective in image generation tasks ). Specifically, we import two discriminators Dis rgb and Dis ir to distinguish real images from the generated ones on RGB and IR modalities, respectively. In contrast, the encoders and decoders aim to make the generated images indistinguishable. The GAN loss can be formulated as below:
L gan =E[logDis rgb (X rgb ) + log(1 − Dis rgb (X ir2rgb ))] +E[logDis ir (X ir ) + log(1 − Dis ir (X rgb2ir ))](7)
Feature Alignment Module Set-Level Feature Alignment. To reduce the modality-gap, most methods attempt to learn a shared feature-space for different modalities by using dual path (Ye et al. 2018a;2018b), or GAN loss (Dai et al. 2018). However, those methods do not explicitly remove the modality-specific information, which may be encoded into the shared feature-space and harms the performance (Chang et al. 2019). In our method, we utilize a set-level encoder E sl to learn set-level aligned features. The weights E sl are shared with the modalityinvariant encoder E i . As we can see, in the cross-modality paired-images generation module, our modality-invariant encoder E i is trained to explicitly remove modality-specific features. Thus, given images X from any modality, we can learn their set-level aligned features M = E sl (X). Instance-Level Feature Alignment. Even so, as we discuss in the introduction, only performing global set-level alignment between the entire RGB and IR sets may lead to misalignment of some instances. To overcome this problem, we propose to perform instance-level alignment by using the cross-modality paired-images generated by the generation module. Specifically, we first utilize instance-level encoder E il to map the set-level aligned features M to a new feature space T , i.e. T = E il (M ). Then, based on the feature space T , we align every two cross-modality paired-images by minimizing their Kullback-Leibler Divergence. Thus, the loss of the instance-level feature alignment can be formulated in Eq. (8).
L align = E (x1,x2)∈(Xir,X ir2rgb ) [KL(p 1 ||p 2 )] + E (x1,x2)∈(X rgb2ir ,X rgb ) [KL(p 1 ||p 2 )](8)
where p 1 = C(t 1 ) and p 2 = C(t 2 ) are the predicted probabilities of x 1 and x 2 on all identities, t 1 and t 2 are the features of x 1 and x 2 in the feature space T , C is a classifier implemented with a fully-connected layer. Identity-Discriminative Feature Learning. To overcome the intra-modality variation, following (Zheng, Yang, and Hauptmann 2016;Hermans, Beyer, and Leibe 2017), we averagely pool the feature maps T in instance-level aligned space T to corresponding feature vectors V . Given real images X, we optimize their feature vectors V with a classification loss L cls of a classifier C and a triplet loss L triplet .
L cls = E v∈V (−log p(v)) (9) L triplet = E v∈V [m − D va,vp + D va,vn ] + (10) where p(·)
is the predicted probability predicted by the classifier C that the input feature vector belongs to the groundtruth, v a and v p are a positive pair of feature vectors belonging to the same person, v a and v n are a negative pair of feature vectors belonging to different persons, m is a margin parameter and [x] + = max(0, x).
Overall Objective Function and Test
Thus, the overall objective function of our method can formulated as below: L = λ cyc L cyc + λ gan L gan + λ align L align + λ reid (L cls + L triplet )
where λ * are weights of corresponding terms. Following , we set λ cyc = 10 and λ gan = 1. λ reid is set 1 empirically and λ align is decided by grid search. During the test stage, only feature learning module F is used. Given images X, we use the set-level alignment encoder E sl and the instance-level encoder E il to extract features, i.e. V = E il ((E sl (X)). Finally, matching is conducted by computing cosine similarities of feature vectors V between the probe images and gallery ones.
Experiment Dataset and Evaluation Protocol
Dataset. We evaluate our model on two standard benchmarks including SYSU-MM01 and RegDB. (1) SYSU-MM01 (Wu et al. 2017) is a popular RGB-IR Re-ID dataset, which includes 491 identities from 4 RGB cameras and 2 IR ones. The training set contains 19,659 RGB images and 12,792 IR images of 395 persons and the test set contains 96 persons. Following (Wu et al. 2017), there are two test modes, i.e. all-search mode and indoor-search mode. For the all-search mode, all images are used. For the indoor-search mode, only indoor images from 1st, 2nd, 3rd, 6th cameras are used. For both modes, the single-shot and multi-shot settings are adopted, where 1 or 10 images of a person are randomly selected to form the gallery set. Both modes use IR images as probe set and RGB images as gallery set. (2) RegDB (Nguyen et al. 2017) contains 412 persons, where each person has 10 images from a visible camera and 10 images from a thermal camera. Evaluation Protocols. The Cumulative Matching Characteristic (CMC) and mean average precision (mAP) are used as evaluation metrics. Following (Wu et al. 2017), the results of SYSU-MM01 are evaluated with official code based on the average of 10 times repeated random split of gallery and probe set. Following (Ye et al. 2018a;2018b), the results of RegDB are based on the average of 10 times repeated random split of training and testing sets.
Implementation Details
In generation module G, following (Radford, Metz, and Chintala 2016), we construct our modality-specific encoders with 2 strided convolutional layers followed by a global average pooling layer and a fully connected layer. For decoders, following (Wang et al. 2017), we use 4 residual blocks with Adaptive Instance Normalization (AdaIN) and 2 upsampling with convolutional layers. Here, the parameters of AdaIN are dynamically generated by the modalityspecific features. In GAN loss, we use discriminator and LS-GAN as in (Mao et al. 2016) to stable the training.
In feature learning module F, for a fair comparison, we adopt the ResNet-50 (He et al. 2016) pre-trained with Im-ageNet (Russakovsky et al. 2015) as our CNN backbone. Specifically, we use the first two layers of the ResNet-50 as our set-level encoder E sl , and use the remaining layers as our instance-level encoder E il . For the classification loss, the classifier C takes the feature vectors V as inputs, followed by a batch normalization, a fully-connected layer and a soft-max layer to predict the inputs' labels.
Please see our code 1 for more implementation details.
Comparision with State-of-the-arts
Results on SYSU-MM01 Datasets. We compare our model with 10 methods including hand-crafted features (HOG (Dalal and Triggs Table 1. Firstly, LOMO only achieves 3.64% and 4.53% in terms of Rank-1 and mAP scores, respectively, which shows that hand-crafted features cannot be generalized to the RGB-IR Re-ID task. Secondly, One-Stream, Two-Stream and Zero-Padding significantly outperform hand-crafted features by at least 8% and 8.3% in terms of Rank-1 and mAP scores, respectively. This verifies Table 1: Comparison with the state-of-the-arts on SYSU-MM01 dataset. The R1, R10, R20 denote Rank-1, Rank-10 and Rank-20 accuracies (%), respectively. The mAP denotes mean average precision score (%). that the classification loss contributes to learning identitydiscriminative features. Thirdly, BCTR and BDTR further improve Zero-Padding by 1.4% in terms of Rank-1 and by 3.2% in terms of mAP scores. This shows that the ranking and classification losses are complementary. Additionally, D-HSME outperforms BDTR by 3.6% Rank-1 and 3.5% mAP scores, which demonstrates the effectiveness of metric learning. In addition, D 2 RL outperform D-HSME by 8.1% Rank1 and 6.0% mAP scores, implying the effectiveness of adversarial training. Finally, Our method outperforms the state-of-the-art method by 9.2% and 7.7% in terms of Rank-1 and mAP scores, showing the effectiveness of our model.
Methods
All-Search
Results on SYSU-RegDB Dataset. We evaluate our model on RegDB dataset and compare it with Zero-Padding (Wu et al. 2017), TONE (Ye et al. 2018b), BCTR (Ye et al. 2018a), BDTR (Ye et al. 2018b) and D 2 RL (Wang et al. 2019c). We adopt visible2thermal and thermal2visible modes. Here, the visible2thermal means that visible images are query set and thermal images are gallery set, and so on. As shown in Table 2, our model can significantly outperform the state-of-the-arts by 4.7% and 5.1% in terms of Rank-1 scores with thermal2visible and visible2thermal modes, respectively. Overall, the results verify the effectiveness of our model.
Model Analysis
Ablation Study. To further analyze effectiveness of the setlevel alignment and the instance-level alignment, we evaluate our method under four different settings, i.e. with or without set-level (SL) and instance-level (IL) alignment. Specifically, when removing set-level alignment, we use separate set-level encoder E sl , i.e. we don't share weights of set-level encoder E sl with modality-invariant encoder E i . When removing instance-level alignment, we set λ align = 0. Moreover, to analyze whether the feature disentanglement contributes to set-level alignment, we remove the disentanglement strategy by using separate set-level encoder E sl and training it with a GAN loss as in (Dai et al. 2018). As shown in Table 3, when removing both SL and IL (index-1), our method only achieve 32.1% Rank-1 score. By adding SL (index-2) or IL (index-3), the performance is improved to 35.1% and 36.0% Rank-1 score, which demonstrate the effectiveness of both SL and IL. When using both SL and IL (index-4), our method achieves the best performance at 38.1% Rank-1 score, which demonstrates that SL and IL can be complementary with each other. Finally, when removing the disentanglement from set-level alignment (index-5), Rank-1 score drops by 1.3%. This illustrates that disentanglement is helpful for set-level alignment.
To better understand set-level alignment (SL) and instance-level alignment (IL), we visualize the distribution of intra-person similarity and inter-person similarity under different variants. The similarity is calculated with cosine distance. Firstly, when comparing with Figure 4 ure 4(b), we can find that even using no SL and IL, model can easily fit training set, while fails to generalize to test set. As we can see in Figure 4(b), the two kind of similarities are seriously overlapped. This shows that the crossmodality variation cannot be well reduced by simply fitting identity information in training set. Secondly, in Figure 4(c), we find that although the similarity of intra-person becomes more concentrated, the similarity of inter-person also become larger. This shows that SL imports some misalignment of instances which may harm the performance. Finally, in Figure 4(c) we can see that, IL boosts intra-person similarity, meanwhile keeps the inter-person similarity unchanged. In summary, experimental results and analysis above show the importance and effectiveness of instance-level alignment.
Parameters Analysis. We evaluate the effect of the weights, i.e. λ align . As shown in Figure 5, we analyze our method with respect to the λ align on SYSU-MM01 dataset under single-shot&all-search mode. We can see that, with different λ align , our method can stably have an significant improvement. The experimental results show that our method is robust to different weights. Figure 6: Comparision among generated images from ours, CycleGAN and StarGAN (Choi et al. 2018). Ours can stably generate paired-images with given real ones, while CycleGAN and StarGAN fail.
Visualization of Images
In this part, we display the generated cross-modality pairedimages from ours, CycleGAN ) and Star-GAN (Choi et al. 2018). From Figure 6(a), we can see that, images of a person in the two modalities are significant different, even human beings cannot easily identify them. In Figure 6(b), our method can stably generate fake images when given cross-modality unpaired-images from a person. For example, in person A, ours can translate her IR images to RGB version with right colors (yellow upper and black bottom clothes). However, in Figure 6(c) and Figure 6(d), CycleGAN and StarGAN cannot learn the right colors even poses. For example, person B should have blue upper clothing. However, images generated by CycleGAN and Star-GAN are red and black, respectively. Those unsatisfying images cannot be used to learn instance-level aligned features.
Conclusion
In this paper, we propose a novel Joint Set-Level and Instance-Level Alignment Re-ID (JSIA-ReID). On the one hand, our model performs set-level alignment by disentangling modality-specific and modality-invariant features. Compared with vanilla methods, ours can explicitly remove the modality-specific information and significantly reduce the modality-gap. On the other hand, given cross-modality unpaired images, we we can generate cross-modality pairedimages by exchanging their features. With the pairedimages, instance-level variations can be reduced by minimizing the distances between every pair of images. Experimental results on two datasets show the effectiveness of our proposed method.
2005), LOMO (Liao et al. 2015)), feature learning with the classification loss (One-Stream, Two-Stream, Zero-Padding) (Wu et al. 2017), feature learning with both classification and ranking losses (BCTR, BDTR) (Ye et al. 2018a), metric learning (D-HSME (Hao et al. 2019)), and reducing distribution divergence of features (cmGAN (Dai et al. 2018), D 2 RL (Wang et al. 2019c)). The results are shown in
Figure 4 :
4Distribution of cross-modality similarities of intra-person and inter-person. The instance-level alignment (IL) can enhance intra-person similarity while keep interperson similarity unchanged, which improves performance. w/ means with and w/o means without.
Figure 5 :
5Rank-1 and mAP scores with different λ align on SYSU-MM01 under single-shot&all-search mode.
Table 2 :
2Comparison with state-of-the-arts on the RegDB
dataset under different query settings.
Methods
thermal2visible visible2thermal
Rank-1 mAP Rank-1 mAP
Zero-Padding
16.7
17.9
17.8
31.9
TONE
21.7
22.3
24.4
20.1
BCTR
-
-
32.7
31.0
BDTR
32.8
31.2
33.5
31.9
D 2 RL
43.4
44.1
43.4
44.1
Ours
48.1
48.9
48.5
49.3
Table 3 :
3Analysis of set-level (SL) and instance-level (IL) alignment. Please see text for more details.index SL IL R1
R10 R20 mAP
1
× × 32.1 75.7 87.0 31.9
2
× 35.1 78.6 88.2 33.8
3
×
36.0 79.8 89.0 35.5
4
38.1 80.7 89.9 36.9
5
-
36.8 80.2 89.4 36.0
https://github.com/wangguanan/JSIA-ReID
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|
[
"https://github.com/wangguanan/JSIA-ReID.",
"https://github.com/wangguanan/JSIA-ReID"
] |
[
"Spin chains and classical strings in rotating Rindler-AdS space",
"Spin chains and classical strings in rotating Rindler-AdS space"
] |
[
"Shou-Huang Dai \nLeung Center for Cosmology and Particle Astrophysics\nNational Taiwan University\n106TaipeiTaiwan\n",
"Shogo Kuwakino \nDepartment of Physics and Center for High Energy Physics\nChung Yuan Christian University\nChung Li CityTaiwan\n",
"Wen-Yu Wen \nLeung Center for Cosmology and Particle Astrophysics\nNational Taiwan University\n106TaipeiTaiwan\n\nDepartment of Physics and Center for High Energy Physics\nChung Yuan Christian University\nChung Li CityTaiwan\n"
] |
[
"Leung Center for Cosmology and Particle Astrophysics\nNational Taiwan University\n106TaipeiTaiwan",
"Department of Physics and Center for High Energy Physics\nChung Yuan Christian University\nChung Li CityTaiwan",
"Leung Center for Cosmology and Particle Astrophysics\nNational Taiwan University\n106TaipeiTaiwan",
"Department of Physics and Center for High Energy Physics\nChung Yuan Christian University\nChung Li CityTaiwan"
] |
[] |
In this paper, we study the spin chain and string excitation in the rotating Rindler-AdS 3 proposed in[12]. We obtain a one-parameter deformed SL(2) spin chain at the fast spin limit.Two-spin GKP-like solutions are studied at short and long string limits. General ansatz for giant magnons and spiky strings are analyzed in detail for various β. At last, we explore its counterpart in analytic continuation and pp-wave limit.
|
10.1007/jhep04(2014)018
|
[
"https://arxiv.org/pdf/1401.6915v1.pdf"
] | 19,935,724 |
1401.6915
|
9ae5b49c3def05d2e94d6ae3f9fc408754b54984
|
Spin chains and classical strings in rotating Rindler-AdS space
27 Jan 2014
Shou-Huang Dai
Leung Center for Cosmology and Particle Astrophysics
National Taiwan University
106TaipeiTaiwan
Shogo Kuwakino
Department of Physics and Center for High Energy Physics
Chung Yuan Christian University
Chung Li CityTaiwan
Wen-Yu Wen
Leung Center for Cosmology and Particle Astrophysics
National Taiwan University
106TaipeiTaiwan
Department of Physics and Center for High Energy Physics
Chung Yuan Christian University
Chung Li CityTaiwan
Spin chains and classical strings in rotating Rindler-AdS space
27 Jan 2014* Electronic address:
In this paper, we study the spin chain and string excitation in the rotating Rindler-AdS 3 proposed in[12]. We obtain a one-parameter deformed SL(2) spin chain at the fast spin limit.Two-spin GKP-like solutions are studied at short and long string limits. General ansatz for giant magnons and spiky strings are analyzed in detail for various β. At last, we explore its counterpart in analytic continuation and pp-wave limit.
I. INTRODUCTION AND SUMMARY
AdS/CFT correspondence [1][2][3] opens a new window for us to understand the fundamental physics of the nature. This conjecture had been tested from many aspects.
One of them is the duality between the integrable spin chains arising from the single trace operators in N = 4 super Yang-Mills (SYM) theory [4] and the rotating strings in AdS 5 × S 5 [5][6][7]. This duality can be checked by the agreement between the spin chain effective action and the rotating string sigma model action [6,7], and from the view point of integrability such that the Bethe equation for the spin chain and that for the classical string sigma model in AdS 5 × S 5 are equivalent [8]. The spiky string solution dual to the giant magnon excitation of the spin chain are also constructed [9,10].
One may wonder the possibility of generalizing the AdS/CFT correspondence to the non-AdS/non-CFT cases. A way to investigate this problem is to consider deformations on both sides and see how the correspondence works. One example in the context of the super Yang-Mills spin chain is the β-deformed N = 4 SYM and its gravity dual [11]. In contrast to [11] where a smooth deformation from S 5 in the original AdS 5 ×S 5 background is accounted for, in this paper we explore the spin chain dual solution in a gravitational background which switches to another conjugacy class of AdS 3 once the deformation is turned on.
One-parameter stationary vacua 1 in AdS 3 belonging to the loxodromic conjugacy class of the Lorentz group (i.e. rota-boosts) were constructed in [12] by twisting the Lorentz group in the embedding flat space. In particular, a combination of two boosts M 01 −βM 23
(where x 0 and x 3 are time-like coordinates) gives rise to the rotating Rindler AdS space, which is the universal cover for the BTZ black hole but with less symmetry. There is an event horizon and an ergosphere due to this rota-boost. As the deformation (or rotation) parameter β vanishes, it reduces to the Rindler spacetime, with the acceleration identified as the inverse of the AdS radius, a ∼ L −1 . On the other hand, a combination of a temporal and a spatial rotations M 03 − βM 12 gives rise to the rotating global AdS, where some region around the center is hidden from a co-rotating observer. As β vanishes, it reduces to the AdS in global coordinates. Although the boundary theory dual to the rotating global AdS still satisfies the Virasoro algebra, the conformal symmetry in static vacuum is broken for uneven deformation in the left and the right sectors. While the computation power seems out of control on the field theory side due to lost symmetry, one is hoping that computation from gravity side is still tractable either analytically (at certain limits) or numerically. As β vanishes, the rotating Rindler AdS 3 falls in the conjugacy class of hyperbolic transformations (i.e. pure boosts), while the global rotating AdS 3 switches to the class of elliptic transformations (i.e. pure rotations). Note that the rota-boosts cannot reduce to the pure boosts or the pure rotations by Lorentz transformations.
The goal of this paper is to study the rotating global AdS vacuum by probing a classical string and observe the effect of deformation to the string excitation and dispersion. We solve the classical string solutions dual to the spin chain in the β-vacua of global rotating AdS 3 × S 1 embedded in AdS 5 × S 5 . At certain limits, we can also obtain the analytic expressions for spin chain model and dispersion relation. Their implications on the dual field theory side, however, remains to be investigated.
The structure of this paper is outlined as follows. In Section II, we derive a deformed SL(2) spin chain Hamiltonian for a fast spinning string, and some simple excitation is examined for nonzero deformation. In Section III, we first study the dispersion relation for GKP string at different limits, and then the general solutions for giant magnons and spiky strings are analyzed in detail. In Section IV, we study the analytically continued version of the geometry and its sin(h)-Gordon model. In Section V, the dispersion relation for a spiky string is studied in the pp-wave limit.
II. A SPIN CHAIN FROM ADS β-VACUA
It was shown in [6] that in the fast spinning limit, one was able to obtain the Heisenberg spin chain by using the sigma model approach, which agrees with the one-loop calculation of anomalous dimensions in N = 4 super Yang-Mills. Although this quantity is no longer protected by the symmetry in the deformed theory, one can still study the effect of deformation to the spin chain Hamiltonian and dynamics from gravity side. In order to reach sensible spinning limit, we will include additional circle in the background metric.
This circular dimension is easily obtained from dimensional reduction, say from type IIB theory in ten dimensions to AdS 3 × S 3 × T 4 . After deforming to the rotating global AdS 3 [12] , the global metrics reads:
ds 2 = −((1 − β 2 ) cosh 2 ρ + β 2 )dt 2 + 2βdtdθ + dρ 2 + ((1 − β 2 ) cosh 2 ρ − 1)dθ 2 + dφ 2 (1)
Note that in this coordinate, cosh 2 ρ < 1 1−β 2 is inaccessible due to the deformation, since in this region the rotation generator ∂ ∂θ becomes time-like. Now we would like to show that at the fast spinning limit, a spin chain Hamiltonian can be obtained from the string worldsheet. Without loss of generality, we apply the following embedded ansatz:
t = κτ, ρ = ρ(τ, σ), θ = θ(τ, σ), φ = φ(τ, σ),(2)
where (τ, σ) are worldsheet coordinates. A change of coordinates
θ → θ + t, φ → φ + (1 − β)t, ρ → 1 2 ρ,(3)
brings the Polyakov action into 2
S = √ λ 4π dτ dσ{ 1 4 (ρ 2 − ρ ′2 ) + [(1 − β 2 ) cosh 2 ρ 2 − 1](θ 2 − θ ′2 ) + 2[(1 − β 2 ) cosh 2 ρ 2 −(1 − β)]κθ + (φ 2 − φ ′2 ) + 2(1 − β)κφ},(4)
where we use X ′ to denote the derivative of X with respect to σ andẊ with respect to τ . Then we take fast spinning limit [6], by sending κ → ∞ andẊ µ → 0, such that κẊ µ remains finite. After taking the limit, the above action simplifies as
S = √ λ 4π dτ dσ{2[(1−β 2 ) cosh 2 ρ 2 −(1−β)]κθ+2(1−β)κφ− 1 4 ρ ′2 −[(1−β 2 ) cosh 2 ρ 2 −1]θ ′2 −φ ′2 }.(5)
Taking account of the Virasoro constraint: G µνẊ µ X ν ′ = 0, that is
[(1 − β 2 ) cosh 2 ρ 2 − (1 − β)]θ ′ + (1 − β)φ ′ = 0,(6)
one reaches the classical action of spin chain
S = √ λ 4π dtdσ{(1 − β)[(1 + β) cosh ρ − (1 − β)]∂ t θ + 2(1 − β)∂ t φ −λ 2L 2 H},(7)
where we defineλ ≡ λ 8π 2 and L ≡ κ √ λ 2π , and the spin chain Hamiltonian density reads 2 The general form of the equations of motion and the Virasoro constraints arising from the Polyakov action is presented in the Appendix.
H = (∂ σ ρ) 2 + (1 + β) 2 sinh 2 ρ(∂ σ θ) 2 .(8)
This will reduce to SL(2) XXX spin chain at the limit β → 0 as expected [7].
To illustrate the effect of deformation, we will examine a rotated string solution against the deformed spin chain action (7). First, the equations of motion are derived:
(1 − β 2 )∂ t cosh ρ −λ L 2 (1 + β) 2 ∂ σ (sinh 2 ρ∂ σ θ) = 0, (1 − β 2 ) sinh ρ∂ t θ +λ L 2 [∂ 2 σ ρ − 1 2 (1 + β) 2 sinh 2ρ(∂ σ θ) 2 ] = 0.(9)
The simplest solution is obtained for θ = ωt and ρ = ρ(σ). Then the equation of motion for ρ can be integrated to give
∂ σ ρ = ± a − b cosh ρ, b = 2L 2 λ (1 − β 2 )ω,(10)
for some constant a. This solution describes that a folded string stretching between ρ = ±ρ max = cosh −1 a b rotates in uniform speed at the center of AdS 3 . The total energy E and spin S, defined as follows, can be written in terms of complete elliptic integral of the first kind K(x) and second kind E(x):
E =λ 4L dσ(∂ σ ρ) 2 = −2 √ 2bλ L {E(x) − (1 − x)K(x)}, S = L 2 dσ(1 − β 2 ) cosh ρ = 2(1 − β 2 )L 2 b {2E(x) − K(x)},(11)
where x = b−a 2b . We plot the energy and spin against ω for several β's in the Figure (1) and find out that deformation increases the energy but slows down the spin.
Several comments are in order: first, the horizon censoring the AdS center seems boosted away in the fast spin limit such that the folded string is able to pass through ρ = 0. Secondly, the equation (10) implies that the apparent string tension is enhanced by a factor (1 −β 2 ) −1/2 . We recall in the earlier studies of turning on the NSNS field B for spinning string in S 3 [13,14], the apparent tension is reduced by a factor (1 − B 2 ) 1/2 . This might be some kind of electric-magnetic duality or strong-weak duality between the metric component G tθ and its analytic continued counterpart G ′ ϕ 1 ϕ 2 , which acts as a nontrivial B ϕ 1 ϕ 2 on the string worldsheet 3 .
III. TWO-SPIN STRINGS
There are three U(1)'s found in the isometry SO(2, 2) × SO(2) for the target space (1).
While a U(1) is identified to the global time generator, one can still turn on maximal two spins charged under the remaining U(1) 2 . In the following sections, we will first study the spinning closed string solution to understand its leading Regge trajectory behavior at short and long string limits. Then we will construct general ansatz for giant magnons and spiky strings and obtain their dispersion relations.
GKP solution
First we consider a two-spin classical solution in β-deformed AdS 3 × S 1 background 4 .
We apply the following ansatz to the metric (1):
t = κτ, ρ = ρ(σ), θ = ωτ, φ = χτ,(12)
where κ, ω and χ are integers. In the conformal gauge, the Polyakov action leads to
S = √ λ 4π dτ dσ −[(1 − β 2 ) cosh 2 ρ + β 2 ]κ 2 + 2βκω − (∂ σ ρ) 2 +[(1 − β 2 ) sinh 2 ρ − β 2 ] ω 2 + χ 2 .(13)
From the Virasoro constraint, we get the following equation
dρ dσ = ± {(κ − βω) 2 − χ 2 } cosh 2 ρ − {(ω − βκ) 2 − χ 2 } sinh 2 ρ.(14)
We will present the solution later. For now, by integrating (14), we obtain the following
relation 2π = 4 ρmax 0 dρ {(κ − βω) 2 − χ 2 } cosh 2 ρ − {(ω − βκ) 2 − χ 2 } sinh 2 ρ = 4 (κ − βω) 2 − χ 2 1 √ 1 + η K( 1 √ 1 + η ),(15)
here ρ max and η are defined as
coth 2 ρ max = (ω − βκ) 2 − χ 2 (κ − βω) 2 − χ 2 = 1 + η.(16)
We calculate three conserved quantities of this system, the energy E and two angular momenta S and J which are associated to the coordinates θ and φ respectively
E = −P t = √ λ 2π 2π 0 κ (1 − β 2 ) cosh 2 ρ + β 2 − βω dσ,(17)S = P θ = √ λ 2π 2π 0 ω((1 − β 2 ) sinh 2 ρ − β 2 ) + βκ dσ,(18)J = P φ = √ λχ.(19)
Using (14), the energy E and the spin S can be evaluated in terms of the elliptic integral
as E = 2 √ λ π 1 (κ − βω) 2 − χ 2 κ(1 − β 2 ) √ 1 + η η E( 1 √ 1 + η ) − β(ω − βκ) √ 1 + η K( 1 √ 1 + η ) ,(20)S = 2 √ λ π 1 (κ − βω) 2 − χ 2 ω(1 − β 2 ) √ 1 + η η E( 1 √ 1 + η ) − ω − βκ √ 1 + η K( 1 √ 1 + η ) .(21)
In the following we consider the short string limit ρ max ≪ 1 and the long string limit ρ max ≫ 1, and evaluate a relation between the energy E and the spins S and J for each case.
A. Short string limit
First we consider the short string limit ρ max ≪ 1. From (16), this corresponds to the limit η ≫ 1. From (15) and (16), by taking the limit, we get the relations
(κ − βω) 2 ∼ χ 2 + 1 η ,(22)(ω − βκ) 2 ∼ χ 2 + 1 + 1 η .(23)
Also, by a suitable combination of (20) and (21), we obtain a simple expansion relation
as S − βE ∼ √ λ(1 − β 2 ) 2 χ 2 + 1 η .(24)
It is natural to identify E − βS and S − βE as the twisted energy and spin in the rotating global AdS background, because they correspond to generators ∂ t − β∂ θ and ∂ θ − β∂ t respectively. By substituting this, we obtain the relation of the twisted energy and the twisted spin for any value of χ as
E − βS ∼ √ λ(1 − β 2 ) χ 2 + 2 √ λ S − βE (1 − β 2 ) χ 2 + 1 +(S − βE) χ 2 + 2 √ λ S−βE (1−β 2 ) √ χ 2 +1 χ 2 + 1 + 2 √ λ S−βE (1−β 2 ) √ χ 2 +1 .(25)
In order to describe this relation in terms of the spin J, let us first consider the limit of χ ≪ 1. At this limit, (24) leads to
S − βE ∼ √ λ(1 − β 2 ) 2 1 η ,(26)
and (25) is simplified to
E − βS ∼ √ λ(1 − β 2 ) χ 2 + 2 √ λ S − βE (1 − β 2 ) .(27)
Then, using (19), we obtain the relation
E − βS 1 − β 2 2 ∼ J 2 + 2 √ λ S − βE 1 − β 2 + · · · .(28)
Next let us see another limit χ ≫ 1. At this limit, (24) leads to
S − βE ∼ √ λ(1 − β 2 ) 2 χ η ,(29)
and (25) becomes
E − βS ∼ √ λ(1 − β 2 )χ + S − βE + S − βE χ 2 ,(30)
or, using (19), this corresponds to the relation
E − βS ∼ (1 − β 2 )J + (S − βE) − 1 2 λ J 2 (S − βE) + · · · .(31)
In the case of β = 0, this result reduces to (3.24) in [17] as expected.
B. Long string limit
Next we consider the case of the long string limit ρ max ≫ 1, i.e. η ≪ 1. From (15) and (16), by taking the limit, we get the relations
(κ − βω) 2 ∼ χ 2 + 1 π 2 ln 2 ( 1 η ),(32)(ω − βκ) 2 ∼ χ 2 + (1 + η) 1 π 2 ln 2 ( 1 η ).(33)
Also, from the expansion of the spin S in (21), we obtain
S ∼ 2(1 + β) √ λ πη χ 2 + 1 π 2 ln 2 1 η 1 π 2 ln 2 1 η .(34)
We find that the spin is large S ≫ 1 for any value of χ. Since it is not easy to spot a dispersion relation in this complicated expression, we consider several limits of χ in the following. First let us consider the case for χ ≪ ln 1 η . For the limit (34) leads to
S ∼ 2 √ λ π (1 + β) η ,(35)
and using (19) we obtain the following relation
E ∼ S + (1 − β) √ λ π ln S √ λ(1 + β) + (1 − β)π 2 √ λ J 2 ln S √ λ(1+β) .(36)
In the case of the opposite limit χ ≫ ln 1 η , (34) leads to
S ∼ 2 √ λ(1 + β) χ η .(37)
From this equation we find that S ≫ χ. Comparing an expansion of the energy (20) with (37), and using (19), we obtain the relation
E ∼ S + (1 − β)J + (1 − β) λ 2π 2 J ln S (1 − β)J 2 ,(38)
which also leads to (3.32) of [17] when β = 0.
Giant magnon/Spiky solution
A. The solution
To describe the 2-spin giant magnon/spiky string solutions, we take the ansatz following [10]:
t = τ + h 1 (y), ρ = ρ(y), θ = ω[τ + h 2 (y)], φ = Ω τ,(39)
where the worldsheet coordinates are changed from (τ, σ) to (τ, y), with y = σ − vτ and 0 < v < 1. In the following, we will set Ω = 1 in our analysis for convenience, and 0 < ω < 1. Then the equations of motion (A.2) and (A.3) are rewritten into the differential equations for h 1 (y), h 2 (y), 5
{vg tt + (1 − v 2 )g tt h ′ 1 + βω(1 − v 2 )h ′ 2 } ′ = 0,(40)
{v
ω g θθ + (1 − v 2 ) ω g θθ h ′ 2 + β(1 − v 2 )h ′ 1 } ′ = 0,(41)
where now ′ stands for d/dy. These equations further reduce to
h ′ 1 = − 1 1 − v 2 βωc 2 + (βωv + c 1 )g θθ − vg tt g θθ (1 − β 2 ) 2 sinh 2 ρ cosh 2 ρ ,(42)h ′ 2 = − 1 1 − v 2 −βc 1 + (βv − ωc 2 )g tt − ωvg tt g θθ ω(1 − β 2 ) 2 sinh 2 ρ cosh 2 ρ ,(43)
where c 1 , c 2 are two integration constants arising from integrating (40), (41). As β = 0,
ρ ′′ = 1 − β 2 (1 − v 2 ) 2 cosh ρ sinh ρ 1 − βωc 2 − (c 1 + βωv + v)β 2 (1 − β 2 ) 2 sinh 2 ρ cosh 2 ρ + c 1 + βωv (1 − β 2 ) cosh 2 ρ 2 −ω 2 1 − −βc 1 /ω + (c 2 − βv/ω − v)β 2 (1 − β 2 ) 2 sinh 2 ρ cosh 2 ρ + c 2 − βv/ω (1 − β 2 ) sinh 2 ρ 2 . (44)
With the string profile ansatz (39), the Virasoro constraints are given by
g tt (1 − vh ′ 1 )h ′ 1 + βω(h ′ 1 + h ′ 2 − 2vh ′ 1 h ′ 2 ) + ω 2 g θθ (1 − vh ′ 2 )h ′ 2 − v(ρ ′ ) 2 = 0,(45)g tt [1 − 2vh ′ 1 + (1 + v 2 )(h ′ 1 ) 2 ] + 2βω[1 − v(h ′ 1 + h ′ 2 ) + (1 + v 2 )h ′ 1 h ′ 2 ] +ω 2 g θθ [1 − 2vh ′ 2 + (1 + v 2 )(h ′ 2 ) 2 ] + (1 + v 2 )(ρ ′ ) 2 + 1 = 0.(46)
Eliminating (ρ ′ ) 2 by equating the LHS of these two equations and substituting in the h ′ 1 , h ′ 2 expressions from (42) and (43), one obtains the following relation
c 1 − ω 2 c 2 + 2βωv + v = 0(47)
for the two integration constants. 5 We set √ λ 2π = 1 in the following numerical analysis of the giant magnon/spiky sting solutions.
In order to proceed to obtain the explicit string solutions, we need to assign specific values to c 1 , c 2 . In this paper we set 6
c 1 = −βωv − v 1 − βω , c 2 = − β 2 v 1 − βω .(49)
This choice yields a constraint
v 2 < 1 − βω (50)
by requiring forward propagation of the strings,
dt dτ = 1 1 − v 2 1 − v 2 (1 − βω)(1 − β 2 ) cosh 2 ρ > 0.(51)
Note that the constraint (50) is regarded as a natural β-deformation from the original v 2 < 1. Moreover, (47) and (49) reduce to the corresponding results in [10] at β = 0.
With the given c 1 ,
c 2 , h ′ 1 , h ′ 2 become h 1 (y) ′ = − v 1 − v 2 1 − 1 (1 − βω)(1 − β 2 ) cosh 2 ρ ,(52)h 2 (y) ′ = − v 1 − v 2 1 − β/ω (1 − βω)(1 − β 2 ) cosh 2 ρ .(53)
Substituting these expressions into the Virasoro constraints, one obtains the differential equation for ρ(y):
ρ ′2 = 1 (1 − v 2 ) 2 (1 − ω 2 )(1 − β 2 ) cosh 2 ρ + v 2 (1 − βω) 2 cosh 2 ρ + (β − ω) 2 − (1 + v 2 ) .(54)
It is straightforward to check that ρ ′2 is indeed an integral of ρ ′′ according to (44) and
(54).
The next step is to solve ρ(y) profile. (54) can be rewritten into 6 As being demonstrated later in this paper, the choice of c 1 , c 2 in (49) gives rise to consistent β = 0 reduction. One may choose other c 1 , c 2 , for example
ρ(y) ′ = ± 1 (1 − v 2 ) cosh ρ f (ρ),(55)f (ρ) = (1 − ω 2 )(1 − β 2 ) cosh 4 ρ + [(β − ω) 2 − (1 + v 2 )] cosh 2 ρ + v 2 (1 − βω) 2 ,c 1 = −v − βωv, c 2 = βv/ω.(48)
But when β is taken to zero, such a choice does not yield the same condition to distinguish the hanging string and the spiky string profiles as the β-free case in [10], despite that β = 0 in (49) and (48) reduce to the c 1 , c 2 choice of [10] . (1), with the corresponding β values, where the strings end. ρ 0 increases with β.
in which one finds ρ ′ → ±∞ as ρ → ∞. In principle, ρ(y) is solved by integrating dy = ±(1 − v 2 )(cosh ρ)dρ/ f (ρ) over some appropriate range on both sides, but the form of f (ρ) leaves ρ(y) no analytic solution. However, it can be solved numerically with the same method.
f (ρ) is a quadratic function in cosh 2 ρ, and analysis shows that within β, ω, v ∈ (0, 1), there always exist two real roots cosh 2 ρ + , cosh 2 ρ − (where ρ + > ρ − ) for f (ρ):
cosh 2 ρ ± = (1 + v 2 ) − (β − ω) 2 ± [(1 + v 2 ) − (β − ω) 2 ] 2 − 4v 2 (1−ω 2 )(1−β 2 ) (1−βω) 2 2(1 − ω 2 )(1 − β 2 ) .(56)
The string solutions only exist for f (y) ≥ 0, i.e. ρ ≥ ρ + and ρ ≤ ρ − . On the other hand, the radial coordinate is physical for ρ ≥ ρ 0 , where cosh 2 ρ 0 = 1 1−β 2 . Therefore the rotating string solutions can be classified by comparing ρ + and ρ 0 :
(1) ρ + > ρ 0 . This type of solution has two subclasses: (a) ρ − ≤ ρ 0 . The strings can only extend between ρ = ρ + and ρ = ∞, but not for ρ 0 ≤ ρ < ρ + . This corresponds to the hanging strings displayed in Fig. 2(a). (b) ρ − > ρ 0 . The string can either be a hanging one like that in Fig. 2(a), or a bulging string confined between ρ 0 ≤ ρ ≤ ρ − in Fig. 2(b). Note that the bulging string solution does not allow β = 0 reduction, and can exist only for some limited range of (β, ω, v). The parameter region for this type of solution is constrained by ρ − > ρ 0 and 1 − βω − v 2 > 0 in (50). (See Fig. 4 for numerical results.)
In this subclass, ρ(y) ′ = 0 at ρ = ρ + or ρ = ρ − corresponds either to the bottom points of the hanging strings, or to the tips of the bulging stings respectively.
(2) ρ + ≤ ρ 0 . The string extends all the way from ρ 0 to the asymptotic infinity, and it is a spiky solution depicted in Fig. 3.
One finds that the range of y for the spiky string becomes finite due to the β- coordinates still have a remaining Weyl symmetry in the Polyakov action, one can rescale σ according to β to get rid of this issue. As for the bulging stings in Fig. 2(b), the length of the string segment increases with β.
The parameter regions of (β, ω, v) ∈ (0, 1) for all three types of the string solutions are shown in Fig. 4. The boundary separating the spiky and the hanging profiles are obtained comparing ρ + and ρ 0 , and as β is infinitely small, it cannot be obtained as a smooth deformation from the profile distinguishing condition at β = 0 derived from (55). The transition from vanishing to non-vanishing β is discontinuous in this aspect. At β = 0, the two roots in for f (ρ) becomes cosh 2 ρ 1 = 1 and cosh 2 ρ 2 = v 2 1−ω 2 , while the origin is at ρ 0 = 0 = ρ 1 . Here ρ 2 can be greater or smaller than ρ 1 . If ρ 2 > 1, i.e. v 2 1−ω 2 > 1, it is a hanging string; otherwise the solution is spiky (for v 2 1−ω 2 ≤ 1), as predicted by [10]. This can be seen in Fig. 4
B. Dispersion relation
The rotating strings in the β-vacua in AdS 3 × S 1 carry the following energy E and spins S, J which associate with θ, φ respectively:
E = √ λ 2π(1 − v 2 ) dσ (1 − β 2 ) cosh 2 ρ + β 2 − βω − v 2 1 − βω ,(57)S = √ λ 2π(1 − v 2 ) dσ ω(1 − β 2 ) sinh 2 ρ − ωβ 2 + β − v 2 β 1 − βω ,(58)J = √ λ 2π dσ.(59)
They satisfy a relation
E − J = S ω + K,(60)
where K is a β-dependent correction term and given by the expression
K = − √ λ 2π(1 − v 2 ) dσ β 2 − βω + β ω 1 − v 2 1 − βω .(61)
K vanishes identically as β = 0, and (60) reduces to E − J = S ω , consistent with [10]. (57)∼(59) diverge as they are integrated to ρ → ∞, and require regularization. However, due to lack of analytic results out of these integrals, we are unable to obtain analytic regularized expressions for E, S and J. We refer to the numerical computation to reveal their dependence on ω, such that the dispersion relation (60) is satisfied. We take a cutoff at Λ = 50 while integrating these quantities over ρ, and denote them by E Λ , S Λ , J Λ , and K Λ . The behaviors of E, S, J against ω remains the same whether they are regularized
ρ → iγ, φ →t, t → ϕ 1 , θ → ϕ 2 ,(62)
v.s. ω for a rotating string FIG. 1: (a) shows the log plot for energy of a rotating string for deformation parameter β = 0(solid black), 0.5(solid gray), 0.7(dashed brown), 0.9(dotted green). (b) shows the spin of a rotating string for the same deformation parameters.
( 42 )
42and (43) reduce to the equations presented in [10]. The equation of motion for ρ in (A.4) becomes
string ρ(y) solutions at various β FIG. 2: (a) shows the hanging string ρ(y) profile at various β values, with ω = 0.8, v = 0.7. As β increases, the width in y decreases; (b) is the bulging string ρ(y) at various β, with ω = 0.7, v = 0.8. The dotted lines are the coordinate origins ρ = ρ 0 of the AdS β-vacua in eqn.
FIG. 3 :
3This figure shows the spiky string solutions ρ(y) at various β, with ω = 0.2, v = 0.4. The dotted lines are the coordinate origins ρ = ρ 0 of the AdS β-vacua for each β.
deformation of the AdS vacua. In fact, the y range of all three types of solutions depends on the value of β. It is revealed in Figures 2(a) and 3 that, for the hanging and the spiky strings, as β increases, the width of ρ(y) decreases, which implies that at constant worldsheet time τ , the range of σ shrinks as β increases. However, since the worldsheetFIG. 4: This figure shows the parameter ranges of (β, ω, v) corresponding to each of the three types of the string solutions. The green surface depicts ρ + = ρ 0 , the brown surface 1 − βω − v 2 = 0, while the purple surface ρ − = ρ 0 . For β = 0, the spiky strings fall in the region A, to the left of the green surface, where ρ + < ρ 0 . The hanging strings fall in the region B, in between the green and the brown surfaces, where ρ + > ρ 0 subject to the constraint 1 − βω − v 2 > 0. There is a small region C in between the purple and the brown surfaces, confined by ρ − > ρ 0 and 1 − βω − v 2 > 0, corresponding to the bulging strings. At β = 0 (the bottom plane), region D (the blue plane) depicts 1 − ω 2 − v 2 > 0 and corresponds to the spiky string solutions, while the region E (the magenta plane) has 1 − ω 2 − v 2 < 0, giving rise to the hanging strings. Note that the spiky string (giant magnon) regions for vanishing and non-vanishing β's (i.e. regions D and A respectively) are disjoint.
FIG. 5 :
5Rotating string profiles in the target space (t, ρ, θ) with β = 0 (red) and β = 0.5 (blue) at τ = 0: (a) hanging strings, with ω = 0.8, v = 0.7; (b) spiky strings, with ω = 0.2, v = 0.4.
that the region A (the spiky strings at β = 0) and region C (the spiky strings at β = 0) are disjoint. This result implies that the string profile classification condition is different when the background belongs to different conjugacy classes, and they may not deform smoothly to each other, as the global rotating AdS space (β = 0) is in the loxodromic transformation class, while the global AdS (β = 0) is in the elliptic transformation one. Comparison of the snapshots (at constant worldsheet time τ = 0) of the spiky and the hanging strings in the AdS 3 β-vacua target space for β = 0 and β = 0.5 are shown in Fig. 5.
FIG. 6 :
6E Λ of the spiky strings against ω, with (a) three values of β, and with (b) β = 0.5 magnified from (a). We've chosen v = 0.4, and the integration cutoff Λ is at ρ = 50. S Λ ω behaves similarly against ω for sufficiently large ω.
FIG. 7 :FIG. 8 :
78J Λ of the spiky strings against ω, with (a) three values of β, and with (b) β = 0 magnified from (a). Here v = 0.4, and Λ is at ρ = 50. K Λ of the spiky strings against ω, with β = 0.5 and β = 0.7, at v = 0.4. The integration cutoff is at ρ = 50. The magnitude of K Λ increases as ω decreases, but K Λ = 0 identically at β = 0. or not, up to a large constant to be subtracted in regularization. The results are given in Figures 6 ∼ 8. IV. ANALYTICAL CONTINUATION AND COMPLEX SIN(H)-GORDON MODEL While a string prapogates in metric (1) is expected to have an equivalent description in terms of sinh-Gordon model, it is entertaining to see its connection to sine-Gordon model via analytic continuation. With the coordinates change:
Note that the deformation parameter of our AdS β-vacua has nothing to do with that of the βdeformed SYM. We just follow the convention in the literature, and remind the readers to mind the possible confusion.
For the analytic continuation of the metric (1), see the change of coordinates (62) in the section IV.
In[16], a spinning string in AdS 3 space was originally discussed. Also a two-spin solution in AdS 3 × S 1 space was discussed in[17].
One obtains the new metricwhose spatial part is a squashed three-sphere7. The deform parameter β changes the geometry to a round sphere at β = 0 and to a 3-tori at β = 1. The spin chain at fast spinning limit after analytic continuation takes the following form:for another coordinates change φ 1 → ϕ 1 + ϕ 2 , φ 2 → ϕ 1 − ϕ 2 , γ → γ/2 and boosting. Utilizing the Virasoro constraint, one obtains a deformed spin chainHamiltonian density:which recovers the SU(2) Heisenberg XXX spin chain in[6]for β = 0. One might wonder what is this deformed spin chain on the dual field theory side. If one embeds the squashed sphere in R 4 as X 1 = cos γ cos (ϕ 1 − βϕ 2 ), X 2 = sin γ cos (ϕ 2 − βϕ 1 ),An academic guess is that the deformed spin chain is still made of the single trace operator T r(ZZZ · · · ) but with a twisted complex scalar Z ≡ X 1 + iX 4 = cos(γ)e i(ϕ 1 −βϕ 2 ) .At last, the complex sine-Gordon model can be obtained by first constructing a new vector K i = ǫ ijkl X j ∂ + X k ∂ − X l out of X i and use them to define two O(4)-invariants φ and χ:It can be shown that the equations of motion of φ and χ are nothing but the complex sine-Gordon equations[21]. By analytically continuing back to deformed AdS 3 × S 1 , onecan also obtain the sinh-Gordon equation as shown in[10].7A different way to deform S 3 is by squashing the Hopf fiber S 1 along the S 2 base. A different spin chain model could be also obtained[18].V. PP-WAVE LIMITIn this section, we consider a spiky string solution on a pp-wave limit of the β-deformed AdS 3 background, in which we especially see the region of ρ → ∞ and θ → t. Following[19], we start with the β-AdS 3 part in the metric (1), and consider the following reparametrization,x ± = e ρ 0 e ∓θ 0 (θ ± t).We take limits of the two parameters as ρ 0 , θ 0 → ∞ with fixing the ratioHere µ > 0 can be regarded as a free parameter. At this limit, we obtain β-AdS pp-wave metricThe boundary is located at z = 0.We shall consider a spiky string solution in the β-AdS 3 -pp-wave with the following ansatz:The Nambu-Goto action in this case is given byFrom the equations of motion, we obtain the differential equationand this can be solved byWe can evaluate conserved quantities associated with x + and x − asHere we set µ = 1 and z 0 = 2(1 + β)σ 0 , and we also introduced a cutoff ǫ that should be taken ǫ → 0. By expanding by ǫ we getFrom these expansions, we obtain the following relationThis relation can be regarded as a β-deformed result from that in[20]. The Polyakov action is given byfrom which the equations of motion can be derived as follows:
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|
[] |
[
"Effective Langevin Equation Approach to the Molecular Diffusion on Optical Lattices",
"Effective Langevin Equation Approach to the Molecular Diffusion on Optical Lattices"
] |
[
"Aliezer Martínez-Mesa \nDynAMoS (Dynamical processes in Atomic and Molecular Systems)\nFacultad de Física\nFacultad de Física\nDynAMoS (Dynamical processes in Atomic and Molecular Systems)\nUniversidad de la Habana\nSan Lázaro y L\nLa Habana 10400Cuba\n",
"Uranga Llinersy \nUniversidad de la Habana\nSan Lázaro y L, La Habana 10400Cuba\n",
"Piña \nUniversidad de la Habana\nSan Lázaro y L, La Habana 10400Cuba\n"
] |
[
"DynAMoS (Dynamical processes in Atomic and Molecular Systems)\nFacultad de Física\nFacultad de Física\nDynAMoS (Dynamical processes in Atomic and Molecular Systems)\nUniversidad de la Habana\nSan Lázaro y L\nLa Habana 10400Cuba",
"Universidad de la Habana\nSan Lázaro y L, La Habana 10400Cuba",
"Universidad de la Habana\nSan Lázaro y L, La Habana 10400Cuba"
] |
[] |
Optical micro-manipulation techniques has evolved into powerful tools to efficiently steer the motion of microscopical particles on periodic and quasi-periodic potentials, driven by the external electromagnetic field. Here, the dynamics of molecular diffusion on optical lattices is analysed within the framework of the theory of open systems, for polar molecules coupled to a transient electromagnetic field. Using the normal mode expansion of the field, we derive an effective, generalised Langevin equation which describes the motion of the system along the molecular degrees of freedom. The present approach is universally applicable (for molecules with non-vanishing permanent dipole moment) and it opens a wide spectrum of applications in the control of the molecular transport mechanisms on optical lattices. The numerical analysis of suitable model external fields demonstrates the feasibility of neglecting memory terms in the resulting Langevin equation.
| null |
[
"https://arxiv.org/pdf/1811.04321v1.pdf"
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1811.04321
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4485efcb06d79c46fd62dc5e0b52741464de5851
|
Effective Langevin Equation Approach to the Molecular Diffusion on Optical Lattices
10 Nov 2018
Aliezer Martínez-Mesa
DynAMoS (Dynamical processes in Atomic and Molecular Systems)
Facultad de Física
Facultad de Física
DynAMoS (Dynamical processes in Atomic and Molecular Systems)
Universidad de la Habana
San Lázaro y L
La Habana 10400Cuba
Uranga Llinersy
Universidad de la Habana
San Lázaro y L, La Habana 10400Cuba
Piña
Universidad de la Habana
San Lázaro y L, La Habana 10400Cuba
Effective Langevin Equation Approach to the Molecular Diffusion on Optical Lattices
10 Nov 2018arXiv:1811.04321v1 [physics.atom-ph]
Optical micro-manipulation techniques has evolved into powerful tools to efficiently steer the motion of microscopical particles on periodic and quasi-periodic potentials, driven by the external electromagnetic field. Here, the dynamics of molecular diffusion on optical lattices is analysed within the framework of the theory of open systems, for polar molecules coupled to a transient electromagnetic field. Using the normal mode expansion of the field, we derive an effective, generalised Langevin equation which describes the motion of the system along the molecular degrees of freedom. The present approach is universally applicable (for molecules with non-vanishing permanent dipole moment) and it opens a wide spectrum of applications in the control of the molecular transport mechanisms on optical lattices. The numerical analysis of suitable model external fields demonstrates the feasibility of neglecting memory terms in the resulting Langevin equation.
I. INTRODUCTION
During the two decades following the experimental realisation of Bose-Einstein condensates (BEC) in atomic gases [1][2][3][4][5], the confinement, cooling and optical manipulation of atoms and molecules in magnetic and optical traps have attracted a lot of attention. Optical trapping schemes lie at the heart of the continuous progress in the field of cooling and confinement of atoms, and for the collimating of atomic beams. These applications triggered the development of a variety of trapping techniques such as optical tweezers (including beam shaping), optical fibber traps, optical binding, etc. [6][7][8][9][10][11][12][13][14][15][16][17][18][19].
Furthermore, the low temperatures attainable and the possibility of fine-tuning the parameters determining the shape and depth of the trap [19], enable the precise monitoring and control of the molecular motion. Likewise, the ability to tune the strength of intermolecular interactions in a continuous way, via Feshback resonances, provided unparalleled experimental access to a rich body of collective atomic phenomena [20,21], and it propitiated the advent of quantum simulators [22].
Optical lattices constitute a periodic generalisation of optical traps. The periodic arrangement of trapping sites is created by counter-propagating laser fields resulting in a standing wave. The three-dimensional periodic pattern resembles the geometry of a crystal, with the additional advantage over real materials of being free of thermal distortions and of structural defects caused by impurities. The key parameters defining the shape of the optical lattices are the well depth, V 0 , and the spatial periodicity λ. They can be controlled by modifying the intensity and the wavelength of the electromagnetic field, respectively.
Optical lattices constitute an ideal tool for the investigation of a wide variety of collective phenomena and quantum phase transitions such as Bose-Einstein condensation, the BCS-BEC crossover and the Mott insulator transition [23][24][25][26][27][28][29][30]. Moreover, these structures have been widely used in the cooling of atoms and molecules down to nanoKelvin temperatures, the synthesis of new molecules [19], and they constitute promising candidates for the design and implementation of quantum information and quantum simulation schemes [31]. Likewise, the spatial localization of the molecules allows to improve the signal to noise ratio, compared to gas phase spectroscopy, as in matrix isolation spectroscopy experiments.
One exciting spin-off of the molecular trapping in optical lattices is the possibility to drive their motion through the influence of an additional time-dependent electromagnetic field [32][33][34][35]. The intensity of this time-dependent electromagnetic field is usually much lower than that of the standing wave. This approach have been pursued to control particle diffusion (e.g., to achieve particle localization via the interaction with the field E(t)) and to tune the emergence of chaotic behaviour [36][37][38][39][40].
In this Letter, we address the modelling of the classical motion of polar molecules motion on optical lattices in presence of time-dependent electromagnetic fields, and we demonstrate that it can be mapped onto a stochastic process governed by a generalised Langevin equation.
To this purpose, we will account for the rapid variations of the electromagnetic field (and their influence on the molecular motion) by using its normal mode decomposition. The electromagnetic field can be regarded as a superposition of plane waves, whose frequencies are typically larger than 1/τ (where τ is the characteristic time scale of the motion of the molecule on the lattice). From this perspective, molecular diffusion on optical lattices belongs to the group of problems with a clear time scale separation (i.e., comprising both fast and slow degrees of freedom), whose study has a long history in physics, and it constitutes a cornerstone in the system-bath separation in statistical mechanics and thermodynamics [41,42].
Although it seems intuitively correct to employ suitable stochastic forces to mimic the influence of transient electromagnetic fields on the molecular diffusion, whether such description remains valid for any waveform is not settled. Up to our knowledge, most studies focused on the control gained over the molecular motion by the manipulation of the coupling between two or more electronic states, induced by a classical or a quantised radiation field [32][33][34][35]. A few investigations addressed the classical description of atomic motion under the influence of a quantised electromagnetic field [43]. Here, we demonstrate that the stochastic model can be derived in the context of classical physics for arbitrary field shapes, and we also provide an analytic formula for the determination of the dissipative memory kernel. Finally, we show that for typical field parameters non-Markovian effects are negligible. Therefore, the numerical analysis of the molecular diffusion can be carried out at a significantly reduced computational cost by using the molecule-field effective friction coefficient.
II. ONE-DIMENSIONAL MODEL
A. Hamiltonian and equations of motion
In the following, we consider the motion of a molecule of mass m and permanent electric dipole moment d, on a one-dimensional, periodic potential energy curve, e.g., V (x) = V 0 cos 2 (λx). The properties of the lasers building up the periodic potential V (x) are not treated explicitly. This information is masked in the controllable parameters V 0 and λ, which allow to take into account the main features of a prototypical optical lattice. The molecule is regarded as a point particle, and the Hamiltonian of the "system" is given by
H S = p 2 2m + V (x).(1)
The conclusions derived within this minimal one-dimensional model can be straightforwardly extended to higher dimensional systems.
Within the dipolar approximation, the interaction between the molecule and a time-
dependent electromagnetic field E(x, t) is H SB = − d · E(x, t).
The field E(x, t) is taken to be a superposition of plane waves, thus the system-bath interaction can be rewritten as
H SB = j α j (x)q j (t) + j β j (x)π j (t),(2)
where q j and π j are the generalised coordinate associated to the j-normal mode of the electromagnetic field and its conjugate momenta, respectively. The coefficients α j (x) and β j (x) appearing in equation (2) are defined as
α j (x) = dω j √ ε 0 L sin(k j x), β j (x) = d √ ε 0 L cos(k j x).(3)
Here, the constants ε 0 and L denote, respectively, the electric permitivity of the vacuum and the quantisation length introduced to define the normal modes of the electromagnetic field.
Likewise, the energy of the field E(t) can be expressed as a superposition of the energies of the normal modes with frequencies ω j :
H B = 1 2 j π 2 j + ω 2 j q 2 j .(4)
The total Hamiltonian
H = p 2 2m + V (x) + 1 2 j (π j + β j ) 2 + ω 2 j q 2 j + α j ω 2 j 2 (5)
can be regarded as a generalisation of the Caldeira-Legget model [44]. The main differences with respect to the standard Caldeira-Legget Hamiltonian are the non-linear character of the couplings (mediated by the position-dependent coupling functions α j (x) and β j (x)), and the presence of a term implying a coupling between the coordinate of the particle x and the momenta of the harmonic bath modes.
As in the standard treatment of explicit bath, the Hamiltonian in eq. (5)
mẍ + dV dx = − j α ′ j q j + β ′ j π j − j β j β ′ j + α j α ′ j ω 2 j .(6)
where α ′ j = dα/dx and β ′ j = dβ/dx. At the same time, for the bath modes, the Hamilton equations take the forṁ
q j = π j + β j , π j = −ω 2 j q j − α j .(7)
The solution of the set of equations (7) can be expressed analytically using the Green's function approach:
q j (t) = Q j (t) + t 0 β ′ j (x(t ′ ))ẋ(t ′ ) − α j (x(t ′ )) ω j sin [ω j (t − t ′ )] dt ′ ,(8)
where
Q j (t) = q j0 cos (ω j t) + π j0 ω j sin (ω j t)(9)
describes the harmonic oscillations in absence of system-bath couplings. From eq. (8), the momenta of the bath oscillators can be computed as π j =q j − β j .
B. Langevin equation
We are interested in the reduced dynamics along the molecular degree of freedom. Inserting the expressions obtained for q j (t) and π j (t) into equation (6), we first note, after some algebra, that the terms depending on products of the coupling functions α j (x), β j (x), and their spatial derivatives α ′ j (x), β ′ j (x), drop out. The resulting equation of motion can be casted in the form of a generalised Langevin equation:
mẍ + dV dx + ξ(x, t) + π| d| 2 c 2 ε 0 L t 0 F (τ )ẋ(t − τ )dτ = 0.(10)
In this formula, c is the speed of light while the force
ξ(x, t) = j α ′ j Q j (t) + α ′ j α j0 ω 2 j cos(ω j t) + β ′ jQ j (t) − β ′ j α j0 ω j sin(ω j t)(11)
describes the influence of the "random" component of the molecule-field interaction on the molecular motion (α j0 = α j (x(t = 0))).
In equation (10), F (τ ) is the inverse Fourier transform of the function ω 2S (ω), wherẽ S(ω) is the even continuation of the spectral density of the electromagnetic field:
S(ω) = |E(ω)| 2 , ω ≥ 0 |E(−ω)| 2 , ω < 0(12)
It can be noticed, that for smooth spectral distributions |E(ω)| 2 , the effective bath modes participate in the frictional kernel F (τ ) with the superhomic weight ∼ ω 2 characteristic of electromagnetic fields.
Let us note in passing, that conversely to the standard case of dissipative dynamics in mechanical systems, the spectral densityS(ω) contains information on the population of the field normal modes but not on the intensity of the system-bath coupling.
Under the assumption of rapidly decaying kernels, Markov approximation holds, and equation (10) can be casted in the form of the standard Langevin equation, for the description of particle-resolved dynamics, or the equivalent Fokker-Planck equation, for the simulation of the time evolution of the probability density in phase space. Within this approximation, the friction coefficient γ is given by
γ = πd 2 c 2 ǫ 0 L ∞ 0 F (τ )dτ.(13)
C. Approximate and numerical evaluation of the friction kernel
To assess the influence of the different parameters determining the shape of the electromagnetic field, on the time-dependence of the friction kernel, we consider a few examples in this section. If the spectral density of the electromagnetic field vanishes outside a vicinity of width ∆ω centred a given frequency ω 0 , the diffusion coefficient takes the form:
F (τ ) = 1 2 ω 0 +∆ω/2 ω 0 −∆ω/2 ω 2 |E(ω)| 2 cos[ω(t)]dω .(14)
Moreover, if the function E(ω) varies smoothly in the interval ω 0 − ∆ω 2 , ω 0 + ∆ω
2 F (τ ) = ω 2 0 |E(ω 0 )| 2 ω 0 +∆ω/2 ω 0 −∆ω/2 cos[ω(t)]dω = ω 2 0 |E(ω 0 )| 2 cos[ω 0 τ ] sin ∆ω 2 τ τ .(15)
It is straightforward to show, by evaluating the autocorrelation function of the stochastic force ξ(x, t), that the diffusion coefficient in this limit is given by:
D = 2| d| 2 c ω 0 | E(ω 0 )| 2 ∆ω .(16)
For radiation fields with a smooth spectral density, the dissipative kernel decays over a period of ∼ 1/∆ω. If the wavepacket is wide enough in the frequency domain, then this time interval may be much shorter than the characteristic time scale of the molecular motion.
We explored numerically the changes in the form of the kernel F (τ ) upon modification of the width and the central frequency of wavepackets of various shapes, in the mid infrared region of the electromagnetic spectrum. In Figure 1, we show the results corresponding to a Lorentzian pulse. They illustrate the general trend of the time evolution of F (τ ) for frequency limited wavepackets, which we also observed in the cases of Gaussian and squarewell pulse shapes and for multiperiodic electromagnetic fields.
It can be seen, that for different Lorentzian pulses centred at ω 0 = 10 14 Hz, the friction kernel decays faster as the wavepacket gets broader in spite of the non-standard system-bath couplings in the present case. This behaviour is analogous to that observed in dissipative systems where local-in-time dissipation is obtained for spectral densities of the bath which extend far beyond the characteristic frequency of the system. On the other hand, the characteristic time scale of the friction kernel is independent of the central frequency of the frequency limited pulses. However, the overall magnitude of the dissipative molecule-field coupling significantly enhances as the centre of the wavepacket is shifted to larger frequencies.
III. CONCLUSIONS
In summary, we have introduced an stochastic method for studying the molecular diffusion on optical lattices in presence of an external time-dependent electromagnetic field, where the influence of the waveform is filtered into the form of the frictional kernel. The approach is based on the normal mode representation of the electromagnetic field, and the field-molecule interaction is described within the dipole approximation. The method allows to obtain numerically converged results for the diffusion dynamics for arbitrary transient external fields.
The present description is appropriate for wide classes of systems and properties (molecules without a permanent dipole moment constitute nevertheless an important example of systems lying outside the domain of applicability of this methodology). In many situations, the rapid decay of the friction kernel indicates that the numerical integration of the equations of motion may be further simplified by treating the molecular diffusion as a Markov process.
In spite of its appeal, to the best of our knowledge, the concept of Langevin dynamics has not been used in the context of laser-driven molecular diffusion on optical lattices.
Although we have focused on an one-dimensional model of the classical motion of molecules on optical lattices, extensions to three-dimensional and quantum systems are straightforward. The present analysis paves the way to employ the mapping of molecular diffusion on optical lattice into a dissipation problem to investigate phenomena such as field-assisted diffusion and tunnelling, taking advantage of the theoretical and computational tools developed over the years to investigate semiclassical (field-free) molecular dynamics.
In particular, this methodology can be applied to the control of the diffusive dynamics by tailoring the time-dependent electromagnetic field.
contains the sum H S + H B + H SB plus a V (x) to be the bare potential along the coordinate x. We now aim to derive an equation that describes the dynamics along the molecular degrees of freedom, upon integration of the normal modes of the transient electromagnetic field. The canonical equations of motion for the molecule can be combined in a single Newton's equation:
FIG. 1 .
1(a) Weights f i (ω) of the bath oscillators for different widths of a Lorentzian laser pulse with a central frequency of 10 1 4 Hz. (b) Inverse Fast Fourier transforms F 1 of the weight functions f i (ω).
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|
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"UNIFYING THE LINEAR TIME-BRANCHING TIME SPECTRUM OF STRONG PROCESS SEMANTICS",
"UNIFYING THE LINEAR TIME-BRANCHING TIME SPECTRUM OF STRONG PROCESS SEMANTICS"
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[
"David De Frutos Escrig \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"Carlos Gregorio Rodríguez \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"Miguel Palomino [email protected] \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"David Romero [email protected] \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"David De Frutos Escrig \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"Miguel Palomino \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"David Romero Hernández \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"Gregorio Rodríguez \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"M Palomino \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n",
"D Romero Hernández \nDepartamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n\n"
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[
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n",
"Departamento de Sistemas Informáticos y Computación\nCC Creative Commons\nUniversidad Complutense de Madrid\n"
] |
[
"Logical Methods in Computer Science"
] |
VanGlabbeek's linear time-branching time spectrum is one of the most relevant work on comparative study on process semantics, in which semantics are partially ordered by their discrimination power. In this paper we bring forward a refinement of this classification and show how the process semantics can be dealt with in a uniform way: based on the very natural concept of constrained simulation we show how we can classify the spectrum in layers; for the families lying in the same layer we show how to obtain in a generic way equational, observational, logical and operational characterizations; relations among layers are also very natural and differences just stem from the constraint imposed on the simulations that rule the layers. Our methodology also shows how to achieve a uniform treatment of semantic preorders and equivalences.ACM CCS: [Theory of computation]:Formal languages and automata theory-Formalisms-Algebraic language theory; Formal languages and automata theory-Semantics and reasoning-Program semantics; Logic; Models of computation-Concurrency.
|
10.2168/lmcs-9(2:11)2013
|
[
"https://arxiv.org/pdf/1304.6574v2.pdf"
] | 16,254,640 |
1304.6574
|
261fd9f8473c28687cd4f310dd720e5eda693ad9
|
UNIFYING THE LINEAR TIME-BRANCHING TIME SPECTRUM OF STRONG PROCESS SEMANTICS
2013
David De Frutos Escrig
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
Carlos Gregorio Rodríguez
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
Miguel Palomino [email protected]
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
David Romero [email protected]
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
David De Frutos Escrig
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
Miguel Palomino
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
David Romero Hernández
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
Gregorio Rodríguez
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
M Palomino
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
D Romero Hernández
Departamento de Sistemas Informáticos y Computación
CC Creative Commons
Universidad Complutense de Madrid
UNIFYING THE LINEAR TIME-BRANCHING TIME SPECTRUM OF STRONG PROCESS SEMANTICS
Logical Methods in Computer Science
92201310.2168/LMCS-9(2:11)2013Submitted Mar. 6, 2012LOGICAL METHODS ÐI N COMPUTER SCIENCE DOI:10.2168/LMCS-9(2:11)2013 c D. de Frutos Escrig, C. 2 D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ Contentsand phrases: process semanticslinear time-branching time spectrumalgebraic languagessimulation semanticslinear semanticsconstrained simulationaxiomatizationsunification acd
VanGlabbeek's linear time-branching time spectrum is one of the most relevant work on comparative study on process semantics, in which semantics are partially ordered by their discrimination power. In this paper we bring forward a refinement of this classification and show how the process semantics can be dealt with in a uniform way: based on the very natural concept of constrained simulation we show how we can classify the spectrum in layers; for the families lying in the same layer we show how to obtain in a generic way equational, observational, logical and operational characterizations; relations among layers are also very natural and differences just stem from the constraint imposed on the simulations that rule the layers. Our methodology also shows how to achieve a uniform treatment of semantic preorders and equivalences.ACM CCS: [Theory of computation]:Formal languages and automata theory-Formalisms-Algebraic language theory; Formal languages and automata theory-Semantics and reasoning-Program semantics; Logic; Models of computation-Concurrency.
Introduction
Since the foundational work by Robin Milner [41,42] and Tony Hoare [32] on process semantics, there has been a multitude of proposals to endow processes with meaning and to define equivalence and preorder relations over them. Among the most relevant work are those of Matthew Hennessy [30], who introduced the testing methodology defining process semantics from test cases, and those of Jan Bergstra and Jan Willen Klop [11], later continued by Jos Baeten and Peter Weijland [10], which were based on an axiomatic approach.
These proposals define algebraic languages for the specification of processes, diverging in subtle details concerning the treatment of non-determinism and parallelism. These aspects are captured by means of certain operators which may (strongly) vary in each particular language.
Focusing on equivalences, it is interesting to note how the pioneering work in this area already established two fundamental notions, bisimulation and traces/failures, that constitute an upper and a lower bound on the natural framework in which other process equivalences can be studied. Hoare-with his characteristic clarity-summarizes the situation in the following paragraph.
CCS makes many distinctions between processes which would be regarded as identical in this book. The reason for this is that CCS is intended to serve as a framework for a family of models, each of which may make more identifications than CCS but cannot make less. To avoid restricting the range of models, CCS makes only those identifications which seem absolutely essential. In the mathematical model of this book [CSP] we have pursued exactly the opposite goal -we have made as many identifications as possible, preserving only the most essential distinctions. [32] In between these two fundamental notions of equivalence-bisimulation and tracesthe last two decades of the 20th century witnessed the surge of a large variety of new equivalences associated to new calculi and process algebras, whose aim was to explore the different needs for expressivity and distinction capabilities in many applications.
The most important taxonomic work on process semantics was carried out by Rob van Glabbeek as part of his doctoral dissertation [54]. In two papers, titled Linear timebranching time spectrum [55,56], he collected the most important of these equivalences establishing, among other results that we will comment on, a classification based on their capability to distinguish processes. The first of the papers concentrate on strong semantics, in the sense that they consider each action processes perform as being observable by their environment. The second paper consider the inclusion in the language of a new and invisible action τ ; process semantics considering this internal action are usually called weak semantics. Figure 1 shows a slightly expanded version of the spectrum proposed by van Glabbeek for the case of strong semantics in [55]. These strong semantics, that do not consider at all the special role of internal actions τ , are the only ones that we consider in this paper. This array of semantics is supported by many authors who claim that there is no single "good" definition. Process theory can be applied in a wide spectrum of contexts and situations and the concrete uses will have a decisive influence in the election of what a suitable semantics should be.
The choice of a suitable semantics may depend on the tools an environment has, to distinguish between certain processes. It is conceivable that a concurrency theory is equipped with different semantics, and has the capacity to express equality on different levels. [57] The possibility to define several and varied semantics can then be considered to be an advantage of the theory, since it allows for the necessary flexibility to reflect different notions of processes and equivalence and preorder relations over them.
Nevertheless, this multiplicity has gone hand in hand in the literature with an individual study of each of the semantics that somehow makes the whole theory less appealing because such a cornucopia can become a handicap both for its study and its practical application. For instance, although most of the semantic notions defined for processes simultaneously induce both a (pre)order and an equivalence, 1 fact that these two notions are mutually intertwined, as we will show later. Likewise, the study of the concrete models has been usually undertaken paying little attention to the other semantics or to the relations among them, even though it is well-known that there exist "families" of semantics-such as the linear semantics-which are undoubtedly related. A unified study of semantics has both methodological and practical implications that have been explored along the last years by the authors of this work, for example in [22,24,25,20,21], and also in work by important researchers in the area [5,4,16,40]. This research shows that a unified view of process semantics is indeed possible. This is precisely the main goal we set to reach with our work: we aim to study process semantics in a generic way, making the equivalence and (pre)order relations our object of study in order to find patterns, to identify families, to search for properties among these relations, so that we obtain generic results that need not be proved again and again for each of the semantics.
We aim, in a nutshell, at a unifying view of process semantics that can be used to understand them both jointly and individually and that allows to continue with their theoretical study in a more focused manner, helping to identify those properties a semantics should have for a particular application.
1.1. Overview of results. This paper contains a consolidated and extended presentation of the unification of observational, equational and the logic process semantics published in [20,19,47] for strong behavioral semantics. 2 We take advantage of the joint and larger presentation of the subject to tighten the connections between the different views. Besides, we make the paper mostly self-contained providing proofs for all the results; we also complete the study with new results not included in [20,19,47]. We have also completed the 2 We comment in Section 10 the work of some of the authors on unification of weak semantics. UNIFYING THE LTBT SPECTRUM 5 revision of the unification of strong process semantics with a section devoted to the unified presentation of the operational semantics.
Next we describe the main results we have obtained. They can be used as a roadmap for reading the paper and to understand the technical details in the following sections.
• One of the most generic results we have proved is the existence of two essential families of semantics: branching semantics and linear semantics. Certainly, this was already hinted by van Glabbeek when he named its spectrum of semantics "linear time-branching time". Our results show that the most representative branching semantics have characterizations as simulation semantics. Moreover, every simulation has a natural family of coarser linear semantics associated to it that inherit some of its properties.
In Figure 11 (page 34), branching semantics are located to the left and each of them defines a layer of "induced" linear semantics, to their right. For example, ready simulation is the branching semantics from which the classic diamond of linear semantics composed of failures, readiness, failure trace, and ready trace semantics is generated. These semantics, as we will later see in detail, inherit some of their axiomatic characterizations directly from the ready simulation. In addition, the same layer also contains the possible worlds semantics, which is a deterministic branching semantics.
Even though the axiomatic characterizations contained in Section 3 already show this dependency between linear and branching semantics-see Figure 5 (page 17)-it is in Section 4 where, using techniques from denotational semantics, the relations between the original branching semantics and the induced linear semantics can be fully appreciated. The relationships among the different linear semantics in the same branching layer are also completely specified in that section.
• Equational characterizations reveal in a very concise manner the basic properties of the different semantics. As a result of our research, we have been able to derive a generic axiomatic characterization of the semantics in the spectrum which shows clearly the relationships among them: the uniformity in the definitions of the branching semantics and the different families of linear semantics becomes apparent, as well as the tight relation between each branching semantics and its associated linear semantics. In Section 3 we present all the details related to these axiomatic characterizations. • Another result that we consider important is that it is indeed possible to establish a clear relationship between the preorder and the equivalence associated to a given semantics.
From an axiomatic point of view, in Section 3 we show how these characterizations are closely related. In fact, there are algorithms that allow to easily obtain the axioms for the equivalence from those of the preorder [4,21], and also the other way around, the axioms for the preorder from those of the equivalence [23,24]. • We also offer a unifying view of the process semantics based on observational (denotational) semantics, according to which we have classified the process semantics in four categories: − bisimulation semantics, which is the finest semantics in the spectrum and the only one that cannot be defined by means of a non-trivial preorder; − the simulation semantics (simulation, complete simulation, ready simulation, nested simulation, . . . ) which are characterized by means of branching observations, that is, labeled trees; − the linear semantics (traces, failures, readiness, . . . ), characterized by linear observations, a degenerated case of branching observations;
− the deterministic branching semantics corresponding to an intermediate class between branching and linear, where observations are deterministic trees. Possible worlds semantics is the only semantics in the original van Glabbeek's spectrum in this class. Besides their linear or branching nature, semantics are characterized by a local observation function that generates the local observations at the states. For the linear case there is also the possibility of observing this local information in a partial way and this is how for each local observer, in principle, up to four different semantics can be obtained. In particular, this gives rise to the classic diamond below the ready simulation semantics formed by failures, failure trace, readiness, and ready trace semantics.
The uniform presentation of the process semantics that we offer in Section 4 clarifies the relationships and hierarchies among all the semantics; moreover, it will make possible the development of generic proofs of their common properties.
• We also present a unified view of the logical semantics. Again, the bisimulation semantics, which is characterized by the Hennessy-Milner logic HML [31], is our starting point, and then we aim for the sublogics that characterize each of the semantics in the spectrum. Guided by our main unification goal, we have not tried to obtain the smallest possible sets of formulas, but have veered for the largest sublogics that characterize each of the semantics. Hence, the finest semantics are characterized by the largest sublogics and in fact we obtained a uniform characterization that informs us about the hierarchy of semantics, by proving that a semantics S 1 is finer than another S 2 if and only if the corresponding logics satisfy L S 2 ⊆ L S 1 . Moreover, the classification into branching and linear time semantics is also reflected in the structural definition of each logic. In particular, the branching semantics are characterized by the free use of negation over the formulas that define the corresponding constraint, while the linear semantics at each layer of the spectrum introduce ever more limitations in the subformulas.
• Finally, we also discuss an operational-like presentation of the semantics in the spectrum; more precisely, we consider an evaluation semantics to derive the appropriate data which characterize them. Those data are quite similar to the ones employed for our observational semantics so that it is not them, but the way in which they are derived, that enhances our understanding of the features of each of the semantics and the relationships between them. These presentations somehow generalize the work by Cleaveland and Hennessy [17] on the characterization of the Testing semantics by means of bisimulation. There is also a clear connection with the work by two of the authors of this paper on (Bi)simulation up-to [25]. • A concomitant, but still important result of our work, is of methodological nature: the semantics are amenable to a working methodology that allows for general results that can be applied to families of semantics as well as to yet to be defined semantics. The requirements we impose on these new semantics are relatively mild. An example of this is shown seen in Section 8, where some new process semantics-indeed two new familiessmoothly integrate into our general theory. In fact, it was nice to discover that one of them is just the revivals semantics, which has been recently developed by Bill Roscoe [48]. • Each of the characterization frameworks-equational, observational, logical or operationalsheds light on the spectrum in different and complementary ways. This has provided us with different ways to study all the previously known semantics and the relationships between them. That complementary nature also sprang up along our unification work when we discovered one by one all the factors that contribute to the structure of the extended spectrum. In particular, when considering simple and natural combinations of axioms we found out the new meet semantics in Section 8, while their dual join semantics was discovered in a natural way when considering the observational characterizations. Finally, the semantics of minimal readies (in Section 6.2.2) appeared when investigating the logical framework. While not too important on its own, our unification work has also revealed a mistake in the classic logical characterization of one of the semantics in the original spectrum (see Section 6.2.3): it was the general and systematic approach that guides our uniform characterization that allowed it.
1.2. Some related work. Naturally, the goal of defining a global or general theory of process semantics has been around for a long time and several relevant authors in the field have already paved the way that we now tread. Despite the methodological differences between Milner's work, based on bisimulation, and Hoare's, on denotational semantics, both of them had in common the search for characterizations-logical, axiomatic, observational-that could shed light from different angles on the world of process semantics.
Hennessy introduced the testing methodology to endow processes with semantics, making the notion of equivalence to spring from the application of the interaction principles for processes expressed within the model. Perhaps one of the most important contributions of his work was what he called the "trinity": processes can be seen as syntactic terms in an algebra, as operational descriptions in labeled transition systems, or as denotational objects in a mathematical model. With our work we have somehow extended this trinity in a generic manner to all the semantics in the extended spectrum. Van Glabbeek's work, the linear time-branching time spectrum aimed at the comparison of most known semantics-at the time he developed his seminal work-by presenting them within common frameworks that would allow a comparative study of their properties. Besides providing uniform definitions over transition systems, van Glabbeek also proposed to characterize the semantics in terms of logical formulas. The set of modal formulas whose satisfaction equivalence identifies the same processes as the corresponding semantics is defined. Because of the compositional definition of the corresponding sets of formulas, this characterization can be considered to be denotational semantics.
Another characterization provided by van Glabbeek is the axiomatic one, for which he defines the BCCSP language that is used in this work (see Definition 2.1). Twelve of the semantics in the spectrum are characterized by means of sets of axioms over syntactic terms for this language. For most of them-except for bisimulation, that has no associated ordertheir characterizations are actually twofold: on the one hand, the natural order relation that defines each semantics- Table 1-and, on the other hand, the induced equivalence- Table 2. Many of these characterizations were previously known but, again, their uniform presentation is one of its main merits.
A deep study-individual as well as comparative-of these axiomatizations and the quest for answers to the new questions that arise from this study has been one of the leading forces behind our research. Actually, some of our most relevant results can be combined into a new way of presenting the spectrum- Figure 11 (page 34)-that allows for a better comprehension of the semantics since it clarifies their relative positions within it and shows the existence of "gaps" that correspond to new semantics whose addition to the graph reflects a desirable regularity that makes it clearer. Hoare's work on the unification of the study of process algebras [33] was also an important influence. Specially, the relationship between similarity and trace refinement, which we have generalized by establishing the connection between branching time and linear time semantics, and the connection between the denotational, the algebraic, and the operational styles proposed by him and He Jifeng in [34].
As already mentioned, Roscoe has contributed in an independent research effort in parallel with ours to the study of new process semantics by proposing his stable revivals model [48]. He relates his new semantics with other well known linear time semantics and the rediscovery of that semantics in our expanded spectrum gave us the opportunity to present those relationships with a unified and generic light.
There is other relevant work in the area of process theory that has inspired us. The number of contributions is too large to cite all of them here. Anyone interested on finding a more exhaustive list of relevant references may collect them, for instance, from [28,7,1,57,50]. There the historic evolution of the area and many of the most important contributions to it are reviewed. To them we can add four recent books on process algebras and related subjects [8,49,6,51], presenting different points of view and some of the semantics studied in this paper. Finally, in our Conclusions, we will discuss a bit the work on the generic study and classification of the weak semantics.
1.3. Paper structure. We have structured this paper as follows. Section 2 introduces all the basic definitions and notation to properly follow the developments in the following sections.
In Section 3 we propose alternative characterizations for the axiomatizations of the semantics in the spectrum, both for orders and equivalences. All these axiomatizations are based on just two parametrical axiom skeletons that clearly highlight the relations among the different semantics.
Section 4 presents a unified observational characterization for process semantics. One of the key ideas is that constrained simulations are uniformly characterized by a branching observation plus a local observation function. From the observations of a given constrained simulation, the linear semantics in its layer are uniformly derived.
In Section 5 we prove that the equations we presented in Section 3 are deduced from the observations defined in Section 4 in a general way, without using at all the already known axiomatizations for the semantics. Section 6 follows the trails of Sections 3 and 4 by introducing a unified logical characterization of process semantics.
In Section 7 we prove that the observational characterizations developed in Section 4 allow for generic proofs for the logical characterizations presented in Section 6. Therefore the "trinity" of equations, observations, and logical formulas is established in a generic way for large families of process semantics. Section 8 is a practical proof of the applicability of our unification proposal. Some new process semantics, that were not listed in the original linear time-branching time spectrum, are easily accommodated in our framework thus getting the corresponding semantic characterizations that we have presented in previous sections.
In Section 9 we conclude the unified presentation of the semantics in the spectrum by developing an operational characterization which mainly produces the information provided by the observational semantics, but inferred in an operational way, using a (unified) set of SOS-like rules.
Finally, in Section 10 we offer some conclusions and lines for future work.
Acknowledgments. We gratefully acknowledge three anonymous referees for their very thoughtful and detailed comments on a previous version of this work, that have greatly helped us to improve the presentation of this material.
Preliminaries
Although the main results in this paper are also valid for infinite processes-as we showed in [22,25]-in order to simplify the presentation of the concepts, we will mainly consider finite processes generated by the basic process algebra BCCSP which contains only the basic process algebraic operators from CCS [42], and CSP [32], but is sufficiently powerful to express all finite synchronization trees [41]. This language has repeatedly been used in unification work, e.g. [4,58].
Definition 2.1. Given a set of actions Act, the set BCCSP(Act) of processes is defined by the following BNF-grammar:
p ::= 0 | ap | p + q where a ∈ Act; 0 represents the process that performs no action; for every action in Act, there is a prefix operator; and + is a choice operator.
The operational semantics for BCCSP terms is defined in Figure 2. As usual, we write p a −→ if there exists a process q such that p a −→ q, and p α =⇒ q if α = a 1 . . . a n and p
a 1 −→ p ′ a 2 −→ . . . an −→ q.
The initial offer of a process is the set I (p) = {a | a ∈ Act and p a −→}. This is a simple, but quite important observation function that plays a central role in the definition of the most popular semantics in the linear time-branching time (ltbt) spectrum. We will also denote by I the relation expressing the fact that two processes have the same initial offers: pIq ⇔ I (p) = I (q).
One way to capture semantics is by means of the equivalence relation induced by it: given a formal semantics [[·]] Z , we say that processes p and q are equivalent iff they have the same semantics, that is,
p ≡ Z q ⇔ [[p]] Z = [[q]] Z .
These semantics can be defined by means of adequate observational scenarios, or by logical characterizations that induce natural preorders ⊑ Z whose kernels are the semantic equivalences. We refer to [58] for the original definition and usual notation for all the semantics in the ltbt spectrum that will be discussed throughout the paper.
To properly express equations or inequations within the process language, we introduce variables from any adequate set V, and consider the extended set BCCCSP(Act, V ) of terms including variables in V .
Some of the semantics in the spectrum are concrete examples of the general notion of constrained simulation semantics that can be defined in a parameterized way.
Definition 2.2. Given a relation N over BCCSP processes, a relation S N is an N -constrained simulation if pS N q implies:
• for every a ∈ Act, if p a −→ p ′ then there exists some q ′ such that q a −→ q ′ and p ′ S N q ′ , and • pN q. We say that process p is N -simulated by process q, or that q N -simulates p, written p ⊑ NS q, whenever there exists an N -constrained simulation S N such that pS N q. We have already studied the constrained simulation semantics in detail in [24], stressing their general properties. In particular, the following constraints were considered:
−→ p p a −→ p ′ p + q a −→ p ′ q a −→ q ′ p + q a −→ q ′(B 1 ) x + y ≃ y + x (B 3 ) x + x ≃ x (B 2 ) (x + y) + z ≃ x + (y + z) (B 4 ) x + 0 ≃ x
• the universal relation U relating all processes, which gives rise to the simulation semantics;
• the relation C, which holds for processes p and q when both, or none, are isomorphic to 0, and that gives rise to the complete simulation semantics; • the relation I relating processes with the same initial offer, which is the constraint for ready simulation; • the relation T , that holds for processes having the same set of traces and gives rise to the trace simulation semantics; • the relation S, the inverse of the simulation equivalence relation, whose associated constrained simulation is the 2-nested simulation. Throughout this paper there appear different order relations. We use ⊑ to denote semantic preorders and, for the sake of simplicity, we use the symbol ⊒ to represent the preorder relation ⊑ −1 . With ≡ we denote the induced equivalence (that is, ⊑ ∩ ⊒). To refer to a specific preorder we shall append the initials of its name as subscripts to the symbol ⊑ (⊑ RS for ready simulation, ⊑ F for failures, and so on). A similar convention applies to the kernels of the preorders (≡ RS , ≡ F , . . . ) and to the bisimulation equivalence ≡ B .
An inequation (respectively, an equation) over the language BCCSP is a formula of the form t u (respectively, t ≃ u), where t, u ∈ BCCSP(Act, V ). An (in)equational axiom system is a set of (in)equations over the language BCCSP. An equation t ≃ u is derivable from an equational axiom system E, written E ⊢ t ≃ u, if it can be proven from the axioms in E using the rules of equational logic (viz. reflexivity, symmetry, transitivity, substitution and closure under BCCSP contexts):
t ≃ t t ≃ u u ≃ t t ≃ u u ≃ v t ≃ v t ≃ u σ(t) ≃ σ(u) t ≃ u at ≃ au t ≃ u t ′ ≃ u ′ t + t ′ ≃ u + u ′
where substitutions σ are defined and applied as usual.
For the derivation of an inequation t u from an inequational axiom system E, the rule for symmetry-that is, the second rule above-is omitted. We write E ⊢ t u if the inequation t u can be derived from E.
It is well-known that, without loss of generality, one may assume that substitutions happen first in (in)equational proofs, i.e., that the fourth rule may only be used when its premise is one of the (in)equations in E. Moreover, by postulating that for each equation in E its symmetric counterpart is also present in E, one may assume that applications of symmetry happen first in equational proofs, i.e., that the second rule is never used. In the remainder of this paper, we shall always tacitly assume that equational axiom systems are closed with respect to symmetry. Note that, with this assumption, there is no difference between the rules of inference of equational and inequational logic. In what follows, we shall consider an equation t ≃ u as a shorthand for the pair of inequations t u and u t.
An inequation t u is sound with respect to a given preorder relation ⊑, if t ⊑ u holds true. An (in)equational axiom system E is sound with respect to ⊑ if so is each (in)equation in E. An (in)equational axiomatization is called ground-complete if it can prove all the valid (in)equivalences relating terms with no occurrences of variables. As in [58], we abbreviate ground-completeness for completeness because this is the only kind we use along the paper.
Bisimilarity, the strongest of the semantics in the spectrum, can be axiomatized by means of the four simple axioms in Figure 3. These axioms state that the choice operator is commutative, associative and idempotent, having the empty process as identity element. These axioms also justify the use of the notation a i ap i a for processes, where the commutativity and associativity of the choice operator is used to group together the summands whose initial action is a. We will also write p| a for the (sub)process we get by projecting all the a-summands of p; that is, if p = a i ap i a , then p| a = i ap i a . Besides the semantics in the spectrum, we are interested in a general study that can be applied to any "reasonable" semantics coarser than bisimilarity. Since we will use preorders to characterize these semantics we introduce the following definitions that state the desired properties of those reasonable preorders.
Definition 2.3. A preorder relation ⊑ over processes is a behavior preorder if
• it is weaker than bisimilarity, i.e., p ≡ B q ⇒ p ⊑ q, and • it is a precongruence with respect to the prefix and choice operators, i.e., if p ⊑ q then ap ⊑ aq and p + r ⊑ q + r. If ⊑ is actually an equivalence, it is said to be a behavior equivalence.
Another way of presenting a semantics is by means of a logical characterization. The Hennessy-Milner logic [31], characterizing the bisimulation semantics is the most popular one.
Definition 2.4 (Hennessy-Milner logic, HML). The set L HM of Hennessy-Milner logical formulas is defined by: if ϕ, ϕ i ∈ L HM for all i ∈ I and a ∈ Act, then i∈I ϕ i , aϕ, ¬ϕ ∈ L HM .
The satisfaction relation |= is defined by:
• p |= aϕ if there exists q such that p a → q and q |= ϕ;
• p |= i∈I ϕ i if for all i ∈ I : p |= ϕ i . • p |= ¬ϕ if p |= ϕ.
Note that i∈∅ ϕ i ∈ L HM , and we have p |= i∈∅ ϕ i for all p. Therefore, in the following we will consider that ⊤ ∈ L HM , where ⊤ is syntactic sugar for i∈∅ ϕ i . The finite version of this logic, L f HM , uses binary conjunction ∧ instead of the general conjunction i∈I . It is well-known that L f HM characterizes the bisimulation semantics between image-finite processes, that are those that do not allow infinite branching for any action a ∈ Act at any state. Van Glabbeek uses L B to refer to L HM in [58].
Equational semantics
On Tables 1 and 2 appear the axiomatic characterizations for the preorders and equivalences in van Glabbeek's spectrum [58]. For each column, the set of axioms marked with "+" are sound and complete with respect to the preorder or equivalence in the head of that column; axioms marked with "v" are valid but not needed. When studying these tables there are several questions that naturally arise: for every semantics, is there any connection between the axioms defining the preorder and those for the equivalence? Can the axiomatizations of some of these semantics be jointly tackled?
In this section we will develop new axiomatizations for all the semantics in the ltbt spectrum that offer a clear answer to the previous questions: even if there was not a systematic procedure that led to produce the axiomatizations of those tables, we can obtain equivalent axiomatizations that do follow a given procedure.
These new axiomatizations are obtained after noticing that every process semantics can be understood as the product of two "design decisions", decisions that define what we have called the "dynamic" and the "static" basis of the semantics. We will show that, besides B 1 -B 4 , we only need a generic simulation axiom (N S)-Proposition 3.1-which characterizes the family of constrained simulation semantics, to axiomatize the whole class of pure branching semantics. Moreover, to characterize the linear time semantics, we only need to add to the corresponding simulation axiom the adequate instantiation of a generic axiom (N D)-see page 14-for reducing the observability of non-determinism in processes, by means of which we introduce the additional identifications induced by each of the linear semantics.
Also the axiomatizations between orders and equivalences are closely related; in fact, in the case of the linear semantics we could just use an equivalence (N D ≡ ) axiom, leaving the order or equivalence aspect to be determined by the use of the order or equivalence axiom of the corresponding branching semantics, see Figure 5.
In order to justify the form of our axiomatizations without leaving the axiomatic framework, in this section we prove our results with separate and ad-hoc proofs for each semantics just comparing the new characterizations with those previously known. This allows us to quickly get the taste of the underlying relations of the process semantics. Once the unified observational characterization of semantics is presented in Section 4, we will provide generic proofs for these results in Section 5 that show the suitability of the new axiomatizations with respect to the observational characterizations of the semantics.
3.1.
A new axiomatization of the most popular semantics. We start our study with a very representative and well-known group of semantics in the spectrum, each of which has been developed and used in important work in the area: ready simulation [38,13], failures [14,32], readiness [43], ready trace [9] and failure traces [44].
3.1.1. Semantic preorders. As already hinted above, the dynamic part of the semantics is inherited from a simulation preorder. As stated in our Introduction, bisimilarity can be axiomatized by the set of axioms B1 − B4. All the other semantics in the spectrum are coarser than it, and therefore also satisfy these axioms. But due to the fact that bisimulations define equivalence relations and not just preorders, we cannot base on them the characterization of any other interesting semantics. But, plain simulations are somehow defined as half-bisimulations, and can indeed be used as support for the characterizations of B RS PW RT FT R F CS CT S T (x + y) + z ≃ x + (y + z) + + + + + + + + + + + x + y ≃ y + x + + + + + + + + + + + x + 0 ≃ x + + + + + + + + + + + x + x ≃ x + + + + + + + + + + + ax ax + ay some interesting semantics, such as trace semantics. Nevertheless, plain similarity becomes too weak, and some other finer class of simulations is needed to support the characterization of the interesting semantics listed above. Next we recall the axiomatizations of plain, ready and general constrained similarity. (1) Plain similarity can be axiomatically defined by means of the axiom (S) x x + y, together with the axioms B 1 -B 4 that define bisimilarity.
+ + + + + + v v v v a(bx + by + z) ≃ a(bx + z) + a(by + z) + v v v v v v I(x) = I(y) ⇒ ax + ay ≃ a(x + y) + v v v v v ax + ay a(x + y) + v v v a(bx + u) + a(by + v) a(bx + by + u) + v v v ax + a(y + z) a(x + y) + v v ax ax + y + + v v a(bx + u) + a(cy + v) ≃ a(bx + cy + u + v) + v x x + y + + ax + ay ≃ a(x + y) +
(2) Ready similarity can be axiomatically defined by means of the conditional axiom (RS) xIy ⇒
x x+y, together with B 1 -B 4 . It can also be axiomatized by means of the axiom scheme ax ax + ay, where a represents any arbitrary action. (3) Whenever N is a behavior preorder, N -similarity can be axiomatically defined by means of the conditional axiom (NS) N (x, y) ⇒ x x + y, together with B 1 -B 4 .
Let us now consider the diamond of semantics coarser than ready similarity in the ltbt spectrum. It consists of the failures, readiness, failure trace, and ready trace semantics. None of them is a simulation semantics, so their classic axiomatizations (see Table 1) contain an additional axiom: Failures: (F ) a(x + y) ax + a(y + w) Readiness: (R) a(bx + by + u) a(bx + u) + a(by + v) failure trace: (F T ) a(x + y) ax + ay ready trace: (RT ) I(x) = I(y) ⇒ ax + ay ≃ a(x + y)
Since we are interested in capturing the reduction of observability of non-determinism, our first candidate for a general axiom covering all cases was (F T ), which captures the fact that by delaying the choices we get "smaller" processes. However, since this axiom characterizes the failure trace semantics and this is finer than failure semantics, a more general axiom is needed: axiom (F ) became our next proposal because failure semantics is the coarsest of the four semantics. More precisely, we expected to achieve the axiomatization of the four semantics in the diamond by adding the adequate instance of the generic This seemed reasonable since the other semantics in the group are finer than failures and by adding a constraint to (F ) we certainly obtain a more restricted axiom that produces a finer preorder. The conjecture turned out to be correct and we found that the semantics in the diamond can be characterized by the following instances:
(ND F ) M F (x, y, w) ⇐⇒ true (ND R ) M R (x, y, w) ⇐⇒ I(x) ⊇ I(y) (ND FT ) M F T (x, y, w) ⇐⇒ I(w) ⊆ I(y) (ND RT ) M RT (x, y, w) ⇐⇒ I(x) = I(y) and I(w) ⊆ I(y)
Since M F is the universal relation containing all triples of processes, the corresponding instance of the conditional axiom (ND ) is clearly equivalent to (F ), and thus adding it to the set {B 1 -B 4 , (RS)} we obtain a ground-complete axiomatization of ⊑ F . Let us now prove that the remaining three semantics are also axiomatized by the corresponding instances of the axiom (ND) together with (RS). (RS), (R)}. By taking x = bx ′ + u, y = by ′ , and w = v we have that (ND R ) implies (R). In the other direction, let x and y be arbitrary closed BCCSP terms with I(y) ⊆ I(x): we will prove, by structural induction on y, that {B 1 -B 4 , (RS), (R)} ⊢ a(x + y) ax + a(y + w), for any term w.
• For y = 0, we have a(x + y) ≃ ax ax + a(y + w), by application of (RS).
• For y = by ′ + y ′′ , it must be x = bx ′ + x ′′ and taking v = y ′′ + w in (R) we obtain a(x + y) = a(bx ′ + by ′ + x ′′ + y ′′ ) a(x + y ′′ ) + a(y + w). Then we have I(y ′′ ) ⊆ I(x) and we can apply the induction hypothesis to get {B 1 -B 4 , (RS), (R)} ⊢ a(x + y) ax + a(y + w). In the other direction, let w and y with I(w) ⊆ I(y), so that a(x + y) ax + ay using (F T ) and, since I(y) = I(y + w), we have y y + w using (RS): hence we conclude, a(x + y) ax + a(y + w). where (RT ) is the axiom M RT (x, y, w) ⇒ ax + ay a(x + y). This follows from the fact that, whenever I(x) = I(y), we can use (RS) to get x x + y and y x + y, and then ax + ay a(x + y). Now, the implication from left to right follows by taking w = 0. From right to left, as above, whenever I(w) ⊆ I(y) we have y y + w and then, if I(x) = I(y) we have a(x + y) ax + ay, and therefore a(x + y) ax + a(y + z). Figure 4 shows the already known relations between the semantics of the spectrum in the ready simulation layer. However, we want to stress the fact that once the new axiomatizations are proved to be correct, those relations became obvious since the four constraints defined above trivially satisfy M RT (x, y, w) ⇒ M F T (x, y, w) ∧ M R (x, y, w) and M F T (x, y, w)∨M R (x, y, w) ⇒ M F (x, y, w). It is even more important that the tight relations and the subtle differences between these semantics clearly stand out by just looking at their axiomatizations.
⊑ RS
I(x) = I(y) ⇒ x x + y ⊑ RT I(x) = I(y) ⇒ x x + y I(x) = I(y) and I(w) ⊆ I(y) ⇒ a(x + y) ax + a(y + w) ⊑ F T I(x) = I(y) ⇒ x x + y I(w) ⊆ I(y) ⇒ a(x + y) ax + a(y + w) ⊑ R I(x) = I(y) ⇒ x x + y I(x) ⊇ I(y) ⇒ a(x + y) ax + a(y + w) ⊑ F I(x) = I(y) ⇒ x x + y a(x + y) ax + a(y + w)
Certainly, if we compare our new axiomatizations and those in Table 1, the use of conditions in our axioms could be on the grounds that complex conditions could be used to hide the complexity of the semantics. However, the conditions that we have introduced for the alternative axiomatizations of the semantics in the spectrum are very simple. In any case, our main interest was to obtain a uniform presentation of the axiomatizations that could be used to simplify their generic algebraic study.
Corollary 3.3. (1) ⊑ F T is axiomatized by the set {B 1 -B 4 , (RS), (ND FT 0 )}, where (ND FT 0 ) is the instance of (ND FT ) where w is 0. (2) ⊑ RT is axiomatized by {B 1 -B 4 , (RS), (ND RT 0 )}, where (ND RT 0 ) is the instance of (ND RT ) where w is 0.
Proof. Note that for the proof of Proposition 3.2 only the case w = 0 is needed.
Even if the simplifications above are possible, we prefer to maintain the general forms of the axioms (ND FT ) and (ND RT ) to keep all axiomatizations as similar as possible, which will come in handy when proving general properties of the semantics. + + + + + + + + + + + x + y ≃ y + x + + + + + + + + + + + x + 0 ≃ x + + + + + + + + + + + x + x ≃ x + + + + + + + + + + + Proof. Note that (ND F ) implies (RS) and therefore (ND R ) implies (RS), by taking y = 0 and w = y.
I(x) = I(y) ⇒ a(x + y) ≃ a(x + y) + ay + v v v v v v v v v a(bx + by + z) ≃ a(bx + z) + a(by + z) + v v v v v v I(x) = I(y) ⇒ ax + ay ≃ a(x + y) + + v v v v ax + ay ≃ ax + ay + a(x + y) + v v v a(bx + u) + a(by + v) ≃ a(bx + by + u) + a(by + v) + + v v ax + a(y + z) ≃ ax + a(x + y) + a(y + z) + v v a(x + by + z) ≃ a(x + by + z) + a(by + z) + v v v a(bx + u) + a(cy + v) ≃ a(bx + cy + u + v) + v a(x + y) ≃ a(x + y) + ay + v ax + ay ≃ a(x + y) +
Equivalences and their preorders.
Let us now study the equivalences and first of all note that the axiom (ND ) controlling the reduction of non-determinism has been presented as an inequational axiom. Certainly, it cannot simply be replaced by the corresponding equation since, in general, it is not true that ax + ay ≃ a(x + y). However, the two dimensions corresponding to (RS) and (ND Z ) that control the "growth" of a process with respect to a preorder are not orthogonal; for example, a(x + y) a(x + y) + ax can be derived either by an application of (ND FT ) or by one of (RS). As a consequence of the relation between these two axioms, once (RS) is assumed then the inequational axiom (ND) can be substituted by its (stronger) equational form (ND ≡ ) M (x, y, w) ⇒ ax + a(y + w) + a(x + y) ≃ ax + a(y + w) .
As above, we write (ND Z ≡ ) for the concrete instances of this axiom for Z ∈ {F, R, FT, RT}. M (x, y, w) ⇒ ax + a(y + w) + a(x + y) ax + a(y + w) . Proof.
(1) We only need to prove the implication from right to left, since the other follows from being a precongruence. For that, from (RS) we get a(x + y) a(x + y) + ax + a(y + w) whence, using (ND + ), a(x + y) ax + a(y + w).
(2) We only need to prove that, if M (x, y, w), then
{B 1 -B 4 , (RS), (ND + )} ⊢ ax + a(y + w) ax + a(y + w) + a(x + y) ,
which follows from (RS).
This result can be interpreted as saying that the only way to "enlarge" a process is by extending its possible behaviors by means of the "dynamic" simulation axioms; the static rules, (ND) and its variants, instead generate new identifications among processes. Actually, any complete axiomatization of a preorder that contains the axiom (RS) can be turned into an equivalent axiomatization by replacing every inequality u v by u+v ≃ v.
= = {B 1 -B 4 , (RS)} ∪ {M ⇒ u + v ≃ v | M ⇒ u v ∈ Q ′ } is also an axiomatization of ⊑.
Proof. Analogous to the particular case considered in Proposition 3.5 above. For the sake of clarity we have preferred to present the particular case before, because it is easily stated and it corresponds to the most important instance of the general result.
Finally, to conclude this section we gather in Table 2 axiomatic characterizations for the semantic equivalences that are an alternative to the classic axioms appearing in [58].
⊑ RS ≡ RS I(x) = I(y) ⇒ x x + y I(x) = I(y) ⇒ a(x + y) ≃ a(x + y) + ay ⊑ RT ≡ RT
I(x) = I(y) ⇒ x x + y I(x) = I(y) ⇒ a(x + y) ≃ a(x + y) + ay I(x) = I(y) and I(w) ⊆ I(y) ⇒ ax + a(y + w) + a(x + y) ≃ ax + a(y + w)
⊑ F T ≡ F T I(x) = I(y) ⇒ x x + y I(x) = I(y) ⇒ a(x + y) ≃ a(x + y) + ay I(w) ⊆ I(y) ⇒ ax + a(y + w) + a(x + y) ≃ ax + a(y + w) ⊑ R ≡ R I(x) = I(y) ⇒ x x + y I(x) = I(y) ⇒ a(x + y) ≃ a(x + y) + ay I(x) ⊇ I(y) ⇒ ax + a(y + w) + a(x + y) ≃ ax + a(y + w) ⊑ F ≡ F
I(x) = I(y) ⇒ x x + y I(x) = I(y) ⇒ a(x + y) ≃ a(x + y) + ay ax + a(y + w) + a(x + y) ≃ ax + a(y + w) Figure 5: Axioms for the ready simulation layer of semantics.
Following the same ideas that we have already discussed for the preorders, a key point is to find the equations that characterize the simulation equivalence that governs each layer. As showed in [24], there is a generic axiom that we can use:
(N S ≡ ) N (x, y) ⇒ a(x + y) ≃ a(x + y) + ay.
We consider the instantiated equation that characterizes the ready simulation equivalence:
(RS ≡ ) I(x) = I(y) ⇒ a(x + y) ≃ a(x + y) + ay, and the rest of the characterization follows by using the equation (N D ≡ ) presented above.
Proposition 3.7. (1) The failure equivalence ≡ F is axiomatized by {B 1 -B 4 , (RS ≡ ), (ND F ≡ )}. (2) The readiness equivalence ≡ R is axiomatized by {B 1 -B 4 , (RS ≡ ), (ND R ≡ )}. (3) The failure trace equivalence ≡ F T is axiomatized by {B 1 -B 4 , (RS ≡ ), (ND FT ≡ )}. (4) The ready trace equivalence ≡ RT is axiomatized by the set {B 1 -B 4 , (RS ≡ ), (ND RT ≡ )}. Proof.
To prove these results we can compare the new and old axiomatizations similarly as we did in the proof of Proposition 3.5 or alternatively make use of the "ready to preorder" algorithm thoroughly studied in [4,18,21].
The results in this section clarify the entanglement between axiomatizations for preorders and equivalences. For example: for the ready simulation and its associated linear semantics, we just need three axioms (RS), (RS ≡ ) and (N D ≡ )-conveniently instantiated-to characterize the 10 relations (orders and equivalences) involved, as summarized in Figure 5.
3.2. The coarsest semantics in the spectrum. The results in Section 3.1 show the relations between the ready simulation and the linear semantics naturally associated to it. The same phenomenon occurs for other simulations. In this section we focus on the bottom part of the spectrum where lie the simulation semantics coarser than ready simulation: plain and complete simulation, and the semantics coarser than these. For the simulation semantics we obtain the corresponding axiomatizations simply by considering the universal constraint for the case of plain simulations and the complete constraint for complete simulations:
Simulation
U (x, y) ⇐⇒ true Complete simulations C(x, y) ⇐⇒ (x = 0 iff y = 0) Trace and completed trace semantics can be defined by simply adding our axiom (ND F ) to the appropriate instance of (NS) N (x, y) ⇒ x x + y. Proof. 3 Note that (S) is equivalent to (US ), the instantation of (NS ) with U as N .
(1) The classic axiomatization of trace semantics is given by {B 1 -B 4 , (S), (T )}, where (T ) is the axiom ax + ay ≃ a(x + y). Note that {B 1 -B 4 , (S), (T )} is logically equivalent to {B 1 -B 4 , (S), (T ⊑ )}, where (T ⊑ ) is the axiom a(x + y) ax + ay, because (S) can be used to obtain ax a(x + y) and ay a(x + y). And it is immediate that (ND F ) implies (T ⊑ ). Also, {(S), (T ⊑ )} ⊢ a(x + y) ax + a(y + w), since a(x + y) ax + ay by (T ⊑ ) and ax + ay ax + a(y + w) by (S).
(2) Analogous to the previous case once we realize that the classic axiom for completed trace, (CT ) a(bx+u)+a(cy+v) ≃ a(bx+cy+u+v), is equivalent to the conditional axiom C(x, y) ⇒ ax+ay ≃ a(x+y). This follows because bx+u and cy+v are two independent patterns describing non-null processes and when the condition is instantiated with x and y equal to 0 the identity is trivial: a0 + a0 ≃ a0.
By an argument analogous to that in Proposition 3.5, we can obtain for ⊑ T the axiomati-
zation {B 1 -B 4 , (S), (ND F ≡ )}. Note that although (ND F ≡ )
is an equation, this axiomatization is not the classic one; obviously, (T ) ax + ay = a(x + y) implies (ND F ≡ ) but the converse is false.
It is easy to check that in the case of trace semantics, the particular instance (ND 0 ) of the axiom (ND) with w equal to 0 is powerful enough to generate the trace preorder. This was certainly not the case when we were under ready simulation, where (ND 0 ) just generates the failure trace preorder instead of the coarser failures preorder.
It is also interesting to note that for the trace semantics the symmetric version of (ND),
(ND vw ) a(x + y) a(x + v) + a(y + w),
is also valid, so we can take both
{B 1 -B 4 , (S), (ND vw )} and {B 1 -B 4 , (S), (ND ≡ vw )}, where (ND ≡ vw ) a(x + v) + a(y + w) + a(x + y) ≃ a(x + v) + a(y + w)
, as alternative axiomatizations of the trace preorder.
Should we expect another diamond of "reasonable" semantics under plain simulation in the spectrum? Were that to be the case, why have we only found the trace semantics?
In order to answer these questions, note that the diamond of semantics under ready simulation was completely governed by the function I, which appears in the constraints of the different instantiations of the axiom (ND). For plain simulations, however, the trivially true predicate U (x, y) corresponds to the observation function that can see nothing. As a consequence, if we substitute U for I in each of the four constraints of the diamond they all collapse into a single one: trace semantics. Nevertheless, an alternative path can be explored to obtain new semantics: let us keep the different axioms (ND Z ) the way they stand and simply replace (RS) by (S). Then we obtain the following results:
Proposition 3.9. {B 1 -B 4 , (S), (ND FT )} is another axiomatization of trace semantics.
Hence, under (S) the failures and the failure trace axioms generate the same preorder, namely the trace preorder.
Proof. {B 1 -B 4 , (S), (ND 0 )} is a complete axiomatization of trace preorder, and (ND 0 ) is a particular case of (ND FT ).
The axioms corresponding to readiness and ready trace, however, give rise to two new semantics that we shall name extended ready and extended ready trace semantics. They are defined by the order obtained by inclusion of the offers of the processes, either just at the end of a trace, or after each action within it: in order to have p ⊑ ER q, for each p α =⇒ p ′ with I(p ′ ) = R we need some q α =⇒ q ′ with I(q ′ ) ⊇ R; the extended ready trace preorder ⊑ ERT is defined analogously, but using ready traces.
Proposition 3.10. (1) The set {B 1 -B 4 , (S), (ND R )} is an axiomatization of ⊑ ER . (2) The set {B 1 -B 4 , (S), (ND RT )} is an axiomatization of ⊑ ERT .
Let us now consider the versions of the axioms (ND R ), (ND FT ), (ND RT ) where the constraint I has been replaced by the completeness condition C defined by C(x) ⇐⇒ x = 0:
(C-ND R ) M CR (x, y, w) ⇐⇒ (C(x) implies C(y)) (C-ND FT ) M CFT (x, y, w) ⇐⇒ (C(y) implies C(w)) (C-ND RT ) M CRT (x, y, w) ⇐⇒ (C(x) iff C(y)) and (C(y) implies C(w))
Once again, we simply obtain three alternative axiomatizations of the completed trace semantics.
Proposition 3.11. The following axiomatizations are equivalent:
(1) {B 1 -B 4 , (CS), (ND F )}. (2) {B 1 -B 4 , (CS), (C-ND R )}. (3) {B 1 -B 4 , (CS), (C-ND FT )}. (4) {B 1 -B 4 , (CS), (C-ND RT )}.
Proof. Clearly, (1) ⇒ (2) ⇒ (3) ⇒ (4) and therefore it is enough to prove that (4) ⇒ (1). If x and y are not 0 we can apply (C-ND RT ) to obtain the inequality in (ND F ). If x is 0 but y is not, we need to obtain ay a0 + a(y + w). By (CS) we have y y + w and then ay a(y + w); applying (CS) again, a(y + w) a(y + w) + a0 and thus ay a0 + a(y + w). If y is 0 but x is not, we need to obtain ax ax + aw, which results from an immediate application of (CS). Finally, if both x and y are 0, a0 a0 + aw.
As before, if we consider the original axioms (ND R ), (ND FT ), and (ND RT ) we obtain, together with an alternative axiomatization of the completed trace semantics, two new semantics. Proof. It is enough to prove that (C-ND FT ) can be derived from
{B 1 -B 4 , (CS), (ND F )}.
• If y is 0 we then have w equal to 0 and can apply (ND FT ).
• If y is not 0 we can apply (ND FT 0 ) to obtain a(x+y) ax+ay and then (CS) to conclude that a(x + y) ax + a(y + w).
By contrast, as happened for plain simulations, under (CS) the axioms of the ready semantics generate two slightly different versions of the extended ready and extended ready trace semantics introduced before, that we call extended complete ready and extended complete ready trace semantics. In order to have p ⊑ ECR q, whenever p α =⇒ p ′ with I(p ′ ) = ∅ we require some q α =⇒ q ′ with I(q ′ ) ⊇ I(p ′ ), but if I(p ′ ) = ∅ then the corresponding q ′ also has to satisfy I(q ′ ) = ∅. The extended complete ready trace preorder ⊑ ECRT is defined in an analogous way, starting from the ready traces of the processes.
As we did in Section 3.1.2, we can prove that the axioms that characterize trace and completed trace preorders reflect the fact that the order relation is inherited from simulation and complete simulation, respectively, and that the role of the static rules is to introduce identifications. As stated in Proposition 3.12 above, the only inequation that we use to axiomatize the trace and completed trace orders is (S), the remaining axioms being equational axioms.
Proposition 3.13. (1) {B 1 -B 4 , (S), (ND F )} is logically equivalent to {B 1 -B 4 , (S), (ND F ≡ )}. (2) {B 1 -B 4 , (CS), (ND F )} is logically equivalent to {B 1 -B 4 , (CS), (ND F ≡ )}.
A similar discussion could have been carried out for trace and completed trace equivalences, and indeed a very natural axiomatization for these relations can be obtained based on the corresponding instantiation of the (N S ≡ ) equation:
(S ≡ ) a(x + y) ≃ a(x + y) + ay (CS ≡ ) C(x, y) ⇒ a(x + y) ≃ a(x + y) + ay . Proposition 3.14. (1) The trace equivalence ≡ T is axiomatized by {B 1 -B 4 , (S ≡ ), (ND F ≡ )}. (2) The completed trace equivalence ≡ CT is axiomatized by {B 1 -B 4 , (CS ≡ ), (ND F ≡ )}.
To conclude this section devoted to the unification of the equational characterizations of process semantics, we present in Figure 6 a condensed view of our new spectrum. This presentation exploits in an expressive way the two dimensions of the picture, which in fact reflects a tridimensional structure. On the lefthand side the constrained simulations and bisimulations appear, totally ordered from top to bottom. Each constrained simulation generates a layer of semantics. Here, we have only detailed the layers corresponding to ready simulation and that of plain simulation. As a matter of fact, the latter degenerates to a single point due to the simplicity of the constraint U governing plain simulations. The naturality of the semantics appearing in this part of the spectrum is illustrated by our generic axiomatization, where a single (constrained) simulation axiom governs all the constrained simulation semantics, whereas adding a single axiom we complete the axiomatizations of each of the linear semantics at the righthand side of the picture.
Observational semantics
Along Section 3 we have presented some views of the axiomatizations for process semantics that highlight the common properties and the subtle differences between them; likewise these views of the axiomatic characterizations point out the similarities between the preorder and the equivalence of a given semantics.
In this section we focus on the characterizations of process semantics based on observations. Indeed, this idea of determining the semantics by means of observations lies deep inside the foundations of process theory.
Our calculus is founded in two central ideas. The first is observation; [. . . ] two systems are indistinguishable if we cannot tell them apart without pulling them apart. We therefore give a formal definition of observation equivalence and investigate its properties. [41] Imagine there is an observer with a notebook who watches the process and writes down the name of each event as it occurs.
[32] Besides the classical references to Milner and Hoare, this idea of observation pervades the Hennessy's testing methodology [30] and most of the work on linear semantics. Observations, in spite of the variations in different proposals, constitute a denotational space closely related to the classical developments of semantics based on denotations for programming languages [52].
In this section we will show how most of the semantics can be characterized with one of the two main families of observations:
• Branching general observations, Section 4.1, that are essentially labeled trees, that characterize the simulation semantics: simulation, complete simulation, ready simulation, nested simulation, . . . • Linear observations, a simplified case of branching observations, Section 4.2, that characterize the linear semantics: traces, failures, readiness, ready trace, . . . We consider also in Section 4.3 a more exotic kind of observations, deterministic branching observations, which are essentially deterministic trees. Possible worlds semantics is the only semantics appearing in the classical spectrum in this class, although, our general approach will show how this kind of observations define new full families of process semantics.
To develop this observational characterization for process semantics allows us to deepen into the ultimate nature of the similarities and differences between them. Along this section we present a thorough study of the local observation functions that generate the local observations of the states, Figure 10. For the linear case, there is also the possibility of observing this local information in a partial way and this is how for each local observer, in principle, up to four different semantics can be obtained. This fact explains the classic diamond below the ready simulation semantics formed by the failures, failure trace, readiness, and ready trace semantics. Again, the generality of our study makes it exportable to other simulation layers enriching and completing the spectrum of semantics, Figure 11.
Finally, from a methodological point of view, the unification of observational semantics that we present in this section introduces all the technical machinery needed to rewrite the proofs of Section 3 in a generic way, proving that the two unification procedures produce characterizations of the same semantics. We will address this topic in Section 5. Let us now concentrate on the observational semantics.
4.1.
Branching general observations. In order to characterize the simulation semantics in an extensional way we need local and branching general observations. (1) A branching general observation (bgo for short) of a process is a finite, non-empty tree whose arcs are labeled with actions in Act and whose nodes are labeled with local observations from L N , for N a constraint; the corresponding set BGO N is recursively defined as:
bgo 1 bgo 2 {a} a } } ④ ④ ④ ④ ④ ④ ④ ④ a 3 3 ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ {b} b {b} b {c} {d} {a} a {b} b } } ④ ④ ④ ④ ④ ④ ④ ④ b 3 3 ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ {c} {d}• l, ∅ ∈ BGO N for l ∈ L N . • l, {(a i , bgo i ) | i ∈ 1.
.n} ∈ BGO N for every n ∈ IN, a i ∈ Act and bgo i ∈ BGO N . (2) The set BGO N (p) of branching general observations of p corresponding to the constraint N is
BGO N (p) = { L N (p), S | S ⊆ {(a, bgo) | bgo ∈ BGO N (p ′ ), p a −→ p ′ }} . (3) We write p ≤ b N q if BGO N (p) ⊆ BGO N (q).
In Figure 7 some simple examples of bgo's for N = I are shown. We represent bgo 1 as
{a}, {(a, {b}, {(b, {c}, ∅ )} ), (a, {b}, {(b, {d}, ∅ )} )} and bgo 2 as {a}, {(a, {b}, {(b, {c}, ∅ ), (b, {d}, ∅ )} )} .
We use braces for the set of children of a node, parentheses to represent a branch of the tree as a pair (initial arc, subtree below), and angular brackets to represent each tree as a pair root, children . Note that the bgo's of a process p described by its transition system can be generated by inductively applying the clauses defining the set BGO N (p), even when p is infinite. For instance, if N = I and we consider the process p :
bgo 1 bgo 2 bgo 3 {a} {a} a {b} b {c, d} d 3 3 ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ c } } ④ ④ ④ ④ ④ ④ ④ ④ ④ ∅ ∅ {a} a } } ④ ④ ④ ④ ④ ④ ④ ④ a 4 4 ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ {b} b {b} b {c} c {c, d} d ∅ ∅:= c.p defining a clock, since ∅ ⊆ {(c, bgo) | bgo ∈ BGO I (p), p c −→ p}, it follows that {c}, ∅ ∈ BGO I (p). But now {(c, {c}, ∅ )} ⊆ {(c, bgo) | bgo ∈ BGO I (p), p c −→ p} and therefore {c}, {(c, {c}, ∅ )} ∈ BGO I (p), and so on.
It is clear that the bgo's of a process have an operational flavor. The nodes of the observations correspond to its states and the arcs to its transitions; this is why we will be able to define the orders associated to the different simulation semantics simply by set inclusion over the sets of bgo's.
Let us also comment on the fact that in all five cases that we have considered in Definition 4.1, which correspond to the five constrained simulation semantics in the spectrum, the local observation functions L N define a representation of the equivalence relation N used to define the constrained simulation relations. This means that we have
L N (p) = L N (q) ⇐⇒ pN q. Example 4.3. For N = I, if x = b(c + d) and y = bc + bd, then for p = a(x + y) we have bgo k ∈ BGO I (p) for k ∈ {1, 2, 3},
where the bgo's are depicted in Figure 8. It is easy to check that all of them are also branching observations of q = a(x + y) + ax. As a matter of fact, we have BGO I (p) = BGO I (q). Note that in order to obtain bgo 3 ∈ BGO I (p) we need to combine two different observations of the process x + y, which is the only p ′ such that a(x + y) a −→ p ′ . In contrast, for p = a(bc + bd) and q = abc + abd, BGO I (q) ⊆ BGO I (p), since for the branching observation bgo 1 in Figure 7 we have bgo 1 ∈ BGO I (q) and bgo 1 ∈ BGO I (p). And also, we have BGO I (p) ⊆ BGO I (q), since for bgo 2 as in Figure 7 we have bgo 2 ∈ BGO I (p), but bgo 2 / ∈ BGO I (q). The key idea is that we can indeed include in a single bgo two separated computations but we cannot "mix" two different ones, even if the labels both in their initial transitions and in the local observations of the reached nodes were the same. This is why bgo 2 / ∈ BGO I (q).
The following simple properties will be immediate consequences of Theorem 4.9 below; we use them here to illustrate the expressive power of each kind of bgo. (1), this produces the result.
Definition 4.4. An axiom t u, respectively t ≃ u, is satisfied in a model BGO N if BGO N (t ′ ) ⊆ BGO N (u ′ ), respectively BGO N (t ′ ) = BGO N (u ′ ), for every possible ground instantiation t ′ u ′ or t ′ ≃ u ′ of the axiom. Proposition 4.5. (1) The axiom (S) x x + y is satisfied in the model BGO U . (2) The axiom (S ≡ ) a(x + y) ≃ a(x + y) + ax is satisfied in the model BGO U . Proof. (1) It is an immediate consequence of the fact that if p a −→ p ′ then p + q a −→ p ′ , and therefore {a | p a −→} ⊆ {a | p + q a −→}. (2) Again, it is a simple exercise to check that BGO U (p) ⊆ BGO U (q) implies BGO U (ap) ⊆ BGO U (aq), and that if BGO U (p), BGO U (q) ⊆ BGO U (r), then BGO U (p+q) ⊆ BGO U (r); in combination withProposition 4.6. BGO I (p) ⊆ BGO I (p + q) iff I(q) ⊆ I(p).
Proof. (⇐) Since I(p + q) = I(p), the root of the bgo's is the same for both processes and obviously p + q has all the observations of p.
(⇒) If I(q) ⊆ I(p), then I(p) = I(p + q) and then no bgo of p is a bgo of p + q because the roots of the observations of both processes are different.
The fact, that we now prove, that the observational semantics BGO N (p) can be defined in a compositional way, is an important property that will simplify the proofs of many of their properties.
• BGO N (ap) = { a L L(p), {(a, bgo) | bgo ∈ B} | B ⊆ BGO N (p)}. • BGO N (p+q) = { L(p)+ L L(q), S 1 ∪S 2 | L(p), S 1 ∈ BGO N (p), L(p), S 2 ∈ BGO N (q)}.
Proof. The first equality is immediate by definition of BGO N (ap). As for the second, we only need to realize that p + q a −→ r iff p a −→ r or q a −→ r: then, the set of children of the root labeled by L N (p + q) at any bgo ∈ BGO N (ap) correspond to the union of the two sets of children that contain some bgo's of processes p i such as p a −→ p i (and then p + q a −→ p i ) or q i such that q a −→ q i (and then p + q a −→ q i ). Note that from the equalities above it follows that BGO N (p) can be computed compositionally.
In particular, BGO N (p) is compositional for any of the constraints considered in Definition 4.1. Proof. The result for U is obvious since it is a degenerate semantics that identifies all processes. By Theorem 4.7 and Theorem 4.9 below we can conclude that the simulation semantics can indeed be denotationally defined. The result for traces is well-known, while I and C can be easily defined denotationally since I(ap) = {a} and I(p + q) = I(p) ∪ I(q).
⊑ NS q iff p ≤ b N q. Proof. (⇒) Let p =
ap i a and q = aq j a ; if p ⊑ NS q, then pN q and therefore L N (p) = L N (q). Now we proceed by induction on p. If p ≡ 0 the result is trivial. Otherwise, for every a ∈ I(p) such that p
a −→ p ′ there exists q a −→ q ′ such that p ′ ⊑ NS q ′ . By induction hypothesis BGO N (p ′ ) ⊆ BGO N (q ′ ) from where, by the definition of BGO N (p), it follows that BGO N (p) ⊆ BGO N (q). (⇐) Let us show that the relation R = {(p, q) | BGO N (p) ⊆ BGO N (q)} is an N - simulation. If (p, q) ∈ R, then L N (p) = L N (q) because L N (p), ∅ ∈ BGO N (q) and thus pN q. Now, for each p a −→ p ′ we have { L N (p), {(a, bgo)} | bgo ∈ BGO N (p ′ )} ⊆ BGO(q) and therefore there must exist some q a −→ q ′ such that BGO N (p ′ ) ⊆ BGO N (q ′ ), so that (p ′ , q ′ ) ∈ R.
Note that for this result to hold it is only required that the local observation function L N satisfies pN q iff L N (p) = L N (q). That is, L N must compute a concrete representative of the equivalence class defined by N and this stresses again the interest of using behavior equivalences N as constraints for the definition of constrained simulations. Let us recall that, in principle, any behavior preorder could be used as such a constraint. For instance, the predicate I ⊇ defined by pI ⊇ q iff I(q) ⊆ I(p) could be used to define I ⊇ -simulations (which in fact coincide with I-simulations). But from I(q) ⊆ I(p) we cannot conclude that L N (p) = L N (q) and, hence, either a more complicated characterization of ⊑ NS in terms of bgo's or an additional argument to show that p ⊑ I ⊆ q implies I(p) ⊆ I(q) would be needed. And although this is obvious for a constraint as simple as I, or even T or S, it could be far from trivial for other, more complex constraints: therefore, it is always advisable to consider equivalence behaviors as constraints.
(p) = L N (q) ⇐⇒ pN q, then p ⊑ NS q iff BGO N (p) ⊆ BGO N (q).
The results above bring forward the fact that despite the resemblance between the bgo's of a process and its computation tree, the possibility of mixing several computations in a single branching observation makes it possible to identify non-bisimilar processes by their sets of branching observations. 4.2. Linear observations and linear time semantics. We introduce the linear observations of a process as a particular (degenerate) case of branching observations: those with a linear structure. (1) The set LGO N of linear general observations (lgo for short) for a local observer L N is the subset of BGO N defined as:
• l, ∅ ∈ LGO N for each l ∈ L N . • l, {(a, lgo)} , whenever a ∈ A and lgo ∈ LGO N . (2) The set of linear general observations of a process p with respect to the local observer
L N is LGO N (p) = BGO N (p) ∩ LGO N .
Since lgo's are linear they can be presented as traces, avoiding the sets of descendants in the bgo's. Therefore, we will consider them as elements of the set L N × (Act × L N ) * . It is also clear that the set of linear observations can be defined recursively without resorting to branching observations. Definition 4.12. The set LGO N (p) of linear general observations of a process p is recursively defined by
LGO N (p) = { L N (p) } ∪ { L N (p), a • lgo | p a −→ p ′ , lgo ∈ LGO N (p ′ )}.
We can also compute LGO N (p) in a compositional way.
L N → L N satisfying L(ap) = a L L(p) and L(p + q) = L(p) + L L(q). Then: • LGO N (ap) = { a L L(p) } ∪ { a L L(p), a • LGO N (p)}. • LGO N (p + q) = { L(p) + L L(q) • t | L(p) • t ∈ LGO N (p) or L(p) • t ∈ LGO N (q)}.
Proof. Just like that of Theorem 4.7.
Obviously, for N = U we have that LGO U is isomorphic to Act * and thus LGO U (p) = Traces(p). By contrast, for N = I, LGO I (p) is the set of ready traces of p, ReadyTraces(p).
Set inclusion of linear observations with respect to a local observer L N gives us the preorder defining the corresponding semantics.
Definition 4.14. A process p is less than or equal to q with respect to the linear observations generated by L N , denoted p ≤ l N q, if LGO N (p) ⊆ LGO N (q). We will denote the corresponding equivalence by = l N .
Proposition 4.15. (1) ≤ l U = ⊑ T ; (2) ≤ l I = ⊑ RT ; (3) ≤ l C = ⊑ CT . Proof. It is trivial, since LGO U (p) = Traces(p),
LGO I (p) = ReadyTraces(p), and LGO C (p) = {(false, a 1 )• . . . •(false, a n , true), (false, a 1 )• . . . •(false, a i , false) | a 1 . . . a n ∈ CompleteTraces(p), i < n}. Proof. The implication follows from Theorem 4.9 and the fact that lgo's are just a particular case of bgo's. To see that the converse is false in general consider
N = U ; we have ⊑ U S = ⊑ S and ≤ l U = ⊑ T , and it is well-known that ⊑ S ⊆ ⊑ T since, for instance, a(b + c) ⊑ S ab + ac, but a(b + c) = T ab + ac.
Therefore, by means of linear observations and set inclusion we can characterize the orders that define some of the semantics in the spectrum which are not simulation semantics. However, there are still some other semantics for which a different way of treating the linear observations is needed. We need to introduce some identifications in the corresponding domain LGO N to obtain their characterizations.
• T ≤ l⊇ I T ′ ⇐⇒ for all X 0 a 1 X 1 . . . X n ∈ T there is some Y 0 a 1 Y 1 . . . Y n ∈ T ′ with X i ⊇ Y i , for all i ∈ 0..n.• T ≤ lf I T ′ ⇐⇒ for all X 0 a 1 X 1 . . . X n ∈ T there is some Y 0 a 1 Y 1 . . . Y n ∈ T ′ with X n = Y n . • T ≤ lf ⊇ I T ′ ⇐⇒ for all X 0 a 1 X 1 . . . X n ∈ T there is some Y 0 a 1 Y 1 . . . Y n ∈ T ′ with X n ⊇ Y n . Then, for each δ ∈ {⊇, f, f ⊇} we write p ≤ lδ I q if LGO I (p) ≤ lδ I LGO I (q)
. Since the definition of ≤ lf I ignores all the intermediate ready sets X i with i < n and requires the final ready sets to coincide, it is obvious that it defines the readiness preorder. Let us now prove that the two semantics based on failures are also characterized by our preorders ≤ lf ⊇ I and ≤ l⊇ I . Proof. The proof is based on the definition of initial failures of a process: we say that p rejects X if and only if X ∩ I(p) = ∅. Then, α, X is a failure of p if and only if p α =⇒ p ′ and p ′ rejects X. Using lgo's, for α = a 1 . . . a n , α, X is a failure of p iff there exists X 0 a 1 . . . X n ∈ T such that X n ∩ X = ∅. Thus:
p ⊑ F p ′ ⇐⇒ Failures(p) ⊆ Failures(p ′ ) ⇐⇒ α, X ∈ Failures(p ′ ) for all α, X ∈ Failures(p) ⇐⇒ X 0 a 1 . . . X n ∈ LGO I (p) with X n ∩ X = ∅ implies that there exists Y 0 a 1 . . . Y n ∈ LGO I (p ′ ) with Y n ∩ X = ∅ , and then p ≤ lf ⊇ I p ′ implies p ⊑ F p ′ . Conversely, assume that p ⊑ F p ′ and recall that p ≤ lf ⊇ I p ′ iff for all t = X 0 a 1 . . . X n ∈ LGO I (p) there exists Y 0 a 1 . . . Y n ∈ LGO I (p ′ ) such that X n ⊇ Y n .
For each set X, let us denote by X c its complement. If t ∈ LGO I (p), we have α, X c n ∈ Failures(p) and therefore α, X c n ∈ Failures(p ′ ), which implies that there exists p ′ α =⇒ p ′′ such that I(p ′′ ) ∩ X c n = ∅. This means that there is some t ′ = Y 0 a 1 . . . a n I(p ′′ ) ∈ LGO I (p ′ ) with I(p ′′ ) ⊆ X n , and therefore we can conclude that p ≤ lf ⊇ I p ′ . The proof for failure trace is very similar and we omit it.
As a matter of fact, the characterization of failures by means of the reverse inclusion of offerings is not a great discovery at all: for instance, the same idea can be found in the definition of acceptance trees [29]. However, our sets of linear observations produce quite a nice characterization and allow us to forget about the notion of failures and consider instead reverse inclusion of offerings. But the most important property of our characterizations in terms of different orders on the set LGO I is that they can be generalized to other local observation functions.
Definition 4.19. For T , T ′ ⊆ LGO N we define the orders ≤ l⊇ N , ≤ lf N , and ≤ lf ⊇ N by: • T ≤ l⊇ N T ′ ⇐⇒ for all X 0 a 1 X 1 . . . X n ∈ T there is some Y 0 a 1 Y 1 . . . Y n ∈ T ′ with X i ⊇ Y i for all i ∈ 0..n. • T ≤ lf N T ′ ⇐⇒ for all X 0 a 1 X 1 . . . X n ∈ T there is some Y 0 a 1 Y 1 . . . Y n ∈ T ′ with X n = Y n . • T ≤ lf ⊇ N T ′ ⇐⇒ for all X 0 a 1 X 1 . . . X n ∈ T there is some Y 0 a 1 Y 1 . . . Y n ∈ T ′ with X n ⊇ Y n . Then, for each δ ∈ {⊇, f, f ⊇} we write p ≤ lδ N q if LGO N (p) ≤ lδ N
LGO N (q). By abuse of notation, we have used the superset inclusion symbol ⊇ in the definitions above for any N . That is indeed the right interpretation for the cases N = I, T ; however, for N = U, C the superset inclusions degenerate to equalities while for N = S it should be interpreted as [[p]] S ≥ S [[q]] S . Then, with the right notation we could have used such an
inequality [[p]] N ≥ N [[q]] N in all the cases.
When defining an observational semantics one expects the order between processes to be plain set inclusion as is the case, for instance, for the classic definition of failures semantics. Fortunately, it is easy to obtain such a characterization for the three semantics considered above by means of some suitable closure operators.
Definition 4.20. For T ⊆ LGO N , the following three closures are defined: , f, f ⊇} and T ,
• T ⊇ = {X 0 a 1 X 1 . . . a n X n | there is some Y 0 a 1 Y 1 . . . a n Y n ∈ T with X i ⊇ Y i for i ∈ 0..n}. • T f = {X 0 a 1 X 1 . . . a n X n | there is some Y 0 a 1 Y 1 . . . a n X n ∈ T }. • T f ⊇ = {X 0 a 1 X 1 . . . a n X n | there is some Y 0 a 1 Y 1 . . . a n Y n ∈ T with X n ⊇ Y n }.T ′ ⊆ LGO N , then T ⊆ T δ and T δ δ = T δ ; also, if T ⊆ T ′ then T δ ⊆ T ′ δ .
Proof. The first and third conditions are immediate from the definitions. As for the second,
let X 0 a 1 X 1 . . . a n X n ∈ T f f . Then, there exists Y 0 a 1 Y 1 .
. . a n X n ∈ T f and thus there exists Z 0 a 1 Z 1 . . . a n X n ∈ T , which implies X 0 a 1 X 1 . . . a n X n ∈ T f ; the inclusion in the other direction follows from monotonicity. Analogously for the other two operators.
Proposition 4.22. For all δ ∈ {⊇, f, f ⊇}, T ≤ lδ N T ′ iff T δ ⊆ T ′ δ .
Proof. It is easy but tedious, so only the case δ = f ⊇ is presented in detail. Assume
T ≤ lf ⊇ N T ′ : for all t = X 0 a 1 X 1 . . . a n X n ∈ T there exists Y 0 a 1 Y 1 . . . a n Y n ∈ T ′ with X n ⊇ Y n and hence t ∈ T ′ f ⊇ and T ⊆ T ′ f ⊇ ; T f ⊇ ⊆ T ′ f ⊇ follows because of the properties of closures. Conversely, from T f ⊇ ⊆ T ′ f ⊇ it follows that T ⊆ T ′ f ⊇ and thus for all X 0 a 1 X 1 . . . a n X n ∈ T there exists Y 0 a 1 Y 1 . . . a n Y n ∈ T ′ with X n ⊇ Y n : therefore T ≤ lf ⊇ N T ′ . Definition 4.23. For each δ ∈ {⊇, f, f ⊇}, p ∈ BCCSP, and N a constraint, we define LGO δ N (p) = LGO N (p) δ .
Let us see which of the semantics in the spectrum are characterized by the orders ≤ lδ N defined above.
Proposition 4.24. For N = U we have ≤ l U = ≤ l⊇ U = ≤ lf U = ≤ lf ⊇ U = ⊑ T .
As a consequence, the only semantics coarser than plain simulation that can be characterized by means of linear observations using L U is the trace semantics.
Proof. The first three equalities are obvious since U provides useless (empty) local information (L U = {·}). The last equality was proved in Proposition 4.15(1).
N = C we have ≤ l C = ≤ l⊇ C = ≤ lf C = ≤ lf ⊇ C = ⊑ CT .
As a consequence, the only semantics coarser than complete simulation that can be characterized by means of linear observations using L C is the completed trace semantics.
Proof. Note that the local information at the intermediate steps of traces in LGO C has to be false, since it corresponds to non-terminated states; thus, only the final states provide real information. Since in this case ⊇ corresponds to Boolean equality, the first three equalities follow; the fourth was proved in Proposition 4.15 (3).
Proposition 4.26. For N = I, ≤ lf ⊇ I
characterizes the failures semantics, ≤ lf I the readiness semantics, ≤ l⊇ I the failure trace semantics, and ≤ l I the ready trace semantics. Therefore, the possible worlds semantics is the only semantics in the ltbt spectrum coarser than ready simulation that cannot be characterized using lgo I 's.
Proof. We have already proved (Propositions 4.15 and 4.18) the four characterizations, while ⊑ P W cannot be characterized using lgo I 's because all the information available in our lgo I 's was needed to capture the ready trace semantic and it is well-known that the possible worlds semantics is strictly finer.
As we will see in Section 4.3, the possible worlds semantics is the only deterministic branching semantics in the spectrum and will require the use of the deterministic branching observations introduced there to be characterized in an observational way. This is not the case, however, for the possible futures semantics (already discussed in [58]), and the impossible futures semantics [59].
p ⊑ IF q if for all S ⊆ P(Act * ), if p α =⇒ p ′ with T (p ′ ) ∩ S = ∅ then there exists q α =⇒ q ′ with T (q ′ ) ∩ S = ∅.
(2) The possible futures semantics is defined as:
p ⊑ P F q if p α =⇒ p ′ then there exists q α =⇒ q ′ with T (q ′ ) = T (p ′ ).
Proposition 4.28.
(1) ≤ lf T is the possible futures preorder.
(2) ≤ lf ⊇ T is the impossible futures preorder. Proof. (1) Obvious. (2) Assume that p ≤ lf ⊇ T q. Then p α =⇒ p ′ , with α = a 1 . . . a n , implies q α =⇒ q ′ with T (q ′ ) ⊆ T (p ′ ). Therefore, if p α =⇒ p ′ with T (p ′ )∩X = ∅ then q α =⇒ q ′ with T (q ′ )∩X = ∅ which implies p ⊑ IF p ′ . Conversely, if p ⊑ IF q, t = X 0 a 0 X 1 . . . X n ∈ LGO T (p) and p α =⇒ p ′ with α = a 1 . . . a n , obviously we have T (p ′ ) ∩ T (p ′ ) c = ∅, where T (p ′ ) c just represent the com- plement of the set T (p ′ ). Now applying the definition of ⊑ IF , we have some q α =⇒ q ′ with T (q ′ ) ∩ T (p ′ ) c = ∅. Hence, there exists t ′ = X ′ 0 a 0 X ′ 1 . . . X ′ n ∈ LGO T (q) with T (q ′ ) ⊆ T (p ′ ), which implies p ≤ lf ⊇ T q.
As a matter of fact, the possible futures semantics is just below the 2-nested simulation semantics in the spectrum only because the trace simulation semantics is missing there.
At this point we are ready to present our first two "missing links", which arise through the remaining two orders: ≤ l T and ≤ l⊇ T .
ExtSimFailures(p) = { α, p ′′ | α ∈ A * , p α =⇒ p ′ , p ′ ⊑ S p ′′ }.
The simulation failures of a process p are defined as (1) We say that a bgo is deterministic if the set of children {(a i , bgo i )} of every node satisfies a i = a j whenever i = j. We denote with dBGO N the set of deterministic observations in BGO N . (2) The set of deterministic branching observations (dbgo for short) of a process p is
SimFailures(p) = { α, B | p α =⇒ p ′ , B ∩ BGO U (p ′ ) = ∅}. We write p ⊑ SF q iff SimFailures(p) ⊆ SimFailures(q).dBGO N (p) = BGO N (p) ∩ dBGO N . (3) We write p ≤ db N q if dBGO N (p) ⊆ dBGO N (q)
. Like the linear observations, the set dBGO N (p) can be defined recursively and the corresponding semantics, compositionally.
Example 4.33. For the two processes p = a(bc + bd) and q = abc + abd we have that both deterministic observations in Figure 9 belong to dBGO I (p) and dBGO I (q). Indeed, that must be the case since it is easy to check that dBGO I (p) = dBGO I (q).
In order to prove that dbgo's for the constraint I characterize the possible worlds semantics we first recall the definition of that semantics in [58]. The set of possible worlds of p is denoted by P W (p). We define the order p ⊑ P W q iff P W (p) ⊆ P W (q). In our proof below we will relate the dbgo's in dBGO I (p) and the possible worlds in P W (p). When necessary, we will consider observations in dBGO I (p) as processes in BCCSP by removing the information from their nodes; by abuse of notation we will also denote with dbgo the process obtained after such a removal. Also, we call complete those observations that, for every node labeled by an offering A, have a branch labeled by each of the actions in A. We also associate to a deterministic process q its universal (complete deterministic) branching observation.
Definition 4.36. For a deterministic process p, its universal deterministic branching observation cdbgo(p) is:
• cdbgo(0) = ∅, ∅ . • cdbgo( a∈A ap a ) = A, {(a, cdbgo(p a )) | a ∈ A} .
The following result is now immediate. Proof. By structural induction on q:
• If q is 0, then p ≡ 0 and ∅, ∅ ∈ cdBGO I (0). • If q is aq a , since q ∈ P W (p) we have q ⊑ RS p. This implies I(q) = I(p) and that, for all a ∈ A, there exists p a −→ p a , q a ⊑ RS p a , so that q a ∈ P W (p a ). By induction hypothesis, cdbgo(q a ) ∈ cdBGO I (p). Now, by definition, cdbgo(q) = A, {(a, cdbgo(q a )) | a ∈ A)} and, from p a −→ p a and I(p) = I(q), we conclude cdbgo(q) ∈ dBGO I (p) and therefore cdbgo(q) ∈ cdBGO I (p).
Lemma 4.39. For every process q such that cdbgo(q) ∈ cdBGO I (p) we have q ⊑ RS p and therefore q ∈ P W (p).
Proof. We will prove that the set S = {(q, p) | cdbgo(q) ∈ cdBGO I (p)} is a ready simulation. Obviously, for (q, p) ∈ S it is I(q) = I(p) and, if q a −→ q a , there exists p a −→ p a with cdbgo(q a ) ∈ cdBGO I (p a ), which shows that (q a , p a ) ∈ S and that S is a ready simulation.
Theorem 4.40. For all processes p 1 , p 2 ∈ BCCSP, p 1 ⊑ P W p 2 iff p 1 ≤ db I p 2 . Proof. (⇐) For q ∈ P W (p 1 ), by Lemma 4.38 we have cdbgo(q) ∈ cdBGO I (p 1 ) and therefore cdbgo(q) ∈ cdBGO I (p 2 ). Now, by Lemma 4.39, q ⊑ RS p 2 and thus q ∈ P W (p 2 ).
(⇒) Let dbgo ∈ dBGO I (p 1 ): by definition of dBGO I (p 1 ) it is clear that we can extend dbgo into some dbgo ′ ∈ cdBGO I (p 1 ). Now, by Lemma 4.39, dbgo ′ ⊑ RS p 1 (taking dbgo ′ as a deterministic process). Therefore, dbgo ′ ∈ P W (p 1 ) and thus dbgo ′ ∈ P W (p 2 ) and, by Lemma 4.38, cdbgo(dbgo ′ ) = dbgo ′ ∈ cdBGO I (p 2 ): hence dbgo ∈ dBGO I (p 2 ) as required.
Remark 4.41. If we consider infinite processes, then our characterization of ⊑ P W by means of ≤ db I only works if we restrict ourselves to image-finite processes. We will continue the discussion on this part when studying the logical characterization of this semantics at Section 6.
Let us briefly consider the remaining new semantics definable by means of deterministic branching observations. It is clear that in all cases the corresponding orders verify ≤ b N ⊆ ≤ db N ⊆ ≤ l N , so that the associated semantics will be situated between the corresponding semantics defined by branching observations in BGO N and linear observations in LGO N , as is the case for the possible worlds semantics, located between the ready simulation semantics and the ready trace semantics.
Admittedly, most of these semantics are rather strange and this is probably the reason why, as far as we know, they have not been previously considered. However, the simplest of them all, that corresponding to N = U , has properties similar to the possible worlds semantics and, in fact, can be defined by simply removing from its definition the "R" in the condition q ⊑ RS p. Hence, we can regard as possible worlds those deterministic implementations where we offer just a part of the action offered by the given process.
Definition 4.42. The partial possible worlds of a process p are those deterministic processes that verify q ⊑ S p. We denote with P W U (p) the set of partial possible worlds of a process p and define p ⊑ UPW q if P W U (p) ⊆ P W U (q).
Proposition 4.43. For all processes p 1 , p 2 ∈ BCCSP, p 1 ⊑ UPW p 2 iff p 1 ≤ db U p 2 . Proof. Similar to Theorem 4.40, simplified by the fact that all dbgo in P W U (p) satisfy dbgo ⊑ S p.
Example 4.44. We have a ⊑ UPW a + b since ·, {(a, ∅)} ∈ dBGO U (a + b). By contrast, for p = ab + ac and q = a(b + c) we have p ⊑ UPW q but q ⊑ UPW p because ·, {(a, ·, {(b, ·, ∅ ), (c, ·, ∅ )} } ∈ dBGO U (q) − dBGO U (p).
Analogously, for any other constraint N we could define the N -possible worlds order ⊑ NPW using ⊑ NS instead of ⊑ S at Definition 4.42. However, it is easy to see that when N is fine enough, e.g. N = T , this order would become totally wrong. Instead, we can still consider the observations in dBGO N and by means of them we define the "reasonable" deterministic branching semantics, for any layer in the spectrum. The extended spectrum can now be depicted as in Figures 11 and 10.
4.4.
Back to branching observations. The orders ≤ lδ N with δ ∈ {⊇, f, f ⊇} that characterize some of the linear semantics studied in Section 4.2, restricted in several ways the use of the local information, when characterizing those semantics. The same scheme can be generalized to the branching observations. This way, for each constraint N we would obtain three new branching semantics based on bgo's in BGO N which, together with the original N -simulation semantics, would constitute a diamond of branching semantics at a higher layer in our extended ltbt spectrum. The introduction of these new semantics also offers a clearer view of the spectrum, with two main levels of branching and linear semantics and an intermediate one of deterministic branching semantics. Although this provides the means for obtaining a host of new semantics, it is also true that most of them are bizarre, in sharp contrast with the fact that the corresponding orders gave rise to interesting semantics when applied to linear observations.
To illustrate the comments above, next we consider in some detail the case N = I, which corresponds to the most interesting semantics. • bgo ≤ f I bgo ′ ⇐⇒ bgo = A 1 , ∅ and bgo ′ = A 1 , ∅ or bgo = A 1 , S 1 and bgo ′ = A 2 , S 2 and
S 1 = {(a i , bgo i ) | i ∈ I} and S 2 = {(a i , bgo ′ i ) | i ∈ I} and for all i ∈ I (bgo i ≤ f I bgo ′ i ) . • bgo ≤ f ⊇ I bgo ′ ⇐⇒ bgo = A 1 , ∅ and bgo ′ = A 2 , ∅ and A 1 ⊇ A 2 or bgo = A 1 , S 1 and bgo ′ = A 2 , S 2 and S 1 = {(a i , bgo i ) | i ∈ I} and S 2 = {(a i , bgo ′ i ) | i ∈ I} and for all i ∈ I (bgo i ≤ f ⊇ I bgo ′ i ) .• B ≤ bδ I B ′ ⇐⇒ for all bgo ∈ B there exists bgo ′ ∈ B ′ with bgo ≤ δ I bgo ′ . Then, we write p ≤ bδ I q if BGO I (p) ≤ bδ I BGO I (q)
. It is somewhat surprising to discover that ≤ b⊇ I = ≤ b I , since this was not the case for their linear "projections" ≤ l⊇ I and ≤ l I .
Proposition 4.47. For all processes p 1 , p 2 ∈ BCCSP, p 1 ≤ b⊇ I p 2 iff p 1 ≤ b I p 2 . Proof. Assume that p 1 ≤ b⊇ I p 2 and let bgo ∈ BGO I (p 1 ): it is clear that it can be extended into a complete cbgo ∈ BGO I (p 1 ). Then, there exists some cbgo ′ ∈ BGO I (p 2 ) with cbgo ≤ ⊇ I cbgo ′ and, since cbgo is complete, cbgo = cbgo ′ and hence bgo ∈ BGO I (p 2 ). The other implication is trivial. Example 4.49. For the processes p and q in Figure 12, p ≤ bf I q but p ≤ b I q. This example shows that it is quite difficult to characterize this semantics as a simulation one. Furthermore, we conjecture that it is not finitely axiomatizable in the classic way (that means using only unconditional axioms). As a matter of fact, we were also unable to find any conditional axiomatization, what we interpret as a "proof" of the fact that these new branching semantics are quite strange.
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Definition 4.50. We say that R ⊆ BGO I × BCCSP is a final-ready simulation when:
• ( A, ∅ , q) ∈ R implies I(q) = A. • ( A, {(a i , bgo i )} , q) ∈ R implies that for all i ∈ I there exists q a i −→ q i such that (bgo i , q i ) ∈ R.
We say that p is final-ready simulated by q when for all bgo ∈ BGO I (p) there exists a final-ready simulation R with (bgo, q) ∈ R, and write p ⊑ fRS q.
Theorem 4.51. For all p, q ∈ BCCSP, p ⊑ fRS q iff p ≤ bf I q.
Example 4.52. It is easy to check that for p and q ′ as in Figure 12 we have p ≤ bf ⊇
I q ′ but p ≤ bf I q ′ .
Final failure simulations are defined exactly like final-ready simulations but substituting I(q) ⊆ Act for I(q) = A in the first clause, giving rise to the order ⊑ fFS between processes.
Theorem 4.53. For all p, q ∈ BCCSP, p ⊑ fFS q iff p ≤ bf ⊇ I q.
As previously noted, these are certainly bizarre semantics but we believe it is interesting to indicate their existence because, by analogy to the linear case, their definitions in terms of branching observations look quite natural. However, it also seems that when dealing with branching observations the introduction of any kind of asymmetry in the treatment of local observations produces quite involved semantics.
Relating the observational and equational frameworks
In this section we tie up all loose ends and show how our unification theory is fully selfcontained. Namely, we prove the results on axiomatic characterizations in Section 3 from the observational semantics developed in Section 4: we show how the equations are deduced from the observations in a general way without resorting to the already existing axiomatizations.
One of the key points of this section is to illustrate how the particular proofs needed in Section 3 for every one of the semantics can be replaced by a generic proof that stands for a whole family of semantics. In fact, we will show in Section 8 that the same proof is still valid for the new semantics suggested in Roscoe's work. 5.1. Semantics coarser than ready simulation. Let us now see how, from this uniform definition of the linear semantics, the proofs of the correctness and completeness of the corresponding axiomatizations can be derived in a uniform way avoiding the case analyses of Sections 3.1 and 3.2. Although this could be done generically, with N ∈ {U, C, I, T }, we prefer to start with the particular case N = I, which corresponds to the most popular semantics already studied in Section 3.1.
To start with, we show how the axiomatizations can be synthetized from the observational characterizations. Our general axiom (ND) for the reduction of non-determinism specifies the hypothesis M (x, y, w) under which the process ax + a(y + w) can be (syntactically) expanded by adding a new summand a(x + y) without changing its semantics. Then, let us compare the two sides of our general axiom. Since I(ax + a(y + w)) = I(ax) = I(a(y + w)) = {a}, we have
LGO I (ax + a(y + w)) = LGO I (ax) ∪ LGO I (a(y + w)),
LGO I (a(y + w))
= However, due to the fact that this axiom will be used in combination with (RS), the following, more restrictive but simpler form, can be used instead:
M RT (x, y, w) ⇐⇒ I(x) = I(y) ∧ I(w) ⊆ I(y).
Clearly, this form is stronger than the condition synthetized above. Reciprocally, a(x + y) ax+a(y+w) can be proved from the assumptions I(y) ⊆ I(x) and I(x) = I(y)∪I(w) using (RS) first to get a(x + y) a(x + y + w), and then (ND) instantiated with M RT to obtain a(x + y + w) ax + a(y + w). • ≤ l⊇ I . We need the inclusion LGO I (a(x + y)) ⊇ ⊆ LGO I (ax + a(y + w)) ⊇ to hold. Since I(x) ∪ I(y) ⊇ I(x), the general observations in a(x + y) that arise from x will be also in
LGO I (ax) l⊇ . For those that arise from y, it is required that I(x) ∪ I(y) ⊇ I(y) ∪ I(w).
Once again, (RS) can be used to simplify this condition into the simpler
M F T (x, y, w) ⇐⇒ I(w) ⊆ I(y).
The less restrictive variant of the axiom can be derived from the stronger one and (RS) as follows. Taking w = 0, since I(0) ⊆ I(y) we obtain a(x + y) ax + ay from (ND FT ); in particular, a(x + y + w) ax + a(y + w). Also, by (RS), x + y (x + y) + (x + y + w), from where it follows a(x + y) a(x + y + w). . Again, we can remove the second condition and define M R (x, y, w) ⇐⇒ I(y) ⊆ I(x) since, whenever I(x) ∪ I(y) = I(y) ∪ I(w), a(x + y + w) ax + a(y + w) can be obtained by taking x = y + w, y = x, and w = 0, and then by applying (RS) we conclude a(x + y) ax + a(y + w).
• ≤ lf ⊇
I . An argument analogous to the previous one leads us to check that I(x)∪I(y) ⊇ I(x) or I(x) ∪ I(y) ⊇ I(y) ∪ I(w), and the first is certainly true.
In order to prove the completeness of our axiomatizations we introduce the following notions of head normal forms. • For a ∈ X 0 , i ∈ I a , and X 1 ⊆ i∈Ia I(p i a ) such that I(p i a ) ⊆ X 1 , we define hnf Z (p, a, i,
X 1 ) = a(p i a + {p j a | X 1 | j = i, M Z (p i a , p j a | X 1 , p j a | X 1 )}). • hnf Z (p) = p + a∈X 0 i∈Ia X 1 ⊆ I(p i a ) hnf Z (p, a, i, X 1 )
. It is clear that several redundancies arise in this definition: for example, if Z = RT then hnf Z (p, a, i, X 1 ) = hnf Z (p, a, i, I(p i a )), so that the argument X 1 would not be needed in this case. We prefer to maintain the generic definition in order to allow a homogeneous treatment of all the semantics. Proof. Let p = a∈X 0 i∈Ia ap i a . Considering the definition of hnf Z (p, a, i, X 1 ), let us consider an enumeration of the set of j's contributing to it:
If J i = {j = i | M Z (p i a , p j 1 a | X 1 , p j 1 a | X 1 )}, we take {j k | k = 1 . . . |J i |} = J i .
Then we can prove by induction on l that for all 0 ≤ l ≤ |J i | we have {B 1 -B 4 , (RS), (ND Z )} ⊢ a(p i a + l h=1 p j h a | X 1 ) ap i a + l h=1 ap j h a . The case of l = 0 is trivial. Assuming the result for l, we prove the result for l + 1.
From M Z (p i a , p j l+1 a | X 1 , p j l+1 a | X 1 ) we can infer M Z (p i a + l h=1 p j h a | X 1 , p j l+1 a | X 1 , p j l+1 a | X 1 ) so that we can derive ⊢ a(p i a + l+1 h=1 p j h a | X 1 ) a(p i a + l h=1 p j h a ) + ap j l+1
a ; and applying the i.h. we conclude
{B 1 -B 4 , (RS), (ND Z )} ⊢ a(p i a + l+1 h=1 p j h a | X 1 | X 1 ) ap i a + l+1 h=1 ap j h a . From this we immediately obtain {B 1 -B 4 , (RS), (ND Z )} ⊢ hnf Z (p, a, i, X 1 ) p| a . Fi- nally, adding all these inequalities we conclude {B 1 -B 4 , (RS), (ND Z )} ⊢ hnf Z (p) p.
Let us define l(F ) = lf ⊇, l(F T ) = l⊇, l(R) = lf , and l(RT ) = l. In order to apply structural induction to prove the completeness of the axiomatizations we need to show that, whenever p = a∈X 0 i∈Ia ap i a and p ≤ l(Z) I q, there is a summand ah k a of hnf Z (q) such that p i a ≤ l(Z) I h k a for each a ∈ Act, i ∈ I a . . In addition, M Z (q j a , q jt a | I(p i a ) , q jt a | I(p i a ) ): • If Z = RT , then t ∈ LGO I (q jt a ) and therefore I(q jt a ) = I(p i a ) = I(q j a ). Hence, condition M RT (q j a , q jt a | I(p i a ) , 0) holds and therefore M RT (q j a , q jt a | I(p i a ) , q jt a | I(p i a ) ).
• If Z = F T , from t ∈ LGO I (q jt a ) ⊇ it follows that I(q jt a ) ⊆ I(p jt a ) and therefore I(q jt a | I(p i a ) ) = ∅ ⊆ I(q jt a | I(p i a ) ). Hence, M F T (q j a , q jt a | I(p i a ) , q jt a | I(p i a ) ).
• If Z = R, from I(p i a ) ∈ LGO I (q j a ) f
we have that I(p i a ) = I(q j a ) and thus I(q jt a | I(p i a ) ) ⊆ I(q j a ) and M R (q j a , q jt a | I(p i a ) , q jt a | I(p i a ) ).
• For Z = F it is trivial since M F (x, y, w) is always true.
Therefore q jt a is one of the summands of h k a and, since t ∈ LGO I (q jt a )
l(Z) , we have p i a ≤ l(Z) I h k a .
5.2.
The semantics that are not coarser than ready simulation. Once we have a clear picture of the semantics that are coarser than ready simulation, it is time to consider the rest of the semantics in the spectrum. Let us start with the possible futures and the impossible futures semantics. Recall that we have shown that they can be described by LGO T observations so that they are defined by ≤ lf ⊇ T and ≤ lf T , respectively. We introduce the T -versions of our (ND Z ) axioms: all of them are instances of our general axiom for the reduction of non-determinism and therefore are defined by the adequate Z (x, y, w). As expected, they are obtained by substituting every occurrence of I in M Z (x, y, w) by the observer T defining the traces of processes.
Definition 5.5. The constraints M T Z that characterize the semantics coarser than Tsimulation semantics are:
(
T-ND F ) M T F (x, y, w) ⇐⇒ true (T-ND R ) M T R (x, y, w) ⇐⇒ T (x) ⊇ T (y) (T-ND FT ) M T F T (x, y, w) ⇐⇒ T (w) ⊆ T (y) (T-ND RT ) M T
RT (x, y, w) ⇐⇒ T (x) = T (y) and T (w) ⊆ T (y) As indicated in Section 4.2 (Definition 4.29), the semantics associated to the last two conditions do not appear in the ltbt spectrum and, as far as we know, they have not been previously studied nor even defined.
Using the same arguments as in Section 5.1, we can prove that ≤ l(Z) T satisfies the axiom (T-ND Z ) for Z ∈ {RT, F T }.
Proposition 5.6. M T Z (x, y, w) implies T (a(x + y)) = T (ax + a(y + w)) for Z ∈ {RT, F T }. However, this is not the case for Z ∈ {R, F } Proof. M T RT implies M T F T , and therefore T (y +w) = T (y), which leads to T (ax+a(y +w)) = T (a(x + y)). Neither M T R nor M T F refer to w and therefore, in general, T (ax + a(y + w)) = T (a(x + y)) in those cases.
Note that when proving the correctness of the corresponding axiom (ND Z ) for ≤ l(Z) I we had I(a(x + y)) = {a} = I(ax + a(y + w)) in all cases. Now, T (a(x + y)) = T (ax + a(y + w)) only under the constraints corresponding to the finer semantics ⊑ F T and ⊑ RT . The properties of the prefixes appearing in all the terms in both sides of the axiom (ND) are not used anymore in the proofs in Section 5.1, so they can be transferred to the T -semantics, thus proving the correctness of (T-ND Z ) for both ≤ l(RT ) T and ≤ l(F T ) T . The introduction of the equational version (ND ≡ ) of the axiom (ND) now becomes crucial in order to preserve the generality of our unifying work. We saw that under (RS) these two axioms were equivalent. However, when observing the set of traces T (x) of any process, instead of just the initial offer I(x) we need to consider T -simulations, that are constrained by the condition T (x, y) ⇐⇒ T (x) = T (y); under the corresponding axiom (T S), things turn out to be different.
Proposition 5.7. T (a(x+ y)+ ax + a(y + w)) = T (ax)∪ T (ay)∪ T (aw) = T (ax + a(y + w)). Proof. To show that ≤ l(Z) T satisfies (T-ND Z + ) we just need to apply Proposition 5.7 and follow the line of thought in the second bullet on page 37, substituting the observer T for I. For the other axiom, from T (a(x + y)) ⊆ T (ax + a(y + w)) it follows that {(T S)} ⊢ ax + a(y + w) (ax + a(y + w)) + a(x + y).
As a consequence, for (T-ND
Notice that for Z ∈ {RT, F T } we can also obtain the correctness of (T-ND Z ≡ ) from that of (T-ND Z ) and vice versa, as a consequence of the following fact. ) and, since T (w) ⊆ T (y) implies T (a(x+y)) = T (ax + a(y + w)) and then we have {B 1 -B 4 , (T S)} ⊢ a(x + y) a(x + y) + (ax + a(y + w)).
To prove
{B 1 -B 4 , (T S), (T-ND Z + )} equivalent to {B 1 -B 4 , (T S), (T-ND Z ≡ )} we only need to show that {B 1 -B 4 , (T S), (T-ND Z + )} ⊢ (M T Z (
x, y, w) ⇒ ax + a(y + w) ax + a(y + w) + a(x + y)), but we have that for Z ∈ {RT, F T }, M T Z (x, y, w) implies T (w) ⊆ T (y), so that T (a(x + y)) = T (ax + a(y + w)) and therefore {(T S)} ⊢ ax + a(y + w) (ax + a(y + w)) + a(x + y).
The important fact about the obtained sets of correct axioms for the semantics ≤ T . However, the notion of head normal form for N = I uses the fact that the summands hnf Z (q, a, i, X 1 ) can be defined in terms of the offers X 1 ⊆ P(Act), which correspond to the values produced by the observer I. For an arbitrary N , a more general definition of hnf's, valid for every observer, is needed. Definition 5.10. For p = a∈X 0 i∈Ia ap i a , its totally expanded Z-head normal form tehnf Z N (p) is that given by: • For a ∈ X 0 , i ∈ I a , and K a ⊆ I a we consider a decomposition p k a = p k 1 a + p k 2 a such that M N Z (p i a , p k 1 a , p k 2 a ). Then, tehnf Z N (p, a, i, (p k 1 a , p k 2 a ) k∈Ka ) = a(p i a + k∈Ka p k 1 a ). • tehnf Z N (p) = tehnf Z N (p, a, i, (p k 1 a , p k 2 a ) k∈Ka ). It is clear that for K ′ a ⊆ K a , or any decomposition p k a = p k 3 a + (p k 4 a + p k 2 a ) with p k 1 a = p k 3 a + p k 4 a , the corresponding tehnf Z N (. . .) is a subterm of tehnf Z N (p, a, i, p k 1 a , p k 2 a k∈Ka ) and thus contributes nothing to the expanded normal form. This is the reason why we preferred the more compact definition of hnf Z (p) for semantics coarser than ready simulation.
Logical characterization of semantics
The third and a very natural alternative to associate a semantics to processes lies in the logical framework. This is indeed quite a natural way to do it. We have a language to express properties of processes and a way to check whether a process satisfies a formula of the language: then, two processes are equivalent with respect to this semantics if and only if they satisfy the same set of formulas. In fact, the semantics can also be defined in terms of ν ν Table 3: Van Glabbeek's logical characterizations for the semantics in the ltbt spectrum. the induced preorder that indicates whether a process satisfies more formulas than another one.
ν ν Xϕ ∈ LZ ϕi ∈ LZ ∀i ∈ I ⇒ • • • • • i∈I ϕi ∈ LZ X ⊆ Act, ϕa ∈ LP W ∀a ∈ X ⇒ • ν ν ν a∈X aϕa ∈ LZ ϕi, ϕj ∈ LT ∀i ∈ I ∀j ∈ J ⇒ • ν ν i∈I ϕi ∧ j∈J ¬ϕj ∈ LZ ϕ ∈ LS ⇒ • ν ¬ϕ ∈ LZ ϕ ∈ LZ ⇒ • ¬ϕ ∈ LZ
Each subset L of L HM induces a semantics as stated in the following definition.
Definition 6.1. Any subset L of L HM induces a logical semantics for processes, given by the preorder ⊑ L : p ⊑ L q whenever for all ϕ ∈ L, if p |= ϕ then q |= ϕ. We say that L and L ′ are equivalent, and we write L ∼ L ′ , if they induce the same semantics, that is ⊑ L = ⊑ L ′ .
Let us start with a look at Table 3, which contains the logical characterization of each of the semantics in van Glabbeek's spectrum. L Z with Z ∈ {T, CT, F, F T, R, RT, P F, S, CS, RS, 2S, P W, B} denotes each of the logics; the dots indicate the clauses needed to obtain the corresponding languages; and the boxes marked with ν correspond to rules that could be added to L Z , but would only introduce redundant formulas. The following constructs, which appear in the table but are not in L HM , can be obtained as syntactic sugar:
X := a∈X ¬a⊤ Xϕ ′ := X ∧ ϕ ′ 0 := Act ϕ 1 ∧ ϕ 2 := i∈{1,2} ϕ i X := a∈X a⊤ ∧ a ∈X ¬a⊤ Xϕ ′ := X ∧ ϕ ′ a := ¬a⊤
Disjunction does not appear in L HM and therefore neither in any of the logics L Z characterizing the semantics in the ltbt spectrum. It is probably folklore that it can be added in all cases without affecting the expressive power of each of these logics, but since we have not found a clear statement in this direction in any of our references, next we establish the result and comment on its proof. Proposition 6.2. Let us define L ∨ Z , with Z ∈ {T, CT, F, F T, R, RT, P F, S, CS, RS, 2S, P W, B}, by adding the clause i∈I ϕ i ∈ L ∨ Z if ϕ i ∈ L ∨ Z for all i ∈ I to the clauses that define L Z , replacing L Z by L ∨ Z in the other clauses, and making p |= ϕ i iff there exists i ∈ I with p |= σ i . Then, L ∨ Z ∼ L Z . Proof. It is interesting to observe that even if the result is valid for all the semantics, the reason behind is not the same as for bisimulation. In that case, we only need to apply the De Morgan laws to get the "definition" of ∨ as a combination of ¬ and ∧. However, for the rest of the semantics we do not have negation as "constructor", but ∨ distributes over ∧ and the prefix operator (that is aϕ i = a ϕ i ), while negation is never applied to a formula ϕ ′ ∈ L ∨ Z . Therefore, by floating to the top any ∨, using those distribution laws, a formula in L ∨ Z becomes equivalent to a disjunction of formulas within the corresponding language L Z , and the equivalence of both logics follows.
As we will see in this section, each of our logics is defined by a set of rules and, as usual, only the formulas that can be obtained by finite application of these rules belong to the logics. One important feature of our approach is that instead of focusing on small sets of formulas characterizing each of the semantics, we somewhat follow the opposite approach by including all the formulas, from a certain family, that are preserved by each of the semantics. This choice has many interesting side effects. In particular, we will not need to look for adequate formulas reflecting the characteristics of each of the semantics, but instead pick up from our "repository" of possible formulas those that are preserved by the current semantics. Thus, we characterize each of the semantics by means of the formulas that "see" the kind of observations that define it. As a consequence, we know whether a semantics is coarser than another by checking whether the logic characterizing the former is included in the logic characterizing the latter. Moreover, by using a larger logic we may find a formula expressing some property that is preserved by the corresponding semantics, while if we settle on a smaller logic we might need a collection of formulas to express a simple property.
Formally speaking, for each semantics defined by a preorder ≺ we have a language L ⊆ L HM characterizing it: ϕ ∈ L iff ((p ≺ q ∧ p |= ϕ) implies q |= ϕ). However, it is not easy (nor specially illustrative) to capture the whole set of formulas characterizing the semantics. Instead, we will consider sufficiently large families defined in a simple way that provide natural characterizations of the different semantics and show the relationship between them so that, as stated above, whenever a semantics is finer than another, the logic characterizing the first will contain that for the latter.
As will become clear when we introduce our new logical characterizations, Table 3 readily presents the features that allow us to classify the semantics in the spectrum in four categories:
• Bisimulation semantics, characterized by HML, that is closed under negation (¬), so that the preorder defined is an equivalence (the bisimulation). The remaining semantics are defined by non-trivial preorders, i.e., the preorders are not equivalences and their logical characterizations are, of course, not closed under negation. • Simulation semantics (S, CS, RS, . . . ), characterized by branching observations, which will be reflected by the unrestricted use of the operator in the formulas. • Linear semantics (T, F, R, . . . ), characterized by linear observations. We will get them by severely restricting the use of and the use of the negation. • Deterministic branching semantics, corresponding to an intermediate class between branching and linear semantics, where determinism appears restricting the use of the operator in combination with the prefix operator. The only semantics in this class in the classical spectrum is PW. As already happened in Sections 3 and 4, our unified logical semantics will provide an enlarged spectrum- Figure 11. In particular, we will show the logical characterization of revivals semantics, introduced by Roscoe in [48] and already axiomatized in [19].
6.1.
A new logical characterization of the most popular semantics. Again, we start with the best known classical semantics, that is, those at the layer of ready simulation in the spectrum. All of them use in some way the set of formulas L I = {a⊤ | a ∈ Act} that characterizes the initial offers of a process. In Section 6.2 we will present the logics for the rest of the semantics in a unified way, remarking how they are obtained similarly to those in this section but working from the set L N of formulas associated to the corresponding constraint N .
We will prove the equivalence between each of our logics and the corresponding logical characterization defined by van Glabbeek, thus checking that our new logical characterizations are indeed correct. But one of the intended goals of our unification was to obtain direct and natural proofs. This will be illustrated in Section 7 by showing the equivalence between each of our logical semantics and the corresponding observational semantics of Section 4. This will provide a new, single proof of their correctness without having to resort to the characterizations defined by van Glabbeek. Definition 6.3. Ready simulation semantics. We define the set of formulas L ′ RS for ready simulation semantics by:
• If σ ∈ L I then σ ∈ L ′ RS ; • if σ ∈ L I then ¬σ ∈ L ′ RS ; • if ϕ i ∈ L ′ RS for all i ∈ I then i∈I ϕ i ∈ L ′ RS ; • if ϕ ∈ L ′
RS and a ∈ Act then aϕ ∈ L ′ RS . Ready trace semantics. We define the set of formulas L ′ RT for ready trace semantics by:
• ⊤ ∈ L ′ RT ; • if ϕ ∈ L ′ RT and X 1 , X 2 ⊆ L ′ I then ( a∈X 1 a⊤ ∧ b∈X 2 ¬b⊤) ∧ ϕ ∈ L ′ RT ; • if ϕ ∈ L ′
RT and a ∈ Act then aϕ ∈ L ′ RT . Failure trace semantics. We define the set of formulas L ′ F T for failure trace semantics by:
• ⊤ ∈ L ′ F T ; • if ϕ ∈ L ′ F T and X 1 ⊆ L ′ I then ( a∈X 1 ¬a⊤) ∧ ϕ ∈ L ′ F T ; • if ϕ ∈ L ′
F T and a ∈ Act then aϕ ∈ L ′ F T . Readiness semantics. We define the set of formulas L ′ R for readiness semantics by:
• ⊤ ∈ L ′ R ; • if X 1 ⊆ L ′ I and X 2 ⊆ L ′ I then ( a∈X 1 a⊤ ∧ b∈X 2 ¬b⊤) ∈ L ′ R ; • if ϕ ∈ L ′
R and a ∈ Act then aϕ ∈ L ′ R . Failures semantics. We define the set of formulas L ′ F for failures semantics by:
• ⊤ ∈ L ′ F ; • if X 1 ⊆ L ′ I then ( a∈X 1 ¬a⊤) ∈ L ′ F ; • if ϕ ∈ L ′
F and a ∈ Act then aϕ ∈ L ′ F .
It is immediate that L ′ RS ⊆ L B and hence ready simulation semantics is coarser than bisimulation equivalence. We also have L ′
F ⊆ L ′ R , L ′ F ⊆ L ′ F T , L ′ R ⊆ L ′ RT , L ′ F T ⊆ L ′ RT , and L ′
RT ⊆ L ′ RS , which can be interpreted in a similar way. Let us now focus our attention on the third rule of the definition of L ′ RS : the unrestricted use of conjunction corresponds to the branching nature of the semantics. Moreover, the two first rules allow to fix the set of offers of a process as I-simulations impose. By contrast, the linear semantics only allow the use of conjunction to join those simple formulas that fix the set of offers along a computation (in the case of the readies-based semantics), or their over-approximations (obtained by means of the negated formulas ¬a⊤, in the case of the failures-based semantics). Finally, notice how these simple formulas can only be checked at the corresponding final state, for the two simpler coarser semantics. Now, for Z ∈ {RS, RT, F T, R, F }, each of the logics L ′ Z is a superset of the corresponding logic L Z defined in Table 3. To be precise, for F T and F we need to remove the syntactic sugar used by van Glabbeek as stated below.
Remark 6.4. We have used in Section 4.2 X c to denote the complementary of a set, because previously in Definition 4.20 we used the classic over line notation to refer to closures of sets T ⊆ LGO N . However, since we will not need those closure operators anymore we prefer to used the classic notation referring the complement of a set X by X. Table 3.
Proposition 6.5. (1) L RS L ′ RS . (2) L RT L ′ RT . (3) L ′ F T ⊇ desugared(L F T ),
(1) To prove that L RS ⊆ L ′ RS , it is sufficient to show that each formula ϕ X = a∈X a⊤ ∧ b / ∈X ¬b⊤ corresponding to X ⊆ Act belongs to L ′ RS . Both a⊤ and ¬b⊤ are in L ′ RS and the combination of these formulas with the operator ∧ is also in the set L ′ RS . For the inclusion to be proper, it is sufficient to notice that the formula ¬b⊤ belongs to L ′ RS but not to the set L RS .
(2) To prove that L RT ⊆ L ′ RT it is sufficient to show that for every X ⊆ Act and any ϕ ∈ L RT , the formula ( a∈X a⊤ ∧ b / ∈X ¬b⊤) ∧ ϕ belongs to L ′ RT . Note that b / ∈ X is equivalent to b ∈ X, so taking X 1 = X and X 2 = X we have that the considered formula belongs to L ′ RT . To prove that L RT ⊂ L ′ RT , it is sufficient to note that (¬b⊤)∧ϕ belongs to L ′ RT , by taking X 1 = ∅ and X 2 = {b}, but it does not belong to L RS . (3) In this case the result is trivial, since the definitions of L F T and L ′ F T are almost the same, once the syntactic sugar is removed. The only difference is that ⊥∈ L ′ F T , which obviously does not affect the inclusion. (4) To prove that L R ⊆ L ′ R , it is sufficient to show that for every X ⊆ Act the formula a∈X a⊤ ∧ b / ∈X ¬b⊤ belongs to L ′ R . Note that the condition b / ∈ X is equivalent to b ∈ X, so taking X 1 = X and X 2 = X we have that the considered formula belongs to L ′ R . To check that L R L ′ R , it is sufficient to note that the formula ¬b⊤ belongs to L ′ R by taking X 1 = ∅ and X 2 = {b}, while it does not belong to L R . As stated earlier, in order to obtain more natural characterizations, our logics typically contain large sets of formulas. This is why in most cases our logics contain those proposed by van Glabbeek. In order to prove the equivalence between ours and his, we have to show that our additional formulas are in fact redundant and could be safely removed.
Proposition 6.6. (1) L RS ∼ L ′ RS ; (2) L RT ∼ L ′ RT ; (3) L F T ∼ L ′ F T ; (4) L R ∼ L ′ R ; and (5) L F ∼ L ′ F . Proof.
(1) Any conjunction and negation of formulas in L I can be obtained as the disjunction of the formulas X describing all the "compatible" offers. These are those including the positive and negative information in the corresponding conjunction, i.e., a⊤ ∼ a∈X X; ¬a⊤ ∼ a / ∈X X. Then, by applying Proposition 6.2, we obtain L ′ RS ∼ L RS . (2) We have shown that the formulas in L RT are particular cases of the formulas in L ′ RT : those that completely define the offers at the states along a computation (when we apply the second clause in the definition of L ′ RT with X 2 = X 1 ). In contrast, our more general formulas ( a⊤∈X 1 a⊤ ∧ b⊤∈X 2 ¬b⊤)∧ ϕ, where ϕ ∈ L ′ RT , could provide us with some partial information, combining both positive information a⊤ ∈ X 1 and negative information b⊤ ∈ X 2 , which tells us that we are in an arbitrary state X satisfying X 1 ⊆ X ⊆ X 2 . But we can replace these formulas by the disjunction of all the formulas describing any of these possible offers X. By repeating this procedure at each level of the formula, we finally obtain a disjunction of formulas in L RT . To conclude, it is enough to apply Proposition 6.2.
(3) We know ⊥= ¬⊤ = i∈∅ ϕ i , and applying Proposition 6.2 we get the equivalence. (4) Note that van Glabbeek allowed in L R only "normal form" formulas from L ′ R , which can give us information about the offers at the final state in a computation (when we apply the second clause in the definition of L ′ R ) or simply define these computations by means of the prefix operator (when we apply the third clause in the definition of L ′ R ). However, our more general formulas ( a⊤∈X 1 a⊤ ∧ b⊤∈X 2 ¬b⊤) can also provide us with some partial information about the final state, which could be both positive a⊤ ∈ X 1 and negative b⊤ ∈ X 2 . In the (allowed) case X 1 X 2 = ∅ we have that the formula is unsatisfiable. Otherwise, we are offering the actions a corresponding to formulas a⊤ in any X ⊆ L I that satisfies X 1 ⊆ X and X ⊆ X 2 , and we can replace again the corresponding formula by a disjunction of formulas in L R . (5) Analogous to 3.
In the following, when we consider a logic L Z and the index Z refers to some concrete semantics, as is the case with RS, RT , F T , R, and F above, by abuse of notation we will simply write ⊑ ′ Z instead of ⊑ L ′ Z for the preorder induced by the logic L ′ Z . Theorem 6.7. Proof. It is a consequence of Proposition 6.6 and the results by van Glabbeek collected in Table 3, Theorem 4.9, and Proposition 4.18.
(1) We have already checked that our formulas are equivalent to van Glabbeek's: L ′ RS ∼ L RS . It is easy to show that once we have eliminated the unsatisfiable formulas in L ′
RS
(those that simultaneously make two different offers, or perform an action that was not included in the corresponding offer) the remaining formulas in L ′ RS admit a normal form in the language N (L RS ), which we define as follows:
• if X ⊆ Act, {a i | i ∈ I} ⊆ X, and ϕ i ∈ N (L RS ), then ( b∈X b⊤ ∧ b / ∈X ¬b⊤) ∧ i∈I a i ϕ i ∈ N (L RS ); • if {a i | i ∈ I} ⊆ Act and ϕ i ∈ N (L RS ) then i∈I a i ϕ i ∈ N (L RS ).
Within this set, consider the subset of formulas CN (L RS ) which can be generated using the first clause in the above definition. We can establish an isomorphism between CN (L RS ) and the set of possible branching general observations BGO I . Moreover, it is easy to prove that if for every formula ϕ ∈ CN (L RS ) we define bgo ϕ as the corresponding observation, then ϕ |= p iff bgo ϕ ∈ BGO I (p), from which it immediately follows that CN (L RS ) characterizes the ready simulation semantics defined via BGO I . Now, to conclude the proof it is sufficient to show that N (L RS ) and CN (L RS ) are equivalent. Note that whenever we use the second clause in the definition of N (L RS ), we are ignoring the possibility of specifying the offer X at the state we are. As a consequence, the offer could be any satisfying {a i | i ∈ I} ⊆ X, for the corresponding set {a i | i ∈ I}. Then we can complete the associated formula i∈I a i ϕ i by adding the disjunction {a i /i∈I}⊆X ( b∈X b⊤ ∧ b / ∈X ¬b⊤). Floating all the disjunctions away we obtain a disjunction of formulas in N (L RS ), which ends the proof.
(2) • If Z = RT , we know that L ′ RT ∼ L RT . It is easy to show that eliminating all the unsatisfiable formulas (those that simultaneously offer two different sets of actions, or perform an action a that is not included in the corresponding offer X) the rest of the formulas in L ′ RT admit a normal form in the language N (L RT ), which we define as follows: − if X ⊆ Act then ( b∈X b⊤ ∧ b / ∈X ¬b⊤) ∈ N (L RT ); − if X ⊆ Act, a ∈ X, and ϕ ∈ N (L RT ) then ( b∈X b⊤ ∧ b / ∈X ¬b⊤)∧ aϕ ∈ N (L RT ); − ⊤ ∈ N (L RT ); − if a ∈ Act and ϕ ∈ N (L RT ) then aϕ ∈ N (L RT ). As we did for the case of ready simulation, we could define the corresponding language of complete formulas CN (L RT ). The formulas in L ′ RT that we obtained in the proof of Proposition 6.6, for the case of RT , are indeed in CN (L RT ) because any subformula gives us some partial information about the offers at the corresponding state, which in the worst case could be empty. Therefore, when we translate this information into the language L ′ RT we obtain a disjunction between complete formulas in CN (L RT ). We can easily establish the isomorphism between CN (L RT ) and the domain LGO I , and then prove that for every formula ϕ ∈ CN (L RT ), if we define lgo ϕ as the corresponding observation, we have ϕ |= p iff lgo ϕ ∈ LGO I (p). From here it follows that CN (L RT ) characterizes the ready simulation semantics defined via LGO I . To conclude the proof we need to show that N (L RT ) and CN (L RT ) are equivalent, which is analogous to N (L RS ) and CN (L RS ) above.
• Z = F T . (⇒) Let p and q be such that p ⊑ ′ F T q: we will show that p ≤ l⊇ I q. Given an observation X 0 a 1 X 1 . . . a n X n ∈ LGO I (p), we have a failure trace X 0 a 1 X 1 . . . a n X n
48D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
for the process p. Now, we consider the formulas ϕ n = a∈X ¬a⊤, ϕ i = a∈X i ¬a⊤ ∧ a i+1 ϕ i+1 with i ∈ 0..n − 1, and we have that p |= ϕ 0 . Therefore q |= ϕ 0 , which means that X 0 a 1 X 1 . . . a n X n is a failure trace of q. Then, there is some Y 0 a 1 Y 2 . . . a n Y n ∈ LGO I (p) with Y i X i = ∅ for all i = 0..n or, equivalently, X i ⊇ Y i for all i = 0..n. As a result, LGO I (p) ≤ l⊇ I LGO I (q), which means p ≤ l⊇ I q. (⇐) Let us suppose that for all X 0 a 1 X 1 . . . a n X n ∈ LGO I (p) there exists Y 0 a 1 Y 1 . . . a n Y n ∈ LGO I (q) such that X i ⊇ Y i for all i = 0..n; we want to show that if p |= ϕ then q |= ϕ, for all ϕ ∈ L ′ F T . If p |= ϕ, we can decompose ϕ by means of a sequence of formulas, taking ϕ = ϕ n , ϕ i = a∈X i 2 ¬a⊤ ∧ a i ϕ i−1 for i ∈ 1..n and ϕ 0 = a∈X 0 2 ¬a⊤ . Therefore, X n a n X n−1 . . . a 1 X 0 is a failure trace for the process p, so there exists Z n a n Z n−1 . . . a 1 Z 0 ∈ LGO I (p) with Z i X i = ∅, and using that p ≤ l⊇ I q, there exists some Y n a n Y n−1 . . . a 1 Y 0 ∈ LGO I (q) with Y i ⊆ Z i , so that Y i X i = ∅ and then we get q |= ϕ n .
• If Z = R, using the result in the proof of Proposition 6.6 for the case of R it is enough to show the result for the set of "normal form" formulas N (L R ) defined by: − if X ⊆ Act then ( b∈X b⊤ ∧ b / ∈X ¬b⊤) ∈ N (L R ); − ⊤ ∈ N (L R ); − a ∈ Act and ϕ ∈ N (L R ) then ϕ ∈ N (L R ). (⇒) Let p and q be such that p ⊑ ′ R q: we will show p ≤ lf I q. Given an observation X 0 a 1 X 1 . . . a n X n ∈ LGO I (p), it corresponds to the readiness information a 1 . . . a n X n of p. Now, we consider the formulas ϕ n = a∈X a⊤∧ a / ∈X ¬a⊤; ϕ i−1 = a i ϕ i with i ∈ 1 . . . n − 1, and we have that p |= ϕ 0 . Therefore q |= ϕ 0 , and a 1 . . . a n X n is a readiness information of q and, as a consequence, there is an observation Y 0 a 1 Y 2 . . . a n Y n ∈ LGO I (q) with Y n = X n , proving p ≤ lf I q. (⇐) Let us suppose that for all X 0 a 1 X 1 . . . a n X n ∈ LGO I (p) there exists some Y 0 a 1 Y 1 . . . a n Y n ∈ LGO I (q) such that X n = Y n . We want to show that if p |= ϕ then q |= ϕ for all ϕ ∈ CN (L R ). If p |= ϕ, we can decompose ϕ taking ϕ = ϕ n , ϕ i = a i ϕ i−1 , for all i ∈ 1..n, and ϕ 0 = a∈X 0 a⊤ ∧ a / ∈X 0 ¬a⊤. Then we have that a n a n−1 . . . a 1 X 0 is a readiness information of p, so there exists some Z n a n Z n−1 . . . a 1 X 0 ∈ LGO I (p), and some Y n a n Y n−1 . . . a 1 Y 0 ∈ LGO I (q) with Y 0 = X 0 , from which we conclude that q |= ϕ n . • Z = F . (⇒) Let p and q be such that p ⊑ ′ F q: we will show p ≤ lf ⊇ I q. Given an observation X 0 a 1 X 1 . . . a n X n ∈ LGO I (p), it generates a (maximal) failure a 1 . . . a n X n of the process p. Now, we consider the formulas ϕ n = a∈X ¬a⊤; ϕ i+1 = a i+1 ϕ i with i ∈ 0..n − 1, and we have that p |= ϕ 0 . Therefore, q |= ϕ 0 , so a 1 . . . a n X n is a failure information of q, and there is some Y 0 a 1 Y 2 . . . a n Y n ∈ LGO I (q) with Y n X n = ∅, or equivalently X n ⊇ Y n , proving that p ≤ lf ⊇ I q.
(⇐) Let us suppose that for all X 0 a 1 X 1 . . . a n X n ∈ LGO I (p) there exists some Y 0 a 1 Y 1 . . . a n Y n ∈ LGO I (q) such that X n ⊇ Y n . We want to show that if p |= ϕ then q |= ϕ for all ϕ ∈ L ′ F . If p |= ϕ, we can decompose ϕ taking ϕ = ϕ n , ϕ i = a i ϕ i−1 , with i ∈ 1..n, and ϕ 0 = a∈X 0 ¬a⊤. From p |= ϕ we infer that a n a n−1 . . . a 1 X 0 is a failure information of the process p, so there exists Z n a n Z n−1 . . . a 1 Z 0 ∈ LGO I (p) with Z 0 X 0 = ∅, and then there is some Y n a n Y n−1 . . . a 1 Y 0 ∈ LGO I (q) with Y n ⊆ Z n , so that Y n X n = ∅, obtaining q |= ϕ n .
· b · c · d · c ·
· · Figure 13: A simple example to show the strength of the different logics Example 6.8. Figure 13 shows a collection of examples to illustrate the differences between the semantics in the RS layer of the spectrum. All the following equivalences can be checked by taking any arbitrary formula from the logic defining each of the semantics. For readability, we omit the last ⊤ in all subformulas. Besides, ∼ X (resp. ≁ X ), where X is a set of indexes, represents any ∼ Z (resp. ≁ Z ), with Z ∈ X.
• p 1 ⊑ ′ F p 2 and p 1 ⊑ ′ {R, F T, RT, RS} p 2 because p 1 |= a(¬b ∧ ¬c), but p 2 does not satisfy it. • p 2 ∼ F p 3 , but p 2 ⊑ ′ {R, F T } p 3 and thus p 2 ⊑ ′ {RT, RS} p 3 , since p 2 satisfies a(¬e ∧ c) but p 3 does not.
• p 3 ∼ {F, R} p 4 , but p 3 ⊑ ′ F T p 4 and thus p 3 ⊑ ′ {RT, RS} p 4 , because p 3 satisfies a(¬c∧b(¬e∧d)) but p 4 does not.
• p 5 ∼ {F, F T } p 6 , but p 5 ⊑ ′ R p 6 and thus p 5 ⊑ ′ {RT, RS} p 6 , since p 5 satisfies ab(c ∧ d) but p 6 does not. • p 6 ∼ {F, R, RT, F T } p 7 but p 7 ⊑ ′ RS p 6 , because p 7 satisfies a(bc ∧ bd) but p 6 does not. • p 7 ∼ {F, R, RT, F T, RS} p 8 .
6.2.
Our new unified logical characterizations of the semantics. Inspired by the semantics studied in Section 6.1, next we define the general format for the logics characterizing each of the semantics in the spectrum. We start by enlarging the spectrum yet a bit more. Definition 6.9.
(1) Universal semantics. We define the set L ′ U of universal formulas that characterize the trivial semantics that identifies all the processes by L ′ U = {⊤}. (2) Complete semantics. We define the set L ′ C of complete formulas characterizing the semantics that only distinguishes the terminated processes from the non-terminated ones by L ′ C = {⊤, ¬0}. (3) Initial offer semantics. We define the set L ′ I of initial offer formulas characterizing the semantics that only observers the set of initial actions of a process by L ′
I = {⊤, ¬0} ∪ {a⊤ | a ∈ Act}.
In the definition above the subformula ¬0 is just syntactic sugar for the formula ¬( a∈Act ¬a⊤). Therefore, once again all these new logics are sublogics of L HM and, as a result, we do not need to define their semantics.
Note that L ′ I is a bit larger than the logic L I from Section 6.1. Once again, this is so in order to get a more uniform presentation of our logics: ¬0 is indeed redundant. By including it we immediately obtain that the complete semantics is coarser than the initial offer semantics, because L ′ C ⊆ L ′ I . Based on this result we will also obtain that the complete simulation is coarser than the ready simulation. Certainly, ¬0 is redundant in L ′ I (but not in L ′ C !), because by means of it we can only distinguish a process that cannot execute any action from any other that can execute someone. But using the corresponding a⊤ formula we can also get that.
6.2.1. The simulation semantics. As repeatedly noted, the family of simulation semantics constitute the spine of the new spectrum. All of them are defined in a homogeneous way thanks to the notion of constrained simulation from [24]. Next we present their logical characterization.
Definition 6.10. Given a set of formulas L ′ N defining a semantics N , we define the set of formulas L ′ N S that characterizes the N -constrained simulation semantics by:
• If σ ∈ L ′ N then σ ∈ L ′ N S ; • if σ ∈ L ′ N then ¬σ ∈ L ′ N S ; • if ϕ i ∈ L ′
N S for all i ∈ I then i∈I ϕ i ∈ L ′ N S ; • if ϕ ∈ L ′ N S and a ∈ Act then aϕ ∈ L ′ N S . Taking N ∈ {U, C, I} we obtain L ′ U S , L ′ CS and L ′ IS , that we rewrite as L ′ S and L ′
RS
in the first and last cases to emphasize the classic notation for simulation semantics. From L ′ S we obtain L ′ SS , that we will denote as L ′ 2S . To complete the collection of simulation semantics in the spectrum we need L ′ T S , that will be based on L ′ T , to be defined in the next section.
The definition above differs from the particular case of ready simulation in Definition 6.3 in the two first rules, by means of which we impose that the process will traverse states which are in the corresponding N -equivalence class all along the tree of computations checked by a formula in L ′ N S . Note that the combination of positive and negated formulas allows us to shape each of these classes. Next we state the equivalence between our logics for the simulation semantics and those by van Glabbeek's recalled in Table 3. (3) Once again, the sets generated by L ′ 2S and L 2S are the same. The clause "if σ ∈ L ′ S then σ ∈ L ′ 2S " in L ′ 2S does not generate any new formulas because L S ⊆ L 2S (the formulas in L S are exactly those that can be created using only the last two clauses in the definition of L 2S ).
Remark 6.12. We can use both positive formulas in L ′ C and their negations for defining L ′ CS due to the fact that C-constrained simulation can be built from the equivalence relation defined by C as constraint. However, we could also use ⊑ C as a constraint and then remove the clause "if σ ∈ L ′ C then σ ∈ L ′ S C ", which generates ¬0 ∈ L S C . The other clause, which generates 0 ∈ L ′ S C , is crucial and cannot be removed from the definition. These two facts also concur in the definition of the other simulation semantics in the extended spectrum, for which we also present a logical characterization including the two clauses above.
6.2.2.
Logical characterization of the linear semantics. We start by defining the closure operators by means of which we express the extent to which conjunction and negation can be used in the logical characterizations of each of the linear semantics. Definition 6.13. Given a logical set L ′ N with N ∈ {U, C, I, T, S}, we define:
(1) Its symmetric closure L ≡ N by: if σ ∈ L ′ N then σ ∈ L ≡ N and ¬σ ∈ L ≡ N ; if σ i ∈ L ≡ N for all i ∈ I then i∈I σ i ∈ L ≡ N . (2) Its negative closure L ¬ N by: if σ ∈ L ′ N then ¬σ ∈ L ¬ N ; if σ i ∈ L ¬ N for all i ∈ I then i∈I σ i ∈ L ¬ N . (3) Its positive closure L √ N by: if σ ∈ L ′ N then σ ∈ L √ N ; if σ i ∈ L √
N for all i ∈ I then i∈I σ i ∈ L √ N . Remark 6.14. Obviously these closures make sense for any given logic L, but we prefer to restrict our attention to L ′ N since it will be enough for our goal and gives rise to a simpler notation.
Whenever we have a bag of "good" properties (such as L ′ N above), to assert by means of a single formula which is the subset of properties that a certain element satisfies it is not enough to assert that it satisfies each of them: we also need to assert that it does not satisfy any of the rest. This is why we need formulas in the symmetric closure. By contrast, if the only available formulas belong to the negative (resp. positive) closure, we can only assert that the element has at most (resp. at least) the enumerated properties. Next we present the unified logics for all the linear semantics in the spectrum. ( We have used the negative and symmetric closures for the "failures-based" and"readiesbased" semantics, and we can use the positive closure to define two new semantics that have not been considered earlier in this paper, nor elsewhere as far as we know. For that we need to observe partial offers along a computation, or just at its end, where X is a partial offer of p if X ⊆ I(p). It is clear the duality with respect to the failures semantics, where F is a failure of p if I(p) ⊆ F . We can introduce these two new semantics at each layer of the spectrum through the corresponding partial offers for each N ∈ {U, C, I, T, S}. Definition 6.16.
3) • ⊤ ∈ L ′ ≤ lf N ; • if σ ∈ L ≡ N then σ ∈ L ′ ≤ lf N ; • if ϕ ∈ L ′ ≤ lf N and a ∈ Act then aϕ ∈ L ′ ≤ lf N . (4) • ⊤ ∈ L ′ ≤ lf ⊇ N ; • if σ ∈ L ¬ N then σ ∈ L ′ ≤ lf ⊇ N ; • if ϕ ∈ L ′ ≤ lf ⊇
(1) The semantics of partial offer traces for the constraint N is that defined by the logic L ′ ≤ l⊆ N with:
• ⊤ ∈ L ′ ≤ l⊆ N ; • if ϕ ∈ L ′ ≤ l⊆ N and σ ∈ L √ N then σ ∧ ϕ ∈ L ′ ≤ l⊆ N ; • if ϕ ∈ L ′ ≤ l⊆ N
and a ∈ Act then aϕ ∈ L ′ ≤ l⊆ N .
(2) The semantics of partial offers for the constraint N is that defined by the logic L ′
≤ lf ⊆ N with: • ⊤ ∈ L ′ ≤ lf ⊆ N ; • if σ ∈ L √ N then σ ∈ L ′ ≤ lf ⊆ N ; • if ϕ ∈ L ′ ≤ lf ⊆ N and a ∈ Act then aϕ ∈ L ′ ≤ lf ⊆ N .
Duality between failures and partial offers causes the picture of the complete layer of linear semantics for each N to become two diamonds that share the side corresponding to the readies-based semantics. Now, recalling Theorem 6.7. (2) L ′ F T and L ′ ≤ l⊆ I are incomparable: p ≤ l⊇ I q does not imply p ≤ l⊆ I q and p ≤ l⊆ I q does not imply p ≤ l⊇ I q. Proof. In fact, we have a stronger result by combining these two statements: if we consider p = ab + ac, q = a(b + c), and r = p + q, then p = l⊇ I r but r lf ⊆ I p, and q = l⊆ I r but r lf ⊇ I q.
Similar counterexamples exist for N ∈ {T, S}. However, for N ∈ {U, C}, which produce the trace and the completed trace semantics, respectively, it is easy to prove that the six logics of the layer are equivalent.
Proposition 6.18. (1) L ′ ≤ lf U = L ′ ≤ l U = L ′ ≤ l⊇ U = L ′ ≤ l⊆ U = L ′ ≤ lf ⊇ U = L ′ ≤ lf ⊆ U = L T (2) L ′ ≤ lf ⊇ C = L ′ ≤ lf ⊆ C = L ′ ≤ l⊇ C = L ′ ≤ l⊆ C = L ′ ≤ lf C = L ′ ≤ l C = L CT .
Proof. An interesting result illustrating the generality of our characterizations concerns one of the finest semantics in the classic spectrum: possible futures. Possible futures is located in Figure 1 below 2-nested simulation because the more accurate trace simulation semantics was not yet included in the spectrum; this is corrected in the spectrum in Figure 11. Indeed, for N = T we have the following result.
6.2.3.
Logical characterization of the deterministic branching semantics. Now we consider the deterministic branching semantics. In the classic spectrum the only such semantics is possible worlds but, as we pointed out before, there is one such semantics at each layer of the extended spectrum. Table 4: Logical characterizations of the semantics used as constraints.
❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ Formulas Constraints (N ) U C I T S B ⊤ ∈ L ′ N • • • • ν ν ¬⊤ = ⊥ ∈ L ′ N ν ν ν ν ν ν ¬0 ∈ L ′ N • • ν ν ν a ∈ Act ⇒ a⊤ ∈ L ′ N • ν ν ν ϕ ∈ L ′ N , a ∈ Act ⇒ • • • aϕ ∈ L ′ N ϕi ∈ L ′ N ∀i ∈ I ⇒ • • i∈I ϕi ∈ L ′ N ϕ ∈ L ′ N ⇒ • ¬ϕ ∈ L ′ N
In order to capture determinism we need to consider conjunctive formulas to express the desired branching, but only when it corresponds to a choice between different actions. This leads us to the following scheme:
if X ⊆ Act and ϕ a ∈ L D N for all a ∈ X, then a∈X aϕ a ∈ L D N .
L ′ D N by: • ⊤ ∈ L ′ D N ; • if ϕ ∈ L ′ D N and σ ∈ L ≡ N then σ ∧ ϕ ∈ L ′ D N ; • if X ⊆ Act and ϕ a ∈ L ′
D N for all a ∈ X then a∈X aϕ a ∈ L ′ D N . For N = I we obtain the unified logical characterization of the possible worlds semantics.
Proposition 6.23. L ′ D I ⊇ L P W . Proof. Analogous to the case of ready simulation semantics. Proposition 6.24. L ′ D I ≁ L P W . Proof. This is a consequence of the fact that the original logical characterization of the possible worlds semantics, L P W , was wrong. For instance, taking p = abc + a(bc + d) + ab and q = a(bc + d) + ab then p ≡ P W q but p ∼ L P W q, since L P W cannot "observe" the intermediate offer that makes the possible world abc different from those of q. By contrast, the formula ϕ = a(¬d ∧ bc) ∈ L ′ D I is enough to distinguish p and q, since p |= ϕ and q |= ϕ.
We postpone to Section 7 the proof of the equivalence between our observational and logical characterizations of the possible worlds semantics. As a consequence of this correspondence, we have that a logical characterization only works in the infinite case if we restrict ourselves to image-finite processes.
In Tables 4 and 5 we present our results in a three-dimensional way. Table 5 shows the rules defining the logics characterizing each of the semantics at each layer of the spectrum. On top of it also appears, as example, the classic notation for the corresponding semantics represented when N = I. Table 4 contains the logics that characterize the constraint governing each of these layers. There are two semantics that are included in both tables, in order to emphasize their double role as "main" and "auxiliary" semantics. However they are disguised under different names: this is the case of T = ≤ l U (in fact, it is also equal to the other three linear U -semantics) and S = U S.
❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ Formulas Semantics (YN ) ≤ lf ⊇ N ≤ lf N ≤ l⊇ N ≤ l N DN N S N ∈ {U, C, I, T, S} F R FT RT PW RS when N = I ⊤ ∈ L ′ Y N • • • • • ν ϕ ∈ L ′ Y N , a ∈ Act ⇒ • • • • ν • aϕ ∈ L ′ Y N ϕ ∈ L ¬ N ⇒ • ν ν ν ν ν ϕ ∈ L ′ Y N ϕ ∈ L ≡ N ⇒ • ν ν ν ϕ ∈ L ′ Y N ϕ ∈ L ′ Y N , σ ∈ L ¬ N ⇒ • ν ν ν σ ∧ ϕ ∈ L ′ Y N ϕ ∈ L ′ Y N , σ ∈ L ≡ N ⇒ • • ν σ ∧ ϕ ∈ L ′ Y N X ⊆ Act, ϕa ∈ L ′ Y N ∀a ∈ X ⇒ • ν a∈X aϕa ∈ L ′ Y N ϕi ∈ L ′ Y N ∀i ∈ I ⇒ • i∈I ϕi ∈ L ′ Y N ϕ ∈ LN ⇒ • ϕ ∈ L ′ Y N ϕ ∈ LN ⇒ • ¬ϕ ∈ L ′ Y N
Relating the unified logics and the unified observational model
In this section we will relate our unified logical characterizations and the unified observational semantics. As indicated in Section 2, we have to restrict ourselves to image-finite processes; as a byproduct, the finite parts of each of the corresponding languages, that are obtained by intersection with L f HM , provide us with a pure finite logical characterization of the semantics. However, it is convenient in the first part of this Section to consider still the full (infinitary) logic characterizing each of the semantics. (1) Given a set of formulas L whose outermost operator is not conjunction, the set N (L) of induced normal formulas is defined by:
• ⊤ ∈ N (L); • if Γ 1 , Γ 2 ⊆ L, {a i | i ∈ I} ⊆ Act, and ϕ i ∈ N (L), then ( σ∈Γ 1 σ ∧ σ∈Γ 2 ¬σ) ∧ i∈I a i ϕ i ∈ N (L). (2) For each N ∈ {U, C, I, T, S} and each Y N ∈ {N S, ≤ l N , ≤ l⊇ N , ≤ lf N , ≤ lf ⊇ N , ≤ l⊆ N , ≤ lf ⊆ N , D N } in the spectrum, we define the set of normal formulas N Y N (L ′′ N ) ⊆ L ′ Y N as N Y N (L ′′ N ) = N (L ′′ N ) L ′ Y N ,
where L ′′ N is the set of formulas in L ′ N whose outermost operator is not conjunction.
Remark 7.2. The clause in Definition 7.1.1 is more involved than it appears. Initially, we can apply it with I = ∅ to obtain the first (non-trivial) normal formulas and then recursively to obtain more complex normal formulas; note that the two first subformulas stem always from the original set L. By abuse of notation, when some of the elements in our normal formulas do not appear in the corresponding set L ′ Y N , we assume that these formulas have been extended by conjunction with ⊤ using the fact that σ∈∅ σ is another syntactic form to express ⊤. Also note that infinite conjunction is allowed in the two first subformulas. As a consequence, if we consider the tree-like form of these (possibly infinitary) formulas they could have infinite depth. However, if we define the normal depth of formulas in N (L N ) as that obtained by counting the recursive nesting in the application of Definition 7.1, then any normal formula has finite normal depth, and the set they form can be explored by structural induction.
Theorem 7.3. Each set of normal formulas N Y N (L ′′ N ) associated to the semantics in the spectrum is equivalent to the full set of formulas L ′ Y N . Proof. By structural induction, all the formulas in L ′ Y N admit a normal formula in the sense of Definition 7.1, that is obtained by gathering the subformulas and applying Proposition 6.2.
Definition 7.4. The set of complete normal formulas CN (L) (resp., the set of complete normal formulas associated to each semantics in the spectrum, CN Y N (L ′′ N )) is the set of normal formulas (resp., the set of normal formulas associated to each semantics in the spectrum) for which the rule in Definition 7.1 is applied with Γ 2 = Γ 1 . Now we prove that infinite conjunction in Definition 7.1 can be approximated by finite conjunction.
Theorem 7.5. If we restrict ourselves to image-finite processes, for each denumerable set of formulas L, any complete normal formula ϕ ∈ CN (L) can be approximated by a set of finite normal formulas {ϕ k | k ∈ IN} that only use finite conjunction, that is, p |= ϕ iff p |= ϕ k for all k ∈ IN.
Proof. We define the sequence ϕ k by structural induction on the normal depth of ϕ: • ϕ = ( σ∈Γ 1 σ ∧ σ∈Γ 1 ¬σ). We consider a fixed enumeration of the set L = {σ n | n ∈ IN}, and define L n = {σ j ∈ L | j n}. Then, for each k ∈ IN:
ϕ k = σ∈Γ 1 ∩L k σ ∧ σ∈Γ 1 ∩L k ¬σ.
We have p |= ϕ ⇔ (p |= σ ∀σ ∈ Γ 1 and p |= σ ∀σ ∈ Γ 1 ) and p |= ϕ k ∀k ∈ IN ⇔ (p |= σ ∀σ ∈ Γ 1 ∩ L k and p |= σ ∀σ ∈ Γ 1 ∩ L k ) and the result follows from a the equality
Γ 1 = Γ 1 ∩ L = Γ 1 ∩ ( k∈IN L n ).
• ϕ = ( σ∈Γ 1 σ ∧ σ∈Γ 1 ¬σ) ∧ i∈I a i ϕ i . By structural induction we can assume that the result is true for any subformula ϕ i . Then we define ϕ k = σ∈Γ 1 ∩L k σ ∧ σ∈Γ 1 ∩L k ¬σ ∧ i∈I a i ϕ k i . Now, if we decompose ϕ as ϕ I ∧ ϕ II (taking ϕ II = i∈I a i ϕ i , and analogously for the set of approximations) we have that p |= ϕ k iff p |= ϕ k I and p |= ϕ k II . If p |= ϕ k then p |= ϕ k I for all k ∈ IN and arguing as in the base case above we conclude that p |= ϕ I . Any image-finite process p can be decomposed as p = a i ∈Act m i j=1 a j i p j i , and we have p |= ϕ k II iff for all i there exists j with a i = a j i and p j i |= ϕ k i . Then, if p |= ϕ k II for all k ∈ IN, for each i there exists some j ∈ 1..m i such that p j i |= ϕ k i for infinitely many k, but this means that p j i |= ϕ k i for all k ∈ IN and then, by the induction hypothesis, p j i |= ϕ i thus getting p |= ϕ.
l ∈ L N (p) l n ∈ L N (p n ) l 1 ∈ L N (p 1 )Y N ∈ {N S, ≤ l N , ≤ l⊇ N , ≤ lf N , ≤ lf ⊇ N , ≤ l⊆ N , ≤ lf ⊆ N , D N } in the spectrum, if we restrict ourselves to the set of image-finite processes we have L f Y N ∼ L ′ Y N . Proof.
We only need to apply Theorem 7.5. The only non trivial case is when N = S, where we have to apply twice the Theorem, using also the fact that CN (L ′′
S ) ∼ CN (L f S ), because L ′′ S ∼ L f S .LGO N / ≃ lf N (resp. LGO N / ≃ lf ⊇ N ) are isomorphic, that is, ↔ −1 is injective and ϕ ↔ θ iff θ ≃ lf ⊇ N θ ϕ , for some adequate θ ϕ . Proof.
(1) As can be seen in Figure 14, a branching observation is a labeled tree whose nodes are local observations and whose arcs are labeled by actions. The general form of any complete normal formula in CN N S (L N ) is ( σ∈Γ σ∧ σ / ∈Γ ¬σ)∧ i∈I a i ϕ i , with ϕ i ∈ CN N S (L N ) for all i ∈ I. Since the language L ′ N characterizes the semantics used to get the local observations, we can associate to each complete formula ( σ∈Γ σ ∧ σ / ∈Γ ¬σ) the corresponding local observation l ∈ L N . Then, by structural induction, we obtain the observation associated to each formula ϕ i ∈ CN N S (L N ), thus getting the branching general observation BGO N associated to the given formula. It is easy to see that this correspondence is indeed a bijection. The case for CN D N (L N ) is analogous, but now it is not allowed to have repeated actions in the arcs leaving any node of an observation; this is obviously reflected in the form of the formulas in the corresponding language.
(2) The case for CN ≤ l N (L N ) is similar to the previous one, but now the obtained (degenerated) tree is just a single branch corresponding to a lgo in LGO N .
For CN ≤ l⊇ N (L N ) the general form of a complete normal formula is ϕ = (⊤∧ σ / ∈Γ ¬σ)∧ aϕ ′ , with ϕ ′ ∈ CN ≤ l⊇ N (L N ). If we close the set Γ by derivability obtaining Γ ′ and then consider its complement Γ ′ , we can consider the local observation l that satisfies all the formulas in Γ ′ and none in Γ ′ . The linear general observation lgo corresponding to ϕ is then recursively defined as l, {(a, lgo ′ )} where lgo ′ is the linear general observation corresponding to ϕ ′ .
To proceed in the opposite direction, we just need to take as Γ the complement of the set of formulas in L ′ N satisfied by the local observation l at the root of the given LGO N , and then proceed in a recursive way.
(3) In this case, the general form of a complete normal formula in CN ≤ lf N (L N ) is ϕ = ⊤ ∧ a 1 (. . . (⊤ ∧ a n−1 (⊤ ∧ a n ( σ∈Γ σ ∧ σ / ∈Γ ¬σ) . . .). Now we establish a correspondence between the set of local observations L N and the sets Γ ⊆ L N as done in cases (1) and (2) ∈Γ ¬σ corresponding to the complement I(p). Since in this case the sets of lgo's could be assumed to be closed with respect to the order ≤ l⊇ N in Definition 4.19, soundness is retained after "assuming" that any formula σ / ∈Γ ¬σ "generates" the observation associated to Γ, even though some of the formulas σ ∈ Γ may not be satisfied when the corresponding I(p) is smaller. But for the failures and failure trace semantics we can proceed by closing the set of offers upwards with respect to ⊆ and no new failure is introduced. To finish the proof, we just need to show that N Y N (L N ) and CN Y N (L N ) are equivalent. Any consistent formula in N Y N (L N ) (Γ 1 Γ 2 = ∅) provides only some partial information about the states in a computation, so that the concrete values of these states are characterized by a set Γ with Γ 1 ⊆ Γ ⊆ Γ 2 . Therefore, we can replace Γ 1 and Γ 2 by Γ and Γ, respectively, adding the disjunction over all the possible values of Γ, to characterize the set of processes specified by the formula. Now it is enough to float the disjunction up to obtain a disjunction of formulas in CN Y N (L N ), and applying Proposition 6.2 we get the equivalence between the two sets of formulas. Finally, we only need to apply Corollary 7.7 to conclude. Proof. Since it was proved in Section 4 that any observational semantics characterizes the corresponding (classical) semantics in the (extended) ltbt spectrum, the desired equivalence between our (unified) logical characterizations and the classical semantics is an immediate corollary.
On the real diamond structure
This section is a practical proof of the suitability of our unification work. Some recently proposed semantics that were not in the original ltbt spectrum are nicely included in our extended spectrum, which shows why and how the old spectrum has to be expanded. Our unified approach immediately absorbs these new semantics and the results about the different characterizations are easily extended to cover them. We warmly thank Roscoe for pointing out to us his work on the stable revivals semantics [46,48], where an endeavor for an adequate presentation of the notion of responsiveness for a CSP-like language is made. (Responsiveness had been previously studied by Fournet et al. in [27] for CCS, under the name of stuck-freeness.)
When faced with the diamond shape of the collection of linear semantics that are associated to each simulation semantics in the extended spectrum, it would be natural to expect it to reflect the structure of a lattice. Then, failure semantics would be the greatest lower bound of the readiness and failure trace semantics, while ready trace semantics would be the corresponding lowest upper bound. However, both intuitions turn out to be wrong and a new semantics finer than failures and another one coarser than ready trace can be found: together with readiness and failure trace, they do constitute a lattice.
Let us first consider the case of the lowest upper bound. We postulate that the axiomatization of the associated semantics is obtained by instantiating our general axiom with the conjunction of the two conditions M R and M F T :
M R∧F T (x, y, w) ⇐⇒ I(x) ⊇ I(y) and I(w) ⊆ I(y).
We denote with ⊑ R∧F T the order axiomatized by the corresponding axiom (ND R∧F T ). Proof. ⊑ RT ⊆ ⊑ R∧F T is an immediate consequence of Proposition 3.2 and the fact that condition M RT implies both M R and M F T , and hence also M R∧F T . To show that ⊑ RT ⊆ ⊑ R∧F T , let us take w = 0, y = b, and x = bB ′ + c; then we have:
a(bB + bB ′ + c) p ⊑ R∧F T a(bB ′ + c) + abB q but, if I(B) = I(B ′ ), a(bB + bB ′ + c) ⊑ RT a(bB ′ + c) + abB because {a}a{b, c}bI(B) ∈ ReadyTraces(p) \ ReadyTraces(q).
It is clear that the readiness and failure trace semantics is finer than both the readiness and the failure trace semantics; to show that it is actually the coarsest upper bound we need to prove that ⊑ R∧F T = ⊑ R ∩ ⊑ F T . Even if the axiom (ND R∧F T ) was created with this goal in mind, this cannot be easily shown using only algebraic arguments. Instead, it is trivial to obtain the observational characterization of the desired semantics by gathering together the failure trace and the ready observations. Based on Definition 4.19, we can define the corresponding order ≤ l⊇∧f N by taking T ≤ l⊇∧f N T ′ ⇐⇒ T ≤ l⊇ N T ′ and T ≤ lf N T ′ . A direct characterization can be obtained as follows. We combine both kinds of observations into a single family of decorated traces that we call failure trace with final ready sets, by considering failure sets all along the trace except at the end of it, where we introduce the corresponding ready set.
Definition 8.3. We define the order ≤ l⊇∧f N by T ≤ l⊇∧f N T ′ ⇐⇒ for all X 0 a 1 . . . X n ∈ T there is some Y 0 a 1 . . . Y n ∈ T ′ with X n = Y n and X i ⊇ Y i , for i ∈ 0..n − 1.
Proposition 8.4. The semantics defined by the order ⊑ R∧F T coincides with that defined by ≤ l⊇∧f I and is thus the lowest upper bound of the readiness and failure trace semantics.
Proof. Similar to that of Theorem 5.4.
Let us finally consider the logical characterization of this semantics. It is clear that the conjunction of two semantics should be characterized in a logical way by simply considering the union of the logics that characterize both semantics (although there could possibly be a more compact presentation).
Definition 8.5. We define the set of formulas L ′ ≤ l⊇∧f I as that generated by the clauses: , which is immediate.
• ⊤ ∈ L ′ ≤ l⊇∧f I ; • if ϕ ∈ L ′ ≤ l⊇∧f I and σ ∈ L ¬ I then σ ∧ ϕ ∈ L ′ ≤ l⊇∧f I ; • if σ ∈ L ≡ I then σ ∈ L ′ ≤ l⊇∧f I ; • if ϕ ∈ L ′ ≤ l⊇∧f I
By replacing the I above by the generic N, we get the definitions and results for the general case.
The axiomatic characterization of the greatest lower bound of the readiness and failure trace semantics is much simpler: we simply put together the axioms for the orders defining both semantics. Proposition 8.9. The semantics defined by the order ⊑ R∨F T is the finest semantics that is coarser than both the readiness and the failure trace semantics.
Proof. Obvious since any semantics coarser than the readiness semantics has to satisfy Once again, the semantics defined by ⊑ R∨F T is not included in the ltbt spectrum and neither in our extended one; in particular, it is different from the failures semantics. To prove this fact we make essential use of the notion of revival, as defined by Reed, Roscoe, and Sinclair [46]. Revivals are sequences a 1 , . . . , a n (X, a) where a 1 , . . . , a n is a trace of the corresponding process after which the action a is offered, but the set of actions X is refused.
Proposition 8.10. The meet semantics R ∨ F T is strictly finer than failure semantics.
Proof. The inclusion ⊑ R∨F T ⊑ F is obvious since failures semantics is coarser than both the readiness and the failure trace semantics. To show that the inclusion is strict, note that any two processes related by ⊑ R∨F T do not only have the same failures but also the same revivals. This is indeed the case since all the axioms u v in preserve the revivals, which means Revivals(σ(u)) ⊆ Revivals(σ(v)) for every ground substitution σ, and the revivals order is a precongruence for the operators in BCCSP. For instance, for (ND F T ) we need to prove that Revivals(σ(a(x + y))) ⊆ Revivals(σ(ax)) ∪ Revivals(σ(a(y + w))) whenever I(σ(w)) ⊆ I(σ(x)). It is clear that the only non-trivial case occurs when a(X, b) ∈ Revivals(σ(a(x + y))); then we have (X, b) ∈ Revivals(σ(x + y)) so that X ∈ Failures(σ(x)) ∩ Failures(σ(y)) and b ∈ I(σ(X)) or b ∈ I(σ(y)). In the first case, a(X, b) ∈ Revivals(σ(ax)) whereas, in the second, X ∈ Failures(σ(x + y)) and therefore a(X, b) ∈ Revivals(σ(a(x + y))). The case for (ND R ) is simpler. Once we know that ⊑ R∨F T preserves the revivals we only need to observe that the revivals cannot be obtained from the failures of a process. In particular, we have ab ⊑ F a + a(b + c), but a({c}, b) ∈ Revivals(ab) \ Revivals(a + a(b + c)).
Next we present the characterization of the revivals semantics in terms of our observational framework.
Definition 8.11. We define the order ≤ l⊇∨f LGO I (q).
N by T ≤ l⊇∨f N T ′ ⇐⇒ for all X 0 a 1 . . . X n ∈ T there is {Y 0 a 1 Y 1 . . . Y j n | j ∈ J} ⊆ T ′ such that X n = j∈J Y j n .
Proof. Note that ≤ l⊇∨f I can be equivalently defined as T ≤ l⊇∨f I T ′ ⇐⇒ for all X 0 a 1 . . . X n ∈ T and for all a ∈ X n there is Y 0 a 1 . . . Y n ∈ T ′ with a ∈ Y n and Y n ⊆ X n . Now, since a 1 . . . a n (X, a) ∈ Revivals(p) if and only if there exists X 0 a 1 . . . X n ∈ LGO I (p) such that a ∈ X n and X n ∩ X = ∅, we obtain the desired characterization.
Definition 8.13. Given T ⊆ LGO N , T ⊇∨f is defined as
T ⊇∨f = {X 0 a 1 . . . X n | there is {Y 0 a 1 . . . Y j n | j ∈ J} ⊆ T with X n = j∈J Y j n }.
This clearly indicates that ≤ l⊇∨f I is in between ≤ lf ⊇ I , defining the failures semantics, and ≤ lf I , defining readiness semantics. This is useful for the proof of the axiomatic characterization of the revivals semantics. Proof sketch. It is quite similar to that of Theorem 5.4 for the case of failures semantics and, hence, also similar to the characterization of that semantics by means of acceptance trees [30] (and where the closure of the set of offers with respect to both union and convex closure is a critical argument), and this is why we only sketch it. In connection to that, recall that the application of the particular case of (ND) corresponding to (ND F T ) allowed us to join arbitrary states after the same trace, while that corresponding to (ND R ) allowed us to obtain a common continuation after the same action at any state reachable by the same trace. All this can be done now using (ND R∨F T ); however, we cannot add to an arbitrary state an action offered at another state reachable by the same trace since to do that we needed the unlimited strength of axiom (ND).
Note that for the join semantics R ∧ F T the logical approach was the most direct way of defining it, whereas its equational characterization needed more care. For the meet semantics R∨F T , the situation is just the opposite. As we have seen, R∨F T is axiomatized by putting together the axioms for R and those for F T ; in contrast, the logic characterizing R ∨ F T is obtained by cleverly selecting the common part of the logics characterizing both R and F T . If we had defined the logical semantics by considering all the formulas from HML that are preserved by each semantics, then we could take the intersection of these sets as the logical semantics of any meet semantics. Since we defined our logical semantics by considering only a "basis" that generates the corresponding full set, we cannot simply take their intersection.
Definition 8.15. We define the set of formulas L ′ ≤ l⊇∨f I as that generated by the clauses:
• ⊤ ∈ L ′ ≤ l⊇∨f I ; • if σ, σ j ∈ L ′ I for all j ∈ J then (σ ∧ j∈J ¬σ j ⊤) ∈ L ≤ l⊇∨f I ; • if ϕ ∈ L ′ ≤ l⊇∨f I and a ∈ Act then aϕ ∈ L ′ ≤ l⊇∨f I .
Note that in the second clause of this definition we have relaxed the condition in the definition of L ′ R by considering an arbitrary failure (that defined by the set J), but only a positive offer (the action appearing in σ). This is how the revivals semantics becomes slightly finer than the failures semantics. p |= ϕ and q |= ϕ, which is almost immediate.
Again, by replacing the I above by the generic N, we get the definitions and results for the general case.
We can generalize most of the results obtained for the refusal semantics when N = I to any reasonable local observation function such as T or S, once we interpret ⊆ as the corresponding order and = as the induced equivalence. However, in order to define the adequate observational characterization of the revivals semantics for a local observation (or constraint) N , we should look for the adequate "elements" of the universe of observations. This leads us to traces when N is T , but it is not so clear how to define those "elements" for a non-extensional semantics such as that obtained when N is S.
Let us conclude this section with a look at the picture in Figure 15 showing the real structure of the full (bidimensional!) diamond, that should be included in all the upper levels of the extended ltbt spectrum.
Operational semantics
In this section we explain how to develop the semantics in the spectrum in an operational way. Certainly, this presentation could be argued to be ad-hoc at times since some "highlevel" conditions are required in the SOS-like rules for some of the semantics. Moreover, the style of presentation at this Section is certainly less precise and detailed than in the previous ones. However, we believe it still provides some additional insight on the common properties of the semantics and also establishes a connection with our previous work on (bi)simulations up-to [23,25] as a way to get coinductive characterizations of any "reasonable" process semantics.
Structural operational semantics was introduced by G. Plotkin in 1981, even though his seminal work was not published in a journal until 2004 [45]. In Section 2 we already presented a basic operational semantics for our processes as a starting point for the definition of all the semantics in the spectrum: a small-step semantics that collects the (atomic) actions executed by the processes into the corresponding transition system. By contrast, all the operational semantics in this section will be big-step semantics which directly return the adequate semantic values defining each of the semantics. They are generated by means of SOS-like rules that obtain these values in a compositional way. An extensive presentation of structural operational semantics covering all its variants can be found in [39]. 9.1. Local simulations up-to. In order to characterize all the reasonable behavior preorders in a coinductive way we need to generalize constrained N -simulations (Definition 2.2) with N -simulations up-to an order ⊑. Definition 9.1. Let ⊑ be a behavior preorder and N a relation over processes. We say that a binary relation S over processes is an N -simulation up-to ⊑ if S ⊆ N and S is a simulation up-to ⊑. Or equivalently, in a coinductive way, whenever we have pSq we also have:
• for every a, if p a −→ p ′ a then there exist q ′ , q ′ a such that q ⊒ q ′ a −→ q ′ a and p ′ a Sq ′ a ; • pN q.
We say that process p is N -simulated up-to ⊑ by process q, or that process q N -simulates process p up-to ⊑, written p ❁ ∼ N ⊑ q, if there exists an N -simulation up-to ⊑, S, such that pSq.
We often just write ❁ ∼ N , instead of ❁ ∼ N ⊑ , when the behavior preorder is clear from the context.
We proved in [23] that all the preorders defining the semantics in the ltbt spectrum can be characterized as N -simulations up-to the corresponding equivalence relation ≡, where N is the constraint defining the coarsest simulation semantics finer than the given semantics. For instance, the result for the semantics between failures semantics and ready simulation was the following. Table 6 shows the constraints defining the adequate constrained simulation order finer than each of the semantics in the linear time-branching time spectrum. Obviously, they coincide with the layer of the extended spectrum at which each semantics appear.
T S CT CS F R FT RT PW RS PF 2N C O U U C C I I I I I I V W pU q ⇐⇒ true pV q ⇐⇒ p ≡ T q pCq ⇐⇒ (p = 0 iff q = 0)
pW q ⇐⇒ p ≡ S q pIq ⇐⇒ I(p) = I(q) Table 6: Constraints for the semantics in the ltbt spectrum.
Note that Theorem 9.2 is more subtle that it could appear: it characterizes a given preorder with a constrained simulation upto the preorder itself (Definition 9.1). Therefore, there are several semantics that share the same constraint. This characterization is indeed rather technical and the key point is that it allows to express any behavior preorder in a simulation-like fashion. We have used this characterization to prove many useful statements in our previous work 5 and we will use it again several times in the current Section.
In our proof of the completeness of the axiomatizations for the linear semantics in the spectrum in Section 5 we used a notion of normal form which, roughly, was defined by applying repeatedly to any term p the axiom (ND ≡ ) from right to left, for as long as possible. Propositions 5.2 and 5.3 were then the key results to complete the proof, and also lie behind the intuition for introducing now the notion of local I-simulation up-to. Definition 9.3. For Z ∈ {F, R, F T, RT } and p = a i ap i a , whenever we have a pair of indices i, j and a decomposition p j a = r j a + s j a with M Z (p j a , r j a , s j a ) we say that p is 1-locally Z-equivalent to q = p + a(p i a + r j a ), and we write p ≡ l1 Z q. We say that p and q are locally Z-equivalent when they are related by the reflexive and transitive closure of ≡ l1 Z , and then we write p ≡ l Z q. For Z ∈ {F, R, F T, RT } we refer to the I-simulations up-to ≡ l Z as local I-simulations up-to ≡ Z . We say that process p is locally I-simulated up-to ≡ Z by process q, or that process q locally I-simulates process p up-to ≡ Z , written p ❁ ∼ I ≡ l Z q, if there exists a local I-simulation up-to ≡ Z , S, such that pSq.
Local I-simulations up-to are enough to characterize the linear semantics in {F, R, F T, RT }. Note that we cannot get a local notion of bisimulation up-to equivalent to our unrestricted notion of bisimulation up-to. Proof. The implication from right to left is an immediate consequence of Theorem 9.2. For the other, note that {(p, q) | p ⊑ Z q} is a local I-simulation up-to ≡ Z . Indeed, for any p a −→ p i a we have q ≡ l Z hnf Z (q) and taking hnf Z (q) = a i ah j a there exists some j such that hnf Z (q) a −→ h j a and p i a ⊑ Z h j a . Example 9.5. Let us consider the processes p = abc + abd and q = a(bc + bd). We have p ≡ F q and we can check that p ❁ ∼ I ≡ l F q since p ⊑ RS q. In order to prove that we also have q ❁ ∼ I ≡ l F p, we apply ≡ l F to p to obtain p ≡ l F p + q and then we obtain q ⊑ RS p. By contrast, if we wanted to apply our bisimulation up-to characterization to prove directly that p ≡ F q then we would have to turn q into q + p in order to simulate the transition p a −→ bc. This would correspond to the local application of (ND F ≡ ) combined with that of (RS ≡ ) I(x) = I(y) =⇒ a(x + y) ≃ a(x + y) + ax. But if we replace the action a by a larger prefix a 1 . . . a n then we should also modify the process q ′ = a 1 . . . a n (bc + bd) in a non-local way in order to obtain q ′′ = q ′ + p ′ , so that we could suitably simulate the transition p ′ = a 1 . . . a n bc+a 1 . . . a n bd a 1 −→ a 2 . . . a n bc. Certainly, this is not necessary when checking p ′ ≡ F q ′ by means of local simulations up-to.
The coinductive characterization of the semantics by means of simulations up-to has at least two important advantages over that of using bisimulations up-to. First, we can characterize the orders defining the semantics and not just the induced equivalences; and second, we can use a local variant of the up-to mechanism so that we only need to rely on the equivalence relation ≡ l Z for the up-to part.
9.2.
Operational rules for the linear semantics of processes. In Section 9.1 we have introduced and proved some results that establish the framework using which we achieve our goal: to define for each of the classic linear semantics an operational semantics over BCCSP terms in such a way that we can use constrained simulations to characterize the considered semantics. For instance, if we consider the case of the failures preorder ⊑ F , we are going to define a new operational semantics for BCCSP terms (P, Act, ⇒ F ) such that p ⊑ F q if and only if q ready simulates p in (P, Act, ⇒ F ). Next we will concentrate first on the diamond of linear semantics coarser than ready simulation. All these semantics are based on the observation of the initial set of actions of each process, that can be obtained by application of the SOS-like rules in Figure 16.
The rules in Figure 17 define the transition relation =⇒ Z that induces the operational semantics to characterize each of the Z-semantics. The transition relation ←→ Z is an auxiliary relation that captures the iterated application of the axiom (ND Z ≡ ). Rules (RF) and (TR) define reflexivity and transitivity of the relation ←→ Z . Finally, the rule (CL) 0 −→ I ∅ ap −→ I {a} p −→ I A q −→ I B p + q −→ I A ∪ B Figure 16: Rules that compute the set of initial actions of a process.
(ND) p −→ I A p q −→ I A q r −→ I A r M Z (A p , A q , A r ) ap + a(q + r) + s ←→ Z ap + a(q + r) + a(p + q) + s combines the auxiliary relation ←→ Z and the original operational transition relation −→ (see Figure 2), to define the new labeled transitions =⇒ Z . Definition 9.6. For Z ∈ {F, R, F T, RT }, the operational semantics for BCCSP terms is given by the labeled transition system (P, Act, =⇒ Z ) where the transition relation =⇒ Z is defined by the rules in Figure 17.
(RF) p ←→ Z p (TR) p ←→ Z q q ←→ Z r p ←→ Z r (CL) p ←→ Z p ′ p ′ a −→ q p a =⇒ Z q
By abuse of notation, we have written M Z (A p , A q , A r ) to express that we check M Z (p, q, r) using the initials computed by −→ I .
The relation =⇒ Z has some interesting properties. First, it is an extension of the original transition system. Proposition 9.7. For Z ∈ {F, R, F T, RT }, p and q BCCSP processes, and α a sequence of actions in Act, we have that p α =⇒ q implies p α =⇒ Z q.
Although usually some new transitions appear, the set of initial actions of any process always remains the same. It is also clear that, for any Z ∈ {F, R, F T, RT }, the auxiliary relation ←→ Z preserves the equivalence ≡ Z because the rule (N D) corresponds to the application of axiom (I-ND Z ≡ ), which is sound with respect to ≡ I Z . Proposition 9.9. For Z ∈ {F, R, F T, RT } and any two BCCSP processes p and q, we have p ←→ Z q implies p ≡ Z q. Now we prove the main theorem in this section, that asserts that for each of the semantics in the considered diamond we can define the corresponding operational semantics as stated in Figure 17. Proof. We will apply our characterization of the orders ⊑ Z by means of local I-simulations up-to at Proposition 9.4 to show that p ⊑ ⇒ Z RS q implies p ❁ ∼ I ≡ l Z q. This is because any ready simulation over the transition system =⇒ Z is also a local I-simulation up-to ≡ Z . Indeed, if R is a ready simulation over the transition system =⇒ Z , and pRq, then whenever we have p a −→ p ′ we also have p a =⇒ Z p ′ , and therefore there is some q a =⇒ Z q ′ with p ′ Rq ′ . By definition of the transition system =⇒ Z , there is some process q ′′ such that q ←→ Z q ′′ and q ′′ a −→ q ′ . Then we also have q ≡ l Z q ′′ , and thus R is indeed a local I-simulation up-to ≡ Z . To prove that p ❁ ∼ I ≡ l Z q implies p ⊑ ⇒ Z RS q, we will check that the relation ❁ ∼ N ≡ l Z is a ready simulation over the transition relation =⇒ Z . If p ❁ ∼ N ≡ l Z q, whenever p a =⇒ Z p ′ we have some process p ′′ such that p ←→ Z p ′′ and p ′′ a −→ p ′ . Then we also have p ≡ Z p ′′ , and so p ′′ ❁ ∼ N ≡ l Z q. From p ′′ a −→ p ′ we now obtain that there are processes q ′ and q ′′ such that q ≡ l Z q ′′ , q ′′ a −→ q ′ , and therefore we also have q ←→ Z q ′′ , thus concluding the proof.
As a consequence of our negative results at the end of Section 9.1, it is not possible to obtain an operational semantics locally defined from that which characterizes the linear semantics by means of bisimilarity. However, this can be done if we use mutual similarity instead of bisimulation.
Certainly, the fact that the characterizations in terms of bisimilarity cannot be defined in a local way is related to the fact that the transition systems generated by application of the algorithm in [17] are larger than those generated by our local transformation here. Unfortunately, it is true that our presentation does not magically lead (at least at the theoretical level) to more efficient algorithms to decide the equivalences with respect to the linear semantics (which are known to be quite hard to decide). Obviously, this is related to the fact that simulation is harder than bisimulation [37]. Even so, these are just theoretical worst case bounds, and it is nice to know that in practice we can apply a local transformation to generate the transition systems characterizing those semantics by means of the simulation orders, that in many concrete cases will not be too difficult to decide. 9.3. Characterizing the semantics corresponding to other constraints. Let us start by considering the case of the universal constraint U . As discussed in Section 3.2, if we use U in the condition M Z it is clear that all the semantics in the corresponding diamond collapse into a single one: trace semantics. It is immediate to realize that the transition system to characterize it in terms of plain simulations is the same transition system =⇒ F that we use to characterize the failures semantics by means of ready simulations.
Theorem 9.11. The trace preorder ⊑ T coincides with the simulation order on the transition system =⇒ F , that is, p ⊑ T q iff p ⊑ ⇒ F S q.
Even if this coincidence is a simple fact that reflects the relation between traces and failures semantics, it contributes to clarify it. In plain words, failures semantics is just traces semantics enriched by the observation of initials, so that the plain simulation order that implies the trace order becomes the ready simulation order.
For other, finer observers such as T we can also characterize the corresponding semantic orders, such as possible and impossible futures, in terms of local simulations up-to. We can use that result to justify that the corresponding transition systems =⇒ T Z would characterize the semantic orders ⊑ T Z in terms of T -simulations that preserve the set of traces of the simulated process. In this case the corresponding operational characterization has to include rules for the computation of the set of traces T (p) and this cannot certainly be done for infinite processes. But out of the computation of these sets, the rest of the rules for the generation of the corresponding transition systems =⇒ T Z are also valid, and their local character is still present. 9.4. Application: trace deterministic normal forms. As a simple application we present the example used by Klin in [36], that we already used in [22] to illustrate our coinductive characterization of the behavior preorders by means of our bisimulations up-to.
Definition 9.12. For any process p = a i ap i a the deterministic form of p is defined as Det(p) = a aDet( i p i a ). We wish to prove that p and Det(p) are trace equivalent. We will do it by proving that they are simulation equivalent over the transition system =⇒ F . Proposition 9.13. For any process p we have p ⊑ F Det(p).
Proof. We will prove that p ⊑ ⇒ F S Det(p) by showing that R = {(p, Det(p+q)) | p, q processes} is a simulation for the transition system =⇒ F . For q = a j aq j a we have Det(p + q) = a Det( i p i a + j q j a ). Then, for any p a =⇒ F p ′ we have p = p i a + k r k a , for some index i and p k a = r k a + s k a a decomposition of any of the rest of the summands of p. We have Det(p + q) a −→ Det( i ap i a + j aq j a ) = Det((p i a + k r k a ) + ( k r k a + j q j a )), so that we also have Det(p + q) a =⇒ F Det((p i a + k r k a ) + ( k r k a + j q j a )), with (p i a + k r k a , Det((p i a + k r k a ) + ( k r k a + j q j a ))) ∈ R.
Proposition 9.14. For any process p we have Det(p) ⊑ F p.
Proof. We will prove that Det(p) ⊑ ⇒ F S p by showing that R = {(Det(p), p)} is a simulation for the transition system =⇒ F . Since Det(p) is deterministic for each a ∈ Act there is a unique transition Det(p) a =⇒ F Det( i p i a ). By applying the definition of a =⇒ F we have p a =⇒ F i p i a , and clearly we have (Det( i p i a ), i p i a ) ∈ R. Although this is a very simple example, it is interesting to compare the proof above with that in [22]. This proof is simpler and more natural, mainly because the proof obligations to check bisimulations forced us to remove the sub-terms that were not in the chosen transition when we had to simulate it. This is not necessary for any of the two simulations that are needed to check mutual simulation, as done above. Obviously, this is also related to the impossibility to obtain a notion of local bisimulation up-to characterizing the equivalence under any of the linear semantics.
Conclusions and some future work
Throughout this paper we have provided a global outline of process semantics from different points of view, each of which reveals some of the key ingredients for a more uniform comprehension of those semantics. We have noted that the family consisting of the simulation semantics-constrained simulations, in its generalized version-plays an essential role in the class of process semantics, becoming the cornerstone for sorting and classifying the remaining semantics. From a framework in which, based on observational trees, denotational semantics are assigned -Section 4-we have been able to prove that the spectrum of process semantics can be structured by means of layers that are induced by the simulations. Each layer is dominated by a simulation semantics that determines the finest distinction available for that layer. The remaining semantic families are also described by abstracting or simplifying the observations needed for the corresponding layer. In particular, below each constrained simulation there appear the corresponding versions for each of the classic linear semantics-failures, readiness, failure trace, and ready trace-and, as we saw in Section 8, other semantics are also explained within our framework.
This observational characterization allowed us to offer a new insight into the axiomatic characterization of the semantics-Sections 3 and 5-revealing a uniformity lacking in all previous studies. To characterize any of the orders that define a process semantics, we have proved that it is enough to use two parametric axioms: one of the required axioms is that for the generalized simulation of the corresponding layer while the other, when it is present, has to do with the reduction of non-determinism that is carried out in each semantics.
Analogously, in Sections 6 and 7 we showed how to characterize process semantics by means of sets of Hennesy-Milner logic formulas out of their observational characterization, and finally we have also discussed a unified operational presentation of the semantics in the extended spectrum.
One of the more obvious lines for future work would be to consider those semantics that allow for an inner, non-visible action, known as weak semantics. Actually, some promising results have already been obtained that make clear the regularity and generality present in the domain of weak semantics. In particular, in [16] it is proved that it is possible to apply to weak semantics the algorithm to obtain axiomatic characterizations of semantic equivalences from the axioms for corresponding order [21]. And [2,3] provides a detailed study of the axiomatization of weak simulation semantics.
Let us also cite here the recent work by Anti Valmari [53], where he presents the full catalogue of (weak) linear-time congruences for finite state systems. Certainly, it is interesting to limit somehow the class of "reasonable" semantics for processes, but this has not been so much the intention of our work in this paper. In fact, it is interesting to note that the results in the paper referenced above limit the set of semantics to explore in a quite personal way: for instance, the semantics of failure traces and that of ready traces are not included in the category, because Valmari (implicitly) considers that they are not "linear-time enough".
Another interesting approach consists in the use of coalgebras-following the work, among others, of Jesse Hughes and Bart Jacobs [35]-where powerful categorical techniques allow to connect the idea of simulation with that of bisimulation, which is central in the coalgebraic setting. More concretely, these techniques were successfully used in [26] to relate classic and probabilistic bisimulation.
Figure 1 :
1Linear time-branching time spectrum.
Figure 2 :
2Operational semantics for BCCSP terms.
Figure 3 :
3The axiomatization for the (strong) bisimulation equivalence.
12D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
14D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ constrained conditional axiom (ND) M (x, y, w) ⇒ a(x + y) ax + a(y + w) .
The readiness preorder ⊑ R is axiomatized by{B 1 -B 4 , (RS), (ND R )}. (2) The failure trace preorder ⊑ F T is axiomatized by {B 1 -B 4 , (RS), (ND FT )}.(3) The ready trace preorder is axiomatized by the set {B 1 -B 4 , (RS), (ND RT )}. Proof. (1) Let us show that the set {B 1 -B 4 , (RS), (ND R )} is logically equivalent to {B 1 -B 4 ,
( 2 )
2Let us show that the sets {B 1 -B 4 , (RS), (ND FT )} and {B 1 -B 4 , (RS), (F T )} are logically equivalent. The implication from left to right follows by taking w = 0.
( 3 )
3Let us show that the set {B 1 -B 4 , (RS), (ND RT )} is logically equivalent to {B 1 -B 4 , (RS), (RT )}. We first note that {B 1 -B 4 , (RS), (RT )} is equivalent to {B 1 -B 4 , (RS), (RT )},
Figure 4 :
4Inclusion relation for the ready simulation preorder and its associated linear semantics.
Corollary 3.4. (1) ⊑ F can be axiomatized by the axioms {B 1 -B 4 , (ND F )}. (2) ⊑ R can be axiomatized by the axioms {B 1 -B 4 , (ND R )}.
16D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ B RS PWRT FT R F CS CT S T (x + y) + z ≃ x + (y + z)
The set {B 1 -B 4 , (RS), (ND)} is logically equivalent to {B 1 -B 4 , (RS), (ND + )}, where (ND + ) is the axiom
( 2 )
2{B 1 -B 4 , (RS), (ND + )} is logically equivalent to {B 1 -B 4 , (RS), (ND ≡ )}.
Proposition 3 . 6 .
36Let Q = {B 1 -B 4 , (RS)} ∪ Q ′ be an axiomatization of an order ⊑ such that ⊑ ⊆ I. Then, the equational variant of Q, Q
18D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
⊑ T is axiomatized by the axioms 3 {B 1 -B 4 , (S), (ND F )}. (2) ⊑ CT is axiomatized by the axioms {B 1 -B 4 , (CS), (ND F )}, where (CS) is the instantiation of (NS) taking C(x, y) as N (x, y).
20D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
Proposition 3 . 12 .
312The set {B 1 -B 4 , (CS), (ND FT )} is logically equivalent to {B 1 -B 4 , (CS), (ND F )}. Hence, under (CS), the failures and the failure trace axioms generate the same semantics.
Figure 6 :
6New view of the linear time-branching time spectrum.
Figure 7 :
7Two branching observations.
Definition 4. 1 .
1The sets L N of local observations corresponding to each of the constrained simulations in the spectrum, and L N (p) of observations associated to a process p, are defined as follows: • Universal (or Plain) simulation: L U = {·}; L U (p) = ·. • Ready simulation: L I = P(Act); L I (p) = I(p). • Complete simulation: L C = Bool; L C (p) is true if I(p) = ∅ and false otherwise. • Trace simulation 4 : L T = P(Act * ); L T (p) = T (p), the set of traces of p. • 2-nested simulation: L S = {[[p]] S | p ∈ BCCSP}; L S (p) = [[p]] S , where [[p]] S represents the equivalence class of p with respect to the simulation equivalence.
Figure 8 :
8Three branching observations.
Theorem 4 . 7 .
47Let L be a function used as a local observation function and let us also denote by L the range of L, as done in Definition 4.1. If there exist semantic functions + L : L × L → L and a L : L → L satisfying L(ap) = a L L(p) and L(p + q) = L(p) + L L(q), then:
Proposition 4 . 8 .
48For N ∈ {U, I, C, T, S}, L N can be defined in a compositional way over the terms in BCCSP.
26D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ Now we show that bgo's characterize N -simulation semantics in all cases.
Theorem 4. 9 .
9For all N ∈ {U, I, C, T, S} and any two processes p and q, p
Corollary 4 . 10 .
410For any constraint N that is a behavior equivalence, whenever we have as local observation function L N the quotient function L N (p) = [[p]] N or any concrete representation of it satisfying L N
Proposition 4 . 13 .
413Let L be a local observation function such that there exist semantic functions + L : L N × L N → L N and a L :
Proposition 4 . 16 .
416For N ∈ {U, C, I, T, S}, if p ⊑ NS q then p ≤ l N q, but the converse is false in general.
Definition 4 . 17 .
417For T , T ′ ⊆ LGO I we define the orders ≤ l⊇ I , ≤ lf I , and ≤ lf ⊇ I by:
28D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
Proposition 4 . 18 .
418The preorder ≤ lf ⊇ I generates the failures preorder and ≤ l⊇ I generates the failure trace preorder.
Proposition 4 . 21 .
421All the operators in Definition 4.20 are indeed closures: if δ ∈ {⊇
impossible futures semantics is defined as:
Definition 4 . 29 .
429The possible futures trace semantics is defined by lgo T 's related by ≤ l T and the impossible futures trace semantics is defined by ≤ l⊇ T . Let us complete this part of the new extended spectrum by introducing the diamond generated by lgo S 's. This produces four new semantics coarser than 2-nested semantics. For instance, for the case of failures we obtain the following definition.
Definition 4 . 30 .
430The extended simulation failures of a process p are defined as
It can be proved that the inclusion SimFailures(p) ⊆ SimFailures(q) holds if and only if ExtSimFailures(p) ⊆ ExtSimFailures(q). Thus, simulation failures are essentially defined by translating the characterization of ordinary failures with the closure of readiness.
Proposition 4.31. ≤ lf ⊇ S = ⊑ SF .Proof. Analogous to the characterization of ⊑ F in terms of ≤ lf ⊇ I .4.3. Deterministic branching observations.Definition 4.32.
Definition 4 .
434. A deterministic process p is a possible world of a process q if p ⊑ RS q.
Figure 9 :
9Deterministic branching observations. When defining the possible worlds of a process we have to solve all the non-deterministic choices in it, each choice leading to one of its possible worlds. The same idea supports the selection of dbgo's to characterize this semantics: the non-deterministic branching observations in BGO N (p) are not present in dBGO N (p), where we have instead all the possible deterministic subtrees of every branching observation.
Definition 4 . 35 .
435The set of complete deterministic branching observations for the local observation function L I is the set cdBGO I ⊆ dBGO I recursively defined as: • ∅, ∅ ∈ cdBGO I . • A, {(a, cdbgo a ) | a ∈ A} ∈ cdBGO I for every a ∈ A and cdbgo a ∈ cdBGO I . For each p ∈ BCCSP we define its set of complete deterministic branching observations cdBGO I (p) = dBGO I (p) ∩ cdBGO I .
Proposition 4 . 37 .
437For every p ∈ BCCSP, cdbgo(p) ∈ cdBGO I (p).
Lemma 4 . 38 .
438For every q ∈ P W (p), cdbgo(q) ∈ cdBGO I (p).
Figure 10 :Figure 11 :
1011Basic layer in the linear time-branching time spectrum. Semantics in the new linear time-branching time spectrum.
Definition 4 . 45 .
445For bgo, bgo ′ ∈ BGO I we define:• bgo ≤ ⊇ I bgo ′ ⇐⇒ bgo = A 1 , S 1 and bgo ′ = A 2 , S 2 and A 1 ⊇ A 2 and S 1 = {(a i , bgo i ) | i ∈ I} and S 2 = {(a i , bgo ′ i ) | i ∈ I} and for all i ∈ I (bgo i ≤ ⊇ I bgo ′ i ) .
Definition 4 . 46 .
446For B, B ′ ⊆ BGO I and δ ∈ {⊇, f, f ⊇}, we define the orders ≤ bδ I by:
Example 4 . 48 .
448For p 1 = a(b + c) and p 2 = ab + ac, p 1 ≤ l⊇ I p 2 but p 1 ≤ l I p 2 . However,p 1 ≤ b⊇ I p 2 since for bgo = {a}, (a, {b, c}, {(b, ∅), (c, ∅)} ) ∈ BGO I (p 1 )there is no bgo ′ ∈ BGO I (p 2 ) with bgo ≤ l⊇ I bgo ′ . By contrast, the branching semantics defined by ≤ bf I and ≤ bf ⊇ I are indeed new.
Figure 12 :
12Three processes.36D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
{ {a} } ∪ { {a}, a, I(y + w) • S | {a}, a, I(y) • S ∈ LGO I (ay) ∨ {a}, a, I(w) • S ∈ LGO I (aw)}.Notice then that the observations of a(y + w) are exactly those of ay + aw simply replacing I(y) or I(w), respectively, by I(y + w) = I(y) ∪ I(w). Analogously,LGO I (a(x + y)) = { {a} } ∪ { {a}, a, I(x + y) • S | {a}, a, I(x) • S ∈ LGO I (ax) ∨ {a}, a, I(y) • S ∈ LGO I (ay)}.Now, in order to get the adequate condition M Z (x, y, w) for each of the semantics, let us examine the formulas that define the preorders ≤ lY I :• ≤ l I . To have LGO I (a(x + y)) ⊆ LGO I (ax) ∪ LGO I (a(y + w)) it is enough to require { {a}, a, I(x) ∪ I(y) • S | I(x) • S ∈ LGO I (x)} ⊆ { {a}, a, I(x) • S | I(x) • S ∈ LGO I (x)}and { {a}, a, I(x)∪I(y) •S | I(y) •S ∈ LGO I (y)} ⊆ { {a}, a, I(y)∪I(w) •S | I(y) • S ∈ LGO I (y)}. Thus, a first proposal for M RT would be I(y) ⊆ I(x) ∧ I(x) = I(y) ∪ I(w).
• ≤ lf I . We consider the inclusion LGO I (a(x + y)) f ⊆ LGO I (ax + a(y + w)) f . Weonly have to consider the lgo {a}, a, I(x) ∪ I(y) in LGO I (a(x + y)) f and show that it also belongs to LGO I (ax + a(y + w)) f , since all lgo's of length greater than 1 start with the prefix {a}, a . For that, either I(x) ∪ I(y) = I(x) or I(x) ∪ I(y) = I(y) ∪ I(w), that is, 38D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ I(y) ⊆ I(x) or I(x) ∪ I(y) = I(y) ∪ I(w)
i a and Z ∈ {F, R, FT, RT}, its Z-head normal form hnf Z (p) is:
Proposition 5 . 2 .
52For Z ∈ {RT, F T, R, F }, {B 1 -B 4 , (RS), (ND Z )} ⊢ hnf Z (p) p.
Proposition 5 . 3 .
53Let Z ∈ {F, F T, R, RT }, and let p = a∈X 0 i∈Ia ap i a , and q = a∈X 0 j∈Ja aq j a . If p ≤ l(Z) I q then there exists a summand ah k a of hnf Z (q) such that p i a ≤ l(Z) I h k a . Proof. Using Definition 4.19 and Proposition 4.22, we need to show that there exists LGO I (p i a ) l(Z) ⊆ LGO I (h k a ) l(Z) but, due to the fact that ( ) l(Z) is a closure operator (Proposition 4.21), it is enough to prove that LGO I (p i a ) ⊆ LGO I (h k a ) l(Z) . For I(p i a ) ∈LGO I (p is some q j a such that I(p i a ) ∈ LGO I (q j a ) l(Z) ; we then consider hnf Z (q, a, j, I(p i a )) = ah k a . If t ∈ LGO I (p i a ) then I(p), a • t ∈ LGO I (p) ⊆ LGO I (q) l(Z) and there exists j t such that t ∈ LGO I (q jt a ) l(Z)
Theorem 5. 4 (
4Soundness and completeness). For Z ∈ {RT, F T, R, F }: {B 1 -B 4 , (RS), (ND Z )} ⊢ p q. Proof. (Soundness) The axiomatizations are sound because of the way they have been derived. (Completeness) By structural induction on p. • Let p be 0. As usual, we can consider terms up to bisimulation since B 1 -B 4 are equations needed for all the semantics. If p ≤ l(Z) I q, then q must be 0 (or bisimilar to 0) because the set of local observations of 0 is empty and cannot contain any observations (see Definition 4.19.) • If p = a∈X 0 i∈Ia ap i a then, by Proposition 5.3, p ≤ l(Z) I q implies that there exists a summand ah k a of hnf Z (q) such that p . By induction hypothesis, {B 1 -B 4 , (RS), (ND Z )} ⊢ p i a h k a and therefore {B 1 -B 4 , (RS), (ND Z )} ⊢ ap i a ah k a ; adding all these inequalities and using (RS), which is allowed because I(p) = I(q), it follows that {B 1 -B 4 , (RS), (ND Z )} ⊢ p hnf Z (q) and, by Proposition 5.2, {B 1 -B 4 , (RS), (ND Z )} ⊢ p q.
40D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ constraint M T
.
Z + ) and (T-ND ≡ ) we can apply the same arguments used in Section 5.1 to show that (ND Z ) was satisfied by ≤ For Z ∈ {RT, F T, R, F }, the preorder ≤ l(Z) T satisfies the axiom (T-ND Z + ) and also (T-ND Z ≡ ).
Proposition 5. 9 .
9The axiomatization {B 1 -B 4 , (T S), (T-ND Z )} is equivalent to the axiomatization {B 1 -B 4 , (T S), (T-ND Z ≡ )} for Z ∈ {RT, F T }. Proof. Let us first show that {B 1 -B 4 , (T S), (T-ND Z )} is equivalent to {B 1 -B 4 , (T S), (T-ND Z + )}. This holds because (T-ND Z ) implies (T-ND Z +
, although our proofs of completeness for the axiomatizations {B 1 -B 4 , (RS), (ND Z )} considered the inequational axioms (ND Z ), they are also valid for the axiomatizations {B 1 -B 4 , (RS), (ND Z ≡ )}. The steps in the procedure that leads to the completeness of {B 1 -B 4 , (RS), (ND Z )} can be adapted by substituting each reference to the observer I by T , thus obtaining a proof of the completeness of {B 1 -B 4 , (RS), (T-ND Z ≡ )} for ≤ l(Z)
Theorem 5. 11 .
11For Z ∈ {RT, F T, R, F }, {B 1 -B 4 , (T S), (T-ND Z ≡ )} ⊢ p q if and only if p ≤
where the desugaring function removes the syntactic sugar used in L F T .(4) L R L ′ R . (5) L ′ F ⊇ desugared(L F ),where the desugaring function removes the syntactic sugar used in L F . Proof. Recall the definition of L Z in
( 1 )
1The logical semantics ⊑ ′ RS induced by the logic L ′ RS is equivalent to the observational branching semantics defined by ≤ b I , generated by the set of branching general observations BGO I .(2) For Z ∈ {F, F T, R, RT }, the logical semantics ⊑ ′ Z induced by the logic L ′ Z is equivalent to the observational linear semantics ≤ l(Z) I in Definitions 4.14 and 4.19.
50D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
Proposition 6 .
611. (1) L ′ S ∼ L S ; (2) L ′ CS ∼ L CS ; and (3) L ′ 2S ∼ L 2S . Proof. (1) The clauses defining L ′ S and L S produce the same set of formulas. The first two clauses in L ′ S only add the two trivial formulas ⊤ and ¬⊤ because in L ′ U = {⊤}. (2) Again, the sets of formulas produced by L ′ CS and L CS are the same because the two first clauses of L ′ CS can only generate ⊤, ¬⊤, 0 and ¬0 from L ′ C = {⊤, ¬⊤}. 0 is needed to reflect the second clause in the definition of L CS , while ¬0 ≡ a∈Act a⊤ so that any formula containing ¬0 can be rewritten into a disjunction of formulas in L CS .
Definition 6 . 15 .•
615Inspired by the orders ≤ l N , ≤ l⊇ N , ≤ lf N , and ≤ lf ⊇ N , we define the set of formulas L a ∈ Act then aϕ ∈ L if ϕ ∈ L ′ ≤ l⊇ N and σ ∈ L ¬ N then σ ∧ ϕ ∈ L
for the coarsest semantics (i.e. those corresponding to plain refusals and plain readiness when N = I) we only check for N at the "end" of the formula because there are no conjunctions in the corresponding languages L corresponding closures L ≡ N and L ¬ N . The other two logics do introduce additional conjunctions that allow to observe N along the computations.
Proposition 6 .
617. (1) L ′ F and L ′ ≤ lf ⊆ I are incomparable: p ≤ lf ⊇ I q does not imply p ≤ lf ⊆ I q and p ≤ lf ⊆ I q does not imply p ≤ lf ⊇
( 1 )
1Trivial, since the sets of clauses defining L ′ ≤ lf U and L T are almost the same. Note that the clause "if σ ∈ L ≡ U then σ ∈ L ′ ≤ lf U " does not give rise to new formulas because L ≡ U = {⊤}. (2) Note that the sets of clauses defining L ′ ≤ lf ⊇ C and L CT are the same but for the clause"if σ ∈ L ¬ C then σ ∈ L ′ ≤ lf ⊇ C ". On the one hand, this causes ¬⊤ ∈ L ′ ≤ lf ⊇ C (which adds nothing) because ⊤ ∈ L ′ C and thus ¬⊤ ∈ L ¬ C . On the other hand, we also have 0
which does not appear explicitly in that of L P F because it corresponds to the conjunction of an empty set of formulas.
54D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
Definition 6 . 22 .
622For each N ∈ {U, C, I, T, S}, we define the formulas of
Definition 7 . 1 (
71Normal formulas N (L)).
56D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
Figure 14 :
14A branching observation.Definition 7.6. For each N ∈ {U, C, I, T, S} and eachY N ∈ {N S, ≤ l N , ≤ l⊇ N , ≤ lf N , ≤ lf ⊇ N , ≤ l⊆ N , ≤ lf ⊆ N , D N } in the spectrum, we define the finite logic for the semantics L f Y N as L ′ Y N ∩L f HM .Corollary 7.7. For each N ∈ {U, C, I, T, S} and each
Theorem 7. 8 .
8For each N ∈ {U, C, I, T, S} and each Y N ∈ {N S, ≤ l N , ≤ l⊇ N , ≤ lf N , ≤ lf ⊇ N , ≤ l⊆ N , ≤ lf ⊆ N , D N } in the spectrum there exists a correspondence between the set of complete normal formulas CN Y N (L ′′ N ) and the corresponding domain of observations ΩGO N with Ω ∈ {B, L}.This correspondence ↔ satisfies that ϕ ↔ θ implies that (p |= ϕ iff θ ∈ ΩGO N (p)). Moreover:(1) The set of complete normal formulas CN N S (L ′′ N ) (resp. CN D N (L ′′ N )) and the domain of branching general observations BGO N (resp. dBGO N ) are isomorphic, that is, ↔ is one to one.(2) The set of complete normal formulas CN ≤ l N (L ′′ N ), CN ≤ l⊇ N (L ′′ N ) and the domain of linear general observations LGO N are isomorphic, that is, ↔ is one to one. (3) The set of complete normal formulas CN ≤ lf N (L ′′ N ) (resp. CN ≤ lf ⊇ N (L ′′ N )) and the quotient domain
58D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
above, and then define the correspondence ↔ by ignoring the values of all the intermediate local observations in the considered lgo, keeping only the local observation at the end. For CN ≤ lf ⊇ N (L N ) we just need to apply the same procedure above combined with the ideas along the proof for CN ≤ l⊇ N (L N ).
Remark 7 . 9 .
79It came as a surprise to notice that the lgo ′ s in LGO N are in a bijective relation both with the complete normal formulas in N ≤ l N (L N ) and those in N ≤ l⊇ N (L N ), so let us consider the case N = I to explain this fact. A cnf in N ≤ l I (L I ) specifies the corresponding local observation I(p) ⊆ P(Act) by means of a formula ( σ∈Γ σ ∧ σ / ∈Γ ¬σ), where the formulas in Γ are just the elements of the corresponding set I(p) while those in Γ correspond to its complement. When considering the failure trace semantics, the formulas in N ≤ l⊇ I (L I ) only contain the part σ /
Theorem 7 . 10 .
710For each N ∈ {U, C, I, T, S} and each Y N ∈ {N S, ≤l N , ≤ l⊇ N , ≤ lf N , ≤ lf ⊇ N , ≤ l⊆ N , ≤ lf ⊆ N , D N } in the spectrum, if we restrict ourselves to image-finite processes, the logical semantics ⊑ f Y N induced by the logic L f Y N ,is equivalent to the corresponding observational semantics in Definitions 4.2, 4.11 and 4.32. In order to unify our notation, here we will denote by GO N the corresponding semantic domain.Proof. By Theorem 7.3, L ′ Y N ∼ N Y N (L N ), and from Theorem 7.8 we get the isomorphism between the set CN Y N (L N ) and the corresponding set of general observations GO N .
The unified logical semantics in Definition 6.10 is equivalent to the N -simulation semantics.(2) The unified logical semantics in Definition 6.15.1 is equivalent to the N -ready trace semantics.(3) The unified logical semantics in Definition 6.15.2 is equivalent to the N -failure trace semantics. (4) The unified logical semantics in Definition 6.15.3 is equivalent to the N -readiness semantics. (5) The unified logical semantics in Definition 6.15.4 is equivalent to the N -failure semantics. (6) The unified logical semantics in Definition 6.22 is equivalent to the N -deterministic branched semantics. Moreover, if we restrict ourselves to image-finite processes we have also an equivalence with the corresponding finite logical semantics.
Definition 8 . 1 .
81The readiness and failure trace semantics, or join semantics R ∧ F T , is that defined by the order ⊑ R∧F T generated by the set of axioms {B 1 -B 4 , (RS), (ND R∧F T )}.Proposition 8.2. The ready trace semantics is strictly finer than the readiness and failure trace semantics.
Definition 8 . 7 .
87The meet semantics R ∨ F T is that defined by the order ⊑ R∨F T generated by the set of axioms{B 1 -B 4 , (RS), (ND R ), (ND F T )}. If we define M R∨F T as M R ∨ M F T , that is, M R∨F T (x, y, w) holds if I(x) ⊇ I(y) or I(w) ⊆ I(y), we have the following characterization of ⊑ R∨F T . Proposition 8.8. The order ⊑ R∨F T is that generated by the set of axioms {B 1 -B 4 , (RS), (ND R∨F T )}, where (ND R∨F T ) is the instantiation of the generic axiom (ND) with the condition M R∨F T .
{B 1 -B 4 , (RS), (ND R )}, any one coarser than failure trace must satisfy {B 1 -B 4 , (RS), (ND F T )}, and M R∨F T is equivalent to M R ∨ M F T .
{B 1 -B 4 , (RS), (ND R ), (ND F T )} 62D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
Theorem 8. 14 .
14The revivals semantics defined by ⊑ l⊇∨f I is axiomatized by {B 1 -B 4 , (RS), (ND R∨F T )}.
.
Then, to prove that it defines R ∨ F T it is enough to check that p ⊑ l⊇∨f I q implies that there exists ϕ in L
Figure 15 :
15The real diamond below ready simulation.
Theorem 9.2 ([23]). For every behavior preorder ⊑ satisfying the axiom (RS) and ⊑ ⊆ I, we have p ⊑ q if and only if p ❁ ∼ I ⊑ q.
Proposition 9 . 4 .
94For Z ∈ {F, R, F T, RT } we have p ⊑ Z q if and only if p ❁ ∼
Figure 17 :
17Operational semantics characterizing the linear semantics.
Corollary 9. 8 .
8For Z ∈ {F, R, F T, RT } and for any BCCSP process p, we have I → (p) = I ⇒ Z (p).
Theorem 9 . 10 .
910For Z ∈ {F, R, F T, RT } and any two BCCSP processes p and q, we havep ⊑ Z q ⇐⇒ p ⊑ ⇒ Z RS q.68D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
70D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
Table 1 :
1Axiomatization for the preorders in the linear time-branching time spectrum.
Table 2 :
2Axiomatization for the equivalences in the linear time-branching time spectrum.
Table 5 :
5Our new logical characterizations for the semantics at each level of the spectrum.
and a ∈ Act then aϕ ∈ L ′ Proof. We just need to check that L ′≤ l⊇∧f
I
.
Proposition 8.6. The logical semantics ⊑ ′
≤ l⊇∧f
I
induced by the logic L ′
≤ l⊇∧f
I
is equivalent
to the observational semantics defined by ≤ l⊇∧f
I
.
≤ l⊇∧f
I
= L ′
≤ l⊇
I
∪ L ′
≤ lf
I
Proposition 8.12. For all p, q ∈ BCCSP, Revivals(p) ⊆ Revivals(q) if and only if LGO I (p) ≤ l⊇∨f I
A remarkable exception, however, is the bisimulation notion, for which no non-trivial order relation is known.4 D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
22D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
Trace simulations are the only ones in this list that do not appear in[58]. They can be defined as T -simulations, with T (x, y) ::= T (x) = T (y), and the general theory about constrained simulations in[24] applies to them. In particular, they can be axiomatized as stated in Proposition 3.1(3), page 13.24D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
For instance in[24] (Theorem 10) we provided an axiomatization for any behavior preorder starting from the equations of the corresponding equivalence.66D. DE FRUTOS ESCRIG, C. GREGORIO RODRÍGUEZ, M. PALOMINO, AND D. ROMERO HERNÁNDEZ
· a x x q q q q q q q q q q q q q q a a 8 8· · · a y y r r r r r r r r r r r r r
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This work is licensed under the Creative Commons Attribution-NoDerivs License. Marc Voorhoeve, Sjouke Mauw, Information Processing Letters. 801Suite. or Eisenacher Strasse 2, 10777Marc Voorhoeve and Sjouke Mauw. Impossible futures and determinism. Information Processing Letters, 80(1):51-58, 2001. This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/2.0/ or send a letter to Creative Commons, 171 Second St, Suite 300, San Francisco, CA 94105, USA, or Eisenacher Strasse 2, 10777 Berlin, Germany
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"Heralded Entanglement of Arbitrary Degree in Remote Qubits",
"Heralded Entanglement of Arbitrary Degree in Remote Qubits"
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"U Schilling \nInstitut für Optik\nInformation und Photonik\nMax-Planck Forschungsgruppe\nUniversität Erlangen-Nürnberg\n91058ErlangenGermany\n",
"C Thiel \nInstitut für Optik\nInformation und Photonik\nMax-Planck Forschungsgruppe\nUniversität Erlangen-Nürnberg\n91058ErlangenGermany\n",
"E Solano \nDepartamento de Química Física\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 64448080BilbaoSpain\n",
"T Bastin \nInstitut de Physique Nucléaire\nAtomique et de Spectroscopie\nUniversité de Liège\n4000LiègeBelgium\n",
"J Von Zanthier \nInstitut für Optik\nInformation und Photonik\nMax-Planck Forschungsgruppe\nUniversität Erlangen-Nürnberg\n91058ErlangenGermany\n"
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[
"Institut für Optik\nInformation und Photonik\nMax-Planck Forschungsgruppe\nUniversität Erlangen-Nürnberg\n91058ErlangenGermany",
"Institut für Optik\nInformation und Photonik\nMax-Planck Forschungsgruppe\nUniversität Erlangen-Nürnberg\n91058ErlangenGermany",
"Departamento de Química Física\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 64448080BilbaoSpain",
"Institut de Physique Nucléaire\nAtomique et de Spectroscopie\nUniversité de Liège\n4000LiègeBelgium",
"Institut für Optik\nInformation und Photonik\nMax-Planck Forschungsgruppe\nUniversität Erlangen-Nürnberg\n91058ErlangenGermany"
] |
[] |
Incoherent scattering of photons off two remote atoms with a Λ-level structure is used as a basic Young-type interferometer to herald long-lived entanglement of an arbitrary degree. The degree of entanglement, as measured by the concurrence, is found to be tunable by two easily accessible experimental parameters. An estimate of the variation of the degree of entanglement due to uncertainties in an experimental realization is given.PACS numbers: 03.67. Bg, 42.50.Ar, 42.50.Ct Quantum interference [1] and entanglement [2] are two of the most stunning consequences of quantum mechanics. Although these phenomena are usually studied separately, both have quantum parallelism as a common origin: Quantum interference deal with the coherent superposition of multiple quantum paths, typically for a single system, while entanglement is inherent to the nonseparable character of linear superpositions in the multipartite case. This common origin leads to the possibility of constructing tight links between both phenomena. For example, Jakob and Bergou [3, 4] derived a relation between the entanglement of two qubits and the visibility of the interference pattern generated by one of the qubits in a Ramsey-type interferometer. Another way to link these two properties is to couple interfering quantum paths to remote physical systems. With this ansatz, Scholak et al.[5]showed that the interference pattern of a single photon probing two spatially separated atomic systems can witness their mutual entanglement.With a similar approach, by detecting the interference pattern of scattered photons, it is also possible to create entanglement among the particles[6,7,8,9,10,11,12,13,14]. Hereby, the atoms may be separated by arbitrary distances, as there is no need for a particle interaction. This should be contrasted with other schemes entangling massive particles, which require some kind of interaction, be it Coulomb-like[15,16,17,18]or mediated by photons[19,20,21,22,23]. Furthermore, at variance with photon entanglement, usually achieved by parametric down conversion[24,25], the entanglement of electronic ground states of atoms can be preserved over long periods of time[17,18]. This may prove to be useful for diverse applications in quantum communication and quantum computation [2], where long-lived entanglement plays a crucial role.In this paper, we present a proposal involving a simple scheme to operationally tune the amount of long-lived entanglement present in two remote atomic qubits. We will demonstrate that our scheme allows to create heralded entangled states, where the degree of entanglement between the atomic qubits can be tuned at will. The exact value is determined by adjusting two easily accessi-ble experimental parameters, namely the position of two photodetectors and the relative orientation of two polarizers.The proposed setup is based on a Young-type interferometer realized by two localized atoms [26] with an internal Λ-level structure (seeFig. 1). The atoms, representing the double-slit of the interferometer, are excited by a laser pulse and subsequently scatter photons in their deexcitation process. Without loss of generality, we will assume that the upper state (denoted |e ) decays to the lower states (denoted |± ) by emitting a σ ∓ polarized photon, respectively. These photons are registered in the far field with photon detectors, which are additionally equipped with polarization filters in front of them. The far-field detection is a simple method to erase the which-way information of the photons propagating from the atoms to the detector. The atoms are projected by the measurement of the two photons into a given state, depending on the position of the detectors and the detected polarizations[13,14]. For an arbitrary two-qubit pure state |ψ = a| + + + b| + − + c| − + + d| − − the concurrence reads(1)where |ψ = (σ y,A ⊗ σ y,B )|ψ , with σ y,X the usual σ y Pauli matrix of the qubit formed by the two lower states of atom X (X = A, B) [27]. Omitting proportionality factors, the detection of a photon scattered off two Λ-level atoms A and B, with a detector D i at position r i behind a polarization filter aligned along ε i , is described by the projection operator [28]where the sum runs over the two ground states |± . Here, d ±e is the dipole moment of the transition |e → |± and the phase difference δ( r i ) is given by δ( r i ) = k( R B − R A ) · e( r i ),
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10.1103/physreva.80.022312
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0901.2592
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003c8ffc1193db8deec8251ef4ab3168b694f2a6
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Heralded Entanglement of Arbitrary Degree in Remote Qubits
24 Nov 2009
U Schilling
Institut für Optik
Information und Photonik
Max-Planck Forschungsgruppe
Universität Erlangen-Nürnberg
91058ErlangenGermany
C Thiel
Institut für Optik
Information und Photonik
Max-Planck Forschungsgruppe
Universität Erlangen-Nürnberg
91058ErlangenGermany
E Solano
Departamento de Química Física
Universidad del País Vasco -Euskal Herriko Unibertsitatea
Apdo. 64448080BilbaoSpain
T Bastin
Institut de Physique Nucléaire
Atomique et de Spectroscopie
Université de Liège
4000LiègeBelgium
J Von Zanthier
Institut für Optik
Information und Photonik
Max-Planck Forschungsgruppe
Universität Erlangen-Nürnberg
91058ErlangenGermany
Heralded Entanglement of Arbitrary Degree in Remote Qubits
24 Nov 2009(Dated: November 24, 2009)
Incoherent scattering of photons off two remote atoms with a Λ-level structure is used as a basic Young-type interferometer to herald long-lived entanglement of an arbitrary degree. The degree of entanglement, as measured by the concurrence, is found to be tunable by two easily accessible experimental parameters. An estimate of the variation of the degree of entanglement due to uncertainties in an experimental realization is given.PACS numbers: 03.67. Bg, 42.50.Ar, 42.50.Ct Quantum interference [1] and entanglement [2] are two of the most stunning consequences of quantum mechanics. Although these phenomena are usually studied separately, both have quantum parallelism as a common origin: Quantum interference deal with the coherent superposition of multiple quantum paths, typically for a single system, while entanglement is inherent to the nonseparable character of linear superpositions in the multipartite case. This common origin leads to the possibility of constructing tight links between both phenomena. For example, Jakob and Bergou [3, 4] derived a relation between the entanglement of two qubits and the visibility of the interference pattern generated by one of the qubits in a Ramsey-type interferometer. Another way to link these two properties is to couple interfering quantum paths to remote physical systems. With this ansatz, Scholak et al.[5]showed that the interference pattern of a single photon probing two spatially separated atomic systems can witness their mutual entanglement.With a similar approach, by detecting the interference pattern of scattered photons, it is also possible to create entanglement among the particles[6,7,8,9,10,11,12,13,14]. Hereby, the atoms may be separated by arbitrary distances, as there is no need for a particle interaction. This should be contrasted with other schemes entangling massive particles, which require some kind of interaction, be it Coulomb-like[15,16,17,18]or mediated by photons[19,20,21,22,23]. Furthermore, at variance with photon entanglement, usually achieved by parametric down conversion[24,25], the entanglement of electronic ground states of atoms can be preserved over long periods of time[17,18]. This may prove to be useful for diverse applications in quantum communication and quantum computation [2], where long-lived entanglement plays a crucial role.In this paper, we present a proposal involving a simple scheme to operationally tune the amount of long-lived entanglement present in two remote atomic qubits. We will demonstrate that our scheme allows to create heralded entangled states, where the degree of entanglement between the atomic qubits can be tuned at will. The exact value is determined by adjusting two easily accessi-ble experimental parameters, namely the position of two photodetectors and the relative orientation of two polarizers.The proposed setup is based on a Young-type interferometer realized by two localized atoms [26] with an internal Λ-level structure (seeFig. 1). The atoms, representing the double-slit of the interferometer, are excited by a laser pulse and subsequently scatter photons in their deexcitation process. Without loss of generality, we will assume that the upper state (denoted |e ) decays to the lower states (denoted |± ) by emitting a σ ∓ polarized photon, respectively. These photons are registered in the far field with photon detectors, which are additionally equipped with polarization filters in front of them. The far-field detection is a simple method to erase the which-way information of the photons propagating from the atoms to the detector. The atoms are projected by the measurement of the two photons into a given state, depending on the position of the detectors and the detected polarizations[13,14]. For an arbitrary two-qubit pure state |ψ = a| + + + b| + − + c| − + + d| − − the concurrence reads(1)where |ψ = (σ y,A ⊗ σ y,B )|ψ , with σ y,X the usual σ y Pauli matrix of the qubit formed by the two lower states of atom X (X = A, B) [27]. Omitting proportionality factors, the detection of a photon scattered off two Λ-level atoms A and B, with a detector D i at position r i behind a polarization filter aligned along ε i , is described by the projection operator [28]where the sum runs over the two ground states |± . Here, d ±e is the dipole moment of the transition |e → |± and the phase difference δ( r i ) is given by δ( r i ) = k( R B − R A ) · e( r i ),
Incoherent scattering of photons off two remote atoms with a Λ-level structure is used as a basic Young-type interferometer to herald long-lived entanglement of an arbitrary degree. The degree of entanglement, as measured by the concurrence, is found to be tunable by two easily accessible experimental parameters. An estimate of the variation of the degree of entanglement due to uncertainties in an experimental realization is given. Quantum interference [1] and entanglement [2] are two of the most stunning consequences of quantum mechanics. Although these phenomena are usually studied separately, both have quantum parallelism as a common origin: Quantum interference deal with the coherent superposition of multiple quantum paths, typically for a single system, while entanglement is inherent to the nonseparable character of linear superpositions in the multipartite case. This common origin leads to the possibility of constructing tight links between both phenomena. For example, Jakob and Bergou [3,4] derived a relation between the entanglement of two qubits and the visibility of the interference pattern generated by one of the qubits in a Ramsey-type interferometer. Another way to link these two properties is to couple interfering quantum paths to remote physical systems. With this ansatz, Scholak et al. [5] showed that the interference pattern of a single photon probing two spatially separated atomic systems can witness their mutual entanglement.
With a similar approach, by detecting the interference pattern of scattered photons, it is also possible to create entanglement among the particles [6,7,8,9,10,11,12,13,14]. Hereby, the atoms may be separated by arbitrary distances, as there is no need for a particle interaction. This should be contrasted with other schemes entangling massive particles, which require some kind of interaction, be it Coulomb-like [15,16,17,18] or mediated by photons [19,20,21,22,23]. Furthermore, at variance with photon entanglement, usually achieved by parametric down conversion [24,25], the entanglement of electronic ground states of atoms can be preserved over long periods of time [17,18]. This may prove to be useful for diverse applications in quantum communication and quantum computation [2], where long-lived entanglement plays a crucial role.
In this paper, we present a proposal involving a simple scheme to operationally tune the amount of long-lived entanglement present in two remote atomic qubits. We will demonstrate that our scheme allows to create heralded entangled states, where the degree of entanglement between the atomic qubits can be tuned at will. The exact value is determined by adjusting two easily accessi-ble experimental parameters, namely the position of two photodetectors and the relative orientation of two polarizers.
The proposed setup is based on a Young-type interferometer realized by two localized atoms [26] with an internal Λ-level structure (see Fig. 1). The atoms, representing the double-slit of the interferometer, are excited by a laser pulse and subsequently scatter photons in their deexcitation process. Without loss of generality, we will assume that the upper state (denoted |e ) decays to the lower states (denoted |± ) by emitting a σ ∓ polarized photon, respectively. These photons are registered in the far field with photon detectors, which are additionally equipped with polarization filters in front of them. The far-field detection is a simple method to erase the which-way information of the photons propagating from the atoms to the detector. The atoms are projected by the measurement of the two photons into a given state, depending on the position of the detectors and the detected polarizations [13,14]. For an arbitrary two-qubit pure state |ψ = a| + + + b| + − + c| − + + d| − − the concurrence reads
C = | ψ |ψ | = 2|ad − bc|.(1)
where |ψ = (σ y,A ⊗ σ y,B )|ψ , with σ y,X the usual σ y Pauli matrix of the qubit formed by the two lower states of atom X (X = A, B) [27].
Omitting proportionality factors, the detection of a photon scattered off two Λ-level atoms A and B, with a detector D i at position r i behind a polarization filter aligned along ε i , is described by the projection operator [28]
D i =D i ( r i ) = m=± ( ε i · d me ) |m A e| + e −iδ( ri) |m B e| ,(2)
where the sum runs over the two ground states |± . Here, d ±e is the dipole moment of the transition |e → |± and the phase difference δ( r i ) is given by where k is the wavenumber of the detected photon, R A,B is the position of the respective atom, and e( r i ) is the unit vector pointing from the atoms towards the detector D i at r i . Hereby, the far-field detection ensures that e( r i ) is identical for both atoms.
δ( r i ) = k( R B − R A ) · e( r i ),(3)
The far-field detection scheme provides for the loss of which-way information of the scattered photons. The same can be accomplished by using optical fibers guiding the photons from the atoms to the detectors [12,29,30]. In this case, the phase difference δ( r i ) is given by
δ( r i ) = k (w B ( r i ) − w A ( r i )) ,(4)
where w A,B ( r i ) is the optical path length from the respective atom to the detector D i at position r i via the corresponding optical fiber (cf. Fig. 1). In this configuration the atoms can be separated by arbitrary distances, i.e., they are truly remote. By applying the operatorD 1 andD 2 to the initial double excited state of the two atoms, |ψ (i) Λ = |ee , we find the normalized atomic state after the detection of two photons to be
|ψ (f ) Λ =D 1D2 |ψ (i) Λ ψ (i) Λ |D † 2D † 1D 1D2 |ψ (i) Λ = ζ 1 + e −iδ21 (ε 2− ε 1− | + + + ε 2+ ε 1+ | − − ) + ε 2+ ε 1− + e −iδ21 ε 2− ε 1+ | − + + e −iδ21 ε 2+ ε 1− + ε 2− ε 1+ | + − . (5)
Here, the abbreviation ε i± = ε i · d ∓e is used, where without loss of generality we assume |ε i+ | 2 + |ε i− | 2 = 1, δ 21 is given by the phase difference
δ 21 = δ( r 2 ) − δ( r 1 ),(6)
depending on the two detector positions r 1 and r 2 , and ζ is a normalization factor.
Using Eq. (1), the concurrence of the pure state Eq. (5) can be explicitly calculated. One obtains
C(δ 21 , V 12 ) = |ε 2+ ε 1− − ε 2− ε 1+ | 2 1 + | ε 2 · ε * 1 | 2 cos δ 21 = 1 − V 12 1 + V 12 cos δ 21 ,(7)
where the parameter V 12 is given by
V 12 = | ε 2 · ε * 1 | 2 .(8)
According to Eq. (7), the long-lived entanglement generated between the ground states of the two Λ-level atoms only depends on the relative phase δ 21 and on the relative orientation of the two polarization filters V 12 (see Fig. 2) [31]. In order to obtain a certain amount of entanglement between the two atoms, both parameters have to be tuned to suitable values and the excitation of the atoms has to be repeated until both detectors register a photon. By postselection we then know that the atomic pair contains exactly the desired amount of entanglement as described by Eq. (7). Taking a look at the extremal values of C with respect to δ 21 , we obtain
C min = 1 − V 12 1 + V 12 if cos δ 21 = 1 , C max = 1 if cos δ 21 = −1 .(9)
These expressions show that, depending on the value of V 12 , any amount of concurrence between 1−V12 1+V12 and 1 can be achieved. In particular, by choosing δ 21 to be an odd multiple of π, it is always possible to generate a state with maximal (unit) concurrence, independent of the explicit value of V 12 , i.e., independent of the relative orientation of the two polarization filters (see Fig. 2).
The extrema of the concurrence with respect to V 12 are given by C = 1 for all V 12 , if cos δ 21 = −1 C min = 0 for V 12 = 1, C max = 1 for V 12 = 0, if cos δ 21 = −1. (10) Thus, if the phase difference is not fixed to an odd multiple of π, it is always possible to use V 12 as a single parameter to tune the concurrence to any desired value. In particular, by choosing δ 21 an odd multiple of π/2, we find in Eq. (7) a linear relation between the concurrence and the parameter V 12 . In this case, when linear polarizers are used, by keeping one of them fixed and turning the other by a relative angle α, we are able to implement a fully tunable concurrence (0 < C < 1)
C = 1 − V 12 = sin 2 α,(11)
yielding an analog to the Malus' Law [32]. In its classical version, it says that the intensity of the same light beam passing consecutively through two linear polarizers is proportional to the square of the cosine of the relative angle between the polarizers. Here, we find that the concurrence, a measure characterizing the entanglement of two qubits, behaves in a similar way. Even though each of the two indistinguishable photons passes a different polarizer, the degree of entanglement generated between the atoms upon detection of the photons is determined by the relative angle between the two polarizers. This result can be seen as an operational implementation of a tunable measure of entanglement between matter qubits following a simple and intuitive law of classical optics. Note that the parameter V 12 intervenes also in the second order correlation function G (2) (δ 21 ), which is proportional to the measured signal. The second order correlation function reads [28]
G (2) (δ 21 ) = ψ (i) Λ |D † 2D † 1D 1D2 |ψ (i) Λ = 2 (1 + V 12 cos δ 21 ) ,(12)
In this expression V 12 appears as the visibility of the G (2) (δ 21 )-function, revealing again the close relationship between quantum interference and entanglement. In the following, we will give an estimate of the variation of the concurrence due to experimental uncertainties (see also [13,14]). The probability to detect a scattered photon is proportional to the solid angle subtended by the detector divided by 4π. By extending the detection area, the detection probability will increase, though the accumulated phase will be less well defined. Thus, there is a trade-off between the count rate of the scattered photons and the error in the concurrence generated in the final state. For estimating errors, we will assume identical rectangular detectors. Let α D be the azimuthal angular extension of each detector in direction of θ i , with θ i the azimuth angle between e( r i ) and the axis connecting the two atoms, and ϕ D the polar angle subtended by each detector perpendicular to the plane of α D . Then, for small α D , the probability to detect a randomly emitted photon with one of the two detectors can be approximated to
P (α D , ϕ D ) = α D ϕ D 4π .(13)
The count rate R of two-photon detection events is thus given by
R = 2r · P (α D , ϕ D ) 2 · G (2) (δ 21 ),(14)
where r is the repetition rate of the experiment. A factor of 2 appears since either detector, D 1 or D 2 , might register the first photon. The count rate is thus maximal if the condition for constructive interference of the second order correlation function G (2) (δ 21 ) is fulfilled, i.e., if δ 21 is an even multiple of π.
The uncertainty in the concurrence ∆C is defined by the difference in the concurrence of the density matrix of the state ρ (g) actually generated and the pure target state ρ (f ) = |ψ
(f ) Λ ψ (f ) Λ |: ∆C = |C(ρ (g) ) − C(ρ (f ) )|.(15)
∆C is essentially determined by the uncertainty in the orientation of the polarization filters ∆V and the uncertainty in the phase ∆δ 21 . With current experimental technology, ∆V can be suppressed to the order of 10 −10 [33]. Thus, ∆V is negligible compared to the uncertainty imposed by the phase and will be neglected in the following. The uncertainty in the phase ∆δ 21 is governed by two contributions: the solid angle subtended by the detector (determined by α D and ϕ D ) and the finite confinement µ of the atoms in the trap. To calculate ρ (g) , we have to integrate over the whole relevant parameter space: (16) where w(δ 21 ) is a weight factor determined by the geometry of the setup and normalization. To minimize the deviation from the desired final state, we have to minimize ϕ D and α D , while θ i should be close to π 2 . However, ϕ D and α D are bound from below by the requirement of an acceptable count rate R. In addition, there is a lower boundary to ∆C due to the finite confinement of the atoms.
A,µ w(δ 21 ) |ψ (f ) Λ ψ (f ) Λ |(δ 21 ) dA dµ
For realistic experimental parameters, d = 5 µm, µ = 10 nm, α D = 5 mrad, ϕ D = π 6 , θ ≈ π 2 , and photons of wavelength λ = 650 nm, this results in ∆C max < 0.025 for all δ 21 ∈ [− π 2 , π 2 ] and all V. Within this parameter range, the fidelity of the final state ρ (g) always remains above 95 %, while for a repetition rate r of a few Mhz the count rate amounts to a few events per second. These estimates include a detector efficiency of about 30 % and a dark count rate of up to a few 100 Hz.
The protocol presented here is capable of producing heralded entangled states with a high fidelity. The count rate, on the other hand, is relatively low in the analysed case. Modifications in the setup concerning the detector shape and the number of detectors are possible, as well as the use of fibers or cavities to increase the detection probability of the scattered photons without curtailing the fidelity. These suggested modifications do not change the principal results of this paper, but they might contribute to a better implementation of the presented basic ideas.
In conclusion, we have shown that with a simple and realistic setup, it is possible to create heralded entanglement of any degree between two remote atoms with a Λ-type level structure. As the atoms are entangled by projective measurements requiring no atomic interaction, the atomic distances in a given experiment are arbitrary. In particular, instead of using a far-field measurement to erase the which-way information of photons, the use of optical fibers could provide a more practical approach to reach similar goals. We expect that our results inspire and stimulate further research in operational and realistic methods for the generation and measure of entanglement in different experimental contexts.
U.S. thanks the Elite Network of Bavaria for financial support. E.S. thanks funding from Ikerbasque Foundation, EU EuroSQIP, and UPV-EHU Grant GIU07/40. C.T. and J.v.Z. gratefully acknowledge financial support by the Staedtler foundation. The authors thank G. S. Agarwal for fruitful discussions.
PACS numbers: 03.67.Bg, 42.50.Ar, 42.50.Ct
FIG. 1 :
1Scheme of two atoms with internal Λ-level structures using two detectors in the far field with polarization filters in front to register the photons emitted by the atoms. The inset shows the same configuration using optical fibers.
FIG. 2 :
2The concurrence as a function of the phase δ21 (scaled in multiples of π) and the parameter V12. The thick orange lines mark constant δ21 = (n + 1/2)π, where the dependence of C on V12 becomes linear.
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. J Volz, M Weber, D Schlenk, W Rosenfeld, J Vrana, K Saucke, C Kurtsiefer, H Weinfurter, Phys. Rev. Lett. 9630404J. Volz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, K. Saucke, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. Lett. 96, 030404 (2006).
. S Gerber, D Rotter, M Hennrich, R Blatt, F Rohde, C Schuck, M Almendros, R Gehr, F Dubin, J Eschner, arXiv: quant-ph/0810.1847S. Gerber, D. Rotter, M. Hennrich, R. Blatt, F. Rohde, C. Schuck, M. Almendros, R. Gehr, F. Dubin, and J. Es- chner, arXiv: quant-ph/0810.1847.
) only holds if V12 and cos δ21 are not simultaneously equal to 1 and −1, respectively, since in this case no two-photon signal is detected (cf. Eq. (12)) so that the projection. Note that Eqdescribed in Eq. (5) does not occurNote that Eq. (7) only holds if V12 and cos δ21 are not simultaneously equal to 1 and −1, respectively, since in this case no two-photon signal is detected (cf. Eq. (12)) so that the projection described in Eq. (5) does not occur.
. Eugene Hecht. Addison WesleyOpticsEugene Hecht, Optics (Addison Wesley, San Francisco, California, USA, 2002).
. F Tamburini, B A Bassett, C Ungarelli, Phys. Rev. A. 7812114F. Tamburini, B. A. Bassett, and C. Ungarelli, Phys. Rev. A 78, 012114 (2008).
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[
"Estimating Subagging by cross-validation",
"Estimating Subagging by cross-validation"
] |
[
"CRESTMatthieu Cornec "
] |
[] |
[] |
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for subagged estimators, both for classification and regressor. General loss functions and class of predictors with both finite and infinite VC-dimension are considered. We slightly generalize the formalism introduced by [DUD03] to cover a large variety of cross-validation procedures including leave-one-out cross-validation, k-fold cross-validation, hold-out cross-validation (or split sample), and the leave-υ-out cross-validation.An interesting consequence is that the probability upper bound is bounded by the minimum of a Hoeffding-type bound and a Vapnik-type bounds, and thus is smaller than 1 even for small learning set. Finally, we give a simple rule on how to subbag the predictor.
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[
"https://arxiv.org/pdf/1011.5142v1.pdf"
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1011.5142
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d53c924e40c736de09f10e4939943d654b65ec9e
|
Estimating Subagging by cross-validation
23 Nov 2010 November 24, 2010
CRESTMatthieu Cornec
Estimating Subagging by cross-validation
23 Nov 2010 November 24, 2010Cross-validationgeneralization errorconcentration inequalityoptimal splittingre- sampling
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for subagged estimators, both for classification and regressor. General loss functions and class of predictors with both finite and infinite VC-dimension are considered. We slightly generalize the formalism introduced by [DUD03] to cover a large variety of cross-validation procedures including leave-one-out cross-validation, k-fold cross-validation, hold-out cross-validation (or split sample), and the leave-υ-out cross-validation.An interesting consequence is that the probability upper bound is bounded by the minimum of a Hoeffding-type bound and a Vapnik-type bounds, and thus is smaller than 1 even for small learning set. Finally, we give a simple rule on how to subbag the predictor.
Introduction and motivation
One of the main issue of pattern recognition is to create a predictor (a regressor or a classifier) which takes observable inputs in order to predict the unknown nature of an output. Typical applications range from predicting the figures of a digitalized zip code to predicting the chance of survival from clinical measurements. Formally, a predictor φ is a measurable map from some measurable space X to some measurable space Y. When Y is a countable set (respectively R m ), the predictor is called a classifier (respectively a regressor). The strategy of Machine Learning consists in building a learning algorithm Φ from both a set of examples and a class of methods. Typical class of methods are empirical risk minimization or k-nearest neighbors rules. The set of examples consists in the measurement of n observations (x i , y i ) 1≤i≤n . Thus, formally, Φ is a measurable map from X × ∪ n (X × Y) n to Y. One of the main issue of Statistical Learning is to analyse the performance of a learning machine in a probabilistic setting. (x i , y i ) 1≤i≤n are supposed to be observations from n independent and identically distributed (i.i.d.) random variables (X i , Y i ) 1≤i≤n with distribution P. (X i , Y i ) 1≤i≤n is denoted D n in the following and called the learning set. In order to analyse the performance, it is usual to consider the conditionnal risk of a machine learning Φ denotedR n , so called the generalization error. It is defined by the conditional expectation of L(Y, Φ(X, D n )) given D n where (X, Y ) ∼ P is a random variable independent of D n , i.e.R n := E X,Y (L(Y, Φ(X, D n ))|D n ) with L a cost function from Y 2 −→ R + . Notice that R n is a random variable measurable with respect to D n .
Bagging, to be defined formally below, is a procedure building an estimator by a resample and combine technique. Bagging [bootstrap aggregating] was introduced by [?] to reduce the variance of a predictor. From an original estimator, a bagged regressor is produced by averaging several replicates trained on bootstrap samples, a bagged classifier is produced by voting at the majority. It is one of the recent and successful computationally intensive methods for improving unstable estimation or classification schemes. It is extremely useful for large, high dimensional data set problems where finding a good model or classifier in one step is impossible because of the complexity and scale of the problem. Regarding prediction error, the method often compares favorably with the original predictor, and also, in situations with substantial noise, with other ensemble methods such as boosting or randomization. Hence it is very important to understand the reasons for its successes, and also for its occasional failures. However, even if it has attracted much attention and is frequently applied, important questions remain unanswered theoretically. In this article, we study a variant of bagging called Subagging [Subsample aggregating] that has appeared in [?] and [?]. It is more accessible for analysis and has also substantial computational advantages. The subagged estimator will be denoted by Φ B (X, D n ) or Φ B n (X) in the following.
Important questions are: Is the generalization error of a subagged predictor lower than the original predictor, i.e R n (Φ B n ) ≤ R n (Φ)? The distribution P of the generating process being unknown, can we estimate the generalization error of a subagged predictor? Our strategy is the following: after briefly emphasizing the difficulty to provide a general answer to the first question, we will concentrate on the second question. To estimate the generalization error of a subagged predictor, we propose to use an adapted cross-validation estimator denoted by R Out CV (Φ).
[?] aggregates regression trees to build random forest and calls this process bagging. [?] prove that the bagged functional is always smooth in some sense.
[?] also show that bagging can increase both bias and variance. [?] prove that (in the limit of infinite samples) bagging reduces the variance of nonlinear components of the Taylor decomposition while leaving the linear part unaffected.
[?] consider non-differentiable and discontinuous predictors and concentrate on the asymptotic smoothing effect of bagging on neighborhood of discontinuities of decision surfaces. [?] brings new argument to explain bagging effect: bagging's improvement/deteriations are explained by the goodness/badness of highly influential examples. [?] prove the effect of bagging on the stability of a learning method and derive non asymptotic bounds for the approximation error of the bagging predictor. An interesting asymptotic result was derived in [?] : asymptotically, bagging of weak predictors can produce a strong learner, namely the bayes classifier. However, a general answer to the following non-asymptotic question R n (Φ B n ) ≤ R n (Φ)? seems hard to reach in a general framework. Using Gauss-Markov theorem, [?] shows that both bagged and unbagged predictor are unbiased, thus the variance of the unbagged predictor is lower than the variance of the bagged one.
[?] exhibit general quadratic statistics for which the bagged predictor increase both variance and bias. Thus, we propose to estimate directly the generalization error of the subagged predictor by an adapted cross-validation procedure. The latter is inspired by [?], who proposed to use the left-out example of the bootstrap samples.
In the general setting, the cross-validation procedures include leave-one-out cross-validation, k-fold cross-validation, hold-out cross-validation (or split sample), leave-υ-out cross-validation (or Monte Carlo cross-validation or bootstrap cross-validation). With the exception of [BUR89], theoretical investigations of multifold cross-validation procedures have first concentrated on linear models ( [Li87] ; [SHAO93] ; [ZHA93]). Results of [DGL96] and [GYO02] are discussed in Section 3. The first finite sample results are due to Wagner and Devroye [DEWA79] and concern k-local rules algorithms under leave-one-out and hold-out cross-validation. More recently, [HOL96,HOL96bis] derived finite sample results for υ-out cross-validation, k−fold cross-validation, and leave-one-out cross-validation for ERM over a class of predictors with finite VC-dimension in the realisable case (the generalization error is equal to zero). [BKL99] have emphasized when k−fold can beat υ-out cross-validation in the particular case of k-fold predictor. [KR99] has extended such results in the case of stable algorithms for the leave-one-out cross-validation procedure. [KEA95] also derived results for hold-out crossvalidation for ERM, but their arguments rely on the traditional notion of VC-dimension. In the particular case of ERM over a class of predictors with finite VC-dimension but with general crossvalidation procedures, [?] derived probability upper bounds.
[?] derived upper bounds for general cross-validation estimate of the generalization error of stable predictors that do no make reference to VC-dimension. However, these bounds obtained are called "sanity check bounds" since they are not better than classical Vapnik-Chernovenkis's bounds.
We introduce our main result for symmetric cross-validation procedures (i.e. the probability for an observation to be in the test set is independent of its index) in the special case of empirical risk minimization (ERM). We divide the learning sample into two samples: the training sample and the test sample, to be defined below. We denote by p n the percentage of elements in the test sample. Suppose that H holds, to be defined below. Suppose also that φ n is an empirical risk minimizer. Then, we have for all ε > 0,
Pr( R n (Φ B n ) −R Out CV ≥ ε)
≤ min(B ERM (n, p n , ε), V ERM (n, p n , ε)) < 1, with • B ERM (n, p n , ε) = min((2np n + 1) 4VC/pn exp(−nε 2 ), (2n(1 − p n ) + 1)
4V C 1−pn exp(−nε 2 /9)) • V ERM (n, p n , ε) = exp(−2np n ε 2 ).
The term B(n, p n , ε) is a Vapnik-Chernovenkis-type bound controlled by the size of the training sample n(1 − p n ) whereas the term V (n, p n , ε) is the minimum between a Hoeffding-type term controlled by the size of the test sample np n , a polynomial term controlled by the size of the training sample. This bound can be interpreted as a quantitative answer to a trade-off issue. As the percentage of observations in the test sample p n increases, the term V (n, p n , ε) decreases but the term B(n, p n , ε) increases. Other similar bounds are derived for infinite VC-dimension machine learning in the stability framework. The main interest of the previous results is in the following • our bounds are valid for machine learning with both finite and infinite VC-dimension. In the latter, it is sufficient that the machine learning satisfies some stablity property as introduced in chapter 2. As a motivation, we quote the following list of algorithms satisfying stability properties: regularization networks, ERM, k-nearest rules, boosting.
• our bounds are strictly less than 1 for any size of learning set. Thus it is also valid for small samples.
Using these probability bounds, we can then deduce that the expectation of the difference between the generalization error and the cross-validation estimate
E Dn R n (Φ B n ) −R Out CV ≤ min( 1/np n , 6 V C (ln(n(1 − p n )) + 2) n(1 − p n ) ).
Eventually, we define a splitting rule on how to chose the percentage of elements p ⋆ n in the test sample in order to get both a low generalization error together with a good approximation rate. We derive for this optimal choice of p ⋆ a bound of the form
Pr( R n (Φ B,⋆ n ) −R Out CV (p ⋆ n ) ≥ ε) = O n ((n + 1) 8VC exp(−2n(ε − 2 √ 2V 1/2 C ln(n)/n) 2 /(1 − exp(−2ε 2 )).
The paper is organized as follows. We detail the main cross-validation procedures and we summarize the previous results for the estimation of generalization error. In Section 3, we introduce the main notations and definitions. Finally, in Section 4, we introduce our results, in terms of concentration inequalities.
Main notations
In the following, we follow the notations of cross-validation introduced in [?]. We will consider the following shorter notations inspired by the literature on empirical processes. In the sequel, we will denote Z := X × Y, and (Z i ) 1≤i≤n := ((X i , Y i )) 1≤i≤n the learning set. For a given loss function L and a given class of predictors G, we define a new class F of functions from Z to R + by F := {ψ ∈ R Z + |ψ(Z) = L(Y, φ(X)), φ ∈ G}. For a machine learning Φ, we have the natural definition Ψ(Z, D n ) = L(Y, Φ(X, D n )). With these notations, the conditional risk R n is the expectation of Ψ(Z, D n ) with respect to P conditionally on D n : R n := E Z [Ψ(Z, D n ) | D n ] with Z ∼ P independent of D n . In the following, if there is no ambiguity, we will also allow the following notation ψ(X, D n ) instead of Ψ(X, D n ). To define the accurate type of cross-validation procedure, we introduce binary vectors. Let V n = (V n,i ) 1≤i≤n be a vector of size n. V n is a binary vector if for all 1 ≤ i ≤ n, V n,i ∈ {0, 1} and if n i=1 V n,i = 0. Consequently, we can define the subsample associated with it: D Vn := {Z i ∈ D n |V n,i = 1, 1 ≤ i ≤ n}. We define a weighted empirical measure on Z P n,Vn := 1
n i=1 V n,i n i=1 V n,i δ Zi ,
with δ Zi the Dirac measure at {Z i }. We also define a weighted empirical error P n,Vn ψ where P n,Vn ψ stands for the usual notation of the expectation of ψ with respect to P n,Vn . For P n,1n , with 1 n the binary vector of size n with 1 at every coordinate, we will use the traditional notation P n . For a predictor trained on a subsample, we define ψ Vn (.) := Ψ(., D Vn ).
With the previous notations, notice that the predictor trained on the learning set ψ(., D n ) can be denoted by ψ 1n (.). We will allow the simpler notation ψ n (.). The learning set is divided into two disjoint sets: the training set of size n(1 − p n ) and the test set of size np n , where p n is the percentage of elements in the test set. To represent the training set, we define V tr n a random binary vector of size n independent of D n . V tr n is called the training vector. We define the test vector by V ts n := 1 n − V tr n to represent the test set.
The distribution of V tr n characterizes all the subagging procedures described in the previous section. Using our notations, we can now define the bagged predictor.
Definition 1 (Subagged regressor) The subagged predictor build from φ n denoted φ B n is defined by: φ B n (.) := E V tr n φ V tr n (.). In the case of classifiers, the bagging rule corresponds to the vote by majority. We suppose in this case that Y = {1, . . . , M }.
Definition 2 (Subagged classifier) Cross-validated subagged classifiers of φ B n defined by:
φ B n (X) := arg min k∈{1,...,M} E V tr n L(k, Φ(X, D V tr n ))
We can now define the cross-validation estimator.
Definition 3 (Cross-validated subagged estimator) Cross-validated subagged estimates of φ B n denoted can be defined in two different ways by:
R Out CV (Φ B n ) := E V tr n P n,V ts n (ψ V tr n ) and R In CV (Φ B n ) := E V tr n P n,V tr n (ψ V tr n ) Remark 4 Recall that E V tr n P n,V ts n (ψ V tr n )
is the conditional expectation of P n,V ts n (ψ V tr n ) with respect to the random vector V tr n given D n .
Remark 5
The cross-validated subagged estimate differs from the usual cross-validation estimate of R Out CV (ψ B n ) which is equal to E U tr n P n,U ts n (ψ B U tr n ) with U tr n the training vector as defined in chapter 1.
We will give here a few examples of distributions of V tr n to show we retrieve subagging procedures described previously. Suppose n/k is an integer. The k-fold subagging procedure divides the data into k equally sized folds. It then produces a predictor by training on k − 1 folds. This is repeated for each fold, and the trained predictors are averaged to form the subagged predictor. We provide another popular example: the leave-one-out cross-validation. In leave-one-out crossvalidation, a single sample of size n is used. Each member of the sample in turn is removed, the full modeling method is applied to the remaining n − 1 members, and the fitted model is applied to the hold-backmember.
Example 7 (leave-one-out cross-validation)
Pr(V tr n = (0, 1, . . . , 1)) = 1 n Pr(V tr n = (1, 0, 1, . . . , 1)) = 1 n . . .
Pr(V tr n = (1, . . . , 1, 0)) = 1 n .
3 Results for the cross-validated subagged regressor 3.1 VC Framework
Notations and definition
We denote by R opt the minimal generalization error attained among the class of predictors C, R opt = inf φ∈C R(φ). In the sequel, we suppose that φ n belongs to some C. Notice that R opt is a parameter of the unknown distribution P (X,Y ) whereas R n is a random variable.
At last, recall the definitions of:
Definition 8 (Shatter coefficients) Let A be a collection of measurable sets. For (z 1,..., z n ) ∈ {R d } n , let N A (z 1,..., z n ) be the number of differents sets in {{z 1 , . . . , z n } ∩ A; A ∈ A} The n-shatter coefficient of A is S(A, n) = max (z1,...,zn)∈{R d } n N A (z 1,..., z n )
That is, the shatter coefficient is the maximal number of different subsets of n points that can be picked out by the class of sets A. and Definition 9 (VC dimension) Let A be a collection of sets with A ≥ 2. The largest integer k ≥ 1 for which S(A,k) = 2 k is denoted by V C , and it is called the Vapnik-Chernovenkis dimension (or VC dimension) of the class A. If S(A,n) = 2 n for all n, then by definition V C = ∞.
A class of predictors
C is said to have a finite VC-dimension V C if the dimension of the collection of sets {A φ,t : φ ∈ C, t ∈ [0, 1]} is equal to V C , where A φ,t = {(x, y)/L(y, φ(x)) > t}.
Results
In the sequel, we suppose that the cross-validation is symmetric (i.e. Pr(V n,i = 1) is independent of i) and the number of elements in the training set is constant and equal to np n , that the training sample and the test sample are disjoint and that the number of observations in the training sample and in the test sample are respectively n(1 − p n ) and np n . Moreover, we suppose also that φ n belongs to a class of predictor with finite VC-dimension. Suppose also that L is bounded in the following way: L(Y, φ(X)) ≤ C(h(Y, φ(X)) with C convex function -bounded itself by 1 on the support of h(Y, φ V tr n (X)) for simplicity-, and h such that for any 0 < λ < 1, we have h(y, λφ(x 1 ) + (1 − λ)φ(x 2 ) ≤ λh(y, φ(x 1 ) + (1 − λ)h(y, φ(x 2 ). We will also suppose that the predictors are symmetric according to the training sample, i.e. the predictor does not depend on the order of the observations in D n . We denote these hypotheses by H.
Remark 10 Typical upperbounding convex cost functions are : the hinge loss C(x) = (1 + x) + , the exponential loss C(x) = e x , the logit loss C(x) = log 2 (1 + e x ).
We will show upper bounds of the kind Pr( R n (Φ B n ) −R Out CV ≥ ε) ≤ min(B(n, p n , ε), V (n, p n , ε)) with ε > 0. The term B(n, p n , ε) is a Vapnik-Chernovenkis-type bound whereas the term V (n, p n , ε) is a Hoeffding-type term controlled by the size of the test sample np n . This bound can be interpreted as a quantitative answer to a trade-off question. As the percentage of observations in the test sample p n increases, the V (n, p n , ε) term decreases but the B(n, p n , ε) term increases.
Theorem 11 (Absolute error for symmetric cross-validation) Suppose that H holds. Then, we have for all ε > 0,
Pr( R n (Φ B n ) −R Out CV ≥ ε) ≤ min(B sym (n, p n , ε), V sym (n, p n , ε)) < 1 with • B sym (n, p n , ε) = (2np n + 1) 4VC /pn e −nε 2 • V sym (n, p n , ε) = exp(−2np n ε 2 ).
Remark 12 We do not require φ n to be an empirical risk minimizer.
Proof.
We have R n (Φ B n ) = Pψ B n = PL(Y, E V tr n φ V tr n (X)
). Since C is a convex function -bounded itself by 1 on the support of h(Y, φ V tr n (X))-, and h linear in the second variable, we get
R n (Φ B n ) ≤ PC(h(Y, E V tr n φ V tr n (X)) ≤ E V tr n PC(h(Y, φ V tr n (X)) Then, we split according to E V tr n P n,V ts n C(h(Y, φ V tr n (X)): R n (Φ B n ) ≤ E V tr n P n,V ts n C(h(Y, φ V tr n (X)) + E V tr n (P − P n,V ts n )C(h(Y, φ V tr n (X)) =R Out CV + E V tr n (P − P n,V ts n )C(h(Y, φ V tr n (X)
Thus, we obtain:
Pr( R n (ψ B n ) −R Out CV ≥ ε) ≤ Pr(E V tr n (P−P n,V ts n )C(h(Y, φ V tr n (X) ≥ ε).
To prove our result, we proceed now in two steps. For this, we consider
E V tr n (P n,V ts n C(h(Y, φ V tr n (X)) − PC(h(Y, φ V tr n (X)
)) in two different ways 1. using conditional Hoeffding's inequality, 2. using Vapnik-Chernovenkis-type inequality to bound the supremum over a class.
1. First, by conditional Hoeffding arguments (for a proof, see e.g. chapter 1),
Pr( R n (Φ B n ) −R Out CV ≥ ε) ≤ exp(−2np n ε 2 ).
2. Secondly, we derive the bound:
Pr( R n (Φ B n ) −R Out CV ≥ ε) ≤ Pr(E V tr n (P − P n,V ts n )C(h(Y, φ V tr n (X)) ≥ ε) ≤ Pr(E V tr n sup φ∈C (P − P n,V ts n )C(h(Y, φ(X)) ≥ ε).
Recall a useful lemma (for the proof, see Appendices).
Lemma 13 Under the assumptions H, we have for all, ε > 0,
Pr (E V tr n sup φ∈C (P−P n,V tr n )C(h(Y, φ(X)) ≥ ε) ≤ (S(2np n , C)) 4/pn e −nε 2 .
and we also have (for the proof, see e.g. [DGL96]): ∀n, S(n, C) ≤ (n + 1) VC .
Thus, it follows that Pr( R n (Φ B n ) −R Out CV ≥ ε) ≤ (2np n + 1) 4VC /pn e −nε 2 . Putting altogether, we get Pr( R n (Φ B n ) −R Out CV ≥ ε) ≤ min(exp(−2np n ε 2 ), (2np n + 1) 4VC/pn e −nε 2 ).
Theorem 14 (Absolute error for symmetric cross-validation) Suppose that H holds. Then, we have for all ε > 0,
Pr( R n (Φ B n ) −R In CV ≥ ε) ≤ min(B sym (n, p n , ε), V sym (n, p n , ε)) < 1 with • B sym (n, p n , ε) = (2n(1 − p n ) + 1) 4V C 1−pn e −nε 2
• V sym (n, p n , ε) = exp(−2np n ε 2 ).
Proof.
We proceed as previously:
R n (Φ B n ) = PΦ B n = PL(Y, E V tr n φ V tr n (X)) ≤ PC(h(Y, E V tr n φ V tr n (X)) ≤ E V tr n PC(h(Y, φ V tr n (X)
. We then split this quantity according to E V tr n P n,V tr
n C(h(Y, φ V tr n (X) R n (Φ B n ) ≤ E V tr n P n,V tr n C(h(Y, φ V tr n (X) + E V tr n (P − P n,V ts n )C(h(Y, φ V tr n (X) =R In CV + E V tr n (P − P n,V tr n )C(h(Y, φ V tr n (X)). Thus, we get Pr( R n (Φ B n ) −R In CV ≥ ε) ≤ Pr(E V tr n (P−P n,V tr n )C(h(Y, φ V tr n (X)) ≥ ε) ≤ Pr(E V tr n sup φ∈C (P−P n,V tr n )C(h(Y, φ(X)) ≥ ε).
Recall two useful results (for the proof, see e.g. chapter 1)
Lemma 15 Under the assumptions H, we have for all ε > 0,
Pr (E V tr n sup φ∈C (P(φ) − P n,V tr n (φ)) ≥ ε) ≤ (S(2n(1 − p n ), C)) 4/(1−pn) e −n(1−pn)ε 2 .
In the special case of empirical risk minimization, we can obtain a stronger result.
Theorem 16 (Absolute error for symmetric cross-validation) Suppose that H holds. Suppose also that φ n is based on empirical risk minimization. But instead of minimizing R n (φ), we suppose φ n minimizes 1
n n i=1 C(h(Y i , φ(X i ))
. For simplicity, we suppose the infimum is attained i.e. φ n = arg min φ∈C
1 n n i=1 C(h(Y i , φ(X i )). Then, we have for all ε > 0, Pr( R n (Φ B n ) −R Out CV ≥ ε) ≤ min(B ERM (n, p n , ε), V ERM (n, p n , ε)) < 1, with • B ERM (n, p n , ε) = min((2np n + 1) 4VC/pn exp(−nε 2 ), (2n(1 − p n ) + 1) 4V C 1−pn exp(−nε 2 /9)) • V ERM (n, p n , ε) = exp(−2np n ε 2 ).
Remark 17 1. The assumption φ n = arg min φ∈C
1 n n i=1 C(h(Y i , φ(X i ))
is not so restrictive, since in practice in order to numerically minimizes 1
n n i=1 L(Y i , φ(X i )), one looks for C convex such that for all x, y, L(y, φ(x)) ≤ C(h(y, φ(x)).
2. Thanks to the Hoeffding's part, the bound is always smaller than 1, so it remains valid for small samples. For bigger samples, we will prefer the Vapnik-Chernovenkis's part.
Proof.
Appying the previous result, we have Pr
( R n (Φ B n )−R Out CV ≥ ε) ≤ min(exp(−2np n ε 2 ), (2np n +1) 4VC/pn exp(−nε 2 )). Recall that R n (Φ B n ) −R Out CV ≤ E V tr n (PC(h(Y, φ V tr n (X)) − P n,V ts n C(h(Y, φ V tr n (X))).
We need the following lemma (for a proof, see chapter 1):
E V tr n P n,V ts n C(h(Y, φ V tr n (X)) ≥ P n C(h(Y, φ n (X)) since φ n = arg min φ∈C 1 n n i=1 C(h(Y i , φ(X i )).
Denote ψ(Z) := C(h(Y, φ(X))) with Z := (X, Y ). We have the following natural notation ψ V tr n (Z) := C(h(Y, φ V tr n (X))).
We thus get
Pr( R n (Φ B n ) −R Out CV ≥ 3ε) ≤ Pr(E V tr n (Pψ V tr n −P n,V ts n ψ V tr n ) ≥ 3ε) ≤ Pr(E V tr n (Pψ V tr n −P n ψ n ) ≥ 3ε
) and by splitting according to Pψ opt , we have:
Pr( R n (Φ B n ) −R Out CV ≥ 3ε) ≤ Pr(E V tr n (Pψ V tr n −P n,V tr n ψ V tr n +P n,V tr n ψ V tr n − Pψ opt +Pψ opt −P n ψ n ) ≥ 3ε) ≤ Pr(E V tr n sup ψ∈F (Pψ−P n,V tr n ψ) ≥ ε) + Pr( sup ψ∈F (P n,V tr n ψ−Pψ) ≥ ε) + Pr( sup ψ∈F (Pψ−P n ψ) ≥ ε).
Recall the following lemma (for the proof, see e.g.chapter 1),
Lemma 18 Under the assumption of Proposition ??, we have for all ε > 0,
Pr (E V tr n sup ψ∈F (P n,V tr n ψ−Pψ) ≥ ε) ≤ (S(2n(1 − p n ), C)) 4 1−pn e −nε 2
and symmetrically
Pr (E V tr n sup ψ∈F (Pψ−P n,V tr n ψ) ≥ ε) ≤ (S(2n(1 − p n ), C)) 4 1−pn e −nε 2 .
Then, we get
Pr( R n (ψ B n ) −R Out CV ≥ 3ε) ≤ 2(S(2n(1 − p n ), C)) 4 1−pn e −nε 2 + (S(2n, C)) 4 e −nε 2 ≤ 3(2n(1 − p n ) + 1) 4V C 1−pn e −nε 2 .
This implies in turn that
Pr( R n (ψ B n ) −R Out CV ≥ ε) ≤ (2n(1 − p n ) + 1) 4V C 1−pn exp(−nε 2 /9).
Putting altogether, we get
Pr( R n (ψ B n ) −R Out CV ≥ ε) ≤ min(exp(−2np n ε 2 ), (2np n + 1) 4VC/pn e −nε 2 , (2n(1 − p n ) + 1) 4V C 1−pn exp(−nε 2 /9))
Theorem 19 Suppose that H holds. Suppose also and that n/k is an integer. Then, we have also for all ε > 0, Pr( R n (ψ B n ) −R Out CV ≥ ε) ≤ min(B k (n, p n , ε), V k (n, p n , ε)) with • B k (n, p n , ε) = (2n/k + 1) 4kVC exp(−nε 2 )
• V k (n, pn, ε) = min exp(−2n/kε 2 ), 2 1 pn exp − nǫ 2 64( VC ln(2(2n/k + 1)) + 2) .
Proof.
The proofs starts as previously. We have
Pr(R Out CV − R n (ψ B n ) ≥ ε) ≤ Pr(E V tr n (P n,V ts n ψ V tr n −Pψ V tr n ) ≥ ε) ≤ exp(−2np n ε 2 ) but we also have Pr(R Out CV − R n (ψ B n ) ≥ ε) ≤ Pr(E V tr n ( sup ψ∈F (P n,V ts n ψ−Pψ) ≥ ε) ≤ 2 1 pn exp − nǫ 2
64( V C ln(2(2np n + 1)) + 2) .
according to chapter 1.
Following the previous results, we can obtain results for the expectation of the difference R n (ψ B n ) − R Out CV Theorem 20 (L 1 error) Suppose that H holds. Suppose also and that n/k is an integer. Then, we have also for all ε > 0, E Dn R n (ψ B n ) −R Out CV ≤ 1/np n Furthermore, suppose also that φ n is based on empirical risk minimization. But instead of minimizing R n (φ), we suppose φ n minimizes 1
n n i=1 C(h(Y i , φ(X i ))
. For simplicity, we suppose the infimum is attained i.e. φ n = arg min φ∈C
1 n n i=1 C(h(Y i , φ(X i ))
. Then, we have,
E Dn R n (ψ B n ) −R Out CV ≤ min( 1/np n , 6 V C (ln(n(1 − p n )) + 2) n(1 − p n ) )
Proof. We just need to apply the previous results together with the following useful lemma (for a proof, see e.g. [DGL96]):
Lemma 21 Let X be a nonnegative random variable. Let K, C nonnegative real such that C ≥ 1. Suppose that for all ε > 0, P(X ≥ ε) ≤ C exp(−Kε 2 ). Then, we have EX ≤ ln(C) + 2 K .
Stability framework
Introduction to stability
To avoid the traditional analysis in the VC framework, notions of stability have been intensively worked through in the late 90's [KEA95], [BE01], [BE02], [KUT02], and [KUNIY02]. The object of stability framework is the learning algorithm rather than the space of classifiers. The learning algorithm is a map (effective procedure) from data sets to classifiers. An algorithm is stable at a learning set D n if changing one point in D n yields only a small change in the output hypothesis. Several different notions of algorithmic stability are described. The attraction of such an approach is that it avoids the traditional notion of VC-dimension, and allows to focus on a wider class of learning algorithms than empirical risk minimization. For example, this approach provides generalization error bounds for regularization-based learning algorithms that have been difficult to analyze within the VC framework such as boosting. If a map is stable, exponential bounds on generalization error may be obtained. As a motivation, we quote the following list of algorithms satisfying stability properties: regularization networks, ERM, k-nearest rules, boosting.
Definitions and notations of stability
The basic idea is that an algorithm is stable at a training set D n if changing one point in D n yields only a small change in the output hypothesis. Formally, a learning algorithm maps a weighted training set into a predictor space. Thus, stability can be translated into a Lipschitz condition for this mapping with high probability.
To be more formal, following [?], we define a distance between two weighted empirical errors:
Definition 22 (Total variation) Let P n,Vn and P n,Un be two empirical measures on Z with respect to the binary vectors V n and U n . We do not assume their support to be equal. The distance between them is defined as their total variation: Example 23 In the case of leave-one-out (i.e. n i=1 U n,i = n − 1), we have:
||P n,Un − P n || = 2 n .
In the case of leave-ν-out, we get:
||P n,Un − P n || = 2ν n .
At least, we need a distance d on the set F . Let us quote three important examples. Let ψ 1 , ψ 2 ∈ F . The uniform distance is defined by:
d ∞ (ψ 1 , ψ 2 ) = sup Z∈Z |ψ 1 (Z) − ψ 2 (Z)|, the L 1 -distance by: d 1 (ψ 1 , ψ 2 ) = P|ψ 1 − ψ 2 | , the error-distance d e (ψ 1 , ψ 2 ) = |P(ψ 1 − ψ 2 )|.
It is important to notice that what matters here is not an absolute distance between the original class of predictors G seen as functions but the distance with the respect to the loss or/and the distribution P. In particular, for the L 1 -distance, we do not care about the behavior of the original predictors φ 1 and φ 2 outside the support of P. At last, notice that we always have d e ≤ d 1 ≤ d ∞ .
We are now in position to define the different notions of stability of a learning algorithm which cover notions introduced by [KUNIY02]. We begin with the notion of weak stability. In essence, it says that for any given resampling vectors, the distance between two predictors is controlled with high probability by the distance between the resampling vectors. As a motivation, notice that algorithms such as Adaboost ( [KUNIY02]) satisfies this property. With the previous notations, we have:
Definition 24 (Weak stability) Let D n = (Z i ) 1≤i≤n be a learning set. Let λ, (δ n,pn ) n,pn be nonnegative real numbers. A learning algorithm Ψ is said to be weak (λ, (δ n,pn ) n,pn , d) stable if for any training vector U n whose sum is equal to n(1 − p n ):
Pr(d(ψ Un , ψ n ) ≥ λ||P n,Un − P n ||) ≤ δ n,pn .
Notice that in the former definition Pr stands for P ⊗n . Indeed, ψ n is trained with n observations, drawn independently from P. A stronger notion is to consider ψ n trained with n−1 observations drawn independently from P and an additionnal general observation z. We consider the stronger notion of strong stability. As a motivation, notice that algorithms such as Empirical Risk Minimization with finite VC dimension ( [KUNIY02]) satisfies this property.
Definition 25 (Strong stability) Let z ∈ Z. Let D n = D n−1 ∪ {z} be a learning set. Let λ, (δ n,pn ) n,pn be nonnegative real numbers. A learning algorithm Ψ is said to be strong (λ, (δ n,pn ) n,pn , d) stable if for any training vector U n whose sum is equal to n(1 − p n ):
Pr(d(ψ Un , ψ n ) ≥ λ||P n,Un − P n ||) ≤ δ n,pn .
What we have in mind for classical algorithms is δ n,pn = O n (p n exp(−n(1−p n )). We can state the last definition in other words. Let V tr n be a training vector with distribution Q such that the number of elements in the training set is constant and equal to n(1 − p n ). Notice then that the former definition also implies that sup Un∈support(Q) P( d(ψU n ,ψn) ||Pn,U n −Pn|| ≥ λ) ≤ δ n,pn , where support(Q) stands for the support of Q. The previous notion stands for any U n having the same support of Q. A stronger hypothesis would be that the previous probability stands uniformly over U n in support(Q). This leads formally to the notion of cross-validation stability. To be more accurate:
Definition 26 (Cross-validation weak stability) Let D n = (Z i ) 1≤i≤n a learning set. Let V tr n a training vector with distribution Q. Let λ, (δ n,pn ) n,pn be nonnegative real numbers. A learning algorithm Ψ is said to be weak (λ, (δ n,pn ) n,pn , d, Q) stable if it is weak (λ, (δ n,pn ) n,pn , d) stable and if:
Pr( sup
Un∈support(Q)
d(ψ Un , ψ n ) ||P n,Un − P n || ≥ λ) ≤ δ n,pn .
As before, we also define the following stronger notion:
Definition 27 (Cross-validation strong stability) Let z ∈ Z. Let D n = D n−1 ∪ {z} a learning set. Let V tr n a cross-validation vector with distribution Q. A learning algorithm Ψ is said to be strongly (λ, (δ n,pn ) n,pn , d, Q) stable if it is strong (λ, (δ n,pn ) n,pn , d) stable and if:
Pr( sup Un∈support(Q) d(ψ Un , ψ n ) ||P n,Un − P n || ≥ λ) ≤ δ n,pn .
Remark 28 If the cardinal of the support of Q is denoted κ(n), then a learning algorithm which is weak (λ, (δ n,pn ) n,pn , d, Q)-stable is also strong (λ, (κ(n)δ n,pn ) n , d, Q)-stable.
As seen in the following error stability [KEA95] To motivate this approach, we also quote a list of class of predictors satisfying the previous stability conditions.
E P [L(Y, φ(X, D n )) | D n ] Resubstitution error R n 1 n n i=1 L(Y i , φ n (X i , D n )) Cross-validation error R CV E V tr n P n,V ts n ψ V tr n
Main results
Let D n be a learning set of size n. Let V tr n ∼ Q be a training vector independent of D n such that the cross-validation is symmetric and the number of elements in the training set is constant and equal to np n . Let d be a distance among d e , d 1 , d ∞ . At last, we suppose that the loss function L is bounded by 1. We derive the following general results that stands for general cross-validation procedures and stable algorithms.
Theorem 29 (Cross-validation Strong stability) Suppose that H holds. Let Ψ a machine learning which is strong (λ, (δ n , p n ) n,pn , Q) stable with respect to the distance d. Then, for all ε ≥ 0, we have:
Pr ( R n (Φ B n ) −R Out CV ≥ ε) ≤ exp(−2np n ε 2 ) Furthermore, if d is the uniform distance d ∞ , then we have for all α > 0: Pr ( R n (Φ B n ) −R Out CV ≥ ε) ≤ min(exp(−2np n ε 2 ), 2(exp(− ε 2 8n(8λnp n + α) 2 ) + n α δ n,pn ))
Thus, if we choose α = 8λnp n ,
Pr ( R n (Φ B n ) −R Out CV ≥ ε) ≤ min(exp(−2np n ε 2 ), 2(exp(− ε 2 8(16λ) 2 np 2 n ) + n 8λp n δ n,pn ))
Proof.
On the one hand, we have as before by conditional Hoeffding's inequality (for a proof, see e.g. chapter 1):
Pr( R n (Φ B n ) −R Out CV ≥ ε) ≤ Pr(E V tr n (Pψ V tr n −P n,V ts n ψ V tr n ) ≥ ε) ≤ exp(−2np n ε 2 )
On the other hand, notice that P ⊗n E V tr n (Pψ V tr n −P n,V ts n ψ V tr n ) = 0
Denote f (Z 1 , Z 2 , . . . , Z n ) := E V tr n (Pψ V tr n −P n,V ts n ψ V tr n ). Let z ∈ Z. Now denote:
B := { sup Un∈support(Q) d(ψ Un , ψ n+1 ) ||P n,Un − P n+1 || ≥ λ} with ψ n+1 trained on D n+1 = {Z 1 , . . . , Z i−1 , Z i , Z i+1 , .
. . , Z n , z}. Under our assumptions, we have Pr(B)≤ δ n+1,pn+1 .
We want to show that with high probability there exist constants c i such that for all i ∈ {1, . . . , n}, for all z ∈ Z,
∆ i := |f (Z 1 , . . . , Z i , . . . , Z n ) − f (Z 1 , . . . , Z i−1 , z, Z i+1 , . . . , Z n )| ≤ c i .
Notice that: So, first, let us bound the first term,
|∆ i | = |E V tr n (Pψ V tr n −P n,V ts n ψ V tr n ) − (E V tr n Pψ ′ V tr n − P ′ n,V ts n Pψ ′ V tr n )| ≤ |E V tr n P(ψ V tr n − ψ ′ V tr n )| + |E V tr n (P n,V ts n ψ V tr n −P ′ n,V ts n Pψ ′ V tr n )| with P ′ n,V|E V tr n P(ψ V tr n − ψ ′ V tr n )| ≤ E V tr n |P(ψ V tr n − ψ n+1 )| + E V tr n |P(ψ n+1 − ψ ′ V tr n )|. Thus, on B ⊂ , we have |E V tr n P(ψ V tr n − ψ ′ V tr n )| ≤ 4λ n+1 .
To upper bound the second term, notice that:
|E V tr n P n,V ts n ψ V tr n − E V tr n P ′ n,V ts n ψ ′ V tr n | = |E V tr n (P n,V ts n (ψ V tr n − ψ ′ V tr n )|V tr n,i = 1) × (1 − p n ) + E V tr n ((P n,V ts n − P ′ n,V ts n )ψ V tr n |V ts n,i = 1) × p n |
We always have for any ψ, |(P n,V ts n − P ′ n,V ts n )ψ| ≤ 1/np n thus |E V tr n ((P n,V ts n − P ′ n,V ts n )ψ V tr n , V ts n = 1) × p n | ≤ 1/n We still have to bound |E V tr n (P n,V ts n (ψ V tr n −ψ ′ V tr n )|V tr n,i = 1)| which is always smaller than E V tr n (d ∞ (ψ V tr n , ψ ′ V tr n )|V tr n,i = 1) in the special case of the most stable kind of stability namely the uniform stability.
On B ⊂ , we get d ∞ (ψ V tr n , ψ ′ V tr n ) ≤ d ∞ (ψ V tr n , ψ n+1 ) + d ∞ (ψ n+1 , ψ ′ V tr n ) ≤ 4λp n . Thus, on B ⊂ , we derive E V tr n (d ∞ (ψ V tr n , ψ ′ V tr n )|V tr n,i = 1) ≤ 4λp n .
Putting all together, with probability at least 1 − δ n,pn , we get sup 1≤i≤n,z∈Z
|f (Z 1 , . . . , Z i , . . . , Z n ) − f (Z 1 , . . . , z, . . . , Z n )| ≤ 4λ n + 1 + 4λp n (1 − p n ) ≤ 8λp n .
Applying theorem ??, we obtain that for all ε ≥ 0:
Pr(E V tr n (Pψ V tr n −P n,V ts n ψ V tr n ) ≥ ε) ≤ 2(exp(− ε 2 8n(8λp n + α) 2 ) + n α δ ′ n,pn ) ≤ 2(exp(− ε 2 8(16λ) 2 np 2 n ) + n 8λp n δ ′ n,pn ) by taking α = 8λp n
Theorem 30 (Cross-validation Weak stability) Suppose that H holds. Let Ψ be a machine learning which is weak (λ, (δ n,pn ) n,pn , Q) stable with respect to the distance d. Then, for all ε ≥ 0, we have
Pr ( R n (Φ B n ) −R Out CV ≥ ε) ≤ exp(−2np n ε 2 )
. Furthermore, if the distance is the uniform distance d ∞ , we have for all ε ≥ 0: Proof.
Pr ( R n (Φ B n ) −R Out CV ≥ ε) ≤Denote f (Z 1 , Z 2 , . . . , Z n ) :=R Out CV − R n and B := {sup Un∈support(Q) d(ψU n ,ψn+1) ||Pn,U n −Pn+1|| ≥ λ} with ψ n+1 trained on D n+1 = {Z 1 , . . . , Z i−1 , Z i , Z i+1 , . . . , Z n , Z ′ i }.|∆ i | = |E V tr n (Pψ V tr n −P n,V ts n ψ V tr n ) − (E V tr n Pψ ′ V tr n − P ′ n,V ts n Pψ ′ V tr n )| ≤ E V tr n |P(ψ V tr n − ψ ′ V tr n )| + E V tr n |(P n,V ts n ψ V tr n −P ′ n,V ts n Pψ ′ V tr n )|.
with P So, first, let us bound the first term,
|E V tr n P(ψ V tr n − ψ ′ V tr n )| ≤ E V tr n |P(ψ V tr n − ψ n+1 )| + E V tr n |P(ψ n+1 − ψ ′ V tr n )| Thus, on B ⊂ , we have |E V tr n P(ψ V tr n − ψ ′ V tr n )| ≤ 4λ n+1 .
To upper bound the second term, notice that:
|E V tr n P n,V ts n ψ V tr n −E V tr n P ′ n,V ts n ψ ′ V tr n |= |E V tr n (P n,V ts n (ψ V tr n −ψ ′ V tr n ), V tr n = 1) × (1 − p n ) +E V tr n ((P n,V ts n −P ′ n,V ts n )ψ V tr n , V ts n =1) × p n |.
We always have for all ψ, |(P n,V ts n − P ′ n,V ts n )ψ| ≤ 1/np n thus we get |E V tr n ((P n,V ts n − P ′ n,V ts n )ψ V tr n , V ts n = 1) × p n | ≤ 1/n.
We still have to bound |E V tr n (P n,V ts
n (ψ V tr n − ψ ′ V tr n ), V tr n = 1)| ≤ E V tr n (d ∞ (ψ V tr n , ψ ′ V tr n )
, V tr n = 1) in the special of the uniform stability.
On B ⊂ , we derive d ∞ (ψ V tr n , ψ ′ V tr n ) ≤ d ∞ (ψ V tr n , ψ n+1 ) + d ∞ (ψ n+1 , ψ ′ V tr n ) ≤ 4λp n , thus on B ⊂ E V tr n (d ∞ (ψ V tr n , ψ ′ V tr n ), V tr n = 1) ≤ 4λp n .
Putting all together, with probability at least 1 − δ n,pn ,
|f (Z 1 , . . . , Z i , . . . , Z n ) − f (Z 1 , . . . , Z i ′ , . . . , Z n )| ≤ 8λp n .
Following the previous results, we can obtain results for the expectation of the difference R n (Φ B n ) − R Out CV .
Theorem 31 In the case of classification, we can bound the excess risk by E Dn ( R n (Φ B n ) −R Out CV ) ≤ 1/np n Furthermore, if d is the uniform distance d ∞ , then we have for all α > 0:
E Dn ( R n (Φ B n ) −R Out CV ) ≤ min( 1/np n , √ 16 3 nλp n + n 4λp n δ n,pn )
Similar results can be derived in the context of the weak stability.
Proof
It is sufficient to apply the previous probability upper bounds together with the lemma 21.
Results for the cross-validated subagged classification
In the case of subagging of classifiers (i.e. the majority vote), we can obtain the following results:
Theorem 32 For any subbaged classifier, we can bound the excess risk.
Pr( R n (Φ B n ) −
2R
Out CV ≥ ε) ≤ exp(−8np n ε 2 /9) and also Pr( R n (Φ B n ) − lR Maj CV ≥ ε) ≤ l exp(−2np n ε 2 /9)
where N denotes the total number of training vectors in the cross-validation and l denotes [(N −1)/2]+1 that is the strict majority of the subbaged classifiers andR Maj CV the cross-validated estimate of this majority. Furthermore, in the particular case of binary classification we also have
Pr( R n (Φ B n ) − (R Out CV /2 − 1/2)) ≤ −ε) ≤ exp(−2np n ε 2 /9) and Pr( R n (Φ B n ) − (lR Maj CV − l + 1) ≤ −ε) ≤ l exp(−2np n ε 2 )
Proof.
We consider a ghost sample i.i.d. of size m: (X
m m i=1 L(Y ′ i , φ V tr n (X ′ i )) (respectively e a m := E V tr n [ 1 m m i=1 L(Y ′ i , φ V tr n (X ′ i ))])
is the average number of the mistakes of φ V tr n (respectively the weighted average number of mistakes of the family of predictors φ V tr n ).
Denote by
1. L 1 := R n (Φ B n ) − 1 2R Out CV 2. L 2 := R n (Φ B n ) − e B m 3. L 3 := e B m − e a m /2 4. L 4 := 1 2 [e a m − E X,Y E V tr n L(Y, φ V tr n (X))] 5. L 5 := 1 2 [E X,Y E V tr n L(Y, φ V tr n (X)) −R Out CV ]
We have Pr(L 1 ≥ 3ε) ≤ Pr(L 2 ≥ ε) + Pr(L 3 ≥ 0) + Pr(L 4 ≥ ε) + Pr(L 5 ≥ ε) By Hoeffding's inequality, we have:
Pr(L 2 ≥ ε) ≤ exp(−2mε 2 ).
and also Pr(L 4 ≥ ε) ≤ exp(−2m(2ε) 2 ) By conditionnal Hoeffding's inequality (for a proof, see e.g.
[?]), we deduce
Pr(L 5 ≥ ε) ≤ exp(−2np n (2ε) 2 )
By conditionnal Hoeffding's inequality, we also have
Pr(e a m − E X,Y E V tr n L(Y, φ V tr n (X)) ≥ ε) ≤ exp(−2mε 2 ). since for fixed v tr n Pr( 1 m m i=1 L(Y ′ i , φ v tr n (X ′ i )) − E X,Y L(Y, φ v tr n (X)) ≥ ε) ≤ exp(−2mε 2 )
We suppose here that Pr(V tr n = v n ) are rational numbers whose smallest multiplicator is denoted by N . Thus e a m can be seen as a simple average number of mistakes of a family of predictors (φ j ) 1≤j≤N on the ghost sample.
First notice, that if e a m is small then e B m must be small either. Indeed,we have
e a m = 1 N N j=1 1 m m i=1 L(Y ′ i , φ j (X ′ i )) = 1 N 1 m N 1≤j≤N,1≤i≤m ǫ i,j with ǫ i,j := L(Y ′ i , φ j (X ′ i )) ∈ {0, 1}
. We thus deduce that the total number of mistakes on the ghost sample of the family of predictors (φ j ) 1≤j≤N is equal to N me a m . Notice that if the number of mistakes of the family (φ j ) 1≤j≤N on the i-th observation is less that ⌊(N − 1)/2⌋ (i.e. N j=1 ǫ i,j ≤ ⌊(N − 1)/2⌋) then it means that a strict majority of predictors have classified correctly Y ′ i , which in turns tells us that a strict majority of predictors have the same output Y
′ i = φ j (X ′ i ). We thus have φ B n (X ′ i ) = Y ′ i which implies η j = L(Y ′ i , φ B n (X ′ i )) = 0.
Denoting by κ = me B m the number of mistakes of the subbaged classifier on the ghost sample, we necessarly have Thus, for binary classification, we can even obtain an probability upper bound for Pr(| R n (Φ B n ) −
m i=1 N j=1 ǫ i,j ≥ κ(⌊(N − 1)/2⌋ + 1) = κ(⌊(N + 1)/2⌋). It follows that e B m ≤ N ⌊(N + 1)/2⌋ e a m < e a m /2. Thus Pr(L 3 ≥ 0) = 0 We conclude Pr( R n (Φ B n ) − 1 2R Out CV ≥ 3ε) ≤ exp(−2np n (2ε) 2 ) + exp(−2m(2ε) 2 ) + exp(−2mε 2 ). If we let m → ∞, Pr( R n (Φ B n ) − 1 2R Out CV ≥ ε) ≤ exp(−
2R
Out CV | ≥ ε) not only for Pr( R n (Φ B n ) − 1
2R
Out CV ≥ ε). Indeed, denote by
1. L ′ 1 := R n (Φ B n ) − N ⌊N/2+1⌋ (R Out CV − 1/2) 2. L ′ 2 := R n (Φ B n ) − e B m 3. L ′ 3 := e B m − ( N ⌊N/2+1⌋ e a m − 1/2) 4. L ′ 4 := ( N ⌊N/2+1⌋ e a m − 1/2) − ( N ⌊N/2+1⌋ E X,Y E V tr n L(Y, φ V tr n (X)) − 1/2) 5. L ′ 5 := ( N ⌊N/2+1⌋ E X,Y E V tr n L(Y, φ V tr n (X)) − 1/2) − ( N ⌊N/2+1⌋R Out CV − 1/2)
We get
Pr(L ′ 1 ≤ −3ε) ≤ Pr(L ′ 2 ≤ −ε) + Pr(L ′ 3 < 0) + Pr(L ′ 4 ≤ −ε) + Pr(L ′ 5 ≤ −ε) ≤ exp(−2mε 2 ) + 0 + exp(−2m( N ⌊N/2 + 1⌋ ε) 2 ) + exp(−2np n ( N ⌊N/2 + 1⌋ ε) 2 )
Taking m → ∞, and noticing that N/⌊N/2 + 1⌋ > 1
Pr( R n (Φ B n ) − (R Out CV /2 − 1/2)) ≤ −ε) ≤ Pr( R n (Φ B n ) − (R Out CV /2 − 1/2) ≤ −ε) ≤ Pr(L ′ 1 ≤ −ε) ≤ exp(−2np n ε 2 /9)
For binary classification, we can eventually obtain that In the same way, denote by µ j := E X,Y L(Y, φ j (X)) the risk of the j-th classifier. We introduce now a cross-validation estimate of the average risk 1 l l j=1 µ (j) of the l best classifiers:R Maj CV . For this, recall that each φ j corresponds to some φ v tr n thus we can define an out sample error for the predictor j :r j := P n,v ts n (L(Y, φ j (X)). And we defineR Maj CV := 1 l l j=1r (j) We have
Pr(| R n (Φ B n ) − 1 2 (R Out CV − 1/2)| ≥ ε) ≤ exp(−1. R 1 := R n (Φ B n ) − lR Maj CV 2. R 2 := R n (Φ B n ) − e BPr(R 1 ≥ 3ε) ≤ Pr(R 2 ≥ ε) + Pr(R 3 > 0) + Pr(R 4 ≥ ε) + Pr(R 5 ≥ ε)
By Hoeffding's inequality, we have:
Pr(R 2 ≥ ε) ≤ exp(−2mε 2 ).
We also derive
Pr(R 4 ≥ ε) = Pr(e G m − 1 l l j=1 µ (j) ≥ ε/l) = Pr( l j=1 ǫ (j) − l j=1 µ (j) ≥ ε)
There exist permutations σ and σ ′ such that ǫ (j) = ǫ σ(j) and µ (j) = µ σ ′ (j) . Thus, we get
Pr(R 4 ≥ ε) ≤ Pr( l j=1 ǫ σ(j) − µ σ ′ (j) ≥ ε) ≤ Pr( l j=1 ǫ σ ′ (j) − µ σ ′ (j) ≥ ε)
by definition of ǫ (j) . It follows that
Pr(R 4 ≥ ε) ≤ l j=1 Pr(ǫ σ ′ (j) − µ σ ′ (j) ≥ ε) ≤ l exp(−2mε 2 ).
In the same way, we deduce Pr(R 5 ≥ ε) ≤ l exp(−2np n ε 2 ). By conditional Hoeffding's inequality (for a proof, see e.g.
[?]), we deduce Pr(L 5 ≥ ε) ≤ exp(−2np n (2ε) 2 ) and also for a fixed v tr n Pr(|e v tr n m − E X,Y L(Y, φ v tr n (X))| ≥ ε) ≤ 2 exp(−2mε 2 ). By conditional Hoeffding's inequality (for a proof, see e.g.
[?]), we also have
Pr(|e a m − E X,Y E V tr n L(Y, φ V tr n (X))| ≥ ε) ≤ 2 exp(−2mε 2 )
. Notice that if all the l best classifiers classify correctly the i-th observation (i.e. ǫ i,(j) = 0 for all j ∈ {1, ..., M }), then the subbaged classification classifies also correctly. Thus η i = 0. Let κ be the number of mistakes of the subbaged classifier on the ghost sample and let x the number of observations correctly classified by all the l classifiers. Then we obtain that the number of correctly classified observations by the subagging is greater that x, i.e. m − κ ≥ x. On the other hand, there is at least one predictor that makes a mistake on each of the remaining m − x observations. Thus m − x is less that the total number of mistakes made by the l best classifiers
(m − x) ≤ mle G m . From which, it follows that e B m ≤ le G m .
Thus Pr(R 3 > 0) = 0.
Putting altogether, we have
Pr( R n (Φ B n ) − lR Maj CV ≥ 3ε) ≤ exp(−2mε 2 ) + l exp(−2mε 2 ) + l exp(−2np n ε 2 ). If we let m → ∞, Pr( R n (Φ B n ) − lR Maj CV ≥ ε) ≤ l exp(−2np n ε 2 /9).
Once again, in the particular case of binary classification, we have by symmetry 1 − e B m ≤ l(1 − e G m ) which leads to e B m ≥ 1 − l(1 − e G m ). In the same way, we have a symmetrical result for binary classification:
Pr( R n (Φ B n ) − (lR Maj CV − l + 1) ≤ −3ε) ≤ exp(−2mε 2 ) + l exp(−2mε 2 ) + l exp(−2np n ε 2 ) ≤ l exp(−2np n ε 2 ).
which gives Pr(| R n (Φ B n ) − (lR Maj CV − l + 1)| ≥ ε) ≤ 2l exp(−2np n ε 2 /9).
In the case of subagging of classifiers (i.e. the majority vote) whose VC dimension is finite, we can obtain a stronger result:
Theorem 33 Suppose H holds and that the machine learning is based on empirical risk minimization. We can bound the excess risk.
Pr( R n (Φ B n ) −
2R
Out CV ≥ ε) ≤ min(exp(−8np n ε 2 /9), (2n(1 − p n ) + 1) 4VC /(1−pn) e −4n(1−pn)ε 2 ).
and also Pr( R n (Φ B n ) − lR Maj CV ≥ ε) ≤ l exp(−2np n ε 2 /9) with the l := [(N − 1)/2]+ 1 the strict majority of the subagged classifiers andR Maj CV the cross-validated estimate of this majority. Furthermore, in the particular case of binary classification we also have
Pr( R n (Φ B n ) − (R Out CV /2 − 1/2)) ≤ −ε) ≤ min(exp(−2np n ε 2 /9), (2n(1 − p n ) + 1) 4VC /(1−pn) e −4n(1−pn)ε 2 )
and
Pr( R n (Φ B n ) − (lR Maj CV − l + 1) ≤ −ε) ≤ l exp(−2np n ε 2 )
Proof.
We use again the lemma (for a proof, see chapter 1):R Out CV ≥ P n L(Y, φ n (X)) since
φ n = arg min φ∈C 1 n n i=1 L(Y i , φ(X i )).
Following the last proof, we can bound L 5 in another way.
Pr(L 5 ≥ 3ε) ≤ Pr(E V tr n [E X,Y L(Y, φ V tr n (X)) − P n L(Y, φ n (X))] ≥ 6ε) ≤ Pr(E V tr n [E X,Y L(Y, φ V tr n (X)) − P n L(Y, φ n (X))] ≥ 6ε)
Then as in proof, we split according to PL(Y, φ opt (X)) and we obtain by lemma 21
Pr(L 5 ≥ ε) ≤ (2n(1 − p n ) + 1) 4VC/(1−pn) e −n(1−pn)(2ε) 2
Results for the subagged predictor selection
The remaining important question is: in practice, how should we choose p n ? We give a hint for this question. First, suppose that the final user wants to have an accuracy equal to a certain level η.
Then we need to provide him a rule to chose an optimal p ⋆ n and to upper bound the probability of excess risk Pr( R n (φ
B,p ⋆ n n )−R Out CV (p ⋆ n ) ≥ η).
Previous bounds tell us that for any fixed p n , Pr( R n (φ B n )− R Out CV (p n ) ≥ ε) ≤ min(B(n, p n , ε), V (n, p n , ε)). Notice that min(B(n, p n , ε), V (n, p n , ε)) seen as a function of ε is a continuous non-increasing function. Thus, we can define an inverse denoted by f .
The previous probability bound becomes for any p n : Pr( R n (φ B n ) −R Out CV (p n ) ≥ f (n, p n , δ) ≤ δ. For each k, define δ n,k by f (n, k/n, δ n,k ) = η, i.e. δ n,k = min(B(n, k/n, η), V (n, k/n, η)). Denote k ⋆ n := arg min k∈{1...n−1}R
Out CV (k/n) + f (n, k/n, δ n,k ) and denote by p ⋆ n := k ⋆ n /n. Thus, we obtain:
Theorem 34 (Subbaging selection) Suppose that H holds. Suppose also that φ n is based on empirical risk minimization. But instead of minimizing R n (φ), we suppose φ n minimizes 1
n n i=1 C(h(Y i , φ(X i ))
. For simplicity, we suppose the infimum is attained i.e. φ n = arg min φ∈C
1 n n i=1 C(h(Y i , φ(X i )).
In this context, we have:
• if δ ≥ δ n f (n, p n , δ) = ln(1/δ) 2np n • and if δ < δ n , f (n, p n , δ) = 3 4V C ln(2n(1 − p n ) + 1)/(1 − p n ) + ln(1/δ) n with δ n := (2n(1 − p n ) + 1) − 4pn V C
(1−pn )(1/9−2pn ) . Furthermore, we have for all ε > 0:
Pr( R n (φ B,p ⋆ n n ) −R Out CV (p ⋆ n ) ≥ ε) = O n ((n + 1) 8VC exp − 2n(ε − 2 √ 2V 1/2 C ln(n)/n) 2 1 − exp(−2ε 2 )
).
Proof
We have:
Pr( R n (φ B,p ⋆ n n ) −R Out CV (p ⋆ n ) ≥ η) = Pr( R n (φ B,p ⋆ n n ) −R Out CV (p ⋆ n ) ≥ f (n, p ⋆ n , δ n,k ⋆ n )) ≤ k∈{1...n−1} Pr( R n (φ B,p k n ) ≥R Out CV (p k ) + f (n, k/n, δ n,k )).
It follows that:
Pr( R n (φ B,p ⋆ n n ) −R Out CV (p ⋆ n ) ≥ η) ≤ k∈{1...n−1} Pr( R n (φ B,p k n ) −R Out CV (p k ) ≥ η) ≤ k∈{1...n−1}
min(B(n, k/n, η), V (n, k/n, η)).
Thus, using previous bounds we get:
Pr( R n (φ B,p ⋆ n n ) −R Out CV (p ⋆ n ) ≥ η) ≤ min k0∈{1...n−1} ( k0−1 k=1 (2n(1 − k/n) + 1) 4VC/(1−k/n) exp(−2nη 2 ) + n−1 k=k0 exp(−2kη 2 )) ≤ min k0∈{1...n−1} (k 0 (2n + 1) 4VC/(1−k0/n) exp(−2nη 2 ) + exp(−2k 0 η 2 ) 1 − (exp(−2kη 2 )) n−k0 1 − exp(−2η 2 ) ) ≤ min k0∈{1...n−1} ((2n + 1) 4VC /(1−k0/n) α n + α k0 1 − α ) with α := exp(−2η 2 )
We look for k 0 in {(1 − z n )n, 0 < z n < 1 and z n → n∞ 0} Pr( R n (φ B,p ⋆ n n ) −R Out CV (p ⋆ n ) ≥ η) ≤ min zn (2n + 1) 4VC /zn α n + α (1−zn)n 1 − α
We look for z n such that (2n + 1) 4VC/zn ∼ n∞ α −znn 1−α Let us even find z n such that (2n + 1) 4VC/zn = α −znn 1−α . It is thus equivalent to: −n ln(α)z 2 n − ln(1 − α)z n − 4V C ln(2n + 1) = 0 We have ∆ = ln(1 − α) 2 − 16V C ln(2n + 1)n ln(α) > 0 since |α| < 1 Since 0 < z n < 1, we have necesseraly z n the non negative root of the previous equation which leads to: Let us now find the expression of f the inverse of min ε (B(n, p n , ε), V (n, p n , ε)) with
• B(n, p n , ε) = min((2n(1 − p n ) + 1)
4V C 1−pn exp(−nε 2 /9))
• V (n, p n , ε) = exp(−2np n ε 2 ).
In the case of ERM algorithm, exp(−2np n ε 2 ) ≤ (2n(1 − p n ) + 1) if and only if −2np n ε 2 ≤ 4VC 1−pn ln(2n(1 − p n ) + 1) − nε 2 /9 which is equivalent to n(1/9 − 2p n )ε 2 ≤ 4V C ln(2n(1 − p n ) + 1) 1 − p n and also ε ≤ 4VC ln(2n(1−pn)+1) n(1−pn)(1/9−2pn) := ε n .
Thus if ε ≤ ε n , it follows that min(B(n, p n , ε), V (n, p n , ε)) = exp(−2np n ε 2 ), thus if δ = exp(−2np n ε 2 ) we deduce that ε = ln(1/δ) npn . If ε > ε n , min(B(n, p n , ε), V (n, p n , ε)) = (2n(1−p n )+1)
4V C 1−pn exp(−nε 2 /9).
Thus if δ = (2n(1−p n )+1)
4V C 1−pn exp(−nε 2 /9), we then deduce that ε = 3 4VC ln(2n(1−pn)+1)/(1−pn)+ln(1/δ) n .
Denote δ n = exp(−2np n ε 2 n ) = exp(− 4pnVC ln(2n(1−pn)+1) (1−pn)(1/9−2pn) ) = (2n(1 − p n ) + 1) − 4pnV C
(1−pn )(1/9−2pn ) .
In conclusion, if δ ≥ δ n , we have:
f (n, p n , δ) = ln(1/δ) 2np n and if δ < δ n , f (n, p n , δ) = 3 4V C ln(2n(1 − p n ) + 1)/(1 − p n ) + ln(1/δ) n ≤ 6 V C ln(2n + 1) + ln(1/δ) n(1 − p n ) .
In summary, the probability of the deviation between the out-of-bag cross-validation estimate and the generalization error is bounded by the minimum of a Hoeffding-type bound and a Vapnik-Chernovenkis-type bounds, and thus it is smaller than 1 even for small learning sets. Finally, we also give a simple rule on how to subbag the predictor. However, in the case of classification, we show that subagging strong learners can give a strong learner. It would be more interesting to answer the following question : can we obtain a similar result with the subagging of weak learners ?
Appendices
We will use the definition of strong difference bounded introduced by [KUT02] and a corollary of his main theorem inspired by [McD89].
Definition 35 (Kutin [KUT02]) Let Ω 1 , . . . , Ω n be probability spaces. Let Ω = n k=1 Ω k and let X a random variable on Ω. We say that X is strongly difference bounded by (b, c, δ) if the following holds: there is a "bad" subset B ⊂ Ω, where δ = P(B). If ω, ω ′ ∈ Ω differ only in k-th coordinate, and ω / ∈ B, then |X(ω) − X(ω ′ )| ≤ c Furthermore, for any ω, ω ′ ∈ Ω, |X(ω) − X(ω ′ )| ≤ b
We will need the following theorem. It says in substance that a strongly difference bounded function of independent variables is closed to its expectation with high probability.
Theorem 36 (Kutin [KUT02]) Let Ω 1 , . . . , Ω n be probability spaces. Let Ω = n k=1 Ω k and let X a random variable on Ω, which is strongly difference bounded by (b, c, δ). Assume b ≥ c ≥ 0 and α > 0. Let µ = E(X). Then, for any τ > 0, Pr(X − µ ≥ τ ) ≤ 2(exp(− τ 2 8n(c + bα) 2 ) + n α δ)
We will use the definition of weak difference bounded introduced by [KUT02] and a corollary of his main theorem.
Definition 37 (Kutin) Let Ω 1 , . . . , Ω n be probability spaces. Let Ω = n k=1 Ω k and let X a random variable on Ω. We say that X is weakly difference bounded by (b, c, δ) if the following holds: for any k,
∀ δ (ω, v) ∈ Ω × Ω k , P(|X(ω) − X(ω ′ )|) ≤ c
where ω ′ k = v and ω ′ i = ω i for i = k. and the notation ∀ δ ω, Φ(ω) means "Φ(ω) holds for all but but a δ fraction of Ω" |X(ω) − X(ω ′ )| ≤ c Furthermore, for any ω, ω ′ ∈ Ω, differing only one coordinate:
|X(ω) − X(ω ′ )| ≤ b
We will need the following theorem. It says in substance that a weakly difference bounded function of independent variables is closed to its expectation with probability.
Theorem 38 (Kutin) Let Ω 1 , . . . , Ω n be probability spaces. Let Ω = n k=1 Ω k and let X a random variable on Ω.which is weakly difference bounded by (b, c, δ). Assume b ≥ c ≥ 0 and α > 0. Let µ = E(X). Then, for any ε > 0 Pr(|X − µ| ≥ ε) ≤ 2 exp(− ε 2 10nc 2 (1 + 2ε 15nc ) 2 ) + 2nbδ 1/2 c exp( εb 4nc 2 )) + 2nδ 1/2
||P n,Un − P n,Vn || = sup A∈P(Z) |(P n,Un − P n,Vn )(A)|.
empirical measure on the sampleE n = {Z 1 , . . . , Z i−1 , z, Z i+1 , . . . , Z trained on E V tr n .
We want to show that for all i, there exists constant c i such |∆ i | := |f (Z 1 , . . . , Z i , . . . , Z n )−f (Z 1 , . . . , Z ′ i , . . . , Z n )| ≤ c i with high probability where Z 1 , . . . , Z i , . . . , Z n , Z ′ i are i.i.d. variables.
weighted empirical measures of the sample D ′ n = {Z 1 , . . . , Z ′ i , . . . , Z n } and ψ ′ n the predictor built on D ′ n .
i corresponds to the average number of mistakes of Φ B n on the ghost sample. In the same way,
8np n ε 2 / 9 )
29Notice that in the particular case of the binary classification, we have by symmetry, 1 − e
8np n ε 2 /9) + exp(−2np n ε 2 /9) ≤ 2 exp(−2np n ε 2 /9) Denote by ǫ j := 1 m m i=1 ǫ i,j the average number of mistakes by predictors j on the ghost sample. We can order them by increasing order: ǫ (1) , ..., ǫ (N ) . Let l := ⌊N/2 + 1⌋ be the strict majority. An interesting case is when we know that a strict majority of classifiers are very good. average error of the first l best classifiers on the ghost sample.
)
z n = ln(1 − α) + ln(1 − α) 2 − 16V C ln(2n + 1)n ln(α) −2n ln(α)We can inject z n in (2n + 1) 4VC/zn α n + α −R Out CV (p ⋆ n ) ≥ η) = O n ((n + 1) 8VC exp(−2n(η − 2 √ 2V1/2 C ln(n)/n) 2 )/(1 − exp(−2η 2 ))
Table 1 :
1Main notations
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On the uniform convergence of relative frequencies of events to their probabilities. V N Vapnik, A Y Chervonenkis, Theory of Probability and its Applications. 16V. N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264-280,1971.
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The nature of statistical learning theory. V Vapnik, SpringerV. Vapnik. The nature of statistical learning theory. Springer, 1995.
Statistical learning theory. V Vapnik, Wiley-Interscience PublicationNew YorkV. Vapnik. Statistical learning theory. John Wiley and Sons Inc., New York, 1998. A Wiley- Interscience Publication.
Consistency of Cross Validation for Comparing Regression Procedures. Y Yang, Annals of Statistics. Y. Yang, Consistency of Cross Validation for Comparing Regression Procedures. Accepted by Annals of Statistics.
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A leave-one-out cross validation bound for kernel methods with applications in learning. T Zhang, 14th Annual Conference on Computational Learning Theory. SpringerT Zhang . A leave-one-out cross validation bound for kernel methods with applications in learning. 14th Annual Conference on Computational Learning Theory, 2001 -Springer.
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[
"Activity-dependent self-wiring is a basis of structural plastisity in neural networks",
"Activity-dependent self-wiring is a basis of structural plastisity in neural networks"
] |
[
"Fail M Gafarov s:[email protected] \nDepartment of Theoretical Physics\nTatar State University of Humanity and Pedagogic\nTatarstan Street420021Kazan2 Russia\n"
] |
[
"Department of Theoretical Physics\nTatar State University of Humanity and Pedagogic\nTatarstan Street420021Kazan2 Russia"
] |
[] |
Dynamical wiring and rewiring in neural networks are carried out by activity-dependent growth and retraction of axons and dendrites, guided by gudance molecules, released by target cells. Experience-dependent structural changes in cortical microcurcuts lead to changes in activity, i.e. to changes in information encoded. Specific pattens of external stimulation can lead to creation of new synaptical connections between neurons. Calcium influxes controlled by neuronal activity regulates processes of neurotrophic factors release by neurons, growth cones movement and synapse differentiation in developing neural system, therefore activity-dependent self-wiring can serve as a basis of structural plasticity in cortical networks and can be considered as a form of learning.
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"https://arxiv.org/pdf/q-bio/0607021v1.pdf"
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q-bio/0607021
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3e82bc648034f499ccaa05b3fe8d1eb0da45bb58
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Activity-dependent self-wiring is a basis of structural plastisity in neural networks
17 Jul 2006
Fail M Gafarov s:[email protected]
Department of Theoretical Physics
Tatar State University of Humanity and Pedagogic
Tatarstan Street420021Kazan2 Russia
Activity-dependent self-wiring is a basis of structural plastisity in neural networks
17 Jul 2006
Dynamical wiring and rewiring in neural networks are carried out by activity-dependent growth and retraction of axons and dendrites, guided by gudance molecules, released by target cells. Experience-dependent structural changes in cortical microcurcuts lead to changes in activity, i.e. to changes in information encoded. Specific pattens of external stimulation can lead to creation of new synaptical connections between neurons. Calcium influxes controlled by neuronal activity regulates processes of neurotrophic factors release by neurons, growth cones movement and synapse differentiation in developing neural system, therefore activity-dependent self-wiring can serve as a basis of structural plasticity in cortical networks and can be considered as a form of learning.
I. INTRODUCTION
The neural system is a complex self-wiring system, which consists of a huge number of interconnected individual cells (neurons). Two main types of signaling exist in a neural system: short-range synaptic signaling between neurons, provided by neurotransmitters, which acts on postsynaptic neuron's state and long(short)-range signaling by chemicals that acts on neuron's geometrical properties (position of cell, dendrites, axons) [1]. Each function of the mature neural system depends on the actions of distinct neuronal circuits and therefore proper functioning of the neural net depends on correctness of axonal projections and interneuronal connections [2].
In in vitro experiments it is shown that neurons self-organize into homogeneous or clustered networks and correlations between neurons activity emerges [3]. Different models had been proposed for investigation of connection structure emergence between initially disconnected neurons [4,5].
It is currently accepted that cortical maps are dynamic constructs that are remodelled in response to external input. Two types of plasticity in neural system is known: synaptical plasticity and structural plasticity. Synaptical plasticity involves activity-dependent weight changes between previously connected neurons [6]. Structural plasticity includes remodelling of axons and dendrites, synapse formation and elimination [7]. Neuronal activity controls metabolic process in neurons through calcium influx through voltage dependent calcium channels. Calcium transients plays a cental role in axon guidance, neuron differentiation and synaptical plasticity (LTP,LTD) [8,9].
In this paper, using the framework presented in [10] is demonstrated that activity-dependent self-wiring could provide sufficient basis for structural plasticity in the developing neural system. The model proposed in [10] is based on the following experimental data (i)-(iii) and assumption (iv)-(v): (i) Development of neuronal connectivity dependents on the neurons activity [12] ; (ii) Direction of motion of the growth cones is controlled by diffusible chemicals -axon guidance molecules (AGM) [1]; (iii) Axon's growth rate dependent on the neuron's activity [8]; (iv) Depolarization causes neurons to release axon guidance molecules [11]; (v) Type of neuronal connectivity is determined postsynaptically during synaptogenesis [12,13].
II. THE MODEL AND NUMERICAL RESULTS
Axon growth is a complex process in which a growth cone, located at the tip of growing neurite travels through surrounding media, controlled by chemical released by targets. The growth cone is able to detect gradients of AGM as small as a difference of a single molecule across its structure [1]. In the model suggested we assumed that, if travelling growth cone's soma is at inactive state (not depolarization and calcium influx), is guided only by AGM, which causes the local calcium influx and the growth cone changes the direction of motion. When the cell generates an action potential, the depolarization of its membrane leads to opening of the growth cone's voltage-gated calcium channels and inhibition of its growth rate [8]. Therefore the rate of axon's growth depends on AGM concentration gradient ∇C(g n k , t) at the growth cone's position [1] and axon's firing rate ν(t) [8], and growth cone's equation of motion can be written in the form
dg n k dt = λF (∇C(g n k , t), ν(t)),(1)
where λ is a coefficient describing axon's sensitivity and motility. Here we used the most simple case of the function describing axon outgrowth
dg n k dt = λ 1 + e −bν(t) ∇C(g n k , t)(2)
In [10] was investigated self-wiring between binary neurons. Here we used Spike-Response Model (SRM) [14] for description of neuron's activity. Following [14,15] the membrane potential u j (t) of j-th neuron at time t is defined as
u i (t) = η(t −t j ) + j w ij f ǫ 0 (t − t (f ) j ) + ∞ 0 k(s)I ext i (t − s)ds.(3)
The response kernels η, ǫ 0 , k that describe the effects of spike generation, spike reception and external input on the membrane potential are described by the following functions [15]:
η(t −t) = −v − η 0 exp t −t τ ref r Θ(t −t) (4) ǫ 0 (t − t i ) = 1 1 − (τ s /τ m ) exp − t − t i τ m − exp − t − t i τ s Θ(t − t i ) (5) k(s) = exp − s τ e Θ(s) (6) If u j (t) crosses from below ( du dt > 0) a threshold θ at a moment t (f ) j
then a spike is generated. Release of some neurotrophic factors can be triggered by external stimulation and neuron's electrical activity [11]. In [10] was proposed a hypothesis that neurons release AGM at firing time. The activity dependent release of AGM is a key point in our model. As far as we know in neurobiological literature there is no complete proof of the activity dependent AGM release and we consider this point as an hypothesis. We suppose that all neurons release the unit amount of the one type AGM which causes only attraction of growth cones. Therefore, the concentration of AGM, c ij , released by the i-th cell at the moment t j can be found as the solution of the diffusion equation
∂c ij ∂t = D 2 ∆c ij − kc ij ,(7)
with the initial conditions c ij (r, r i , t j ) = δ(r − r i )δ(t − t j i ) (point-like sources). Here D and k are AGM's diffusion and degradation coefficients in the intracellular medium. We consider here the case without boundary conditions. The solution of this equation, describing the concentration of AGM released by a single spike of i − th neuron is
c ij (r, r i , t, t (f ) i ) = Θ(t − t (f ) i ) (2D π(t − t (f ) i )) 3 exp −k(t − t (f ) i ) − |r − r i | 2 4D 2 (t − t (f ) i ) .(8)
The total concentration and gradient of concentration of AGM at the point r can be found by summation of concentrations and and gradients of concentration of AGM which were released by each cell [10]
C(r, t) = N i=1 ∞ f =1 c ij (r, r i , t, t (f ) i ),(9)∇C(r, t) = N i=1 ∞ f =1 ∇c ij (r, r i , t, t (f ) i ),(10)
where If the position of some growth cone is close to the another cell's soma, i.e if |g n k − r i | < ε (ε can be considered as the soma's geometrical radius) then synaptic connection between these neurons will be established.
∇c ij (r, r i , t, t (f ) i ) = Θ(t − t (f ) i )(r − r i ) 16D 5 π 3/2 (t − t (f ) i ) 5/2 exp −k(t − t (f ) i ) − |r − r i | 2 4D 2 (t − t (f ) i ) .(11)
Certain neurons choose the neurotransmitter they use and synapse type in an activity-dependent manner, and different trophic factors are involved in this phenotypic differentiation during development. Regulation of transmitter expression occurs in a homeostatic manner. Suppression of activity leads to an increased number of neurons expressing excitatory transmitters and a decrease number of neurons expressing inhibitory transmitters and vice versa [13]. In the model, we supposed in framework of our approach that each neuron's different terminals (branches of the same axon) can release different neurotransmitters and can establish different types of synaptic connections (inhibitory or excitatory). The type of synapse can be determined by state of presynaptic or/and postsynaptic neuron. For simplification we assumed also that the type of a synaptic connection between cells depends on the state of postsynaptic cell at synaptogenesis process. This assumption can be changed for special neurons, according to experimental data. For simplicity the type of the neuronal connections between i-th and n-th neurons we describe by using static synaptic weights w n,i (w n,i = 1 means excitatory and w n,i = −1 inhibitory connections). According experimental data [13], in the model the type of synaptic connection established between neurons depends on the state of postsynaptic cell at synaptogenesis moment (if V i (t) > V tr then w n,i = −1, else w n,i = −1).
The model gives a closed set of equations describing AGM's release and diffusion, and axons growth and synaptical connections establishment as well as the net's electrical activity dynamics. The concentration of AGM in the extracellular space is controlled by neurons activity. Growth and movement of growth cones is managed by the concentration gradients of AGM and neuronal activity. Growth cones can make synaptic connections with other neurons and change the network's connections structure which change the network's activity. Numerical simulation of the model were performed using the net which consist of N=8 neurons, placed at tops of a cube (Fig.1). Initially all neurons has no synaptic connections and all axons placed near the the soma. Different values of parameters gives different connectivity patterns between neurons, because these parameters characterize neuron's electrical properties,and growth cone's movement speed, and AGM's acting distance, and etc. The figures presented here have been obtained using the following values of parameters: η 0 = 0.06, τ ref r = 0.1, v = 0.03, θ = −0.02, τ s = 0.01, τ m = 0.2, τ e = 0.1 and D = 0.8, a = 0.2, λ = 0.00005. External current were taken in the form I ext i = 0.5 sin(0.2(i − 4)t + i) + 1, where i is the number of neuron. After simulation start, growth cones began moving in the direction of concentration gradients (Fig. 1). Several studies show that neural activity affects individual axonal branching in vivo. Increasing of neuronal activity leads to appearance of new branches [9,16]. In the model, branching of growth cone is dependent on activity (new branch added only if V i > 10 c −1 ). Metabolic constrains requires, that axon cannot infinitely make new branches, to When the growth cone reaches the soma of another neuron, two neurons became connected and axon's connected branch is depicted by thick curve (Fig. 2). The synapse type (inhibitory or excitatory) is depicted by color of a curve (bright -excitatory, dark-inhibitory). When the branch of the same axon reaches the another cell, which has already connected with it, this branch will be deleted. Structural changes leads to changes in electrical activity of neurons, i.e to changes in the time dependence of membrane potentials u j (t) (Fig. 3).
III. CONCLUSIONS
Neural activity plays a central role in experience-dependent rewiring of cortical microcircuits. Activity-dependent structural plasticity can be considered as a special case of synaptical plasticity in fully connected network. Both of these types of plasticity are dependent on activity pattern of neurons. But in contrast to synaptical plasticity structural plasticity imply changes in connection map. Real neural net are sparsely connected and in no circumstances we cannot consider them as fully connected network. Activity-dependent self-wiring involves establishment of new synaptic connections between previously unconnected cells, and this conception is important for investigation of learning in real networks. In the model developed here, finding of appropriate partnership between pre-and postsynaptic neuron is controlled by activity of neurons, therefore different patterns of external input I ext i through regulation of neuronal activity will lead to functionally distinct circuits, and changes in connections structure can lead to changes in activity (Fig. 3) of the wholly network. For the further theoretical and computational investigations of structural plasticity in neural networks the model presented here, can be sophisticated on the basis of new experimental findings, for example:
• Cells can release also repellant in activity-dependent or activity-independent manner [1], therefore a model where at inactive state cells release a repellant, at active state -an attractant, or vice versa, can be considered.
• A set of cells releasing different types of AGM, and different types of growth cones regulated by different AGM can be considered [1].
• Depending on the cell's level of activity, a growth cone can be repelled or attracted by the same AGM [8].
• Growth cones themselves can release chemicals and influence other growth cones movement [17].
• A real neurons has also dendrites. For simplification of model we did not consider them, and supposed that axons connect directly to a soma. Incorporation into model also guidance of dendrites [18] by different chemicals, can cause discovery of a new interesting properties of structural plasticity.
• Hebbian learning can be incorporated [6].
We believe that the model developed here may help in investigations of fundamental problems in neural networks self-organization in vivo and in vitro [3]. Specially, this model can be used in construction of novel biosensors and hybrid neural-computer systems.
FIG. 1 :
1Three dimensional picture of the state of the net at the simulation beginning t = 5.5 c. Individual neurons (whilst without neuronal connections) depicted as spheres, the firing rates depicted by brightness (bright -hight firing rate, dark-low). The growing axons are depicted as thin curves. One can see from this figure, how axons grow toward active cells and that some axons began branching (new branches pointed by arrows).
FIG. 2 :
2The state of the net at t = 23.3 c. The net has weighs w5,7 = 1, w2,4 = 1,w8,4 = −1, w3,4 = −1, w1,6 = 1, w7,6 = −1, w6,7 = −1.
FIG. 3 :
3The time dependence of membrane potential of 5-nd neuron. One can see postsynaptic potentials, and firing rate increasing after t = 6.35 c because of new synaptic connection between 7-nd and 5-nd neurons.agree with it in the model a new branch is added only after some time (0.4 c) after previous branch addition.
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|
[] |
[
"Rotational spectroscopy of isotopic oxirane, c-C 2 H 4 O",
"Rotational spectroscopy of isotopic oxirane, c-C 2 H 4 O"
] |
[
"Holger S P Müller \nAstrophysik/I. Physikalisches Institut\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany\n",
"Jean-Claude Guillemin \nUniv Rennes\nEcole Nationale Supérieure de Chimie de Rennes\nCNRS\nISCR\n",
"Frank Lewen \nAstrophysik/I. Physikalisches Institut\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany\n",
"Stephan Schlemmer \nAstrophysik/I. Physikalisches Institut\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany\n",
"\nUMR 6226\n35000RennesFrance\n"
] |
[
"Astrophysik/I. Physikalisches Institut\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany",
"Univ Rennes\nEcole Nationale Supérieure de Chimie de Rennes\nCNRS\nISCR",
"Astrophysik/I. Physikalisches Institut\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany",
"Astrophysik/I. Physikalisches Institut\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany",
"UMR 6226\n35000RennesFrance"
] |
[] |
We studied the rotational spectrum of oxirane in a sample of natural isotopic composition in selected regions between 158 GHz and 1093 GHz. Investigations of the isotopologs with one 13 C or one 18 O were the primary focus in order to facilitate searches for them in space. We also examined the main isotopic species mainly to look into the performance of Watson's A and S reductions both in an oblate and in a prolate representation. Even though oxirane is a rather asymmetric oblate rotor, the A reduction in the III l representation did not yield a satisfactory fit, as was observed frequently earlier for other molecules. The other three combinations yielded satisfactory fits of similar quality among each other; the A reduction in the I r representation required two parameters less than both S reduction fits.
|
10.1016/j.jms.2022.111584
|
[
"https://arxiv.org/pdf/2201.09266v2.pdf"
] | 246,240,447 |
2201.09266
|
d14461dd5223a4bc4ac0cce2e8ad93a07388de4d
|
Rotational spectroscopy of isotopic oxirane, c-C 2 H 4 O
Holger S P Müller
Astrophysik/I. Physikalisches Institut
Universität zu Köln
Zülpicher Str. 7750937KölnGermany
Jean-Claude Guillemin
Univ Rennes
Ecole Nationale Supérieure de Chimie de Rennes
CNRS
ISCR
Frank Lewen
Astrophysik/I. Physikalisches Institut
Universität zu Köln
Zülpicher Str. 7750937KölnGermany
Stephan Schlemmer
Astrophysik/I. Physikalisches Institut
Universität zu Köln
Zülpicher Str. 7750937KölnGermany
UMR 6226
35000RennesFrance
Rotational spectroscopy of isotopic oxirane, c-C 2 H 4 O
rotational spectroscopysubmillimeter spectroscopyinterstellar moleculecyclic moleculecentrifugal distortionreduced Hamiltonian
We studied the rotational spectrum of oxirane in a sample of natural isotopic composition in selected regions between 158 GHz and 1093 GHz. Investigations of the isotopologs with one 13 C or one 18 O were the primary focus in order to facilitate searches for them in space. We also examined the main isotopic species mainly to look into the performance of Watson's A and S reductions both in an oblate and in a prolate representation. Even though oxirane is a rather asymmetric oblate rotor, the A reduction in the III l representation did not yield a satisfactory fit, as was observed frequently earlier for other molecules. The other three combinations yielded satisfactory fits of similar quality among each other; the A reduction in the I r representation required two parameters less than both S reduction fits.
Introduction
Oxirane, c-C 2 H 4 O, also know as ethylene oxide, oxacyclopropane, epoxyethane, and dimethylene oxide, is a molecule of astrochemical interest that was detected first toward the prolific high-mass Galactic center source Sagittarius (Sgr) B2(N) [1]. It was found subsequently in several other high-mass star-forming regions [2,3]. It was also observed in three rotationally cold, but kinetically warm, Galactic center sources [4] and more recently toward the prototypical low-mass protostellar source IRAS 16293−2422 [5] and toward a prestellar core [6].
Observations of molecules containing 13 C are viewed as important diagnostic tools because the 12 C/ 13 C ratio in the interstellar medium (ISM) differs from the terrestrial value of 89 [7]. It is around 20 to 25 in the Galactic center region [8,9,10,11,12], increases to about 68 in the Solar neighborhood and even further in the outskirts of the Milky Way [8,12,13,14]. Even lower 12 C/ 13 C ratios were found in the envelopes of some late-type stars, such as K4−47 [15,16].
Numerous detections of isotopologs containing 13 C were reported in recent years [8,11,17,18,12,19] and some even much earlier. Variations in the 12 C/ 13 C ratios within one source, such as in IRAS 16293−2422 [18], may provide clues on the formation pathways of these molecules. Some of these observations benefited greatly from recent or concomitant laboratory investigations of 13 C containing isotopologs, such as ethanol [20], acetaldehyde [21], or ethyl cyanide with two 13 C [17]. Other recently studied isotopologs await detection in space. These include methylamine [22], methyl isocyanate [23], methyl mercaptan [24], and cyclopropenone [25].
The relative abundance of 18 O is less favorable; the terrestrial 16 O/ 18 O ratio is almost exactly 500 [7], and the ratio in the Galactic center of ∼200 [11,13,26,27] is only slightly lower. Nevertheless, some complex organic molecules containing 18 O have been detected in recent years, among them methyl formate [28] and formamide [29]. Cyclopropenone is an example of a molecule detected in the ISM whose 18 O isotopolog was studied recently, but was not yet detected [25].
The first studies of the rotational spectrum of oxirane date back to the early days of microwave spectroscopy and include the determination of structural parameters and of its dipole moment through Stark effect measurements [30]. Its dipole moment was determined even earlier through investigations of its dielectric properties along with those of several other molecules [31].
More accurate transition frequencies of several isotopic species were determined later in the microwave and lower millimeter-wave regions [32,33,34]. Extensive data pertaining to the main isotopolog were obtained more recently between 260 and 360 GHz [35] and between 15 and 73 cm −1 [36]. Results of a millimeterwave and far-infrared study on c-C 2 H 3 DO were reported very recently [37]. Unlabeled atoms in formulae designate 12 C and 16 O here and in the following. We also mention a high-resolution IR study [38] and intensity measurements on c-C 2 H 4 O [39].
Our present work describes the investigations of c-13 CCH 4 O and c-C 2 H 4 18 O employing samples of oxirane in natural isotopic composition to facilitate their detection in space. We also measured transition frequencies of the main isotopic species mainly to test the performances of Watson's S and A reductions of the rotational Hamiltonian in combination with an oblate and a prolate representation.
Experimental details
Our measurements were carried out at room temperature using two different spectrometers. Pyrex glass cells with an inner diameter of 100 mm were employed. Both spectrometer systems used VDI frequency multipliers driven by Rohde & Schwarz SMF 100A microwave synthesizers as sources. Schottky diode detectors were utilized below 250 GHz, whereas a closed cycle liquid Hecooled InSb bolometer (QMC Instruments Ltd) was applied above 340 GHz. Frequency modulation was used throughout. The demodulation at 2 f causes an isolated line to appear close to a second derivative of a Gaussian, as can be seen in Fig. 1.
A 5 m long double pass cell equipped with Teflon lenses was used for measurements between 158 and 250 GHz. The pressures were about 1 Pa. Further information on this spectrometer is available elsewhere [40]. We achieved frequency accuracies of 5 kHz for this spectrometer in a study of 2-cyanobutane [41], a molecule that displays a much richer rotational spectrum. Measurements between 340 and 505 GHz and between 758 and 1093 GHz were carried out employing a 5 m long single pass cell at pressures of ∼2 Pa. A pressure slightly below 1 Pa was chosen for measurements of medium strong lines of the main isotopolog. Our studies on isotopic formaldehyde [42] or thioformaldehyde [43] demonstrate that accuracies of 10 kHz can be reached quite easily for very symmetric lines with good signal-to-noise ratio up to 1.5 THz. We assigned uncertainties of mostly 5 or 10 kHz for the best lines up to 50 kHz for less symmetric lines or lines fairly close to other lines.
Results
Spectroscopic properties of oxirane
Oxirane is a very asymmetric rotor of the oblate type with κ = (2B − A − C)/(A − C) equals 0.4093, 0.3557, and 0.6389 for c-C 2 H 4 O, c-13 CCH 4 O, and c-C 2 H 4 18 O, respectively. The main isotopolog as well as the one with 18 O have C 2v symmetry, and the four equivalent H atoms lead to para and ortho spin-statistics with a 3 : 5 intensity ratio, see Fig. 1. The para and ortho levels are described by K a + K c being odd and even, respectively. These spin-statistics are absent in c-13 CCH 4 O because the molecule has only C S symmetry.
The dipole moment of 1.90 D of c-C 2 H 4 O is along the b-axis [30]. This value includes a small increase of 0.02 D caused by the difference of the OCS reference value used in [30] compared to more modern values [44,45]. The lower symmetry of c-13 CCH 4 O is associated with a small a-dipole moment component of ∼0.17 D.
The rotational spectrum of oxirane is sparse at the level of the strongest lines because the rotational parameters are fairly large. However, the spectrum is much richer at the level of the weakest transitions in the line list which are about 4.5 orders of magnitude weaker, leading to somewhat increased accidental overlap of lines.
Observed spectrum and determination of spectroscopic parameters of c-C 2 H 4 O
The initial calculation of the rotational spectrum of c-C 2 H 4 O was based on the second version of the Cologne Database for Molecular spectroscopy, CDMS, [46,47,48] entry of the oxirane main isotopic species from July 2012, which, in turn, is based on the FIR study from that year [36]. This study employed Watson's A reduced Hamiltonian in the prolate I r representation, as was done in the first version of the c-C 2 H 4 O CDMS catalog entry of March 2000, on which the FIR study built. The same combination of reduction and representation was used in an IR study [38], whereas Pan et al. employed Watson's A reduction in the oblate III l representation using 25 spectroscopic parameters in their fit [35]. The initial CDMS fit, in contrast, used only 14 spectroscopic parameters, and the FIR study as well as the second CDMS fit used only one additional spectroscopic parameter.
We scrutinized the existing line lists in order to approach the best possible spectroscopic parameters as closely as possible. The FIR transition frequencies had uncertainties of 0.00020 cm −1 assigned initially [36]. These uncertainties were judged rather conservatively given their rms value was 0.00011 cm −1 in the fit of the FIR study [36]. In addition, one pair of transitions, J = 64 − 63, K c = 62 − 61 at 61.468827 cm −1 , was omitted from our fit because of a large residual of ∼0.0012 cm −1 between measured and calculated frequencies, and one pair of transitions, J = 61 − 60, K c = 61 − 60 supposedly at 57.435516 cm −1 , was reassigned to a much stronger transition (37 26,12 − 36 25,11 ) much closer to the calculated position. The FIR rms was reduced to 0.00009 cm −1 after these two corrections, and we assigned 0.00010 cm −1 as uncertainties to the FIR transition frequencies. The assignments of four more transition frequencies were extended eventually because more individual transitions contributed substantially to each measured line than was indicated in the initial line list [36]; further details are available in our line list which is part of the Supplementary Material. These corrections, however, had only very small effects on the fit. It is important in this context to emphasize that in the case of two or more overlapping transitions, our fitting program SPFIT only judges the intensity-weighted average position of all overlapping transitions.
One set of MW transition frequencies [34] was reportedly accurate to 0.01 MHz, but several lines displayed larger residuals between calculated and measured frequencies. In fact, the rms of this data set was very close to 0.02 MHz, and we attributed this value to this set of transition frequencies. Uncertainties of 0.05 MHz and 0.15 MHz, respectively, were employed for the remaining datasets [33,35], as was done earlier [36].
Spectroscopic parameters were determined for this as well as various intermediate line lists using the A reduction in the I r representation and employing the S reduction in the III l and I r representations. The A reduction in the III l representation was only used for the final line list as this combination of reduction and representation appeared to require many more spectroscopic parameters than the other combinations. Pickett's SPFIT program [49] was applied for all fitting, and the SPCAT program was utilized for calculations of all rotational spectra.
We recorded individual transitions or pairs thereof if transitions happened to be close in frequency throughout our study. Series in particular combinations of K a and K c were followed in all instances. Asymmetry doublets with significant, but not complete splitting were avoided as their positions are usually determinable with less accuracy compared to well collapsed or well re- The newly recorded transition frequencies deviated by modest amounts from the calculations based on the previous FIR study [36], the largest deviation was 4.32 MHz. The deviations were largest for transitions with high values of J and with K c = J. The rms of the new lines fit with the old parameters was 580 kHz, which corresponded to 38 times the experimental uncertainties on average. The signed deviation was 15 kHz, which means that deviations to higher frequencies were only marginally more important than deviations to lower frequencies.
h 3 × 10 9 −49.529 (4) −6.868 (3) −1.730 (7) −57.773 (50) φ J × 10 9 L K × 10 12 −3.325 (25) −14.450 (79) −0.62 (9) −132.5 (38) L K × 10 12 L KK J ×
We test the need for additional spectroscopic parameters usually by adding one parameter at a time and evaluating which of the new parameters improves the fit the most. This procedure ensures that the parameter set is about as small as possible and that it is fairly unique. The procedure works usually very well for prolate rotors, though correlation may in some cases require sets of two parameters to be tested. Fitting oblate type rotors was often less straightforward. In all combinations of reductions and representations we tested sets of two parameters to fit the c-C 2 H 4 O transition frequencies; we even tested sets of three and four parameters in the case of the A reduction in the III l representation.
The previous FIR study [36] employed 15 parameters using the A reduction in the I r representation; these were a nearly complete set of parameters up to sixth order, except Φ J , plus l K . Our corresponding new fit consists of seven more parameters and is described as a nearly complete set of parameters up to eighth order, except L J and l J , as summarized in Table 1. The weighted rms (or rms error) of the fit is 0.946 with modest scatter for the individual data sets: 1.087 and 1.057 for 19 [34] and 40 microwave lines [33], respectively; 1.000 for 607 millimeter and submillimeter transitions [35]; 0.872 for 1185 FIR transitions [36]; and 0.957 for our 1091 transitions with an rms of 20.3 kHz.
The quality of the fits utilizing the S reduction in the III l or the I r representation were similar, but required two parameters more in most fits; the difference was in some intermediate fits larger, in few smaller. The final spectroscopic parameters are also listed in Table 1. We tested the predictive power of the three parameter sets occasionally and found that there was no persistent preference for either set.
Obtaining a somewhat satisfactory fit applying the A reduction in the III l representation with the final line list was challenging. A complete set of parameters up to the eighth order plus several decic parameters yielded a fit that had an rms error more than 20% worse than any of the other fits; the parameters of this fit are also given in Table 1. We tested all reasonable parameters individually, all or nearly all combinations of two of these along with several combinations of three and even some four parameter combinations. These improved the rms error only marginally, ∼1.12 was the best we were able to achieve. This modest refinement was discarded as it came at the expense of many more spectroscopic parameters.
Observed spectrum and determination of spectroscopic parameters of c-13 CCH 4 O and c-C 2 H 4 18 O
Only 11 and 15 transitions were reported by Creswell and Schwendeman for c-13 CCH 4 O and c-C 2 H 4 18 O, respectively. The uncertainties were reportedly 10 kHz, a value that was used in early fits, but later increased to 20 kHz because of the rms of these data, as in the case of the main isotopolog, see section 3.2. Data reported by Hirose [32] were not considered initially even though slightly more transition frequencies were reported extending to slightly higher quantum numbers. The reasons were the larger uncertainties and in part large residuals between measured and calculated frequencies already in the publication, especially for c-C 2 H 4 18 O. The number of transition frequencies was small for both isotopologs. Therefore, the spectroscopic parameters of the main isotopic species in the A reduction and I r representation from the previous FIR study [36] were taken as starting values for both isotopic species. The rotational parameters were adjusted in a first step. Through several trial fits, δ K and δ J were determined as the parameters which improved the rms error of the c-13 CCH 4 O fit the most. The corresponding parameters for c-C 2 H 4 18 O were also δ K and δ J and additionally ∆ JK . Minor adjustments of some further parameters, as frequently done [25,42,43], improved the fit only slightly.
We started our investigations of the two minor isotopic species in the 340−505 GHz region followed by 158−250 GHz and 758−1092 GHz, as for the main isotopic species. There was no caution required to avoid strong transitions of these isotopologs as their abundances in natural isotopic composition are lower than those of the main species by factors of ∼45 (there are two structural identical C atoms in the molecule) and ∼500, respectively. The quality of the extrapolations to submillimeter wavelengths was unclear because of the limited data set. Moreover, only one R-branch transition frequency was reported for either isotopolog, and this was the J = 1 − 0 transition. Since patterns of two transitions are easier to recognize than individual transitions, we searched for transitions with small, but resolved asymmetry splitting. In the case of c-13 CCH 4 O, these were transitions with K a = J and J = 9, 8, and 7 as well as transitions having K c = J − 4 and J = 14, 13, and 12. Both types of transitions were also searched for in the case of c-C 2 H 4 18 O, but with slightly different values of J because of the very different κ value, see section 3.1, namely 10 to 7 for the nearly prolate paired transitions and 11 to 9 for the nearly oblate paired transitions plus the oblated paired J = 14 in this series. After almost all of these transitions were found to be unblended, the spectroscopic parameters of both isotopologs were updated, and more transitions with similar combinations of quantum numbers were recorded. Subsequently, further series were sought, including large series of Q-branch transitions until there were no more transitions that had enough intensity to be recorded and a calculated uncertainty at least of order of the achievable measurement accuracy.
The As the A reduction in the I r representation required the least number of spectroscopic parameters to fit the transition frequencies of the main isotopic species, we only tried this combination of reduction and representation. Additional parameters for the main isotopolog were subsequently transferred as fixed parameters to the fits of the minor isotopologs. Higher order distortion parameters of these, which were kept fixed in the fit, were adjusted if lower order parameters differed markedly from those of the main isotopic species. The changes were applied by evaluating trends in related parameters, such as ∆ K , Φ K , and L K .
Our new transition frequencies of the two minor isotopic species deviated much more from the initial calculations than in the case of the main isotopolog because of the much smaller initial line lists for both minor isotopologs which resulted in substantially less reliable spectroscopic parameters. The rms of the new lines fit with the initial parameters is 29.9 MHz, the average shift is 17.3 MHz, and the rms corresponds to more than 2500 times the experimental uncertainties on average for c-13 CCH 4 O. In the case of c-C 2 H 4 18 O, that rms is 63.4 MHz, the average shift is −14.5 MHz, and the rms corresponds to almost 3450 times the experimental uncertainties on average.
Spectroscopic parameters derived from the main isotopolog or new spectroscopic parameters were tested as was done for the main isotopic species. That is, we searched for the parameter whose floating or inclusion improved the fit the most in each fitting round. An additional prerequisite to keep a new parameter in the fit or to keep a previously fixed parameter floated was that the parameter was determined sufficiently well, meaning that its uncertainty should be less than one fifth of its magnitude.
In the final fitting round, we tested the transition frequencies reported by Hirose [32]; 15 and 18 lines were included with uncertainties of 0.05 MHz for c-13 CCH 4 O and c-C 2 H 4 18 O, respectively, and three respectively four lines were omitted because of large residuals. The final sets of spectroscopic parameters are given in Table 2.
The weighted rms (or rms errors) of the fits are 0.907 and 0.906 for the isotopologs containing one 13 C and one 18 O, respectively. The rms errors for 11 lines from Creswell and Schwendemann [34], 15 lines from Hirose [32], and 1111 lines from our present study are 0.898, 0.805, and 0.909, respectively for c-13 CCH 4 O; the rms of our lines is 22.1 kHz. The corresponding values for 16 lines from Creswell and Schwendemann [34], 18 lines from Hirose [32], and 667 lines from our present study are 1.056, 0.978, and 0.897, respectively; the rms of our lines is 24.4 kHz.
Discussion
The main interest for our study of the oxirane main isotopolog was the test of various combinations of reduction and representation. We point out that we did not have any program at our disposal to try out the II r or II l representation, but these are very rarely used in published fits anyhow. A rather large set of spectroscopic parameters was needed to obtain a somewhat satisfactory fit in the III l representation using the A reduction, similar to results from a millimeter and submillimeter study [35]. This combination of reduction and representation would be considered by many spectroscopists to be the natural choice as they commonly use the A reduction for traditional reasons and as oxirane is an asymmetric top of the oblate type that is with κ = 0.4093 quite far from the symmetric limit.
Considerably fewer parameters, however, are required in the III l representation using the S reduction, as can be seen in Table 1, emphasizing again the greater versatility of the S reduction compared to the A reduction if the representation is used that fits to the asymmetry type of the molecule. We may, of course, also use a different representation. The S reduction in the I r representation yields a fit of similar quality and with the same number of parameters as the S reduction in the III l representation. It is interesting to note that the A reduction in the I r representation requires two parameters fewer still to produce a fit of about the same quality. Thus, while the S reduction is the prime choice if one only considers the III l representation, the A reduction in the I r representation is preferable if all these four combinations are considered.
The poor performance of the A reduction in an oblate representation (III l and III r differ only in some signs of off-diagonal distortion parameters) is not unique to oxirane, but is at least somewhat more widespread. Yamada and Klee [50] carried out an FIR study on H 2 S and fit their data together with rotational data of microwave accuracy, employing full parameter sets of eighth order and diagonal decic parameters. Although H 2 S is an asymmetric rotor of the oblate type (κ = 0.5234), they encountered convergence problems using the A reduction in the III r representation. A satisfactory fit resulted from a fit in which the S reduction in the same representation was applied. The authors also tried fits in the I r representation. In this case, the A reduction gave a fit that was noticeably worse than the S and III r combination, whereas the S reduction (in I r ) yielded the best fit.
Another example is the lowest energy conformer of 2-cyanobutane, which is a very asymmetric rotor of the oblate type κ = 0.1404 [41]. The combination of A reduction and III l representation required the most parameters; the best result was achieved with the S reduction in the III l representation while both reductions in the I r representation required an intermediate number of parameters.
In a study of SO 2 isotopologs containing one and two 18 O, asymmetric top molecules somewhat close to the prolate limit, Margulès et al. [51] reemphasized Watson's recommendation to favor the S reduction over the A reduction because of the smaller correlation coefficients [52]. The authors added that the S reduction may be a better choice even in cases in which more spectroscopic parameters are needed because of the more favorable condition numbers [51]. The condition number is the ratio of the largest singular value of the Jacobian matrix over the smallest and indicates the degree of ill-conditioning; a large value may indicate that the determined spectroscopic parameters are not reliable. Among the condition numbers of the fits of the main isotopolog, the value is very large, almost 15000, only in the case of the III l representation and the A reduction; the values are much smaller and fairly similar, 150, 91, and 92, for the combinations III l & S, I r & A, and I r & S, respectively. Therefore, our preference of I r & A is supported from the viewpoint of the condition number.
Substitution of one 12 C with 13 C or of one 16 O with 18 O reduces the rotational parameters by modest amount, except for B of the c-C 2 H 4 18 O isotopolog. Substitution of an atom on the symmetry axis does not affect the equilibrium rotational parameter in the Born-Oppenheimer approximation. The corresponding ground state rotational parameter of a heavy-atom substitution is frequently slightly larger because of effects of anharmonicity, as is observed in the case of oxirane. The quartic distortion parameters of both minor isotopic species are quite close to those of the main species, those of the 13 C species more so than those of the 18 O, which is commensurate with the changes in rotational parameters. The situation is more complex among the higher order parameters, in part possibly because fewer parameters were varied as a consequence of the smaller data sets extended to lower quantum numbers. Some parameters of c-13 CCH 4 O display large changes compared to the main isotopolog, possibly as a consequence of the different symmetry. The Φ J value of the main species appears to be accidentally very small in magnitude compared with almost all of the remaining sextic distortion parameters. The Φ J value of c-13 CCH 4 O is larger in magnitude and has an opposite sign; it was even necessary to fit the related L J for this isotopolog. On the other hand, φ J is much smaller in magnitude than that of the main species. The values of some of the octic distortion parameters will depend on the values of parameters that were kept fixed in the fit.
We were interested if c-13 CCH 4 O, as the more promising of the two minor isotopologs, could be found in existing astronomical data. The 12 C/ 13 C ratio is particularly favorable in the Galactic center. Among its sources, Sagittarius (Sgr) B2 is among the most promi-nent sources in the whole Galaxy. A search toward Sgr B2(N1S) and Sgr B2(N2) in the ReMoCA data, a molecular line survey at 3 mm with the Atacama Large Millimeter Array [29], was unsuccessful. Assuming a 12 C/ 13 C ratio of 20 [9,10,11], all lines of c-13 CCH 4 O were heavily blended with other, stronger lines or did not exceed the 3σ average noise level of the survey (A. Belloche, private communication, 2021).
Conclusion
We have obtained greatly improved spectroscopic parameters for the c-13 CCH 4 O and c-C 2 H 4 18 O isotopologs which should be sufficiently accurate for all radio astronomical observations. The investigations into the performance of various combinations of representations and reductions in fitting an extended data set of the oxirane main isotopolog revealed once more the poor performance of the A reduction in an oblate representation, suggesting that these combinations should be avoided. In the more general case of oblate-as well as prolatetype asymmetric rotors we conclude that the S reduction should be preferred for fits employing a natural choice of representation. Other choices are only recommended if all common combinations of representation and reduction were tested.
Concerning c-13 CCH 4 O in space, it appears as if dedicated searches are required to detect this isotopolog in the interstellar medium.
Note added in proof
After acceptance of our manuscript, we became aware of an interesting article on dimethylsulfoxide [53] that also discusses reductions and representations. The findings are in agreement with the findings in publications discussed in our article.
CRediT authorship contribution statement
Figure 1 :
1Sections of the rotational spectrum of oxirane displaying the presence of 3 : 5 para to ortho spin-statistics for c-C 2 H 4 O (a) and c-C 2 H 4 18 O (c) as well as the absence in c-13 CCH 4 O (b).
a
Numbers in parentheses are one standard deviation in units of the least significant figures. b Dimensionless. c See Section 3.2 for additional details on the numbers of transitions and the maximum quantum numbers. d Weighted standard deviation of the fit. solved asymmetry doublets. The strongest transitions of the main isotopolog were avoided because their uncertainties were already small such that their impact in the fit would be small. Moreover, these transitions are affected by opacity issues. The 340−506 GHz region was studied extensively with 842 transitions in the final line list. Additional measurements were made between 160−250 GHz and 764−1046 GHz, where 61 and 188 further transitions, respectively, were recorded and retained in the final line list. The number of different frequencies is 727; 728 transitions represent unresolved asymmetry doublets. The maximum quantum numbers J, K a , and K c of our new transition frequencies are 55, 43, and 52, respectively.
final line lists consisted of 553 and 274 transitions between 340−505 GHz, 211 and 186 transitions between 158−250 GHz, and 341 and 207 transitions between 758−1092 GHz for the isotopologs containing one 13 C and one 18 O, respectively. 680 of the 1111 and 468 of the 667 transitions correspond to unresolved asymmetry doublets for c-13 CCH 4 O and c-C 2 H 4 18 O, respectively, resulting in 771 and 433 different transition frequencies. The maximum quantum numbers J, K a , and K c of our new transition frequencies are 45, 34, and 39 for c-13 CCH 4 O and 38, 30, and 38 for c-C 2 H 4 18 O.
a
Watson's A reduction was used in the I r representation. Numbers in parentheses are one standard deviation in units of the least significant figures. Spectroscopic parameters without uncertainties were evaluated from the main isotopic species and kept fixed in the fits, see section 3.3. b Dimensionless. c See Section 3.3 for additional details on the numbers of transitions and the maximum quantum numbers. d Weighted standard deviation of the fit.
Holger S.P. Müller: Conceptualization, Investigation, Methodology, Formal analysis, Validation, Data curation, Writing − Original Draft, Writing − review & editing. Jean-Claude Guillemin: Resources, Writing − review & editing. Frank Lewen: Resources, Writing − review & editing. Stephan Schlemmer: Funding acquisition, Resources, Writing − review & editing.
Table 1 :
1Spectroscopic parameters a (MHz) of the main isotopolog of oxirane employing Watson's S and A reduction of the rotational Hamiltonian both in the oblate representation III l and in the prolate representation I r .S reduction
A reduction
Parameter
III l
I r
I r
III l
Parameter
A
25483.88987 (11) 25483.88751 (10)
25483.86273 (9)
25483.88386 (10) A
B
22120.83638 (9)
22120.82714 (9)
22120.87303 (7)
22120.84394 (9)
B
C
14097.83590 (11) 14097.85210 (11)
14097.82581 (8)
14097.83400 (16) C
D K × 10 3
26.07490 (13)
2.95932 (23)
27.58930 (22)
27.63421 (33) ∆ K × 10 3
D JK × 10 3
−68.65174 (30)
50.46597 (26)
20.91041 (21)
−70.50171 (68) ∆ JK × 10 3
D J × 10 3
50.83976 (24)
15.75678 (24)
20.68240 (14)
51.14486 (26) ∆ J × 10 6
d 1 × 10 3
9.02067 (5)
−6.21001 (6)
18.10824 (9)
3.39130 (43) δ K × 10 3
d 2 × 10 3
−0.15233 (2)
−2.46284 (1)
6.21005 (2)
−9.01957 (5)
δ J × 10 3
H K × 10 9
−191.41 (11)
2712.32 (23)
2.2415 (2)
−10.8512 (10)
Φ K × 10 6
H K J × 10 9
373.61 (21)
−3535.95 (28)
−2.6387 (3)
16.0605 (16)
Φ K J × 10 6
H JK × 10 9
−198.52 (24)
1096.22 (23)
624.58 (18)
−5321.27 (97)
Φ JK × 10 9
H J × 10 9
17.92 (19)
−46.58 (18)
−1.797 (69)
110.91 (22)
Φ J × 10 9
h 1 × 10 9
−8.211 (44)
5.268 (45)
−0.3278 (1)
−18.0197 (18)
φ K × 10 6
h 2 × 10 9
46.364 (24)
22.470 (14)
277.30 (10)
1546.63 (67)
φ JK × 10 9
Table 2 :
2Spectroscopic parameters a (MHz) of the oxirane isotopologs containing one 13 C or one18 O.Parameter
c-13 CCH 4 O
c-C 2 H 4
18 O
A
25291.99778 (17) 23992.43450 (20)
B
21597.97459 (13) 22121.17197 (14)
C
13825.77925 (14) 13628.24006 (9)
∆ K × 10 3
27.62104 (47)
24.65010 (70)
∆ JK × 10 3
20.47198 (31)
17.45625 (69)
∆ J × 10 6
20.03716 (31)
20.48033 (8)
δ K × 10 3
18.29406 (30)
16.34518 (30)
δ J × 10 3
6.05179 (5)
6.31363 (4)
Φ K × 10 6
1.9510 (4)
2.0748 (8)
Φ K J × 10 6
−2.2614 (6)
Declaration of competing interestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgmentsIt is our great pleasure to dedicate this work to the memory of J.K.G. Watson for his many and invaluable contributions to the theory of high-resolution spectroscopy. We will also keep him in mind for his great sense of humor and for his Scottish accent.We thank Arnaud Belloche for communicating results of his search for c-13Appendix A. Supplementary MaterialSupplementary data associated with this article can be found, in the online version, at https://doi.org/101016/j.jms. ...
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Extreme 13 C, 15 N and 17 O isotopic enrichment in the young planetary nebula K4-47. D R Schmidt, N J Woolf, T J Zega, L M Ziurys, 10.1038/s41586-018-0763-1Nature. 5647736D. R. Schmidt, N. J. Woolf, T. J. Zega, L. M. Ziurys, Ex- treme 13 C, 15 N and 17 O isotopic enrichment in the young plan- etary nebula K4-47, Nature 564 (7736) (2018) 378-381. doi: 10.1038/s41586-018-0763-1.
Spectroscopic study and astronomical detection of doubly 13 C-substituted ethyl cyanide. L Margulès, A Belloche, H S P Müller, R A Motiyenko, J C Guillemin, R T Garrod, K M Menten, 10.1051/0004-6361/201628309arXiv:1604.03256Astron. Astrophys. 59093L. Margulès, A. Belloche, H. S. P. Müller, R. A. Motiyenko, J. C. Guillemin, R. T. Garrod, K. M. Menten, Spectroscopic study and astronomical detection of doubly 13 C-substituted ethyl cyanide, Astron. Astrophys. 590 (2016) A93. arXiv: 1604.03256, doi:10.1051/0004-6361/201628309.
The ALMA-PILS survey: isotopic composition of oxygen-containing complex organic molecules toward IRAS 16293-2422B. J K Jørgensen, H S P Müller, H Calcutt, A Coutens, M N Drozdovskaya, K I Öberg, M V Persson, V Taquet, E F Van Dishoeck, S F Wampfler, 10.1051/0004-6361/201731667arXiv:1808.08753doi:10.1051/ 0004-6361/201731667Astron. Astrophys. 620170J. K. Jørgensen, H. S. P. Müller, H. Calcutt, A. Coutens, M. N. Drozdovskaya, K. I.Öberg, M. V. Persson, V. Taquet, E. F. van Dishoeck, S. F. Wampfler, The ALMA-PILS sur- vey: isotopic composition of oxygen-containing complex or- ganic molecules toward IRAS 16293-2422B, Astron. Astro- phys. 620 (2018) A170. arXiv:1808.08753, doi:10.1051/ 0004-6361/201731667.
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Laboratory spectroscopic study of isotopic thioformaldehyde, H 2 CS, and determination of its equilibrium structure. H S P Müller, A Maeda, S Thorwirth, F Lewen, S Schlemmer, I R Medvedev, M Winnewisser, F C De Lucia, E Herbst, 10.1051/0004-6361/201834517arXiv:1812.01554doi:10. 1051/0004-6361/201834517Astron. Astrophys. 621143H. S. P. Müller, A. Maeda, S. Thorwirth, F. Lewen, S. Schlem- mer, I. R. Medvedev, M. Winnewisser, F. C. De Lucia, E. Herbst, Laboratory spectroscopic study of isotopic thioformaldehyde, H 2 CS, and determination of its equilibrium structure, Astron. Astrophys. 621 (2019) A143. arXiv:1812.01554, doi:10. 1051/0004-6361/201834517.
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[
"Structure and magnetism of S = 1/2 kagome antiferromagnets NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2",
"Structure and magnetism of S = 1/2 kagome antiferromagnets NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2"
] |
[
"Yue-Sheng Li \nDepartment of Physics\nRenmin University of China\n100872BeijingP. R. China\n",
"Qing-Ming Zhang \nDepartment of Physics\nRenmin University of China\n100872BeijingP. R. China\n"
] |
[
"Department of Physics\nRenmin University of China\n100872BeijingP. R. China",
"Department of Physics\nRenmin University of China\n100872BeijingP. R. China"
] |
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We have successfully synthesized S = 1/2 kagome antiferromagnets MCu 3 (OH) 6 Cl 2 (M = Ni and Co) by hydrothermal method with rotating pressure vessel. Structural characterization shows that both compounds have the similar crystal structure as ZnCu 3 (OH) 6 Cl 2 with R-3m symmetry. Similar to ZnCu 3 (OH) 6 Cl 2 , the compounds show no obvious hysteresis at 2 K. A spin glass transition is found in both NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 at low temperatures (6.0 and 3.5 K respectively) by AC susceptibility measurements. It indicates no long-range magnetic order and a strong spin frustration. The substitution of Zn 2+ by magnetic ions Ni 2+ or Co 2+ , effectively enhances interlayer exchange coupling and changes the ground state of the kagome spin system.
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10.1088/0953-8984/25/2/026003
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"https://arxiv.org/pdf/1112.4117v3.pdf"
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1112.4117
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81e6760b80223eae0f79de5be5e07e8ef313816b
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Structure and magnetism of S = 1/2 kagome antiferromagnets NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2
Yue-Sheng Li
Department of Physics
Renmin University of China
100872BeijingP. R. China
Qing-Ming Zhang
Department of Physics
Renmin University of China
100872BeijingP. R. China
Structure and magnetism of S = 1/2 kagome antiferromagnets NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2
2 Corresponding author.spin frustrationkagome antiferromagnetspin glass
We have successfully synthesized S = 1/2 kagome antiferromagnets MCu 3 (OH) 6 Cl 2 (M = Ni and Co) by hydrothermal method with rotating pressure vessel. Structural characterization shows that both compounds have the similar crystal structure as ZnCu 3 (OH) 6 Cl 2 with R-3m symmetry. Similar to ZnCu 3 (OH) 6 Cl 2 , the compounds show no obvious hysteresis at 2 K. A spin glass transition is found in both NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 at low temperatures (6.0 and 3.5 K respectively) by AC susceptibility measurements. It indicates no long-range magnetic order and a strong spin frustration. The substitution of Zn 2+ by magnetic ions Ni 2+ or Co 2+ , effectively enhances interlayer exchange coupling and changes the ground state of the kagome spin system.
Introduction
In a spin frustrated system, a spin cannot find a proper orientation to favor all the interactions with its neighboring spins. The frustration is caused either by competing interactions or by spin geometric configuration with antiferromagnetic (AF) nearest-neighbor interaction. Many interesting and unexpected consequences caused by spin frustration, are not well understood so far [1]. Among spin frustrated systems, low-dimensional S = 1/2 antiferromagnet with highly geometric frustration has attracted particular attention [2]. For this kind of antiferromagnets, novel quantum states such as resonating-valence-bond (RVB) and "spin-liquid" ground state have been proposed. The novel concepts are considered to have an intimate connection with high-temperature superconductivity in cuprates [3]. Structurally perfect S = 1/2 kagome compounds are rare. S = 1/2 kagome antiferromagnet ZnCu 3 (OH) 6 Cl 2 was successfully synthesized, which has been regarded as the first "structure-perfect" S = 1/2 frustrated compound in some aspects [4][5][6]. Recently synthesized β-Vesignieite BaCu 3 V 2 O 8 (OH) 2 was reported to be structurally perfect and exhibits long-range order at 9 K due to Dzyaloshinki-Moriya (DM) interaction [7]. In ZnCu 3 (OH) 6 Cl 2 , kagome layers are separated by non-magnetic and non-Jahn-Teller-active Zn 2+ ions, which predominantly occupy interlayer triangular sites. Structurally perfect kagome layers with R-3m space group are determined to be preserved even down to 2 K [8]. No magnetic transition was observed at least down to 50 mK and the magnetism was considered as quantum paramagnetic state at low temperature [9,10], which is in accord with many theoretical results using approximations and numerical simulations. On the other hand, there are many debates on the occupancy of Zn in kagome planes in ZnCu 3 (OH) 6 Cl 2 [11][12][13][14]. Non-magnetic defects in kagome planes can partially release the geometric frustration while Cu 2+ ions entering into interlayer triangular sites bring interlayer magnetic coupling. These will substantially affect spin ground state. The question is what we can expect for the kagome system with a stronger interlayer magnetic coupling.
In this paper, we performed a comparative study on the structures and magnetic properties of ZnCu 3 (OH) 6 Cl 2 and herbertsmithite-like kagome antiferrromagnets MCu 3 (OH) 6 Cl 2 (M = Ni and Co). Ni 2+ and Co 2+ are non-Jahn-Teller-active magnetic ions with S = 1 and 3/2 respectively and prefer interlayer sites [4]. It is expected that the two magnetic ions may introduce novel interactions and spin states which may open a new window for exploring kagome antiferromagnets. In fact, several groups have tried to synthesize NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 [15][16][17]. Unfortunately the attempts with a conventional static hydrothermal method failed or CuO impurities cannot be removed completely.
By employing hydrothermal method with rotating pressure vessel, so-called dynamic hydrothermal method, we have successfully synthesized NiCu 3 (OH) 6
Materials and methods
ZnCu 3 (OH) 6 Cl 2 , NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 powder samples were synthesized under dynamic and hydrothermal conditions provided by a homogeneous reactor. A 30 mL teflon liner was charged with 498 mg of CuCO 3 · Cu(OH) 2 · xH 2 O(4.5 mmol Cu), 10 mL of water, and 307 mg (1: 2.25 mmol Zn) of ZnCl 2 , 535 mg (2: 2.25 mmol Ni) NiCl 2 · 6H 2 O, or 535 mg (3: 2.25 mmol Co) CoCl 2 · 6H 2 O. The liner was capped and placed into a stainless steel pressure vessel. The pressure vessel was fixed to the ratation axis of a homogeneous reactor. The rotation rate of pressure vessel was fixed to be 35 r/min all the time. The vessel was heated to 205 o C at a rate of 1 o C/min, and its temperature was maintained for 48 hr. and then cooled down to room temperature at a rate of 0.1 o C/min. A blue-green polycrystalline powder was found at the bottom of each vessel, isolated from the liner by filtration, washed with water, and dried over by a loft drier (at 70 o C). By the procedure, 610 mg of ZnCu 3 (OH) 6 Cl 2 , 590 mg of NiCu 3 (OH) 6 Cl 2 and 610 mg of CoCu 3 (OH) 6 Cl 2 were obtianed, which correspond to the yields of 95%, 93% and 96% respectively, with respect to the starting material CuCO 3 · Cu(OH) 2 · xH 2 O. The product does not include any agglomerate pieces, which suggests a homogenous and complete reaction process. We compared ZnCu 3 (OH) 6 Cl 2 samples synthesized by dynamic and static method. They have a very close quality.
Magnetization measurements demonstarte that the sample synthesized by dynamic method has a slightly lower occupancy of Cu at interlayer sites than that by static method (~ 17% vs. ~ 21%). X-ray diffraction measurements were carried on a Shimadzu XRD-7000 diffractometer using Cu K α radiation (λ = 1.5403 Å). The diffraction data were processed and fit using Rietveld techniques with the GSAS program [20] using the same crystal structure model as ZnCu 3 (OH) 6 Cl 2 (R-3m) [4]. Inductively coupled plasma (ICP) measurements were made with a HORIBA Jobin Yvon Ultima 2 ICP system. The measured ratio of M:Cu is very close to 1:3 in all the three samples (1.03 (4)
Results and discussion
X-ray powder diffraction patterns and Rietveld refinements are shown in figure 2. No additional peak is seen, which implies that CuO and other impurity phases are negligible in the samples. The refinement results suggest that the crystal grains in powder samples are well crystallized. The grain size is estimated to be ~200 nm for the three samples from full width at half maximum (FWHM) [20]. Due to the chemical similarity among Zn 2+ , Ni 2+ and Co 2+ , the structural parameters are very close (table 1), as well as the site exchange. This is confirmed by very similar diffraction patterns, relative intensities and FWHMs of the three samples. The crystal symmetry of R-3m is kept in both NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 , indicating the occupation fraction of Cu at triangular sites is smaller than a critical value. For example, the symmetry change from rhombohedral (R-3m, x > 0.33) to monoclinic (P 21/n, x < 0.33) was observed in Zn x Cu 4-x (OH) 6 Cl 2 [4]. (11) 96.48(9) a Considering the insignificant scattering contribution of proton, the positional parameters of proton are fixed during the refinements according to Ref [4].
The temperature dependence of magnetization (or DC susceptibility) from 2 to 300 K with an applied field of 2000 Oe, is shown in figure 3 (a). All the magnetization curves exhibit a Curie-like tail at low temperatures (7 K ~ 100 K), which are attributed to the magnetic ions (or defects) at interlayer sites [21,22]. CoCu 3 (OH) 6 Cl 2 shows the largest magnetization in the whole temperature range. And the magnetization of NiCu 3 (OH) 6 Cl 2 and ZnCu 3 (OH) 6 Cl 2 decreases gradually. This can be understood through spin moments of Co 2+ (S = 3/2) and Ni 2+ (S = 1). The inset of figure 3(a) shows the Curie-Weiss fitting at low temperatures, the fitting constants are c ~ 0.520(2) Kcm 3 /mol Cu; θ ~ -12.17(2) K and c ~ 0.991(3) Kcm 3 /mol Cu; θ ~ -9.19(4) K for NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 , respectively. Supposing a full occupation of interlayer sites by Ni or Co, the Curie constant c gives effective spin moment μ eff ~ 3.5μ B for Ni 2+ and 4.9μ B for Co 2+ . The effective moments are consistent with the reported ones [23]. Both the Curie constant c and the Weiss temperature θ are much larger than those of ZnCu 3 (OH) 6 Cl 2 (c ~ 0.0287(1) Kcm 3 /mol Cu; θ ~ -1.08(1) K) [21]. This suggests a much stronger interlayer coupling in NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 . The interlayer coupling will substantially affect magnetism at low temperatures (T < |θ|) in the two compounds. Figure 3(b) shows Curie-Weiss fit at high temperatures (200 ~ 300 K). The Weiss temperatures given by high-T mean field fitting exhibit a monotonic evolution with effective moments of interlayer ions. And the Weiss temperatures of NiCu 3 (OH) 6 Cl 2 (-100 K) and CoCu 3 (OH) 6 Cl 2 (-40 K) are much lower than that of ZnCu 3 (OH) 6 Cl 2 (-380 K). The high-temperature susceptibilities contain the contributions from both interlayer and kagome ions. So the Weiss temperatures are no longer a good measure for interlayer or intralayer coupling. Interestingly, the monotonic decrease of the Weiss temperatures was also observed in Zn x Cu 4-x (OH) 6 Cl 2 and Mg x Cu 4-x (OH) 6 Cl 2 when increasing the interlayer magnetic ions Cu 2+ [5,24]. The Weiss temperatures given by high-temperature susceptibilities may be a rough indicator of paramagnetic-like magnetizations of interlayer magnetic ions. High-T fitting also gives the Curie constants of 0.79, 0.87 and 1.28 Kcm 3 /mol Cu for ZnCu 3 (OH) 6 Cl 2 , NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 , respectively. Approximately they are consistent with spin quantum numbers, g-factors, and the composition of the Cu, Ni and Co ions. As mentioned above, high-temperature susceptibilities are contributed by both interlayer and kagome magnetic ions. It is difficult to separate the total bulk susceptibilities into the two parts. So a precise consistency is limited. Most interestingly, a deviation from Curie-Weiss law ("kink") is clearly observed in the inset of figure 3(a) in MCu 3 (OH) 6 Cl 2 (M = Ni and Co). In order to get deeper insight into this kink, we further compare susceptibilities under ZFC and FC from 2 to 10 K, as shown in figure 4. For ZnCu 3 (OH) 6 Cl 2 , there is no difference between FC and ZFC in the whole temperature range, which is consistent with the reported results and reflects a quantum paramagnetic state [6]. While an obvious splitting develops below a characteristic temperature T s ~ 6.0 K and 3.5 K for NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 , respectively. These exactly correspond to the kinks in figure 3(a). Surprisingly, the transition temperatures vary little compared with Clinoatacamite (Cu 2 (OH) 3 Cl) [5,26,27]. We notice that clinoatacamite Cu 2 (OH) 3 Cl, NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 , have almost identical ∠MOCu angles (M refers to interlayer magnetic ions), which are very close to the critical angle of 95 degrees.
We think this may cause an interlayer coupling at the same level of ~10 K in the three compounds and hence similar transition temperatures. For Cu 2 (OH) 3 Cl it is more complicated because the slight distortions of kagome planes by interlayer Jahn-Teller-active Cu 2+ ions need to be taken into account. While for non-Jahn-Teller-active Ni 2+ and Co 2+ , perfect kagome planes are preserved. So it can be safely said that the transition is driven by interlayer magnetic coupling considering that no magnetic transtion was observed in herbertsmithite above 2 K. Moreover, one can see that the kink temperature under ZFC is a little higher than that under FC in CoCu 3 (OH) 6 Cl 2 , as shown in figure 4. It is a hint for spin-glass behaviour and we will come back to this point later. M-H measurements are shown in figure 5. No obvious hysteresis can be seen at 2 K in all the samples, which excludes the possibility that ferromagnetic components exist in the compounds. This is clearly different from the Mg-and Zn-compounds with smaller x [17,24,25], in which a Cu 2 (OD) 3 Cl-like distortion around interlayer Cu 2+ may develop at low temperatures due to Jahn-Teller effect. And it is reported that a ferromagnetic transition occurs at ~ 6 K in Cu 2 (OD) 3 Cl [26,27,28]. These confirm that interlayer sites are dominantly occupied by non-Jahn-Teller-active magnetic ions M 2+ (M = Zn, Ni and Co) in these three samples. The low-temperature kink/splitting is a prominent feature of the Ni-and Co-compounds.
The most possible origin is AF ordering or spin-glass transition. In figure 6, we present AC susceptibilities in CoCu 3 (OH) 6 Cl 2 and NiCu 3 (OH) 6 Cl 2 . For CoCu 3 (OH) 6 Cl 2 , the kink/splitting position shifts to a higher temperature with increasing measurement frequencies, which is a typical spin-glass behaviour and rule out the possibility of AF ordering. The shift in NiCu 3 (OH) 6 Cl 2 is a little obscure but still visible. At present we do not exactly know which kind of spins participate in spin-glass transition:
interlayer or in-plane or even both spins? However it is certain that spin-glass state in NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 reflects a subtle change of spin configuration in kagome layers, which is ultimately connected with interlayer magnetic spins in some way. It is plausible that the spin-glass transition may occur on interlayer triangular sites. But a direct interaction between interlayer spins seems less possible because the distance between interlayer spins is very large and there is no anion to bridge the exchange interaction between them. The chemical disorder may not be the driving force of spin-glass transition in this spin system. In fact, spin-glass transition has been reported in S = 5/2 kagome antiferromagnet hydronium jarosite (H 3 O)Fe 3 (SO 4 ) 2 (OH) 6 [29], in which it was argued that planar anisotropy rather than chemical disorder drives spin-glass transition [30]. Further theoretical and experimental efforts are highly required to understand the novel spin-glass state in NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 .
Conclusions
In conclusion, we have successfully synthesized single-phased kagome compounds NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 . X-ray diffraction refinements reveal that the compounds have a crystal structure similar to that of ZnCu 3 (OH) 6 Cl 2 . Magnetic measurements show that a kink or FC/ZFC splitting appears at several Kelvins in NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 .
No obvious magnetization hysteresis loop accompanies the kink. Further AC susceptibility measurements demonstrate that the kink position shifts to higher temperatures with increasing measurement frequencies, which is a typical spin-glass behaviour. On one hand it means no long-range magnetic order develops down to 2 K, which suggests that NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 are still good candidates for studying strong spin frustration. On the other hand, the substitution of Zn by magnetic ions effectively enhances interlayer coupling in the kagome systems and brings out new physics. The kagome compounds and the novel spin-glass state in them will inspire further experimental and theoretical efforts in the future.
ray diffraction patterns and Rietveld refinements show that the compounds are well single-phased without any sign of CuO and other impurities. Structural characterizations reveal that the crystal structures of both compounds are similar to that of ZnCu 3 (OH) 6 Cl 2 . The interlayer triangular sites are dominantly occupied by M 2+ ions and kagome sites by Cu 2+ ions (figure 1). As the bond angle ∠MOCu is close to the critical value of 95 o [18,19] (M=Ni, Co), the exchange interaction between interlayer Ni 2+ /Co 2+ and nearest-neighbor Cu 2+ in kagome planes falls in the critical region of AF and ferromagnetic (FM) coupling and is much smaller than in-plane Cu-Cu AF exchange interactions. In this sense, the compounds NiCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 still preserve the essences of 2D kagome antiferromagnets in both structure and magnetism, and will play an important and unique role in exploring the effects of spin frustration.
Figure 1 .
1The crystal structure of MCu 3 (OH)6 Cl 2 (M = Zn, Ni or Co).
Figure 2 .
2Powder X-ray diffraction Rietveld refinements in (a), (b) and (c) for NiCu 3 (OH)
Figure 3 .
3(a) Temperature dependence of magnetization under an applied field of 2000 Oe for the three samples. The inset shows the data in the low temperature range (2 ~ 13 K) and the dashed lines are extrapolated from Curie-like tails above magnetic freezing temperature (T > T s ). (b) Fitting high-temperature data (200 ~ 300 K) to Curie-Weiss law.
Figure 4 .
4(a) Temperature dependence of DC susceptibility under zero-field-cooling (ZFC) and field-cooling (FC) condition (H cooling = H measure = 100 Oe). The colored arrows mark the kink/peak temperatures (T p (FC) and T p (ZFC)) for CoCu 3 (OH) 6 Cl 2 .
Figure 5 .
5Magnetic field dependence of magnetization measured at 2 K. For ZnCu 3 (OH) 6 Cl 2 and CoCu 3 (OH) 6 Cl 2 , intact loops were measured at a constant rate of 100 Oe/sec. For NiCu 3 (OH) 6 Cl 2 , the applied field was first increased to 14 T, then decreased to 0 T at a constant rate of 50 Oe/sec.
Figure 6 .
6AC susceptibilities are measured under zero applied field. (a) and (b) The real and imaginary parts of AC susceptibilities of NiCu 3 (OH) 6 Cl 2 . The dashed lines mark the corresponding peak temperatures, 2.9, 3.0 and 3.2 K for 13 Hz, 1333Hz and 9333 Hz respectively; (c) and (d) The real and imaginary parts of AC susceptibilities of CoCu 3 (OH) 6 Cl 2 .
:2.97 in ZnCu 3 (OH) 6 Cl 2 ; 1.02(4):2.98 in NiCu 3 (OH) 6 Cl 2 ; 1.06(4):2.94 in CoCu 3 (OH) 6 Cl 2 ). Magnetic measurements were made with a Physical Property Measurement System (PPMS) by Quantum Design.
Table 1 .
1X-ray refinement results a
MCu 3 (OH) 6 Cl 2 (R -3 m)
M = Ni
M = Co
M = Zn
a (Å)
6.8505(8)
6.8416(6)
6.8418(4)
c (Å)
13.9288(18)
14.0934(14)
14.0954(8)
M
Uiso*100
0.36(11)
0.57(11)
1.77(7)
Cu
Uiso*100
1.46(7)
1.79(7)
1.56(5)
O
x = -y
0.20421(30)
0.20578(30)
0.20542(23)
z
0.06475(25)
0.06286(26)
0.06066(20)
Uiso*100
1.26(19)
1.27(19)
1.35(15)
Cl
z
0.19411(21)
0.19448(22)
0.19513(17)
Uiso*100
0.78(12)
1.43(12)
1.70(9)
M-O (Å)
2.0887(33)
2.1035(33)
2.1285(26)
Cu-O (Å)
2.0104(20)
1.9963(20)
1.9839(15)
Cu-Cl (Å)
2.7697(21)
2.7803(22)
2.7741(17)
∠Cu-O-Cu (deg)
116.83(19)
117.92(19)
119.12(15)
∠M-O-Cu (deg)
96.12(11)
96.90
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[
"Young pre-Low-Mass X-ray Binaries in the propeller phase Nature of the 6.7-hour periodic X-ray source 1E 161348-5055 in RCW 103",
"Young pre-Low-Mass X-ray Binaries in the propeller phase Nature of the 6.7-hour periodic X-ray source 1E 161348-5055 in RCW 103"
] |
[
"Harshal Bhadkamkar [email protected] \nDepartment of Astronomy & Astrophysics\nTata Institute of Fundamental Research\n400 005MumbaiIndia\n",
"Pranab Ghosh [email protected] \nDepartment of Astronomy & Astrophysics\nTata Institute of Fundamental Research\n400 005MumbaiIndia\n"
] |
[
"Department of Astronomy & Astrophysics\nTata Institute of Fundamental Research\n400 005MumbaiIndia",
"Department of Astronomy & Astrophysics\nTata Institute of Fundamental Research\n400 005MumbaiIndia"
] |
[] |
Context. Discovery of the 6.7-hour periodicity in the X-ray source 1E 161348-5055 in RCW 103 has led to investigations of the nature of this periodicity. Aims. To explore a model for 1E 161348-5055 wherein a fast-spinning neutron star with a magnetic field ∼ 10 12 G in a young pre-Low-Mass X-ray Binary (pre-LMXB) with an eccentric orbit of period 6.7 hr operates in the "propeller" phase. Methods. The 6.7-hr light curve of 1E 161348-5055 is modeled in terms of orbitally-modulated mass transfer through a viscous accretion disk and subsequent propeller emission. Formation of eccentric binaries in supernovae and their subsequent tidal evolution are studied. Results. The light curve of 1E 161348-5055 can be quantitatively accounted for by models of propeller torques of both Illarionov-Sunyaev type and Romanova-Lovelace et al. type, and spectral and other properties are also in agreement. Formation and evolution of model systems are shown to be in accordance both with standard theories and with X-ray observations of 1E 161348-5055. Conclusions. The pre-LMXB model for 1E 161348-5055 and similar sources agrees with observation. Distinguishing features between this model and the recently-proposed magnetar model need to be explored.
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10.1051/0004-6361/200912552
|
[
"https://arxiv.org/pdf/0909.0159v1.pdf"
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Young pre-Low-Mass X-ray Binaries in the propeller phase Nature of the 6.7-hour periodic X-ray source 1E 161348-5055 in RCW 103
1 Sep 2009 September 1, 2009
Harshal Bhadkamkar [email protected]
Department of Astronomy & Astrophysics
Tata Institute of Fundamental Research
400 005MumbaiIndia
Pranab Ghosh [email protected]
Department of Astronomy & Astrophysics
Tata Institute of Fundamental Research
400 005MumbaiIndia
Young pre-Low-Mass X-ray Binaries in the propeller phase Nature of the 6.7-hour periodic X-ray source 1E 161348-5055 in RCW 103
1 Sep 2009 September 1, 2009Received; acceptedAstronomy & Astrophysics manuscript no. 12552X-rays: binaries -Stars: neutron -Stars: evolution -Accretion, accretion disks -ISM: supernova remnants -X-rays: general
Context. Discovery of the 6.7-hour periodicity in the X-ray source 1E 161348-5055 in RCW 103 has led to investigations of the nature of this periodicity. Aims. To explore a model for 1E 161348-5055 wherein a fast-spinning neutron star with a magnetic field ∼ 10 12 G in a young pre-Low-Mass X-ray Binary (pre-LMXB) with an eccentric orbit of period 6.7 hr operates in the "propeller" phase. Methods. The 6.7-hr light curve of 1E 161348-5055 is modeled in terms of orbitally-modulated mass transfer through a viscous accretion disk and subsequent propeller emission. Formation of eccentric binaries in supernovae and their subsequent tidal evolution are studied. Results. The light curve of 1E 161348-5055 can be quantitatively accounted for by models of propeller torques of both Illarionov-Sunyaev type and Romanova-Lovelace et al. type, and spectral and other properties are also in agreement. Formation and evolution of model systems are shown to be in accordance both with standard theories and with X-ray observations of 1E 161348-5055. Conclusions. The pre-LMXB model for 1E 161348-5055 and similar sources agrees with observation. Distinguishing features between this model and the recently-proposed magnetar model need to be explored.
Introduction
The point soft X-ray source 1E 161348-5055 (henceforth 1E) near the center of the young (∼ 2000 yr old) supernova remnant (SNR) RCW 103 has attracted much attention lately, following the discovery of a strong 6.67 hr periodic modulation in 1E by de Luca et al. (2006, henceforth dL06) from a deep XMM-Newton observation of the source in 2005. 1E was discovered in 1980 (Touhy & Garmire 1980) as a soft Einstein X-ray source. The original interpretation as an isolated neutron star was found to be untenable in view of subsequent discovery by other X-ray satellites (e.g., ROSAT, ASCA, Chandra) of the large variability of 1E on the timesacle of a few years (dL06). A periodicity at ∼ 6 hr was first hinted at by Chandra observations, but the first clear, strong detection came from the above 2005 observations of dL06, who also showed the existence of this periodicity in the data from earlier 2001 observations of 1E with XMM-Newton, when the source luminosity was higher by a factor ∼ 6 during the course of its sequence of several-year timescale outbursts nentioned above, documented by these authors from archival data.
The nature of the above 6.67 hr periodicity is an interesting question, on which preliminary discussions were reported in dL06. Recently, Pizzolato et al. (2008, henceforth P08) have proposed a model for the 1E system wherein it is a close binary consisting of a magnetar, i.e., a neutron star with a superstrong magnetic field ∼ 10 15 G, and a low-mass companion. The 6.67 hr periodicity is identified in this model with the spin period of the neutron star, to which this young neutron star has been spun down in such a short time by the torques associated with its enormous magnetic field. This period has also been proposed by P08 to be in close synchronism with with the orbital period of the binary, in analogy with what is believed to be happening in Polar Cataclysmic Variables or AM Her-type systems. The observed X-ray emission from 1E is that from the magnetar in this model.
In this paper, we explore an alternative model for the 1E system wherein it is a close binary system consisting of a young neutron star with a canonical magnetic field ∼ 10 12 G, and a low-mass companion, i.e., a pre-Low-mass X-ray Binary (henceforth pre-LMXB), such as are believed to be the standard progenitors of Low-mass X-ray Binaries (henceforth LMXBs). Such pre-LMXBs are born after the common-envelope (CE) evolution phase of the original progenitor binary system consisting of a massive star and a low-mass companion, which leads to the formation of a binary consisting of the He-core of the original massive star and the low-mass companion (Ghosh 2007 and references therein). The He-star susequently explodes in a supernova, leading to a neutron star in orbit with a low-mass companion, i.e., the pre-LMXB referred to above. This is the standard He-star supernova scenario for the formation of LMXBs (Ghosh 2007 and references therein). The 6.67 hr periodicity is identified in our model with the orbital period of the binary. In our model the young neutron star is still spinning very rapidly, with a canonical spin period ∼ 10 − 100 ms, and is operating in the "propeller" regime, wherein any matter approaching the fast-rotating magnetosphere of the neutron star is expelled by the energy and angular momentum deposited into it through its interaction with the magnetospheric boundary (Illarionov & Sunyaev 1975, henceforth IS75, Davies et al. 1979, Illarionov & Kompaneets 1990, Mineshige et al. 1991, Illarionov et al. 1993, Ghosh 1995 and references therein, henceforth G95, Lovelace et al. 1999, henceforth LRB99 Romanova et al. 2004, Romanova et al. 2005, henceforth RUKL05, Ustyugova et al. 2006, henceforth UKRL06).
The observed X-ray emission from 1E in our model is that from the propeller: indeed, it is well-known that soft X-ray transients (SXRTs) like Aquila X-1 and others (see Sec. 7.1) go through low/quiescent states during the decay of their outbursts, during which their luminosities and spectral properties are very similar to those of 1E, and the neutron stars in them are believed to be in the propeller regime (Campana et al. 1998, Stella et al. 2000. The observed 6.67 hr periodicity in our model is due to the orbital modulation of the supersonic propeller, which is caused by the orbital modulation of the mass-transfer rate in the eccentric binary orbit of a young system like 1E. It is well-known that young post-SN binaries with low-mass companions like 1E are almost certain to have eccentric orbits, due to the large eccentricities produced in such systems in the SN explosion (see Sec. 6.1) and the duration of the subsequent tidal circularization compared to the ages of systems like 1E (see Sec. 6.2). By contrast, SXRTs are believed to be old LMXB systems with circular orbits, where such modulation will not occur.
We show in this work that the 6.67 hr light curve of 1E can be accounted for quantitatively by our model for propeller torques of both Illarionov-Sunyaev type and , and that the observed spectral and other characteristics are also in general agreement with our overall picture. Thus, further diagnostic features need to be explored in order to distinguish between our model and the magnetar model as a viable description of this and similar sources.
Propeller phase in pre-LMXBs
In a pre-Low-Mass X-ray Binary (pre-LMXB: see above), the newborn, fast-rotating neutron star is unable at first to accrete the matter that is being transferred from the companion through the inner Lagrangian point L 1 , because of the fast rotation of the neutron star (IS75, Davies et al. 1979, Illarionov & Kompaneets 1990, Mineshige et al. 1991, Illarionov et al. 1993, G95, LRB99, Romanova et al. 2004. Because of its large angular momentum, this matter forms an accretion disk and reaches the magnetospheric boundary of the magnetized neutron star, whereupon this ionized matter interacts with the fast-rotating neutron star's magnetic field, and the energy and angular momentum deposited in it by magnetic stresses associated with this fast-rotating magnetic field expel it. This is the propeller phase of the system (IS75), during which the neutron star spins down as it loses angular momentum and rotational energy. During this propeller phase, the disk matter at the magnetospheric boundary is shockheated as the "vanes" of the supersonic propeller (IS75) hit it, and the hot matter emits in the soft X-ray band. This emission appears unmodulated at the neutron-star spin frequency (as opposed to the X-ray emission from canonical accretion-powered pulsars, which comes from the neutron-star surface) to a distant observer, who sees only the total emission from the heated matter at the magnetospheric boundary. Observations of transient low-mass X-ray binaries (i.e., the soft X-ray transients or SXRTs) like Aquila X-1 (Campana et al. 1998) and SXJ1808.4-3658 (Stella et al. 2000 in quiescence, when the neutron stars in them are thought to be operating in the propeller phase, amply confirm this point.
The propeller luminosity L during the above phase is given by L = Nω, where N is the propeller torque acting on the neutron star and ω is its spin angular velocity. The propeller torque N was first estimated by IS75 in their pioneering suggestion of this mechanism, and subsequent work over approximately the next two decades considered variations of this torque under different circumstances, as summarized in G95. These works addressed themselves largely to quasi-spherical accretion, and we shall call this kind of propeller torque the Illarionov-Sunyaev type (or IStype for short) torque, which was widely used in that time-frame in propeller spindown calculations. In the 2000s, Romanova, Lovelace and co-authors reported a series of calculations of the propeller effect for disk-accreting magnetic stars based on their numerical MHD simulations (Romanova et al. 2004, RUKL05, UKRL06; also see the analytic estimates in LRB99). We shall call the propeller torque obtained from this line of work the Romanova-Lovelace et al. type (or RUKL-type for short) torque. In this work, we shall consider both IS-type and RUKL-type propeller torques for the problem at hand.
Consider IS-type torques first. For such fast-rotating neutron stars as we are concerned with in this work, the propeller operates in the supersonic regime, and its torque is given by (G95 and the references therein),
N = 1 6 µ 2 ω 2 GM x Ω(r m ) ω .(1)
In Eq. (1), Ω(r m ) is the Keplerian angular velocity at the magnetospheric radius r m , µ is the magnetic moment of neutron star and M x is its mass. Combining this equation with the standard expression for the magnetospheric radius (Ghosh 2007), viz.,
r m = µ 2 M √ 2GM x 2 7 ,(2)
whereṀ is the rate at which transferred matter arrives at the magnetospheric boundary, we obtain the following expression for the propeller luminosity:
L 35 ≈ 5Ṁ 3 7 14 (P spin /0.1s) −2 µ 8 7 30 m −2 7 x .(3)
In Eq. (3),Ṁ 14 isṀ in units of 10 14 g s −1 , L 35 is L in the units of 10 35 erg s −1 , P spin is the neutron-star spin period, µ 30 is the neutron-star magnetic moment in units of 10 30 G cm 3 , and m x is the neutron-star mass in units of solar mass. As neutron stars are thought to have P spin ∼ 0.01 − 0.1 s at birth, and as the propeller phase is thought to end when the spin period is longer than P spin ∼ 0.1 − 1 s, we have made the canonical choice for the expected scale of P spin in systems like 1E. Equation (3) clearly shows how the propeller luminosity scales with the mass-arrival rateṀ, and essential neutron-star properties, namely, its spin period P spin , its magnetic moment µ, and its mass M x . Now consider RUKL-type torques. These authors summarized the results of some of their extensive MHD simulations in RUKL05 and UKRL06 in terms of power-law fits to these results, showing that the scaling of the total propeller torque N with the magnetic moment µ and the spin rate ω of the neutron star was
N ∝ µ 1.1 ω 2 .(4)
However, the scaling of N withṀ was not available from the above references, because only the parameters µ and ω (and also the turbulence and magnetic diffusivity parameters of the disk: see below) seem to have been varied in the series of simulations reported in these references. In order to estimate the scaling of N withṀ for RUKL-type torques, we proceeded in the following way. First, we did an analytic estimate in the following manner. In their analytic study, LRB99 argued that the radius r d of the inner edge of the disk should depend on the stellar rotation rate ω in addition to the parameters µ andṀ that r m (see above) depended upon. The scaling with ω, µ, andṀ that these authors derived was revised in UKRL06, the final result being given as r d ∝ µ 1/2Ṁ−1/4 ω −1/4 . (Note the closeness of the scalings with µ andṀ with those which apply to r m , as given above.)
In a simple first approach, if we argue that a reasonable estimate of the torque scalings may be obtained by replacing r m with r d in Eq. (1) for disk accretion, we arrive at the scaling N ∝ µ 5/4 ω 11/8Ṁ3/8
for RUKL-type torques. Noticing the qualitative similarity of the the scalings with µ and ω in Eq. (5) with those of the actual RUKL-type torque given in Eq. (4), and furthermore the quantitative closeness for the scaling with µ, we argued that the best estimate would be to use the scalings of Eq. (4) for µ and ω, and the scaling of Eq. (5) forṀ, thus arriving at a suggested scaling for the RUKL-type torque as
N ∝ µ 1.1 ω 2Ṁ3/8 .(6)
Before proceeding further, we recognized that RUKL-type torques may arise from more complicated interactions than are describable by the above arguments, and so attempted to verify the aboveṀ scaling by further comparison with RUKL results. To this end, we noted the correlated variations of N andṀ recorded in Figure 4 of Romanova et al. 2004, and fitted the two prominent peaks in N andṀ at the extreme right of this figure to a power law. This gave an exponent ≈ 0.37, coincident with that in Eq. (6) within errors of determination. With this support, we use the scalings of Eq. (6) for the RUKL-type torque in this work, deferring further considerations to future publications.
In order to obtain the dimensional values of the RUKL-type propeller torques and related varaiables, we now insert the reference units for the RUKL simulations given in RUKL05 and UKRL06, thus obtaining for the torque:
N 33 ≈ 0.87µ 1.1 30 (P spin /0.01s) −2Ṁ3/8 14 .(7)
Here, N 33 is the propeller torque in units of 10 33 g cm 2 s −2 , the units of other variables are as before, and we have kept the values of the turbulence and magnetic diffusivity parameters of the accretion disk in RUKL-type models at the canonical values given in RUKL05 and UKRL06. The RUKL propeller luminosity L is then obtained in a straightforward manner as
L 35 ≈ 5.5Ṁ 3 8 14 (P spin /0.01s) −3 µ 1.1 30 .(8)
In Eq. (8), the units of all variables are as before.
Comparison of Eqs.
(3) and (8) immediately leads to the following conclusions about IS-type and RUKL-type propeller luminosities. First, the scalings with µ andṀ are almost identical for the two types. Secondly, the scaling with the neutron-star spin period P spin is stronger (-3 instead of -2) for the RUKL-type than for the IS-type. Finally, for identical values of the variables µ,Ṁ, and P spin , the RUKL-type propeller luminosity is about three orders of magnitude lower than the IS-type propeller luminosity. Conversely, at fixed values of µ andṀ, roughly equal luminosities are given by the two types if the spin-rate for the RUKL type is about an order of magnitude higher than that for the IS type.
As indicated earlier, in this work we are exploring the properties of such propellers as described above during the relatively early stages of post-supernova binaries containing pre-LMXBs, when the binary orbits are expected to be appreciably eccentric, as explained in Sec. 6.1. In such a system, the mass-transfer ratė M tr through the inner Lagrangian point L 1 is expected to vary periodically with the orbital phase, as detailed below in Sec. 3. This flow of matter forms an accretion disk because of its large specific angular momentum, as explained above, and slow viscous effects in the disk modify the profile of the above periodic modulation (making it less sharp), and the resultant periodic profile is that which is shown by the mass-arrival rateṀ at the neutron star. The propeller luminosity then follows suite, showing a periodic modulation, as described by Eq. (3) for the IS-type torque or Eq. (8) for the RUKL-type torque. In this scenario, therefore, we identify the 6.67 hour period of 1E with the binary period of a young, eccentric pre-LMXB, which is expected to turn much later into a standard LMXB after passing through further intermediate phases (see Sec. 6.4). In the next section, we give details of the expected nature of the mass-transfer modulationṀ tr (θ) at the orbital period.
Orbital modulation of mass transfer
The problem of orbital modulation of mass transfer in eccentric orbits has been studied by a number of authors over almost three decades now, adopting various approaches appropriate for various aspects of the problem they have studied. These aspects have covered a considerable range, from a scrutiny of the concept of the Roche lobe in an eccentric orbit (Avni 1976), to a study of test-particle motion through numerical integration of the restricted three-body problem at or near periastron passage (Lubow & Shu 1975), to explicit calculations of orbital phasedependent flow through L 1 from a suitably-modeled stellar envelope (Joss & Rappaport 1984 and references therein). For our purposes here, we have adopted the results of the calculations described by Brown and Boyle (Brown & Boyle 1984, hereafter BB): these authors described the flow through L 1 from the atmosphere of the lobe-filling companion with a scale height H as a sort of nozzle flow through the inner Lagrangian point, integrating over a Maxwellian distribution of velocities (characterized by thermal velocity scale v T ) for the stellar matter. Their final result for the rate of mass transfer as a function of the true anomaly θ is given by:
M tr (θ) =Ṁ 0 γ γ p 1 + e 1 + e cos θ exp −γβe 1 − cos θ 1 + e cos θ .(9)
In Eq. (9), e is the orbital eccentricity, and the dimensionless function γ(θ) is the ratio of the phase-dependent equivalent Roche-lobe radius R(θ) of the companion to the phase-dependent orbital distance d(θ) in the eccentric orbit, γ p being the value of γ at periastron (θ = 0). From standard geometry of ellipses, d(θ) is given by:
d(θ) = p 1 + e 1 + e cos θ ,(10)
where p ≡ a(1 − e) is periastron distance. Finally, β ≡ p/H is the the dimensionless scale-height parameter introduced by BB.
It is convenient to work in terms of the ratio γ as it varies relatively slowly with orbital phase (and is, in fact, independent of this phase for a non-rotating companion: see below). The other properties of the binary system that γ depends on are (a) the mass ratio Q ≡ M c /M x , M c being the mass of the low-mass companion, and (b) the rate of rotation Ω c of the companion, usually expressed in units of the orbital angular velocity Ω p at periastron as λ ≡ Ω c /Ω p . The scaleṀ 0 = √ 2πγ p pHv T ρ 0 of the mass-transfer rate in Eq. (9) is set by the above velocity scale v T , the scale-size pH for the effective cross-section of the above "nozzle", and the basic density scale ρ 0 in the stellar atmosphere.
Prescriptions for γ have been given in the 1970s and '80s; we use here the generalized Joss-Rappaport (Joss & Rappaport 1984) expressions adopted by BB, namely,
γ = A − B log Q + C(log Q) 2 ,(11)
where the coefficients in γ are given by:
A = 0.398 − 0.026K + 0.004K 3/2 B = −0.264 + 0.052K − 0.015K 3/2 C = −0.023 − 0.005K (12)
and the variable K depends on the above rotation parameter λ and the orbital phase as:
K = λ 2 (1 + e) 4 (1 + e cos θ) 3 .(13)
From Eqs. (11)-(13), it is clear that, for a non-rotating companion with K = 0, γ is independent of the orbital phase, and depends only on the mass ratio Q. Thus, for a given Q, R(θ) simply scales with d(θ) as the eccentric orbit is traversed. It is stellar rotation which modifies the Roche potential in such a way that this simple scaling is broken, and γ depends on orbital phase. The phase-dependent factor in K goes back to the original work of Avni (1976). In our present work, we study the limits of (a) no stellar rotation, λ = 0, and (b) synchronous stellar rotation, λ = 1, to cover a range of possibilities (see below). The estimated accuracy in the above prescription for determining equivalent Roche-lobe radii is ∼ 2%.
Detailed models with the mass transfer profileṀ tr (θ) given by Eq. (9) are described below. From general considerations, it is clear that this profile peaks at the periastron and that the sharpness of the peak depends on the quantity γβe. Since γ ∼ 1, and typical values of β for the current problem are in the range 10 2 − 10 3 (BB), we see that the profile is expected to be sharply peaked at the periastron even for realtively low values of eccentricity, such as e ∼ 0.2.
Viscous flow in accretion disks
Matter transferred through L 1 into the Roche lobe of the neutron star first forms a ring around the neutron star, the radius r ring of this ring being related to the specific angular momentum ℓ tr of the transferred matter as (Pringle 1981):
r ring = ℓ 2 tr /(GM x ).(14)
Through effective viscous stresses, this ring spreads into an accretion disk, wherein matter slowly spirals inward towards the neutron star as the viscous stresses remove angular momentum from it. The accretion disk extends from its outermost radius r out inward upto the magnetospheric boundary r m , where the propeller torques expel the matter by depositing energy and angular momentum in it, as explained above.
The rateṀ at which the matter drifting radially inward through the accretion disk arrives at r m depends, therefore, both on the profile of mass supplyṀ tr at L 1 , as described above, and on the rate of viscous radial drift through the accretion disk, which occurs on a timescale t visc .
In a quasi-steady state, the relation between the two profileṡ M(t) andṀ tr (t) is of the forṁ
M(t) = t t−N * P orbṀ tr (t 0 ) f (τ)dt 0 , where τ ≡ t − t 0 t visc .(15)
The convolution integral in Eq. (15) describes the viscous drift with the timescale t visc of the mass supplied to the disk at earlier times t 0 at the rateṀ tr (t 0 ), as indicated above. In principle, the integral extends over all previous times, but in practice it is sufficient to keep track of only about N orbital periods in the past (as the lower limit of integration indicates). This is so because of the rapid fall of of the viscous-evolution profile f (τ) of the accretion disk at large values of τ (see below). Viscous-evolution profiles have been calculated analytically and numerically at various levels of approximation by several authors (Lynden-Bell & Pringle 1974, Lightman 1974. For our purposes here, we have adopted an analytic approximation of the generic form
f (τ) = τ −n exp −n τ(16)
introduced and utilized by Pravdo and Ghosh (Pravdo & Ghosh 2001, hereafter PG). This reference has discussions of earlier analytical and numerical investigations. The generic PG profile in Eq. (16) reaches its maximum at τ = 1, and decays subsequently as τ −n . Clearly, therefore, most of the contribution to the above convolution integral comes from those orbital cycles which are closest to the earlier time t 0 = t − t visc , and N is determined by the sharpness of the fall of the profile, i.e., n. In our computations, we estimated the optimal values of N by running test cases with increasing values of N until the desired accuracy was obtained. For example, in the best-fit case reported below, we found that N = 9 gave an accuracy of ≈ 10%, while N = 15 gave an accuracy of ≈ 1%. Given the error bars on the data points in the observed light curve, further accuracy was unnecessary.
The following generic feature of viscous evolution of accretion disks is a key aspect of the phenomenon we are exploring here. Whereas the orbital modulation of the mass-supply ratė M tr (t) to the disk at its outer radius r out is expected to sharply peaked at periastron for typical values of the scale height in the companion's atmosphere, as above, the viscous drift of matter through the accretion disk would decrease the sharpness of this modulation, since variations on timescales much shorter than t visc tend to be "washed out" by viscous diffusion. This is what makes the orbital modulation of the mass-arrival rateṀ(t) at the disk's inner radius r m gentler, and therefore also the modulation of the propeller luminosity L(t), leading naturally to light curves of the form observed in 1E. Quantitative details follow.
Model light curves
We constructed model light curves for 1E by combining the model of mass transfer described in Sec. 3 with that of viscous flow through the accretion disk described in Sec. 4. We then fitted these models to the observed light curve of 1E in 2005 (dL06). The fitting parameters were (β, e), which come from the above BB mass-transfer model in elliptic orbit, and also (t visc , n), which come from the above PG parametrized description of viscous evolution of accretion disks. In this introductory work, we kept β constant at a canonical value of β = 10 3 (BB), and varied the parameters e, t visc , and n to obtain acceptable fits. For the viscous timescale, we found it more convenient to work in terms of the ratio κ ≡ t visc /P orb of this timescale to the known period P orb ≈ 6.67 hr of the system, which we of course identify with the orbital period in this model. The ratio κ is of immediate physical significance, since it measures the relative importance of viscous diffusion to orbital modulation in the system. For κ ≪ 1, viscous diffusion would be so rapid as to enable the disk flow to adjust to the orbital modulation of mass-supply rate, and flow-rate would essentially follow the supply rate. For κ ≫ 1, on the other hand, the viscous diffusion would be so slow as to wash out any rapid variations in the mass-supply rate, and the modulation would be essentially determined by the disk viscosity. As we see below, κ values of a few seem to describe the 1E system, indicating comparable importance of the two effects in this system. We fitted model light curves corresponding to both IS-type and RUKL-type torques to the data on 1E, the best-fit values of the parameters being given in Table 1 and Table 2. In each case, we have considered both a non-rotating secondary (λ = 0) and a synchronously-rotating secondary (λ = 1), as indicated. Note that the best-fit values for the two types of torques are very close to each other, as may have been expected. This is so because the closeness of the scaling of L withṀ between the two types, as discussed in Sec. 2, since only this aspect of the torque is relevant for fitting the profile of the light curve. Other aspects, e.g., the fact that the RUKL-type propeller luminosity is about three orders magnitude below the IS-type propeller lumnosity for identical vaues of µ,Ṁ and ω, have important consequences elsewhere, as detailed in Sec. 6.3, but not in this matter. Further, the absolute values of the observed luminosities in the light curves are easily accounted for, e.g., by having the stellar spin rate higher for RUKL-type torques by about a factor of 10 than that for IS-type torques, and µ,Ṁ identical for the two types, as the scalings in Eqs. (8) and (3) show. This implies neutron-star spin periods in the range P spin ∼ 0.01 − 0.1 s, i.e., the canonical range for propellers, for both type of torques, as explained in Sec. 1.
This closeness of best-fit parameters is reflected in the bestfit light curves, which are visually essentially identical for the two types of torques. In Fig. 1, we display this common best-fit light curve, superposed on the data on 1E.
Our inferred best-fit value of κ in the above tables indicates that the dominant contribution to the convolution integral described in the last section comes from the second and third orbits preceding the time of observation. The corresponding viscous timescale t visc ≈ 17.3 hr is consistent with a rather thick disk with h/r ∼ 0.1 − 0.5 and a canonical value ∼ 0.1 − 1 for the disk viscosity parameter (Shakura & Sunyaev 1973). This seems consistent with the results of the RUKL numerical simulations. Note also that the best-fit value of the viscous-profile index n is consistent with the range of values n ∼ 4 − 5 generally expected for neutron-star systems, as per the discussion given in PG. Indeed, we found that values of n in the above range generally worked for the 1E system. Regarding the orbital eccentricity e, the best-fit values are as given in the tables, and we found that values of the eccentricity e in the range ∼ 0.35 − 0.45 genearlly worked for the 1E system: we discuss this in the next section. It is clear, therefore, that the model explored in this paper can account quantitatively for the observed 1E light curve in 2005, for both IS-type and RUKL-type torques. We discuss in Sec. 7.2 possible reasons for the apparently different, "jagged" light curve hinted at by the 2001 observations of this system (dL06).
Formation & evolution of prototype systems
As indicated in Sec. 1, we are exploring in this work a model for systems like 1E wherein the binary system of a He-star and a low-mass star (left after completion of the CE evolution phase in which the extensive envelope of the evolved primary has been expelled and its He-core left behind) produces the pre-LMXB when the He-star explodes in a supernova (SN), leading to a newborn neutron star with a low-mass companion. Essential features of the formation and subsequent evolution of such systems are, therefore, essential components of this model. We now discuss these features in brief, considering in this section first the immediate post-SN status of the system, and then the evolution of this system with the low-mass companion in an eccentric orbit at or near the point of Roche-lobe contact at periastron, producing a system like 1E where orbitally-modulated mass transfer proceeds through the inner Lagrangian point, and the newborn, fast-spinning neutron star is operating in the propeller regime, expelling this matter instead of accreting. Subsequently, we summarize further evolution of such systems.
Immediate post-SN systems
A major question that concerns us here is the expected eccentricity of systems formed by the SN in the above scenario, since this eccentricity is crucial for the proposed mechanism. Qualitatively, it is obvious that the immediate post-SN system is almost guaranteed to be highly eccentric, as the mass loss from a typical pre-SN system of, say, a 3M ⊙ He-star and a M c = 0.4M ⊙ low-mass companion (see below) in forming the post-SN system of M x = 1.4M ⊙ neutron star with its M c = 0.4M ⊙ lowmass companion is 1.6M ⊙ , which is close enough to maximum allowed value of mass loss ( = half of the initial total mass of 3.4M ⊙ for zero kick velocity) to ensure that the post-SN orbit would be very eccentric. We shall use these values for the stellar masses throughout the rest of this paper.
To see this quantitatively, we can adapt the extensive calculations of Kalogera, who computed the probability of the formation of X-ray binaries as a funtion of orbital parameters (Kalogera 1996). In the following, we shall use the same masses for the pre-and post-SN system as given above for illustrative purposes. The probability density from Kalogera's work is:
G(α, e) = ζ 2πξ 2 3/2 2πe α(1 − e 2 ) 1/2 × (17) α − 1 1 + e 1 1 − e − α −1/2 × exp − 1 2ξ 2 ζ 2α − 1 α + 1 I o (z) Here, z ≡ ζ α (1 − e 2 ) 1/2 ξ 2 ,
and I o is the modified Bessel function of zeroth order. Further, α is the ratio of semimajor axes of the pre-and post-SN orbits, ζ is the ratio of the total mass of the post-SN binary to that of the pre-SN one, and ξ ≡ σ/V r , σ being the velocity dispersion in the SN kick-velocity, and V r the orbital velocity of the exploding star relative to its low-mass companion just before the SN (Kalogera 1996).
In the problem we are studying here, the semimajor axis of the post-SN binary is determined by Kepler's third law from our assumed stellar masses above, and the known orbital period of 1E. However, when there is a kick associated with the SN, the inferred semimajor axis of the pre-SN binary is not determined uniquely by the semimajor axis and the eccentricity of the post-SN binary: rather, there is a range of values corresponding to the range of the kick-velocity. Thus, there is a range in the values of α: it is well-known that the allowed range for α is limited from 1/(1+e) to 1/(1−e), these limits being first identified by Flannery and van den Heuvel (1975). Thus, for our purposes, it is aprropriate to integrate G(α, e) over the above allowed range of α, and display the resultant probability density G(e) ≡ G(α, e)dα as a function of the eccentricity e. We show this in Fig. 2 for various typical values of σ as indicated. In this figure, we have used the symbol vk 5 there to denote σ in units of 10 5 m s −1 = 100 km s −1 , the typical scale for the SN kick dispersion, and we have normalized the probability density G(e) so that G(e)de = 1 in each case. As explained above, the closeness of the value of ζ ≈ 0.53 in this typical case to its lower limit for no binary destruction in the SN (this limit is 0.5 for zero kick velocity) ensures that the probability peaks at a high value of e, as Fig. 2 shows. It is clear, therefore, that such a pre-LMXB would generically have a considerable eccentricity at the time of its formation in the SN.
Tidal-evolution phase of pre-LMXBs
The above newly-formed pre-LMXB undergoes tidal evolution, wherein three simultaneous processes occur, namely, (1) tidal circularization, i.e., decrease in the orbital eccentricity e, (2) tidal orbit-shrinkage or hardening, i.e., decrease in the orbital semimajor axis a, and (3) tidal synchronization, whereby the rotation frequency Ω c of the low-mass companion approaches the orbital angular frequency Ω ≡ 2π/P orb . These processes happen through tidal torques, and their quantitative descriptions pioneered by Zahn (1977Zahn ( , 1978 are widely used for calculations: we use them here, as have P08. Complete equations are given in Zahn (1977), and an Erratum was published by Zahn (1978). We have found a further algebraic or transcription error in the original paper, which we describe below, and which seems to have gone unnoticed so far.
Complete formulations for the rates of change of e, a, and Ω c are given in Zahn (1977), but for our work here we shall utilize a widely-used simplification which comes naturally out of these formulations, namely, that the timescale for tidal synchronization comes out to be much shorter than that for tidal circular-ization and tidal hardening (see, e.g., Meibom & Mathieu 2005, P08). This is appropriate, since we shall be interested in this work only in phenomena which occur on the timescales of tidal circularizatuion or longer. Under such circumstances, we can look upon the system as being roughly synchronous at all times, and describe tidal circularization and tidal hardening respectively by Zahn's (1977) equation (4.7) and the appropriately simplified (i.e., synchronized) version of Zahn's equation (4.3), thereby obtaining:
− 1 e de dt = 63 4 k 2 t F q(1 + q) R a 8 ,(18)
and − 1 a
da dt = 114 k 2 t F q(1 + q) R a 8 e 2 .(19)
In equations (18) and (19), q ≡ 1/Q in terms of the mass ratio Q ≡ M c /M x defined above in Sec 3, k 2 is the apsidal motion constant for the low-mass companion, and t F is the "friction time" of Zahn (1977), which, for stars with convective envelopes (as in the present case) is given by Zahn's (1977) pioneering prescription of the turbulent eddy-viscosity timescale t EV :
t F ∼ t EV = (M c R 2 c /L c ) 1/3 .(20)
Equations (18) and (19) describe simultaneous tidal circularization and hardening of close binaries, but before presenting our results, we need to correct two errors related to them. First, if we define a circularization timescale t circ ≡ −e/(de/dt) in the usual way, we get from eq. (18) the result:
t circ = 4 63 1 k 2 q(1 + q) M c R 2 c L c 1/3 a R 8 ,(21)
which would be identical to Zahn's (1977) equation (4.13), except that the factor of 4 on the right-hand side is missing in Zahn (1977). Unfortunately, this error has propagated over the years into numerous papers, e.g., in P08, in their equation (2) 1 We have corrected this now. Secondly, in an erratum published in 1978, Zahn corrected a few other (generally smaller) numerical errors, of which the one relevant to our work is that the numerical coefficient on the right-hand side of our eq. (18) should be 21 instead of 63/4. In all calculations reported here, we have made these corrections. We have integrated eqs. (18) and (19) numerically for close binary systems like 1E, with values of initial post-SN semimajor axes and eccentricities, a i and e i , chosen over a range of plausible values for such systems. We find that, in all cases, the systems circularize and harden in a way that, in the (e vs. a) plane, the circularization point is approached in a "cut off" like manner. This is shown in Fig. 3 for a possible prototype 1E-like system, so chosen that the parameters of it evolve to those roughly corresponding to 1E in ∼ 2000 years. This cut-off approach is similar to what Meibom and Mathieu (2005) found. Of course, our detailed shape is slightly different from that of these authors, since they fitted their results to an assumed parameterized distribution shape applicable to observations on a collection of "normal" binaries. These details wil be given in a separate publication. For Fig. 3. Tidal evolution of a prototype 1E-like system in the e vs. a plane. Semimajor axis a in units of solar radius. Note the "cut off" like approach to the circularization point (see text).
our purposes here, we note that the total time τ circ taken to reach this circularization point (Meibom & Mathieu 2005) can be expressed roughly as:
τ circ ≈ τ 0 a i R c 8 e −2.55 i ,(22)
where a i and e i are the initial semimajor axis and eccentricity of the immediate post-SN orbit, and the scale parameter τ 0 is given by:
τ 0 ≈ 1 2k 2 q(1 + q) M c R 2 c L c 1/3 .(23)
Equation (22) is a rough analytic fit to the mumerical results, adequate for our purposes. Note that the scale parameter τ 0 depends on the companion mass M c , its value being τ 0 ≈ 1 yr for the inferred companion mass of 1E. It is clear from eq. (22) that circularization is faster for orbits which are born more compact and more eccentric. The scaling with a is straightforward from the above equations of tidal evolution; the scaling with e is more complicated (although inspection of the same equations gives some clue), involving details of the numerical solution.
The lifetime τ circ of the eccentric phase of the pre-LMXB is obviously also the lifetime of its orbital-modulation phase which we are investigating in this work. The sensitive dependence of this lifetime on the initial post-SN orbital parameters and the companion mass (through the scale parameter τ 0 and due to the mass-dependence of R c in eq.[22]) makes for a wide range of possible values of this lifetime, ∼ 10 3 − 10 8 years.
A crucial point is, of course, that if the companion is at or close to filling its Roche lobe at periastron in the post-SN orbit, it must remain so throughout most of this eccentric phase in order for the scenario to be self-consistent. The size of the Roche lobe at periastron is simply p = a(1 − e) multiplied by a well-known function of the mass ratio q. Since the latter does not change significantly during this phase, we need only study the evolution of the former. Our integration of the tidal-evolution equations show that, while a and e both decrease during this phase, p = a(1 − e) decreases slowly through most of this phase, reaching a minimum and increasing thereafter at late stages. This is shown in Fig. 4 for the prototype 1E-like system displayed in Fig. 3 (see above). Thus, if the companion is initially at or close to filling its Roche lobe at periastron, it will remain so over most of this phase, and if it is inside its Roche lobe initially, it is likely to fill its Roche lobe later during this phase. It is also seen that Rochelobe contact ends at the last parts of this phase (when the orbit is nearly circular), since p increases and becomes roughly constant there.
Fig. 4. Evolution of periastron distance p = a(1 − e) during tidal evolution of a prototype 1E-like system (see text). Shown is p in units of the solar radius vs. time in years.
Thus, this tidal-evolution phase is a rough measure of the lifetime of Roche-lobe contact and orbital modulation of the propeller output. After this, the pre-LMXB becomes detached, and remains so until angular-momentum loss through gravitational radiation and/or magnetic braking brings it back to Roche-lobe contact on a long timescale of 10 8 − 10 9 yrs. We discuss this phase below in Sec. 6.4.
Duration of propeller phase
When the above tidal-evolution phase ends, is the neutron star still operating in the propeller phase? To answer this question, we consider the spindown of the neutron star from an initial spin period P i spin to a final, longer spin period P f spin under the action of the propeller torque given by either the IS-type torque (Eq. [1]) or the RUKL-type torque (Eq. [7]). In each case, this spindown is decsribed bẏ
ω ω = N Iω = 1 t prop(24)
where I is the moment of inertia of the neutron star and t prop is the propeller spindown timescale. First consider IS-type torques, for which t prop is given by x I 45 yr,
t prop = 3 √ 2 (2GM
where I 45 is I in units of 10 45 gm cm 2 , and other units are as before. Equation (24) can be integrated readily in this case, the total spindown time τ prop from P i spin to P f spin being:
τ prop = 2.303t prop log(P f spin /P i spin ).(26)
As discussed earlier, the ratio P f spin /P i spin is believed to be in the range 10−100 (Ghosh 1995 and references therein), and its exact value does not matter because of the logarithmic dependence. On taking m X = 1.4 and the corresponding moment of inertia for a standard modern EOS, we arrive at τ prop ≈ 2 × 10 6 yr (27) for canonical values ofṀ, µ and I. Now consider RUKL-type torques, for which t prop is given by
t prop = t 0 (P spin /0.01s), where t 0 ≈ 2.3 × 10 7Ṁ − 3 8 14 µ −1.1 30 I 45 yr.(28)
Equation (24) can be integrated readily in this case also, the total spindown time τ prop from P i spin to P f spin being:
τ prop = t 0 (P f spin − P i spin )/(0.01s) ≈ t 0 (P f spin /0.01s),(29)
the second equality in the above equation coming from the fact that the ratio P f spin /P i spin is believed to be in the range 10 − 100, as indicated above. The numerical value of τ prop in this case is thus
τ prop ≈ 2.3 × 10 7Ṁ − 3 8 14 µ −1.1 30 I 45 (P f spin /0.01s) yr,(30)
which implies that, for canonical range P f spin ∼ 0.1 − 1 s, as indicated in Sec. 2, we arrive at τ prop ≈ 2 × 10 8 − 2 × 10 9 yr (31)
for canonical values of the variablesṀ, µ and I. In comparing the total spindown times τ prop given by the two types of torques, we notice that the time taken by the RUKLtype torque is 2-3 orders of magnitude longer than that taken by the IS-type torque for identical values ofṀ, µ and I. This reflects the relative weakness of the RUKL-type torque discussed in Sec. 2. Next, comparing the values of τ prop given by the above two types of propeller torques with the lifetime τ circ of the eccentric phase given in the previous section, we reach the following conclusions. For the IS-type torque, we find that, over most of the parameter space, the neutron star would still be in the propeller phase at the end of the above tidal-evolution phase of the binary. For the RUKL-type torque, we find that this conclusion is valid over the entire parameter space. Thus, the RUKL-type torque makes the conclusion stronger.
As shown above, the companion has moved out of Rochelobe contact by the time that the tidal-evolution phase of the binary reaches conclusion, so that mass transfer stops, and so does the propeller action and its consequent soft X-ray production. Accordingly, throughout this first Roche-lobe contact phase, we expect the system to be in the propeller phase.
Re-contact with Roche lobe & LMXB phase
After orbit circularization and the loss of its first Roche-lobe contact, as described above, the pre-LMXB thus ceases to be an X-ray source. But its orbit shrinks (i.e., the binary hardens) on a long timescale (∼ 10 8 − 10 9 yr) due to two mechanisms of angular momentum loss from the system, viz., graviational radiation and magnetic braking (Ghosh 2007 and references therein). These are the standard mechanisms through which short-period pre-LMXBs are believed to harden, until Roche-lobe contact is regained and mass transfer restarts. But the transferred mass is now accreted by the neutron star, because its spin has been slowed down sufficiently over this long time that it acts as an accretor and not a propeller at the (large) mass-transfer rates that occur at this second Roche-lobe contact in the circularized binary. The system thus turns on as a canonical LMXB now, emitting strongly (L ∼ 10 37 − 10 38 erg s −1 ) in the canonical X-ray band characteristic of emission from the neutron-star surface, rather than the soft X-ray band characteristic of propeller emission from the vicinity of the magnetospheric boundary.
The timescale t GR of orbit shrinkage due to gravitational radiation is given by (see, e.g., Faulkner 1971, Banerjee & Ghosh 2006:
t GR ≈ 2 × 10 9 m 1/3 T m x m c P orb 6 h .7 8/3 yr(32)
where m T ≡ m x + m c , and all masses are in solar units. In this equation, we have scaled P orb to the value for 1E, and substitution of the masses we have used above for this system gives t GR ≈ 4 × 10 9 yr. Generally, 1E-like systems with shorter periods and/or somewhat different companion masses will have t GR ∼ 10 8 − 10 9 yr. Magnetic braking is believed to be comparable or weaker in strength to shrinkage by gravitational radiation at these orbital periods, so that the above estimate is a reasonable one for the 1E-type systems we have in mind here. Thus, the system become a canonical, bright LMXB with a circular orbit and P orb in the range of, say, 2 -10 hours. It is wellknown that systems with P orb exceeding about 12 hours cannot come into Roche lobe contact by the above orbit-shrinkage mechanisms, since the time requied would exceed the Hubble time, as eq. (32) readily shows. However, these long-period systems also come into Roche-lobe contact eventually, as the lowmass companion completes its main-sequence evolution and expands. These systems thus also become canonical long-period LMXBs with circular orbits. The lifetime of this standard, bright LMXB phase is t LMXB ∼ 10 8 − 10 9 yr.
Discussion
In this work, we have explored a pre-LMXB model of 1E, wherein the eccentric orbit of the very young pre-LMXB causes an orbital modulation in the mass-transfer rate, and the newborn, fast-rotating neutron star operates in the propeller regime, the propeller emission in soft X-rays following the above modulation after viscous smoothening in the accretion disk. In this section, we first discuss first some essential spectral and luminosity-dependent features of 1E, and their connections with corresponding features in old, low-mass, soft X-ray transients (SXRTs) in their low/quiescent states, the prime example of this class being Aquila X-1 (Campana et al. 1998). Note that the well-known transient accretion-powered millisecond pulsar SAX J1808.4-3658 also shows a similar behavior (Stella et al. 2000). In these classes of low-mass X-ray binaries with old neutron stars, the neutron star is thought to operate in the propeller regime when the sources are in their low/quiescent states during decays of their outbursts. We then compare our model with the magnetar model which has been proposed recently for 1E (P08), and discuss how distinction between the two kinds of models might be attempted in future. Finally, we summarize our conclusions.
X-ray spectra
The XMM-Newton/EPIC (0.5-8 keV) X-ray spectra of 1E have been described by dL06. The time-averaged spectra from the 2005 low-state observations, when the source luminosity was L ∼ 10 33 erg s −1 , can be fitted by a two-component model consisting of a blackbody (BB) of temperature kT bb ∼ 0.5 keV and an equivalent blackbody radius R bb ∼ 0.6 km, plus a power-law (PL) of index Γ ∼ 3, with ∼ 70% of the total flux coming from the blackbody component. Alternatively, the second component can also be a blackbody with a higher temperature. A re-analysis of the earlier 2001 XMM-Newton data, when 1E had a higher luminosity (by a factor ∼ 6), yielded a similar two-component (BB+PL) model with essentially the same blackbody temperature kT bb and power-law index Γ, but a larger equivalent blackbody radius R bb ∼ 1.3 km, and a higher contribution from the PL component (the blackbody contribution was ∼ 50% of the total flux as opposed to the above ∼ 70%), which made the overall spectrum harder (dL06).
We stress the remarkable similarity of the above observations with those of the spectra of SXRTs in their low/quiescent states (when the neutron stars in them are believed to be functioning in the propeller regime), taking the well-known source Aquila X-1 as the example. A detailed analysis of the BeppoSAX observations of Aquila X-1 in 1997 (Campana et al. 1998) has yielded the following results. At the lowest state, with source luminosity L ∼ 0.6 × 10 33 erg s −1 , the (BB+PL) fit had a BB of temperature kT bb ∼ 0.3 keV and an equivalent blackbody radius R bb ∼ 0.8 km, plus a power-law (PL) of index Γ ∼ 1, with ∼ 60% of the total flux coming from the blackbody component. As the luminosity increased by a factor ∼ 150 to L ∼ 9 × 10 34 erg s −1 , the (BB+PL) fit yielded a BB of temperature kT bb ∼ 0.4 keV and an equivalent blackbody radius R bb ∼ 2.6 km, plus a power-law (PL) of index Γ ∼ 1.9, with ∼ 20% of the total flux coming from the blackbody component. Remembering that the total range of luminosities in these Aquila X-1 low-state observations during outburst decay is roughly 10 33 − 10 35 erg s −1 (Campana et al. 1998), essentially identical to that of the 1E observations reported by dL06, the correspondence is very suggestive.
SXRTs are believed to be old systems with a neutron star and a low-mass companion in a close circular orbit, undergoing outbursts due to instabilties either in the accretion disk or in the mass supply from the low-mass companion. In their low/quiescent states during decays of outbursts, the fast-spining neutron star (spun up by accretion as per standard LMXB scenario) is believed to operate in the propeller regime. What we suggest in this work is that 1E-like systems are very young systems in the same regime: the young systems can show orbital modulation because of the orbital eccentricity, while the old systems are in circular orbit and cannot show such orbital modulation. However, the spectral signatures are very similar at similar luminosities, which supports our basic suggestion. We note that the timescales associated with 1E outburst appear to be ∼ 2 − 3 years while those associated with Aquila X-1 outbursts appear to be ∼ 30 − 70 days. It is possible that the basic phenomenon is rather similar in the two cases, and that the difference in detail is caused by the fact that accretion onto the neutron-star surface (with attendant high luminosities and hard X-ray spectra) does occur at the high states during the outbursts for old systems like Aquila X-1, but not for young systems like 1E.
A comprehensive theory of the emission spectra of propeller sources appears to be lacking, though Illarionov and co-authors have studied some effects of Comptonization in propellers in wind-accreting massive X-ray binaries (Illarionov & Kompaneets 1990, Illarionov et al. 1993). Attempts at constructing such a theory for propellers in pre-LMXBs and in old LMXBs in low/quiescent states is clearly beyond the scope of this paper, and we shall confine ourselves here to the comment that the importance of Compton heating, considered in the above works on propellers in massive binaries, is also likely to be crucial for the systems we are focusing on in this work, as the observed power-law tails in the spectra at low luminosities suggest. These tails are particularly prominent in the low-state spectra of Aquila X-1 (Campana et al. 1998).
Luminosity dependence of light curve
dL06 have compared the 1E light curve in the 2005 low-state observations with that during the 2001 observations when the source luminosity was a factor ∼ 6 higher. While the former light curve is relatively smooth with some cycle-to-cycle variations, the latter one shows more complex, somewhat "jagged" structure, with an occasional dip. Further, the pulsed fraction decreses from ∼ 43% to ∼ 12% as the luminosity increases. We discuss qualitatively how such features may arise. First, a propeller system is inherently more fluctuating than an accreting system, because of a variety of fluctuations possible at the site of shock-heating and outflow. As mass-supply rate through the accretion disk increases, these fluctuations may increase, causing more complex profiles. Secondly, accretion disks in low-mass systems like LMXBs and CVs are thought to develop structures at their outer edges, which obscure emission from the compact object, and lead to dips. If these obscuring structures increase in size as mass-supply rate through the accretion disk increases, this would provide a natural explanation for the above appearance of the dips. Thirdly, as the mass-arrival rateṀ at r m increases, r m decreases (see Sec. 2), matter at the magnetospheric boundary becomes hotter, and the propeller becomes less supersonic, ultimately becoming subsonic. Now, it is well-known that the subsonic propeller torque N sub ∼ µ 2 Ω 2 s /GM x is independent ofṀ (see Ghosh 1995 and references therein), and so will not follow the modulations ofṀ. Hence, asṀ and L increase, the following phenomenon is likely to happen. As the upper limit of the excursions inṀ goes beyond the critical cross-over point from supersonic to subsonic propeller regime, the pulsed fraction will decrease because that part ofṀ which is above this critical point will not contribute to the pulsed flux, and this decrease will increase with increasingṀ. This may be a natural explanation for the above observation of reduced pulsed fraction at higher luminosity. More quantitative considerations will be given elsewhere.
Comparison with magnetar model
In a recent paper, P08 have described a model in which 1E is a magnetar, i.e., a neutron star with a superstrong magnetic field ∼ 10 15 G with a low-mass companion. The 6.7 hr period is interpreted in this model as the spin period of the neutron star, the idea being that a neutron star with such strong magnetic field as above can be spun down to such long spin period, or such low spin frequency, in ∼ 2000 yrs. Magnetars are a fascinating possibility, and their relevance to soft gamma repeaters (SGRs) and possibly to anomalous X-ray pulsars (AXPs) has been the subject of much recent study. P08 have invoked an analogy with polars or AM Her-type cataclysmic variables containing white dwarfs with unusually strong magnetic fields, wherein torques acting on the magnetar spin it down in a short time to spin periods in close synchronism with the binary orbital period. In this analogy, they have been inspired by the similarity of the shape the 1E light curve to those of AM Her systems.
We have desribed in this work a model which does not require a neutron star with a superstrong magnetic field, but rather interprets the 6.7 hr period as the orbital period of the binary system consisting of a neutron star with a canonical magnetic field of ∼ 10 12 G with a low-mass companion, the newborn, fast-rotating neutron star being in the propeller phase, and the propeller emission being modulated in the eccentric orbit of a young post-SN binary. We find that the observed 1E light curve can be quantitatively accounted for by our model. Our analogy is with propeller regimes of SXRTs like Aquila X-1 in their low/quiescent states, which we consider to be old, circularized analogues of 1E which are no longer orbitally modulated, but which have remarkably similar spectral properties. In this analogy, we have been inspired by the similarity between 1E and the SXRTs in both the spectral characteristics and their changes with source luminosity, as well as the shapes of the outbursts and the way in which propeller-like properties emerge at low luminosities during outburst decays.
An interesting question is that of possible discriminators between the above two models. It appears to us that if all observed properties of 1E and similar systems can be accounted for by known characteristics of early stages of pre-LMXBs born according to the standard CE evolution and He-star supernova scenario, such as we have described in this paper (or by other possible models involving standard evolutionary scenarios), there would not be any compelling need for invoking exotic objects like magnetars for this class of objects. On the other hand, if one finds unique observed features in this class of objects that cannot be explained at all within the framework of standard evolutionary scenarios, presence of magnetars in such objects may well be hinted at. However, answering this question is beyond the scope of this paper: we are pursuing the matter, and the results will be reported elsewhere.
Conclusions
The work reported here suggests that 1E-type systems are early stages of pre-LMXBs born in the SN of He-stars in binaries of (He-star + low-mass star) produced by common-envelope (CE) evolution. As long as the post-SN binary is eccentric, and the neutron star is in the propeller regime, soft X-ray emission modulated at the orbital period may be expected to occur. As the orbit circularizes, modulation would stop, and as the low-mass companion moves out of Roche-lobe contact, the source would not be observed in X-rays. The companion would come into Rochelobe contact again on a long timescale due to orbit shrinkage by emission of gravitational radiation and magnetic braking, and/or by the evolutionary expansion of the companion. This would lead to a standard LMXB: an old neutron star in circular orbit with a low-mass companion. Thus, steady-state arguments, with lifetimes of 1E-type systems estimated at ∼ 10 6 − 10 7 yrs and those of LMXBs estimated at ∼ 10 8 − 10 9 yrs, would lead us to expect ∼ 1 1E-type systems per ∼ 100 LMXBs, which is consistent with current observations. However, we must be careful here, as these are overall arguments for the whole population. If one specifically investigates young supernova remnants (SNRs), the chances of finding such systems may be considerably higher, since eccentric binary systems are to be found preferentially in such SNRs. More detailed considerations will be given elsewhere.
The lifetime of the eccentric-binary phase may be increased by an effect we have not included in this introductory work. The effect is that of an enhancement of eccentricity when mass and angular momentum are lost from a binary system which is already eccentric. This dynamical effect is well-known in the literature (see, e.g., Huang 1963) and its applications to compact X-ray binaries have been made earlier (Ghosh et al. 1981). For an eccentric compact binary with the neutron star in the propeller regime leading to the loss of both mass and angular momentum from the system, such considerations are applicable. However, it is possible that, at the rates of mass transfer and loss inferred for 1E-type systems, this effect is a minor one.
Several lines of further investigation are naturally suggested by the considerations we have given in this paper. Foremost among them is a theory of the spectral characteristics of propeller emission in disk-fed propeller systems. This would help clarify the remarkable spectral similarity (including changes in spectral parameters with luminosity) between 1E and SXRTs like Aquila X-1 in their low/quiescent state, as described in Sec. 7.1. A search for point soft X-ray sources in other young SNRs would clarify the observational situation greatly. We note that these sources may or may not be periodically modulated, as we have argued in Sec. 7.2 that such modulations may decrease and disappear in certain luminosity states. However, the spectral characteristics would still be a most valuable diagnostic. These and other investigations are under way, and results will be reported elsewhere.
Fig. 1 .
1X-ray light curve of 1E. Shown is the observed light curve from dL06, superposed on the (common) best-fit model light curve for IS-type and RUKL-type propellers. Left panel: model curve for λ = 0 (nonrotating companion). Right panel: same for λ = 1 (synchronously rotating companion).
Fig. 2 .
2Formation probability-density G(e) of immediate post-SN binaries as a function of eccentricity e for various values of the dispersion σ in the SN kick velocity (see text). Curves labeled by the value of vk 5 = σ in units of 100 km s −1 . Each curve so normalized that G(e)de = 1.
x ) 2/7 I µ 8/7Ṁ3/7 ≈ 2.5 × 10 5Ṁ −
Table 1 .
1Best Fit Model Parameters: IS-type torqueParameter
Best fit value (λ=0) Best fit value (λ=1)
κ
2.6
2.6
viscous-profile index n
5.04
5.04
eccentricity
0.405
0.406
χ 2
1.013
1.012
Table 2 .
2Best Fit Model Parameters: RUKL-type torqueParameter
Best fit value (λ=0) Best fit value (λ=1)
κ
2.6
2.6
viscous-profile index n
5.02
5.02
eccentricity
0.400
0.401
χ 2
1.003
1.006
Harshal Bhadkamkar and Pranab Ghosh: Young pre-Low-Mass X-ray Binaries in the propeller phase
Because of this, the parameters adopted for 1E by P08 and by ourselves in this work actually give t circ ∼ 10 4 yr. In our work, we have used the values of the apsidal-motion constant k 2 given byLandin et al. (2009).
Acknowledgements. It is a pleasure to thank A. de Luca for sending data on the light curves, to thank E. P. J. van den Heuvel and L. Stella for stimulating discussions, and to thank the referee for comments which improved the paper considerably.
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|
[] |
[
"Giant spin-orbit splitting of point defect states in monolayer WS 2",
"Giant spin-orbit splitting of point defect states in monolayer WS 2"
] |
[
"Wun-Fan Li \nSoft Condensed Matter\nDebye Institute for Nanomaterials Science\nUtrecht University\nPrincetonplein 53584CCUtrechtThe Netherlands\n",
"Changming Fang \nSoft Condensed Matter\nDebye Institute for Nanomaterials Science\nUtrecht University\nPrincetonplein 53584CCUtrechtThe Netherlands\n",
"Marijn A Van Huis \nSoft Condensed Matter\nDebye Institute for Nanomaterials Science\nUtrecht University\nPrincetonplein 53584CCUtrechtThe Netherlands\n"
] |
[
"Soft Condensed Matter\nDebye Institute for Nanomaterials Science\nUtrecht University\nPrincetonplein 53584CCUtrechtThe Netherlands",
"Soft Condensed Matter\nDebye Institute for Nanomaterials Science\nUtrecht University\nPrincetonplein 53584CCUtrechtThe Netherlands",
"Soft Condensed Matter\nDebye Institute for Nanomaterials Science\nUtrecht University\nPrincetonplein 53584CCUtrechtThe Netherlands"
] |
[] |
The spin-orbit coupling (SOC) effect has been known to be profound in monolayer pristine transition metal dichalcogenides (TMDs). Here we show that point defects, which are omnipresent in the TMD membranes, exhibit even stronger SOC effects and change the physics of the host materials drastically. In this Article we chose the representative monolayer WS2 slabs from the TMD family together with seven typical types of point defects including monovacancies, interstitials, and antisites. We calculated the formation energies of these defects, and studied the effect of spin-orbit coupling (SOC) on the corresponding defect states. We found that the S monovacancy (VS) and S interstitial (adatom) have the lowest formation energies. In the case of VS and both of the WS and WS2 antisites, the defect states exhibit giant splitting up to 296 meV when SOC is considered. Depending on the relative position of the defect state with respect to the conduction band minimum (CBM), the hybrid functional HSE will either increase the splitting by up to 60 meV (far from CBM), or decrease the splitting by up to 57 meV (close to CBM). Furthermore, we found that both the WS and WS2 antisites possess a magnetic moment of 2 µB localized at the antisite W atom and the neighboring W atoms. All these findings provide new insights in the defect behavior under SOC point to new possibilities for spintronics applications for TMDs. arXiv:1606.02906v1 [cond-mat.mes-hall]
|
10.1103/physrevb.94.195425
|
[
"https://arxiv.org/pdf/1606.02906v1.pdf"
] | 33,775,499 |
1606.02906
|
8aee258dbb984fd12115ba5ef8c926a50e7a0c50
|
Giant spin-orbit splitting of point defect states in monolayer WS 2
(Dated: June 10, 2016)
Wun-Fan Li
Soft Condensed Matter
Debye Institute for Nanomaterials Science
Utrecht University
Princetonplein 53584CCUtrechtThe Netherlands
Changming Fang
Soft Condensed Matter
Debye Institute for Nanomaterials Science
Utrecht University
Princetonplein 53584CCUtrechtThe Netherlands
Marijn A Van Huis
Soft Condensed Matter
Debye Institute for Nanomaterials Science
Utrecht University
Princetonplein 53584CCUtrechtThe Netherlands
Giant spin-orbit splitting of point defect states in monolayer WS 2
(Dated: June 10, 2016)
The spin-orbit coupling (SOC) effect has been known to be profound in monolayer pristine transition metal dichalcogenides (TMDs). Here we show that point defects, which are omnipresent in the TMD membranes, exhibit even stronger SOC effects and change the physics of the host materials drastically. In this Article we chose the representative monolayer WS2 slabs from the TMD family together with seven typical types of point defects including monovacancies, interstitials, and antisites. We calculated the formation energies of these defects, and studied the effect of spin-orbit coupling (SOC) on the corresponding defect states. We found that the S monovacancy (VS) and S interstitial (adatom) have the lowest formation energies. In the case of VS and both of the WS and WS2 antisites, the defect states exhibit giant splitting up to 296 meV when SOC is considered. Depending on the relative position of the defect state with respect to the conduction band minimum (CBM), the hybrid functional HSE will either increase the splitting by up to 60 meV (far from CBM), or decrease the splitting by up to 57 meV (close to CBM). Furthermore, we found that both the WS and WS2 antisites possess a magnetic moment of 2 µB localized at the antisite W atom and the neighboring W atoms. All these findings provide new insights in the defect behavior under SOC point to new possibilities for spintronics applications for TMDs. arXiv:1606.02906v1 [cond-mat.mes-hall]
I. Introduction
The transition metal dichalcogenides (TMDs) are a member of the layered 2D van der Waals (vdW) materials, in which the atoms are bound by intra-layer chemical bonding and interlayer vdW bonding. Among many other TMDs, the molybdenum dichalcogenides and tungsten dichalcogenides (MX 2 , M=Mo or W, and X= S, Se, or Te) are the group 6 branch of the whole TMD family and have attracted much scientific attention. Theoretically, the most stable structure of MX 2 consists of one layer of transition metal atoms sandwiched by two layers of chalcogen atoms with a prismatic coordination, forming the so-called 1H form 1 . Due to the weak inter-layer vdW interaction, TMDs can be exfoliated from bulk into the few-layer or monolayer (ML) forms. When reducing the number of layers from bulk to ML, the band gap of TMDs evolves from an indirect band gap to a direct band gap with an increased gap size due to quantum confinement 2,3 . The layerdependent tunability of the electronic structure together with other distinct physical properties of ML TMDs make them promising candidates of applications in fields like electronics, optoelectronics, spintronics and valleytronics, sensing, and catalysis 4-7 . There are two effects governing the band structure (BS) of MX 2 , namely crystal field (CF) splitting and spin-orbit (SO) splitting (∆ SO ). These two effects strongly affect the electronic properties of MX 2 and influence in particular the d bands of the transition metal.
According to crystal field theory, the five formerly degenerate d bands of the transition metal will split in energy if the transition metal is bonded to other ligands (the chalcogen atoms in our case), and the pattern of the energy splitting is dependent on the metal-ligand coordination geometries. For ML MX 2 in the 1H phase, the transition metal is surrounded by six chalcogen atoms in a trigonal prismatic coordination (Fig. 1). Consequently, the d bands split according to their orientations -the more they are along the direction d x 2 of the M-X bond, the higher in energy they will be due to the electron-electron repulsion with the X orbitals. As shown in Fig. 1, the d z 2 orbital is the lowest in energy, and the d x 2 −y 2 and d xy orbitals are higher in energy. The d xz and d yz orbitals are the highest in energy 8,9 . The Supplemental Material (SM) shows the decomposed band structures of both bulk and ML WS 2 which illustrate the CF splitting of the d bands ( Figures S3 and S4). The order of increasing energy is d z 2 < d x 2 −y 2 = d xy < d xz = d yz for both bulk and ML WS 2 , as expected. The spin-orbit coupling (SOC) effect in MX 2 materials has been discovered in the last few years [10][11][12][13] . In bulk MX 2 , the system possesses both the space inversion symmetry (E ↓ ( k) = E ↓ ( −k))and time inversion symmetry (E ↓ ( k) = E ↑ ( −k)). The net result is spin degeneracy in reciprocal space when no external magnetic field is present:
E ↓ ( k) = E ↑ ( k).
However, in the case of ML MX 2 , because of the lack of space inversion symmetry, the spin states are expected to split under SOC. Especially, the band splitting can be as large as 463 meV for the valence band maximum (VBM) of ML WSe 2 at the K point in the first Brillouin zone 13 . Based on symmetry arguments 10,13 , for ML MX 2 only the orbitals with magnetic quantum number m l = 0 will participate SO splitting. Furthermore, because the X atoms are rather light, their p orbitals are not affected by the SOC effect. Lastly, as indicated in the BSs of ML WS 2 in the SM (Fig. S4), the VBM and conduction band minimum (CBM) are dominated by the d z 2 (m l = 0), d xy (m l = −2) and d x 2 −y 2 (m l = 2) orbitals. As a result, only the d xy and d x 2 −y 2 orbitals will have the SO splitting.
Besides the novel physical properties of pristine TMDs, atomic point defects are omnipresent in the materials. Furthermore, adatom adsorption and doping on ML MX 2 is especially achievable by virtue of their 2D surface nature. Both the naturally occurring and chemically or physically introduced point defects in MX 2 will extensively modulate the physical properties such as charge transport, magnetism, optical absorption, and absorbability [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] 34 . Among the single vacancy, vacancy complexes and antisite complexes, they found that the V S is the predominant point defect. First principles calculations confirmed that V S has the lowest formation energy among all the defect kinds. Hong et al. did a systematic study which shows the route-dependency of predominant point defect types 35 . In ML MoS 2 synthesized by CVD and mechanical exfoliation (ME), V S is the only dominating point defect, whereas in ML MoS 2 fabricated by physical-vapor-deposition (PVD), the antisites Mo S2 and Mo S are the dominant point defects. They also found that the Mo S antisite possesses a local magnetic moment around the Mo defect site. From the theoretical perspective, several exhausive works have been done to study the point defects systematically by virtue of the density functional theory (DFT) [36][37][38] . Their results predict that in ML MX 2 , the V S and sulfur interstitial S i have the lowest formation energy.
Despite the significance of SOC and point defects for ML MX 2 systems, to the best of our knowledge thus far no study has been conducted on the SOC effect on the electronic structure of defective ML MX 2 . Therefore, here we investigate how the SOC effect will change the band structure (BS) of ML MX 2 when different types of point defects are present. We chose systematically three categories of point defects: monovacancies (V S and V W ), interstitials (S i and W i ), and antisites (S W , W S , and W S2 ). For conciseness, the ML WS 2 slabs containing these defects are abbreviated as: V S -WS 2 , V W -WS 2 , S i -WS 2 , W i -WS 2 , S W -WS 2 , W S -WS 2 , and W S2 -WS 2 , respectively. The relaxed structure of each point defect is shown in Figure 2. We chose WS 2 as a representative of the MX 2 family as the physical and chemical properties of all the MX 2 members are very similar, and thus the results of WS 2 are expected to be applicable to other MX 2 systems.
After describing the computational settings, we will first discuss the formation energies of the selected defect species. We then chose V S , S i , W S , and W S2 for further investigation of the SO defect state splitting. We found that SOC causes giant defect state splitting in the cases of V S and W S2 , with the magnitude of the band splitting up to 194 meV for V S and 167 meV for W S2 respectively. In addition, we also found that both W S and W S2 antisites possess a magnetic moment around the antisite W atom, which is contrary to the previous study of MoS 2 35 . The findings in this work provide a deeper insight in the point defect physics of MX 2 and will help developing potential applications of MX 2 in electronics and spintronics.
II. Computational details
All calculations were performed using the DFT code VASP 39-41 within the Projector-Augmented Wave (PAW) framework 42 .
The exchange and correlation energies were described using the Generalized Gradient Approximation (GGA) formulated by Perdew, Burke and Ernzerhof (PBE) 43,44 . The cut-off energy of the wave functions and the augmentation functions were 400 eV and 550 eV, respectively. The van der Waals correction with the optB88-vdW density functional 45 was used as at the beginning of this study the bulk WS 2 was also included 46 . The supercell size of the ML WS 2 was 6 × 6 in the x − y plane, and the vacuum along the z direction was larger than 16Å. These dimensions of the supercell were sufficiently large to avoid the artificial defectdefect interaction. A Γ-centered 2 × 2 × 1 k-mesh was used. The thresholds of energy convergence and force convergence were 10 −4 eV and 10 −2 eV/Å, respectively. We examined the SOC effect and found that it does not affect the structure but only influences the electronic properties of WS 2 , therefore we only included SOC after geometry relaxation to obtain the band structure (BS) and DOS for the systems. We first performed the geometry relaxation and total energy calculation with only vdW correction included (without SOC). Then we turn on SOC, and exclude vdW correction for calculating the electronic properties (BS and DOS) of the relaxed geometry. Spin polarized (SP) calculations were performed for every point defect species, and only the W S and W S2 antisites were found to be magnetic due to their unpaired electrons. The initial geometry of each point defect configuration was chosen based on previous theoretical studies 37,38 . The stringent setting described above guarantees a good convergence of defect formation energy within 0.01 eV.
In addition to standard DFT calculations, we also performed the more advanced hybird functional (HSE06) 47 calculations for the defective ML WS 2 which shows defect state splitting under SOC (the V S -WS 2 , W S -WS 2 , and W S2 -WS 2 ). The goal of these HSE+SOC calculations is to investigate how HSE will affect the defect state splitting. The HSE calculations were performed on the DFT-relaxed geometries and we found that HSE relaxation gave almost identical geometries compared to traditional DFT. We set the fraction of Hartree-Fock exchange functional to 0.168 by fitting the calculated band gap of ML WS 2 to the experimental value. This fraction gives us a band gap of 2.04 eV, which is very close to the ex- perimental value of 2.05 eV 48 . In the HSE+SOC calculations only the Γ point was included as we did a test for V S -WS 2 and W S -WS 2 and found that a 2 × 2 × 2 k-mesh only improves the band gap for 7 meV for V S -WS 2 , and for 13 meV for W S -WS 2 . Therefore we believe that Γ is sufficient in our case. Our SO splitting of the top valence bands of perfect ML WS 2 calculated by DFT is 430 meV, which is perfectly matching the previous DFT result of 433 meV 13 . HSE increases this splitting considerably to 517 meV.
(g) W S2 (f) W S (c) S i (a) V S (b) V W (d) W i (e) S W
III. Results and discussion
Defect formation energy
The formation energy E f of a neutral defect is defined as 49
E f = E def ect − E perf ect + i n i µ i .(1)
In Eq. 1, E def ect is the total energy of the defective system, E perf ect is the total energy of the perfect system, n i is the number of atoms being added (plus) or removed (minus) from the perfect system, and µ i is the chemical potential of the added or removed atom. The added/removed atom is imagined to be taken from/put to an atomic reservoir, thus the energy required for creating a defect inside a system does not depend only on the system itself, but also on the atomic resevoir, or the surrounding environment. Chemical potentials µ i are therefore needed to reflect the chemical environment surrounding the system. µ i is not fixed, but a variable for which we could determine its boundaries by considering the formation reaction of a material. We give the derivation of the boundaries of µ i in the case of WS 2 in the SM 50 The final expressions of the boundaries of µ W and µ S which are used to calculate the defect formation energies are: The calculated defect formation energies are listed in Table I dependent on W-rich or S-rich chemical potentials. The next step is to choose relevant defect types for further study of the effect of SOC on electronic properties of the defective ML WS 2 slabs. Table I provides a simple criterion in terms of defect formation energy: V S and S i possess the lowest formation energies in both the W-rich and S-rich conditions, thus it is sensible to select them for more detailed study. Although the W S and W S2 antisites have a higher formation energy, it has been reported that the Mo S and Mo S2 antisites are the predominant point defects in MoS 2 synthesized by physical-vapor-deposition (PVD). Therefore the W S and W S2 antisites are also included in the present study 35 .
E W S2 − 2E S ≤ µ W ≤ E W (2a) 1 2 (E W S2 − E W ) ≤ µ S ≤ E S .(2b)
Defect state splitings under SOC
As seen in Ref. 51 and Fig. S4 in the SM, the valence bands of MX 2 are composed of the p x and p y orbitals of the X atoms (here: S atoms), and the d xy , d x 2 −y 2 and d z 2 orbitals of the M atoms (here: W atoms). The d xz , d yz orbitals are far from the band gap region. Furthermore, Fig. 4 indicates that the top valence bands and the bottom conduction bands consist mainly of the d orbitals of W atoms. The only p orbital present is the p z orbital from the S atoms, and it does not split under SOC. The calculated BSs with and without SOC are shown in Fig. 3. We can see from Fig. 3 and Fig. 4 that irrespective of the type of point defects, the VBM of WS 2 always splits into two bands under SOC.
V S
As discussed in the Introduction, only the W d xy and d x 2 −y 2 orbitals will undergo SO splitting. This is the case for V S . The defect states are composed of the the linear combinations of W d xy and d x 2 −y 2 orbitals, which are formerly degenerate are now split into two bands. The magnitude of the SO splitting for V S is 194 meV. The HSE+SOC calculation gave a SO splitting of 252 meV, which is 58 meV larger than the DFT+SOC value. This substantial energy difference shows the necesity of hybrid functionals in calculating the SO splitting of the defect states.
S i
In the case of S i , the only defect state is composed of the p x and p y orbitals of the interstitial S atom, which do not split under SOC. This defect state is hidden in the top valence bands.
W S
For W S , the defect states are also composed of W d xy and d x 2 −y 2 , but they do not split when SOC is included in the calculations. Further eigenstate analysis shows that the reason for the defect states to be kept degenerate is that the spin projections of these states in the SOC BS are all on the m x − m y plane (m x , m y and m z are the magnetization axes), in contrast to the defect states of the other three defect kinds where the spin projections are either mostly on along the m z axis (in the case of W S2 , +m z for spin-up and −m z for spin-down). As a result, the spin states are not split even when SOC is present. We performed a second calculation in which the magnetization was constrained along the m z axis and thus the defect states indeed split. This allows us to examine the effect of the orientation of magnetization on the defect state splitting. We also found that the m z -constrained magnetic configuration is 38.9 meV higher in energy (for HSE, the value is 58.4 meV) than the m x − m y -relaxed magnetic ground state. This finding suggests that the W S -WS 2 is a magnetically anisotropic material and that the easy axis lies on the m x − m y plane.
In Figs. 3 and 4 we show the BS and the band energies at Γ of the m z -constrained W S -WS 2 . There are six defect states for W S -WS 2 as shown in Fig. 4 (d). Three of these states are spin-up, and the other three are spin-down. For each spin species, the two degenerate states with a lower energy are composed of d xy and d x 2 −y 2 of the antisite W atom, and the state higher in energy originates from the d z 2 orbital. It is worth mentioning that the spin-up d xy and d x 2 −y 2 orbitals are occupied by two unpaired electrons which are the source of the magnetic moment of W S -WS 2 as will be discussed in next section. Under SOC, the d xy and d x 2 −y 2 orbitals split into two bands and each of these bands is a linear combination of d xy and d x − y 2 . For spin-up, this splitting is 296 meV, which is the highest ∆ SO among all the WS 2 defects studied in this paper. For spin-down, the splitting is 87 meV. The smaller ∆ SO for spin-down may be related to the fact that the spindown defect states are much higher in energy than the spin-up states, thus they are closer to the CBM which are the d z 2 orbitals that do not exhibit SO splitting. The consequence is that the spin-down defect states are hybridized with the d z 2 conduction bands and thus their ∆ SO is reduced. This argument is supported by the wave function analysis, which shows that both the d xy and d x 2 −y 2 orbitals approximately have a 1 3 d z 2 character.
The ∆ SO from HSE+SOC are 356 meV and 62 meV for spin-up and spin-down, respectively. With HSE, the SO splitting of the spin-up defect states ubcreases significantly (60 meV) similar to the case of V S -WS 2 . However, for the spindown defect states, with HSE the SO splitting decreases by 25 meV. The reason for the decreased ∆ SO for spin-down defect states is that HSE pushes these states further into the conduction band region, thereby enhancing the mixing with the d z 2 orbitals. W S2 W S2 is the most complicated case among the chosen defects. It involves ten defect states -five are spin-up and five are spindown. As indicated in Fig. 4 (e), without SOC, the five defect states for each spin type can be categorized into three groups: two groups of doubly degenerate states which are lower in energy, and a single d z 2 orbital higher in energy. The mixing of the conduction d z 2 band with the spin-down d xy and d x 2 −y 2 defect bands is even worse in the case of W S2 -WS 2 as the spindown defect d z 2 state is already in the conduction band region. The two sets of doubly degenerate states are composed of the linear combinations of the d xy and d x 2 −y 2 orbitals of the antisite W atom, and will split into four states if SOC is present. Thus, for W S2 -WS 2 , there are four sets of SO splittings. The ∆ SO of each split set is 121 meV, 105 meV, 167 meV, and 138 meV, respectively, with ascending energy.
In contrast to DFT, HSE calculation for W S2 -WS 2 relaxed the magnetization onto the m x − m y plane. Therefore we again constrained the magnetization along the m z axis. The constrained configuration is less stable than the relaxed one by 23.5 meV. For the magnetically constrained W S2 -WS 2 , HSE again enchances the splittings which are not close to CBM (the first three splittings in Fig. 4 (e)). The increments are 46 meV, 38 meV, 33 meV, respectively. In contrast, for the fourth splitting HSE decreases ∆ SO by 57 meV. One noteworhty feature is that the spin-up splittings are always larger than the spin-down splittings.
Magnetic moments of the WS and WS2 antisites
We found that both W S and W S2 defects possess a magnetic moment of 2 µ B . This is different from the result of Ref. 35, which indicated that for MoS 2 , only Mo S -MoS 2 has a magnetic moment but not Mo S2 -MoS 2 . These magnetic moments are generated by the unpaired spin-up electrons residing on the d xy and d x 2 −y 2 defect states, as indicated by Fig. 4 (d) and (e). These states split under SOC. We defined the spin density as the difference between the spin-up charge density and the spin-down charge density: ρ = ρ ↑ − ρ ↓ to visualize the magnetic moment distribution around the defect site. The resulting spin density plots are presented for both antisite defects in Fig. 5. At first glance, the magnetic moment seems to be fully localized on the antisite W atom, however for both W S and W S2 , the d orbitals of the neighboring W atoms contribute to the magnetic moment as well, and to a lesser extent also the next-nearest-neighboring (NNN) W atoms. are involved. For W S2 , the magnetic moment spreads to both the nearest-neighboring (NN) and NNN W atoms. We compared the ratio between the magnetic moment at the defect W atom and the total magnetic moment (µ r = µ(W def ) µ(all) ) to give a semi-quantitative description of the distribution of the magnetic moment. We used the VASP default atomic radii for W (1.455Å) and S (1.164Å) to perform the spherical integration of the spin density. We calculated µ r using DFT (spin-polarized), DFT+SOC, and HSE+SOC methods. For W S , µ r (DFT)= 88.4%, µ r (DFT+SOC)= 88.0%, and µ r (HSE+SOC)= 98%, respectively. For W S2 , the corresponding values were lower at 53.1%, 53.5%, and 66.6%, respectively. In addition, we also found that the magnetic moment distribution shown in Fig. 5 has a triangular shape with a side length of around 6.4Åin both cases. Therefore these two antisite defects could also be named magnetic 'superatoms' 35 .
Therefore one can conclude that, first, for W S the magnetic moment is almost solely localized on the defect W atom, yet for W S2 the magnetic moment is centered at the defect W atom, but half of it spreads to the NN and NNN W atoms. Second, with the HSE hybrid functional, the magnetic moment is more localized on the dedect atom, yielding a higher µ r .
In order to trace back the origin of these magnetic moments, we compared the total energy and the density of states (DOS) of both the non-spin-polarized (NSP) and spin-polarized (SP) solutions of W S -WS 2 and W S2 -WS 2 . It was found that the NSP solutions are significantly higher in energy than the SP counterparts. The energy difference E(SP) -E(NSP) is 402 meV for W S -WS 2 and 151 meV for W S2 -WS 2 . Therefore both antisite configurations are indeed spin-polarized and are magnetic. The DOS plots of both the NSP and SP solutions for W S -WS 2 and W S2 -WS 2 in Fig. 6 show clearly the magnetism. By combining Fig. 6, Fig. 4 and the projected DOSs (PDOSs) (a) Perfect
-2.0 -1.5 -1 -0.5 NSP d z 2 + p z d z 2 + p z d xy + d x 2 −y 2 d xy + d x 2 −y 2 d z 2 E F SOC Energy (eV) (b) V S -2.0 -1.5 -1 -0.5 NSP d z 2 + p z d z 2 + p z d xy d x 2 −y 2 d xy + d x 2 −y 2 d xy + d x 2 −y 2 d z 2 E F SOC ∆ 1 ∆ 1 = 194 meV (252) Energy (eV) (c) S i -2.0 -1.5 -1 -0.5 NSP p x p y d z 2 + p z d z 2 + p z d xy + d x 2 −y 2 d xy + d x 2 −y 2 d z 2 E F SOC Energy (eV) (d) W S -2.0 -1.5 -1 -0.5 0 spin-up d z 2 + p z d xy + d x 2 −y 2 d xy d x 2 −y 2 d z 2 d z 2 E F SOC ∆ 1 ∆ 2 ∆ 1 = 296(356) meV ∆ 2 = 87(62) meV spin-down d z 2 + p z d xy + d x 2 −y 2 d xy d x 2 −y 2 d z 2 d z 2 Energy (eV) (e) W S2 -2.0 -1.5 -1 -0.5 0 spin-up d z 2 + p z d xy + d x 2 −y 2 d xy + d z 2 d x 2 −y 2+ d z 2 d xy + d z 2 d x 2 −y 2+ d z 2 d z 2 d z 2 E F SOC ∆ 1 ∆ 2 ∆ 3 ∆ 4 ∆ 1 = 121(167) meV ∆ 2 = 105(143) meV ∆ 3 = 167(190) meV ∆ 4 = 138(81) meV spin-down d z 2 + p z d xy + d x 2 −y 2 d xy + d xy + d z 2 d z 2 d x 2 −y 2 + d z 2 d x 2 −y 2 + d z 2 d xy + d xy + d z 2 d z 2 d z 2 d z 2
Energy (eV) S2 in the SM), we performed a thorough eigencharacter analysis of the defect states, revealing that these states are composed of the d orbitals of the antisite W atom which are numbered for each antisite in Fig. 6. For W S -WS 2 , group 1 is composed of the d xy and d x 2 −y 2 orbitals and group 2 is characterized by the d z 2 orbital. For W S2 -WS 2 there are three groups of defect states. Group 1 and 2 are both composed of the d xy and d x 2 −y 2 orbitals. However, they are now mixed with the d z 2 orbital to different extents. Group 2 is more heavily mixed with the d z 2 orbital than group 1. Group 3 is simply the d z 2 orbital. Furthermore, for both antisite defects, only the spin-up part of peak 1 is under the Fermi level and is occupied by two electrons from the d xy and d x − y 2 orbitals of the antisite W atom. Therefore the magnetism and its origin is confirmed.
IV. Conclusion
In this study we calculated the formation energies of seven different configurations of point defects including monovacancies, interstitials and antisites. We found that among the point defects, V S and S i possess the lowest formation energies; E f (V S ) =1.689 eV in a W-rich chemical environment, and E f (S i ) = 1.211 eV under a S-rich chemical environment. We selected the V S , S i , W S and W S2 defects to investigate the SOC band splitting of the defect states. We have shown that the SO splitting depends on both the orbital constitution and the orientation of magnetization of the defect states. The states having the d xy and d x 2 −y 2 character will undergo significant SO splitting when the magnetization is oriented along the m z magnetization axis. The as-generated SO splittings are 194 meV for V S , 296 meV and 87 meV for W S , and 121 meV, 105 meV, 171 meV, and 138 meV for W S2 . The hybrid functional HSE enhances the SO splitting up to 60 meV if the defect state is not close to CBM. However, it decreases the SO splitting up to 57 meV if the defect state is close to CBM. For S i no SO splitting was found as the defect state is composed solely by the d z 2 and p z orbitals. We also found that not only W S , but also the W S2 defect possesses a local magnetic moment of 2 µ B around the antisite W atom due to the two unpaired spin-up electrons occupying the d xy and d x 2 +y 2 defect states. The antisite W atom together with its NN and NNN W atoms thus form the so-called superatom.
The results presented in this Article provide new insights into the SOC behavior of the ML WS 2 containing the most common point defects. These results are expected to be extendable to other ML MX 2 systems. In particular, the controllability of these SO split states are worth further investigation as they are highly promising in spintronics applications. For instance, it would be interesting to examine whether the spins can flip when an electric field is applied. Also, considering the frequent occurrence of the M X2 antisites generated during the PVD synthesis of the ML MX 2 membranes 35 , it will be interesting to increase the concentration of M X2 antisite defects and examine the interaction of the magnetic moments and their arrangement over space. Further development of this topic is beyond the scope of the present paper and will be addressed in future works.
V. Acknowledgements
This project is financially supported by the Dutch science foundation NWO via a VIDI grant (grant number 723.012.006). W.F. Li acknowledges Torbjörn Björkman and Hugo Aramberri for their discussion on the SOC calculations, and Jyh-Pin Chou for his practical instruction on VASP settings and insight of interpreting the SOC band structures.
− y 2 FIG. 1 :
21Schematic of the energy splitting of the transition metal d bands under the crystal field. The coordination is trigonal prismatic.
FIG. 2 :
2The relaxed structures of all the defective ML WS 2 supercells. The vacancies are denoted by light blue circles. The defect sufur atoms are marked in red, and defect tungsten atoms in blue. The arrows indicate the directions and magnitudes of the relaxations.
FIG. 3 :
3The band structures calculated with or without SOC for the selected WS 2 slabs. NSP stands for non-spin-polarized non-SOC calculations, and spin-up and spin-down stand for the spin-polarized calculations, respectively. Here the Fermi level is marked in red. The defect state splitting can be clearly seen in the case of V S and WS 2 . However, the splitting is supressed for S i .
FIG. 4 :
4The energy level diagram of the WS 2 systems at the Γ point. The valence bands are colored in red, defect states in green, and conduction bands in blue. The Fermi level, E F , is marked in cyan. The electrons which contribute to magnetism for W S and W S2 antisites are labeled in light green. The major orbital components of each band are indicated, where the orbitals in bold are the most predominant ones. The dotted lines show the SO splittings of the energy bands. The amount of the SO splitting (∆) is also shown in magenta, the values for ∆in parentheses were calculated by HSE+SOC.
FIG. 5 :FIG. 6 :
56Spin density plots of (a) W S and (b) W S2 antisites calculated by DFT. The Spin-up charge density is marked in red and the spin-down density in green. TDOS plots for both the non-spin-polarized (NSP) and spin-polarized (SP) W S and W S2 antisites. The vertical blue solid lines indicate the Fermi level. The colored dotted lines map the NSP → SP splitting of the defect bands (Fig.
, thus control the applicability of the material. The crucial role of point defects has triggered many studies to investigate their behavior in ML MX 2 . Liu et al. identified the atomic defects and visualized their migrations on ML MoS 2 31 . Komsa et al. found that electron beam irradiation generates sulfur monovacancies (V S ) and also cause these defects to migrate and aggregate 32,33 . Zhou and et al. carried out a joint experiment and theory study and investigated several types of defects and their influence on the electronic structure of ML MoS 2 synthesized by chemicalvapor-deposition (CVD)
TABLE I :
IFormation energies (in eV) of the defects selected
in this study
W-rich
S-rich
VS
1.689
2.897
VW
6.345
3.928
Si
2.419
1.211
Wi
5.317
7.733
SW
8.219
4.594
WS
5.380
9.005
WS2
6.838
11.671
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|
[] |
[
"Isocausal spacetimes may have different causal boundaries",
"Isocausal spacetimes may have different causal boundaries"
] |
[
"J L Flores ",
"J Herrera ",
"M Sánchez \nDepartamento de Geometría y Topología\nFacultad de Ciencias\nUniversidad de Granada\nAvenida Fuentenueva s/n18071GranadaSpain\n",
"\nDepartamento deÁlgebra\nFacultad de Ciencias\nGeometría y Topología\nUniversidad de Málaga\nCampus Teatinos29071MálagaSpain\n"
] |
[
"Departamento de Geometría y Topología\nFacultad de Ciencias\nUniversidad de Granada\nAvenida Fuentenueva s/n18071GranadaSpain",
"Departamento deÁlgebra\nFacultad de Ciencias\nGeometría y Topología\nUniversidad de Málaga\nCampus Teatinos29071MálagaSpain"
] |
[] |
We construct an example which shows that two isocausal spacetimes, in the sense introduced recently in [11], may have c-boundaries which are not equal (more precisely, not equivalent, as no bijection between the completions can preserve all the binary relations induced by causality). This example also suggests that isocausality can be useful for the understanding and computation of the c-boundary.
|
10.1088/0264-9381/28/17/175016
|
[
"https://arxiv.org/pdf/1103.2083v2.pdf"
] | 119,583,553 |
1103.2083
|
c6f5a9b7644e809006552f36b6f9106a97245bf9
|
Isocausal spacetimes may have different causal boundaries
J L Flores
J Herrera
M Sánchez
Departamento de Geometría y Topología
Facultad de Ciencias
Universidad de Granada
Avenida Fuentenueva s/n18071GranadaSpain
Departamento deÁlgebra
Facultad de Ciencias
Geometría y Topología
Universidad de Málaga
Campus Teatinos29071MálagaSpain
Isocausal spacetimes may have different causal boundaries
causal boundarycausal mappingsisocausalityconformal struc- ture of spacetimesCausality 2010 MSC: 83C7553C5083C20
We construct an example which shows that two isocausal spacetimes, in the sense introduced recently in [11], may have c-boundaries which are not equal (more precisely, not equivalent, as no bijection between the completions can preserve all the binary relations induced by causality). This example also suggests that isocausality can be useful for the understanding and computation of the c-boundary.
Introduction
The causal boundary, or c-boundary for short, is a well-known tool for the study of the conformal structure of a spacetime and related topics such as event horizons or singularities. The first approximation to this boundary was introduced four decades ago by Geroch, Kronheimer and Penrose (GKP) in the seminal paper [15]. Since then, a long series of redefinitions and new contributions has been carried out, and a renewed interest comes from the recent contributions by Harris [16,17,18] and Marolf and Ross [19,20] (see the review in [23] for complete references). Recently, the authors have carried out an extensive revision of both, the notion of c-boundary and the tools for its computation [5,7,8]. So, the c-boundary can be regarded now as a useful and consistent notion, which is well related to other geometric objects. Along this paper we understand by c-boundary the last redefinition in [7]. Nevertheless, the properties to be considered here appear at a much more basic level (say, whenever Harris' universal properties for the partial boundaries are satisfied [16]). So, they are valid for any definition of the c-boundary obtained by using the basic ingredients in the seminal GKP construction and, in particular, for all the previous redefinitions of the c-boundary along the literature.
Some years ago, García-Parrado and Senovilla [11] introduced the notions of causal mapping, causal relation and isocausality for two spacetimes V, V . Namely, V is causally related to V , denoted V ≺ V , if there exists a diffeomorphism Φ : V → V which is a causal mapping, that is, such that all the future-directed causal vectors of V are mapped by the differential of Φ into future-directed causal ones of V . Then, V is isocausal to V if V ≺ V and V ≺ V . In that article and subsequent developments [12,13,10], many applications and properties of such notions were carried out. Recall that isocausality is a generalization of conformal equivalence, adding more flexibility. This flexibility yields appealing properties, as the fact that any spacetime is locally isocausal to Lorentz-Minkowski one, even if it is not conformally flat. So, isocausality preserves some relevant global properties associated to the conformal structure, but not all of them -as stressed in [10] for the case of two levels of the causal ladder of spacetimes.
It was also suggested in [11,Sect. 6] that causal mappings could be used to obtain causal extensions and boundaries for spacetimes as a generalization of the (Penrose) conformal boundary. Concretely, a causal extension is an embedding of the spacetime in a larger one such that the former is isocausal to its image in the larger. Clearly, a boundary can be then naturally associated to such an extension (here, we will avoid the name causal boundary for this last boundary, in order to avoid confusions with the c-boundary). Recall that, in spite of its generalized usage in General Relativity, the conformal boundary has serious problems of existence and uniqueness. The problems come from the fact that, in order to find a reasonable conformal boundary, one has to find an appropriate open conformal embedding of the spacetime in some (aphysical) spacetime. It is not clear when such an embedding will exist and, in this case, if the properties of the corresponding boundary will be independent of the embedding 1 . The flexibility of causal mappings and isocausal properties, allows to check their existence much more easily than their conformal counterparts, even though with a cost of uniqueness.
In the present note, we explore the connections between the c-boundary and the notion of isocausality by means of a concrete example. This example firstly shows that two isocausal spacetimes may have different c-boundaries. That is, even though the c-boundary relies on the global conformal structure of the spacetime, it is not an object naturally invariant by isocausality. At a first glance, this property would seem a drawback for the notion of isocausality. On one hand, isocausality would be insufficient to distinguish between two spacetimes with different asymptotic causal behaviors. On the other, the boundaries obtained by using different causal extensions appear as extremely non-unique -as the causal extensions of isocausal spacetimes with different c-boundaries, may look very different. However, a deeper study suggests that, when the causal extensions are compared with the conformal ones, these properties are not a disadvantage. Notice that, essentially, the conformal boundary becomes interesting when it agrees with some intrinsic element of the spacetime, and conditions to ensure this agreement are commonly imposed (see [2,7]). But the most important intrinsic element of the spacetime which may match with the conformal boundary is the c-boundary; so, basically, the conformal boundary becomes useful as an auxiliary tool to compute the more general c-boundary. On the contrary, the properties which remain true for all the elements of a class of isocausal spacetimes (in particular, the possible similarities of their c-boundaries or of the boundaries obtained through causal extensions), become a genuinely new type of information, which reveals new connections among non-conformally related spacetimes. In fact, a closer look at our example in this article, suggests that causal mappings and isocausality may yield a very valuable information on the c-boundary.
Here, we explain this possibility only for our particular example, in order to provide a natural intuitive picture. By using the machinery introduced in [8], this idea will be developed technically in a further work by the authors.
The example
Typical background and terminology in Lorentzian Geometry as in [3,21,22] and on causal boundaries as in [3,7,14] will be used. From the technical viewpoint, our example will be very simple, and the c-boundary will be handled at a very elementary level. Basically, the idea to construct the c-boundary ∂V of a (strongly causal) spacetime V starts by defining its future causal boundary∂V and the dual past one∂V . The former is the set of all the TIP's (terminal indecomposable past subsets) of V , where any TIP can be regarded as the chronological past I − [γ] of some future-directed inextensible timelike curve γ (an obvious dual definition appears for the elements of∂V , or TIF's). A long-standing problem for the definition of the c-boundary appears when one realizes that, eventually, some points in∂V must be paired with some others in∂V . Even though this problem can be solved satisfactorily [7], we will not worry about it. In fact, our example will be robust, in the sense that even the partial boundary∂V will not be preserved by isocausality. Moreover, our concrete example is bidimensional, and the TIPs can be also generated as the chronological past of (piecewise smooth) lightlike curves, instead of timelike ones -this will be straightforward here, however, one can find in [9, Proposition 2] and [7, Sect. 3.5] a precise justification. So, the picture of the cboundary is simplified, as in dimension two the (smooth) lightlike curves must lie in two families of geodesics.
Abstract properties.
Our aim is to endow the manifold V = R × (−∞, 0) with three metrics g cl , g, g op satisfying the following properties:
(i) g cl ≺ 0 g ≺ 0 g op where the symbol ≺ 0 means that the future causal cones of the metric at the left-hand side are included in the ones of the metric at the right-hand one (i.e., the identity in V is a causal mapping from V endowed with the left metric to V endowed with the right one).
(ii) g cl and g op are conformally related. So the future causal boundaries∂ cl V,∂ op V for, resp., V cl := (V, g cl ) and V op := (V, g op ) agree and, taking into account property (i), (V, g) is isocausal to V cl (and V op ). Moreover g cl and g op will be simple standard static metrics, so that its causal boundary will be easily computable.
(iii) g presents a future causal boundary∂V "strictly greater" than the one of g cl (or g op ), in a precise sense explained below. Essentially, a segment of causally but not chronologically related points, appears for∂V where only a point (in a timelike part of the boundary) existed for∂ cl V and∂ op V .
Note that the non-preservation of the c-boundary by isocausality follows from these properties. So, once the metrics are achieved, we will pass to discuss the interplay between the c-boundary and isocausality.
Explicit construction.
Define the metrics g cl , g, g op on V = R × (−∞, 0) in the following way:
g cl = −dt 2 + dx 2 , g = −dt 2 + β(t/x)dx 2 , g op = −dt 2 + (1/4) dx 2 ,
where β : R → (0, ∞) is a smooth function which satisfies:
• β(u) ≡ 1/4 if u(= t/x) ≤ 1/2, that is, g = g op in the region x ≤ 2t.
• β(u) ≡ 1 if u ≥ 1, that is, g = g cl in the region t ≤ x(< 0).
• β increases strictly from 1/4 to 1 on the interval 1/2 ≤ u ≤ 1, so that the causal cones of g cl (resp. of g) are strictly contained in the ones of g (resp. of g op ) in the region 2x < 2t < x.
Note that the announced property (i) becomes clear from the properties of β. About (ii), the conformal relation between g cl and g op is also obvious. Moreover, the future causal boundary∂ cl V can be represented by two lines T , J + with a common endpoint i + , which is the TIP equal to all V (see [18,1,6,8] for much more general computations, which include the c-boundary of all the standard static spacetimes). More precisely, the TIPs which constitute T are the chronological past of all the future-directed lightlike geodesics ρ with endpoint at x = 0. T is timelike in the sense that any two distinct TIPs P, P ∈ T satisfy either P P or P P , where the extended chronological relation can be defined here as: P P if and only if there exists some p ∈ P such that p p for all p ∈ P . It is also clear that, for the (future) chronological topology on∂V (which here reduces to the point set convergence of the corresponding TIP's as subsets of V , see [6,7,8]) T will be homeomorphic to R. That is, in the following, T will be identified with R × {0} (each P ∈ T is identified with the endpoint in R×{0} of the lightlike geodesic whose past is equal to P ), and this identification holds at the point set, chronological and topological levels. The TIPs which constitute J + are the chronological pasts of all the future-directed lightlike ρ as above which goes to infinity (reaching arbitrarily large values of −x). We will not pay attention to this line J + , but we point out that it is horismotic. This means that any two distinct TIPs P, P ∈ J + are horismotically related, i.e. they satisfy either P ⊂ P or P ⊂ P , but neither P P nor P P . For the property (iii), let us focus on the timelike line T , identified with R×{0}. Our aim is to prove that, in addition to this timelike line, the future causal boundary∂V of (V, g) contains other boundary points P = I − [ρ] such that (0, 0) is the endpoint of the generating future-directed lightlike curve ρ. Consider the lightlike vector field X(t, x) = ( β(t/x), 1) for g. All the integral curves of X can be written as γ t (s) = (r t (s), s), with s < 0 and r t : (−∞, 0) → R satisfying:
ṙ t (s) = β rt(s) s r t (−1) = t (2.1) (see Figure 1). Note the following properties of the curves γ t :
(a) For t 1 < t 2 , necessarily r t1 (s) < r t2 (s), as r t1 (−1) = t 1 < t 2 = r t2 (−1) and γ t1 does not intersect γ t2 .
(b) γ −1/2 (s) = (s/2, s) and γ −1 (s) = (s, s) for all s < 0, and thus, any intermediate γ t satisfies: lim
s→0 γ t (s) = (0, 0) ∀t ∈ [−1, −1/2]. (c) I − [γ t1 ] I − [γ t2 ] for all t 1 < t 2 .
In fact, (a) and (b) are direct consequences of the definition of γ t . The property (c) is a consequence of (a) and the following characterization:
I − [γ t ] = {(t , s) : t < r t (s)} ∀t ∈ R. (2.2)
The inclusion ⊃ follows because, for the metric g, t < r t (s) implies (t , s) (r t (s), s). For ⊂, recall that V \{γ t (s) : s < 0} has two connected components, and the right-hand side of (2.2) is equal to one of them. Any past-directed timelike curve α starting at a point p on γ t must enter initially in this region (as any tangent vector in the past timelike cone at p, points to it). Moreover, α cannot touch γ t at a distinct (first) point q, as α and γ t would intersect transversally and, so, the velocity α would point out to the future on q. As consequence α remains totally contained in that region up to the initial point p. Fig. 2). Recall that all the points in the strain are horismotically related. So, T Str differs from T from the chronological viewpoint (there exists no bijection from T Str in T which preserves the chronologically and horismotically related points). Nevertheless, if one replaces the whole strain by any of its elements, this bijection appears naturally. Summing up, the claimed property (iii), as well as the non-equivalence of∂V and∂ cl V , are justified in a precise way.
Final discussion.
We can understand the behavior of the causal boundary in the previous example as follows.
Consider two causally related spacetimes V 1 ≺ 0 V 2 (we will write I − 1 , I − 2 instead of I − in each spacetime). A natural map between the future boundariesĵ :∂V 1 →∂V 2 can be defined by taking into account that if P ∈∂V 1 then I − 2 (P ) ∈∂V 2 . In fact, if P = I − 1 [γ] for some inextendible future-directed timelike curve γ, then γ must be timelike also for V 2 , and I − 2 [γ] = I − 2 (P ). So, we can define:
j(P ) := I − 2 (P ), ∀P ∈∂V 1 .
Nevertheless,ĵ may be very bad-behaved, even if V 1 and V 2 are isocausal. Concretely, our example above shows that the mapĵ cl :∂V cl →∂V associated to V cl ≺ 0 V cannot be continuous (nor surjective), as it induces a map T → T Str whereĵ(0, 0) chooses just the point I − (γ −1 ) of the strain. Even more, the map j op :∂V →∂ op V associated to V ≺ 0 V op is continuous, but it is not injective, as all the strain is mapped into (0, 0). It is worth pointing out that, in spite of these properties, the compositionĵ op •ĵ cl :∂ cl V →∂ op V is an isomorphism (a homeomorphism which also preserves the chronological relation). Our example shows that, this nice last property does not imply a straightforward good relation between∂V and∂ cl V . However, the example suggests another possibility. Assume that all the elements in the strain of∂V were identified to a single one. Then∂ cl V would be naturally embedded in this quotient space (in this particular example, they would be naturally Figure 1: Computed with the metric g, the curves γ −1/2 , γ t and γ −1 are lightlike and define different TIPs. These TIPs are naturally associated to the point (0, 0). However, the point (0, 0) is associated only to one TIP when the metric g op or g cl is considered (recall that these two metrics are conformal). isomorphic). In this sense, the boundary∂ cl V yields an important information about the boundary∂V , namely:∂ cl V represents the quotient of a part of∂V (alternatively,∂V can be seen as an enlargement of∂ cl V ). At what extent is this property generalizable? We will prove that it can be extended to a wide family of spacetimes which are isocausal to the standard stationary ones. However, the computation of such boundaries requires the machinery on Finsler metrics and Busemann functions developed in [8]. So, it is postponed to a forthcoming paper.
γ −1/2 γ −1 γ t V t x g cl gop − 1 2 −1 −1
Appendix
Our example can be understood more clearly as the spacetime (V, g) is conformal (thus, isocausal) to the following open region of Minkowski spacetime:
V = L 2 \ ({x ≥ a} ∪ {t + x ≥ 0, 0 ≤ x ≤ a}),
where a = (π/2)−arctan(1/2). A conformal map f : (V, g) → (V , g 0 ) is represented in Figure 3, and can be described as follows.
The spacetime (V, g) is divided in three regions: (A) the wedge (i.e., the region between γ −1 and γ −1/2 ), (B) the region above the wedge (above γ −1/2 ), and (C) the region below the wedge (below γ −1 ). Accordingly, the spacetime (V , g 0 ) is also divided in three regions:
(A') {(t, x) ∈ V : −2a ≤ t − x ≤ 0}, (B') {(t, x) ∈ V : t − x ≥ 0} and (C') {(t, x) ∈ V : t − x ≤ −2a}.
Given a point p A of the region (A), there exist two lightlike geodesics γ p A , σ p A passing through it, which are integral curves of the lightlike vector fields X(t, x) = ( β(t/x), 1), Y (t, x) = ( β(t/x), −1), resp. These curves determine the parameters r p A (the natural Euclidean distance from σ p A ∩ γ −1/2 to the origin) and α p A (the Euclidean angle between the velocities of γ −1/2 and γ p A at the origin), as indicated in the figure. Then, the image f (p A ) is defined as the point in the region (A ) which lies in the line t − x = −2α p A at the natural Euclidean distance r p A from (−α p A , α p A ). Next, given a point p B in region (B), it is clearly determined by the parameters t p B (where (t p B , 0) is the future endpoint of the integral curve of the lightlike vector field X through p B ) and r p B (Euclidean distance to this endpoint from p B ). Then, the image f (p B ) is defined as the point in region (B ) determined by the analogous parameters for an integral curve of ∂ t + ∂ x , as indicated in the figure. Finally, for any p C belonging to region (C) we proceed similarly to obtain parameters r p C , t p C , and define f (p C ) in the region (C ) of (V , g 0 ) as the point determined by t p C − a (which selects an integral curve of ∂ t + ∂ x ) and r p C (which selects a point in this curve).
Recall that this map f is obviously continuous and piecewise smooth. Its conformal character is ensured as it clearly maps lightlike curves in (V, g) into lightlike curves in (V , g 0 ).
V αp A rp A p A p B (tp B , 0) rp B (tp C , 0) p C rp C (tp B , 0) rp B (−αp A , αp A ) rp A rp C (tp C − a, a) V f (p B ) f (p A ) f (p C )
From the properties (b) and (c), different TIPs P t := I − [γ t ], with t ∈ [−1, −1/2], become naturally associated to the point (0, 0) (which was identified with a point of∂ cl V ). This implies the required property (iii). In fact, the description of the the boundary∂V for g is similar to the one of∂ cl V . However, now in the analog to the timelike line T ⊂ ∂ cl V , the boundary point associated to (0, 0) must be replaced by all the TIPs in the strain Str:= {P t : −1 ≤ t ≤ −1/2}. So, we can regard T Str = ((R\{0}) × {0}) ∪ Str, as a part of∂V (see
Figure 2 :
2Structure of the future causal boundary for (V, g). The part of the boundary T Str is composed by two timelike lines and a lightlike one, denoted by Str in the picture, which corresponds with the strain Str= {P t : −1 ≤ t ≤ −1/2}.
Figure 3 :
3This figure represents a conformal map between the spacetime (V, g) (at the left) and an open region V of Minkowski spacetime (at the right).
An example of the difficulties can be found in the recent article[4]. In order to ensure uniqueness, some technical assumptions (which involve any pair of lightlike curves) must be assumed. Remarkably, the existence of a maximal conformal extension is also ensured in[4]. However, this does not exclude the possibility that no extension exists, nor ensures a priori good properties for such an extension.
AcknowledgmentsThe authors would like to acknowledge Prof. Senovilla for very useful discussions and comments. The second-named author also thanks the kind hospitality of Department of Theoretical Physics and History of Science, University of Basque Country, during a research stay associated to this work. All the authors are partially supported by the research projects with FEDER funds MTM2010-18099 (Spanish MICINN) and P09-FQM-4496 (Regional J. Andalucía). Also, the second-named author is supported by Spanish MEC Grant AP2006-02237.
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[] |
[
"WEIGHTED FUNCTIONAL INEQUALITIES: CONSTRUCTIVE APPROACH",
"WEIGHTED FUNCTIONAL INEQUALITIES: CONSTRUCTIVE APPROACH"
] |
[
"Mitia Duerinckx ",
"Antoine Gloria "
] |
[] |
[] |
Consider an ergodic stationary random field A on the ambient space R d . In a companion article we introduced the notion of weighted functional inequalities, which extend standard functional inequalities like spectral gap, covariance, and logarithmic Sobolev inequalities, and we studied the associated concentration properties for nonlinear functions X(A) of the field. In the present contribution we develop a constructive approach to produce random fields that satisfy such weighted functional inequalities. The construction typically relies on devising approximate chain rules for nonlinear and random changes of variables for random fields. This approach allows us to cover Gaussian fields with non-necessarily integrable covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations (Voronoi or Delaunay). These weighted functional inequalities, which we primarily develop here in view of their application to quantitative stochastic homogenization, are of independent interest.
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"https://arxiv.org/pdf/1711.03152v1.pdf"
] | 55,589,022 |
1711.03152
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65859e2376183d0d06b628547ec5fd4eaf1ada8d
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WEIGHTED FUNCTIONAL INEQUALITIES: CONSTRUCTIVE APPROACH
8 Nov 2017
Mitia Duerinckx
Antoine Gloria
WEIGHTED FUNCTIONAL INEQUALITIES: CONSTRUCTIVE APPROACH
8 Nov 2017
Consider an ergodic stationary random field A on the ambient space R d . In a companion article we introduced the notion of weighted functional inequalities, which extend standard functional inequalities like spectral gap, covariance, and logarithmic Sobolev inequalities, and we studied the associated concentration properties for nonlinear functions X(A) of the field. In the present contribution we develop a constructive approach to produce random fields that satisfy such weighted functional inequalities. The construction typically relies on devising approximate chain rules for nonlinear and random changes of variables for random fields. This approach allows us to cover Gaussian fields with non-necessarily integrable covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations (Voronoi or Delaunay). These weighted functional inequalities, which we primarily develop here in view of their application to quantitative stochastic homogenization, are of independent interest.
In the companion article [7] we introduced the notion of weighted functional inequalities, which are generalizations of standard functional inequalities like spectral gap, covariance, or logarithmic Sobolev inequalities, and imply strong concentration properties. The aim of the present contribution is to complete this work by developing a constructive approach that generates random fields that do satisfy weighted functional inequalities.
In many fields of mathematical analysis, complex objects in a low-dimensional space can be described as the projection of simpler objects of a higher-dimensional space. A prototypical example is given by quasi-periodic structures. Conversely, suitable projections can be a powerful way to generate many (possibly complex) lower-dimensional objects from simpler higher-dimensional objects while preserving some essential properties, which is a useful point of view for modeling. For quasi-periodic functions, the simple high-dimensional objects are periodic functions (on a high-dimensional torus), the projection corresponds to the composition with a winding matrix, and the preserved essential property is some quantitative averaging property. In this contribution, we apply this idea to functional inequalities.
Consider a random field A = Φ(A 0 ) on R d obtained as the image by some "projection" Φ of some higher-dimensional random field A 0 on R d ×R l . In this article we argue that standard functional inequalities satisfied by A 0 can be transferred to A in the form of the weighted functional inequalities introduced in the companion article [7]. To this aim, we develop in Section 2 an abstract yet constructive approach to such inequalities, which amounts to making suitable assumptions on the "projection operator" Φ. In Section 3, we make use of this constructive approach to prove the validity of weighted functional inequalities for various examples of random fields considered in the literature.
To conclude this introduction, we describe three classes of random fields A that satisfy weighted functional inequalities.
(I) Gaussian-like fields: A is (possibly the image by a Lipschitz function of) the convolution of some white noise with some deterministic kernel, which leads to Gaussian fields with arbitrary covariance function. (II) Independent coloring of random geometric patterns: A is characterized by a random geometric pattern completed by an independent product structure. The random geometric pattern is typically constructed starting from a point process (e.g. Poisson, random parking, or Matérn-type processes) by considering inclusions centered at the points, or (Voronoi or Delaunay) tessellations. The associated product structure then determines the values of A on the cells of the random pattern, or even completes the description of the random pattern (e.g. conferring random sizes and shapes to the inclusions). This leads to possibly long-range correlations of the geometric pattern. (III) Dependent coloring of random geometric patterns: This corresponds to (II) for a coloring that does not come from a product structure but from a field that is itself correlated (e.g. of the class (I)). This leads to possibly long-range correlations of the colors of the inclusions (in the sense of e.g. value of A, size, or orientation of the inclusions), on top of the correlations of the geometric pattern.
Details are provided in Section 3. The above three classes of random fields encompass all the examples considered in [26], a reference textbook on random heterogeneous structures for materials sciences, which brings the use of functional inequalities (in their weighted versions) in stochastic homogenization to the state-of-the-art of materials science.
Notation.
• d is the dimension of the ambient space R d ;
• C denotes various positive constants that only depend on the dimension d and possibly on other controlled quantities; we write and for ≤ and ≥ up to such multiplicative constants C; we use the notation ≃ if both relations and hold; we add a subscript in order to indicate the dependence of the multiplicative constants on other parameters; • the notation a ≪ b (or equivalently b ≫ a) stands for a ≤ 1 C b for some large enough constant C ≃ 1; • Q k := [−1/2, 1/2) k denotes the unit cube centered at 0 in dimension k, and for all
x ∈ R d and r > 0 we set Q k (x) := x + Q k , Q k r := rQ k and Q k r (x) := x + rQ k ; when k = d or when there is no confusion possible on the meant dimension, we drop the superscript k; • we use similar notation for balls, replacing Q k by B k (the unit ball in dimension k); • the Euclidean distance between subsets of R d is denoted by d(·, ·); • B(R k ) denotes the Borel σ-algebra on R k ; • E [·] denotes the expectation, Var [·] the variance, and Cov [·; ·] the covariance in the underlying probability space (Ω, A, P), and the notation E [· ·] stands for the conditional expectation; • for all subsets A of a reference set B, we let A c := B \ A denote the complement of A in B; • for all a, b ∈ R, we set a ∧ b := min{a, b}, a ∨ b := max{a, b}, and a + := a ∨ 0;
• for all matrices F , we denote by F t its transpose matrix; • ⌈a⌉ denotes the smallest integer larger or equal to a; • F denotes Fourier transformation.
Constructive approach to weighted functional inequalities
In this section we consider random fields that can be constructed as transformations of product structures. Under suitable assumptions we describe how the standard spectral gaps, covariance inequalities, and logarithmic Sobolev inequalities satisfied by "hidden product structures" are deformed into weighted functional inequalities for the random fields of interest. The analysis of the examples mentioned in the introduction is postponed to Section 3.
Weighted functional inequalities.
We start by recalling the definition of weighted functional inequalities introduced in the companion article [7]. Let A : R d × Ω → R be a jointly measurable random field on R d , constructed on some probability space (Ω, A, P). A spectral gap in probability for A is a functional inequality that allows one to control the variance of any function X(A) in terms of its local dependence on A, that is, in terms of some "derivative" of X(A) with respect to local restrictions of A. In the continuum setting that we consider here, there is no canonical choice of such a (wide-sense) derivative with respect to the field A, and we recall below three such possible notions.
• The oscillation ∂ osc is formally defined by
A ′ ∈ Mes(R d ; R), A ′ | R d \S = A| R d \S − inf ess X(A ′ ) : A ′ ∈ Mes(R d ; R), A ′ | R d \S = A| R d \S , (2.1)
where the essential supremum and infimum are taken with respect to the measure induced by the field A on the space Mes(R d ; R) (endowed with the cylindrical σalgebra). This definition (2.1) of ∂ osc A,S X(A) is not measurable in general, and we rather define
∂ osc A,S X(A) := M[X A| R d \S ] + M[−X A| R d \S ]
in terms of the conditional essential supremum M[· A R d \S ] given σ(A| R d \S ), as introduced in [1]. Alternatively, we may simply define ∂ osc A,S X(A) as the measurable envelope of (2.1).
• The (integrated) functional (or Malliavin) derivative ∂ fct is the closest generalization of the usual partial derivatives commonly used in the discrete setting. Let us denote by M ⊂ L ∞ (R d ) some open set such that the random field A takes its values in M . Given a σ(A)-measurable random variable X(A), and given an extensionX
: M → R, its Fréchet derivative ∂X(A)/∂A ∈ L 1 loc (R d ) is defined for any compactly supported perturbation δA ∈ L ∞ (R d ) by lim t→0X (A + tδA) −X(A) t =ˆR d δA(x) ∂X (A) ∂A (x) dx,
if the limit exists. Since we are interested in the local averages of this derivative, we rather define for all bounded Borel subset S ⊂ R d ,
∂ fct A,S X(A) =ˆS ∂X(A) ∂A (x) dx.
This derivative is additive with respect to the set S: for all disjoint Borel subsets
S 1 , S 2 ⊂ R d we have ∂ fct A,S 1 ∪S 2 X(A) = ∂ fct A,S 1 X(A) + ∂ fct A,S 2 X(A).
It is clear by definition that the oscillation dominates the Glauber derivative. Henceforth we use the notation∂ for any of the above-defined (wide-sense) derivatives with respect to the random field A. We define weighted functional inequalities as follows.
Definition 2.1. Given an integrable function π : R + → R + , we say that A satisfies the weighted spectral gap (∂-WSG) with weight π if for all σ(A)-measurable random variable X(A) we have
Var [X(A)] ≤ E ˆ∞ 0ˆR d ∂ A,B ℓ+1 (x) X(A) 2 dx (ℓ + 1) −d π(ℓ) dℓ ;
it satisfies the weighted covariance inequality (∂-WCI) with weight π if for all σ(A)-measurable random variables X(A) and Y (A) we have
Cov [X(A); Y (A)] ≤ˆ∞ 0ˆR d E ∂ A,B ℓ+1 (x) X(A) 2 1 2 E ∂ A,B ℓ+1 (x) Y (A) 2 1 2 dx (ℓ + 1) −d π(ℓ) dℓ;
it satisfies the weighted logarithmic Sobolev inequality (∂-WLSI) with weight π if for all σ(A)-measurable random variable Z(A) we have
Ent Z(A) 2 := E Z(A) 2 log Z(A) 2 − E Z(A) 2 log E Z(A) 2 ≤ E ˆ∞ 0ˆR d ∂ A,B ℓ+1 (x) Z(A) 2 dx (ℓ + 1) −d π(ℓ) dℓ .
Standard functional inequalities (spectral gap (SG), covariance (CI), and logarithmic Sobolev inequality (LSI)) are recovered by taking a compactly supported weight π (or equivalently, skipping the integral over ℓ). Classical arguments yield the following sufficient criterion for standard functional inequalities. A standard proof is included for completeness in Appendix A and will be referred to at several places in this contribution. (Note that the logarithmic Sobolev inequality (LSI) is only established here with the oscillation ∂ osc , while the version with the Glauber derivative ∂ G is well-known to be much more restrictive, crucially depending on the law of the underlying product structure.) Proposition 2.3. Let A 0 be a random field on R d with values in some measurable space such that restrictions A 0 | S and A 0 | T are independent for all disjoint Borel subsets S, T ⊂ R d . Let A be a random field on R d that is an R-local transformation of A 0 , in the sense that for all S ⊂ R d the restriction A| S is σ(A 0 | S+B R )-measurable. Then, the field A satisfies (∂ G -CI) and (∂ osc -LSI) with radius R + ε for all ε > 0.
Note that any field satisfying the assumption in the above criterion has finite range of dependence. Conversely any field that satisfies (CI) has necessarily finite range of dependence (cf. [7, Proposition 2.3]). One does not expect, however, finite range of dependence to be a sufficient condition for the validity of (SG) in general (compare indeed with the constructions in [5,3]).
Although the Glauber derivative ∂ G and the functional derivative ∂ fct are particularly convenient measures of sensitivity of a random variable X(A) with respect to local restrictions of A, they are essentially only adapted to product structures and to Gaussian-like random fields, respectively. On the other hand, the oscillation ∂ osc is adapted to a much larger variety of random fields (cf. Section 2.3), but it involves taking (essential) suprema, which might be difficult to control in various applications (and in particular in stochastic homogenization, cf. [8]).
In the course of the article, we consider various classes of random fields on R d that can be constructed as (possibly random) projections of random fields having a product structure in a higher-dimensional space R d × R l . Such projections naturally allow one to "deform" the underlying Glauber derivative in a way that cannot be strictly speaking written as a Glauber derivative, but which shares important properties (and in particular avoids taking suprema). The following definition (which can be skipped at the first reading) gives such a proxy for the Glauber derivative, which can typically be used in functional inequalities with loss of integrability.
Definition 2.4. Given l ≥ 0, let X be some random field on R d × R l with values in some measure space, and assume that the random field A under consideration is σ(X )measurable, A = A(X ). Choose X ′ an i.i.d. copy of the field X , and for all x, t let the perturbed field X x,t be defined by X
x,t | (R d ×R l )\(Q d (x)×Q l (t)) = X | (R d ×R l )\(Q d (x)×Q l (t)) and X x,t | Q d (x)×Q l (t) = X ′ | Q d (x)×Q l (t) .
We use the short-hand notation
∂ dis ℓ,x,t X(A) := (X(A) − X(A(X x,t ))1 A| R d \Q 2ℓ+1 (x) =A(X x,t )| R d \Q 2ℓ+1 (x) ,(2.2)
which we abusively call a discrete derivative, and we define a spectral gap with loss (∂ dis -WSG') as follows: given a family (π λ ) λ of integrable functions π λ : R l × R + → R + , the spectral gap with loss (∂ dis -WSG') with weights (π λ ) λ is said to hold if for all σ(A)measurable random variables X(A) and all λ ∈ (0, 1) we have
Var [X(A)] ≤ˆ∞ 0ˆR dˆRl E ∂ dis ℓ,x,t X(A) 2 1−λ 1−λ π λ (t, ℓ)dtdxdℓ.
2.2.
Transformation of product structures. Let the random field A on R d be σ(X )measurable for some random field X defined on some measure space X and with values in some measurable space M . Assume that we have a partition X = x∈Z d ,t∈Z l X x,t , on which X is completely independent, that is, the family of restrictions (X | Xx,t ) x∈Z d ,t∈Z l are all independent.
In the sequel, the case l = 0 (that is, the case when there is no variable t) is referred to as the non-projective case, while the case l ≥ 1 is referred to as the projective case. Note however that the non-projective case is a particular case of the projective one, simply defining X x,0 = X x and X x,t = ∅ for all t = 0. The random field X can be e.g. a random field on R d ×R l with values in some measure space (choosing X = R d ×R l , X x,t = Q d (x)×Q l (t), and M the space of values), or a random point process (or more generally a random measure) on
R d ×R l ×X ′ for some measure space X ′ (choosing X = Z d ×Z l ×X ′ , X x,t = {x}×{t}×X ′ , and M the space of measures on Q d × Q l × X ′ ).
Let X ′ be some given i.i.d. copy of X . For all x, t, we define a perturbed random field X x,t by setting X x,t | X\Xx,t = X | X\Xx,t and X x,t | Xx,t = X ′ | Xx,t . By complete independence, the random fields X and X x,t (resp. A = A(X ) and A(X x,t )) have the same law. Arguing as in the proof of Proposition 2.3 (cf. (A.3) and (A.4) in Appendix A), the complete independence assumption ensures that X satisfies the following standard functional inequalities. Proposition 2.5. For all σ(X )-measurable random variables Y (X ) and Z(X ), we have
Var [Y (X )] ≤ 1 2 x∈Z d t∈Z l E Y (X ) − Y (X x,t ) 2 , (2.3) Ent[Y (X )] ≤ 2 x∈Z d t∈Z l E sup ess X ′ Y (X ) − Y (X x,t ) 2 , (2.4) Cov [Y (X ); Z(X )] ≤ 1 2 x∈Z d t∈Z l E Y (X ) − Y (X x,t ) 2 1 2 E Z(X ) − Z(X x,t ) 2 1 2 . (2.5)
2.3. Abstract criteria and action radius. We now describe general situations for which the functional inequalities for the hidden product structure X are deformed into weighted inequalities for the random field A. We distinguish the following two cases:
• deterministic localization, that is, when the random field A is a deterministic convolution of some product structure, so that the dependence pattern is prescribed deterministically a priori; it leads to weighted functional inequalities with the functional derivative ∂ fct ; • random localization, that is, when the dependence pattern is encoded by the underlying product structure X itself (and therefore may depend on the realization, whence the terminology "random"); the localization of the dependence pattern is then measured in terms of what we call the action radius; it leads to weighted inequalities with the derivatives ∂ osc and ∂ dis , and generalizes the idea of local transformations of Proposition 2.3.
The case of deterministic localization essentially concerns Gaussian fields, which have been thoroughly studied in the literature. Weighted functional inequalities for such random fields then follow from standard functional inequalities (typically formulated in terms of Malliavin calculus on Wiener space, see e.g. [12,13,19]) combined with a deterministic radial change of variables to reformulate the RHS (extracting a 1D weight from Hilbert norms encoding the covariance structure, see the proof of Theorem B.2 below). The RHS of weighted functional inequalities is indeed more explicit (and flexible when it turns to estimates -see e.g. bounds by duality in [8]). A self-contained approach to deterministic localization is included in Appendix B.
In the rest of this section we focus on the more original setting of random localization (which involves a random change of variable, due to the randomness of the dependence pattern). More precisely, we introduce the notion of action radius as a probabilistic measure of the localization of the dependence pattern. General criteria for weighted spectral gaps are then obtained in terms of the properties of this action radius. Various examples that are included in this framework are described in Section 3 below.
We use the notation of Section 2.2: A is a σ(X )-measurable random field on R d , where X is a completely independent random field on some measure space X = x∈Z d ,t∈Z l X x,t with values in some measurable space M . The following definition is inspired by the notion of stabilization radius first introduced by Lee [15,16] and crucially used in the works by Penrose, Schreiber, and Yukich on random sequential adsorption processes [22,21,23,25] (see also [14]).
Definition 2.6. Given an i.i.d. copy X ′ of the field X , an action radius for A with respect to X on X x,t (with reference perturbation X ′ ), if it exists, is defined as a nonnegative σ(X , X ′ )-measurable random variable ρ such that we have a.s.,
A(X x,t ) R d \(Q(x)+Bρ) = A(X )| R d \(Q(x)+Bρ) ,
where as before the perturbed random field X x,t is defined by X x,t | X\Xx,t := X | X\Xx,t and X x,t | Xx,t := X ′ | Xx,t .
Note that if X = A 0 is a random field on R d , and if for some R > 0 the random field A is an R-local transformation of A 0 in the sense of Proposition 2.3, then the constant ρ = R is an action radius for A with respect to A 0 on any set. Reinterpreted in the case when X = P is a random point process on R d × R l × X ′ for some measure space X ′ , the above definition takes on the following guise: given a subset E × F ⊂ R d × R l and given an i.i.d. copy P ′ of P, an action radius for A with respect to P on E × F , if it exists, is a nonnegative random variable ρ such that we have a.s.,
A P \ (E × F × X ′ ) P ′ ∩ (E × F × X ′ ) R d \(E+Bρ) = A(P)| R d \(E+Bρ) .
We display two general results, Theorems 2.7 and 2.9 below. The first result is a general criterion for the validity of weighted spectral gaps in terms of the properties of an action radius, whereas the second result is based on more elaborate properties of action radii and is useful to avoid loss of integrability in some situations. Note that the condition for the validity of the weighted logarithmic Sobolev inequality below is rather stringent (see Section 3 for examples).
Theorem 2.7. Let the fields A, X be as above. Given an i.i.d. copy X ′ of the field X , assume that:
(a) For all x, t, there exists an action radius ρ x,t for A with respect to X in X x,t . (b) The transformation A of X is stationary, that is, the random fields A(X (· + z, ·)) and A(X )(· + z) have the same law for all z ∈ Z d . Moreover, the law of the action radius ρ x,t is independent of x.
Then the following holds.
(i) Setting π(t, ℓ) :
= P ℓ − 1 ≤ ρ 0,t < ℓ, A(X 0,t ) = A(X ) ,
we have for all σ(A)-measurable random variable Z(A) and all λ ∈ (0, 1),
Var [Z(A)] ≤ 1 2 x∈Z d ∞ ℓ=1 t∈Z l π(t, ℓ) λ E ∂ dis ℓ,x,t Z(A) 2 1−λ 1−λ (2.6) and Cov [Y (A); Z(A)] ≤ 1 2 x∈Z d t∈Z l ∞ ℓ=1 π(t, ℓ) λ E ∂ dis ℓ,x,t Y (A) 2 1−λ 1−λ 1 2 × ∞ ℓ ′ =1 π(t, ℓ ′ ) λ E ∂ dis ℓ ′ ,x,t Z(A) 2 1−λ 1−λ 1 2 , (2.7)
where ∂ dis ℓ,x,t Z(A) is the notation defined in (2.2), that is,
∂ dis ℓ,x,t Z(A) := Z(A) − Z(A(X x,t ) 1 A| R d \Q 2ℓ+1 (x) =A(X x,t )| R d \Q 2ℓ+1 (x) .
In particular, for all λ ∈ (0, 1), if we set
π λ (ℓ) := (ℓ + 1) d t∈Z l P ℓ − 1 ≤ ρ 0,t < ℓ, A(X 0,t ) = A(X ) λ ,
we obtain for all σ(A)-measurable random variables Z(A),
Var [Z(A)] ≤ 1 2 ∞ ℓ=1 (ℓ + 1) −d π λ (ℓ) x∈Z d E ∂ osc A,Q 2ℓ+1 (x) Z(A) 2 1−λ 1−λ .
(2.8)
If in addition the random variable ρ x,t is σ(X )-measurable for all x, t, then we have
Ent[Z(A)] ≤ 2 ∞ ℓ=1 (ℓ + 1) −d π λ (ℓ) x∈Z d E ∂ osc A,Q 2ℓ+1 (x) Z(A) 2 1−λ 1−λ .
(2.9)
(ii) Assume that for all x, t the action radius ρ x,t is independent of
A| R d \(Q(x)+B f (ρ x,t ) )
for some influence function f : R + → R + with f (u) ≥ u for all u. Then, with the convention 0/0 = 0, if we set
π(t, ℓ) := P X 0,t = X P ℓ − 1 ≤ ρ 0,t < ℓ X 0,t = X P [ρ 0,t < ℓ] , π(ℓ) := (ℓ + 1) d t∈Z lπ (t, ℓ),
we have for all σ(A)-measurable random variables Z(A),
Var [Z(A)] ≤ 1 2 ∞ ℓ=1 (ℓ + 1) −d π(ℓ) x∈Z d E ∂ osc A,Q 2f (ℓ)+1 (x) Z(A) 2 (2.10) and Cov [Y (A); Z(A)] ≤ 1 2 x∈Z d t∈Z l ∞ ℓ=1π (t, ℓ) E ∂ osc A,Q 2f (ℓ)+1 (x) Y (A) 2 1 2 × ∞ ℓ ′ =1π (t, ℓ ′ ) E ∂ osc A,Q 2f (ℓ ′ )+1 (x) Z(A) 2 1 2 . (2.11)
If in addition the random variable ρ x,t is σ(X )-measurable for all x, t, then we have
Ent[Z(A)] ≤ 2 ∞ ℓ=1 (ℓ + 1) −d π(ℓ) x∈Z d E ∂ osc A,Q 2f (ℓ)+1 (x) Z(A) 2 .
(2.12)
Remark 2.8. The covariance inequalities (2.7) and (2.11) are not in the canonical form of Definition 2.1. However note that ifπ(t, ℓ) is non-increasing with respect to ℓ then the inequality (2.11) (and likewise for (2.7)) easily leads to
Cov [Y (A); Z(A)] ≤ x∈Z d ∞ ℓ=1 (ℓ + 1) −d ℓ ℓ ′ =1 π(ℓ ′ ) E ∂ osc A,Q 2f (ℓ)+1 (x) Y (A) 2 1 2 × E ∂ osc A,Q 2f (ℓ)+1 (x) Z(A) 2 1 2 ,
which is now in the correct form, although the weight ℓ ℓ ′ =1 π(ℓ ′ ) seems to be suboptimal whenever π has algebraic decay.
We now turn to a more complex situation when the dependence pattern is intricate but sufficiently well controlled in terms of a family of action radii. The aim of the following is to avoid the loss of integrability which would follow from Theorem 2.7(i) in the case of the random parking process and of Poisson tessellations.
Theorem 2.9. Let A = A(X ) be a σ(X )-measurable random field on R d , where X is a completely independent random field on some measure space X = x∈Z d X x with values in some measurable space M . For all x ∈ Z d , ℓ ∈ N, set X ℓ x := y∈Z d :|x−y|∞≤ℓ X y .
Given an i.i.d. copy X ′ of the field X , let the perturbed field X x,ℓ be defined by
X x,ℓ | X\X ℓ x = X | X\X ℓ x , and X x,ℓ | X ℓ x = X ′ | X ℓ
x , and assume that:
(a) For all x, ℓ, there exists an action radius ρ ℓ x for A with respect to X in X ℓ x , that is, a nonnegative random variable ρ ℓ x such that we have a.s.,
A(X x,ℓ )| R d \(Q 2ℓ+1 (x)+B ρ ℓ x ) = A(X )| R d \(Q 2ℓ+1 (x)+B ρ ℓ x ) . (b)
The transformation A of X is stationary, that is, the random fields A(X (· + z, ·)) and A(X )(· + z) have the same law for all z ∈ Z d . Moreover, the law of the action radius ρ ℓ x is independent of x. Further assume that
(c) For all x, ℓ, the random variable ρ ℓ x is σ X X ℓ+ρ ℓ x x \X ℓ x -measurable.
(In particular, for all x, ℓ, R, given the event ρ ℓ x ≤ R, the random variables ρ ℓ x and ρ ℓ+R x are independent.)
Let R ≥ 1 be chosen large enough so that
sup ℓ≥R P ρ ℓ x ≥ ℓ ≤ 1 4 , (2.13)
let π 0 : R + → R + be a non-increasing function such that P ℓ/4 ≤ ρ ℓ 0 x < ℓ ≤ π 0 (ℓ) holds for all 0 ≤ ℓ 0 ≤ ℓ/4, and define the weight
π(ℓ) := (ℓ + 1) d 1, if ℓ ≤ 4R; 8ℓ −1 π 0 (ℓ/4), if ℓ > 4R.
Then for all σ(A)-measurable random variables Y (A), Z(A), we have
Var [Z(A)] ≤ 1 2ˆ∞ 0ˆR d E ∂ osc A,B √ d(2ℓ+3) (x) Z(A) 2 dx (ℓ + 1) −d π(ℓ)dℓ, (2.14) Ent[Z(A)] ≤ 2ˆ∞ 0ˆR d E ∂ osc A,B √ d(2ℓ+3) (x) Z(A) 2 dx (ℓ + 1) −d π(ℓ)dℓ, (2.15) Cov [Y (A); Z(A)] ≤ 1 2ˆRd ˆ∞ 0 E ∂ osc A,B √ d(2ℓ+3) (x) Y (A) 2 (ℓ + 1) −d π(ℓ)dℓ 1 2 × ˆ∞ 0 E ∂ osc A,B √ d(2ℓ+3) (x) Z(A) 2 (ℓ + 1) −d π(ℓ)dℓ 1 2 dx. (2.16)
Remark 2.10. We briefly address the claim contained in Remark 2.2 in the context of examples of random fields with random localization. By definition, for all L ≥ 1, an action radius for A with respect to X on X 0,t is always also an action radius for the rescaled field A L := A(L·) with respect to X on X 0,t . This proves that in Theorems 2.7 and 2.9 any result stated for the field A also holds in the very same form (with the same constants and weights) for A L with L ≥ 1.
We start with the proof of Theorem 2.7, and then turn to the proof of Theorem 2.9.
Proof of Theorem 2.7. Recall that for all x, t the perturbed random field X x,t is defined by X x,t | X\Xx,t = X | X\Xx,t and X x,t | Xx,t = X ′ | Xx,t . By complete independence of X , the fields X and X x,t (hence A = A(X ) and A(X x,t )) have the same law. The strategy of the proof consists in deforming the functional inequalities of Proposition 2.5 with respect to the transformation A(X ) in terms of the action radius. We split the proof into four steps.
Step 1. Proof of the spectral gap (2.6).
Conditioning the RHS of (2.3) with respect to the values of the action radius ρ x,t , applying the Hölder inequality, and using the stationarity assumption (b) to recognize the weight π(t, ℓ), we obtain for all 0 < λ < 1,
Var [Z(A)] (2.3) ≤ 1 2 x∈Z d t∈Z l E Z(A) − Z(A(X x,t )) 2 = 1 2 x∈Z d ∞ ℓ=1 t∈Z l E Z(A) − Z(A(X x,t )) 2 1 ℓ−1≤ρx,t<ℓ (2.17) ≤ 1 2 x∈Z d ∞ ℓ=1 t∈Z l π(t, ℓ) λ E Z(A) − Z(A(X x,t )) 2 1−λ 1 ρx,t<ℓ 1−λ .
Noting that the event ρ x,t < ℓ entails that
A| R d \Q 2ℓ+1 (x) = A(X x,t )| R d \Q 2ℓ+1 (x)
, the above can be rewritten as follows,
Var [Z(A)] ≤ 1 2 x∈Z d ∞ ℓ=1 t∈Z l π(t, ℓ) λ E ∂ dis ℓ,x,t Z 2 1−λ 1−λ ,
that is, (2.6).
Step 2. Proof of the spectral gap (2.10). For all x, t, conditioning with respect to the values of ρ x,t , we may decompose
E Z(A) − Z(A(X x,t )) 2 = g 1 x (t) + g 2 x (t), (2.18) 1 X | X x,t =X ′ | X x,t ℓ − 1 ≤ ρ x,t < ℓ P [ℓ − 1 ≤ ρ x,t < ℓ] .
By definition, given ρ x,t < ℓ, the restriction A| R d \Q 2f (ℓ)+1 (x) is independent of X | Xx,t and X ′ | Xx,t . The above thus yields
g 1 x (t) ≤ ∞ ℓ=2 E ∂ osc A,Q 2f (ℓ)+1 (x) Z(A) 2 ℓ − 1 ≤ ρ x,t < ℓ × P ℓ − 1 ≤ ρ x,t < ℓ, X | Xx,t = X ′ | Xx,t .
By assumption in item (ii), the restriction A| R d \Q 2f (ρ x,t )+1 (x) is independent of ρ x,t , so that we may deduce
g 1 x (t) ≤ ∞ ℓ=2 E ∂ osc A,Q 2f (ℓ)+1 (x) Z(A) 2 ρ x,t < ℓ P ℓ − 1 ≤ ρ x,t < ℓ, X | Xx,t = X ′ | Xx,t .
To simplify notation, we set for all ℓ ≥ 1,
Y ℓ := ∂ osc A,Q 2f (ℓ)+1 (x) Z(A) 2 . Estimating E [ Y ℓ ρ x,t < ℓ] ≤ E [Y ℓ ] P [ρ x,t < ℓ] ,
and using the stationarity assumption (b) for the action radius, we may conclude
g 1 x (t) ≤ ∞ ℓ=2 E [Y ℓ ] P [ρ x,t < ℓ] P ℓ − 1 ≤ ρ x,t < ℓ, X | Xx,t = X ′ | Xx,t = ∞ ℓ=2 E [Y ℓ ] P [ρ 0,t < ℓ] P ℓ − 1 ≤ ρ 0,t < ℓ, X | X 0,t = X ′ | X 0,t = ∞ ℓ=2 E ∂ osc A,Q 2f (ℓ)+1 (x) Z(A) 2 P X | X 0,t = X ′ | X 0,t × P ℓ − 1 ≤ ρ 0,t < ℓ X | X 0,t = X ′ | X 0,t P [ρ 0,t < ℓ] . (2.19)
We now turn to the estimate of the term g 2 x (t).
Since the influence function f satisfies f (u) ≥ u for all u, we find
g 2 x (t) = E Z(A) − Z(A(X x,t )) 2 1 X | X x,t =X ′ | X x,t 1 ρx,t<1 ≤ E ∂ osc A,Q 2f (1)+1 (x) Z(A) 2 1 X | X x,t =X ′ | X x,t ρ x,t < 1 P [ρ x,t < 1] . By definition, given ρ x,t < 1, the restriction A| R d \Q 2f (1)+1 (x) is independent of X | Xx,t and X ′ | Xx,t .
The above thus yields
g 2 x (t) ≤ E ∂ osc A,Q 2f (1)+1 (x) Z(A) 2 P X | Xx,t = X ′ | Xx,t ρ x,t < 1 = E ∂ osc A,Q 2f (1)+1 (x) Z(A) 2 P X | Xx,t = X ′ | Xx,t P ρ x,t < 1 X | Xx,t = X ′ | Xx,t P [ρ x,t < 1] .
Using the stationarity assumption (b) again, and combining this with (2.18) and (2.19), the conclusion (2.10) follows.
Step 3. Proof of the logarithmic Sobolev inequalities (2.9) and (2.12).
Conditioning the RHS of (2.4) with respect to the values of the action radius ρ x,t , we obtain
Ent[Z(A)] ≤ 2 x∈Z d ∞ ℓ=1 t∈Z l E sup ess X ′ Z(A(X )) − Z(A(X x,t )) 2 1 ℓ−1≤ρx,t<ℓ ≤ 2 x∈Z d ∞ ℓ=1 t∈Z l E ∂ osc A,Q 2ℓ+1 (x) Z(A) 2 sup ess X ′ 1 ℓ−1≤ρx,t<ℓ .
Hence, if for all x, t the random variable ρ x,t is σ(X )-measurable, we may deduce
Ent[Z(A)] ≤ 2 x∈Z d ∞ ℓ=1 t∈Z l E ∂ osc A,Q 2ℓ+1 (x) Z(A) 2 1 ℓ−1≤ρx,t<ℓ .
The result (2.9) follows from the Hölder inequality, while the result (2.12) follows as in
Step 2.
Step 4. Proof of the covariance inequalities (2.7) and (2.11).
Conditioning the RHS of (2.5) with respect to the values of the action radius ρ x,t , we obtain
Cov [Y (A); Z(A)] ≤ 1 2 x∈Z d t∈Z l ∞ ℓ=1 E Y (A) − Y (A(X x,t )) 2 1 ℓ−1≤ρx,t<ℓ 1 2 × ∞ ℓ ′ =1 E Z(A) − Z(A(X x,t )) 2 1 ℓ ′ −1≤ρx,t<ℓ ′ 1 2 .
Now the sums over ℓ, ℓ ′ are estimated exactly as in Steps 1 and 2, and the results (2.7) and (2.11) follow.
We now prove Theorem 2.9.
Proof of Theorem 2.9. We only prove the spectral gap (2.14). The proof of the logarithmic Sobolev inequality (2.15) and of the covariance inequality (2.16) is similar, based on (2.4) and (2.5), respectively. For all x, let the field X x be defined by X x | X\Xx = X | X\Xx and X x | Xx = X ′ | Xx , and recall that the spectral gap (2.3) for X takes the form
Var [Z(A)] ≤ 1 2 x∈Z d E Z(A) − Z(A(X x )) 2 .
The conclusion (2.14) then follows provided we prove that for all x ∈ Z d ,
E Z(A) − Z(A(X x )) 2 ≤ˆ∞ 0 E ∂ osc A,Q 2ℓ+1 (x) Z(A) 2 (ℓ + 1) −d π(ℓ)dℓ. (2.20)
Without loss of generality, it suffices to consider the case x = 0. Moreover, by an approximation argument, we may assume that the random variable Z(A) is bounded. For simplicity, we set ρ(r) := r + ρ r 0 and ∂ osc r :=∂ osc A,Q 2r+1 . Note that the choice (2.13) of R then takes the form
sup ℓ≥R P ρ(ℓ) ≥ 2ℓ ≤ 1 4 . (2.21)
We split the proof into two steps.
Step 1. Conditioning argument. In this step, we prove for all r 2 ≥ 2r 1 ≥ 2R,
E ∂ osc r 2 Z(A) 2 1 1 2 r 2 ≤ρ(r 1 )<r 2 ≤ 2 P 1 2 r 2 ≤ ρ(r 1 ) < r 2 × E ∂ osc 2r 2 Z(A) 2 + ∞ ℓ=2 E ∂ osc 2 ℓ r 2 Z(A) 2 1 2 ℓ−1 r 2 ≤ρ(r 2 )<2 ℓ r 2 . (2.22)
Conditioning the LHS with respect to the value of ρ(r 2 ), we decompose
E ∂ osc r 2 Z(A) 2 1 1 2 r 2 ≤ρ(r 1 )<r 2 ≤ E ∂ osc r 2 Z(A) 2 1 1 2 r 2 ≤ρ(r 1 )<r 2 1 ρ(r 2 )<2r 2 + ∞ ℓ=2 E ∂ osc r 2 Z(A) 2 1 1 2 r 2 ≤ρ(r 1 )<r 2 1 2 ℓ−1 r 2 ≤ρ(r 2 )<2 ℓ r 2 . (2.23)
We estimate each of the RHS terms separately. For that purpose, note that the definition of ρ and assumption (c) ensure that, given ρ(ℓ 1 ) ≤ ℓ 2 and ρ(ℓ 2 ) ≤ ℓ 3 , the random variable
ρ(ℓ 1 ) is independent of ∂ osc ℓ 3 Z(A). This observation directly yields E ∂ osc r 2 Z(A) 2 1 1 2 r 2 ≤ρ(r 1 )<r 2 1 ρ(r 2 )<2r 2 ≤ E ∂ osc 2r 2 Z(A) 2 1 ρ(r 1 )≥ 1 2 r 2 ρ(r 1 ) < r 2 , ρ(r 2 ) < 2r 2 P ρ(r 1 ) < r 2 , ρ(r 2 ) < 2r 2 ≤ E ∂ osc 2r 2 Z(A) 2 P 1 2 r 2 ≤ ρ(r 1 ) < r 2 P [ρ(r 1 ) < r 2 , ρ(r 2 ) < 2r 2 ] ≤ E ∂ osc 2r 2 Z(A) 2 P 1 2 r 2 ≤ ρ(r 1 ) < r 2 1 − P [ρ(r 1 ) ≥ r 2 ] − P [ρ(r 2 ) ≥ 2r 2 ]
.
For r 2 ≥ 2r 1 ≥ 2R, the choice (2.21) of R yields P [ρ(r 1 ) ≥ r 2 ] + P [ρ(r 2 ) ≥ 2r 2 ] ≤ P [ρ(r 1 ) ≥ 2r 1 ] + P [ρ(r 2 ) ≥ 2r 2 ] ≤ 1 2 ,
so that the above takes the simpler form
E ∂ osc r 2 Z(A) 2 1 1 2 r 2 ≤ρ(r 1 )<r 2 1 ρ(r 2 )<2r 2 ≤ 2 E ∂ osc 2r 2 Z(A) 2 P 1 2 r 2 ≤ ρ(r 1 ) < r 2 . (2.24)
On the other hand, further recalling that assumption (c) ensures that given ρ(ℓ 1 ) ≤ ℓ 2 the random variables ρ(ℓ 1 ) and ρ(ℓ 2 ) are independent, we similarly obtain
E ∂ osc r 2 Z(A) 2 1 1 2 r 2 ≤ρ(r 1 )<r 2 1 2 ℓ−1 r 2 ≤ρ(r 2 )<2 ℓ r 2 ≤ E ∂ osc 2 ℓ r 2 Z(A) 2 1 ρ(r 2 )≥2 ℓ−1 r 2 ρ(r 1 ) < r 2 , ρ(r 2 ) < 2 ℓ r 2 ×P ρ(r 1 ) ≥ 1 2 r 2 ρ(r 1 ) < r 2 , ρ(r 2 ) < 2 ℓ r 2 P ρ(r 1 ) < r 2 , ρ(r 2 ) < 2 ℓ r 2 ≤ E ∂ osc 2 ℓ r 2 Z(A) 2 1 2 ℓ−1 r 2 ≤ρ(r 2 )<2 ℓ r 2 P 1 2 r 2 ≤ ρ(r 1 ) < r 2 P [ρ(r 1 ) < r 2 , ρ(r 2 ) < 2 ℓ r 2 ] ≤ E ∂ osc 2 ℓ r 2 Z(A) 2 1 2 ℓ−1 r 2 ≤ρ(r 2 )<2 ℓ r 2 P 1 2 r 2 ≤ ρ(r 1 ) < r 2 1 − P [ρ(r 1 ) ≥ r 2 ] − P [ρ(r 2 ) ≥ 2 ℓ r 2 ] .
With the choice (2.21) of R, for r 2 ≥ 2r 1 ≥ 2R and ℓ ≥ 1, this turns into
E ∂ osc r 2 Z(A) 2 1 1 2 r 2 ≤ρ(r 1 )<r 2 1 2 ℓ−1 r 2 ≤ρ(r 2 )<2 ℓ r 2 ≤ 2 E ∂ osc 2 ℓ r 2 Z(A) 2 1 2 ℓ−1 r 2 ≤ρ(r 2 )<2 ℓ r 2 P 1 2 r 2 ≤ ρ(r 1 ) < r 2 .
Combining this with (2.23) and (2.24), the conclusion (2.22) follows.
Step 2. Proof of (2.20).
Conditioning the LHS of (2.20) with respect to the value of the action radius ρ(0), we obtain
E Z(A) − Z(A(X x )) 2 ≤ E ∂ osc R Z(A) 2 + ∞ ℓ=1 E ∂ osc 2 ℓ R Z(A) 2 1 2 ℓ−1 R≤ρ(0)<2 ℓ R .
We now iteratively apply (2.22) to estimate the last RHS terms: with the short-hand notation π(ℓ 2 ; ℓ 1 ) := P 1 2 ℓ 2 ≤ ρ(ℓ 1 ) < ℓ 2 , we obtain for all n ≥ 1,
E Z(A) − Z(A(X x )) 2 ≤ E ∂ osc R Z(A) 2 + 2 ∞ ℓ 1 =1 π(2 ℓ 1 R; 0) E ∂ osc 2 ℓ 1 +1 R Z(A) 2 + 2 2 ∞ ℓ 1 =1 π(2 ℓ 1 R; 0) ∞ ℓ 2 =ℓ 1 +2 π(2 ℓ 2 R; 2 ℓ 1 R) E ∂ osc 2 ℓ 2 +1 R Z(A) 2 + . . . +2 n ∞ ℓ 1 =1 π(2 ℓ 1 R; 0) ∞ ℓ 2 =ℓ 1 +2 π(2 ℓ 2 R; 2 ℓ 1 R) . . . ∞ ℓn=ℓ n−1 +2 π(2 ℓn R; 2 ℓ n−1 R) E ∂ osc 2 ℓn+1 R Z(A) 2 + 2 n ∞ ℓ 1 =1 π(2 ℓ 1 R; 0) ∞ ℓ 2 =ℓ 1 +2 π(2 ℓ 2 R; 2 ℓ 1 R) . . . ∞ ℓn=ℓ n−1 +2 π(2 ℓn R; 2 ℓ n−1 R) × ∞ ℓ n+1 =ℓn+2 E ∂ osc 2 ℓ n+1 R Z(A) 2 1 2 ℓ n+1 −1 R≤ρ(2 ℓn R)<2 ℓ n+1 R .
With the choice (2.21) of R in the form
sup ℓ 0 ≥0 ∞ ℓ=ℓ 0 +2 π(2 ℓ R; 2 ℓ 0 R) = sup ℓ 0 ≥0 P ρ(2 ℓ 0 R) ≥ 2 ℓ 0 +1 R ≤ 1 4 ,
the definitionπ(ℓ) := sup ℓ 0 :0≤ℓ 0 ≤ℓ/4 π(ℓ; ℓ 0 ) of the weight, and recalling that the random variable Z(A) is bounded, we deduce
E Z(A) − Z(A(X x )) 2 ≤ E ∂ osc R Z(A) 2 + 2 n−1 m=0 2 −m ∞ ℓ=1π (2 ℓ R) E ∂ osc 2 ℓ+1 R Z(A) 2 + 2 −n−2 Z L ∞ .
Letting n ↑ ∞, we thus obtain
E Z(A) − Z(A(X x )) 2 ≤ E ∂ osc R Z(A) 2 + 4 ∞ ℓ=1π (2 ℓ R) E ∂ osc 2 ℓ+1 R Z(A) 2 .
Comparing sums to integrals and using the definition of π, the conclusion (2.20) follows.
Local operations.
In this subsection, we describe two typical operations on random fields that do preserve functional inequalities: local transformations and gluing of independent random fields with respect to an independent pattern. These operations allow one to generate many variations around the examples of Section 3.
Local transformations. Given a random field
A 0 on R d , we say that a random field A on R d is a R-local transformation of A 0 (as in Proposition 2.3) if A| S is σ(A| S+B R )- measurable for all Borel subsets S ⊂ R d .
Important particular cases are local smoothing (e.g. by convolution with a smooth kernel with bounded support) and truncation (e.g. by applying a Lipschitz function).
Lemma 2.11. If A 0 , A are two random fields on R d , and if A is a R-local transformation of A 0 , then we have for all Borel subsets S ⊂ R d and all σ(A)-measurable random variables
X(A), ∂ osc A 0 ,S X(A(A 0 )) ≤ ∂ osc A,S+B R X(A) and ∂ fct A 0 ,S X(A(A 0 )) ≤ R d ∂A ∂A 0 L ∞ ∂ fct A,S+B R X(A),
so that functional inequalities for A 0 with the oscillation or the functional derivative imply the corresponding functional inequalities for A with the oscillation or the functional derivative (provided ∂A/∂A 0 is bounded if the functional derivative is used).
Proof. By assumption,
A| R d \(S+B R ) is σ(A 0 | R d \S )-measurable, so that the sub-σ-algebra σ(A| R d \(S+B R ) ) is contained in σ(A 0 | R d \B )
, and the inequality follows.
2.4.2. Independent gluing. The following result shows how independent localized fields can be glued together. Since it is a direct consequence of the standard tensorization arguments used e.g. in the proof of Proposition 2.3, details are omitted. 1], and consider the "glued" random field A :
Lemma 2.12. Let A 1 , A 2 , and A 3 be three independent random fields on R d . Assume that |A 1 − A 3 | ≤ C a.s. for some deterministic constant C > 0, that A 2 has values in [0,= A 2 A 1 + (1 − A 2 )A 3 . If A 1 , A 2 ,
and A 3 satisfy different forms of weighted spectral gaps (resp. covariance inequality, resp. logarithmic Sobolev inequality), then the random field A satisfies the worst of these spectral gaps (resp. covariance inequality, resp. logarithmic Sobolev inequality), that is, with the RHS replaced by the sum of the corresponding RHSs.
Examples
In this section we consider four representative examples: Gaussian fields, tessellations associated with a Poisson point process, random parking bounded inclusions, and Poisson or random parking inclusions with unbounded radii. The main results are summarized in the table below.
Example of field Key property Functional inequalities
Gaussian random field covariance function C sup B(x) |C| ≤ c(|x|) (∂ fct -WSG), (∂ fct -WLSI) weight π(ℓ) ≃ (−c ′ (ℓ)) + Poisson tessellations (Voronoi/Delaunay) σ(X )-measurable action radius (∂ osc -WSG), (∂ osc -WLSI) weight π(ℓ) ≃ e − 1 C ℓ d Random parking bounded inclusions σ(X )-measurable action radius & exponential stabilization (∂ osc -WSG), (∂ osc -WLSI) weight π(ℓ) ≃ e − 1 C ℓ Poisson random inclusions with random radii radius law V γ(ℓ) := P [ℓ − 4 ≤ V < ℓ + 2] (∂ osc -WSG) weight π(ℓ) ≃ (ℓ + 1) d γ(ℓ) (and (∂ osc -LSI) if V bounded)
3.1. Gaussian random fields. Gaussian random fields are the main examples of deterministically localized fields as introduced in Section 2.3 (and studied in Appendix B). Note that this result is a weighted reformulation of the "coarsened" functional inequalities used in the first version of [9] for Gaussian fields.
R + → R + , then A satisfies (∂ fct -WSG) and (∂ fct -WLSI) with weight π(ℓ) ≃ (−c ′ (ℓ)) + . If FC ∈ L 1 (R d ) and if sup B(x) |F −1 ( √ FC)| ≤ r(|x|)
holds for some non-increasing Lipschitz function r : R + → R + , then A satisfies (∂ fct -WCI) with weight π(ℓ) ≃ (ℓ + 1) d r(ℓ)(−r ′ (ℓ)).
As shown in the companion article [7, Proposition 2.3], this result is sharp: each sufficient condition is (essentially) necessary.
Proof. Let W denote a Gaussian white noise with intensity 1, that is, a random noise W on R d such that for all bounded Borel subsets E ⊂ R d the random variable W (E) has a centered Gaussian law with variance E[W (E) 2 ] = |E|. As shown in [6, Section XI.8], a stationary Gaussian random field A on R d can be rewritten as a convolution (B.1) with a Gaussian white noise whenever the field A has an absolutely continuous spectral measure, or equivalently, whenever the Fourier transform of the covariance function C is in L 1 (R d ). Under such a restriction on C, since Gaussian random variables satisfy the standard spectral gap (B.3) and logarithmic Sobolev inequality (B.7) (cf. [11]), we can directly apply Proposition B.1 and Theorem B.2 to establish the validity of weighted spectral gaps, covariance, and logarithmic Sobolev inequalities.
It remains to show that this restriction on C can be relaxed in the case of spectral gaps and logarithmic Sobolev inequalities. To this end, it suffices to prove that the conclusion of Proposition B.1 (that is, the validity of Brascamp-Lieb type inequalities) always holds for any jointly measurable Gaussian stationary random field A. This is achieved by an approximation argument. We focus on the Brascamp-Lieb inequality (B.4), while the argument is analogous for (B.8). As an approximation argument shows, it is enough to establish (B.4) for those random variables X(A) that depend on A only via their spatial averages on the partition {Q ε (z)} z∈B R ∩εZ d with ε, R > 0. Let us introduce the following notation for these averages,
A ε (z) := Qε(z) A, for z ∈ εZ d . (3.1)
In this case, the Fréchet derivative { ∂X ∂A (x)} x∈R d and the partial derivatives
{ ∂X ∂Aε(z) } z∈εZ d of X(A) are related via ε d ∂X ∂A (x) = ∂X ∂A ε (z)
,
for x ∈ Q ε (z), z ∈ εZ d .(3.2)
We infer from (3.1) that {A ε (z)} z∈εZ d is a discrete centered Gaussian random field (which is now stationary with respect to the action of εZ d ), characterized by its covariance
C ε (z − z ′ ) := Qε(z) Qε(z ′ ) C(x − x ′ )dx ′ dx. (3.3)
By the discrete result (B.9) obtained in the proof of Proposition B.1 (based on the standard spectral gap for Gaussian random variables [11]), we deduce for all ε, R > 0 and all random variables X(A) that depend on A only via its spatial averages on the partition
{Q ε (z)} z∈B R ∩εZ d , Var [X] ≤ C z∈B R ∩εZ d z ′ ∈B R ∩εZ d |C ε (z − z ′ )| E ∂X ∂A ε (z) ∂X ∂A ε (z ′ ) .
Injecting (3.2) and (3.3), the conclusion (B.4) follows.
3.2.
Poisson random tessellations. In this section, we consider random fields that take i.i.d. values on the cells of a tessellation associated with a stationary random point process P on R d . Such random fields can be formalized as projections of decorated random point processes. Given a point process P on R d and given a random element G with values in some measurable space X, we call decorated random point process associated with P and G a point processP on R d × X defined as follows: choose a measurable enumeration P = {X j } j , pick independently a sequence (G j ) j of i.i.d. copies of the random element G, and setP := {X j , G j } j (that is, in measure notation,P := j δ (X j ,G j ) ). Note that by definitionP is completely independent whenever P is completely independent.
We focus here on the case when the underlying point process P is some Poisson point process P = P 0 on R d with intensity µ = 1. Choose a measurable random field V on R d , corresponding to the values on the cells. We study both Voronoi and Delaunay tessellations.
(1) Voronoi tessellation: LetP 1 := {X j , V j } j denote a decorated point process associated with the random point process P 0 := {X j } j and the random element V (hence (V j ) j is a sequence of i.i.d. copies of the random field V ). We define a σ(P 1 )-measurable random field A 1 as follows,
A 1 (x) = j V j (x)1 C j (x),
where {C j } j denotes the partition of R d into the Voronoi cells associated with the Poisson points {X j } j , that is,
C j := {x ∈ R d : |x − X j | < |x − X k |, ∀k = j}.
(2) Delaunay tessellation: LetṼ := (Ṽ ζ ) ζ denote a family of i.i.d. copies of the random element V , indexed by sets ζ of d + 1 distinct integers. We define a random field A 2 as follows,
A 2 (x) = jṼ ζ(D j ) (x)1 D j (x),
where {D j } j denotes the partition of R d into the Delaunay d-simplices associated with the Poisson points {X j } j (the Delaunay triangulation is indeed almost surely uniquely defined), and where ζ(D j ) denotes the set of the d+1 indices i 1 , . . . , i d+1 of the vertices X i 1 , . . . , X i d+1 of D j .
Since large holes in the Poisson process have exponentially small probability, large cells in the corresponding Voronoi or Delaunay tessellations also have exponentially small probability. This allows one to prove the following weighted functional inequalities with stretched exponential weights.
Proposition 3.2. For s = 1, 2, the above-defined random field A s satisfies (∂ osc -WSG), (∂ osc -WLSI), and (∂ osc -WCI) with weight π(ℓ) = e − 1 C ℓ d for some constant C > 0. Moreover for all σ(A s )-measurable random variables Z(A s ) and all λ ∈ (0, 1) we have
Var [Z(A s )] ≤ C x∈Z d ∞ ℓ=1 e − λ C ℓ d E ∂ dis ℓ,x Z(A s ) 2 1−λ 1−λ , with the notation ∂ dis ℓ,x Z(A s ) defined in (2.2) (with l = 0). Proof.
We focus on the case of the Voronoi tessellation (the argument for the Delaunay tessellation is similar). We shall appeal to Theorem 2.9, and need to construct and control action radii, which we do in two separate steps. (The weighted spectral gap with loss and discrete derivative follows from Theorem 2.7(i).)
Step 1. Definition and properties of the action radius. Let x ∈ R d , ℓ ∈ N be fixed. Changing the point configuration ofP 1
= {X j , V j } j inside Q 2ℓ+1 (x) × R R d only modifies the Voronoi tessellation (hence the field A 1 ) inside the set V P 0 ,ℓ (x) := y ∈ R d : ∃z ∈ Q 2ℓ+1 (x) such that |y − z| ≤ |y − X| for all X ∈ P 0 \ Q 2ℓ+1 (x) .
An action radius for A 1 with respect toP 1 on Q 2ℓ+1 (x) × R R d is thus given by ρ ℓ
x := 2 diam V P 0 ,ℓ (x) + 1 − ℓ, and property (a) of Theorem 2.9 is proved. The stationarity property (b) follows by construction, and it remains to prove the measurability property (c). In particular, we need to prove that ρ ℓ x is σ(P 0 | Q 2(ℓ+ρ ℓ x )+1 (x)\Q 2ℓ+1 (x) )-measurable. Since ρ ℓ x is σ(P 0 | R d \Q 2ℓ+1 (x) )measurable by construction, it remains to prove it is σ(P 0 | Q 2(ℓ+ρ ℓ x )+1 (x) )-measurable. To this aim, letP be an arbitrary locally finite point set and consider the compound point set P 0,ℓ (x) = P 0 | Q 2(ℓ+ρ ℓ x )+1 (x) ∪P| R d \Q 2(ℓ+ρ ℓ x )+1 (x) . The claimed measurability then follows from the identity VP 0,ℓ (x),ℓ (x) = V P 0 ,ℓ (x). We start with the proof that V P 0 ,ℓ (x) ⊂ VP 0,ℓ (x),ℓ (x). Let y ∈ V P 0 ,ℓ (x). Then for all X ∈P 0,ℓ (x)| R d \Q 2(ℓ+ρ ℓ x )+1 (x) we have by the triangle inequality
|X − y| ≥ |X − x| − |x − y| ≥ ℓ + ρ ℓ x − diam V P 0 ,ℓ (x) = diam V P 0 ,ℓ (x) + 1 ≥ |x − y|, so that y ∈ VP 0,ℓ (x),ℓ (x).
Let us turn to the converse inclusion. By definition, V P 0 ,ℓ (x) and VP 0,ℓ (x),ℓ (x) are convex, and thus simply connected. Set η = 1 2 and consider y ∈ (B η + V P 0 ,ℓ (x)) \ V P 0 ,ℓ (x) (the η-fattened boundary of V P 0 ,ℓ (x)). By definition we have y / ∈ V P 0 ,ℓ (x), so that for all z ∈ Q 2ℓ+1 (x) there exists X ∈ P 0 \ Q 2ℓ+1 (x) such that |y − z| > |y − X|. Let us argue that X ∈ Q 2(ℓ+ρ ℓ x )+1 (x). Indeed, by the triangle inequality,
|X − x| ≤ |X − y| + |y − x| < |y − z| + |y − x| ≤ diam V P 0 ,ℓ (x) + η + diam V P 0 ,ℓ (x) + η = ρ ℓ x + ℓ.
Hence, we deduce X ∈P 0,ℓ (x), which in turn implies y / ∈ VP 0,ℓ (x),ℓ (x). This proves the inclusion VP 0,ℓ (x),ℓ (x) ⊂ V P 0 ,ℓ (x) ∪ (R d \ (B η + V P 0 ,ℓ (x))). Combined with the inclusion V P 0 ,ℓ (x) ⊂ VP 0,ℓ (x),ℓ (x) and the fact that both sets are simply connected, this yields the desired identity V P 0 ,ℓ (x) = VP 0,ℓ (x),ℓ (x) and therefore proves the claimed measurability property (c). We then appeal to Theorem 2.9, and it remains to estimate the weights.
Step 2. Control of the weight. By scaling and change of intensity, it is enough to consider ℓ = 0 (we omit the sub-and superscripts ℓ in the notation) and a Poisson point process P 0 of general intensity µ > 0. Denote by C i = {x ∈ R d : x i ≥ 5 6 |x|} the d cones in the canonical directions e i of R d , and consider the 2d cones C ± i := ±(2e i + C i ). By an elementary geometric argument, for some constant C ≃ 1 the following implication holds: for all L > C,
♯ P 0 ∩ C ± i ∩ {x : C ≤ |x i | ≤ L} > 0 for all i and ± =⇒ diam V P 0 (0) ≤ CL.
A union bound then yields for all L > C,
P [diam V P 0 (0) ≥ L] ≤ P ∃1 ≤ i ≤ d, ∃± : ♯ P 0 ∩ C ± i ∩ {x : |x i | ≤ 1 C L} = 0 ≤ 2d e − µ C L d .
Combined with the definition of the action radius in Step 1, this implies the desired estimate.
3.3. Random parking process. In this section we let P be the random parking point process on R d with given radius R > 0. As shown by Penrose [20] (see also [10, Section 2.1]), the random parking point process P can be constructed as a transformation P = Φ(P 0 ) of a Poisson point process P 0 on R d × R + with intensity 1. Let us recall the graphical construction of this transformation Φ. We first construct an oriented graph on the points of P 0 in R d × R + , by putting an oriented edge from (x, t) to (x ′ , t ′ ) whenever B(x, R) ∩ B(x ′ , R) = ∅ and t < t ′ (or t = t ′ and x precedes x ′ in the lexicographic order, say). We say that (x ′ , t ′ ) is an offspring (resp. a descendant) of (x, t), if (x, t) is a direct ancestor (resp. an ancestor) of (x ′ , t ′ ), that is, if there is an edge (resp. a directed path) from (x, t) to (x ′ , t ′ ). The set P := Φ(P 0 ) is then constructed as follows. Let F 1 be the set of all roots in the oriented graph (that is, the points of P 0 without ancestor), let G 1 be the set of points of P 0 that are offsprings of points of F 1 , and let H 1 := F 1 ∪ G 1 . Now consider the oriented graph induced on P 0 \ H 1 , and define F 2 , G 2 , H 2 in the same way, and so on. By construction, the sets (F j ) j and (G j ) j are all disjoint and constitute a partition of P 0 . We finally define P := Φ(P 0 ) := ∞ j=1 F j .
In this setting we show that there exists an action radius with exponential moments for P with respect to P 0 . The proof follows from the exponential stabilization results of [25].
Proposition 3.3.
For all x ∈ Z d and ℓ ≥ 0, the random parking point process P with radius R > 0 as constructed above admits an action radius ρ ℓ x with respect to P 0 on Q 2ℓ+1 (x)×R + , which satisfies for all L ≥ 0,
P[ρ ℓ x ≥ L] ≤ C R (ℓ + 1) d e −L/C R ,
and which is σ P 0 | ((Q 2(ℓ+ρ ℓ x )+1 (x)\Q 2ℓ+1 (x))×R + -measurable.
In particular, the point process P satisfies (∂ osc -WSG), (∂ osc -WLSI), and (∂ osc -WCI) with weight π(ℓ) =: e −ℓ/C R .
Proof. The proof relies on the notion of causal chains defined in the proof of [25, Lemma 3.5] to which we refer the reader. Note that for all consecutive points (x, t) and (y, s) in a causal chain we necessarily have |x − y| < 2R. By definition, it follows that an action radius for P given P 0 on Q 2ℓ+1 (x) × R + can be defined by the maximum of the distances 2R + d(y, Q 2ℓ+1 (x)) on the set of points (y, s) ∈ P 0 such that there exists a causal chain between a point of P 0 in ((Q 2ℓ+1 (x) + B 2R ) \ Q 2ℓ+1 (x)) × R + and (y, s). We denote by ρ ℓ x this maximum. By construction, we note that this random variable ρ ℓ x is σ P 0 | ((Q 2ℓ+1 (x)+B ρ ℓ x )\Q 2ℓ+1 (x))×R + -measurable. It remains to estimate the decay of its probability law. First, note that by definition the event ρ ℓ x > L entails the existence of some (y, s) ∈ P 0 with y ∈ (Q 2ℓ+1 (x) + B L+2R ) \ (Q 2ℓ+1 (x)+B L ) and of a causal chain between a point of ((Q 2ℓ+1 (x)+B 2R )\Q 2ℓ+1 (x))×R + and (y, s). Second, the exponential stabilization result of [25,Lemma 3.5] states that for all z ∈ R d and all L > 0 the probability that there exists (y, s) ∈ Q(z) × R + and a causal chain from a point outside (Q(z) + B L ) × R + towards (y, s) is bounded by
C R e −L/C R . For L ≥ R, covering (Q 2ℓ+1 (x) + B L+2R ) \ (Q 2ℓ+1 (x) + B L ) with C(L + ℓ) d−1 R unit cubes and covering Q 2ℓ+1 (x) + B 2R with C(R + ℓ) d unit
cubes, a union bound then yields
P ρ ℓ x > L ≤ C R L d−1 + ℓ d e −L/C R ≤ C R (ℓ + 1) d e −L/C R .
All the assumptions of Theorem 2.9 are then satisfied with π(ℓ) = C R e −ℓ/C R , and the conclusion follows.
3.4.
Random inclusions with random radii. We consider typical examples of random fields on R d taking random values on random inclusions centered at the points of some random point process P. The inclusions are allowed to have i.i.d. random shapes (hence in particular i.i.d. random radii). For the random point process P, we consider projections Φ(P 0 ) of some Poisson point process P 0 on R d × R l with intensity µ > 0, and shall assume that for all x ∈ Z d the process P admits an action radius ρ x with respect to P 0 on Q(x)×R l .
We turn to the construction of the random inclusions. Let V be a nonnegative random variable (corresponding to the random radius of the inclusions). In order to define the random shapes, we consider the set Y of all nonempty Borel subsets E ⊂ R d with sup x∈E |x| = 1, and endow it with the σ-algebra Y generated by all subsets of the form {E ∈ Y : x 0 ∈ E} with x 0 ∈ R d . Let S be a random nonempty Borel subset of R d with sup x∈S |x| = 1 a.s., that is, a random element in the measurable space Y . (Note that V and S need not be independent.) LetP 0 := {X j , V j , S j } j be a decorated point process associated with the random point process P 0 = {X j } j and the random element (V, S). The collection of random inclusions is then given by {I j } j with I j := X j + V j S j .
It remains to associate random values to the random inclusions. Since inclusions may intersect each other, several constructions can be considered; we focus on the following three typical choices.
(1) Given α, β ∈ R, we setP 1 :=P 0 , and we consider the σ(P 1 )-measurable random field A 1 that is equal to α inside the inclusions, and to β outside. More precisely,
A 1 := β + (α − β)1 j I j .
The simplest example is the random field A 1 obtained for P a Poisson point process on R d with intensity µ = 1, and for S the unit ball centered at the origin in R d ; this is referred to as the Poisson unbounded spherical inclusion model. (2) Let β ∈ R, let f : R → R be a Borel function, and let W be a measurable random field on R d . LetP 2 := {X j , V j , S j , W j } be a decorated point process associated witĥ P 0 and W . We then consider the σ(P 2 )-measurable random field A 2 that is equal to f ( j:x∈I j W j ) at any point x of the inclusions, and to β outside. More precisely,
A 2 (x) := β + f j W j (x)1 I j (x) − β 1 j I j (x).
(Of course, this example can be generalized by considering more general functions than simple sums of the values W j ; the corresponding concentration properties will then remain the same.) (3) Let β ∈ R, let W be a measurable random field on R d , and let U denote a uniform random variable on [0, 1]. LetP 3 := {X j , V j , S j , W j , U j } be a decorated point process associated withP 0 and (W, U ). Given a σ(V S, W )-measurable random variable P (V S, W ), we say that inclusion I j has the priority on inclusion
I i if P (V j S j , W j ) < P (V i S i , W i ) or if P (V j S j , W j ) = P (V i S i , W i ) and U j < U i .
Since the random variables {U j } j are a.s. all distinct, this defines a priority order on the inclusions on a set of maximal probability. Let us then relabel the inclusions and values {(I j , V j )} j into a sequence (I ′ j , V ′ j ) j in such a way that for all j the inclusion I ′ j has the j-th highest priority. We then consider the σ(P 3 )-measurable random field A 3 defined as follows,
A 3 := β + j (W ′ j − β)1 I ′ j \ i:i<j I ′ i .
(Note that this example includes in particular the case when the priority order is purely random (choosing P ≡ 0), as well as the case when the priority is given to inclusions with e.g. larger or smaller radius (choosing P (V S, W ) = V or −V , respectively).)
In each of these three examples, s = 1, 2, 3, the random field A s is σ(P s )-measurable, for some completely independent random point processP s on R d × R l × R + × Y s and some measurable space Y s (the set R d ×R l stands for the domain of the point process P 0 = {X j } j , and the set R + stands for the domain of the radius variables {V j } j ). In order to recast this into the framework of Section 2.2, we may define X s (x, t, v) := P s | Q(x)×Q(t)×Q(v)×Ys , so that X s is a completely independent measurable random field on the space
X = Z d × Z l × Z with values in the space of (locally finite) measures on Q d × Q l × Q 1 × Y s .
Rather than stating a general result, we focus on the representative examples of the Poisson and of the random parking point processes. For the latter, a refined analysis is needed to avoid a loss of integrability. Note that the proof below yields slightly more general statements than contained in the proposition (and can easily be adapted to various other situations).
Proposition 3.4. Set γ(v) := P[v − 1/2 ≤ V < v + 1/2].
(i) Assume that P = P 0 is a Poisson point process on R d with constant intensity µ (hence l = 0). Then, for each s = 1, 2, 3, the above-defined random field A s satisfies (∂ osc -WSG) and (∂ osc -WCI) with weight ℓ → µ (ℓ + 1) d sup 0≤u≤2 γ( 1 √ d ℓ − u). In addition, for all λ ∈ (0, 1),
Cov [Y (A s ); Z(A s )] ≤ (2µ) λ 2 x∈Z d ∞ v=0 γ(v) λ E ∂ dis v+1,x,v Y 2 1−λ 1−λ 2 E ∂ dis v+1,x,v Z 2 1−λ 1−λ 2 , (3.4)
where ∂ dis v+1,x,v Z is the notation defined in (2.2), that is,
∂ dis v+1,x,v Z := Z(A s ) − Z(A s (X x,v ) 1 As| R d \Q 2v+3 (x) =As(X x,v )| R d \Q 2v+3 (x) = Z(A s ) − Z(A s (X x,v )).
In the case when the radius law V is almost surely bounded by a deterministic constant, the standard logarithmic Sobolev inequality (∂ osc -LSI) holds. (ii) Assume that P is a random parking point process on R d with radius R > 0 as constructed in Section 3.3. Then, for each s = 1, 2, 3, the above-defined random field A s satisfies (∂ osc -WSG) with weight π R (ℓ) := C R e −ℓ/C R + (ℓ + 1) d γ(ℓ) . More generally it satisfies the following covariance inequality: for all σ(
A s )-measurable random variables Y (A s ), Z(A s ) we have Cov [Y (A s ); Z(A s )] ≤ˆR d ˆ∞ 0 E ∂ osc As,B 2ℓ+1 (x) Y (A s ) 2 (ℓ + 1) −d π R (ℓ)dℓ 1 2 × ˆ∞ 0 E ∂ osc As,B 2ℓ+1 (x) Z(A s ) 2 (ℓ + 1) −d π R (ℓ)dℓ 1 2 dx. (3.5)
In addition, for all λ ∈ (0, 1),
Cov [Y (A s ); Z(A s )] ≤ x∈Z d ∞ v=0 ∞ ℓ=1 π R (v, ℓ) λ E ∂ dis ℓ,x,v Y 2 1−λ 1−λ 1 2 × ∞ ℓ ′ =1 π R (v, ℓ ′ ) λ E ∂ dis ℓ ′ ,x,v Z 2 1−λ 1−λ 1 2 , (3.6)
where we have set
π R (v, ℓ) := C R γ(v)1 ℓ−1≤v<ℓ + γ(v) ∧ e −ℓ/C R + sup r≥ℓ/2 γ(r) ,
and where ∂ dis ℓ,x,v Z is the notation defined in (2.2), that is,
∂ dis ℓ,x,v Z := Z(A s ) − Z(A s (X x,v ) 1 As| R d \Q 2ℓ+1 (x) =As(X x,v )| R d \Q 2ℓ+1 (x) .
In the case when the radius law V is almost surely bounded by a deterministic constant, the logarithmic Sobolev inequality (∂ osc -WLSI) holds with weight ℓ → C R e −ℓ/C R .
Proof. We split the proof into three steps. We first apply the general results of Theorem 2.7, and then treat more carefully the case of the random parking point process in order to avoid the loss of integrability.
Step 1. Proof of the covariance estimates with loss. Assume for simplicity that the transformation Φ of P 0 into P = Φ(P 0 ) does not add points and does not move points in the direction of R d : more precisely, this means that for any locally finite sequence (x j ) j ⊂ R d × R l we have Φ((x j ) j ) = (p(x j )) j∈I for some subset I of indices (depending on (x j ) j ), where p : R d × R l → R d denotes the projection onto the first factor. Further assume that for all locally finite (
x j ) j ⊂ R d × R l , denoting Φ((x j ) j ) = (p(x j )) j∈I , we have Φ((x j ) j∈J ) = (p(x j )
) j∈I for all subset J ⊃ I. In this step, we show that, for each s = 1, 2, 3, the random field A s satisfies for all σ(A s )-measurable random variables Y (A s ), Z(A s ) and all λ ∈ (0, 1),
Cov [Y (A s ); Z(A s )] ≤ 1 2 x∈Z d ∞ v=0 ∞ ℓ=1 π(v, ℓ) λ E ∂ dis ℓ,x,v Y 2 1−λ 1−λ 1 2 × ∞ ℓ ′ =1 π(v, ℓ ′ ) λ E ∂ dis ℓ ′ ,x,v Z 2 1−λ 1−λ 1 2 , (3.7)
where we have set
π(v, ℓ) := 2 E [♯(P ∩ Q)] + 1 × γ(v)1 ℓ−1≤v<ℓ + γ(v) ∧ E [♯(P ∩ Q 2ρx+1 (x)) P [ ℓ − 1 ≤ ρ x + V < ℓ ρ x ]] ,
and where ∂ dis ℓ,x,v Z is the notation defined in (2.2). Applying this in the case of the random parking process together with Proposition 3.3, the weight becomes
π(v, ℓ) ≤ C R γ(v)1 ℓ−1≤v<ℓ + γ(v) ∧ˆℓ 0 γ(ℓ − r) e −r/C R dr ,(X j ) belongs to (Φ(P 0 ) ∪ Φ(P x,v 0 )) ∩ (Q(x) + B r ) \ Q(x)
. Given the assumptions on the transformation Φ, an action radius for A s with respect to X s on {x} × {v} (or equivalently, with respect toP s on Q(x) × Q(v) × Y s ) is then given by
ρ s x,v := v ∨ (ρ x + max{V j : j ∈ J v (x, ρ x )}) 1 Xs =X x,v s .
In order to prove (3.7), by Theorem 2.7(i), it remains to estimate the corresponding weights. First, for all ℓ ≥ 0, a union bound yields
P [ℓ − 1 ≤ ρ x + max{V j : j ∈ J v (x, ρ x )} < ℓ] ≤ E [♯J v (x, ρ x ) P [ ℓ − 1 ≤ ρ x + V < ℓ ρ x ]] ≤ E [♯((Φ(P 0 ) ∪ Φ(P x,v 0 )) ∩ Q 2ρx+1 (x)) P [ ℓ − 1 ≤ ρ x + V < ℓ ρ x ]] ≤ 2 E [♯(P ∩ Q 2ρx+1 (x)) P [ ℓ − 1 ≤ ρ x + V < ℓ ρ x ]] .
Let us now define I v (x) as the set of all indices j such that either p(X j ) or p(X ′ j ) belongs to (Φ(P 0 ) ∪ Φ(P x,v 0 )) ∩ Q(x). Given the assumptions on the transformation Φ, we may then compute, in terms of the probability law γ
(v) = P [V ∈ Q(v)], P [A s (X x,v s ) = A s (X )] ≤ P [∃j ∈ I v (x) : V j ∈ Q(v)] ≤ γ(v) E [♯I v (x)] ≤ 2γ(v) E [♯(P ∩ Q)] .
Combining the above estimates, we conclude
P ℓ − 1 ≤ ρ s x,v < ℓ, A(X x,v s ) = A(X s ) ≤ 2γ(v) E [♯(P ∩ Q)] ∧ 1 ℓ−1≤v<ℓ + P [ℓ − 1 ≤ ρ x + max{V j : j ∈ J v (x, ρ x )} < ℓ] ≤ 2 E [♯(P ∩ Q)] + 1 × γ(v)1 ℓ−1≤v<ℓ + γ(v) ∧ E [♯(P ∩ Q 2ρx+1 (x)) P [ ℓ − 1 ≤ ρ x + V < ℓ ρ x ]] .
The result (3.7) then follows from Theorem 2.7(i).
Step 2. Proof of (i). We repeat the analysis of Step 1 in the particular case of a Poisson point process P = P 0 on R d with constant intensity µ > 0. In this case, we have ρ x = 0, hence J v (x, r) = ∅, so that the action radius ρ s x,v takes the simpler form
ρ s x,v = v 1 Xs =X x,v s . Estimating P ℓ − 1 ≤ ρ s x,v < ℓ, A s (X x,v s ) = A s (X s ) ≤ P ℓ − 1 ≤ ρ s x,v < ℓ, X x,v s = X s ≤ P [X x,v s = X s ] 1 ℓ−1≤v<ℓ ≤ 2µγ(v) 1 ℓ−1≤v<ℓ ,
the conclusion (3.4) follows from Theorem 2.7(i). It remains to prove (∂ osc -WCI). Since
obviously P ρ s x,v < ℓ = 1 if v < ℓ, we compute for all x ∈ Z d , v ≥ 0, ℓ ≥ 1, P ℓ − 1 ≤ ρ s x,v < ℓ, X x,v s = X P ρ s x,v < ℓ ≤ 2µγ(v) 1 ℓ−1≤v<ℓ P ρ s x,v < ℓ = 2µγ(v) 1 ℓ−1≤v<ℓ ,
and Theorem 2.7(ii) with influence function f (u) = u then yields
Cov [Y (A s ); Z(A s )] ≤ µ x∈Z d ∞ v=0 γ(v) E ∂ osc As,Q 2v+3 (x) Y (A s ) 2 1 2 E ∂ osc As,Q 2v+3 (x) Z(A s ) 2 1 2 .
The desired covariance estimate (∂ osc -WCI) follows by taking local averages.
Step 3. Proof of (ii). In this step, we consider the case when the stationary point process P satisfies a hard-core condition ♯(P ∩ Q) ≤ C a.s. for some deterministic constant C > 0, and also satisfies the following covariance inequality (resp. the corresponding (∂ osc -WSG)) with some integrable weight π 0 : for all σ(P)-measurable random variables Y (P), Z(P),
Cov [Y (P); Z(P)] ≤ˆR d ˆ∞ 0 E ∂ osc P,B ℓ+1 (x) Y (P) 2 (ℓ + 1) −d π 0 (ℓ)dℓ 1 2 × ˆ∞ 0 E ∂ osc P,B ℓ+1 (x) Z(P) 2 (ℓ + 1) −d π 0 (ℓ)dℓ 1 2 dx,
We then show that, for each s = 1, 2, 3, the random field A s satisfies the following covariance inequality (resp. the corresponding (∂ osc -WSG)): for all σ(
A s )-measurable random variables Y (A s ), Z(A s ) we have Cov [Y (A s ); Z(A s )] ≤ CˆR d ˆ∞ 0 E ∂ osc As,B 2ℓ+1 (x) Y (A s ) 2 (ℓ + 1) −d π(ℓ)dℓ 1 2 × ˆ∞ 0 E ∂ osc As,B 2ℓ+1 (x) Z(A s ) 2 (ℓ + 1) −d π(ℓ)dℓ 1 2 dx, (3.8)
where we have set π(ℓ) := π 0 (ℓ)
+ (ℓ + 1) d P [ℓ − 1 ≤ V < ℓ].
In particular, combined with Proposition 3.3, this implies the covariance inequality (3.5) in the case of the random parking point process.
To simplify notation, we only treat the case of the spectral gap inequality. Consider a measurable enumeration of the point process P = {Z j } j , let {Z j , V j , Y s,j } be a decorated point process associated with P and the decoration law (V, Y s ), and let D := {V j , Y s,j } j denote the decoration sequence. Since P and D are independent, the expectation E splits
into E = E P E D , where E P [·] = E[· D]≤ 1 2 k E Z(A s ) − Z(A k s ) 2 +ˆ∞ 0ˆR d E ∂ osc P,B ℓ+1 (x) E D [Z(A s )] 2 dx (ℓ + 1) −d π 0 (ℓ)dℓ, (3.9)
where A k s corresponds to the field A s with the decoration (V k , Y s,k ) replaced by an i.i.d. copy (V ′ k , Y ′ s,k ). We separately estimate the two RHS terms in (3.9), and we begin with the first. For all x ∈ R d , we define the following two random variables,
N (x) := ♯(P ∩ B(x)), R(x) := max{V j : Z j ∈ B(x)}.
Let R 0 ≥ 1 denote the smallest value such that P [V < R 0 ] ≥ 1 2 , which implies in particular by a union bound and by the hard-core assumption
P [R(x) < R 0 ] = E P [V < R 0 ] N (x) ≥ E 2 −N (x) ≥ 2 −C .
(3.10)
Conditioning with respect to the value of R(x), we obtain
k E Z(A s ) − Z(A k s ) 2 ˆ∞ R 0ˆR d k E Z(A s ) − Z(A k s ) 2 1 Z k ∈B(x) 1 ℓ−1≤R(x)<ℓ dx dℓ +ˆR d k E Z(A s ) − Z(A k s ) 2 1 Z k ∈B(x) 1 R(x)<R 0 dx ≤ˆ∞ R 0ˆR d E ∂ osc As,B ℓ+1 (x) Z(A s ) 2 N (x) 1 ℓ−1≤R(x)<ℓ dx dℓ +ˆR d E ∂ osc As,B R 0 +1 (x) Z(A s ) 2 N (x) dx =ˆ∞ R 0ˆR d E ∂ osc As,B ℓ+1 (x) Z(A s ) 2 N (x) 1 R(x)≥ℓ−1 R(x) < ℓ P [R(x) < ℓ] dx dℓ +ˆR d E ∂ osc As,B R 0 +1 (x) Z(A s ) 2 N (x) dx.
Using the hard-core assumption in the form N (x) ≤ C a.s., and noting that given
R(x) < ℓ the random variable R(x) is independent of A s | R d \B ℓ+1 (x) , we deduce k E Z(A s ) − Z(A k s ) 2 ˆ∞ R 0ˆR d E ∂ osc As,B ℓ+1 (x) Z(A s ) 2 P [ℓ − 1 ≤ R(x) < ℓ] P [R(x) < ℓ] dx dℓ +ˆR d E ∂ osc As,B R 0 +1 (x) Z(A s ) 2 dx.
Estimating by a union bound P [ℓ − 1 ≤ R(x) < ℓ] ≤ C P [ℓ − 1 ≤ V < ℓ], and making use of the property (3.10) of the choice of R 0 ≥ 1, we conclude
k E Z(A s ) − Z(A k s ) 2 ˆ∞ R 0ˆR d E ∂ osc As,B ℓ+1 (x) Z(A s ) 2 P [ℓ − 1 ≤ V < ℓ] dx dℓ +ˆR d E ∂ osc As,B R 0 +1 (x) Z(A s ) 2 dx. (3.11)
It remains to estimate the second RHS term in (3.9). The hard-core assumption for P yields by stationarity ♯(P ∩ B ℓ (x)) ≤ Cℓ d a.s. Further noting that a union bound gives
P r − 1 ≤ max 1≤j≤Cℓ d V j < r ≤ Cℓ d j=1 P V j ≥ r − 1, and V k < r ∀1 ≤ k ≤ Cℓ d = Cℓ d P [V < r] Cℓ d −1 P [r − 1 ≤ V < r] ,
and hence for all r ≥ R 0 ,
P r − 1 ≤ max 1≤j≤Cℓ d V j < r P max 1≤j≤Cℓ d V j < r ≤ Cℓ d P [r − 1 ≤ V < r] P [V < r] ≤ 2Cℓ d P [r − 1 ≤ V < r] ,
we find, arguing similarly as above,
∞ 0ˆR d E ∂ osc P,B ℓ (x) E D [Z(A s )] 2 dx (ℓ + 1) −d π 0 (ℓ)dℓ ˆ∞ 0ˆ∞ R 0ˆR d E ∂ osc As,B ℓ+r (x) Z(A s ) 2 dx P [r − 1 ≤ V < r] dr π 0 (ℓ)dℓ +ˆ∞ 0ˆR d E ∂ osc As,B ℓ+R 0 (x) Z(A s ) 2 dx π 0 (ℓ)dℓ. (3.12)
Combining this with (3.9) and (3.11), the conclusion (3.8) follows in variance form.
3.5. Dependent coloring of random geometric patterns. Up to here, besides Gaussian random fields, all the examples of random fields that we have been considering corresponded to random geometric patterns (various random point processes constructed from a higher-dimensional Poisson process or random tessellations) endowed with an independent coloring determining e.g. the size and shape of the cells and the value of the field in the cells. In the present subsection, we turn to the examples of type (III) mentioned at the end of the introduction, and consider dependent colorings of the random geometric patterns. The random field A is now a function of both a product structure (typically some decorated Poisson point processP), and of a random field G (e.g. a Gaussian random field) which typically has long-range correlations but is assumed to satisfy some weighted functional inequality. In other words, this amounts to mixing up all the previous examples. Rather than stating general results in this direction, we only treat a number of typical concrete examples in order to illustrate the robustness of the approach.
(1) The first example A 1 is a random field on R d corresponding to random spherical inclusions centered at the points of a Poisson point process P of intensity µ = 1, with i.i.d. random radii of law V , but such that the values on the inclusions are determined by some random field G 1 with long-range correlations. More precisely, we letP 1 := {X j ,Ṽ j ,Ũ j } j denote a decorated point process associated with P and (V, U ), where U denotes an independent uniform random variable on [0, 1]. Independently ofP 1 we choose a jointly measurable stationary bounded random field G 1 on R d , with typically long-range correlations. The collection of random inclusions is given by {Ĩ j 1 } j withĨ j 1 :=X j +Ṽ j B. As in the third example of Section 3.4, we choose a σ(V, U )-measurable random variable P (V, U ), and we say that the inclusioñ I j 1 has the priority on inclusionĨ i
1 if P (Ṽ j ,Ũ j ) < P (Ṽ i ,Ũ i ) or if P (Ṽ j ,Ũ j ) = P (Ṽ i ,Ũ i ) andŨ j <Ũ i .
This defines a priority order on the inclusions on a set of maximal probability, and we then relabel the inclusions and the points ofP 1 into a sequence (I j 1 , X j , V j , U j ) j such that for all j the inclusion I j 1 has the j-th highest priority. Given β ∈ R, we then consider the σ(P 1 , G 1 )-measurable random field A 1 defined as follows,
A 1 := β + j G 1 (X j ) − β 1 I j 1 \ i:i<j I i 1 .
(2) The second example A 2 is a random field on R d corresponding to random inclusions centered at the points of a Poisson point process P of intensity µ = 1, with i.i.d. random radii of law V , but with orientations determined by some random field G 2 with long-range correlations. More precisely, we letP 2 := {X j , V j } j denote a decorated point process associated with P and V , we choose a reference shape S ∈ B(R d ) with 0 ∈ S, and independently ofP 2 we choose a jointly measurable stationary bounded random field G 2 on R d with values in the orthogonal group O(d) in dimension d, and with typically long-range correlations. The collection of random inclusions is then given by {I j 2 } j with I j 2 := X j + G 2 (X j )S. Given α, β ∈ R, and given a function φ : R → R with φ(t) = 1 for t ≤ 1 and φ(t) = 0 for t ≥ 2, and with φ ′ L ∞ 1, we then consider the σ(P 2 , G 2 )-measurable random field A 2 defined as follows,
A 2 (x) := β + (α − β) φ d x , ∪ j I j 2 .
(Note that the smoothness of this interpolation φ between the values α and β is crucial for the arguments below.) (3) The third example A 3 is a random field on R d corresponding to the Voronoi tessellation associated with the points of a Poisson point process P of unit intensity, such that the values on the cells are determined by some random field G 3 with long-range correlations.
More precisely, we letP 3 := P = {X j } j , and we let {C j } j denote the partition of R d into the Voronoi cells associated with the Poisson points {X j } j . Independently ofP 3 we choose a jointly measurable stationary bounded random field G 3 on R d . We then consider the σ(P 3 , G 3 )-measurable random field A 3 defined as follows,
A 3 (x) := j G 3 (X j )1 C j .
For each of these examples, we show functional inequalities with as derivative the supremum of the functional derivative ∂ fct , which we define as Note that provided A is bounded we have ∂ osc , ∂ fct ∂ sup . From the proofs in the companion article [7], it is clear that weighted functional inequalities with ∂ sup imply the same concentration properties as the corresponding functional inequalities with ∂ osc . (i) For s = 1, 2, the above-defined random field A s satisfies the following weighted spectral gap: for all σ(A s )-measurable random variable Z(A s ) we have
Var [Z(A s )] ˆ∞ 0ˆ∞ 0ˆR d E ∂ sup A,B ℓ+C(v+1) (x) Z(A s ) 2 dx (ℓ + 1) −d ∧ γ(v) π s (ℓ)dvdℓ. (3.13)
In the case when the random variable V is almost surely bounded by a deterministic constant, we rather obtain (3.14) and if the random field G s further satisfies (∂ fct -WLSI) with weight π s , then the corresponding logarithmic Sobolev inequality also holds, that is,
Var [Z(A s )] ˆR d E ∂ osc As,B C (x) Z(A s ) 2 dx +ˆ∞ 0ˆR d E ∂ fct As,B ℓ+C (x) Z(A s ) 2 dx (ℓ + 1) −d π s (ℓ)dℓ,Ent[Z(A s )] ˆR d E ∂ osc As,B C (x) Z(A s ) 2 dx +ˆ∞ 0ˆR d E ∂ fct As,B ℓ+C (x) Z(A s ) 2 dx (ℓ + 1) −d π s (ℓ)dℓ.
(ii) The above-defined random field A 3 satisfies (∂ sup -WSG) with weight π(ℓ) := C(π 3 (ℓ)+ e − 1 C ℓ d ). If the random field G 3 further satisfies (∂ fct -WLSI) with weight π 3 , then A 3 also satisfies (∂ sup -WLSI) with weight π.
The proof of Proposition 3.5 is quite robust and many variants of the above results could be considered.
Proof. For s = 1, 2, 3, sinceP s and G s are independent, the expectation E splits into , and only treat the case of the variance in the proof. Since the random field G s is assumed to satisfy (∂ fct -WSG) with weight π s , we obtain
E = EP s E Gs , where EP s [·] = E[· G s ] denotesVar Gs [EP s [Z(A s )]] ≤ EP s [Var Gs [Z(A s )]] ≤ E ˆ∞ 0ˆR d ∂ fct Gs,B ℓ+1 (x) Z(A s ) 2 dx (ℓ + 1) −d π s (ℓ)dℓ . (3.16)
The chain rule yields
∂ fct Gs,B ℓ+1 (x) Z(A s ) =ˆB ℓ+1 (x) ∂Z(A s (P s , G s )) ∂G s (y) dy ≤ˆB ℓ+1 (x)ˆR d ∂Z(A s ) ∂A s (z) ∂A s (P s , G s )(z) ∂G s (y) dzdy. Since A s is σ(P s , {G s (X j )} j )-measurable, we obtain ∂ fct Gs,B ℓ+1 (x) Z(A s ) ≤ j 1 X j ∈B ℓ+1 (x)ˆR d ∂Z(A s ) ∂A s (z) ∂A s (P s , G s )(z) ∂G s (X j ) dz (3.17)
in terms of the usual partial derivative of A s (P s , G s )(z) with respect to G s (X j ). We now need to compute this derivative in each of the considered examples. We claim that
∂A s (P s , G s )(z) ∂G s (X j ) ≤ C1 R j s (z), (3.18) where R j s := I j 1 \ i:i<j I i 1 , if s = 1; x : 0 < d(x, I j 2 ) < 2 ∧ d(x, I k 2 ), ∀k = j , if s = 2; C j , if s = 3.
This claim (3.18) is obvious for s = 1 and s = 3. For s = 2, the properties of φ and the definition of R j 2 yield
∂A 2 (P 2 , G 2 )(z) ∂G 2 (X j ) ≤ |α − β| φ ′ d z , ∪ k I k 2 1 R j 2 (z) = |α − β| φ ′ d(z, I j 2 ) 1 R j 2 (z),
which indeed implies (3.18). Now injecting (3.18) into (3.17), and noting that in each case the sets {R j s } j are disjoint, we obtain
∂ fct Gs,B ℓ+1 (x) Z(A s ) ≤ C j 1 X j ∈B ℓ+1 (x)ˆR j s ∂Z(A s ) ∂A s = Cˆ j:X j ∈B ℓ+1 (x) R j s ∂Z(A s ) ∂A s ≤ CˆB Ds(ℓ,x) (x) ∂Z(A s ) ∂A s , (3.19) with D s (ℓ, x) := sup d(y, x) : y ∈ j:X j ∈B ℓ+1 (x) R j s .
For s = 1, 2 with radius law V almost surely bounded by a deterministic constant R > 0, we obtain D 1 (ℓ, x) ≤ ℓ + R + 1 and D 2 (ℓ, x) ≤ ℓ + R + 3, and injecting (3.19) into (3.16) directly yields the result (3.14).
We now consider the cases s = 1, 2 with general unbounded radii. Without loss of generality we only treat s = 1, in which case
D 1 (ℓ, x) ≤ ℓ + 1 +D 1 (ℓ, x),D 1 (ℓ, x) := max V j : X j ∈ B ℓ+1 (x) .
Noting that the restriction A 1 | R d \B ℓ+1+D 1 (ℓ,x) (x) is by construction independent ofD 1 (ℓ, x),
we obtain, conditioning on the values ofD 1 (ℓ, x) and arguing as in Step 2 of the proof of Theorem 2.7,
E ˆB ℓ+1+D 1 (ℓ,x) (x) ∂Z(A 1 ) ∂A 1 2 ≤ ∞ i=1 P i − 1 ≤D 1 (ℓ, x) < i E ˆB ℓ+i+1 (x) ∂Z(A 1 ) ∂A 1 2 i − 1 ≤D 1 (ℓ, x) < i ≤ ∞ i=1 P i − 1 ≤D 1 (ℓ, x) < i E sup ess A 1 ,B ℓ+i+1 (x) ˆB ℓ+i+1 (x) ∂Z(A 1 ) ∂A 1 2 D 1 (ℓ, x) < i ≤ ∞ i=1 P i − 1 ≤D 1 (ℓ, x) < i P D 1 (ℓ, x) < i E sup ess A 1 ,B ℓ+i+1 (x) ˆB ℓ+i+1 (x) ∂Z(A 1 ) ∂A 1 2 .
(3.20)
Now by definition of the decorated Poisson point processP 1 , we may compute for all i ≥ 1,
P D 1 (ℓ, x) ≥ i − 1 = P ∃j : V j ≥ i − 1 and X j ∈ B ℓ+1 (x) = e −|B ℓ+1 | ∞ n=0 |B ℓ+1 | n n! 1 − (1 − P [V ≥ i − 1]) n = 1 − e −|B ℓ+1 | P[V ≥i−1] ,
and hence
P i − 1 ≤D 1 (ℓ, x) < i P D 1 (ℓ, x) < i = 1 − e −|B ℓ+1 |P[i−1≤V <i] ≤ 1 ∧ C(ℓ + 1) d P [i − 1 ≤ V < i] .
Combining this computation with (3.16), (3.19) and (3.20), we obtain, setting γ(v) :
= P [v − 2 ≤ V < v + 1], Var G 1 [EP 1 [Z(A 1 )]] E ˆ∞ 0 ∞ i=1ˆR d sup ess A 1 ,B ℓ+i+1 (x) ˆB ℓ+i+1 (x) ∂Z(A 1 ) ∂A 1 2 dx × (ℓ + 1) −d ∧ P [i − 1 ≤ V < i] π s (ℓ)dℓ ≤ E ˆ∞ 0ˆ∞ 0ˆR d sup ess A 1 ,B ℓ+v+2 (x) ˆB ℓ+v+2 (x) ∂Z(A 1 ) ∂A 1 2 dx (ℓ + 1) −d ∧ γ(v) π s (ℓ)dvdℓ ,
and the conclusion (3.13) follows.
We now turn to the case s = 3, for which
D 3 (ℓ, x) ≤ ℓ + 1 +D 3 (ℓ, x),D 3 (ℓ, x) := max diam(C j ) : X j ∈ B ℓ+1 (x) .
Noting that the restriction A 3 | R d \B ℓ+1+2D 3 (ℓ,x) (x) is by construction independent ofD 3 (ℓ, x)
we obtain, after conditioning on the values ofD 3 (ℓ, x) and arguing as in (3.20),
E ˆB ℓ+1+D 3 (ℓ,x) (x) ∂Z(A 3 ) ∂A 3 2 ≤ E sup ess A 3 ,B 3ℓ+1 (x) ˆB 3ℓ+1 (x) ∂Z(A 3 ) ∂A 3 2 + ∞ i=2ℓ P i − 1 ≤D 3 (ℓ, x) < i P D 3 (ℓ, x) < i E sup ess A 3 ,B ℓ+i+1 (x) ˆB ℓ+i+1 (x) ∂Z(A 3 ) ∂A 3 2 . (3.21)
Similar computations as in Step 2 of the proof of Proposition 3.2 yield
P D 3 (ℓ, x) ≥ i ≤ Ce − 1 C (i−ℓ) d + .
Combining this with (3.16), (3.19) and (3.21), we obtain
Var G 3 [EP 3 [Z(A 3 )]] E ˆ∞ 0ˆR d sup ess A 3 ,B 3ℓ+1 (x) ˆB 3ℓ+1 (x) ∂Z(A 3 ) ∂A 3 2 dx (ℓ + 1) −d π 3 (ℓ)dℓ +E ˆ∞ 0 ∞ i=2ℓ e − 1 C i dˆR d sup ess A 3 ,B 2i+1 (x) ˆB 2i+1 (x) ∂Z(A 3 ) ∂A 3 2 dx (ℓ + 1) −d π 3 (ℓ)dℓ E ˆ∞ 0ˆR d sup ess A 3 ,B 3ℓ+1 (x) ˆB 3ℓ+1 (x) ∂Z(A 3 ) ∂A 3 2 dx (ℓ + 1) −d π 3 (ℓ) + e − 1 C ℓ d dℓ ,
and the result follows.
Appendix A. Proof of the criterion for standard functional inequalities
In this appendix, we give a proof of Proposition 2.3.
Proof of Proposition 2.3. Let ε > 0 be fixed, and consider the partition (Q z ) z∈Z d of R d defined by Q z = εz + εQ. Choose an i.i.d. copy A ′ 0 of the field A 0 , and for all z define the random field A z 0 by A z 0 | R d \Qz := A 0 | R d \Qz and A z 0 | Qz := A ′ 0 | Qz . We split the proof into three steps.
Step 1. Tensorization argument. Choose an enumeration (z n ) n of Z d , and for all n let Π n and E n denote the linear maps on L 2 (Ω) defined by In this step, we make use of a martingale argument à la Lu-Yau [17] to show the following tensorization identities for the covariance and for the entropy: for all σ(A 0 )-measurable random variables X(A 0 ) and Y (A 0 ), we have
Π n [X] := E X A 0 | n k=1 Qz k , E n [X] := E X A 0 | R d \Qz n .|Cov [X(A 0 ); Y (A 0 )] | ≤ ∞ k=1 E Cov k Π k [X(A 0 )]; Π k [Y (A 0 )] , (A.1) Ent X(A 0 ) 2 ≤ ∞ k=1 E Ent k Π k [X(A 0 ) 2 ] . (A.2)
First note that for all σ(A 0 )-measurable random variables X(A 0 ) ∈ L 2 (Ω), the properties of conditional expectations ensure that Π n [X(A 0 )] → X(A 0 ) in L 2 (Ω) as n ↑ ∞. We then decompose the covariance into the following telescopic sum
Cov [Π n [X(A 0 )]; Π n [Y (A 0 )]] = n k=1 E [Π k [X(A 0 )]Π k [Y (A 0 )]] − E [Π k−1 [X(A 0 )]Π k−1 [Y (A 0 )]] = n k=1 E Cov k Π k [X(A 0 )]; Π k [Y (A 0 )] ,
so that the result (A.1) follows by taking the limit n ↑ ∞. Likewise, we decompose the entropy into the following telescopic sum
Ent Π n [X(A 0 ) 2 ] = n k=1 E Π k [X(A 0 ) 2 ] log(Π k [X(A 0 ) 2 ]) − E Π k−1 [X(A 0 ) 2 ] log(Π k−1 [X(A 0 ) 2 ]) = n k=1 E Ent k Π k [X(A 0 ) 2 ] ,
and the result (A.2) follows in the limit n ↑ ∞.
Step 2. Preliminary versions of (CI) and (LSI). In this step, we prove that for all σ(A 0 )-measurable random variables X(A 0 ) and Y (A 0 ) we have
|Cov [X(A 0 ); Y (A 0 )] | ≤ 1 2 ∞ k=1 E Π k X(A 0 ) − X(A z k 0 ) Π k Y (A 0 ) − Y (A z k 0 ) ≤ 1 2 z∈Z d E X(A 0 ) − X(A z 0 ) 2 1 2 E Y (A 0 ) − Y (A z 0 ) 2 1 2 , (A.3)
and
Ent[X(A 0 )] ≤ 2 z∈Z d E sup ess A ′ 0 X(A 0 ) − X(A z 0 ) 2 . (A.4)
We first prove (A.3): we appeal to (A.1) in the form
|Cov [X(A 0 ); Y (A 0 )] | ≤ 1 2 ∞ k=1 E E k Π k [X(A 0 ) − X(A z k 0 )] Π k [Y (A 0 ) − Y (A z k 0 )] ≤ 1 2 ∞ k=1 E Π k [X(A 0 ) − X(A z k 0 )] Π k [Y (A 0 ) − Y (A z k 0 )] ,
which directly yields (A.3) by Cauchy-Schwarz' inequality. Likewise, we argue that (A.4) follows from (A.2). To this aim, we have to reformulate the RHS of (A.2): using the inequality a log a − a + 1 ≤ (a − 1) 2 for all a ≥ 0, we obtain for all k ≥ 0,
Ent k Π k [X(A 0 ) 2 ] ≤ E k [Π k [X(A 0 ) 2 ]] E k Π k [X(A 0 ) 2 ] E k [Π k [X(A 0 ) 2 ]] − 1 2 = Var k Π k [X(A 0 ) 2 ] E k [Π k [X(A 0 ) 2 ]] = E k (Π k [X(A 0 ) 2 ] − Π k [X(A z k 0 ) 2 ]) 2 2 E k [Π k [X(A 0 ) 2 ]] = E k (Π k [(X(A 0 ) − X(A z k 0 ))(X(A 0 ) + X(A z k 0 ))]) 2 2 E k [Π k [X(A 0 ) 2 ]] ≤ E k Π k [(X(A 0 ) − X(A z k 0 )) 2 ] Π k [(X(A 0 ) + X(A z k 0 )) 2 ] 2 E k [Π k [X(A 0 ) 2 ]]
.
Since (A 0 , A z k 0 ) and (A z k 0 , A 0 ) have the same law by complete independence, the above implies, using the inequality (a + b) 2 ≤ 2(a 2 + b 2 ) for all a, b ∈ R,
Ent k Π k [X(A 0 ) 2 ] ≤ 2 E k Π k [(X(A 0 ) − X(A z k 0 )) 2 ] Π k [X(A z k 0 ) 2 ] E k [Π k [X(A z k 0 ) 2 ]] ≤ 2 sup ess A ′ 0 | Qz k Π k [(X(A 0 ) − X(A z k 0 )) 2 ] ≤ 2 Π k sup ess A ′ 0 | Qz k (X(A 0 ) − X(A z k 0 )) 2 .
Estimate (A.4) now follows from (A.2).
Step 3. Proof of (CI) and (LSI). We start with the proof of (CI).
Since A = A(A 0 ) is σ(A 0 )-measurable, (A.3) yields for all σ(A)-measurable random variables X(A) and Y (A),
associates a random variable W (E) to any bounded Borel subset E ⊂ R d , in such a way that
(i) E [W (E)] = 0 and E |W (E)| 2 < ∞ for all bounded Borel subset E ⊂ R d ; (ii) if (E n ) n is a family of disjoint Borel subsets of R d , then W ( ∞ n=1 E n ) = ∞ n=1 W (E n ) in the L 2 -sense; (iii) (W (x + E), W (x + E ′ )
) has the same law as (W (E), W (E ′ )) for any two bounded
Borel subsets E, E ′ ⊂ R d and any x ∈ R d ; (iv) W (E 1 ), . . . , W (E n ) are independent for any disjoint Borel sets E 1 , . . . , E n ⊂ R d and any n ∈ N.
Stationarity implies in particular that the Borel measure E[|dW | 2 ] is proportional to the Lebesgue measure: E[|dW | 2 ] = λdx, for some constant λ ≥ 0 that is called the intensity of the random noise W .
Given a (deterministic) nonnegative Borel function F ∈ L 2 (R d ) and a constant m ∈ R d , we now define a measurable random field A on R d by the following convolution,
A(y) = m +ˆR d F (y − z)dW (z), (B.1)
the covariance function of which is then given by
C(x) := Cov [A(x); A(0)] = λˆR d F (x − z)F (z)dz. (B.2)
The following result (which is rather standard) shows that a Brascamp-Lieb inequality holds for such random fields whenever the random noise W satisfies a standard spectral gap, thus mimicking the well-known situation of Gaussian fields. (For Gaussian fields, a discrete version of the Brascamp-Lieb inequality (B.4) below was first due to [4], while a discrete version of the inequality in covariance form (B.5) and in entropy form (B.8) is due to [18] and to [2, Proposition 3.4], respectively.)
Proposition B.1 (Brascamp-Lieb type inequalities). Let W be a random noise on R d with intensity λ, let the stationary random field A on R d be given by (B.1), and let C denote its covariance function.
(i) Assume that for all η > 0 the random variable W (ηQ) satisfies the following spectral gap: for any smooth function φ, Var [φ(W (ηQ))] ≤ Cλη d E φ ′ (W (ηQ)) 2 . (ii) Assume that for all η > 0 the random variable W (ηQ) satisfies the corresponding logarithmic Sobolev inequality: for any smooth function φ, In the following theorem, we show that Brascamp-Lieb inequalities imply weighted functional inequalities, using a suitable radial change of variables. Note that in item (ii), the weights obtained for (∂ fct -WSG) and (∂ fct -WCI) typically have the same scaling.
Theorem B.2. Let A be a jointly measurable stationary random field on R d , let C denote its covariance function. Assume that A satisfies the Brascamp-Lieb inequality (B.4) (resp. in logarithmic Sobolev form (B.8)).
(i) If the map x → sup B(x) |C| is integrable, then the field A satisfies (∂ fct -SG) (resp. (∂ fct -LSI)) for any radius R > 0. (ii) If sup B(x) |C| ≤ c(|x|) holds for some non-increasing Lipschitz function c : R + → R + , then the field A satisfies (∂ fct -WSG) (resp. (∂ fct -WLSI)) with weight π(ℓ) ≃ (−c ′ (ℓ)).
If the field A further satisfies the Brascamp-Lieb inequality in covariance form (B.5), and if sup B(x) |F −1 ( √ FC)| ≤ r(|x|) holds for some non-increasing Lipschitz function r : R + → R + , then A satisfies (∂ fct -WCI) with weight π(ℓ) ≃ (ℓ+1) d r(ℓ)(−r ′ (ℓ)).
Remark B.3. We briefly address the claim contained in Remark 2.2 in the context of examples of random fields with deterministic localization. More precisely, we consider a random field A as in the statement of Theorem B.2 above, and we assume that sup B(x) |C| ≤ c(|x|) holds for some non-increasing Lipschitz function c : R + → R + . By definition, for all L ≥ 1, the rescaled field A L := A(L·) has covariance C L := C(L·) and for |x| ≥ 1 it satisfies sup B(x) |C L | = sup B L (Lx) |C| ≤ c((L|x| − L + 1) + ) ≤ c(|x|) since c is non-increasing. This shows that the same conclusions as for A in Theorem B.2 also hold for A L uniformly with respect to L ≥ 1.
We start with the proof of Proposition B.1, and then turn to the proof of Theorem B.2.
Proof of Proposition B.1. For all ε > 0, consider the following approximations of the random field A,
A ε (x) := y,z∈εZ d 1 Qε(z) (x) W (Q ε (y)) Qε(z) Qε(y) F (z ′ − y ′ )dz ′ dy ′ .
By an approximation argument, we may reduce the proof of the proposition to the proof of the following discrete counterpart: given a random vector W := (W 1 , . . . , W N ) with N independent components, and given a linear transformation F ∈ R N ×N , the transformed random vector A := (A 1 , . . . , A N ) := F W satisfies:
Ent φ(W j ) 2 ≤ CE φ ′ (W j ) 2
for all smooth functions φ : R → R, then the random vector A satisfies for all smooth functions X : R N → R,
Ent X(A) 2 ≤ C N i,j=1 |(F F t ) ij | E ∂X(A) ∂A i ∂X(A) ∂A j . (B.11)
We start with the proof of item (i'). Using the tensorization identity (A.1), the spectral gap assumption yields
Var [X(A)] ≤ N i=1 E [Var [ X(A) (W j ) j:j =i ]] ≤ N i=1 E ∂X(A) ∂W i 2 ,
and hence, by the chain rule,
Var [X(A)] ≤ N i=1 E N j=1 ∂X(A) ∂A j F ji 2 = E ∇X(A) · (F F t )∇X(A) (B.12) ≤ N i,j=1 |(F F t ) ij | E ∂X(A) ∂A i ∂X(A) ∂A j .
In covariance form, using again the tensorization identity (A.1), the spectral gap assumption yields ∂X(A) ∂W i (W j ) j:j≤i
2 ≤ C N i=1 E ∂X(A) ∂W i 2 .
Now arguing as in (B.12), the result of item (ii') follows.
We now prove Theorem B.2.
Proof of Theorem B.2. We focus on items (i) and (ii) for the variance and the covariance (the arguments for the entropy are similar). Assume that A satisfies the Brascamp-Lieb inequality (B.4). If x → sup B(x) |C| is integrable, the inequality |ab| ≤ (a 2 + b 2 )/2 for a, b ∈ R directly yields for all σ(A)-measurable random variables X(A) and all R > 0 (after taking local averages),
Var [X(A)] ≤ C E ˆR dˆRd ∂X(A) ∂A (z) ∂X(A) ∂A (z ′ ) |C(z − z ′ )|dzdz ′ ≤ 2C sup B 2R (·) |C| L 1 E∂X R (A) ∂A (u ′ ) du ′ ℓ d−1 c(ℓ)dℓdσ(u)dz = CE ˆR d ∂X R (A) ∂A (z) ˆS d−1ˆ∞ 0ˆℓ 0 B(z+su) ∂X R (A) ∂A (u ′ ) du ′ s d−1 ds(−c ′ (ℓ))dℓdσ(u)dz ≤ CE ˆR d ∂X R (A) ∂A (z) ˆ∞ 0 ˆB ℓ+1 (z) ∂X R (A) ∂A
"
=" sup ess X(A ′ ) :
Remark 2. 2 .
2In each of the examples considered in the sequel, if the functional inequality (∂-WSG), (∂-WCI), or (∂-WLSI) is proved to hold with some weight π, then for all L ≥ 1 the rescaled field A L := A(L·) satisfies the same functional inequality with the same weight π. See Remarks 2.10 and B.3 for detail.
Corollary 3. 1 .
1Let A be a jointly measurable stationary Gaussian random field on R d with covariance function C(x) := Cov [A(x); A(0)]. (i) If x → sup B(x) |C| is integrable, then A satisfies (∂ fct -SG) and (∂ fct -LSI) with any radius R > 0. (ii) If sup B(x) |C| ≤ c(|x|) holds for some Lipschitz function c :
and estimating the last integral leads to the desired result(3.6). Let X ′ s denote an i.i.d. copy of the field X s , and letP ′ s := {X ′ j , V ′ j , Y ′ j,s } j denote the corresponding i.i.d. copy ofP s := {X j , V j , Y j,s } j .For all x, v, let the perturbations X xdefined as usual, and let P x,v 0 be the corresponding projected point process on R d × R l . Let us consider J v (x, r) the set of all indices j such that the projection p
denotes the expectation with respect to P, and where E D [·] = E[· P] denotes the expectation with respect to D. By tensorization of the variance as in (3.15), the spectral gap assumption for P and the standard spectral gap (2.3) for the i.i.d. sequence D then yields for all random variables Z = Z(A s ), Var [Z(A s )] = E P Var D [Z(A s )] + Var P E D [Z(A s )]
Proposition 3. 5 .
5For s = 1, 2, 3, assume that the random field G s satisfies (∂ fct -WSG) for some integrable weight π s . For s = 1, 2, set γ(v) := P [v − 4 ≤ V < v + 4]. Then the following holds.
the expectation with respect toP s , and where E Gs [·] = E[· P s ] denotes the expectation with respect to G s . The variance and the entropy also tensorize: for all σ(A s )-measurable random variables Z(A s ), Var [Z(A s )] = Var Gs [EP s [Z(A s )]] + E Gs [VarP s [Z(A s )]], (3.15) Ent[Z(A s )] = Ent Gs [EP s [Z(A s )]] + E Gs [EntP s [Z(A s )]]. In each of the examples under consideration, the estimate on the terms VarP s [Z(A s )] and EntP s [Z(A s )] (with G s "frozen") follows from the same arguments as in the proof of Propositions 3.2 and 3.4(i). We therefore focus on the estimates of Var Gs [EP s [Z(A s )]] and Ent Gs [EP s [Z(A s )]]
Then the random field A satisfies the following Brascamp-Lieb inequality: for all σ(A)-measurable random variables X(A),Var [X(A)] ≤ CE ˆR dˆRd ∂X(A) ∂A (z) ∂X(A) ∂A (z ′ ) |C(z − z ′ )|dzdz ′ .(B.4) Moreover, the following Brascamp-Lieb inequality in covariance form holds: for all σ(A)-measurable random variables X(A), Y (A) we have Cov [X(A); Y (A)] − z ′ )dzdz ′ , (B.6) in terms ofC (x) :=ˆ|F −1 ( √ FC)(x − y)||F −1 ( √ FC)(y)|dy.
Ent φ(W (ηQ)) 2 ≤ Cλη d E φ ′ (W (ηQ)) 2 . (B.7)Then the random field A satisfies the corresponding Brascamp-Lieb inequality in logarithmic Sobolev form: for all σ(A)-measurable random variables X(A),Ent[X(A)] ≤ CE ˆR dˆRd ∂X(A) ∂A (z) ∂X(A) ∂A (z ′ ) |C(z − z ′ )|dzdz ′ . (B.8)
(i') If for all 1 ≤ j ≤ N the random variable W j satisfies the standard spectral gapVar [φ(W j )] ≤ CE φ ′ (W j ) 2for all smooth functions φ : R → R, then the random vector A satisfies for all smooth functions X, Y : R N → R ') If for all 1 ≤ j ≤ N the random variable W j satisfies the standard logarithmic Sobolev inequality
Now assume that the covariance function C is not integrable, and that sup B(x) |C| ≤ c(|x|) for some Lipschitz function c : R + → R + . Given a σ(A)-measurable random variable X(A), we consider the projection X R (A) := E[X(A) A| B R ], for R > 0. Taking local averages, using polar coordinates, and integrating by parts (note that there is no boundary term since the Fréchet derivative ∂X R (A)/∂A is compactly supported in B R )
Cov n [X; Y ] := E n [XY ] − E n [X]E n [Y ],Var n [X] := Cov n [X; X], Ent n [X 2 ] := E n X 2 log(X 2 /E n [X 2 ]) .Also define
Appendix B. Abstract criteria for deterministically localized fieldsIn this appendix, we discuss general criteria for weighted functional inequalities in the case when the random field A is deterministically localized in the sense of Section 2.3. To be precise we focus on the typical example of a convolution of a random noise. In this case we prove the validity of a Brascamp-Lieb inequality from which the desired weighted functional inequalities follow. Although Gaussian random fields are the most prominent examples of this framework, we develop the general argument in a slightly more abstract setting. (Note that we choose to argue by approximation and reduce to discrete fields, rather than appeal to Malliavin calculus and associated functional analysis.)Let W be a random noise on R d , that is, a mean-zero stationary completely independent second-order random Borel measure on R d (see e.g.[24,Section 2]). More precisely, W
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The central limit theorem for Euclidean minimal spanning trees. I. S Lee, Ann. Appl. Probab. 74S. Lee. The central limit theorem for Euclidean minimal spanning trees. I. Ann. Appl. Probab., 7(4):996-1020, 1997.
The central limit theorem for Euclidean minimal spanning trees. S Lee, II. Adv. in Appl. Probab. 314S. Lee. The central limit theorem for Euclidean minimal spanning trees. II. Adv. in Appl. Probab., 31(4):969-984, 1999.
Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. S L Lu, H.-T Yau, Comm. Math. Phys. 1562S. L. Lu and H.-T. Yau. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys., 156(2):399-433, 1993.
A Brascamp-Lieb type covariance estimate. G Menz, Electron. J. Probab. 1978G. Menz. A Brascamp-Lieb type covariance estimate. Electron. J. Probab., 19:no. 78, 15, 2014.
Normal approximations with Malliavin calculus. I Nourdin, G Peccati, Cambridge Tracts in Mathematics. 192Cambridge University PressFrom Stein's method to universalityI. Nourdin and G. Peccati. Normal approximations with Malliavin calculus, volume 192 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2012. From Stein's method to uni- versality.
Random parking, sequential adsorption, and the jamming limit. M D Penrose, Comm. Math. Phys. 2181M. D. Penrose. Random parking, sequential adsorption, and the jamming limit. Comm. Math. Phys., 218(1):153-176, 2001.
Multivariate spatial central limit theorems with applications to percolation and spatial graphs. M D Penrose, Ann. Probab. 335M. D. Penrose. Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab., 33(5):1945-1991, 2005.
Limit theory for random sequential packing and deposition. M D Penrose, J E Yukich, Ann. Appl. Probab. 121M. D. Penrose and J. E. Yukich. Limit theory for random sequential packing and deposition. Ann. Appl. Probab., 12(1):272-301, 2002.
Normal approximation in geometric probability. M D Penrose, J E Yukich, Stein's method and applications. SingaporeSingapore Univ. Press5M. D. Penrose and J. E. Yukich. Normal approximation in geometric probability. In Stein's method and applications, volume 5 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 37-58. Singapore Univ. Press, Singapore, 2005.
Series on Multivariate Analysis. M M Rao, World Scientific Publishing Co. Pte. Ltd9Hackensack, NJRandom and vector measuresM. M. Rao. Random and vector measures, volume 9 of Series on Multivariate Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
Gaussian limits for multidimensional random sequential packing at saturation. T Schreiber, M D Penrose, J E Yukich, Comm. Math. Phys. 2721T. Schreiber, M. D. Penrose, and J. E. Yukich. Gaussian limits for multidimensional random sequential packing at saturation. Comm. Math. Phys., 272(1):167-183, 2007.
Microstructure and macroscopic properties. S Torquato, address: [email protected] (Antoine Gloria) Sorbonne Université, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005. New York; Brussels, Belgium E-mail; Paris, France; Brussels, Belgium E-mailSpringer-Verlag16Université Libre de Bruxelles (ULBMitia Duerinckx) Université Libre de Bruxelles (ULB)S. Torquato. Random heterogeneous materials. Microstructure and macroscopic properties, volume 16 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York, 2002. (Mitia Duerinckx) Université Libre de Bruxelles (ULB), Brussels, Belgium E-mail address: [email protected] (Antoine Gloria) Sorbonne Université, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France Université Libre de Bruxelles (ULB), Brussels, Belgium E-mail address: [email protected]
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"The language preservation problem is undecidable for parametric event-recording automata ✩",
"The language preservation problem is undecidable for parametric event-recording automata ✩"
] |
[
"Etienne André \nUMR 7030\nUniversité Paris 13\nLIPN\nCNRS\nF-93430VilletaneuseFrance\n",
"Shang-Wei Lin [email protected] \nNanyang Technological University\n50 Nanyang Avenue639798Singapore\n"
] |
[
"UMR 7030\nUniversité Paris 13\nLIPN\nCNRS\nF-93430VilletaneuseFrance",
"Nanyang Technological University\n50 Nanyang Avenue639798Singapore"
] |
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Parametric timed automata (PTA) extend timed automata with unknown constants ("parameters"), at the price of undecidability of most interesting problems. The (untimed) language preservation problem ("given a parameter valuation, can we find at least one other valuation with the same untimed language?") is undecidable for PTAs. We prove that this problem remains undecidable for parametric event-recording automata (PERAs), a subclass of PTAs that considerably restrains the way the language can be used; we also show it remains undecidable even for slightly different definitions of the language, i. e., finite sequences of actions ending in or passing infinitely often through accepting locations, or just all finite untimed words (without accepting locations).
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10.1016/j.ipl.2018.03.013
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"https://arxiv.org/pdf/1812.08948v1.pdf"
] | 51,877,546 |
1812.08948
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9651744994167cdd7df010eea08fab5ccca4cde1
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The language preservation problem is undecidable for parametric event-recording automata ✩
21 Dec 2018
Etienne André
UMR 7030
Université Paris 13
LIPN
CNRS
F-93430VilletaneuseFrance
Shang-Wei Lin [email protected]
Nanyang Technological University
50 Nanyang Avenue639798Singapore
The language preservation problem is undecidable for parametric event-recording automata ✩
21 Dec 201810.1016/j.ipl.2018.03.013Preprint submitted to Information Processing Letters December 24, 20182010 MSC: 68Q45, 68Q60 ✩ This is the author version of the article of the same name published in Information Processing Letters, volume 136, pages 17-20 (2018). The final version is available atparametric timed automata, event-recording automata
Parametric timed automata (PTA) extend timed automata with unknown constants ("parameters"), at the price of undecidability of most interesting problems. The (untimed) language preservation problem ("given a parameter valuation, can we find at least one other valuation with the same untimed language?") is undecidable for PTAs. We prove that this problem remains undecidable for parametric event-recording automata (PERAs), a subclass of PTAs that considerably restrains the way the language can be used; we also show it remains undecidable even for slightly different definitions of the language, i. e., finite sequences of actions ending in or passing infinitely often through accepting locations, or just all finite untimed words (without accepting locations).
Introduction
Timed automata (TAs) [AD94] are a useful formalism to model and formally verify systems involving timing hard constraints and concurrency. TAs benefit from numerous decidability results, including the emptiness of the accepted language. However, the universality and the language inclusion are undecidable for timed automata [AD94]. Therefore, subclasses have been proposed. The language inclusion becomes decidable for event-recording automata (ERAs) [AFH99].
Parametric timed automata (PTAs) [AHV93] extend TAs with timing parameters: this very expressive formalism can model systems where timing constants are uncertain or unknown, at the cost of most interesting problems to be undecidable [And18]. The mere emptiness of the valuation set for which a given location is reachable ("reachability-emptiness") is undecidable [AHV93].
Restricting the syntax of a formalism may bring decidability: the language inclusion undecidable for TAs [AD94] becomes decidable for ERAs [AFH99]. In contrast, the reachability emptiness problem for PTAs remains undecidable for a subclass of PTAs with only open inequalities [Doy07].
In [AL17], we proposed parametric event-recording automata (PERAs), and showed that the reachability-emptiness problem remains undecidable for PERAs. Although it seems that our proof idea can be extended to most problems where the language (i. e., the transition labels) does not play a role (which would include unavoidability-emptiness [JLR15]), it remains open whether language problems undecidable for PTAs become or not decidable for PERAs.
In [AM15], we showed that the following language preservation problem is undecidable for PTAs: "given a PTA and a reference parameter valuation, does there exist another valuation with the same untimed language?". This problem has connections with the robustness of timed systems, as it asks whether other valuations of the timing constants may lead to the same discrete behavior.
The set of valuations with the same untimed language can also be used to perform optimization of some constants without impacting the system's (untimed) behavior.
We show here that the language preservation problem is undecidable for PERAs, and remains undecidable for different definitions of the language. This quite surprising result comes in contrast with the difference of decidability between TAs and ERAs in the non-parametric setting.
Preliminaries
Clocks, parameters and constraints
Let N, Z, Q + and R + denote the sets of non-negative integers, integers, non-negative rational and non-negative real numbers respectively.
Throughout this paper, we assume a set X of clocks, i. e., real-valued variables that evolve at the same rate. A clock valuation is a function µ : X → R + .
We write 0 for the clock valuation that assigns 0 to all clocks. Given d ∈ R + ,
µ + d denotes the valuation such that (µ + d)(x) = µ(x) + d, for all x ∈ X.
Given x ∈ X, we define the reset of a valuation µ, denoted by [µ] x , as follows:
[µ] x (x ′ ) = 0 if x ′ = x, and [µ] x (x ′ ) = µ(x ′ ) otherwise.
We assume a set P of parameters, i. e., unknown rational-valued constants.
A parameter valuation v is a function v : P → Q + .
A guard g is a constraint over X∪P defined by a conjunction of inequalities of the form x ⊲⊳ η, where η is either a parameter or a constant in Z, and ⊲⊳ ∈ {<, ≤ , ≥, >}. Given v, v(g) denotes g where all occurrences of each parameter p i ∈ P have been replaced by v(p i ).
Parametric Event-Recording Automata
Parametric event-recording automata (PTAs) extend event-recording automata with parameters within guards and invariants in place of integer constants [AL17]; they can also be seen as a syntactic subclass of parametric timed automata [AHV93].
Definition 1 (PERA). A parametric event-recording automaton (hereafter PERA) A is a tuple (Σ, L, l 0 , P, I, E), where: i) Σ is a finite set of actions, ii) L is a finite set of locations, iii) l 0 ∈ L is the initial location, iv) P is a finite set of parameters, v) I is the invariant, assigning to every l ∈ L a guard I(l), vi) E is a finite set of edges e = (l, g, a, l ′ ) where l, l ′ ∈ L are the source and target locations, a ∈ Σ, and g is a guard.
A PERA has a one-to-one mapping between clocks and actions. Given a set of actions Σ, let X Σ denote the set of associated clocks. Similarly, we denote by
• discrete transitions: (l, µ) e → (l ′ , µ ′ ), if (l, µ), (l ′ , µ ′ ) ∈ S, there exists e = (l, g, a, l ′ ) ∈ E, µ ′ = [µ] xa , and µ |= v(g).
• delay transitions:
(l, µ) d → (l, µ + d), with d ∈ R + , if ∀d ′ ∈ [0, d], (l, µ + d ′ ) ∈ S.
A run is a sequence ρ = s 0 γ 0 s 1 γ 1 · · · s n γ n · · · such that ∀i, s i γi → s i+1 . We consider as usual that runs strictly alternate delays d i and discrete transitions e i and we thus write runs in the form ρ = s 0
(d0,e0) → s 1 (d1,e1)
→ · · · . The corresponding timed word is (a 0 , t 0 ), (a 1 , t 1 ), · · · where a i is the action of e i and
t i = i j=0 d i .
The corresponding untimed word is a 0 a 1 · · · . A maximal run is a run that is either infinite, or that cannot be extended by a discrete transition.
As in [AM15], we define the language of v(A), denoted by UL(v(A)) as the set of all untimed words associated with a maximal run of A.
Encoding a 2-counter machine into a PERA
We propose in this section an encoding of a 2-counter machine (2CM) into a PERA. This encoding is adapted from the PTA encoding of [AM15] to our setting of PERAs, and is therefore not a main contribution of this work.
Fix a deterministic 2CM M. Recall that such a machine has two nonnegative counters C 1 and C 2 (the value of which is initially 0), and a finite number of states and of transitions of the form:
• when in state s i , increment C k and go to s j ;
• when in state s i , if C k = 0 then go to s j else decrement C k and go to s j ′ .
The machine starts in state s 0 and halts when it reaches a particular state s halt . The halting problem for 2-counter machines is undecidable [Min67].
Our encoding requires a single parameter p and four actions a t , a 1 , a 2 , and a z (associated with clocks t, x 1 , x 2 and z respectively). Clock t serves as a tick (it is reset exactly every p time units). We encode a configuration of the 2CM as follows: whenever t = 0, then x 1 = c 1 and x 2 = c 2 , where c 1 , c 2 are the current values of C 1 , C 2 . Finally, z is used to count the number of steps of the 2CM:
we use p to bound the length of the computation of the 2CM. The PERA A associated with M is defined as follows:
• its set of locations has two copies of the set S of states of M: for each s i ∈ S, there is a main location l i , and an intermediary location namedl i ;
• Each location of A carries three self-loops, associated with each of the three clocks t, x 1 , and x 2 , and resetting that clock when it reaches value p, i. e., associated with actions a t , a 1 , and a 2 respectively. This requires a global invariant enforcing that all four clocks t, x 1 , x 2 , and z remain below p.
Then each transition (s i , c k + +, s j ) incrementing counter C k in M gives rise to a transition from location l i tol j , with guard x k = p − 1, and labeled with a k (therefore resetting x k ) (see Fig. 1a). Each transition of the form (s i , c k − −, s j , s j ′ ) is handled similarly, but gives rise to two
l ilj l j x k = p − 1 a k z = p − 1 a z (a) Incrementing c k l il j l j l j ′ l j ′ t = 0 ∧ x k = 0 a t t = 1 ∧ x k = 1 a k z = p − 1 a z z = p − 1 a z (b) Decrementing c k
Undecidability of the language preservation
Main result
We now show our main result below.
Theorem 1. The language preservation problem for PERAs is undecidable.
The proof of undecidability of [AM15] for PTAs strongly relies on the fact that all transitions were labeled with the same action a. This reasoning cannot be kept here, as the transitions of our modified encoding in Section 3 are labeled with different actions so as to reset different clocks. Therefore, it is not possible to know in advance the language of the accepting run of the 2CM (if any).
For example, assume a run of the 2CM made of the following two instructions: when in state s 0 , increment C 1 and go to s 3 ; when in state s 3 , if C 2 = 0 then go to s 5 else decrement C 2 and go to s 0 . The sequence of states and actions in our PERA will be (l 0 , µ 0 ) µ 4 ), for some µ 0 , · · · , µ 4 . (In this sequence, we write a i instead of (d i , e i ) to make clear the action a i used in edge e i ). Therefore, the untimed language corresponding to these two instructions is a 1 a z a t a z .
a1 → (l 3 , µ 1 ) az → (l 3 , µ 2 ) at → (l 5 , µ 3 ) az → (l 5 ,
In order to prove Theorem 1, we reduce from the halting problem of a 2CM, using the encoding of Section 3. We will allow all possible infinite (untimed)
words for the reference valuation, and will rely on the difference between finite and infinite words to perform a distinction between the halting or the nonhalting case.
Our proof relies on the PERA A given in Fig. 2 iff M halts. First, let us study v 0 (A): this TA can take the transitions to either l 1 acc (which is ungarded) or l 2 acc (which requires p = 0), but not that to l 0 as the guard requires p > 0. Therefore, UL(v 0 (A)) = (a t |a 1 |a 2 |a z ) ω , i. e., the language made of exactly all infinite words, hereafter Σ ω . Now, assume the machine halts after n steps. There exists a parameter valuation v (typically s.t. v(p) > n) s.t. the machine is correctly simulated.
The (unique) run going through the gadget A 2CM is non-blocking and reaches location l halt , from which it goes to l 2 acc and can perform any action an infinite number of times. The corresponding possible runs are therefore included into Σ ω . Since this valuation can also take the transition from l start to l 1 acc , then the language is Σ ω . Hence there exists v = v 0 s.t. UL(v 0 (A)) = UL(v(A)).
Assume the machine does not halt, and consider any valuation v = v 0 . As Figure 2: Undecidability of the language preservation problem for PERAs in the previous case, the transition from l start to l 1 acc can be taken, giving (at least) Σ ω . In addition, for this valuation, the transition to l 0 can be taken (since v(p) > 0), and the 2CM starts to be simulated. However, from our encoding in Section 3, this run will stop after v(p) steps, and will block (without reaching l halt as the 2CM does not halt). This blocking run is a finite blocking run, therefore is a maximal run, and is therefore part of the language. Hence the language contains all infinite runs (Σ ω ) plus one finite blocking run-which was not part of v 0 (A). Therefore, for all
l start l 0 l halt l 1 acc l 2 acc A 2CM x a < p ∧ x a = 0 Σ Σ x a = p ∧ x a = 0 Σ Σ Σ Σv = v 0 , UL(v 0 (A)) UL(v(A)).
This gives that there exists v = v 0 s.t. UL(v 0 (A)) = UL(v(A)) iff M halts.
Varying the definition of language
We can wonder whether the undecidability comes from our definition of the language (consistent with [AM15]). We briefly discuss other cases to show that it does not. To prove undecidability of the three cases below, we need to perform a common modification to the 2CM encoding. From any intermediary location l of A 2CM , we add a transition guarded with t = p ∧ z = p − 1 leading to a new location l sink with actions Σ. This transition can only be taken after exactly v(p) steps of the 2CM, i. e., instead of blocking after v(p) steps, the run goes to l sink .
Büchi condition or reachability condition
Let us redefine the untimed language as the set of all words associated with a run passing infinitely often through an accepting location. In that case, our scheme in Fig. 2 can be kept with only mild modifications: let l 1 acc , l 2 acc and l sink be the accepting locations. Let us add a self-loop on l sink with a new action, say a 3 . For v 0 , the untimed language is Σ ω (the notation Σ remains unchanged, i. e., does not contain a 3 ). If the 2CM halts, some valuations will reach l halt , and the untimed language is Σ ω . If the 2CM does not halt, Σ ω is part of the language but, for all v = v 0 , the run will block, and end in l sink where it will perform an infinite word with suffix (a 3 ) ω , which differs from the discrete behavior of v 0 (A).
The case of the language defined as the set of finite words ending in an accepting location is similar.
Safety untimed language
Finally, let us redefine the untimed language as the set of untimed words associated with any finite run (no accepting locations are considered). In that case, we relabel the transitions to l sink with the fresh action a 3 . For v 0 , the language becomes Σ * . If the 2CM halts, the language is again Σ * for some valuations. However, if the 2CM does not halt, for v = v 0 the (unique) run will block, and end in l sink with a 3 as suffix-yielding a word not part of UL(v 0 (A)).
Conclusion
We proved that the language preservation remains undecidable for a subclass of PTAs, namely parametric event-recording automata. We believe that the L/U-automata restrictions considered in the additional undecidability results of [AM15] could apply to our setting, and undecidability would still hold for "L/U-PERAs". A more challenging future work is to study the trace preservation problem of [AM15] that considers not only the actions but also the locations.
x a the clock associated with a. On any transition labeled with a, x a is implicitly reset.Given a PERAA and a parameter valuation v, we denote by v(A) the nonparametric event-recording automaton where all occurrences of a parameter p i have been replaced by v(p i ), for each p i ∈ P . Definition 2 (Concrete semantics of an ERA). Given a PERA A = (Σ, L, l 0 , P, I, E), and a parameter valuation v, the concrete semantics of v(A) is given by the timed transition system (S, s 0 , →), with S = {(l, µ) ∈ L × R |XΣ| + | µ |= v(I(l))}, s 0 = (l 0 , 0), and → consists of the discrete and (continuous) delay transition relations:
Figure 1 :
1Encoding a 2-counter machine into a PERA transitions: one transition from l i tol j with guard t = 0 ∧ x k = 0 (the actual 0-test for C k ), and one transition from l i tol j ′ with guard x k = 1 and labeled with a k (therefore resetting x k ). Then, from each locationl of A, there is a transition to the corresponding location l with guard z = p − 1 and resetting z due to action a z (seeFig. 1b).Clock z counts the number of steps (when considering the value of this clock when t = 0, it encodes a counter that is incremented at every transition of M).Notice that clock z counts, but for the moment, it does not impose any constraint on the length of the simulation. Let us now add condition 0 < t < p to the guards z = p − 1 of the transitions leaving the intermediary locations. This way, when z (seen as a counter) has value p − 1 (when t = 0 or p), no transition is available from any locations, so that the encoding stops after mimicking p − 1 steps of the execution of M. As a consequence, our encoding v(A) only encodes properly the v(p) first steps of M, and then blocks (therefore steps beyond v(p) steps are not encoded at all).
, that contains the encoding of a 2CM M, denoted by A 2CM . A transition labeled with Σ denotes 4 transitions labeled with a t , a 1 , a 2 , a z respectively. Let v 0 be the valuation s.t. v 0 (p) = 0. Given A and v 0 , we will show that there exists v = v 0 s.t. UL(v 0 (A)) = UL(v(A))
A theory of timed automata. Rajeev Alur, David L Dill, Theoretical Computer Science. 1262Rajeev Alur and David L. Dill. A theory of timed automata. Theo- retical Computer Science, 126(2):183-235, April 1994.
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What's decidable about parametric timed automata?. Étienne André, International Journal on Software Tools for Technology. To appearÉtienne André. What's decidable about parametric timed automata? International Journal on Software Tools for Technology, 2018. To appear.
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Integer parameter synthesis for timed automata. Aleksandra Jovanović, Didier Lime, Olivier H Roux, IEEE Transactions on Software Engineering. 415Aleksandra Jovanović, Didier Lime, and Olivier H. Roux. Integer pa- rameter synthesis for timed automata. IEEE Transactions on Software Engineering, 41(5):445-461, 2015.
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We propose an efficient protocol for secure comparison of integers when both integers are shared between two parties. Such protocols are useful for implementing secure auctions. The proposed protocol's computational complexity is roughly half the complexity of the best known efficient protocol. The efficiency of the proposed protocol stems from the removal of the XOR computation which is a time consuming operation.
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80caf507df71e0afeab9aaac4b5e3158f3f397ee
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Index Terms-Integer comparisonsecure protocolsecure auctionsexclusive-ORprivacy
We propose an efficient protocol for secure comparison of integers when both integers are shared between two parties. Such protocols are useful for implementing secure auctions. The proposed protocol's computational complexity is roughly half the complexity of the best known efficient protocol. The efficiency of the proposed protocol stems from the removal of the XOR computation which is a time consuming operation.
I. INTRODUCTION
ECURE comparison can be used to perform secure auctions as follows. The two parties performing the auction are A and B. A can be thought of as the auction house and B as an accounting company. A and B have shares of the current highest bid X. A's share is X A and B's share is X B . The next bidder C sends the shares, Y A and Y B of his bid Y to A and B respectively. A and B then conduct a secure protocol to decide if Y > X or not. If Y > X then Y becomes the current highest bid and the next bidder sends the shares of her bid to A and B. The comparison protocol is again repeated to check if the new bid is greater than the current highest bid. We assume that A and B are semi-honest (or they follow the protocol) because the actions of A and B can be checked after the auctions have completed. The proposed protocol is a modification of the most efficient known protocol of [1,2]. Our contributions are: The main reason why the protocol of [1] is computationally complex is because the computation of X⊕Y with shares of bits of X and Y requires 2l encryptions and 2l decryptions, where l is the number of bits in X or Y. The reason why X⊕Y is needed for comparison is to find out which bits of X and Y are the same. We achieve this by simply computing the difference, D, between the individual bits of X and Y. The digits of D can therefore be 0, 1, or -1. To check if a given subset of digits of D, are 0, we check if the sum of the products of the digits of that subset and consecutive powers of 2 is 0. Note that if a subset of digits of D are all 0 then the corresponding bits of X and Y are equal. Related work: Secure comparison protocols have been implemented using Yao's garbled circuits [3], using encryption of bits as quadratic and non-quadratic residues modulo an RSA modulus [4], homomorphic encryption R. S. Katti is with North Dakota State University, Fargo, ND 58108 USA (phone: 701-231-7369; fax: 701-231-8677; e-mail: rajendra.katti@ ndsu.edu).
C. Ababei is with North Dakota State University, Fargo, ND 58108 USA (e-mail: [email protected]). [1,2,5], and other adhoc techniques [6,7]. The most efficient amongst these is the protocol of [1] because it makes use of a smaller plaintext space for the encryption scheme. We use the same encryption scheme of [1] and describe it next. Homomorphic encryption: To generate the keys for encryption, parameters k, t, and l are defined such that k > t > l. k is the number of bits in an RSA modulus n, such that n = pq, where p and q are primes. u and v p , and v q are another set of primes that are chosen such that v p | (p-1) and v q | (q-1) [2]. u has at least l+2 bits and v = v p v q has at least t bits. The shares of X and Y are in Z u . We choose random elements ݃, ݄ ∈ ܼ * such that the multiplicative order of h is v = v p v q and g has order uv. The public key is now pk = (n, g, h, u) and the secret key is sk = (p, q, v p , v q ). The plaintext space is Z u and the ciphertext space is ܼ * . To encrypt a message m ∈ Z u , we choose a random 2t-bit integer r, to obtain the ciphertext c, as follows, ܧ ሺ݉, ݎሻ ൌ ݃ ݄ ݀݉ ݊. The decryption of c can be done as follows, ܿ ௩ ൌ ݃ ௩ ݄ ௩ ൌ ݃ ௩ . Since g v has order uv and u is very small, one can build a table containing values of ݃ ௩ mod n and corresponding values of m. In our protocol we just need to check if c encrypts the message 0. This can easily be done by checking if c v mod n = 1. The following equations make the above encryption scheme homomorphic.
ܧ ሺ݉, ܧሻݎ ሺ݉ ᇱ , ݎ ᇱ ሻ ݀݉ ݊ ൌ ܧ ሺ݉ ݉ ᇱ ݀݉ ,ݑ ݎ ݎ ᇱ ሻ. ܧ ሺ݉, ݎሻ ௦ ݀݉ ݊ ൌ ܧ ሺ݉ݏ ݀݉ ,ݑ .ሻݏݎ
Let the l-bit integers being compared be X = (x l , …, x 1 ) and Y = (y l , …, y 1 ). Parties A and B have additive shares of each bit of X and Y. Therefore A has X A = (x lA , …, x 1A ) and Y A = (y lA , …, y 1A ) and B has X B = (x lB , …, x 1B ) and Y B = (y lB , …, y 1B ) such that x i = x iA + x iB mod u, and y i = y iA + y iB mod u, where x iA , x iB , y iA ,y iB ∈ Z u , i = 1, …, l. Note that x i and y i are bits. A sharing of bit y is written as [y] and denotes y's shares with A and B; namely y A and y B , such that y = y A +y B mod u. The protocol in [1] is given below for completeness. Protocol 1: Secure comparison. Input: A has X A and Y A and B has X B and Y B . Output: Y > X or Y ≤ X. Note that if c i = 0 then ߛ is an encryption of 0, otherwise it is a random non-zero value. B sends these encryptions to A in randomly permuted order. 5. A uses his secret key to check if any of the received encryptions are encryptions of 0. If this is the case he outputs Y > X, otherwise he outputs Y ≤ X. The problem with the above method is that step 1 is computationally intensive. We illustrate this by giving a protocol for computing p⊕q, where p,q ∈ {0,1}. A has shares p A and q A and B has shares p B and q B of p and q respectively such that p = p A +p B mod u, and q = q A +q B mod u. Therefore p⊕q = p + q -2pq = (p A +p B ) + (q A +q B ) -2(p A +p B ) (q A +q B ) = (p A +q A ) + (p B +q B ) -2(p A q A +p B q B + p A q B + p B q A ) (1) A can compute (p A +q A ) and p A q A and B can compute (p B +q B ) and p B q B . However computing p A q B and p B q A is not straight forward. Protocol 2 below shows how to compute p A q B . Protocol 2: Compute p A q B . Input: A has p A and and B has q B . Output: A has p A q B -r and B has r. 1. A sends E pk (p A ) to B. 2. B chooses r ∈ Z u and computes E pk (p A q B -r mod u) using the homomorphic property of the encryption scheme and sends it to A. 3. A decrypts the received encryption to get p A q B -r. Therefore the shares of p A q B are (p A q B -r, r). p B q A can be similarly computed and hence p⊕q.
II. PROPOSED PROTOCOL
We present a method that eliminates the computation of x i ⊕y i in step 1 of Protocol 1. This reduces the complexity of Protocol 1 by half. Our protocol is given below. Protocol 3: Secure comparison. Input: A has X A and Y A and B has X B and Y B . Output: Y > X or Y ≤ X. , it follows that w ≠ 0 if any of its digits is nonzero. Therefore all the digits ݀ must be 0 for w to be 0.
Remark: The largest power of 2 used in step 2 of Protocol 3 is 2 l-1 . Since u has l+2 bits it is always greater than 2 l-1 .
III. CONCLUSION
We have proposed a protocol for comparison of two integers, X and Y, when two parties A and B have additive shares of X and Y. Our protocol has half the computational complexity of the most efficient known protocol of [1]. Our protocol achieves efficiency by eliminating the computation of XOR which is a time consuming task. The communication complexity of our protocol is the same as the protocol of [1]. The proof of security of our protocol when A and B are semihonest consists of a simulator that is given the inputs and outputs of the corrupted party and produces messages that are identically distributed to the messages it receives in a real run of the protocol.
1. A and B compute shares [d i ], i = 1, …, l, where d i = x i + y i -2x i y i = x i ⊕y i . Note d i ∈ {0,1}. 2. A and B compute shares [c i ], i = 1, …, l, where c i = x i -y i + 1 + ∑ ݀ ୀାଵ .If there exists i such that all the bits (x l , …, x i+1 ) are identical to the bits (y l , …, y i+1 ) and x i -y i + 1 = 0, then Y > X. Note that (x l , …, x i+1 ) and (y l , …, y i+1 ) are the same if and only if (d l , …, d i+1 ) are all 0 or
1. A and B compute shares [d i ], i = 1, …, l, where d i = x i -y i . Note d i ∈ {0,1, -1}. 2. A and B compute shares [c i ], i = 1, …, l, where c i = x i -y i + 1 + ∑ ݀ ୀାଵ 2 ିାଵ . The rest of the protocol is the same as Protocol 1. Protocol Correctness: If there exists i such that all the bits (x l , …, x i+1 ) are identical to the bits (y l , …, y i+1 ) and x i -y i + 1 = 0, then Y > X. This is equivalent to: If there exists i such that all the bits (d l , …, d i+1 ) are 0 and d i + 1 = 0, then Y > X (d i 's are from step 1 of Protocol 3). Let w i = ∑ ݀ ୀାଵ 2 ିାଵ = 2݀ 2 ଶ ݀ ିଵ ⋯ 2 ି ݀ ାଵ , where ݀ ∈ ሼ0,1, െ1ሽ. Note that if we had left w i = ∑ ݀ ୀାଵ (like in step 1 of Protocol 1), then w i can be 0 even if all the d j 's are not all 0, is 0 if and only if each d j , j = i+1, …, l is 0 (see Lemma 1 below). Therefore if there exists i such that w i = 0 and d i + 1 = 0, then Y > X. Thus a comparison can be performed without explicitly computing any XOR. Lemma 1: Let w = ∑ ݀ ୀଵ 2 , where ݀ ∈ ሼ0,1, െ1ሽ. w = 0 if and only if ݀ ൌ 0, ∀݆ ൌ 1, … , ݈. Proof: If all ݀ ൌ 0 then it follows that w = 0. Now we show that if w = 0 then all ݀ ൌ 0. Consider the digits (d l , …, d 1 ), where ݀ ∈ ሼ0,1, െ1ሽ. Let the i th digit, d i be the leftmost nonzero digit in (d l , …, d 1 ). Therefore d i ∈ {1, -
Secure Comparison Without Explicit XOR Rajendra S. Katti and Cristinel Ababei Department of ECE, North Dakota State University, Fargo, North Dakota, USA Let ߙ i and ߚ i be the shares of c i that A and B have locally computed. A computes encryptions E pk (ߙ i , r i ) and sends them all to B. 4. B chooses random s i ∈ Z u * and ݏ ᇱ as a 2t bit integer and computes a random encryption of the form ߛ ൌ ሺܧ ሺߙ , ݎ ሻ݃ ఉ ሻ ௦ ݄ ௦ ᇲ ݀݉ ݊.S
equivalently ∑
݀
ୀାଵ
ൌ 0.
3.
ିାଵ , where d and d are shares of d . Note that the sum ∑ d ୀାଵ 2 ିାଵ ൏ ݑ because d ∈ ሼ0,1, െ1ሽ and ∑ 2 ൏ 2 ିଵ ୀଵTherefore
w i
=
∑
ሺd d ሻ
ୀାଵ
2 ିାଵ ݀݉ ݑ ൌ
∑
d
ୀାଵ
2 ିାଵ ݀݉ ݑ ൌ ∑
d
ୀାଵ
2 .
Homomorphic encryption and secure comparison. I Damgard, M Geisler, M Kroigard, Int. J. Applied Cryptography. 11I. Damgard, M. Geisler, and M. Kroigard, "Homomorphic encryption and secure comparison," Int. J. Applied Cryptography, Vol. 1, No. 1, pp. 22-31, February 2008.
A correction to "Efficient and secure comparison for on line auctions. I Damgard, M Geisler, M Kroigard, Int. J. Applied Cryptography. 14I. Damgard, M. Geisler, and M. Kroigard, "A correction to "Efficient and secure comparison for on line auctions"," Int. J. Applied Cryptography, Vol. 1, No. 4, pp. 323-324, August 2009.
Privacy preserving auctions and mechanism design. M Naor, B Pinkas, R Sumner, EC. 99ACM PressM. Naor, B. Pinkas, and R. Sumner, "Privacy preserving auctions and mechanism design," EC'99, New York: ACM Press, pp. 129-139, 1999.
A cost-effective pay-per-multiplication comparison method for millionaires. M Fishlin, CT-RSA'01 and Lecture Notes in Computer Science. Springer2020M. Fishlin, "A cost-effective pay-per-multiplication comparison method for millionaires," in CT-RSA'01 and Lecture Notes in Computer Science, Springer, Vol. 2020, pp. 457-472, 2001.
Practical and secure solutions for integer comparison. J Garay, B Schoenmakers, J Villagas, Lecture Notes in Computer Science. 4450SpringerJ. Garay, B. Schoenmakers, and J. Villagas, "Practical and secure solutions for integer comparison," Lecture Notes in Computer Science, Springer, Vol. 4450, pp. 330-342, 2007.
Conditional encrypted mapping and comparing encrypted numbers. I F Blake, V Kolesnikov, AsiaCrypt'04, and Lecture Notes in Computer Science. Springer3329I. F. Blake and V. Kolesnikov, "Conditional encrypted mapping and comparing encrypted numbers," in AsiaCrypt'04, and Lecture Notes in Computer Science, Springer, Vol. 3329, pp. 515-529, 2004.
Strong conditional oblivious transfer and computing on intervals. I F Blake, V Kolesnikov, FC'06, and Lecture Notes in Computer Science. Springer4107I. F. Blake and V. Kolesnikov, "Strong conditional oblivious transfer and computing on intervals," in FC'06, and Lecture Notes in Computer Science, Springer, Vol. 4107, 2006.
|
[] |
[
"Optimal heterogeneity in a simplified highly renewable European electricity system",
"Optimal heterogeneity in a simplified highly renewable European electricity system"
] |
[
"Emil H Eriksen \nDepartment of Physics and Astronomy\nAarhus University\nNy Munkegade 1208000Aarhus CDenmark\n",
"Leon J Schwenk-Nebbe \nDepartment of Physics and Astronomy\nAarhus University\nNy Munkegade 1208000Aarhus CDenmark\n",
"Bo Tranberg \nDepartment of Engineering\nAarhus University\nInge Lehmanns Gade 108000Aarhus CDenmark\n\nDanske Commodities A/S\nVaerkmestergade 38000Aarhus CDenmark\n",
"Tom Brown \nFrankfurt Institute for Advanced Studies (FIAS)\nJohann Wolfgang Goethe Universität\nRuth-Moufang-Straße 160438Frankfurt am MainGermany\n",
"Martin Greiner \nDepartment of Engineering\nAarhus University\nInge Lehmanns Gade 108000Aarhus CDenmark\n"
] |
[
"Department of Physics and Astronomy\nAarhus University\nNy Munkegade 1208000Aarhus CDenmark",
"Department of Physics and Astronomy\nAarhus University\nNy Munkegade 1208000Aarhus CDenmark",
"Department of Engineering\nAarhus University\nInge Lehmanns Gade 108000Aarhus CDenmark",
"Danske Commodities A/S\nVaerkmestergade 38000Aarhus CDenmark",
"Frankfurt Institute for Advanced Studies (FIAS)\nJohann Wolfgang Goethe Universität\nRuth-Moufang-Straße 160438Frankfurt am MainGermany",
"Department of Engineering\nAarhus University\nInge Lehmanns Gade 108000Aarhus CDenmark"
] |
[] |
The resource quality and the temporal generation pattern of variable renewable energy sources vary significantly across Europe. In this paper spatial distributions of renewable assets are explored which exploit this heterogeneity to lower the total system costs for a high level of renewable electricity in Europe. Several intuitive heuristic algorithms, optimal portfolio theory and a local search algorithm are used to find optimal distributions of renewable generation capacities that minimise the total costs of backup, transmission and renewable capacity simultaneously. Using current cost projections, an optimal heterogeneous distribution favours onshore wind, particularly in countries bordering the North Sea, which results in average electricity costs that are up to 11% lower than for a homogeneous reference distribution of renewables proportional to each country's mean load. The reduction becomes even larger, namely 18%, once the transmission capacities are put to zero in the homogeneous reference distribution. Heuristic algorithms to distribute renewable capacity based on each country's wind and solar capacity factors are shown to provide a satisfactory approximation to fully optimised renewable distributions, while maintaining the benefits of transparency and comprehensibility. The sensitivities of the results to changing costs of solar generation and gas supply as well as to the possible cross-sectoral usage of unavoidable curtailment energy are also examined.
|
10.1016/j.energy.2017.05.170
|
[
"https://arxiv.org/pdf/1706.00463v1.pdf"
] | 119,076,893 |
1706.00463
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a1aa54faecdfc91cb9be7ba55d57b845849c2a20
|
Optimal heterogeneity in a simplified highly renewable European electricity system
Emil H Eriksen
Department of Physics and Astronomy
Aarhus University
Ny Munkegade 1208000Aarhus CDenmark
Leon J Schwenk-Nebbe
Department of Physics and Astronomy
Aarhus University
Ny Munkegade 1208000Aarhus CDenmark
Bo Tranberg
Department of Engineering
Aarhus University
Inge Lehmanns Gade 108000Aarhus CDenmark
Danske Commodities A/S
Vaerkmestergade 38000Aarhus CDenmark
Tom Brown
Frankfurt Institute for Advanced Studies (FIAS)
Johann Wolfgang Goethe Universität
Ruth-Moufang-Straße 160438Frankfurt am MainGermany
Martin Greiner
Department of Engineering
Aarhus University
Inge Lehmanns Gade 108000Aarhus CDenmark
Optimal heterogeneity in a simplified highly renewable European electricity system
large-scale integration of renewablessystem designrenewable energy networkswind power generationsolar power generationlevelised system cost of electricityEurope
The resource quality and the temporal generation pattern of variable renewable energy sources vary significantly across Europe. In this paper spatial distributions of renewable assets are explored which exploit this heterogeneity to lower the total system costs for a high level of renewable electricity in Europe. Several intuitive heuristic algorithms, optimal portfolio theory and a local search algorithm are used to find optimal distributions of renewable generation capacities that minimise the total costs of backup, transmission and renewable capacity simultaneously. Using current cost projections, an optimal heterogeneous distribution favours onshore wind, particularly in countries bordering the North Sea, which results in average electricity costs that are up to 11% lower than for a homogeneous reference distribution of renewables proportional to each country's mean load. The reduction becomes even larger, namely 18%, once the transmission capacities are put to zero in the homogeneous reference distribution. Heuristic algorithms to distribute renewable capacity based on each country's wind and solar capacity factors are shown to provide a satisfactory approximation to fully optimised renewable distributions, while maintaining the benefits of transparency and comprehensibility. The sensitivities of the results to changing costs of solar generation and gas supply as well as to the possible cross-sectoral usage of unavoidable curtailment energy are also examined.
Introduction
The ambitious renewable energy targets set by European governments [1] imply that the share of renewables in electricity generation will increase significantly in the years to come. At present, the leading renewable technologies are wind, solar photovoltaics (PV) and hydroelectricity, of which only wind and solar PV have the potential for large scale expansion. The uneven distribution of wind and solar resources across the continent raises the question of how best to exploit these heterogeneous resources. If wind and solar generation capacities are concentrated in those countries with the best resources, this may increase demand for transmission and increase energy imbalances between countries; if wind and solar generation are distributed homogeneously, then the best renewable resources will not be fully used and total system costs may be higher than the heterogeneous optimum. In this paper, the consequences of heterogeneity for the whole electricity system, including backup generation and transmission, will be quantified. Since wind and solar PV are both Variable Renewable Energy Sources (VRES), backup generation is needed if the electrical demand is to be met at all times. Backup generation introduces additional system costs, which depend on the mismatch between VRES generation and load. Using the degrees of freedom associated with the choice of the capacity distributions of VRES for each country, it is possible to smooth out the aggregated temporal generation pattern or even shape it towards the load pattern. As a result, the mismatch and thus the backup requirements is lowered. To decrease the dimensionality of the problem, renewable assets can be assigned homogeneously, proportional to the mean load of each country, with a uniform wind-to-solar mixing factor. This approach is demonstrated in [2,3], where optimal wind-to-solar mixes for Europe are found that minimise balancing and storage costs. Further reductions in backup requirements are possible by extending the transmission network to enable more energy exchange between the countries [4,5]. The implications for total system costs of different homogeneous renewable penentrations, wind-solar mixes and transmission levels were considered in [6], where the cost-optimal design was found to consist of a renewable energy penetration of 50% and a wind fraction of 94%. Other relevant research on the advantages of grid extensions for the integration of renew-ables, including reduced variability and smaller forecast errors, can be found in [7][8][9][10][11][12][13]. In this paper the consequences of moving from a homogeneous spatial distribution of VRES and a uniform wind-tosolar mixing factor to a cost-optimal placement of VRES capacities around Europe are explored. The distribution of VRES plants is determined by at least two considerations. The first consideration is the geographical variation of the VRES quality. The resource quality is quantified through the capacity factor (CF) defined as CF = average generation rated capacity .
The capacity factor is a number between 0 and 1, where 0 means no generation and 1 means maximum generation at all times. Capacity factors for the European countries for onshore wind and solar PV are calculated using (1) and listed in Table 2. The second consideration is the geographical variation of the temporal generation pattern for a given VRES type. This effect is particularly important for wind since Europe is large compared to the correlation length of wind of ≈ 600 km [14][15][16], and wind therefore benefits from smoothing effects across the continent. With these points in mind, the optimal heterogeneous spatial layouts of wind and solar PV across Europe is investigated and compared to the homogeneous layouts. The main point of comparison is the average cost of electricity, which is composed of the VRES, backup and transmission costs. Different approaches to cope with the resulting large number of degrees of freedom are considered. In the literature a common approach for heterogeneous systems is to use linear programming to optimise generation and transmission capacities simultaneously [17][18][19][20], but this has the drawback that only a selection of representative weather conditions can be considered before computation times become infeasible. This makes the results susceptible to over-tuning to the weather selection. Other groups have used genetic algorithms to optimise generation, storage and transmission over a full year in Australia [21] and over three years in Europe, the Middle East and North Africa [22]. In this paper a novel local search algorithm was found to be most effective given the size and non-linear formulation of the optimisation problem, allowing 8 years of hourly weather to be considered. A downside of pure optimisation approaches is that one loses an understanding of why particular solutions are optimal. This makes it hard to justify investment strategies to policy makers and to the public. To counter this downside, more intuitive heuristic methods are developed here to construct layouts based on knowledge of resource quality, which are then compared to layouts obtained through optimisation. Distributions proportional to capacity factors (similar to the approach in [11]) and distributions based on optimal portfolio theory that reduce risk, or standard deviation, of the in-feed (similar to approaches in [23][24][25][26]) are considered and compared. This paper is organised as follows: Section 2 discusses the general modelling of the simplified European electricity system and the key infrastructure measures. Section 3 describes the construction of heterogeneous layouts. In Section 4 the performance of the different layouts and the resulting renewable penetrations for individual European countries are discussed. Section 5 contains an analysis of the sensitivity of the results to variations in component costs. We conclude the paper with a discussion on the results and an outlook on future research.
Methods I: general modeling
Renewable resource assessment
Realistic time series describing the country-specific wind and solar PV power generation and the load are the starting point of the advocated weather-driven modelling of a simplified networked European electricity system. The utilized data set has been released from the Fraunhofer Institute for Wind Energy and Energy System Technology (formerly ISET, now IWES) [27]. This data set covers the eight-year period from January 2000 to December 2007, has a temporal resolution of one hour and a spatial resolution of 50 × 50 km 2 over all of Europe. Fixed countryspecific capacity layouts have been used to first convert the weather data into onshore wind and solar power generation, and then to aggregate the latter over each of the 30 European countries; off-shore wind power generation is not considered. The country-specific load time series have been obtained from publicly available sources, extrapolated to cover missing data, and detrended from an annual growth of around 2% to their year 2007 values. For more specific details see [2,27]. A good alternative description of the conversion modelling is given in [28][29][30].
The obtained wind and solar PV power generation time series have been rescaled to the capacity factors (CFs) from 2014. The latter have been determined in accordance with equation (1) from the EuroStat data for the installed capacities and the total generation for the year 2014 [31][32][33]. The resulting CFs for each country and each technology are listed in Table 2. For some of the countries (particularly smaller countries) no data was available or the calculated result was too uncertain because of too little or no installed capacity. For these countries the CFs are calculated as an average value from surrounding countries. These cases are marked by a star. Some countries with an already high installed capacity have a relatively low capacity factor compared to validated results from [28].
For them the CFs have been raised by a small factor: 8% in Germany, 4% for wind in Spain, 4% for solar in Italy and 2% for wind in Great Britain. The final capacity factors presented in Table 2 are in accordance with [29,30], which presents a critical assessment of current and future national capacity factors. Capacity factors for wind are likely to rise further in the future because of re-powering of wind turbines with more efficient, modern turbines at higher hub heights [34].
The electricity network
The European electricity network is modelled as a simplified 30-node model, where each node represents a country. For each node n the generation from VRES (see Table 1 for a summary of nomenclature),
G R n (t) = G W n (t) + G S n (t),(2)
can be expressed through two parameters. The penetration γ determines the amount of renewable energy generated relative to the mean load of the node,
G R n = γ n L n ,(3)
while the mixing parameter α fixes the wind-to-solar ratio,
G W n = α n G R n ,(4)G S n = (1 − α n ) G R n .(5)
Other forms of renewable power generation are neglected in this simplistic modelling approach. The nodal difference between VRES generation and load
∆ n (t) = G R n (t) − L n (t)(6)
is called the mismatch. To avoid power outages, the demand must be met at all times. Since storage is not considered, any power deficits must be covered by backup generation. Dispatchable resources are not modelled explicitly, but are considered as part of the backup generation. If ∆ n (t) ≥ 0, excess energy C n (t) must be curtailed, while if ∆ n (t) < 0 backup generation G B n (t) is needed. Together the two terms form the nodal balancing B n (t) = C n (t)−G B n (t). It is possible to lower the balancing needs with transmission. Nodes with excess generation export energy E n (t), allowing nodes with an energy deficit to import energy I n (t) to (partly) cover their energy deficit. The nodal injection, E n (t) − I n (t), is denoted P n (t). This leads to the nodal balancing equation,
G R n (t) − L n (t) = B n (t) + P n (t) ,(7)
The vector of nodal injections is called the injection pattern, and fullfills n P n (t) = 0. The actual imports and exports, and thus the injection pattern, depend on the dispatch of the nodal balancing. The synchronised balancing scheme,
B n (t) = L n k L k m ∆ m (t) ,(8)
where all nodes are curtailing/generating backup synchronously (relative to L n ), fulfills two top priorities: it minimises the total backup generation for each time step and it minimises the overall backup capacity [35]. This stylised synchronised balancing scheme has also been chosen in view of the layout optimisation, since the computational time for an update step is much smaller than for other dispatch schemes, like for example the localised flow scheme used in two previous publications [4,5]. The injection pattern is fixed by Eqs. (7) and (8), and determines the power flows on the links l:
F l (t) = n H ln P n (t) .(9)
The linear relationship follows from the DC approximation, which is known to be a good approximation for highvoltage flows. For the Power Transfer Distribution Factors H ln we have assumed unit susceptances [35], allowing its construction from the Moore-Penrose pseudo inverse of the underlying network Laplacian.
Infrastructure measures
Following [6], the energy system cost is calculated based on a few key measures. Besides the cost of the VRES capacities, K W and K S , costs for the backup system and the transmission network are included. The backup system cost is split into two components, the cost of backup capacity K B and the cost of backup energy E B . The backup capacity cost covers expenses related to construction and to keeping the power plants online while the backup energy cost accounts for actual fuel costs. Expressed in units of the average annual load, the backup energy is given by In principle, the backup capacity is fixed by a single extreme event. However with this definition, the results will be highly coupled to the particular data set used. To decrease the coupling, the 99% quantile is used rather than the maximum value,
E B = n t G B n (t) m t L m (t) = n G B n m L m .(10)q n = K B n 0 p n (G B n ) dG B n ,(11)
where p n (G B n ) is the time sampled distribution of backup generation and q n = 0.99. With this choice, the backup system will be able to fully cover the demand 99% of the time. The remaining 1% is assumed to be covered by unmodelled balancing initiatives, e.g. demand side management. Given the nodal values K B n , the overall backup capacity
K B = n K B n(12)
is calculated by summation. In analogy, the transmission capacity K T l is defined so that the flow is met 99% of the time. Transmission can be positive and negative, but since links are assumed bidirectional, only the magnitude (not the sign) of the flow is to be considered. Hence
q l = K T l 0 p l (|F l |) d|F l | ,(13)
where p l (|F l |) is the time sampled distribution of absolute flows and q l = 0.99. Since the link length varies, K T is not calculated directly by summation, but instead as a weighted sum,
K T = l K T l d l ,(14)
where d l denotes the length of link l. Link lengths are estimated as the distance between the country capitals. In this paper E B will be expressed in units of average annual load, K B in units of average hourly load and K T in units of average hourly load × megametre.
Cost modelling
Cost assumptions for the elements of an electricity system vary greatly across the literature. In this study, the cost assumptions published by [6] have been adapted with a single modification. The cost of solar has been reduced by 50% in accordance with near future solar PV panel price projections [36]. The resulting estimates are listed in Table 3. In general, the cost assumptions are in the low end for VRES which reflects the expectation that the cost of VRES will go down in the future as the penetration increases. Backup generation is priced based on the cost of Combined Cycle Gas Turbines (CCGTs). From the VRES penetration, the mixing factor and the mean load, the mean generation of each node can be calculated. Dividing by the associated capacity factor, the capacity is obtained. Except for transmission capacity, the present value of each element can be calculated directly as
V = CapEx + T life t=1 OpEx t (1 + r) t ,(15)
where r is the rate of return assumed to be 4% per year. The transmission capacity cannot be translated directly into cost as the cost depends on the length and the type of the link. Link costs are assumed to be 400e per km per MW for AC links and 1,500e per km per MW for HVDC links. For HVDC links, an additional cost of 150,000e per MW per converter station pair (one at each end) is added [10,11,37]. The layout of AC and HVDC lines has been constructed by [4] according to the existing European network reported by ENTSO-E for the year 2011 [38] and new predicted lines until 2014 [39,40]. It is shown in Figure 10.
To allow for comparison of different system layouts, the Levelised Cost of Electricity (LCOE) is a convenient measure [6,41,42]. The LCOE is the cost that every generated unit of energy consumed during the lifetime of the project has to match the present value of investment [43],
LCOE V = V T life t=1 L EU,t (1+r) t .(16)
Since the life time of the system elements differs, the LCOE is evaluated separately for each system element from each respective present value. The LCOE for the complete system is calculated by summation. Life times of 25 years for solar PV and onshore wind, 30 years for CCGT plants and 40 years for transmission infrastructure were assumed.
Methods II: heterogeneous layouts
The simplest way to distribute the renewable resources is to assign them homogeneously (relative to the mean load of the node) so that γ n = γ EU = 1 and α n = α EU . This homogenous layout is denoted as HOM. However this assignment might not be ideal since the capacity factors vary significantly between the nodes. Three heuristic schemes and a straightforward optimisation for the construction of heterogeneous layouts will be presented in the following four subsections. The naming of the distribution algorithms is summarised in Table 4.
Heuristic layout I: CF proportional (CFprop)
An intuitive first approach, called CFprop, is to assign resources proportional to the CF, or more general to the CF raised to an exponent β. For a wind-only layout, the nodal renewable penetrations γ n are given by
γ W n = CF W n β L EU m CF W m β L m γ EU ,(17)
where γ EU is the overall penetration assumed to be 1. An equivalent expression for the solar-only layout is obtained by the substitution W → S. Examples for β = 1 are shown in Figure 1a for the wind-and solar-only layouts. In the layout illustrations, each bar represents a country. CFprop layouts for any value of α can be constructed as a linear combination of the wind and solar only layouts with
γ n = α EU γ W n + (1 − α EU )γ S n(18)
and
α n = α EU γ W n α EU γ W n + (1 − α EU )γ S n .(19)
For practical reasons, it is not possible to realise extremely heterogeneous layouts. On the one hand the geographical potentials for VRES installations in countries with good renewable resources may be a limiting factor. On the other hand countries with poor renewable resources may not want to become too dependent on imports. To constrain heterogeneity, the heterogeneity parameter K is introduced by requiring
1 K ≤ γ n ≤ K.(20)
With this definition, K = 1 corresponds to a homogeneous layout while K = ∞ represents unconstrained heterogeneity. For the CFprop layouts, each value of K translates into an α-dependent value of β. For a given value of α, the corresponding β value is found by increasing β until the first country violates equation (20). At the mix α = 0.86 the values K = 1, 2, 3 correspond to β = 0.00, 1.92, 2.91, respectively.
Heuristic layout II: extreme K-constrained (CFmax)
Although the overall capacity factor of a CFprop layout for β > 0 is higher than the capacity factor of the homogeneous layout, it is possible to achieve an even higher capacity factor without violating the constraints in equation (20). In the wind-and solar-only cases, the capacity factor is maximised by assigning γ n = K to the countries with the highest capacity factor and γ n = 1 K to the remaining countries, except for a single in-between country which is fixed by the constraint
n γ n L n = L EU .(21)
The wind-and solar-only cases of the CFmax layout constrained by K = 2 are shown in Figure 1b. Similar to the CFprop layouts, the CFmax layouts for arbitrary α EU values can be constructed as linear combinations (18) of the wind-and solar-only layouts.
Heuristic layout III: OPT
The optimal portfolio theory (OPT) is well known in mathematical finance [44]. It discusses different assets obtained from the tradeoff between maximizing their return and minimizing their risk. This concept has also been applied to find optimal deployment of wind and solar energy resources in large-scale energy systems [24][25][26], where the overall capacity factor has been treated as the return and the variance of the renewable power generation as the risk. In modified form, we will use OPT to further explore VRES capacity layouts over Europe with low system cost of electricity. The overall capacity factor of a wind-only (γ W n ) or solaronly (γ S n ) layout is defined as
CF W/S EU = L EU K W/S EU ,(22)
where represents the overall installed capacity. The overall capacity factor is a useful measure of return as this is of high importance for investors of renewable generation capacity. Investors seek to minimise the overall capacity investment, which corresponds to maximising the overall capacity factor.
K W/S EU = n γ W/S n L n CF W/S n(23)
OPT's second measure is risk, for which we select the relative standard deviation σ ∆ / L EU of the overall mismatch
∆ EU (t) = n ∆ n (t)(24)
based on the country-specific mismatches (6). The smaller the risky standard deviation becomes the more likely is the reduced need for a backup infrastructure, which an investor tries to minimise [23]. A possible heterogeneous wind-or solar-only capacity layout is sampled from a Monte Carlo procedure. The countryspecific renewable penetrations γ n are randomly and independently drawn from a Beta distribution defined on the compact support (20). Γ(β) is the Gamma function. The two shape parameters β 1 and β 2 are determined by requiring γ n = 1 and by envoking the maximum entropy principle [45] to maximal smear out the Beta distribution over the interval (20). For K = 2 the two parameters result in β 1 = 0.80, β 2 = 1.61, and for K = 3 they are β 1 = 0.86, β 2 = 2.57. A capacity layout sampled with this procedure does not necessarily meet the requirement (21). For such cases, all γ n are uniformly rescaled upwards or downwards until the requirement is fulfilled. During the rescaling some of the penetration parameters hit the K constraint (20), and are then frozen for the remainder of the rescaling procedure. The wind-only and solar-only portfolios for K = 2 are shown in Figure 2 in blue and yellow, respectively, with the overall mismatch measure on the first axis and the overall capacity factor measure on the second axis. Each of the portfolios consists of 100000 layouts. Due to the elongated shape of the portfolios there is no clear extended Pareto front in the upper left corners. The Pareto front defines a line, for which at the same time the standard deviation of the overall mismatch (risk) can not be reduced further for a fixed overall capacity factor and the overall capacity factor (return) can not be increased further for a fixed standard deviation of the overall mismatch. For both portfolios we identify a single point to characterise minimum risk and maximum return. This is done by extracting a subset of the points which are simultaneously a part of the top 200 capacity factors and bottom 200 standard deviations. For K = 2 this leaves a sample of 28 layouts for wind and 67 layouts for solar to average and to calculate the respective new overall capacity factor and new overall standard deviation. The resulting points are plotted in white on top of the portfolios. The layouts of these two Pareto optimal points are shown in Figure 1c.
p(γ) = Γ(β 1 + β 2 ) Γ(β 1 )Γ(β 2 ) K K 2 − 1 β1+β2−1 γ − 1 K β1−1 (K − γ) β2−1(25)
In order to find an optimal combined layout, we interpolate between the Pareto-optimal wind-only and solar-only layouts according to (18) and (19). This interpolation conserves the constraint (21) and results in the line shown in Figure 2. Apparently some of the interpolated layouts are able to reduce the standard deviation of the global mismatch further. The interpolated layout marked with a black dot comes with the mixing parameter α EU = 0.87.
Optimised layouts
The full optimisation of the layouts is considered, with the objective to minimise the LCOE with respect to the 60 variables γ 1 , ..., γ N , α 1 , ..., α N for the N = 30 countries. Given the high dimensionality of the search space, a number of optimisation algorithms were tested including the Nelder-Mead method [46], simulated annealing [47], genetic algorithms [48] and cuckoo search [49]. It was found that the continuous enforcement of the normalisation criterion (21) generally decreased the performance of the tested algorithms, and for that reason a new hybrid algorithm was developed to address this problem. While being a classical greedy algorithm in the sense that the locally optimal choice is always taken, the renormalisation problem was circumvented by moving only along the axial directions. The algorithm has been denoted Greedy Axial Search (GAS). When a solution is renormalised, all γ values are scaled either up or down. Therefore, it is possible that some γ values end up outside the boundary (20). The γ values are fixed at the boundary and the rescaling is only applied to the remaining free γ values. In general this approach is problematic since it can change the direction of the search. This is circumvented by holding the specific γ value constant that is considered during the step up/down procedure along a given axis. In this way only some γ values are scaled down/up and the feasibility of moving up/down along the considered axis can be determined. This is the underlying principle of Greedy Axial Search (GAS). As any greedy algorithm, the GAS algorithm works by taking the locally optimal choice. Hence the feasibility for each direction is evaluated, but only the best choice is accepted. This process is repeated until a convergence criterion is fulfilled. At this point the step size is reduced and the iterative optimisation procedure repeated until the step size drops below some tolerance. The algorithm structure is sketched in Algorithm 1. The StepUp and StepDown subroutines generate new solutions by stepping a solution (first argument) up/down along axis i (second argument) with some step size (third argument) after which the solution is renormalised as described above. Values of maxStepSize = 1, minStepSize = 5 · 10 −4 and tolerance = 10 −4 were found to be appropriate. All optimised layouts have been obtained using the GAS routine. These layouts will be denoted GAS layouts. Constraining the transmission and thereby reducing the trans- Distribution proportional to a power (CF ) β of the capacity factor CF CFmax Assignment to each country γ n extremised within 1 K ≤ γ n ≤ K depending on CF OPT Distribution using Optimal Portfolio Theory GAS Distribution optimised using Greedy Axial Search algorithm GAS* As GAS, but with optimally constrained transmission GASnoT As GAS but with no transmission between countries, so that each country is selfsufficient at all times mission capacity can lead to an overall lower LCOE. This is discussed in Section 4.4. The layouts resulting from this additional optimisation will be denoted GAS* layouts.
Results
The optimal heuristic layouts CFprop, CFmax, OPT as well as the optimized layouts GAS will be discussed in the next three subsections, first for K = 1, then for K = 2, and finally for K = 3. The fourth subsection focuses on the transmission capacities.
K = 1 layouts
By construction, the layouts CFprop, CFmax and OPT become identical and homogeneous for K = 1. Due to Eq. (20), their respective renewable penetrations are γ n = 1. Moreover, according to Eq. (19) their renewable mixes α n = α EU also turn out to be independent of the country index. For these strictly homogeneous layouts Figure 3 shows the dependence of the key infrastructure measures on α EU as the blue curves. For the backup energy and backup capacity, the optimal mixing parameters are located around α EU = 0.85, which is slightly larger than the values found by [2,3]. For the transmission capacity, the minimum occurs around α EU = 0.45. The main measure of interest, the LCOE, has a minimum at α EU = 0.90. The high cost at α EU = 0 is caused by a combination of high backup energy/capacity costs and the fact that the CF of solar is generally lower than for onshore wind. The cost of producing one unit of energy is thus higher for solar than for onshore wind even though the specific CapEx is lower for solar. The homogeneous layout producing the minimum LCOE at α EU = 0.90 is denoted as the 'HOM' layout. It is illustrated in Figure 4a. wind, solar, backup and transmission parts are listed in the third column of Table 5 and graphed as the second bar in Figure 5. Wind power dominates the overall LCOE. Its contribution amounts to 61%, and is followed by 21% from backup, 10% from solar and 8% from transmission. Contrary to the HOM layout, the K = 1 GAS layout is no longer strictly homogeneous. Of course, all renewable penetrations are still equal to γ n = 1, but as a result of the optimisation the wind-solar mixing parameters become heterogeneous. This is illustrated in Figure 4b. Twothirds of the countries are wind-only with α n = 1. The remaining countries have a significant share of solar. For some of those this was to be expected. Spain, Greece, Italy, Romania and Serbia have very large solar capacity factors. See again Table 2. However, other solar-rich countries, like Portugal, Bulgaria, Bosnia and Croatia, are not amongst them. Instead, Germany is also assigned a sig-nificant share of solar, although its solar capacity factor is only average. By taking a closer inspection of Table 2 we discover the following empirical finding for the K = 1 GAS layout: all countries with α n = 1 come with a ratio between their solar and wind capacity factor which is smaller than CF S n /CF W n < 0.65. The countries with α n < 1 have a larger ratio CF S n /CF W n ≥ 0.65, except for the three smallest countries Estonia, Latvia and Luxembourg. Compared to the HOM layout, the α-heterogeneity of the K = 1 GAS layout is able to reduce the total LCOE by 3%. This is mostly a consequence of the reduced combined component costs for wind and solar power. Note, that the overall mixing parameter α EU = n α n L n / L EU has also slightly reduced from 0.90 (HOM) to 0.84 (GAS). See the fourth column of Table 5 and the third bar of Figure 5. The costs for backup and transmission have not changed much; which is also apparent from the rightmost panel of Table 5: Componentwise LCOE for the optimal CFprop, optimal CFmax, optimal OPT, optimised GAS and optimised GAS* layouts for K = 1 (left), 2 (middle) and 3 (right). Note that the K = 1 layouts CFprop, CFmax and OPT are identical and denoted as HOM. The K = 1 layout GASnoT without transmission is listed as reference. All costs are given in e/MWh.
K = 1 K = 2 K = 3
GASnoT HOM GAS GAS * CFprop CFmax OPT GAS GAS * CFprop CFmax OPT GAS GAS * All K = 1 layouts discussed so far include the transmission infrastructure. It is also interesting to compare them to an optimised layout without transmission. No exports and imports would then be possible and the injection pattern P n (t) would always be zero. No transmission investment would be needed and the respective componentwise LCOE would be zero. However, the countries then have to balance their mismatches all by themselves, and this in turn requires more backup infrastructure with higher respective componentwise LCOE. For the GAS layout without the transmission infrastructure, which for clarity we denote as Figure 5: Componentwise LCOE for the optimal CFprop, CFmax, OPT, GAS and GAS* layouts for K = 1 (left), 2 (middle) and 3 (right). The K = 1 layout GASnoT without transmission is shown as reference.
K = 1 K = 2 K = 3 K W K S K B E B K T
GASnoT, the total LCOE turns out to be 64.5 e/MWh. Compared to the HOM layout, the combined LCOE components for wind and solar power generation are almost the same, but the increase of the LCOE components for the backup power generation and capacity is significantly larger than the disappearance of the transmission component. See again Figure 3, Table 5 and Figure 5. The total LCOE of the GASnoT layout is 8% and 11.5% larger than for the HOM and GAS layout respectively. This clearly demonstrates the benefit of transmission [4,6]. The GAS and GASnoT layouts are obtained from two independent optimisation efforts. This explains why the two layouts are actually quite different in the distribution of the wind and solar resources. Figure 4c illustrates the resulting wind-solar mixing parameters for the GASnoT layout. Contrary to the more extreme GAS layout, the majority of the countries comes with a mix below α n = 1 and well above 0. Only the most northern countries turn out to be wind-only. However, on average the mixing parameter α EU = 0.86 for the GASnoT layout is again close to α EU = 0.84 for the GAS layout.
K = 2 layouts
More heterogeneity is introduced once K is chosen to be larger than one. Figures 6a-c illustrate the optimal heuristic CFprop, CFmax and OPT layouts for K = 2. Their respective α EU values are 0.86 -0.87 (see Table 5), and have been fixed by minimising the LCOE (see Figure 3d). The general α EU -dependence of the other infrastructure measures are illustrated in Figure 3a-c. The backup energies required for the three layouts are quasi identical, and no difference is seen to the K = 1 HOM layout. Also the backup capacities are almost identical for the three layouts, and are slightly less than for the K = 1 HOM layout. Differences are observed for the transmission capacities. The CFprop layout comes with the smallest transmission capacities, followed by the OPT layout. The CFmax layout has the largest transmission capacities because its heterogeneity is the largest. All K = 2 layouts are found to have larger transmission capacities than the respective K = 1 layouts. The total LCOE of the three heuristic K = 2 layouts are inbetween 57.2 -57.9 e/MWh. See columns 6-8 in Table 5 and bars 5-7 in Figure 5. This is very close to the value 57.8 e/MWh found for the K = 1 GAS layout. In this respect, the larger heterogeneity of the K = 2 layouts do not represent a clear cost advantage when compared to the K = 1 GAS layout, which is homogeneous in the renewable penetration parameters γ n . The situation changes once the optimised K = 2 GAS layout is considered, which is exemplified in Figure 6d. It exploits the wind resources over Europe in a more efficient way and reduces the wind component in the LCOE; consult column 9 of Table 5 and bar 8 in Figure 5. This reduces the total LCOE to 55.3 e/MWh. The overall renewable penetration of the K = 2 GAS layout is γ EU = 1; consult again Equation (21). However, the individual renewable penetration parameters now scatter within 0.5 ≤ γ n ≤ 2. As can be seen in Figure 6d, their distribution is extremely heterogenous. For half of the countries they are either γ n = 2 or γ n = 0.5, and for the other countries just somewhere in-between. A more carefull inspection reveals an approximate heuristic law, which expresses the renewable penetration parameters
γ n = 1/K (CF eff n ≤ CF 1 ) (K − 1 K ) CF eff n −CF1 CF2−CF1 + 1 K (CF 1 ≤ CF eff n ≤ CF 2 ) K (CF eff n ≥ CF 2 )(26)
as a continuous and piece-wise linear function of an effective capacity factor
CF eff n = aCF W n + (1 − a)CF S n .(27)
A least-square fit is shown in Figure 7. The overall mixing parameter α EU = 0.83 of the K = 2 GAS layout is almost the same as for the K = 1 GAS layout. Both layouts also have in common that 20 out of the 30 countries come with α n = 1. The five largest of the α n < 1 countries with a non-zero solar component are also identical. GB IT ES SE PL NO NL BE FI CZ AT GR RO BG PT CH HU DK RS IE BA SK HR LT EE SI It is worth to take again a quick look at Figure 2. It shows that for K = 2 the optimal CFprop, CFmax and OPT layouts have more or less the same close-to-minimum standard deviation of the overall mismatch (24) as the optimised GAS layout. This indicates that a minimised mismatch standard deviation serves as a good measure to determine an optimal infrastructure [23]. However, it is still a rough measure, since it does not allow to finetune the minimum-cost infrastructure.
DE FR
K = 3 layouts
For K = 3 the GAS algorithm has more freedom to optimise the heterogeneous layout and to reduce the overall LCOE, see (20). The resulting layout is depicted in Figure 8. It has some similarity to the K = 2 GAS layout, but of course the K = 3 GAS layout is even more extreme. Its overall wind-solar mixing parameter α EU = 0.82 is almost the same as for the K = 2 counterpart. The overall cost reduction turns out to be small. As can be seen in Table 5, the total LCOE for the K = 2 and K = 3 GAS layouts are 55.3 and 54.5e/MWh, respectively. This small cost reduction is mainly caused by the opportunity to allocate more wind resources to the sites with a very high capacity factor, and it is weakened to some extend by slightly increased costs for the transmission component; compare column 14 with column 9 in Table 5.
Bulk results for the optimal heuristic K = 3 layouts CFprop, CFmax and OPT are also listed in Table 5 and Figure 5. Their layouts are also found to be wind-dominated, with nearly the same α EU values as for the respective GAS layout. The LCOE for these three heuristic layouts are larger than for the K = 3 GAS layout. This of course was to be expected. However, their LCOE also turn out to be slightly larger than for the less heterogeneous K = 2 GAS layout.
Another reason that the GAS optimisation might have been better than the heuristic layouts is that the GAS algorithm sees not just the capacity factors at each site, like the heuristic layouts, but also the geographical variation of the temporal generation pattern, which the GAS algorithm can exploit to shape the VRES generation pattern towards the load. However if this was the reason, the backup generation costs would have decreased from the heuristic to the GAS layout, which they do not. This sug-gests that the GAS optimisation's success really lies with the free exploitation of capacity factors.
Transmission capacities
So far, only the total contribution of the transmission capacities to the overall LCOE have been discussed for various system layouts in Table 5 and Figure 5. Its geographic distribution has not yet been specified. This will be done in this Subsection, but not right away. At first we will investigate a procedure which further reduces the overall LCOE by reducing the transmission capacities to some extend.
The transmission capacities defined in Equation (13) have been derived from unconstrained power flows. They are determined by the most extreme flow events, which typically occur between countries with a large energy deficit and others with a large excess. These events are not expected to overlap with other extreme events when all countries face a large energy deficit. The latter determine the required backup capacities. Consequently, it can be expected that a modest reduction of the total transmission capacities will not, or at least not much, affect the total backup capacities and the total backup energy, and will lower the overall LCOE. The synchronised balancing scheme (8) presented in Section 2.2 is based on unconstrained power flows. In order to include constrained power flows, a generalisation is needed:
min B n (B n (t)) 2 L n s.t. n P n (t) = 0 s.t. − K conT l ≤ F l (t) = n H ln P n (t) ≤ K conT l .(28)
The objective is to minimise the expression in the first line, taking into account the two constraints of the second and third line. K conT l denotes the constrained transmission capacity of line l. In the limit of unconstrained flows, where the second constraint can be discarded, the objective (28) can be rewritten as min B n (B 2 n (t)/ L n − λP n ) with the method of Lagrange multipliers and leads to the solution (8). For the following we will downscale the unconstrained transmission capacities from (13) by a uniform scaling parameter ζ to obtain the constrained transmission capacities Figure 9 illustrates the dependence of the LCOE on the transmission constraints by taking the unconstrained transmission capacities of the K = 2 GAS layout and scaling them down by the uniform factor ζ. At first, as ζ decreases, the LCOE also decreases. A minimum is found at ζ = 0.60. For the K = 1 and K = 3 GAS layouts the minimum is found at the optimal values ζ = 0.55 and 0.65, respectively. If the transmission capacities are downscaled further the LCOE starts to increase again due to increasing requirements for backup energy and backup capacity. Table 5 lists also the modified GAS layouts resulting from the optimal scaling parameters. For clarity, we denote them as GAS * layouts. Compared to the GAS layouts, the transmission contribution to the total LCOE is reduced and the backup contributions are slightly increased. The wind and solar components of the GAS and GAS * layouts are of course identical. Compared to the K = 1 GAS layout, the total LCOE of the K = 1 GAS * layout is reduced by 1.2e/MWh in absolute units and by 2.1% in relative units. For K = 2 and K = 3 the reductions are 2.7% and 2.8%, respectively. The reductions are also illustrated in Figure 5. The geographic distribution of the transmission capacities for the K = 2 GAS * layout is shown in Figure 10. The transmission capacities are not homogeneously distributed across the network. By far the strongest links are attached to Spain and Great Britain, which are the two largest countries with severe renewable excess generation. Links to their second neighbours with big deficits in renewable power generation, in particular Germany and Italy, also turn out to be quite strong. The more expensive HVDC transmission lines are utilised less extensively.
K conT l = ζK T l .(29)
Sensitivity analysis
Reduced solar cost
For the optimised GAS layouts as well as for the heuristic CFprop, CFmax and OPT layouts the optimal mixing parameter α EU minimising the overall costs is located in the wind dominated region. This is a consequence of the substantially higher costs of solar generation compared to wind. The future price development of solar photovoltaic systems is rather uncertain. To analyse the sensitivity to future price drops in solar cost, we calculate optimised layouts for solar cost reductions of 25%, 50% and 75%. Cost reductions could come from improved production processes, or alternatively from increasing capacity factors. Based on data from [28], the capacity factor can be increased by up to 40% by applying dual axis tracking compared to the fixed position installation assumed in Table 2, which may offset the higher capital costs of such systems. In addition, studies on increasing the energy conversion efficiency are still being conducted. A recent study suggests a huge decrease in the total system cost of PVs in a far future system [34]. The resulting K = 2 GAS portfolios are visualized in Figure 11. Not surprisingly we find that a decrease in solar cost leads to a continuously increase in totally installed solar capacity. This increase is not found to be equal at all nodes. The main solar electricity supplier, Spain, initially increases its solar capacity, but for the more extreme price reductions decreases it again. It seems more efficient to shift the production to other sites. Spain is the clear leader in terms of solar generation for large solar costs. However, in the case of 50% solar cost reductions Germany almost produces equal amounts as Spain. This might not appear to be intuitive from the figure as the renewable penetration of Germany is always smaller than for Spain, but the mean load of Germany is more than twice as large as the one of Spain. In the 75% scenario Germany passes Spain and becomes the main producer of solar power. In this most extreme scenario almost all countries deploy solar resources.
We illustrate the change in the associated LCOE due to the cost reductions in the cases of K = 1, 2 and 3 in Figure Figure 12: LCOE of the GAS optimised layouts when the solar cost is reduced by 25% (triangle), 50% (square) and 75% (diamond). The 0% scenario (circle) is included as a reference. Different constraints are shown: K = 1 (blue), 2 (yellow) and 3 (green).
is shifted from above α EU = 0.8 to below this point, and the LCOE values drop by around 2e. As the solar cost is reduced by 50% the optimal mix drops further and lies between 0.6 and 0.7. For K = 2 the LCOE is reduced by almost 5e compared to the reference scenario. When reducing the cost of solar by 75%, solar becomes much cheaper than wind, and the optimal mix is shifted below α EU = 0.5, indicating a dominant share of solar. Compared to the reference scenario, the LCOE dropped by around 9e for the case K = 2. We have to be aware that such large cost reductions for solar photovoltaic systems might not be plausible. A cost reduction is mostly to be expected from material and production costs but not from installation costs.
Increased backup cost
The future price developments of fossil fuels, which are likely to increase, will affect the cost of electricity. An increase in the cost of gas used by the CCGT generators leads to an increase in the variable operational expenses associated with backup generation. In principle this will also affect the structure of the optimised layouts, but we expect the structural change to be very small. As Figure 3a reveals, the mixing parameters α EU = 0.82-0.84 of the optimised GAS layouts also produce the minimum of the backup energy. Consequently, the structure of the layouts will more or less not change, but of course their LCOE will increase as the gas price increases. This increase is linear. For the K = 2 GAS layout an increase in backup fuel price to 150% leads to a LCOE of 59.0e/MWh, which is an increase of 6.8%. An increase to 200% of the gas price results in a LCOE of 62.8e/MWh, which equals an increase of 13.6%. The increased backup costs can to some degree be counterbalanced by the sale of curtailment energy. So far we have assumed that curtailed electricity is wasted renewable production. Selling the curtailment energy to other energy sectors like the heating and transportation sector is a promising possibility. The resulting decrease in LCOE depends on the selling price and the amount of electricity sold. Since we are discussing an all-European renewable penetration of γ EU = 1 throughout this paper, the total amount of curtailment energy is identical to the backup energy. Assuming to sell 1/3 of it at a price of 80e/MWh, the LCOE of the K = 2 GAS layout is reduced to 50.2e/MWh, which is a decrease of 9.2%. Note however, that the sale of curtailment energy might have a slightly bigger impact on the structural change of the optimised GAS layouts than increased backup costs. Since, again, the amount of curtailment energy is equal to the backup energy, Figure 3a also illustrates the dependence of the curtailment energy on the mixing parameter α EU . For parameter values below α EU = 0.82-0.84 the curtailment energy increases strongly. Consequently, when taking the sale of curtailment energy into account, a proper layout optimisation will shift to some degree towards smaller mixing parameters.
Interpolations towards more and less heterogeneity
As the heterogeneity parameter changes from K = 1 to 2 and 3, the LCOE of the optimised GAS layouts has decreased further; consult again Table 5 and Figure 5. It is quite natural to ask how much further the LCOE might decrease as K gets even larger. The answer is shown in Figure 13. The LCOE decreases continuously with increasing heterogeneity. However, the benefit of increased heterogeneity becomes smaller and smaller. The increasing cost of transmission leads to a point where it is almost no longer economic beneficial to increase the heterogeneity. The LCOE of 54.5e/MWh for the K = 3 GAS layout is already very close to the asymptotic value of 54e/MWh for very large K.
On the contrary, it might be more politically correct to reduce the heterogeneity. If the optimised GAS layouts were to represent the minimum of a rather shallow cost landscape, then other, more homogeneous layouts could be found in their vicinity without increasing the LCOE too much. Unfortunately, the search space for the exploration is high-dimensional, 60-dimensional to be more precise, as each of the 30 countries comes with its two variables γ n and α n . If for each variable we were to test two smaller and two bigger values around its GAS value, we would end up in testing 5 60 layout explorations. This is infeasible. Instead, we explore simple one-parameter interpolations between the heterogenous GAS layouts and the homogeneous HOM layout:
γ n = (1 − σ)γ HOM n + σγ GAS n , α n = (1 − σ)α HOM n + σα GAS n .(30)
The interpolation parameter is confined to 0 ≤ σ ≤ 1. A value of σ = 1 represents the GAS layout while σ = 0 reproduces the homogeneous layout. Figure 14 illustrates the LCOE of the interpolated layouts. The dependence on σ turns out to be almost linear. It is only weakly convex. This might indicate that the cost landscape around the GAS minimum is not flat, and that it might not be possible Figure 14: LCOE of the layouts interpolated between the HOM and the GAS layouts for K = 1 (blue), 2 (yellow) and 3 (green).
to find more homogeneous layouts without increasing the LCOE too much.
Discussion and outlook
In this paper the heterogeneity of renewable resources in different countries has been explored, but the distribution of wind and solar capacities within each country was fixed. Further heterogeneity of renewables, particularly wind, could be exploited by fine-tuning the distribution of renewables within each country, or by using a finer-scale model of Europe that exposes the locations with high capacity factors. In a recent paper [50] it was shown that the VRES costs in a heterogenous optimisation are up to 10% lower when using a 362 node model of Europe compared to a one-node-per-country model with 37 nodes, because the better exploitation of good sites offsets the increased exposure of grid bottlenecks within each country. Only three generation technologies were considered here: solar PV, onshore wind and natural gas. The inclusion of offshore wind might not improve system costs, given its high LCOE, but the LCOE may be offset by the system benefit of its steadier feed-in profile. In addition, offshore offers other benefits compared to onshore wind which are not accounted for by the cost optimisation, such as higher rates of public acceptance. Given that offshore wind is geographically concentrated along the coastlines of countries, a finer-resolution grid model would be advisable to fully assess the integration of offshore wind. Modelling hydroelectricity, which already supplies 17% of Europe's electricity, would reduce the costs of backup energy and provide extra flexibility to integrate the VRES. Similarly, the incorporation of storage or the use of flexibility from the electrification of transport and heating may alow VRES to be balanced more locally, favouring homogeneous solutions. Finally, while the cost reduction is a strong argument for a heterogeneous VRES layout, the realisation might be a political challenge. Since the optimal placing of resources was derived from a system perspective, a realisation would require full collaboration from all countries. Countries with low capacity factors would no longer be self sufficient, while countries with high shares of renewables, such as the countries bordering the North Sea with good wind sites, may encounter problems finding enough sites or with public acceptance. An unequal distribution of renewable energy generation also raises the question of who should pay for the generation and transmission assets. Current market conditions do not allow renewable generators to recover their capital costs from the energy-only market, forcing countries to subsidise the expansion of renewables. A highly heterogeneous system would therefore require a system for countries to compensate each other for their renewable imbalances. Recent work on the allocation of network flows to users in highly renewable networks [51,52] may provide the basis for an equitable distribution of such costs in a highly heterogeneous system.
Conclusions
In this paper the cost-optimal spatial distribution of VRES in a simplified European electricity system has been investigated for the case where the mean VRES generation equals the mean load (γ EU = 1). A heterogenous distribution of wind and solar capacities has been shown to result in an average electricity cost that is up to 11% lower than a homogeneous distribution of renewables proportional to each country's mean load. This is because the capital costs of wind and solar dominate the total system costs, and allowing the system to build more VRES in countries with better capacity factors means that fewer wind turbines and solar panels need to be built in order to produce the same amount of energy. If the heterogeneity parameter K, which controls the maximum and minimum levels of renewables generation in each country compared to its mean load, is gradually relaxed from K = 1 (homogeneous) to larger values (heterogenous) then there is a clear trend of cost reduction, which is steepest for smaller values of K and flattens out above K = 3. This has the important policy consequence that Europe can profit from the benefits of heterogeneity without allowing renewable imbalances between countries to become excessive. The optimal mixing parameter between wind and solar is remarkably robust as the heterogeneity is increased, favouring a high proportion of wind of between 80% and 90% in the VRES mix. The mixing parameter is, however, sensitive to the relative capital costs of wind and solar, dropping to between 60% and 70% as solar capital costs are decreased by 50% compared to the default cost assumption. While the best results in terms of low total system costs have been obtained here by explicit optimisation, heuristic methods for heterogeneously distributing wind and solar capacities, based for example on capacity factors, produce results that have costs only a few percent higher than the optimal systems. Given the increased comprehensibility and transparency that heuristic methods provide, this may be a price worth paying for policy makers.
Email addresses: [email protected] (Emil H. Eriksen), [email protected] (Leon J. Schwenk-Nebbe), [email protected] (Bo Tranberg), [email protected] (Tom Brown), [email protected] (Martin Greiner)
Figure 1 :
1Examples of heuristic (blue) wind-only and (yellow) solaronly layouts: (a) CFprop with β = 1, (b) CFmax constrained to K = 2, and (c) Pareto optimal OPT layouts obtained with K = 2.
Figure 2 :
2Scatter clouds for (blue) wind-only and (yellow) solaronly capacity layouts. The diagram plots the overall capacity factor(22) vs. the standard deviation of the overall mismatch(24). The distribution(25) and the constraint (20) with K = 2 have been used for the Monte Carlo simulations. The white point in the upper left cloud corners indicate Pareto optimal layouts; see alsoFigure 1c. The line connecting the wind-and solar-only Pareto optimal layouts results from the interpolation between these layouts in(18). The black point marked on this line represents the OPT layout with minimum LCOE. For comparison, the three triangle points mark the (orange) optimal CFprop, (green) optimal CFmax and (blue) optimised GAS layouts for K = 2.
Figure 3 :
3Overview of the infrastructure measures: (a) the backup energy E B (in units of average annual European load), (b) the backup capacity K B (in units of average hourly European load), (c) the transmission capacity K T (in units of average hourly European load times megametre) and (d) the associated LCOE as a function of α EU . The CFprop and CFmax layouts are shown as solid and dashed lines respectively.The dependence of the OPT layouts on α EU is not shown; only the interpolations leading to a LCOE minimum are plotted as asterisks. The GAS layouts are plotted as dots. The blue diamond represents the GASnoT layout. Different constraints are shown: K = 1 (blue), 2 (yellow) and 3 (green).
Figure 4 :
4Comparison of (a) the optimal homogeneous layout HOM with the optimised (b) GAS and (c) GASnoT layouts constrained by K = 1.
Figure 3 .
3Figure 3. All K = 1 layouts discussed so far include the transmission infrastructure. It is also interesting to compare them to an optimised layout without transmission. No exports and imports would then be possible and the injection pattern P n (t) would always be zero. No transmission investment would be needed and the respective componentwise LCOE would be zero. However, the countries then have to balance their mismatches all by themselves, and this in turn requires more backup infrastructure with higher respective componentwise LCOE. For the GAS layout without the transmission infrastructure, which for clarity we denote as
Figure 6 :
6Comparison of different layouts constrained by K = 2: (a) CFprop, (b) CFmax, (c) OPT and (d) GAS.
Figure 7 :
7Renewable penetration parameters γn from the K = 2 GAS layout as a function of the effective capacity factor CF ef f n defined in Equation(27). The continuous and piecewise linear green function represents the heuristic law (26) with least-square-fitted parameters a = 0.596, CF 1 = 0.173 and CF 2 = 0.211.
Figure 8 :
8GAS layout constrained by K = 3.
Figure 9 :
9Non-VRES components of the LCOE as a function of the scaling parameter ζ. The dashed line indicates the minimum leading to the lowest LCOE. The calculations were performed using the K = 2 GAS layout at ζ = 1. The VRES part, which does not depend on ζ, is not shown; it consists of 30.7e/MWh for wind and 6.7e/MWh for solar.
Figure 10 :
10Geographic distribution of the transmission capacities for the K = 2 GAS * layout. AC links are shown in black while HVDC links are shown in red. Link capacities are indicated relative to the highest capacity, which is 68 GW between France and Spain.
Figure 11 :
11GAS optimised layouts constrained by K = 2 for a solar cost reduction of 0%, 25%, 50% and 75%, from left to right.
Figure 13 :
13LCOE of the optimised GAS layouts as a function of the constraint parameter 1 ≤ K ≤ 5.
Table 1 :
1NomenclatureName
Description
N
Set of nodes
n, m
Node index
l
Link index
∆ n
Mismatch (VRES generation minus load)
α n
Wind/solar mix
γ n
Renewable penetration
G
{W,S,B}
n
Generation of wind, solar or backup
G R
n
Total renewable generation
L n
Load
P n
Net power balance
K
{W,S,B}
n
Wind, solar or backup capacity
K T
l
Transmission capacity for link l
E B
Backup energy
C n
Curtailment
B n
Nodal balancing
H
PTDF matrix
F l
Power flow on link l
CF {W,S} Wind/solar capacity factor
x
Average value of x
q
Quantile
K
Heterogeneity parameter
Table 2 :
2Capacity factors CF W n and CF S n for onshore wind and solar PV for the European countries, derived from the EuroStat data[31][32][33]. The countries are sorted by their respective mean load L (in units of GW) over the 2000-2007 time series. *: estimated values, see text for details.L
CF W
n
CF S
n
L
CF W
n
CF S
n
L
CF W
n
CF S
n
DE
54.2
0.18
0.12
FI
9.0
0.20
0.08
RS
3.9
0.20 *
0.14 *
FR
51.1
0.22
0.12
CZ
6.6
0.19
0.11
IE
3.2
0.27
0.09 *
GB
38.5
0.29
0.09
AT
5.8
0.20
0.11
BA
3.1
0.22 *
0.14 *
IT
34.5
0.20
0.14
GR
5.8
0.21
0.17
SK
3.1
0.21 *
0.12
ES
24.3
0.26
0.21
RO
5.4
0.21
0.14
HR
1.6
0.25
0.14 *
SE
16.6
0.25
0.09
BG
5.1
0.22
0.14
LT
1.5
0.25
0.12 *
PL
15.2
0.23
0.12 *
PT
4.8
0.28
0.18
EE
1.5
0.20
0.10 *
NO
13.7
0.29
0.09 *
CH
4.8
0.20 *
0.11 *
SI
1.4
0.20 *
0.13
NL
11.5
0.23
0.09
HU
4.4
0.22
0.13 *
LV
0.7
0.23
0.11 *
BE
9.5
0.27
0.11
DK
3.9
0.31
0.10
LU
0.7
0.16
0.10
Table 3 :
3Cost assumptions for different assets separated into capital
expenditures (CapEx) and fixed/variable operational expenditures
(OpEx) together with their expected life times.
Asset
CapEx OpEx fixed OpEx var Life time
[e/W] [e/kW/y] [e/MWh]
[years]
CCGT
0.90
4.5
56.0
30
Solar PV
0.75
8.5
0.0
25
Onshore wind
1.00
15.0
0.0
25
Table 4 :
4Summary of the algorithms for distributing VRES.Name
Brief description
HOM
Homogeneous distribution proportional to
the mean of each country's load
CFprop
Its total LCOE amounts to 59.7 e/MWh. The componentwise LCOE corresponding to the Algorithm 1 Pseudo code for the greedy axial search (GAS) routine. The Evaluate function calculates the associated cost of each new solution, and all new solutions are thereupon sorted by the Sort function in ascending order.function GreedyAxialSearch
best ← solution selected randomly from within the solution space
deltaCost ← ∞
stepSize ← maxStepSize
while stepSize > minStepSize do
while deltaCost > tolerance do
for index i = 1 to 2N do
trailSolutions[i] ← StepUp(best,i,stepSize)
trailSolutions[i+2N] ← StepDown(best,i,stepSize)
Evaluate(trailSolutions)
Sort(trailSolutions)
deltaCost ← cost of best minus cost of trailSolutions[1]
if deltaCost > 0 then
best ← trailSolutions[1]
stepSize ← stepSize/2
return best
Acknowledgments Tom Brown is funded by the CoNDyNet project, which is supported by the German Federal Ministry of Education and Research under grant no. 03SF0472C. The responsibility for the contents lies solely with the authors.Bibliography
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|
[] |
[
"A characterization of well-founded algebraic lattices",
"A characterization of well-founded algebraic lattices"
] |
[
"Ilham Chakir Mathématiques \nFaculté des Sciences et Techniques\nUFR de Mathématiques\nUniversité Claude-Bernard\n43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France\n",
"Université Hassan \nFaculté des Sciences et Techniques\nUFR de Mathématiques\nUniversité Claude-Bernard\n43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France\n",
"Maurice Pouzet [email protected] \nFaculté des Sciences et Techniques\nUFR de Mathématiques\nUniversité Claude-Bernard\n43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France\n"
] |
[
"Faculté des Sciences et Techniques\nUFR de Mathématiques\nUniversité Claude-Bernard\n43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France",
"Faculté des Sciences et Techniques\nUFR de Mathématiques\nUniversité Claude-Bernard\n43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France",
"Faculté des Sciences et Techniques\nUFR de Mathématiques\nUniversité Claude-Bernard\n43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France"
] |
[] |
We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice L is well-founded if and only if K(L), the join-semilattice of compact elements of L, is well-founded and contains neither [ω] <ω , nor Ω(ω * ) as a join-subsemilattice. As an immediate corollary, we get that an algebraic modular lattice L is well-founded if and only if K(L) is wellfounded and contains no infinite independent set. If K(L) is a joinsubsemilattice of I <ω (Q), the set of finitely generated initial segments of a well-founded poset Q, then L is well-founded if and only if K(L) is well-quasi-ordered. * Supported by INTAS arXiv:0812.2300v1 [math.CO] 12 Dec 2008Posets which are well-founded and have no infinite antichain are said well-partially-ordered or well-quasi-ordered, wqo for short. They play an important role in several areas (see[8]). If P is a wqo join-semilattice then J(P ), the poset of ideals of P , is well-founded and one may assign to every J ∈ J(P ) an ordinal, its height , denoted by h(J, J(P )). This ordinal is defined by induction, setting h(J, J(P )) := Sup({h(J , J(P )) + 1 : J ∈ J(P ), J ⊂ J}) and h(J , J(P )) := 0 if J is minimal in J(P ). The ordinal h(J(P )) := h(P, J(P )) + 1 is the height of J(P ). If P := I <ω (Q), with Q wqo, then J(P ) contains a chain of order type h(J(P )). This is an equivalent form of the famous result of de Jongh and Parikh [6] asserting that among the linear extensions of a wqo, one has a maximum order type.Problem 1.7 Let P be a wqo join-semilattice; does J(P ) contain a chain of order type h(J(P ))?An immediate corollary of Theorem 1.6 is:Corollary 1.8 A join-semilattice P of [ω] <ω contains either [ω] <ω as a join-semilattice or is wqo.
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A characterization of well-founded algebraic lattices
December 15, 2008
Ilham Chakir Mathématiques
Faculté des Sciences et Techniques
UFR de Mathématiques
Université Claude-Bernard
43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France
Université Hassan
Faculté des Sciences et Techniques
UFR de Mathématiques
Université Claude-Bernard
43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France
Maurice Pouzet [email protected]
Faculté des Sciences et Techniques
UFR de Mathématiques
Université Claude-Bernard
43, Bd. du 11 Novembre 191869622Settat, VilleurbanneMaroc, France
A characterization of well-founded algebraic lattices
December 15, 2008
We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice L is well-founded if and only if K(L), the join-semilattice of compact elements of L, is well-founded and contains neither [ω] <ω , nor Ω(ω * ) as a join-subsemilattice. As an immediate corollary, we get that an algebraic modular lattice L is well-founded if and only if K(L) is wellfounded and contains no infinite independent set. If K(L) is a joinsubsemilattice of I <ω (Q), the set of finitely generated initial segments of a well-founded poset Q, then L is well-founded if and only if K(L) is well-quasi-ordered. * Supported by INTAS arXiv:0812.2300v1 [math.CO] 12 Dec 2008Posets which are well-founded and have no infinite antichain are said well-partially-ordered or well-quasi-ordered, wqo for short. They play an important role in several areas (see[8]). If P is a wqo join-semilattice then J(P ), the poset of ideals of P , is well-founded and one may assign to every J ∈ J(P ) an ordinal, its height , denoted by h(J, J(P )). This ordinal is defined by induction, setting h(J, J(P )) := Sup({h(J , J(P )) + 1 : J ∈ J(P ), J ⊂ J}) and h(J , J(P )) := 0 if J is minimal in J(P ). The ordinal h(J(P )) := h(P, J(P )) + 1 is the height of J(P ). If P := I <ω (Q), with Q wqo, then J(P ) contains a chain of order type h(J(P )). This is an equivalent form of the famous result of de Jongh and Parikh [6] asserting that among the linear extensions of a wqo, one has a maximum order type.Problem 1.7 Let P be a wqo join-semilattice; does J(P ) contain a chain of order type h(J(P ))?An immediate corollary of Theorem 1.6 is:Corollary 1.8 A join-semilattice P of [ω] <ω contains either [ω] <ω as a join-semilattice or is wqo.
Introduction and synopsis of results
Algebraic lattices and join-semilattices (with a 0) are two aspects of the same thing, as expressed in the following basic result.
Theorem 1.1 [12], [10] The collection J(P ) of ideals of a join-semilattice P , once ordered by inclusion, is an algebraic lattice and the subposet K(J(P )) of its compact elements is isomorphic to P . Conversely, the subposet K(L) of compact elements of an algebraic lattice L is a join-semilattice with a 0 and J(K(L)) is isomorphic to L. In this paper, we characterize well-founded algebraic lattices by means of forbidden join-subsemilattices of the join-semilattice made of their compact elements. In the sequel ω denotes the chain of non-negative integers, and when this causes no confusion, the first infinite cardinal as well as the first infinite ordinal . We denote ω * the chain of negative integers. We recall that a poset P is well-founded provided that every non-empty subset of P has a minimal element. With the Axiom of dependent choices, this amounts to the fact that P contains no subset isomorphic to ω * . Let Ω(ω * ) be the set [ω] 2 of two-element subsets of ω, identified to pairs (i, j), i < j < ω, ordered so that (i, j) ≤ (i , j ) if and only if i ≤ i and j ≤ j w.r.t. the natural order on ω. Let Ω(ω * ) := Ω(ω * ) ∪ {∅} be obtained by adding a least element. Note that Ω(ω * ) is isomorphic to the set of bounded intervals of ω (or ω * ) ordered by inclusion. Moreover Ω(ω * ) is a join-semilattice ((i, j) ∨ (i , j ) = (i ∧ i , j ∨ j )). The join-semilattice Ω(ω * ) embeds in Ω(ω * ) as a join-semilattice; the advantage of Ω(ω * ) w.r.t. our discussion is to have a zero. Let κ be a cardinal number, e.g. κ := ω; denote [κ] <ω (resp. P(κ) ) the set, ordered by inclusion, consisting of finite (resp. arbitrary ) subsets of κ. The posets Ω(ω * ) and [κ] <ω are well-founded lattices, whereas the algebraic lattices J(Ω(ω * )) and J([κ] <ω ) (κ infinite) are not well-founded (and we may note that J([κ] <ω ) is isomorphic to P(κ)). As a poset Ω(ω * ) is isomorphic to a subset of [ω] <ω , but not as a join-subsemilattice. This is our first result. Proposition 1.2 Ω(ω * ) does not embed in [ω] <ω as a join-subsemilattice; more generally, if Q is a well-founded poset then Ω(ω * ) does not embed as a join-subsemilattice into I <ω (Q), the join-semilattice made of finitely generated initial segments of Q.
Our next result expresses that Ω(ω * ) and [ω] <ω are unavoidable examples of well-founded join-semilattices whose set of ideals is not well-founded. Theorem 1.3 An algebraic lattice L is well-founded if and only if K(L) is well-founded and contains no join-subsemilattice isomorphic to Ω(ω * ) or to
[ω] <ω .
The fact that a join-semilattice P contains a join-subsemilattice isomorphic to [ω] <ω amounts to the existence of an infinite independent set. Let us recall that a subset X of a join-semilattice P is independent if x ≤ F for every x ∈ X and every non-empty finite subset F of X \ {x}. Conditions which may insure the existence of an infinite independent set or consequences of the inexistence of such sets have been considered within the framework of the structure of closure systems (cf. the research on the "free-subset problem" of Hajnal [21] or on the cofinality of posets [9,16] [15] Let κ be a cardinal number; for a join-semilattice P the following properties are equivalent: (i) P contains an independent set of size κ; (ii) P contains a join-subsemilattice isomorphic to [κ] <ω ; (iii) P contains a subposet isomorphic to [κ] <ω ; (iv) J(P ) contains a subposet isomorphic to P(κ); (v) P(κ) embeds in J(P ) via a map preserving arbitrary joins.
Let L(α) := 1 + (1 ⊕ J(α)) + 1 be the lattice made of the direct sum of the one-element chain 1 and the chain J(α), (α finite or equal to ω * ), with top and bottom added.
Clearly J(Ω(ω * )) contains a sublattice isomorphic to L(ω * ). Since a modular lattice contains no sublattice isomorphic to L(2), we get as a corollary of Theorem 1.3: Theorem 1.5 An algebraic modular lattice L is well-founded if and only if K(L) is well-founded and contains no infinite independent set.
Another consequence is this: Theorem 1.6 For a join-semilattice P , the following properties are equivalent:
(i) P is well-founded with no infinite antichain ; (ii) P contains no infinite independent set and embeds as a join-semilattice into a join-semilattice of the form I <ω (Q) where Q is some wellfounded poset.
Let us compare join-subsemilattices of [ω] <ω . Set P ≤ P for two such join-subsemilattices if P embeds in P as a join-semilattice. This gives a quasi-order and, according to Corollary 1.8 in the particular case of a sierpinskisation of α this amounts to the fact that α is well-ordered . As shown in [20], sierpinskisations given by a bijective map ψ : ωα → ω which is order-preserving on each component ω·{i} of ωα are all embeddable in each other, and for this reason denoted by the same symbol Ω(α). Among the representatives of Ω(α), some are join-semilattices, and among them, join-subsemilattices of the direct product ω×α (this is notably the case of the poset Ω(ω * ) we previously defined). We extend the first part of Proposition 1.2, showing that except for α ≤ ω, the representatives of Ω(α) which are join-semilattices never embed in [ω] <ω as join-semilattices, whereas they embed as posets (see Corollary 4.11 and Example 4.12). From this result, it follows that the posets Ω(α) and I <ω (Ω(α)) do not embed in each other as join-semilattices.
These two posets provide examples of a join-semilattice P such that P contains no chain of type α while J(P ) contains a chain of type J(α). However, if α is not well ordered then I <ω (Ω(α)) and [ω] <ω embed in each other as join-semilattices. Problem 1.9 Let α be a countable ordinal. Is there a minimum member among the join-subsemilattices P of [ω] <ω such that J(P ) contains a chain of type α+1? Is it true that this minimum is I <ω (Ω(α)) if α is indecomposable?
Definitions and basic results
Our definitions and notations are standard and agree with [10] except on minor points that we will mention. We adopt the same terminology as in [4]. We recall only few things. Let P be a poset. A subset I of P is an initial segment of P if x ∈ P , y ∈ I and x ≤ y imply x ∈ I. If A is a subset of P , then ↓ A = {x ∈ P : x ≤ y for some y ∈ A} denotes the least initial segment containing A. If I =↓ A we say that I is generated by A or A is cofinal in I. If A = {a} then I is a principal initial segment and we write ↓ a instead of ↓ {a}. We denote down(P ) the set of principal initial segments of P . A final segment of P is any initial segment of P * , the dual of P . We denote by ↑ A the final segment generated by A. If A = {a} we write ↑ a instead of ↑ {a}. A subset I of P is directed if every finite subset of I has an upper bound in I (that is I is non-empty and every pair of elements of I has an upper bound). An ideal is a non-empty directed initial segment of P (in some other texts, the empty set is an ideal). We denote I(P ) (respectively, I <ω (P ), J(P )) the set of initial segments (respectively, finitely generated initial segments, ideals of P ) ordered by inclusion and we set J * (P ) := J(P ) ∪ {∅}, I 0 (P ) := I <ω (P ) \ {∅}. Others authors use down set for initial segment. Note that down(P ) has not to be confused with
I(P ). If P is a join-semilattice with a 0, an element x ∈ P is join-irreducible if it is distinct from 0, and if x = a ∨ b implies x = a or x = b (this is a slight variation from [10]). We denote J irr (P ) the set of join-irreducibles of P . An element a in a lattice L is compact if for every A ⊂ L, a ≤ A implies a ≤ A for some finite subset A of A.
The lattice L is compactly generated if every element is a supremum of compact elements. A lattice is algebraic if it is complete and compactly generated.
We note that I <ω (P ) is the set of compact elements of I(P ), hence J(I <ω (P )) ∼ = I(P ). Moreover I <ω (P ) is a lattice, and in fact a distributive lattice, if and only if P is ↓-closed , that is, the intersection of two principal initial segments of P is a finite union, possibly empty, of principal initial segments. We also note that J(P ) is the set of join-irreducible elements of I(P ); moreover, I <ω (J(P )) ∼ = I(P ) whenever P has no infinite antichain. Notably for the proof of Theorem 4.13, we will need the following results. ( Higman 1952 [11]); c) if P is a well-founded join-semilattice with a 0, then every member of P is a finite join of join-irreducible elements of P (Birkhoff, 1937, see [1]); d) A join-semilattice P with a zero is wqo if and only if every member of P is a finite join of join-irreducible elements of P and the set J irr (P ) of these join-irreducible elements is wqo (follows from b) and c)).
Theorem 2.1 Let P be a poset. a) I <ω (P ) is well-founded if and only if P is well-founded (Birkhoff 1937, see [1]); b) I <ω (P ) is wqo iff P is wqo iff I(P ) well-founded
A poset P is scattered if it does not contain a copy of η, the chain of rational numbers. A topological space T is scattered if every non-empty closed set contains some isolated point. The power set of a set, once equipped with the product topology, is a compact space. The set J(P ) of ideals of a joinsemilattice P with a 0 is a closed subspace of P(P ), hence is a compact space too. Consequently, an algebraic lattice L can be viewed as a poset and a topological space as well. It is easy to see that if L is topologically scattered then it is order scattered . It is a more significant fact, due to M.Mislove [17], that the converse holds if L is distributive.
3 Separating chains of ideals and proofs of Proposition 1.2 and Theorem 1.3 Let P be a join-semilattice. If x ∈ P and J ∈ J(P ), then ↓ x and J have a join ↓ x J in J(P ) and ↓ x J =↓ {x ∨ y : y ∈ J}. Instead of ↓ x J we also use the notation {x} J. Note that {x} J is the least member of J(P ) containing {x} ∪ J. We say that a non-empty chain I of ideals of P is separating if for every I ∈ I \ {∪I} and every x ∈ ∪I \ I, there is some
J ∈ I such that I ⊆ {x} J.
If I is separating then I has a least element implies it is a singleton set.
In P := [ω] <ω , the chain I := {I n : n < ω} where I n consists of the finite subsets of {m : n ≤ m} is separating. In P := ω * , the chain I := {↓ x : x ∈ P } is non-separating, as well as all of its infinite subchains. In P := Ω(ω * ) the chain I := {I n : n < ω} where I n := {(i, j) : n ≤ i < j < ω} has the same property. We may observe that a join-preserving embedding from a join-semilattice P into a join-semilattice Q transforms every separating (resp. non-separating) chain of ideals of P into a separating (resp. non-separating) chain of ideals of Q (If I is a separating chain of ideals of P , then J = {f (I) : I ∈ I} is a separating chain of ideals of Q). Hence the containment of [ω] <ω (resp. of ω * or of Ω(ω * )), as a join-subsemilattice, provides a chain of ideals which is separating (resp. non-separating, as are all its infinite subchains, as well). We show in the next two lemmas that the converse holds. Proof. Let X = {x n : n < ω} be an infinite independent set. Let I n be the ideal generated by X \ {x i : 0 ≤ i ≤ n}. The chain I = {I n : n < ω} is separating. Let I be an infinite separating chain of ideals. Define inductively an infinite sequence x 0 , I 0 , . . . , x n , I n , . . . such that I 0 ∈ I \ {∪I}, x 0 ∈ ∪I \ I 0 and such that: Indeed if x ∈ X then since x = x n for some n, n < ω, condition c n ) asserts that there is some ideal containing X \ {x} and excluding x. As E ∩ I n = ∅ we can select x n ∈ E ∩ I n and by definition of E, we can select some I n+1 ∈ I such that I n+1 ⊂↓ x n+1 . Thus ω * ≤ P . Case (ii). There is some I ∈ I such that I ∩ E = ∅. In particular all members of I included in I are unbounded in I. Since all infinite subchains of I are non-separating then, with no loss of generality, we may suppose that I = A (hence E = ∅). We set I −1 := A and define a sequence x 0 , I 0 , . . . , x n , I n , . . . such that I n ∈ I, x n ∈ I n−1 \ I n and I n ⊆ {x n } I for all I ∈ I, all n < ω. Members of this sequence being defined for all n , n < n, observe that the set I n := {I ∈ I : I ⊆ I n−1 } being infinite is non-separating, hence there are I ∈ I n and x ∈ I n−1 \ I such that I ⊆ {x} J for all J ∈ I n . Set I n := I and x n := x. Next, we define a sequence y 0 := x 0 , . . . , y n , . . . such that for every n ≥ 1: a n ) x n ≤ y n ∈ I n−1 ; b n ) y n ≤ y 0 ∨ y n−1 ; c n ) y j ≤ y i ∨ y n for every i ≤ j ≤ n.
a n ) I n ∈ I; b n ) I n ⊂ I n−1 ; c n ) x n ∈ I n−1 \ ({x 0 ∨ . . . ∨ x n−1 } I n ) for every n ≥ 1.
Suppose y 0 , . . . , y n−1 defined for some n, n ≥ 1. Since I n−1 is unbounded, we may select z ∈ I n−1 such that z ≤ y 0 ∨ . . . ∨ y n−1 . If n = 1, we set
y 1 := x 1 ∨ z. Suppose n ≥ 2. Let 0 ≤ j ≤ n − 2. Since y j+1 ∨ . . . ∨ y n−1 ∈ I j ⊆ {x j } I n−1 we may select t j ∈ I n−1 such that y j+1 ∨. . .∨y n−1 ≤ x j ∨t j . Set t := t 0 ∨ . . . ∨ t n−2 and y n := x n ∨ z ∨ t. Let f : Ω(ω * ) → P be defined by f (i, j) := y i ∨ y j for all (i, j), i < j < ω. Condition c n ) insures that f is join-preserving. Indeed, let (i, j), (i , j ) ∈ Ω(ω * ). We have (i, j) ∨ (i , j ) = (i ∧ i , j ∨ j ) hence f ((i, j) ∨ (i , j )) = f (i ∧ i , j ∨ j ) = y i∧i ∨ y j∨j .
If F is a finite subset of ω with minimum a and maximum b then conditions c n ) force {y n : n ∈ F } = y a ∨ y b . If F := {i, j, i , j } then, taking account of i < j and i < j , we have
f (i, j) ∨ f (i , j ) = y i ∨ y j ∨ y i ∨ y j = y i∧i ∨ y j∨j . Hence f ((i, j) ∨ (i , j )) = f (i, j) ∨ f (i , j ), proving our claim. Next, f is one-to-one. Let (i, j), (i , j ) ∈ Ω(ω * ) such that f (i, j) = f (i , j ), that is y i ∨ y j = y i ∨ y j (1). Suppose j < j . Since 0 ≤ i < j, Condition c j ) implies y i ≤ y 0 ∨ y j . In the other hand, since 0 ≤ j ≤ j − 1, Condition c j −1 ) implies y j ≤ y 0 ∨ y j −1 . Hence y i ∨ y j ≤ y 0 ∨ y j −1 . From (1) we get y j ≤ y 0 ∨ y j −1 , contradicting Condition b j ). Hence j ≤ j.
Exchanging the roles of j, j gives j ≤ j thus j = j . If i < i then, Conditions a i ) and a j ) assure y i ∈ I i −1 and y j ∈ I j −1 . Since
I j −1 ⊆ I i −1 we have y i ∨y j ∈ I i −1 .
In the other hand x i ∈ I i and x i ≤ y i ∨ y j thus y i ∨ y j ∈ I i . From I i −1 ⊆ I i , we have y i ∨ y j ∈ I i −1 , hence y i ∨ y j = y i ∨ y j and i ≤ i. Similarly we get i ≤ i . Consequently i = i .
Proof of Proposition 1.2
If Ω(ω * ) embeds in [ω] <ω then [ω] <ω contains a non-separating ω * -chain of ideals. This is impossible: a non-separating chain of ideals of [ω] <ω has necessarily a least element. Indeed, if the pair x, I (x ∈ [ω] <ω , I ∈ I) witnesses the fact that the chain I is non-separating then there are at most | x | +1 ideals belonging to I which are included in I (note that the set {∪I \ ∪J : J ⊆ I, J ∈ I} is a chain of subsets of x). The proof of the general case requires more care. If Ω(ω * ) embeds in I <ω (Q) as a join-semilattice then we may find a sequence x 0 , I 0 , . . . , x n , I n , . . . such that I n ⊂ I n−1 ∈ J(I <ω (Q)), x n ∈ I n−1 \ I n and I n ⊆ {x n } I m for every n < ω and every m < ω. Set I ω := {I n : n < ω}, I n := ∪I n for every n ≤ ω, Q := Q \ I ω and y n := x n \ I ω for every n < ω. We claim that y 0 , . . . , y n , . . . form a strictly descending sequence in I <ω (Q ). According to Property a) stated in Theorem 2.1, Q , thus Q, is not well-founded. First, y n ∈ I <ω (Q ). Indeed, if a n ∈ [Q] <ω generates x n ∈ I <ω (Q) then, since I ω ∈ I(Q), a n \ I ω generates x n \ I ω ∈ I(Q ). Next, y n+1 ⊂ y n . It suffices to prove that the following inclusions hold:
x n+1 ∪ I ω ⊆ I n ⊂ x n ∪ I ω Indeed, substracting I ω , from the sets figuring above, we get:
y n+1 = (x n+1 ∪ I ω ) \ I ω ⊂ (x n ∪ I ω ) \ I ω = y n
The first inclusion is obvious. For the second note that, since J(I <ω (Q)) is isomorphic to I(Q), complete distributivity holds, hence with the hypotheses on the sequence x 0 , I 0 , . . . , x n , I n , . . .
we have I n ⊆ {{x n } I m : m < ω} = {x n } {I m : m < ω} = {x n } I ω , thus I n ⊂ x n ∪ I ω .
Proof of Theorem 1.3
In terms of join-semilattices and ideals, result becomes this: let P be a joinsemilattice, then J(P ) is well-founded if and only if P is well-founded and contains no join-subsemilattice isomorphic to Ω(ω * ) or to [ω] <ω .
The proof goes as follows. Suppose that J(P ) is not well-founded. If some ω * -chain in J(P ) is separating then, according to Lemma 3.1, P contains an infinite independent set. From Theorem 1.4, it contains a joinsubsemilattice isomorphic to [ω] <ω . If no ω * -chain in J(P ) is separating, then all the infinite subchains of an arbitrary ω * -chain are non-separating. From Lemma 3.2, either ω * or Ω(ω * ) embed in P as a join-semilattice. The converse is obvious. 4 Join-subsemilattices of I <ω (Q) and proof of Theorem 1.6 In this section, we consider join-semilattices which embed in join-semilattices of the form I <ω (Q). These are easy to characterize internally (see Proposi-tion 4.4). This is also the case if the posets Q are antichains (see Proposition 4.10) but does not go so well if the posets Q are well-founded (see Lemma 4.8).
Let us recall that if P is a join-semilattice, an element x ∈ P is join-prime (or prime if there is no confusion), if it is distinct from the least element 0, if any, and if x ≤ a ∨ b implies x ≤ a or x ≤ b. This amounts to the fact that P \ ↑ x is an ideal . We denote J pri (P ), the set of join-prime members of P . We recall that J pri (P ) ⊆ J irr (P ); the equality holds provided that P is a distributive lattice. It also holds if P = I <ω (Q). Indeed: • P is isomorphic to I <ω (Q) for some poset Q;
• P is a join-semilattice with a least element in which every element is a finite join of primes.
Proof.
Observe that the primes in I <ω (Q), are the ↓ x, x ∈ Q. Let I ∈ I <ω (Q) and F ∈ [Q] <ω generating I, we have I = ∪{↓ x : x ∈ F } . Conversely, let P be a join-semilattice with a 0. If every element in P is a finite join of primes, then P ∼ = I <ω (Q) where Q := J pri (P ). Let L be a complete lattice . For x ∈ L , set x + := {y ∈ L : x < y}. We recall that x ∈ L is completely meet-irreducible if x = X implies x ∈ X, or -equivalently-x = x + . We denote (L) the set of completely meet-irreducible members of L. We recall the following Lemma. (i) P embeds in I <ω (Q), as a join-semilattice, for some poset Q;
(ii) P embeds in I <ω (J(P )) as a join-semilattice;
(iii) P embeds in I <ω ( (J(P ))) as a join-semilattice;
(iv) For every x ∈ P , P \ ↑ x is a finite union of ideals.
Proof. (i) ⇒ (iv) Let ϕ be an embedding from P in P := I <ω (Q). We may suppose that P has a least element 0 and that ϕ(0) = ∅ (if P has no least element, add one, say 0, and set ϕ(0) := ∅; if P has a least element, say a, and ϕ(a) = ∅, add to P an element 0 below a and set ϕ(0) := ∅). For J ∈ P(P ), let ϕ −1 (J ) := {x ∈ P : ϕ(x) ∈ J }. Since ϕ is orderpreserving, ϕ −1 (J ) ∈ I(P ) whenever J ∈ I(P ) ; moreover, since ϕ is join-preserving, ϕ −1 (J ) ∈ J(P ) whenever J ∈ J(P ). Now, let x ∈ P . We have ϕ −1 (P \ ϕ(x)) := P \ ↑ x. Since ϕ(x) is a finite join of primes, P \ ↑ ϕ(x) is a finite union of ideals. Since their inverse images are ideals, P \ ↑ x is a finite union of ideals too.
(iv) ⇒ (iii) We use the well-known method for representing a poset by a family of sets. (c) ϕ Q is an order-embedding if and only if for every x, y ∈ P such that
x ≤ y there is some J ∈ Q such that x ∈ J and y ∈ J.
Applying this to Q := (J(P )) we get immediately that ϕ Q is joinpreserving . Moreover, ϕ Q (x) ∈ I <ω (Q) if and only if P \ ↑ x is a finite union of ideals. Indeed, we have P \ ↑ x = ∪ϕ Q (x), proving that P \ ↑ x is a finite union of ideals provided that ϕ Q (x) ∈ I <ω (Q). Conversely, if P \ ↑ x is a finite union of ideals, say I 0 , . . . , I n , then since ideals are prime members of I(P ), every ideal included in I is included in some I i , proving that ϕ Q (x) ∈ I <ω (Q). To conclude, note that if P is a join-semilattice then ϕ Q is join-preserving.
(iii) ⇒ (ii) Trivial.
(ii) ⇒ (i) Trivial.
Corollary 4.6
If a join-semilattice P has no infinite antichain, it embeds in I <ω (J(P )) as a join-subsemilattice.
Proof.
As is well known, if a poset has no infinite antichain then every initial segment is a finite union of ideals (cf [7], see also [8] Corollary 4.7 Let P be a join-semilattice. If for every x ∈ P , P \ ↑ x is a finite union of ideals and (J(P )) is well-founded then P embeds as a join-subsemilattice in I <ω (Q), for some well-founded poset Q.
The converse does not hold:
Example 4.8
There is a bipartite poset Q such that I <ω (Q) contains a join-semilattice P for which (J(P )) is not well-founded.
where min(F ) and min(G) denote the least element of F and G w.r.t. the natural order on N. For each n ∈ N, let I n := {X ∈ P : (n, 0) ∈ X}. Claim 1. Q is bipartite and P is a join-subsemilattice of I <ω (Q). 2. The I n 's form a strictly descending sequence of members of (J(P )).
Proof of the Claim 1. The poset Q is decomposed into two antichains, namely N × {0} and N × {1} and for this raison is called bipartite. Next, P is a subset of I <ω (Q). Indeed, Let X ∈ P . Let F, G such that X = F × {0} ∪ G × {1}. Set G := G × {1}. If min(G) = min(F ) − 1, then X =↓ G whereas if min(G) = min(F ) then X =↓ G ∪ {(min(F ), 0)}. In both cases X ∈ I <ω (Q). Finally, P is a join-semilattice. Indeed, let X, X ∈ P with X :
= F × {0} ∪ G × {1} and X := F × {0} ∪ G × {1}. Obviously X ∪ X = (F ∪ F ) × {0} ∪ (G ∪ G ) × {1}. Since X, X ∈ P , F ∪ F is a non-empty final segment of N and G ∪ G is a non-empty finite subset of N. We have min(G ∪ G ) = min({min(G), min(G )}) ≤ min({min(F ), min(F )}) = min(F ∪ F ) and similarly min(F ∪ F ) − 1 = min{min(F ), min(F )} − 1 = min{min(F ) − 1, min(F ) − 1} ≤ min{min(G), min(G )} = min(G ∪ G ),
proving that inequalities as in (2) hold. Thus X ∪ X ∈ I <ω (Q). 2. Due to its definition, I n is an non-empty initial segment of P which is closed under finite unions, hence I n ∈ J(P ). Let X n := {(n, 1), (m, 0) : m ≥ n + 1} and Y n := X n ∪ {(n, 0)}. Clearly, X n ∈ I n and Y n ∈ P . We claim that I + n = I n {Y n }. Indeed, let J be an ideal containing strictly I n .
Let Y := {m ∈ N : m ≥ p} × {0} ∪ G × {1} ∈ J \ I n . Since Y ∈ I n , we have p ≤ n hence Y n ⊆ Y ∪ X n ∈ J.
It follows that Y n ∈ J, thus I + n ⊆ J, proving our claim. Since I + n = I n , I n ∈ (J(P )). Since, trivially, I + n ⊆ I n−1 we have I n ⊂ I n−1 , proving that the I n 's form a strictly descending sequence.
Let E be a set and F be a subset of P(E), the power set of E. For x ∈ E, set F ¬x := {F ∈ F : x ∈ F } and for X ⊂ F , set X := X. Let F <ω (resp. F ∪ ) be the collection of finite (resp. arbitrary) unions of members of F. Ordered by inclusion, F ∪ is a complete lattice , the least element and the largest element being the empty set and F, respectively. Lemma 4.9 Let Q be a poset, F be a subset of I <ω (Q) and P := F <ω ordered by inclusion.
(a) The map X → X is an isomorphism from J(P ) onto F ∪ ordered by inclusion.
(b) If I ∈ (J(P )) then there is some x ∈ Q such that I = P ¬x .
(c) If ↓ q is finite for every q ∈ Q then I + \ I is finite for every I ∈ J(P ) and the set ϕ (X) := {I ∈ (J(P )) : X ∈ I} is finite for every X ∈ P . If X ∈ I, then X ⊆ I, thus X ⊆ J. Since X ∈ I <ω (Q), and X ⊆ J, there are X 1 , . . . , X n ∈ J such that X ⊆ Y = X 1 ∪ . . . ∪ X n . Since J is an ideal Y ∈ J. It follows that X ∈ J.
(b) Let I ∈ (J(P )). From (a), we have I ⊂ I + . Let x ∈ I + \ I. Clearly P ¬x is an ideal containing I. Since x ∈ P ¬x , P ¬x is distinct from I + . Hence P ¬x = I. Note that the converse of assertion (b) does not hold in general.
(c) Let I ∈ (J(P )) and X ∈ I + \ I. We have {X} I = I + , hence from(a) {X} I = I + . Since {X} I = X ∪ I we have I + \ I ⊆ X. From our hypothesis on P , X is finite, hence I + \ I is finite. Let X ∈ P . If I ∈ ϕ (X) then according to (b) there is some x ∈ Q such that I = P ¬x . Necessarily x ∈ X. Since X is finite, the number of these I's is finite.
Proposition 4.10 Let P be a join-semilattice. The following properties are equivalent:
(i) P embeds in [E] <ω as a join-subsemilattice for some set E;
Proof of Claim 3 With Ramsey's theorem obtain a sequence (I n ) n<ω of non-principal initial segments which is either strictly increasing or strictly decreasing . Separate two successive members by some element x n and apply the first part of Claim 2.
If we pick x ∈ N \ I then it follows from Claim 3 and the second part of Claim 2 that ϕ ∆ (x) is infinite.
Example 4.12
If α is a countably infinite order type distinct from ω, Ω(α) is not embeddable in [ω] <ω as a join-semilattice.
Indeed, Ω(α) is a sierpinskisation of ωα and ω. And if α is distinct from ω, α contains some element which majorizes infinitely many others. Thus β := ωα satisfies the hypothesis of Corollary 4. 11.
Note that on an other hand, for every ordinal α ≤ ω, there are representatives of Ω(α) which are embeddable in [ω] <ω as join-semilattices. 2) F <ω is wqo;
3) J(F <ω ) is topologically scattered; 4) F ∪ is order-scattered; 5) P(ω) does not embed in F ∪ ; 6) [ω] <ω does not embed in F <ω ; 7) F ∪ is well-founded.
Proof. We prove the following chain of implications: 1) =⇒ 2) =⇒ 3) =⇒ 4) =⇒ 5) =⇒ 6) =⇒ 7) =⇒ 1) 1) =⇒ 2). Since Q is well-founded then, as mentioned in a) of Theorem 2.1, I <ω (Q) is well-founded. It follows first that F <ω is well-founded, hence from Property c) of Theorem 2.1, every member of F <ω is a finite join of join-irreducibles. Next, as a subset of F <ω , F is well-founded, hence wqo according to our hypothesis. The set of join-irreducible members of F <ω is wqo as a subset of F. From Property d) of Theorem 2.1, F <ω is wqo 4.1 Proof of Theorem 1.6 (i) ⇒ (ii) Suppose that (i) holds. Set Q := J(P ). Since P contains no infinite antichain, P embeds as a join-subsemilattice in I <ω (Q) (Corollary 4.6). From b) of Theorem 2.1 Q is well-founded. Since P has no infinite antichain, it has no infinite independent set.
(ii) ⇒ (i) Suppose that (ii) holds. Since Q is well-founded, then from a) of Theorem 2.1, I <ω (Q) is well-founded. Since P embeds in I <ω (Q), P is well-founded. From our hypothesis, P contains no infinite independent set. According to implication (iii) ⇒ (i) of Theorem 1.4 , it does not embed [ω] <ω . From implication 6) ⇒ 1) of Theorem 4.13, it has no infinite antichain.
Figure 1 :
1Ω(ω * )
Figure 2 :
2L(ω * )
, the poset corresponding to this quasi-order has a largest element (namely [ω] <ω ), and all other members come from wqo join-semilattices. Basic examples of join-subsemilattices of [ω] <ω are the I <ω (Q)'s where Q is a countable poset such that no element is above infinitely many elements. These posets Q are exactly those which are embeddable in the poset [ω] <ω ordered by inclusion. An interesting subclass is made of posets of the form Q = (N, ≤) where the order ≤ is the intersection of the natural order N on N and of a linear order L on N, (that is x ≤ y if x ≤ y w.r.t. N and x ≤ y w.r.t. L). If α is the type of the linear order, a poset of this form is a sierpinskisation of α. The corresponding joinsemilattices are wqo provided that the posets Q have no infinite antichain;
Lemma 3. 1
1A join-semilattice P contains an infinite independent set if and only if it contains an infinite separating chain of ideals.
The construction is immediate. Indeed, since I is infinite then I \ {∪I} = ∅. Choose arbitrary I 0 ∈ I \ {∪I} and x 0 ∈ ∪I \ I 0 . Let n ≥ 1. Suppose x k , I k defined and satisfying a k ), b k ), c k ) for all k ≤ n − 1. Set I := I n−1 and x := x 0 ∨ . . . ∨ x n−1 . Since I ∈ I and x ∈ ∪I \ I, there is some J ∈ I such that I ⊆ {x} J. Let z ∈ I \ ({x} J). Set x n := z, I n := J. The set X := {x n : n < ω} is independent.
Lemma 3. 2
2A join-semilattice P contains either ω * or Ω(ω * ) as a joinsubsemilattice if and only if it contains an ω * -chain I of ideals such that all infinite subchains are non-separating.Proof. Let I be an ω * -chain of ideals and let A be its largest element (that is A = ∪I). Let E denote the set {x : x ∈ A and I ⊂↓ x for some I ∈ I}. Case (i). For every I ∈ I, I ∩ E = ∅. We can build an infinite strictly decreasing sequence x 0 , .. . , x n , . . . of elements of P . Indeed, let us choose x 0 ∈ E ∩ (∪I) and I 0 such that I 0 ⊂↓ x 0 . Suppose x 0 , .. . , x n and I 0 , . . . , I n defined such that I i ⊂↓ x i for all i = 0, . . . , n.
Remark 3. 3
3One can deduce the fact that Ω(ω * ) does not embed as a joinsemilattice in [ω] <ω from the fact that it contains a strictly descending chain of completely meet-irreducible ideals (namely the chain I := {I n : n < ω} where I n := {(i, j) : n ≤ i < j < ω}) (see Proposition4.10) but this fact by itself does not prevent the existence of some well-founded poset Q such that Ω(ω * ) embeds as a join semilattice in I <ω (Q).
Fact 4. 1
1For an arbitrary poset Q, we have:J irr (I <ω (Q)) = J pri (I <ω (Q)) = down(Q)(1)
Fact 4. 2
2For a poset P , the following properties are equivalent:
Lemma 4. 3
3Let P be a join-semilattice, I ∈ J(P ) and x ∈ P . Then x ∈ I + \ I if and only if I is a maximal ideal of P \ ↑ x.
Proposition 4. 4
4Let P be a join-semilattice. The following properties are equivalent:
Fact 4. 5
5Let P be a poset and Q ⊆ I(P ). For x ∈ P set ϕ Q (x) := {J ∈ Q : x ∈ J}. Then: (a) ϕ Q (x) ∈ I(Q); (b) ϕ Q : P → I(Q) is an order-preserving map;
Proof.
Let 2 := {0, 1} and Q := N × 2. Order Q in such a way that (m, i) < (n, j) if m > n in N and i < j in 2. Let P be the set of subsets X of Q of the form X := F × {0} ∪ G × {1} such that F is a non-empty final segment of N, G is a non-empty finite subset of N and min(F ) − 1 ≤ min(G) ≤ min(F )
Proof. (a) Let I and J be two ideals of P . Then J contains I if and only if J contains I. Indeed, if I ⊆ J then, clearly I ⊆ J. Conversely, suppose I ⊆ J.
Theorem4.13 Let Q be a well-founded poset and let F ⊆ I <ω (Q). The following properties are equivalent:1) F has no infinite antichain;
).A basic result is the following.Theorem 1.4 [4]
Apply part (c) of Lemma 4.9 . (ii) ⇒ (i) Set E := (J(P )). Proof. (i) ⇒ (ii) Let ϕ be an embedding from P in [E] <ω which preserves joins. Set F := ϕ(P ). We have ϕ (x) ∈ [E] <ω . According to Fact 4.5 and Lemma 4.3, the map ϕ : P → [E] <ω is an embedding preserving joinsProof. (i) ⇒ (ii) Let ϕ be an embedding from P in [E] <ω which preserves joins. Set F := ϕ(P ). Apply part (c) of Lemma 4.9 . (ii) ⇒ (i) Set E := (J(P )). We have ϕ (x) ∈ [E] <ω . According to Fact 4.5 and Lemma 4.3, the map ϕ : P → [E] <ω is an embedding preserving joins.
It is obtained as the intersection of two linear orders L, L on the same set and having respectively order type β and ω. We may suppose that the ground set is N and L the natural order. Claim 1 A non-empty subset I is a non-principal ideal of P if and only if this is a non-principal initial segment of L. Proof of Claim 1 Suppose that I is a non-principal initial segment of LLet P be a sierpinskisation of β and ω. It is obtained as the intersection of two linear orders L, L on the same set and having respectively order type β and ω. We may suppose that the ground set is N and L the natural order. Claim 1 A non-empty subset I is a non-principal ideal of P if and only if this is a non-principal initial segment of L. Proof of Claim 1 Suppose that I is a non-principal initial segment of L.
Let x, y ∈ I; since I is non-principal in L, the set A := I∩ ↑ L x∩ ↑ L y of upper-bounds of x and y w.r.t. L which belong to I is infinite; since B :=↓ L x∪ ↓ L y is finite, A \ B is non-empty. An arbitrary element z ∈ A \ B is an upper bound of x, y in I w.r.t. the poset P proving that I is up-directed. Since I is infinite, I cannot have a largest element in P , hence I is a non-principal ideal of P . Conversely, suppose that I is a nonprincipal ideal of P . Let us check that I is an initial segment of L. Then, clearly, I is an initial segment of P . Let us check that I is up-directed. Let x ≤ L y with y ∈ I. Since I non-principal in P , A :=↑ P y ∩ I is infiniteThen, clearly, I is an initial segment of P . Let us check that I is up-directed. Let x, y ∈ I; since I is non-principal in L, the set A := I∩ ↑ L x∩ ↑ L y of upper-bounds of x and y w.r.t. L which belong to I is infinite; since B :=↓ L x∪ ↓ L y is finite, A \ B is non-empty. An arbitrary element z ∈ A \ B is an upper bound of x, y in I w.r.t. the poset P proving that I is up-directed. Since I is infinite, I cannot have a largest element in P , hence I is a non-principal ideal of P . Conversely, suppose that I is a non- principal ideal of P . Let us check that I is an initial segment of L. Let x ≤ L y with y ∈ I. Since I non-principal in P , A :=↑ P y ∩ I is infinite;
A \ B is non-empty. An arbitrary element of A \ B is an upper bound of x and y in I w.r.t. P . It follows that x ∈ I. =↓ L x∪ ↓ L y is finite. since B :=↓ L x∪ ↓ L y is finite, A \ B is non-empty. An arbitrary element of A \ B is an upper bound of x and y in I w.r.t. P . It follows that x ∈ I.
Claim 2 Let x ∈ N. If there is a non-principal ideal of L which does not contain x, there is a maximal one, say I x . If P is a join-semilattice, I x ∈ ∆(P ). Proof of Claim 2 The first part follows from Zorn's Lemma. The second part follows from Claim 1. and Lemma 4.3. Claim 3 If an initial segment I of β contains infinitely many nonIf I has a largest element w.r.t. L then such an element must be maximal in I w.r.t. P , and since I is an ideal, I is a principal ideal, a contradiction. Claim 2 Let x ∈ N. If there is a non-principal ideal of L which does not contain x, there is a maximal one, say I x . If P is a join-semilattice, I x ∈ ∆(P ). Proof of Claim 2 The first part follows from Zorn's Lemma. The second part follows from Claim 1 and Lemma 4.3. Claim 3 If an initial segment I of β contains infinitely many non-
If F <ω is wqo then I(F <ω ) is well-founded (cf. Property (b) of Theorem 2.1). If follows that I(F <ω ) is topologically scattered (cf. =⇒ 3). =⇒ 3). If F <ω is wqo then I(F <ω ) is well-founded (cf. Property (b) of Theorem 2.1). If follows that I(F <ω ) is topologically scattered (cf.[17]);
hence all its subsets are topologically scattered. in particular J(F <ωhence all its subsets are topologically scattered, in particular J(F <ω ).
=⇒ 4). Suppose that F ∪ is not ordered scatered. Let f : η → F ∪ be an embedding. For r ∈ η setf (r) = {f (r ) : r < r}. Let X := {f (r) : r < η}. Clearly X ⊆ F ∪ . Furthermore X contains no isolated point (Indeed, sincef (r) = {f (r ) : r < r},f (r) belongs to the topological closure of {f (r ) : r < r}). Hence F ∪ is not topologically scatered. =⇒ 4). Suppose that F ∪ is not ordered scatered. Let f : η → F ∪ be an embedding. For r ∈ η setf (r) = {f (r ) : r < r}. Let X := {f (r) : r < η}. Clearly X ⊆ F ∪ . Furthermore X contains no isolated point (Indeed, sincef (r) = {f (r ) : r < r},f (r) belongs to the topological closure of {f (r ) : r < r}). Hence F ∪ is not topologically scatered.
Suppose that P(ω) embeds in F ∪ . Since η ≤ P(ω), we have η ≤ F ∪. =⇒ 5). =⇒ 5). Suppose that P(ω) embeds in F ∪ . Since η ≤ P(ω), we have η ≤ F ∪ .
Suppose that [ω] <ω embeds in F <ω , then J([ω] <ω ) embeds in J(F <ω ). Lemma 4.9 assures that J(F <ω ) is isomorphic to F ∪ . In the other hand J([ω] <ω ) is isomorphic to P(ω). Hence P(ω) embeds in F ∪. =⇒ 6). =⇒ 6). Suppose that [ω] <ω embeds in F <ω , then J([ω] <ω ) embeds in J(F <ω ). Lemma 4.9 assures that J(F <ω ) is isomorphic to F ∪ . In the other hand J([ω] <ω ) is isomorphic to P(ω). Hence P(ω) embeds in F ∪ .
Since Q is well-founded, a) of Theorem 2.1 assures I <ω (Q) well-founded, but F <ω ⊆ I <ω (Q), hence F <ω is well-founded. Furthermore, since I <ω (Q) is closed under finite unions, we have F <ω ⊆ I <ω (Q), Proposition 1.2 implies that Ω(ω * ) does not embed in F <ω. =⇒ 7). Suppose F ∪ not well-founded. From Theorem 1.3, we have F <ω not well-founded=⇒ 7). Suppose F ∪ not well-founded. Since Q is well-founded, a) of Theorem 2.1 assures I <ω (Q) well-founded, but F <ω ⊆ I <ω (Q), hence F <ω is well-founded. Furthermore, since I <ω (Q) is closed under finite unions, we have F <ω ⊆ I <ω (Q), Proposition 1.2 implies that Ω(ω * ) does not embed in F <ω . From Theorem 1.3, we have F <ω not well-founded.
is an infinite antichain of members of F, define f (i, j) : [ω] 2 → Q, choosing f (i, j) arbitrary in M ax(F i ) \ F j . Divide [ω] 3 into R 1 := {(i, j, k) ∈ [ω] 3 : f (i, j) = f (i, k)} and R 2 := [ω] 3 \ R 1 . From Ramsey's theorem, cf. [18], there is some infinite subset X of ω such that [X] 3 is included in R 1 or in R 2 . The inclusion in R 2 is impossible since {f (i, j) : j < ω}, being included in M ax(F i ), is finite for every i. For each i ∈ X, set G i := {F j : i ≤ j ∈ X}. . . , =⇒ 1). Clearly, F is well-founded. If F 0. This defines an ω * -chain in F ∪=⇒ 1). Clearly, F is well-founded. If F 0 , . . . , F n . . . is an infinite antichain of members of F, define f (i, j) : [ω] 2 → Q, choosing f (i, j) arbitrary in M ax(F i ) \ F j . Divide [ω] 3 into R 1 := {(i, j, k) ∈ [ω] 3 : f (i, j) = f (i, k)} and R 2 := [ω] 3 \ R 1 . From Ramsey's theorem, cf. [18], there is some infinite subset X of ω such that [X] 3 is included in R 1 or in R 2 . The inclusion in R 2 is impossible since {f (i, j) : j < ω}, being included in M ax(F i ), is finite for every i. For each i ∈ X, set G i := {F j : i ≤ j ∈ X}. This defines an ω * -chain in F ∪ .
14 If F <ω is closed under finite intersections then equivalence between (3) and (4) follows from Mislove's Theorem mentioned in. 17Remark 4.Remark 4.14 If F <ω is closed under finite intersections then equivalence between (3) and (4) follows from Mislove's Theorem mentioned in [17].
or more generally of the form I <ω (Q) where Q is some wellfounded poset, then J(P ) is well-founded if and only if P has no infinite antichain. Remark. If, in Theorem 4.13 above, we suppose that F is well-founded instead of Q, all implications in the above chain hold. A counterexample is provided by Q := ω ⊕ ω * , the direct sum of the chains ω and ω * , and F, the image of Ω(ω * ) via a natural embedding. Corollary 4.15 If P is a join-subsemilattice of a join-semilattice of the form [ω] <ωCorollary 4.15 If P is a join-subsemilattice of a join-semilattice of the form [ω] <ω , or more generally of the form I <ω (Q) where Q is some well- founded poset, then J(P ) is well-founded if and only if P has no infinite antichain. Remark. If, in Theorem 4.13 above, we suppose that F is well-founded instead of Q, all implications in the above chain hold, except 6) ⇒ 7). A counterexample is provided by Q := ω ⊕ ω * , the direct sum of the chains ω and ω * , and F, the image of Ω(ω * ) via a natural embedding.
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[] |
[
"Spherical sampling methods for the calculation of metamer mismatch volumes",
"Spherical sampling methods for the calculation of metamer mismatch volumes"
] |
[
"Michal Mackiewicz *[email protected] \nSchool of Computing Sciences\nUniversity of East Anglia\nNorwichUK\n",
"Hans Jakob Rivertz \nDepartment of Computer Science\nNorwegian University of Science and Technology\nTrondheimNorway\n",
"Graham Finlayson \nSchool of Computing Sciences\nUniversity of East Anglia\nNorwichUK\n"
] |
[
"School of Computing Sciences\nUniversity of East Anglia\nNorwichUK",
"Department of Computer Science\nNorwegian University of Science and Technology\nTrondheimNorway",
"School of Computing Sciences\nUniversity of East Anglia\nNorwichUK"
] |
[] |
In this article, we propose two methods of calculating a theoretically maximal metamer mismatch volumes. Unlike prior art techniques, our methods do not make any assumptions on the shape of spectra on the boundary of the mismatch volumes. Both methods utilise a spherical sampling approach, but they calculate mismatch volumes in two different ways. The first method uses a linear programming optimisation, while the second is a computational geometry approach based on half-space intersection. We show that under certain conditions the theoretically maximal metamer mismatch volume is significantly larger than the one approximated using prior art method.
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10.1364/josaa.36.000096
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[
"https://arxiv.org/pdf/1901.08419v1.pdf"
] | 58,542,080 |
1901.08419
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e4bd4891573a7e5f360719eb79c7f07c47c6d952
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Spherical sampling methods for the calculation of metamer mismatch volumes
23 Jan 2019
Michal Mackiewicz *[email protected]
School of Computing Sciences
University of East Anglia
NorwichUK
Hans Jakob Rivertz
Department of Computer Science
Norwegian University of Science and Technology
TrondheimNorway
Graham Finlayson
School of Computing Sciences
University of East Anglia
NorwichUK
Spherical sampling methods for the calculation of metamer mismatch volumes
23 Jan 201910.1364/JOSAA.36.000096One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited. https://doi.org/10.1364/JOSAA.36.000096
In this article, we propose two methods of calculating a theoretically maximal metamer mismatch volumes. Unlike prior art techniques, our methods do not make any assumptions on the shape of spectra on the boundary of the mismatch volumes. Both methods utilise a spherical sampling approach, but they calculate mismatch volumes in two different ways. The first method uses a linear programming optimisation, while the second is a computational geometry approach based on half-space intersection. We show that under certain conditions the theoretically maximal metamer mismatch volume is significantly larger than the one approximated using prior art method.
Introduction
A spectral power distribution carries the complete information about the light. However, the human visual system (HVS) or a typical camera has only three colour receptors or sensors, which means that colour information is compressed to only three quantities. Human eye as well as typical camera sensors collect light over broad range of wavelengths and this implies that different spectral power distributions can be mapped to the same set of three colour responses. Consequently, there will be physical samples which will reflect lights of different spectral power distribution that will produce the same colour response under one light and different colour responses under another light [1]. Analogously, if we keep illumination unchanged, but change the observer (human or camera), the same phenomenon may occur i.e. two objects can produce the same colour response for one observer and different colour responses for another. This phenomenon is called metamer mismatching and in this paper we will discuss the issue of the extent of such colour mismatch occurrences. More precisely, given the observed colour match under the first condition, we consider the range of possible colour responses that the two objects can produce under the second condition. The set of all such colours is called the metamer mismatch volume.
Metamer mismatching is important from the viewpoint of camera sensor design. Colour sensors producing large mismatch volumes for the change of observer from camera to a human are highly undesirable. Nevertheless, modern cameras exhibit some level of metamer mismatching necessitating colour correction algorithms in the camera processing pipeline. Development of such algorithms can be aided by a better understanding of the phenomenon in question. Lighting design is another field where metamer mismatching plays an important role, particularly in the context of rapidly growing modern LED lighting. Finally, understanding mismatch volumes helps us understand our own vision and therefore it is an interesting question in its own right that was studied from the the emergence of the colour science.
Recent approaches to calculation of the metamer mismatch volumes were underpinned by the assumption that reflectance spectra of all objects can be modelled using a low dimensional linear model [2][3][4]. The reliance on such a model results in only approximate estimation of the real mismatch volume, notably such estimates will always underestimate the size of these objects. More recently, Logvinenko et al. in [5] proposed to address this problem by calculating the full extent of the metamer mismatch volumes by means of establishing their precise boundaries. This approach is motivated by the observation that there is no metamerism at the boundary of the mismatch volume i.e. a colour response at the boundary of the mismatch volume corresponds to a unique spectral reflectance function. Logvinenko et al. began by asking a question: what is the general form of the shape of the reflectance spectra on the boundary of the mismatch volume? They noted that these are elementary step functions of zeros and ones with some limited number of transitions between these two values. They parametrised these functions with respect to the wavelengths where the transitions between 0 and 1 occur and decided to model them with up to five transitions. They conjectured that such a model should provide a good approximation of the theoretically maximal mismatch volumes. However, as we will later show in our experiments, their chosen parametrisation results in significant underestimation of the theoretically maximal volumes of these objects.
The rest of the paper is organised as follows. In Section 2, we introduce the relevant theory of mismatch volumes and notation and briefly describe the Logvinenko et al.'s algorithm [5]. In Section 3, we describe our early related work [6] that underpins our methods, which are subsequently described in Sections 4 and 5. The experiments evaluating the effectiveness of our algorithms and the comparison with the prior art can be found in Section 6.
The preliminary version of our first algorithm was presented at a conference [7]. Here, we present an extended version of this work, which includes the second and significantly more efficient algorithm which does not require optimisation. We also give additional results and discussions.
Metamer Mismatch Volumes Theory
The recent work by Logvinenko et al. in [5] includes an extensive introduction to metamer mismatching theory. We summarise their major points below.
The colour responses of the set of N sensors Φ(r) = (φ 1 (r), φ 2 (r), ..., φ N (r)) to an object with a spectral reflectance function r(λ) illuminated by the light with the spectral power distribution e(λ) are given by the colour formation equation:
φ i (r) = ∫ λ ma x λ mi n r(λ)e(λ)c i (λ) dλ, i = 1, 2, .., N.(1)
where λ min and λ max denote the limits of the visible spectrum and c i are the sensor spectral sensitivities. A reflectance spectrum r(λ) is a function with values between zero and one.
For the human visual system, c i (λ) would denote cone fundamentals or colour matching functions [1] and hence N is 3. Most modern digital cameras would also use three colour sensors in an attempt to emulate the colorimetry of the tristimulus HVS.
The two objects with reflectance functions r(λ) and r ′ (λ) are called metameric if they produce the same colour signals Φ(r) = Φ(r ′ ). Equation 1 tells us that if either illumination or the sensor spectra change these two objects may no longer remain metamers. We call the above two cases illumination-induced or observer-induced metamer mismatching respectively. From Eq. 1 it is clear that metamer mismatching is determined by the product of the illuminant spectrum and the sensor sensitivities, which we will call the colour system i.e. s(λ) = c(λ)e(λ). We substitute the above into Eq. 1 resulting in:
Φ(r) = ∫ λ ma x λ mi n r(λ)s(λ) dλ.(2)
Let us consider another set of colour responses Ψ = (ψ 1 , .., ψ N ) corresponding to the colour system spectra s ′ (λ) = (s ′ 1 (λ), ..., s ′ N (λ)). This new colour system might have resulted from the alteration of the illuminant spectrum, sensor sensitivities or in general both. Logvinenko et al. [5] point out that both Φ and Ψ can be considered as linear maps: X → R N , where X denotes the set of all reflectance functions. The sets of all possible colour system responses of either Φ and Ψ form closed convex sets in R N . These sets are referred to as object colour solids (OCS).
A metamer set with respect to the colour system Φ is defined as the set of all reflectances metameric to a given reflectance r 0 i.e. Φ −1 (Φ(r 0 )) = {r ∈ X|Φ(r) = Φ(r 0 )}. In general, the above metamer set will be mapped by Ψ to a non-singleton set which is referred to as a metamer mismatch volume.
Logvinenko et al. [5] introduced an additional linear map Γ : X → R 2N such that Γ(r) = (z, z ′ ), where z = Φ(r) and z ′ = Ψ(r). Consequently, Γ(X) is an object colour solid in R 2N . The authors observed that for the colour response Φ(r 0 ) = z 0 , the metamer mismatch volume
M(z 0 , Φ, Ψ) = Ψ(Φ −1 (z 0 )) = {z ′ ∈ R N |(z 0 , z ′ ) ∈ Γ(X)} is a cross-section of Γ(X).
The points in the interior of the OCS Φ(X) represent different metameric classes where each class maps to a metamer set comprising infinitely many spectra. The points on the boundary of the OCS are different. Here, each point has only one corresponding reflectance spectrum which we will call optimal. The main property of the optimal spectra is that they are elementary step functions of zeros and ones.
Schrödinger proposed that the optimal spectra on the boundary of the OCS have no more than two transitions (we call these m < 3 spectra) [8]. West & Brill later showed that, the (m < 3) spectra are optimal if and only if the spectrum locus of the chromaticity diagram is strictly convex and well-ordered in wavelength [9]. Another mathematical description of the optimal spectra that would also generalise to any object colour solid was delivered by Logvinenko in [10] who proposed that for the set of N colour systems the optimal spectra are the elementary step functions with the transition wavelengths at λ 1 , ..., λ m if and only if the above set of transition wavelengths are the only zero-crossings of the following equation:
k 1 s 1 (λ) + k 2 s 2 (λ) + ... + k N s N (λ) = 0,(3)
where k 1 , k 2 , ..., k N are a set of arbitrary real numbers, where at least one of them is not equal to zero.
In [5], Logvinenko et al. extended this description from object colour solids to metamer mismatch volumes. The metamer mismatch volume is denoted as M(z 0 , Φ, Ψ) and its boundary as ∂M. The spectra on the boundary of the mismatch volume are called µ−optimal. The authors noted that Eq. 3 determines the number of transitions in the optimal spectra on the boundary of Γ and in the µ-optimal spectra on the boundary of M. Despite that the number of transitions may be large, they chose to approximate the boundaries of the above volumes using optimal or µ-optimal spectra that were constrained to be the elementary step functions with up to five transitions (m < 6). Their choice was most likely motivated by two reasons. First, their parametrisation of the optimal spectra is based on the finite list of wavelengths where the transitions occur. This required the choice to be made how to limit the length of that list of parameters. Analogously to the 3-D HVS OCS boundary conjectured to be described with the optimal spectra with up to 3 − 1 = 2 transitions, they proposed to approximate the boundary of the larger 6-D Γ(X) and its cross-section M with the optimal spectra with up to 6 − 1 = 5 transitions. Following the aforementioned authors, we will denote the above two boundary approximations as ∂Γ(O 5 ) and ∂M 5 respectively.
The idea of describing mismatch volume boundary with m < 6 elementary step functions has already been discussed in literature. Ohta and Wyszecki held the (incorrect) view that the optimal spectra on the boundary of the mismatch volume are precisely such spectra [11]. Logvinenko et al. recognised that this is not the case and admitted that imposing such a constraint on the model of the boundary of these objects would result in a loss of precision. Nevertheless, they adopted this model in their algorithm which is summarised in Section 2.1.
Interestingly, in the earliest work studying the limits of metamerism from the point of view of optimal spectra [12,13], Allen stated that optimal spectra on the boundary of the metamer mismatch volume contain "up to 5 regions of 100 percent reflectance, each surrounded by regions of 0 percent reflectance" that is using our wording up to 10 transitions (m < 11). However, in the next sentence he observed that "in some cases more than five bands may form".
Calculating the boundary of the mismatch volume in practice
Logvinenko et al. proposed the following algorithm for calculation of ∂M 5 (z 0 , Φ, Ψ) [5]. First, a large number of optimal spectra in ∂Γ(O 5 ) i.e m < 6 spectra is generated randomly. Then, for each generated spectrum denoted as r 5 (λ 1 , ..., λ 5 ) the following optimisation is performed:
min λ 1 ,...,λ 5 ||Φ(r 5 ) − z 0 ||.(4)
In the next section we will briefly describe our recently published algorithm for calculation of the optimal spectra on the boundary of the object colour solid. What is important about this algorithm is that it does not put any constraints on the number of optimal spectra transitions. This algorithm and most importantly the alternative parametrisation of the optimal spectra it introduces will underpin the main contributions of this paper that is the two algorithms for calculation of the boundary of the mismatch volume not limited by the number of transitions of µ-optimal spectra.
Calculating the boundary of the object colour solid using spherical sampling
The key insight in this work [6] is the observation that the components of vector k in Eq. 3 have the geometrical meaning i.e. they constitute the normal vector parametrising the boundary of the object colour solid. Since the object colour solid is convex, in the direction k, we can, in closed form, find the unique system response which is maximum. By repeating this procedure for all spherically sampled directions, we can find all points on the OCS.
Formally, we propose the parametric representation of the boundary of the OCS with respect to k.
All colour system responses Φ(r) = (φ 1 (r), φ 2 (r), ..., φ N (r)) from Eq. 2 are projected onto a unit vector k. That is
k · Φ(r) = ∫ λ ma x λ mi n r(λ) k · s(λ) dλ .
It is clear that the maximum value of k · Φ(r) is obtained by
r opt = r(λ; k) = 0, k · s(λ) < 0 1, k · s(λ) ≥ 0 .(5)
The above observation leads to a very efficient algorithm for calculation of the OCS optimal spectra:
We generate a set of M normal vectors in R N and store them in the rows of M × N matrix P [14]. The colour system spectra are stored in N × q matrix S, where q determines the wavelength resolution e.g. for 1nm resolution, λ min = 380 and λ max = 730, the colour system and reflectance spectra will have 351 components i.e. q = 351.
We denote a matrix resulting from multiplication of P by S as A = PS. Finally, the signs of the elements of A determine the set of optimal spectra in matrix R as:
R ij = 0, A ij < 0 1, A ij ≥ 0(6)
The final step of the algorithm is to connect the points on the boundary of the OCS into a volume. This can be done using any convex hull algorithm. We have used the Matlab implementation of the Quickhull algorithm [15].
In Fig. 1, we illustrate with an example how the algorithm works. This example uses the XYZ colour matching functions and the D65 illuminant. The algorithm is very fast as it does not require optimisation. We admit that calculation of the object colour solid boundary is not a new problem and there were a number of algorithms proposed in the past. For example, for the set of three sensors, one could generate a number of sensor responses from the set of randomly generated elementary spectra with two transitions [16,17]. This said, our method presents two significant benefits. First, the spherical sampling allows for describing the OCS with a small number of samples and second, the normal vector parametrisation allows for optimal spectra with any high number of transitions. Further, this algorithm naturally leads to the related algorithms for calculation of the boundary of the metamer mismatch volume which will be introduced in the next two sections.
Calculating the boundary of the metamer mismatch volume using spherical sampling and linear programming.
The algorithm presented in the previous section can be used to calculate either ∂Φ(X) ∈ R N or the larger ∂Γ(X) ∈ R 2N . We have stated previously that the metamer mismatch volume M is a cross-section of Γ(X) and it is convex. Then, we can find ∂M(z 0 , Φ, Ψ) analogously to the algorithm presented in the previous section. We extremize all spherically sampled directions k in R 2N subject to Φ(r) = z 0 and further constraints on the values of the reflectance function. Hence, this optimisation can be written as follows:
max r ∫ λ ma x λ mi n r(λ)k · s(λ),(7)
subject to
Φ(r) = z 0 0 < r(λ) < 1,
where s(λ) are the 2N colour system spectra.
We choose a wavelength sampling resolution and write the above optimisation using vector notation as a linear programming problem:
max r (Sk) T r,(8)subject to S T Φ r = z 0 0 < r i < 1 for i = 1, ..., q,
where S is a q × 2N matrix containing colour system spectra, S Φ is a q × N matrix containing colour system spectra of Φ and r is a q-vector containing a µ-optimal spectrum in ∂M(z 0 , Φ, Ψ) that corresponds to the direction k.
We also give an alternative optimisation which as we shall see will offer some improvements over (8). This new formulation uses the orthonormal set of colour system spectra which has the potential of achieving a more uniform sampling of the boundaries of both OCS and the metamer mismatch volume. We can write this new optimisation as:
max r (Uk) T r,(9)
subject to the same constraints as (8), where U is a q × 2N matrix containing the set of orthonormal colour system spectra which can be obtained from S using singular value decomposition [18]. In Fig. 2 we show six spectra for color matching functions under D65 and A illuminants and their corresponding orthonormal spectra.
Note, that the use of orthonormal spectra (U) with uniform spherical sampling is equivalent to the use of original spectra (S) with certain non-uniform spherical sampling that would emphasise directions in inverse proportion to their corresponding singular values. The singular value decomposition of matrix S can be written as S = UDV T , and thus U = SVD −1 . It is worth noticing that for the above set of six spectra, the ratio of the lowest to the highest singular value is about 10 −2 and hence the resulting spherical sampling can be considered as very much non-uniform.
In Section 6, we will see that the two versions of the method described in this section do indeed allow for accurate calculation of the metamer mismatch volume theoretical limits. This said, an attentive reader may have spotted that the algorithms presented in this section require a repeated application of linear programming optimisation. If the number of samples is not too high, this may not be a problem. Nevertheless, it is appropriate to ask a question -is there a more efficient algorithm that could avoid optimisation altogether? In the next section, we will show that the answer to this question is 'yes' and consequently present an even more efficient alternative algorithm for calculation of the metamer mismatch volumes.
Calculating the boundary of the metamer mismatch volume using spherical sampling without optimisation.
The idea for the calculation of the metamer mismatch volume that would not require optimisation stems from an observation we made in Section 2, that is, a 3-D metamer mismatch volume M(z 0 , Φ, Ψ) is a particular cross-section of the 6-D object colour solid Γ(X). Therefore, we can envisage an algorithm that can be split into two parts -first, building the 6-D OCS and second, calculating its 3-D cross-section. The difficulty here lies in performing these two operations efficiently.
As to the first operation, we have described the algorithm for calculating object colour solids in Section 3. This can be used to obtain their representations, either in 3-D for Φ or in 6-D for (Φ, Ψ) responses. Importantly here, this algorithm returns a vertex representation of the OCS. While such a representation is perfectly valid, it is not ideal from the point of view of calculating the cross-section of the OCS. Here, we would prefer a half-space representation, which as we will later see is particularly suited for our task.
Conversion from vertex to half-space representation is inefficient for a large number of points, particularly in a high dimensional space such as R 6 . It requires a convex hull operation to identify all faces followed by the calculation of the surface normals. We expect the number of points required to accurately represent the 6-D OCS to be significantly higher than in the case of the 3-D OCS and therefore conclude that such an approach would be inefficient.
Having said that, there is another way for calculating the half-space representation of the OCS in R 6 that is very efficient. We notice that a vertex Γ on the surface of the OCS as calculated using the algorithm in Section 3 has a corresponding and known normal vector k that was used in the calculation of this vertex. Then, the half-space representation of the OCS can be written as k i · x ≤ b i where i is an index over all spherically sampled normals k i and x = (z, z ′ ) ∈ R 6 , z ∈ R 3 , z ′ ∈ R 3 . The offset of the i-th half-space can be calculated using corresponding Γ i as b i = k i · Γ i . Finally, the half-space representation of the object colour solid can be written using the following set of inequalities:
K T x ≤ b(10)
where K is a 6 × t matrix containing t spherically sampled unit vectors in its rows and b is a t-vector containing corresponding elements b i .
Notice that this representation overestimates the volume of the OCS as opposed to the earlier vertex representation which underestimated it.
The above has given us the first part of our algorithm that is an efficient to calculate description of the OCS in R 6 . In order to calculate the relevant metamer mismatch volume, we need to intersect the 6-D OCS with the affine subspace of dimension 3 defined by z = z 0 equation. This can be performed efficiently if the OCS is described using the half-space representation.
We know that the solution to the metamer mismatch volume lies in the subspace spanned by the last three coordinate axes and can be written as:
x = Bz ′ + x 0 ,(11)where B = 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 and x 0 = z 0 0 0 0 .
We substitute (11) into (10) and obtain the half-space representation of the metamer mismatch volume:
K T Bz ′ ≤ b − K T x 0 .(12)
In order to calculate the vertices (and the volume) of the metamer mismatch volume, we need to perform the half-space intersection operation. This can be done using tools such as Qhull [15] and the duality transform of Preparata and Shamos [19]. We expect this operation to be efficient as it is performed in a low dimensional space (R 3 ) and also we predict that only a limited number of planes in (12) will bind the intersection.
Experiments and Results
We performed a number of experiments testing the two algorithms presented in the previous sections. First, we tested the two variants of the algorithm that uses an optimisation and compared them with the algorithm presented in [5] that constitutes our benchmark. In all experiments we used CIE 1931 2-deg, XYZ colour matching functions modified by Judd and Vos [20]. We have built metamer mismatch volumes for the change of illuminant for three illuminants: D65, A and F11 [21] for all 6 pairs. Below, we will only show the results of the two conditions for the change of illuminant from D65 to A and F11 to D65, which are representative of all the results.
In our first experiment, we wished to determine the appropriate wavelength sampling for our algorithm. We tested wavelength resolutions from 0.5nm to 10nm. The metamer mismatch volumes were calculated for the flat grey reflectance (r 0 = 0.5) for the increasing number of samples generated using spherical sampling as described in Section 4. Here, we used the variant of the algorithm employing linear programming and the orthonormal colour system spectra. We also compared the volumes generated with this algorithm with those obtained using [5]. Note that the Logvinenko et al.'s method uses parametrisation that does not require us to choose the wavelength sampling resolution. The results of this experiment can be seen in Figures 3 and 4. In Fig. 3, we can see that the volumes obtained for wavelength sampling resolutions of 0.5nm and 1nm are almost identical. The 2nm and 5nm are very similar, whereas the 10nm sampling significantly overestimates the volume. These observations are confirmed in Fig. 4, where again we can see that 0.5 and 1nm sampling produce almost identical results. However, this time the errors for wavelength resolutions above 1nm become more visible as this result has been produced for the change of illuminant involving F11, which has a spectrum that is less smooth Fig. 3. Metamer mismatch volumes calculated using the method presented in Section 4 for the flat grey reflectances with 50% reflectivity for the two sets of orthonormal colour system spectra for the change of illuminant from D65 to A with spectral sampling varying from 0.5nm to 10nm. (L) -method proposed by Logvinenko et al. [5]. Results for 0.5nm and 1nm sampling resolutions are almost the same and hence 0.5nm is hidden under the 1nm plot.
F11 to D65
0.5nm 1nm 2nm 5nm 10nm L than both D65 and A and hence may require a lower sampling resolution. Therefore, we will use the wavelength resolution of 1nm in all subsequent experiments. Figures 3 and 4 also show that the metamer mismatch volumes obtained for the Logvinenko et al. algorithm are significantly smaller (from approximately 25% to 70%). This is particularly visible for a small number of samples and when the F11 illuminant is involved. Figures 5 and 6 show the results of the second experiment where we investigated two further aspects of the linear programming algorithm. First, we analyse the volumes produced by the two versions of the algorithm, for the original and orthonormal colour system spectra. Second, we look at the size of the mismatch volumes along the achromatic line for both variants of the algorithm. As to the first aspect of this experiment, we can see that the orthonormal colour system spectra indeed result in better distribution of the samples describing the mismatch volume and consequently produce larger volumes. More specifically, in Fig. 6 we can see that for the flat grey reflectance (50%), the version of the algorithm utilising the orthonormal spectra with 100 spherical samples produces the mismatch volume estimate that is matched only by the original version of the algorithm with 500 samples.
Regarding the second aspect of the experiment, we can see that as expected the metamer mismatch volumes are the largest in the centre of the OCS. Moreover, we can also see that the Logvinenko et al.'s method significantly underestimates the sizes of the mismatch volumes particularly in the centre of the OCS.
D65 to A
U 50% S 50% L 50% U 70% S 70% L 70% U 90% S 90% L 90% Fig. 5. Metamer mismatch volumes calculated using the method presented in Section 4 for the flat grey reflectors with 50%, 70% and 90% reflectance for the the change of illuminant from D65 to A using the orthormal colour system spectra (U), original colour system spectra (S) with 1nm spectral sampling. (L) -method proposed by Logvinenko et al. [5].
The prior art method [5] uses an inefficient strategy of random initialisation of the transition wavelengths which then tend to converge to certain clusters on the surface of the mismatch volume. Hence, this method requires a very large number of 5-transition optimal spectrapreferably above 10000 samples -in order to estimate the size of M 5 , which we have reasons to expect will still significantly underestimate the size of M. On the other hand, the better performing version of our algorithm usually requires as few as 1000 samples to accurately estimate the volume of the larger mismatch volume M.
In Figures 7 and 8 metamer mismatch volume 10 10
F11 to D65
U 50% S 50% L 50% U 70% S 70% L 70% U 90% S 90% L 90% Fig. 6. As in Fig. 5, but for the change of illuminant from F11 to D65.
is indeed clearly contained within the theoretical limits of the mismatch volume M produced by our algorithm. The reasons behind this phenomena can be better understood from Figures 9 and 11. There, we show the same convex hulls that were created using our algorithm, but this time we highlight with colour the number of transitions of the corresponding optimal spectra. Both figures use 1000 random samples on the surface of each volume. We can see that for the change of illuminant from D65 to A the number of transitions on the surface of the metamer mismatch volume ranges from 3 to 18 and for the F11 to D65 from 5 to 13. We can also see that the optimal spectra with the same number of transitions form regions or curves on the surface of these objects.
Histograms showing the frequency of occurrence of optimal spectra with different number of transitions can be seen in Figures 10 and 12. They clearly show that the vast majority of optimal spectra have a higher number of transitions than 5. The low number of optimal spectra of less than or equal five transitions for the change of illuminant from F11 to D65 is particularly striking which explains why for this condition there is such a large difference between the volumes produced by our algorithm and the prior art.
In the final experiment, we calculated metamer mismatch volumes using our final method requiring no optimisation that was presented in Section 5. Here, we also used 1nm sampling. We present our results for the same two conditions: change of illuminat from D65 to A (see Figure 13) and from F11 to D65 (Figure 14). Analogously to the previous experiment, we plot the volumes for the flat grey reflectance of 50%, 70% and 90% reflectance and using two sensor sets: original and orthonormal. We compare the metamer mismatch volumes to those obtained in the previous experiment for the largest number of samples (10 4 , see Figs 5 and 6) -these are plotted as black dashed lines.
A reader should notice that the range of spherically sampled points we tested is substantially different comparing to the previous experiment. Earlier, we stopped at 10 4 samples and here we performed our calculations for the range of samples from 10 5 to 10 9 . In Section 5, we predicted that the number of samples needed to cover the 6-D OCS will be significantly higher than for the 3-D mismatch volume and this is reflected in our figures.
Our next observation from these figures is that as predicted the approximated mismatch volumes are overestimated and as the number of samples increases, the volumes tend to those obtained using our earlier linear programming method.
An interesting observation that is different to what we have seen earlier is the difference in performance for the two sensor sets. While we have seen better approximation with orthonormal sensors for the linear programming method, here the gap in performance of the two sensor sets is much wider, particularly for the change of illuminant from D65 to A where using the orthonormal sensors the volumes are more accurately represented with as few as 10 5 samples than using the original sensor set with as many as 10 8 samples. Fig. 7. Comparison of the metamer mismatch volumes calculated using the method presented in Section 4 for the flat grey reflectance with 50% reflectance for the change of illuminant from D65 to A using 1nm spectral sampling with the corresponding volume calculated by the method proposed in [5]. Both methods use 10000 samples. Plotted in the CIE XYZ colour space. Fig. 10. A histogram illustrating the frequency of occurrence for different number of transitions in optimal spectra. Corresponds to data in Fig. 9.
As to the change of illuminant from F11 to A (see Fig. 5), the orthonormal sensors also provide much better approximations of real mismatch volumes. This said, here we need to use a larger number of samples to obtain a comparable accuracy.
Our final comment concerns the major advantage of the method discussed here, namely speed of execution. Linear programming approach required approximately 500s to calculate the metamer mismatch volume using 10 4 samples and proportionally less time if less points were required. The method presented here is much faster. First, we note that the algorithm can be split into two parts: the calculation of the 6-D object colour solid for a given colour system Fig. 10, but for the change of illuminant from F11 to D65. Corresponds to data in Fig. 11 and the calculation of the intersection with the 3-D affine subspace corresponding to a required metamer. The latter takes less time than the former. If an application requires a number of mismatch volumes to be calculated for the same colour system, then we need to calculate the 6-D OCS only once. This step takes approximately 170s for 10 8 samples and proportionally less if less sampling points are required i.e. approximately 1.7s for 10 6 samples. The second step is faster than the first and takes approximately 70s and 0.7s for 10 8 and 10 6 samples respectively. While different conditions may require different numbers of samples for any of the two methods, it is clear that the half-space intersection method is significantly faster e.g. for the change of illuminant from D65 to A, an accurate mismatch volumes can be calculated in less than a second for 10 6 samples, whereas an approximation of this volume with 10 3 samples metamer mismatch volume 10 8 D65 to A U 50% S 50% U 70% S 70% U 90% S 90% lin. prog baselines Fig. 13. Metamer mismatch volumes calculated for the flat grey reflectors with 50%, 70% and 90% reflectance using the method presented in Section 5 for the the change of illuminant from D65 to A using the orthonormal colour system spectra (U), original colour system spectra (S) with 1nm spectral sampling. Dashed lines correspond to the respective volumes (for 10 4 samples) in Fig. 5. using linear programming would require 50s. All experiments were performed on a PC running Matlab 2014a on Intel i7-4790 at 3.6GHz and 32GB of memory.
Conclusions
Metamer mismatching is an important phenomenon in colour science. Here, we proposed two novel algorithms for calculation of the theoretically maximal metamer mismatch volumes. To our knowledge, they are the first algorithms capable of calculating a precise maximum extent of these volumes. Our figures when compared with those produced by the earlier methods show that the 5-transition approximation results in significantly smaller mismatch volumes (sometimes above 50%!). Our two algorithms are computationally efficient due to simple formulations (either as a linear programming optimisation or as a half-space intersection) and a relatively small number of spherical samples required to provide precise mismatch volume estimates. Importantly, we note that both algorithms take advantage of the small number of samples as a result of their use of orthonormal sensors which is equivalent to a certain non-uniform sampling of these objects.
Fig. 1 .
1An illustration how to find an optimal spectrum for a given vector k. XYZ colour matching functions multiplied by the D65 illuminant (top). The linear combination (with k coefficients) of the above three functions (middle). Corresponding optimal spectrum (bottom).
Fig. 2 .
2Spectra of colour matching functions under illuminant D65 and A (top) and their orthonormal representation (bottom). The latter are ordered according to their corresponding singular values decrease.
, we can see the graphical comparison of the mismatch volumes produced by our algorithm and the Logvinenko et al. method. The mismatch volume approximation M 5
Fig. 8 .
8As inFig. 7, but for the change of illuminant from F11 to D65.
Fig. 9 .
9Random subset of 1000 points on the surface of the larger convex hull in Fig. 7. The points are coloured according to the number of transitions of the corresponding optimal spectra.
Fig. 11 .
11As in inFig. 9, but for the larger convex hull inFig. 8.
Fig. 12 .
12As in in
Fig. 14 .
14As inFig. 13, but for the change of illuminant from F11 to D65. Dashed lines correspond to the respective volumes (for 10 4 samples) inFig. 6.
G Wyszecki, W Stiles, Color Science: Concepts and Methods, Quantative Data and Formulae. NYJohn Wiley and SonsG. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantative Data and Formulae (John Wiley and Sons, NY, 1982).
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The quickhull algorithm for convex hulls. C Barber, D P Dobkin, H T Huhdanpaa, ACM Trans. on Mathematical Software. 22C. Barber, D. P. Dobkin, and H. T. Huhdanpaa, "The quickhull algorithm for convex hulls," ACM Trans. on Mathematical Software 22, 469-483 (1996). Http://www.qhull.org.
XYZ to ADL: Calculating Logvinenko's Object Color Coordinates. C Godau, B Funt, Proceedings of the 18th Color and Imaging Conference (CIC). the 18th Color and Imaging Conference (CIC)C. Godau and B. Funt, "XYZ to ADL: Calculating Logvinenko's Object Color Coordinates," in "Proceedings of the 18th Color and Imaging Conference (CIC)," (2010).
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|
[] |
[
"Cascades of phase transitions in spiral magnets caused by dipolar forces",
"Cascades of phase transitions in spiral magnets caused by dipolar forces"
] |
[
"O I Utesov \nNational Research Center \"Kurchatov Institute\" B.P. Konstantinov Petersburg Nuclear Physics Institute\n188300GatchinaRussia\n",
"A V Syromyatnikov \nNational Research Center \"Kurchatov Institute\" B.P. Konstantinov Petersburg Nuclear Physics Institute\n188300GatchinaRussia\n\nSt. Petersburg State University\n7/9 Universitetskaya nab199034St. PetersburgRussia\n"
] |
[
"National Research Center \"Kurchatov Institute\" B.P. Konstantinov Petersburg Nuclear Physics Institute\n188300GatchinaRussia",
"National Research Center \"Kurchatov Institute\" B.P. Konstantinov Petersburg Nuclear Physics Institute\n188300GatchinaRussia",
"St. Petersburg State University\n7/9 Universitetskaya nab199034St. PetersburgRussia"
] |
[] |
We present a mean-field theory describing the influence of long-range dipolar forces on the temperature transition from the paramagnetic to ordered phases in frustrated Heisenberg spiral magnets. It is shown that the dipolar interaction produces a cascade of first-and second-order phase transitions between the paramagnetic and the spiral states upon temperature decreasing. Depending on system parameters, the following intermediate phases can arise: an incommensurate and a commensurate sinusoidally modulated states, spiral phases in which perpendicular spin components have different amplitudes and are modulated with the same and with different wave vectors. We distinguish six possible sequences of phase transitions upon temperature decreasing at least four of which were observed before experimentally in specific compounds. It is found that the action of dipolar forces cannot always be modeled even qualitatively by small one-ion anisotropic spin interactions. We demonstrate that the dipolar interaction is responsible for successive phase transitions in the triangular-lattice multiferroic MnI2: almost all available experimental findings are described quantitatively within the mean-field theory by taking into account the exchange, the dipolar and small symmetry-allowed anisotropic spin interactions.
|
10.1103/physrevb.95.214420
|
[
"https://arxiv.org/pdf/1703.03824v1.pdf"
] | 119,066,931 |
1703.03824
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3409c5064d72798514bbbad4d8a54a43c20ee754
|
Cascades of phase transitions in spiral magnets caused by dipolar forces
O I Utesov
National Research Center "Kurchatov Institute" B.P. Konstantinov Petersburg Nuclear Physics Institute
188300GatchinaRussia
A V Syromyatnikov
National Research Center "Kurchatov Institute" B.P. Konstantinov Petersburg Nuclear Physics Institute
188300GatchinaRussia
St. Petersburg State University
7/9 Universitetskaya nab199034St. PetersburgRussia
Cascades of phase transitions in spiral magnets caused by dipolar forces
(Dated: September 28, 2018)numbers: 7530-m7530Kz7510Jm7585+t
We present a mean-field theory describing the influence of long-range dipolar forces on the temperature transition from the paramagnetic to ordered phases in frustrated Heisenberg spiral magnets. It is shown that the dipolar interaction produces a cascade of first-and second-order phase transitions between the paramagnetic and the spiral states upon temperature decreasing. Depending on system parameters, the following intermediate phases can arise: an incommensurate and a commensurate sinusoidally modulated states, spiral phases in which perpendicular spin components have different amplitudes and are modulated with the same and with different wave vectors. We distinguish six possible sequences of phase transitions upon temperature decreasing at least four of which were observed before experimentally in specific compounds. It is found that the action of dipolar forces cannot always be modeled even qualitatively by small one-ion anisotropic spin interactions. We demonstrate that the dipolar interaction is responsible for successive phase transitions in the triangular-lattice multiferroic MnI2: almost all available experimental findings are described quantitatively within the mean-field theory by taking into account the exchange, the dipolar and small symmetry-allowed anisotropic spin interactions.
We present a mean-field theory describing the influence of long-range dipolar forces on the temperature transition from the paramagnetic to ordered phases in frustrated Heisenberg spiral magnets. It is shown that the dipolar interaction produces a cascade of first-and second-order phase transitions between the paramagnetic and the spiral states upon temperature decreasing. Depending on system parameters, the following intermediate phases can arise: an incommensurate and a commensurate sinusoidally modulated states, spiral phases in which perpendicular spin components have different amplitudes and are modulated with the same and with different wave vectors. We distinguish six possible sequences of phase transitions upon temperature decreasing at least four of which were observed before experimentally in specific compounds. It is found that the action of dipolar forces cannot always be modeled even qualitatively by small one-ion anisotropic spin interactions. We demonstrate that the dipolar interaction is responsible for successive phase transitions in the triangular-lattice multiferroic MnI2: almost all available experimental findings are described quantitatively within the mean-field theory by taking into account the exchange, the dipolar and small symmetry-allowed anisotropic spin interactions.
I. INTRODUCTION
Frustration can have a dramatic impact on properties of magnetic systems leading to novel phenomena which have being extensively studied in recent years: various spin-liquid phases, novel phase transitions, and order-by-disorder phenomena, to mention just a few. 1 In particular, frustration changes the type of transitions to magnetically ordered phases in Heisenberg antiferromagnets (HAFs) on a (stacked) triangular lattice and in frustrated HAFs with a spiral magnetic ordering. The order parameter acquires additional symmetry elements that leads to changing the type of the phase transition in three-dimensional (3D) systems (the continuous transition in non-frustrated magnets vs. the firstorder one in frustrated systems), to a novel pseudo-universal behavior in 3D XY systems, and to the stabilization of a chiral spin-liquid phase upon cooling before the onset of Berezinskii-Kosterlitz-Thouless transition in 2D systems. 2,3 Weak low-symmetry spin interactions, which are always present in real materials, complicate further the behavior of frustrated systems upon temperature decreasing. They can lead, for example, to a crossover to another critical behavior near the critical point, to a changing the type of the phase transition, and to a splitting of the phase transition into a sequence of different phase transitions. In particular, it is well known that dipolar forces, which are always present in real compounds, lead to the splitting of the transition to the ordered state with 120 • magnetic structure into three successive transitions in XY HAFs on the stacked triangular lattice. 4,5 Three successive transitions take place upon the temperature decreasing: the second-order transition from the paramagnetic (PM) phase to an incommensurate sinusoidally-modulated (ICS) state, the second-order transition to an incommensurate phase in which two components of magnetic moments are modulated with different wave vectors and have different amplitudes (an elliptic phase), and, finally, the first-order transition occurs to the commensurate phase with the conventional 120 • magnetic structure. The difference between temperatures of these three transitions is governed by the ratio of the characteristic dipolar energy ω 0 and the exchange coupling constant J which is usually small in real materials. However three successive phase transitions with these two incommensurate intermediate phases were really observed in particular triangular XY HAFs (e.g., in RbFeCl 3 ) with J ∼ ω 0 ∼ 1 K (see Refs. 4,5 ).
Frustrated Heisenberg magnets in which the spiral magnetic ordering arises due to the competition between different exchange interactions fall into the same universality classes as triangular HAFs. 2 To the best of our knowledge, the impact of the dipolar interaction on transitions to magnetically ordered phases has not been discussed yet in such models. On the other hand, such investigation would be of particular interest due to the great attention devoted in recent years to multiferroics with spiral magnetic orderings appearing due to frustrated exchange interactions. 6 This attention is stimulated by a possible application of such compounds in the spin-related electronics. Multiferroics MnI 2 (Refs. [6][7][8][9][10][11] ) and MnWO 4 (Refs. 6,[12][13][14] ) are promising candidates for such analysis because their exchange coupling constants are small ( 1 K). Besides, the magnetocrystalline anisotropy is expected to be very small because Mn 2+ ions are in spherically symmetric states with the orbital and the spin moments L = 0 and S = 5/2, respectively. Then, the dominating low-symmetry interaction in these compounds is the dipolar one. It was found experimentally that these materials show the cascade of phase transitions upon temperature decreasing with the ICS and elliptical intermediate phases.
We develop a mean-field theory in Sec. II describing frustrated spiral HAFs (including HAFs on the triangular lattice) with dipolar forces near the transition from the PM phase. Phases which can arise in this model are described: the ICS phase, the commensurate and the incommensurate spiral states, elliptical phases in which two components of the order parameter are modulated with the same and with different vectors. Six possible sequences of transitions to these phases are established which are summarized in Fig. 1. Phase transitions in MnBr 2 , MnWO 4 , and in XY HAFs on the stacked triangular lattice follow one of these six scenarios. It is shown that the transition from the PM state takes place to the ICS phase. Then, we extend in Sec. II available theories devoted solely to the role of the dipolar interaction in MnBr 2 16 and in triangular XY HAFs 4 . It is always tempting to model the action of dipolar forces by some short-range anisotropic spin interactions in theoretical considerations due to slow convergence of dipolar sums that requires using the inconvenient special resummation technique. We consider in Sec. III the possibility to reproduce all six scenarios obtained in Sec. II by replacing the dipolar interaction (which is a source of a biaxial anisotropy in a system) by a short-range biaxial spin anisotropy. We find that only three scenarios can be reproduced by the single-ion anisotropy whereas all six scenarios arise in the case of the exchange biaxial anisotropy.
We describe quantitatively phase transitions in MnI 2 in Sec. IV within the mean-field approach. It is shown that dipolar forces are indispensable for a proper description of available experimental data 7,8 , but small symmetryallowed easy axis and hexagonal anisotropies should be also taken into account. Besides, our analysis shows that Dzyaloshinskii-Moriya interaction (DMI) should arise in the spiral phase which is responsible for ferroelectric properties in this phase. The latter result is in accordance with recent experimental findings. 11 MnI 2 follows one of six scenarios described in Sec. II which is somewhat complicated by the small anisotropic interactions.
We present a summary of the results and our conclusion in Sec. V. The mean-field expansion of the free energy and DMI in the spiral ferroelectric phase of MnI 2 are discussed in appendixes.
II. PHASE TRANSITIONS IN SPIRAL HEISENBERG MAGNETS WITH DIPOLAR FORCES
In this section, we discuss how dipolar forces change the transition from the PM to the ordered phase in frustrated spiral HAFs. We assume that the order parameter is small and develop the mean-field (Landau) theory. The corresponding mean-field energy reads as
E = 1 2 i,j J ij s i · s j + D αβ ij s α i s β j ,(1)
where the first and the second terms describe the exchange and the dipolar interactions, respectively, s i is a mean magnetic moment which depends on T and which is always smaller than the spin value S, and the summation is implied over repeated Greek letters which denote Cartesian components x, y, z. The dipolar tensor in Eq. (1) has the
form D αβ ij = ω 0 v 0 4π 1 R 3 ij − 3R α ij R β ij R 5 ij ,(2)
where v 0 is the unit cell volume and
ω 0 = 4π (gµ B ) 2 v 0(3)
is the characteristic dipolar energy. Within the mean-field approach, we obtain by expanding the free energy F up to the fourth order in s (see Appendix A)
F = E + AT i s 2 i + BT i s 4 i ,(4)
where A and B are values depending on S only which are given by Eqs. (A7) and (A8), respectively. Introducing the Fourier transform s i = 1 √ N q s q e iqRi , we rewrite energy (1) near the transition from the PM phase as
E = q J q δ αβ + 1 2 D αβ q s α q s β −q = q H αβ q s α q s β −q ,(5)
where J q = j =0 J 0j e iqRj and D αβ q = j =0 D αβ 0j e iqRj . Slowly convergent lattice sums in the latter expression are calculated below numerically by rewriting the sums in fast convergent forms (see Ref. 15 ).
Tensor H αβ q has three generally different eigenvalues λ 1,2,3 (q) which are functions of q. As it is seen from Eqs. (4) and (5), the smallest eigenvalue λ 1 (q = q sin ) and the corresponding eigenvector determine the free energy and the spin ordering in the ordered phase near the critical temperature. We consider below a typical situation of different minimum values of λ 1,2,3 (q) (assuming that the smallest and the largest eigenvalues are λ 1 and λ 3 , respectively) and an incommensurate value of q sin . Notice that q sin ≈ Q at small dipolar interaction, where q = Q minimizes J q . The second-order transition from the PM phase to the ordered one takes place within the mean-field theory at a temperature T N 1 at which the bilinear term in the free energy changes the sign. Then, one obtains from Eqs. (4) and (5)
T N 1 = − λ 1 (q sin ) A .(6)
The spin texture near T = T N 1 is determined by the eigenvector corresponding to λ 1 (q = q sin ) which gives an incommensurate sinusoidally-modulated structure
s i = a 1 sin qR i + a 2 cos qR i(7)
with a 1 ||a 2 , |a 1,2 | ∝ s, and q = q sin . Minimization of the free energy gives for its value and for the order parameter in the ICS phase
F ics = − (λ 1 (q sin ) + AT ) 2 6BT = − A 2 (T N 1 − T ) 2 6BT ,(8)s = 2A 3B T N 1 − T T ,(9)
where Eq. (6) is taken into account.
The model behavior at T < T N 1 depends strongly on values of its parameters. Let us consider possible ordered phases which can arise at T < T N 1 . The first-order transition can happen from the ICS phase to that with the spiral order which is described by Eq. (7) with |a 1 | = |a 2 |, a 1 ⊥ a 2 , and q = q sp . The free energy of this state (denoted below as SP phase) and the transition temperature read as
F sp = − ([λ 1 (q sp ) + λ 2 (q sp )]/2 + AT ) 2 4BT ,(10)T sp = T N 1 − 1 + 2 3 S(S + 1)(λ 1 (q sp ) + λ 2 (q sp ) − 2λ 1 (q sin )),(11)
where λ 1 (q) + λ 2 (q) reaches its minimum at q = q sp which is smaller than the minimum of λ 1 (q) + λ 3 (q). Directions of a 1 and a 2 are determined by eigenvectors corresponding to λ 1 (q sp ) and λ 2 (q sp ). It might happen that a commensurate vector q cs lies not far from q sin such that 2q cs or 4q cs are equal to a reciprocal lattice vector. Although λ 1 (q) does not reach a minimum at q = q cs , the free energy of the sinusoidallymodulated commensurate structures (CS) with q = q cs can become lower at some T < T N 1 than that of the ICS state. It can happen because summations over the lattice give different results at q = q cs and at an incommensurate q after substitution of Eq. (7) to Eqs. (1) and (4). Thus, one obtains for the free energy in this case
F cs = − (λ 1 (q cs ) + AT ) 2 4BT .(12)
Notice the smaller numerical factor in the denominator of Eq. (12) as compared to that in Eq. (8) which makes possible the considered first-order transition from the ICS structure at the critical temperature
T cs = T N 1 − 2 1 + 2 3 S(S + 1)(λ 1 (q cs ) − λ 1 (q sin )).(13)
A second-order transition can take place from the ICS to an elliptic structure described by Eq. (7) with |a 1 | = |a 2 |, a 1 ⊥ a 2 , and q = q sin . Henceforth, it is called EL1 phase. One finds for the free energy of this state and the transition temperature
F el1 = − 3(λ 1 (q sin ) + AT ) 2 − 2(λ 1 (q sin ) + AT )(λ 2 (q sin ) + AT ) + 3(λ 2 (q sin ) + AT ) 2 16BT ,(14)T el1 = T N 1 − S(S + 1)(λ 2 (q sin ) − λ 1 (q sin )).(15)
An elliptical structure in which two orthogonal spin components have different modulation vectors can arise also via a second-order transition from the ICS phase:
s i = a 1 sin q sin R i + a 2 cos q 2 R i ,(16)
where |a 1 | = |a 2 |, a 1 ⊥ a 2 , and q 2 = q sin . Vector q 2 corresponds to the smallest eigenvalue of H αβ q among eigenvectors which are perpendicular to the spin polarization in the ICS phase. Henceforth, this state is called EL2 phase. The free energy and the transition temperature read in this case
F el2 = − 3(λ 1 (q sin ) + AT ) 2 − 4(λ 1 (q sin ) + AT )(λ 1 (q 2 ) + AT ) + 3(λ 1 (q 2 ) + AT ) 2 10BT ,(17)T el2 = T N 1 − 2S(S + 1)(λ 1 (q 2 ) − λ 1 (q sin )).(18)
At T T N 1 , CS and SP phases are stable if λ 1 (q cs ) < (λ 1 (q sp ) + λ 2 (q sp ))/2 and λ 1 (q cs ) > (λ 1 (q sp ) + λ 2 (q sp ))/2, respectively. These conditions are equivalent to T cs > T sp and T cs < T sp , correspondingly. Conditions for transitions from the ICS to other phases mentioned above can be formulated in terms of inequalities between values of T cs , T sp , T el1 , and T el2 given by Eqs. (11), (13), (15), and (18). The following six different scenarios can be distinguished which are schematically shown in Fig. 1.
(i) T cs > T sp , T el1 , T el2 .
There is a first-order transition from the ICS to the CS state. The sequence of the phase transitions under temperature decreasing is the following: PM → ICS → CS. Phase transitions in MnBr 2 follow this scenario. 16 (ii) T sp > T cs , T el1 , T el2 . It is possible only if q sp = q sin . Then, there is a first-order transition from the ICS to the SP phase. The corresponding sequence is PM → ICS → SP. This scenario appears in MnI 2 which is complicated by small anisotropic spin interactions leading to an additional transition splitting the ICS state into two different ICS phases (see below).
(iii) T el1 > T cs , T sp , T el2 and T cs > T sp . There is a second-order transition from the ICS to the EL1 structure and a first-order transition from the EL1 to the CS order. The corresponding sequence is PM → ICS → EL1 → CS. This succession of phase transitions was experimentally observed in MnWO 4 . 17 (iv) T el1 > T cs , T sp , T el2 and T sp > T cs . There is a second-order transition from the ICS to the EL1 structure. The subsequent transition from the EL1 to the SP phase is of the first-order type if q sp = q sin and of the second-order type if q sp = q sin . The corresponding sequence is PM → ICS → EL1 → SP.
(v) T el2 > T cs , T sp , T el1 and T cs > T sp . There is a second-order transition from the ICS to the EL2 structure and a first-order transition from the EL2 to the CS phase. The corresponding sequence is PM → ICS → EL2 → CS.
(vi) T el2 > T cs , T sp , T el1 and T sp > T cs . There is a second-order transition from the ICS to the EL2 structure and a first-order transition from the EL2 phase to the spiral order. The corresponding sequence is PM → ICS → EL2 → SP. This scenario is realized in XY HAFs on the stacked triangular lattice. 4,5 Notice that some fine details can be omitted in the picture just described. For instance, a small third harmonic of the modulation vector q can arise in Eq. (7) which leads to a weak temperature dependence of q in the ICS state as it was observed 16 in MnBr 2 . However we believe that apart from such fine details the above picture reflects all the possible phases and phase transitions which can arise in the considered model. Notice also that small anisotropic short-range spin interactions can complicate the above scenarios as it is demonstrated below by the example of MnI 2 .
III. SHORT-RANGE ANISOTROPIC SPIN INTERACTIONS
In this section, we discuss the possibility to describe at least qualitatively the influence of the long-range dipolar interaction by some short-range spin interactions. We show first that although dipolar forces act as a source of lowsymmetry biaxial anisotropy in a system, six scenarios of phase transitions discussed in Sec. II cannot be reproduced by the one-ion biaxial anisotropy of the form
E an = i E (s x i ) 2 − (s y i ) 2 − G(s z i ) 2 .(19)
Let us assume for definiteness that z is the easy axis and x is the hard one:
G > E > 0, δA = G − E.(20)
Particular analysis shows that the EL2 structure is always less energetically favorable than the EL1 state. Then, only relations between eigenvalues at q = q sin and the lowest eigenvalue among commensurate points λ 1 (q cs ) determine the sequence of phase transitions. By the energy reason, the modulation vector in the SP and in the EL1 phases should be equal to q sin . As a result, the system can follow three different scenarios.
(i) A "strong anisotropy" scenario is realized when T cs > T el1 (see Eqs. (13) and (15)) that reads as
λ 2 (q sin ) − λ 1 (q sin ) = δA > 2 1 + 2 3 (λ 1 (q cs ) − λ 1 (q sin ))(21)
(notice that T el1 is always larger than T sp given by Eq. (11) in the considered model with biaxial anisotropy (19)). Thus, scenario (i) described in Sec. II is realized.
(ii) A "moderate anisotropy" case implies 2(λ 1 (q cs ) − λ 1 (q sin )) < δA < 2 1 + 2 3 (λ 1 (q cs ) − λ 1 (q sin )) (22) that leads to the scenario (iii) described in Sec. II. Thus, the phase diagram very similar to that of MnWO 4 is obtained recently theoretically in Ref. 18 in a spin model containing single-ion anisotropy (19) and not containing the dipolar interaction.
(iii) A "weak anisotropy" case implies that
δA < 2(λ 1 (q cs ) − λ 1 (q sin )) ⇔ λ 1 (q sin ) + λ 2 (q sin ) 2 < λ 1 (q cs )(23)
and scenario (iv) described in Sec. II is realized. However the last first-order transition (from the EL1 to the SP phase) occurs at small temperature beyond the range of the mean-field theory validity: it follows from Eqs. (10) and (14) that F sp does not cross F el1 because
F sp − F el1 = (λ 2 (q sin ) − λ 1 (q sin )) 2 8BT > 0,(24)
where we replace q sp by q sin as it is noted above. The very existence of the transition from the EL1 to the SP phase follows from the fact that the SP state is stable at T = 0 in the considered "weak anisotropy" regime.
We point out that all six scenarios described in Sec. II can be obtained using a small anisotropic short-range exchange interaction of the form (cf. Eq. (19)) It happens because Fourier components of E ij and G ij become momentum-dependent that enriches the model behavior.
E an2 = 1 2 i,j E ij s x i s x j − s y i s y j − G ij s z i s z j .(25)
IV. PHASE TRANSITIONS IN MnI2
MnI 2 crystallizes in a hexagonal-layered structure shown in Fig. 2 with lattice parameters a = 4.146Å and c = 6.829Å. 8 Mn 2+ ions have spin S = 5/2 and g-factor g ≈ 2. Three successive phase transitions were identified upon temperature decreasing. 7 At T N 1 = 3.95 K, a second-order transition occurs from the paramagnetic state to the incommensurate sinusoidal phase with the modulation vector q sin = (0.1025, 0.1025, 0.5). At T N 2 = 3.8 K, a second-order transition occurs to another incommensurate sinusoidal phase in which the modulation vector moves continuously from q sin towards q sp = (0.181, 0, 0.439) upon temperature decreasing. At T N 3 = 3.45 K, a jump takes place to a proper screw helical order with the spiral vector q sp . Spins remain perpendicular to the modulation vectors at T < T N 1 . Then, in the helical phase, spins lie in a plane which is canted from the triangular basal abplane. One notes that a modified scenario (ii) described in Sec. II is realized in MnI 2 (as compared to scenario (ii), the additional transition arises in MnI 2 separating two ICS phases). We demonstrate below that small one-ion anisotropic interactions are responsible for this modification.
A. Basic equations
For the mean-field description of the successive phase transitions in MnI 2 , we use a model which is based on those proposed before for MnI 2 9 and for the isostructural compound MnBr 2 possessing a collinear low-temperature phase rather than the spiral one 16 . The latter model includes the magnetic dipole interaction, three in-plane exchange interactions and three exchange couplings between spins from neighboring planes (see Fig. 2). Notice that interaction J nnc is included because of its straight superexchange path via iodide atoms. This is the only exchange interaction which lowers the sixfold rotational symmetry around the c-axis to the threefold one. We take into consideration also small anisotropy terms which are allowed by symmetry: a single-ion easy-axis anisotropy, an in-plane hexagonal anisotropy, and DMI. DMI arises only in the spiral phase (which is ferroelectric in MnI 2 ) due to displacements of iodide atoms removing the inversion symmetry 11 (see also Appendix B).
The corresponding mean-field energy reads as
E = 1 2 i,j J ij (s i s j ) + 1 2 i,j D αβ ij s α i s β j − Y i (s z i ) 2 − Z i (s y i ) 2 (s y i ) 2 − 3(s x i ) 2 2 + E DM ,(26)
where the first two terms describe the exchange and the dipolar interactions, the third and the fourth terms are the one-ion and the sixfold in-plane anisotropies, respectively, the last term stands for the DMI energy which is discussed below in detail, and a Cartesian coordinate system is implied whose y and z axes coincide with crystallographic b and c ones (see Fig. 2
F = E + AT i s 2 i + BT i s 4 i + CT i s 6 i ,(27)
where A, B, and C are given by Eqs. (A7)-(A9). The Fourier transform of the exchange interaction has the form J q = 2 J 1 (cos q a + cos q b + cos(q a + q b )) + J 2 (cos 2q a + cos 2q b + cos 2(q a + q b )) + J c cos(q c ) + J ab (cos(2q a + q b ) + cos(q a + 2q b ) + cos(q a − q b )) + 2J nc cos q c (cos q a + cos q b + cos(q a + q b )) (28) + J nnc (cos(2q a + q b − q c ) + cos(q a + 2q b + q c ) + cos(q a − q b + q c )) .
B. Sinusoidal phases
As it is explained above, the transition takes place from the PM phase to the ICS one at T = T N 1 which is given by Eq. (6). Let us consider the spin ordering at T < T N 1 . It depends strongly on the values of the model parameters. However, the range of possible values of the exchange constants is reduced considerably by the requirement that J q should have a minimum at q ≈ q sin . Then, we try to reproduce the experimental data by slightly varying the exchange constants and including the small interactions. Our analysis shows that the behavior of λ 1 (q) and the corresponding eigenvector are quite simple at a moderate easy-axis anisotropy constant Y < ω 0 /2. Dipolar forces make the first eigenvector to be always perpendicular to q and to lie in the ab-plane. This finding is in agreement with experimental data observed in ICS phases 7 . The difference λ 1 (q) − J q is almost independent of q z and it depends slightly on the value of the q projection on the ab-plane. Then, at temperatures slightly below T N 1 , we obtain a spin texture of the form (7) with q = q sin and a 1 = s (1, −1, 0), a 2 = 0. The corresponding free energy is given by
F (1) ics = s 2 2 (λ 1 (q sin ) + AT ) + 3 8 BT s 4 − 5 16 Zs 2 y (s 2 y − 3s 2 x ) 2 + 5 16 CT s 6 .(29)
Notice that the last two terms are negligible in Eq. (29) at T ≈ T N 1 because they are of the sixth order in s. However they come into play at lower T upon s growing up. They are indispensable for the description of the experimentally obtained transition at T = T N 2 < T N 1 to another ICS phase in which the modulation vector q moves continuously from q sin towards q sp upon the temperature decreasing at T N 2 > T > T N 3 . The reason for this moving is simple: the sixfold anisotropy makes directions [100], [010], and [110] to be easy directions for the magnetization. On the other hand, s is directed along the hard [110] direction at T ≈ T N 1 . As a result, the magnetization (7) starts to rotate as it is shown in Fig. 3 from [110] to one of the nearest easy directions ([010] or [100]) at some temperature T = T N 2 < T N 1 , when the value of the third term in Eq. (29) becomes large enough. The second-order transition at T = T N 2 is related with the breaking of the two-fold rotational symmetry in the first ICS phase (the magnetization is directed along the twofold symmetry axis of the magnetic subsystem in the first ICS phase, as it is seen from Fig. 3). To demonstrate this, let us consider the correction δF to free energy (29) which arises due to small deviations of s and q from s sin and q sin , respectively,
s 2 (2Z + CT ) ,(30)
where s = s sin + δs, δs ⊥ s sin , |s sin | = s is given by Eq. (9), q = q sin + δq, and c 1,2,3 are some coefficients which are positive in MnI 2 . Minimization of Eq. (30) with respect of δq gives δq = δsc 2 /(c 1 s). Substituting the latter equality to Eq. (30), one finds that the coefficient before δs 2 becomes negative at T < T N 2 signifying the second-order transition at T = T N 2 , where
T N 2 ≈ T N 1 1 + 2B 2 (AT N 1 − κ) 5A 2 Z −1 ,(31)
κ = J qsin − c 3 − c 2 2 /c 1 and we neglect terms proportional to C which are negligible in MnI 2 as specific calculations show. The modulation vector q remains perpendicular to the magnetization in both ICS states in order to minimize the exchange and the dipolar energy.
C. The proper screw spiral phase
The first-order transition is observed in MnI 2 from the second ICS phase to the proper screw spiral phase. The plane in which spins lie in the SP phase does not coincide with the ab-plane. The free energy of this phase reads as
F sp = s 2 J(q) + 1 4 D αβ q v α sp v β * sp − Y 2 sin 2 θ + AT + BT s 4 − Zs 6 f (θ, ϕ) + CT s 6 + E DM ,(32)
where θ and ϕ are spherical angles determining the normal to the plane in which spins lie, v sp = (cos θ cos ϕ + i sin ϕ, cos θ sin ϕ − i cos ϕ, − sin θ), f (θ, ϕ) = (294 + 171 cos 2θ + 42 cos 4θ + 5 cos 6θ + 160 cos 6ϕ sin 6 θ)/1024, and the spin ordering of the form (7) is assumed with a 1 ⊥ a 2 and |a 1 | = |a 2 |. The easy-axis anisotropy Y produces the canting of the plane in which spins lie from the ab-plane (spins would lie in the ab-plane in the spiral phase if Y was zero). In Appendix B, we carry out a phenomenological consideration of DMI in MnI 2 based on available experimental data and show that E DM in Eq. (32) has the form
E DM = −2s 2 D sin √ 3 2 q x cos θ.(33)
D. Results of numerical calculations
We obtain the following set of parameters using which the above theory reproduces quantitatively almost all the essential features of phase transitions in MnI 2 :
J 1 = −0.13, J 2 = 0.1, J ab = −0.04, J c = 0.04, J nc = −0.0084, J nnc = 0.0036, Y = 0.05, Z = 0.015,(34)
where all values are in Kelvins. Eqs. (6) and (31) reproduce accurately transition temperatures to both ICS phases T N 1 = 3.95 K and T N 2 = 3.8 K. The trajectory of the modulation vector q in the second ICS phase is almost straight in the reciprocal space. It can be described as Unfortunately, the above formulas failed to describe quantitatively the experimentally observed first-order transition at T N 3 ≈ 3.45 K to the spiral phase which would take place as a result of the free energies F ics and F sp crossing. The reason is that our theory is actually based on the expansion in powers of s/S whereas this parameter reaches the value of 0.6 at T ≈ T N 3 . We find by minimizing energy (26) at T = 0 that the following set of parameters gives the proper screw spiral ordering with q sp = (0.166, 0, 0.428) (the latter is very close to the experimentally observed value of (0.181, 0, 0.439)):
q ≈ (1 − X(T ))q sin + X(T )q f ,(35)
where all values are in Kelvins. Notice that DMI plays a minor role in the stabilization of the experimentally observed spiral structure at small T .
V. SUMMARY AND CONCLUSION
To summarize, we discuss within the mean-field theory the impact of the dipolar interaction on critical properties of frustrated Heisenberg spiral antiferromagnets. We demonstrate that dipolar forces turn the single second-order temperature transition from the paramagnetic phase to the spiral one into a sequence of phase transitions of the first and of the second orders. We distinguish six possible scenarios of the successive phase transitions and possible intermediate phases which are summarized in Fig. 1. To the best of our knowledge, at least four of these scenarios were observed before experimentally in specific compounds (e.g., MnBr 2 , MnI 2 , MnWO 4 , and RbFeCl 3 ). We find that not all of these scenarios and intermediate phases can be obtained by replacing the long-range dipolar forces by one-ion anisotropy interactions. In contrast, all the essential features obtained can be reproduced qualitatively by proper short-range exchange anisotropy terms in the Hamiltonian.
We examine using the mean-field theory phase transitions in multiferroic MnI 2 showing incommensurate spiral ordering at T = 0. We reproduce quantitatively the majority of experimental findings observed in this compound. It is shown that the dipolar interaction plays the crucial role in producing the sequence of phase transitions found experimentally. However small symmetry-allowed short-range anisotropic interactions should be also taken into account which lead also to a modification of the corresponding scenario of phase transitions: the additional second-order transition arises separating two different ICS phases. infers that the inverse Dzyaloshinskii-Moriya mechanism is responsible for the electric polarization. 11 A contribution to the polarization from a couple of neighboring spins reads as 6
p ij ∝ e ij × [s i × s j ],(B1)
where e ij = r ij /r ij and r ij is a vector connecting sites i and j. Let us consider the grey iodide ion lying on the x axis shown in Fig. 5(a) and calculate contributions to the polarization p from three spin pairs adjacent to this iodide ion which are presented in Fig. 5(a). Using Eq. (B1), we find that one spin pair does not contribute to p because spins are collinear in this pair whereas one has for the rest two spin pairs
s i × s j = s 2 sin √ 3 2 q x n,(B2)
where n = (sin θ cos ϕ, sin θ sin ϕ, cos θ) is a unite vector that is normal to the plane in which spins lie. Then, one obtains from Eqs. (B1) and (B2) for the contribution to p related to one iodide ion p ∝ s 2 sin √ 3 2 q x ( √ 3/2, 1/2, 0) × n + ( √ 3/2, −1/2, 0) × n ∝ e y s 2 sin
√ 3 2 q x cos θ,(B3)
where e y is a unite vector directed along the y axis. Eq. (B3) is nonzero for the incommensurate proper screw spin helix whose plane is canted from the ab plane. It can be shown that Eq. (B3) is valid for all iodide ions. Thus, we obtain that Eq. (B1) describes correctly the direction of p observed experimentally in MnI 2 . Then, due to the C 2 symmetry of the y axis and the translational invariance, iodide ions should shift as it is shown in Fig. 5(a). As DMI vector D ij in DMI between a pair of spins related with one iodide ion is proportional to r ij × v (see Fig. 5(b)), impacts to D ij from two iodide ions (shown in Fig. 5 in grey and white) would cancel each other if there were no these displacements. Then, iodide ions shift produces the electric polarization and D ij ∝ r ij × e y which is parallel to the c axis. As a result, one comes to Eq. (33) for E DM .
PACS numbers: 75.30.-m, 75.30.Kz, 75.10.Jm, 75.85.+t
FIG. 1 :
1Possible sequences of temperature phase transitions in frustrated spiral Heisenberg antiferromagnets with dipolar forces. PM, ICS, CS and SP stand for the paramagnetic, incommensurate sinusoidally modulated, commensurate sinusoidally modulated and spiral phases, correspondingly. EL1 and EL2 are elliptical phases in which perpendicular spin components have different amplitudes and are modulated with the same (EL1) and with different (EL2) wave vectors. Transitions of the first and of the second order are shown by solid and by dashed lines, respectively. The transition from the EL1 phase to the SP one can be either of the first or of the second order depending on the model parameters (see the text).
FIG. 2 :
2Crystal structure of MnI2. Exchange interactions J are also shown.
FIG. 3 :
3), respectively. The characteristic dipolar energy in MnI 2 is ω 0 ≈ 0.31 K. Due to the sixfold anisotropy in Eq. (26), one has to expand the free energy F up to the sixth order in s with the Spin polarization s and the projection q ⊥ of the modulation vector on the ab-plane in two incommensurate sinusoidal phases of MnI2. At TN1 > T > TN2, q = qsin and s = ssin. At TN2 > T > TN3, s and q ⊥ rotate continuously upon the temperature decreasing from ssin and q ⊥ sin to s f and q ⊥ f , respectively. The clockwise rotation presented and the corresponding anticlockwise rotation are equally possible. The spin polarization remains perpendicular to the modulation vector in both incommensurate sinusoidal phases. result (cf. Eq.(4))
FIG. 4 :
4where q sin = (0.1025, 0.1025, 0.5) and q f = (0.167, 0, 0.442) are the initial and the final modulation vectors, correspondingly (seeFigs. 3 and 4). This behavior of q is in a good quantitative agreement with experimental data from Ref.7 . One finds for coefficients in Eqs. (30) and (31): c 1 ≈ 0.08 K, c 2 ≈ 0.045 K, c 3 ≈ 0.007 K, and κ ≈ 0.62 K. Plot of X(T ) which parametrizes the evolution of the modulation vector q upon the temperature decreasing in incommensurate sinusoidal phases of MnI2 (see Eq.(35)). At TN3 < T < TN2, the trajectory of q is almost straight in the reciprocal space which starts at qsin = (0.1025, 0.1025, 0.5) and finishes at q f = (0.167, 0, 0.442). Experimental data taken from Ref.7 are shown by circles.
J 1 =
1−0.105, J 2 = 0.095, J ab = −0.025, J c = 0.06, J nc = −0.0008, J nnc = 0.03, Y = 0.122, Z = 0.015, D = 0.001,
in the ferroelectric spiral phase in which the electric polarization is parallel to y axis. The projection of the spiral vector q ⊥ sp on the ab plane is shown which is directed along the x axis. White and gray big circles are iodide ions which lie above and below the ab plane, correspondingly (see alsoFig. 2). Shifts of iodide ions are depicted by arrows which are discussed in the text. (b) Illustration of how these shifts break the inversion symmetry between two manganese ions and lead to the Dzyaloshinskii-Moriya interaction.
. 4 3 . 5 3 . 6 3 . 7 3 . 8 3 . 9 4 . 0
AcknowledgmentsWe thank D.N. Aristov for discussion. This work is supported by Russian Science Foundation (grant No. 14-22-00281).Appendix A: Mean-field expansion of the free energyThe mean-field expansion of the free energy F in powers of s can be carried out in Heisenberg antiferromagnets with dipolar forces as it is done, e.g., in Ref.19. The effective mean-field Hamiltonian reads aswhere H i is the effective field (see Eq.(1))One obtains from the partition function Z = Sp e −H ef f /T for the magnetization at i-th sitewhere E is given by Eq. It is obtained experimentally that the phase with the spiral magnetic order is ferroelectric in MnI 2 .11The helical magnetic order breaks almost all symmetry elements: as soon as the in-plane projection of q sp is directed along the x axis, only the twofold rotational symmetry with respect to the y axis remains (seeFig. 5(a)). This symmetry element allows the electric polarization to be directed along the y axis. This conclusion is in agreement with the experimental observation of Ref.11. Bearing in mind also that the spin-orbit coupling leads to the ferroelectricity in MnI 2 , 9 one
* Electronic address: [email protected] † Electronic address: [email protected]. * Electronic address: [email protected] † Electronic address: [email protected]
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|
[] |
[
"On impedance conditions for circular multiperforated acoustic liners",
"On impedance conditions for circular multiperforated acoustic liners"
] |
[
"Kersten Schmidt \nFachbereich Mathematik\nTechnische Universität Darmstadt\nAG Numerik und Wissenschaftliches Rechnen\nDolivostrasse 1564293DarmstadtGermany\n",
"Adrien Semin \nBranderburgische\nInstitut für Mathematik\nTechnische Universität Cottbus-Senftenberg\nPlatz der Deutschen Einheit 103046CottbusGermany\n",
"Anastasia Thöns-Zueva \nInstitut für Mathematik\nTechnische Universität Berlin\nStraße des 17. Juni 13610623BerlinGermany\n",
"Friedrich Bake \nInstitute of Propulsion Technology\nGerman Aerospace Center\nMüller-Breslau-Straße 810623BerlinGermany\n"
] |
[
"Fachbereich Mathematik\nTechnische Universität Darmstadt\nAG Numerik und Wissenschaftliches Rechnen\nDolivostrasse 1564293DarmstadtGermany",
"Branderburgische\nInstitut für Mathematik\nTechnische Universität Cottbus-Senftenberg\nPlatz der Deutschen Einheit 103046CottbusGermany",
"Institut für Mathematik\nTechnische Universität Berlin\nStraße des 17. Juni 13610623BerlinGermany",
"Institute of Propulsion Technology\nGerman Aerospace Center\nMüller-Breslau-Straße 810623BerlinGermany"
] |
[] |
Background The acoustic damping in gas turbines and aero-engines relies to a great extent on acoustic liners that consists of a cavity and a perforated face sheet. The prediction of the impedance of the liners by direct numerical simulation is nowadays not feasible due to the hundreds to thousands repetitions of tiny holes. We introduce a procedure to numerically obtain the Rayleigh conductivity for acoustic liners for viscous gases at rest, and with it define the acoustic impedance of the perforated sheet.ResultsThe proposed method decouples the effects that are dominant on different scales: (a) viscous and incompressible flow at the scale of one hole, (b) inviscid and incompressible flow at the scale of the hole pattern, and (c) inviscid and compressible flow at the scale of the wave-length. With the method of matched asymptotic expansions we couple the different scales and eventually obtain effective impedance conditions on the macroscopic scale. For this the effective Rayleigh conductivity results by numerical solution of an instationary Stokes problem in frequency domain around one hole with prescribed pressure at infinite distance to the aperture. It depends on hole shape, frequency, mean density and viscosity divided by the area of the periodicity cell. This enables us to estimate dissipation losses and transmission properties, that we compare with acoustic measurements in a duct acoustic test rig with a circular cross-section by DLR Berlin.Conclusions A precise and reasonable definition of an effective Rayleigh conductivity at the scale of one hole is proposed and impedance conditions for the macroscopic pressure or velocity are derived in a systematic procedure. The comparison with experiments show that the derived impedance conditions give a good prediction of the dissipation losses.
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10.1186/s13362-018-0057-0
|
[
"https://arxiv.org/pdf/1801.04147v2.pdf"
] | 54,798,922 |
1801.04147
|
8c1614fb92d1a5855a88df25d2a46a4bd90d112e
|
On impedance conditions for circular multiperforated acoustic liners
Kersten Schmidt
Fachbereich Mathematik
Technische Universität Darmstadt
AG Numerik und Wissenschaftliches Rechnen
Dolivostrasse 1564293DarmstadtGermany
Adrien Semin
Branderburgische
Institut für Mathematik
Technische Universität Cottbus-Senftenberg
Platz der Deutschen Einheit 103046CottbusGermany
Anastasia Thöns-Zueva
Institut für Mathematik
Technische Universität Berlin
Straße des 17. Juni 13610623BerlinGermany
Friedrich Bake
Institute of Propulsion Technology
German Aerospace Center
Müller-Breslau-Straße 810623BerlinGermany
On impedance conditions for circular multiperforated acoustic liners
Acoustic linerPerforated platesMultiscale analysisRayleigh conductivityImpedance con- ditions MSC classification codes 35Q3035B2774Q1576M50
Background The acoustic damping in gas turbines and aero-engines relies to a great extent on acoustic liners that consists of a cavity and a perforated face sheet. The prediction of the impedance of the liners by direct numerical simulation is nowadays not feasible due to the hundreds to thousands repetitions of tiny holes. We introduce a procedure to numerically obtain the Rayleigh conductivity for acoustic liners for viscous gases at rest, and with it define the acoustic impedance of the perforated sheet.ResultsThe proposed method decouples the effects that are dominant on different scales: (a) viscous and incompressible flow at the scale of one hole, (b) inviscid and incompressible flow at the scale of the hole pattern, and (c) inviscid and compressible flow at the scale of the wave-length. With the method of matched asymptotic expansions we couple the different scales and eventually obtain effective impedance conditions on the macroscopic scale. For this the effective Rayleigh conductivity results by numerical solution of an instationary Stokes problem in frequency domain around one hole with prescribed pressure at infinite distance to the aperture. It depends on hole shape, frequency, mean density and viscosity divided by the area of the periodicity cell. This enables us to estimate dissipation losses and transmission properties, that we compare with acoustic measurements in a duct acoustic test rig with a circular cross-section by DLR Berlin.Conclusions A precise and reasonable definition of an effective Rayleigh conductivity at the scale of one hole is proposed and impedance conditions for the macroscopic pressure or velocity are derived in a systematic procedure. The comparison with experiments show that the derived impedance conditions give a good prediction of the dissipation losses.
Introduction
The safe and stable operation of modern low-emission gas turbines and aero-engines crucially depends on the acoustic damping capability of the combustion system components. Hereby, so called bias flow liner -consisting of a cavity and a perforated face sheet with additional cooling air flow -play a significant role. Since decades the damping performance prediction of these bias flow liner under all possible flow conditions remains a major challenge. However, due to the higher tendency of low-emission, lean burn combustion concepts for combustion instabilities the prediction of the acoustic bias flow liner impedance and therewith its damping performance is a very important prerequisite for the engine design process. Several analytical and semi-empirical models for the impedance description of bias flow liner were developed in the past (see also [1]). This work focuses on the numerical simulation of the acoustic characteristics of bias flow liner applying multi-scale modeling. In principal all theoretical approaches are based on the formulation of the Rayleigh conductivity K R [2,3] which describes the ratio of the fluctuating volume flow Q(t) through a hole to the driving pressure difference P − (t) − P + (t) across the hole:
K R := ρ 0 ∂ t Q(t) P − (t) − P + (t) ,(1)
and has the dimensions of length. One major challenge in the model description of the Rayleigh conductivity represents the definition or the specification of the pressure difference since, above and below the perforated liner face-sheet the pressure is not necessarily constant rather a function of the distance from the hole. Here, the present work applying a multiscale asymptotic model will provide an exactly defined solution. More precisely, the Rayleigh conductivity of a single hole in an array of holes is distributed over the whole liner area. In this way the effective Rayleigh conductivity
k R = K R A δ(2)
as quotient of the Rayleigh conductivity of one hole and the area A δ of one periodicity cell of the array is introduced that has the dimensions of one over length. Using the effective Rayleigh conductivity the liner impedance can be determined like later shown for example in equation (21).
Methods
We consider acoustic liner that consist of a wall or part of a wall with a periodic dense array of equisized and equishaped holes with an characteristic periodicity that is proportional to a small parameter δ > 0. The holes may not be of cylindrical shape and even tilted in general. For sake of simplicity we consider the perforated wall Ω δ liner with a circular cross-section of inner radius R d , while noting that the proposed procedure to define the Rayleigh conductivity and impedance conditions do not depend on the choice of the cross-section, but only on the hole pattern and hole shape and can be directly transfered to other cross-sections like rectangular.
To derive the impedance conditions we let the parameter δ of the hole period tending zero -so the number holes increases accordingly -while the inner and outer diameter of the cross section are scaled like δ 2 as well as the thickness of the perforated wall, see Fig. 1. As δ → 0, the holes merge and the domain Ω δ liner degenerates to an interface Γ liner on which we will prescibe the impedance conditions representing the correct disspation behaviour of the acoustic liner. For the circular liner the limit interface domain Γ liner is an cylinder of radius R d . As it simplifies the derivation and impedance condition greatly we assume that the area of the periodicity cell of the periodic array A δ = δ 2 .
This liner shall be embedded in a duct domain Ω and the computational domain is Ω δ := Ω \ Ω δ liner for every δ > 0, i.e., , the duct domain without the multi-perforated wall. On this domain we introduce as viscoacoustic model the linearized compressible Navier-Stokes equations in frequency domain in a uniform and stagnant media for a source term f (t, x) = Re(f (x) exp(−ıωt)) with an angular frequency ω > 0:
−ıωv δ + 1 ρ0 ∇p δ − ν(δ) ∆ v δ − ν (δ)∇ div v δ = f , in Ω δ , (3a) −ıωp δ + ρ 0 c 2 div v δ = 0, in Ω δ , (3b) v δ = 0, on ∂Ω δ ,(3c)
with the acoustic velocity v δ , the acoustic pressure p δ , the mean density ρ 0 > 0, the speed of sound c, the kinematic and secondary viscosities ν(δ), ν (δ) > 0. We scale the viscosities for δ → 0 like δ 4 such that the size of the viscous boundary layers remain asymptotically the same at the scale of a single hole. If the duct is modelled to be of infinite extend then additional conditions at infinity have to be posed, e.g., , for a channel of constant cross section with infinite extend in z ± ∞ these conditions are
lim z→±∞ p δ = 0. (3d)
Moreover, we assume the souce to be located away from the perforated wall such that f = 0 in a neighbourhood.
In the following section we study the solution of the viscoacoustic model in three different geometrical scales beginning at the scale of one hole, pursuing with the scale of one period of the hole array and concluding with the macroscopic scale on which the impedance conditions follow.
Microscopic scale: the near field around one hole
In the vicinity of one hole that tends to a point x Γ on the interface Γ liner we use the local coordinate X := (r, y, z) = ((r − R d )/δ 2 , rθ/δ 2 , z/δ 2 ). As δ → 0, the hole variable X occupies the whole unbounded domain Ω defined by (see Fig. 2a)
Ω = (r, y, z) ∈ R 3 such that r < 0 or r > h 0 } ∪ Ω hole(4)
where Ω hole is the scaled domain representing one hole, and we assume 0 ∈ Ω hole . For instance a vertical cylindrical hole of diameter d 0 δ 2 can be represented by Ω hole = {(r, y, z) ∈ R 3 such that 0 ≤ r ≤ h 0 and y 2 + z 2 < 1 4 d 2 0 }. Close to one hole of the perforated liner, we represent the solution (v δ , p δ ) of (3) as
v δ = v 0 (x Γ , X) + O(δ) , p δ = p 0 (x Γ , X) + δ 2 p 1 (x Γ , X) + O(δ 3 ) ,(5)
where the near field corrector terms (v 0 , p 0 , p 1 ) do not depend on δ. The scaling of the second corrector for the pressure as δ 2 is due to the associated scaling of the size of the holes. Now, inserting expansion (5) into the viscoacoustic model (3) and identifying formally terms of same powers of δ results first in the fact that p 0 (x Γ , X) is a constant function in X, and then in a product representation of the near-field corrector
(v 0 (x Γ , X), p 1 (x Γ , X)) = c(x Γ ) (ṽ(X),p(X)) ,
where c(x Γ ) allows for a slow variation of near field velocity and pressure along the wall.
The near field profiles (ṽ,p) are solution of the instationary Stokes problem (c) Near-field velocity for ω = 2π × 306 s −1 .
−ıωṽ + 1 ρ0 ∇p − ν 0 ∆ṽ = 0, in Ω,(6a)divṽ = 0, in Ω,(6b)v = 0, on ∂ Ω,(6c)Γ + (S) Γ − (S) h 0 d 0 e r Ω(S)
(d) Near-field velocity for ω = 2π × 30 s −1 . Table 1) at ω = 2π × 306 s −1 using S = 40. (c),(d) The near field velocity (imaginary part) for the same configuration as (b) and for ω = 2π × 30 s −1 .
Here, the color coding corresponds to the amplitude and the arrows to the direction of the velocity.
where ∇, div and ∆ are the gradient, divergence and Laplace operator in X (cf.
that act as a excitation from far away and will be used for the matching with the mesoscopic scale (see Sec. 2.2). Here, Γ ± (S) = (r, y, z) ∈ Ω, ±r > r ± and (r − r ± ) 2 + y 2 + z 2 = S 2 ,
with r − = 0 and r + = h 0 , are the two half-spheres (see Fig. 2a) that are moved towards infinity. Note that in problem (6) the term −ν 0 ∇ divṽ that would appear in the first line cancels out due to the divergence free condition (6b). Moreover, note that the term −ν 0 ∆ṽ can be replaced by ν 0 curl curlṽ and so only the vorticity part of the velocityṽ will exhibit a viscosity boundary layer as we will see later.
Problem (6) is a classical saddle-point problem and admits a unique solution stated by the following
Proposition 2.1. There exists a unique solution (ṽ,p) ∈ (H 1 ( Ω)) 3 × V( Ω) of (6), where V( Ω) = P ∈ H 1 loc ( Ω) such that ∇P L 2 ( Ω) < ∞ .
Note, that the pressure space V( Ω) allows for a constant behavior towards infinity.
With the near field velocity profileṽ defined by (6) we can define in analogy to the Rayleigh conductivity a posteriori the quantity
k R := lim S→∞ ıωρ 0 2 Γ+(S)ṽ · n − Γ−(S)ṽ · n(8)
using the volume flux towards infinity in a symmetric way. Here, n is the outer normal vector. In this way, the quantity k R is a mapping of a constant near field pressure at infinity to the flux at infinity. To see the analogy it suffices to consider time harmonic fields varying like exp(−ıωt), the volume flux Q(t) through the aperture counted positively along the direction of the e r axis to be the same as the volume flux through the surface Γ + (S) (respectively Γ − (S)), counted positively (resp. negatively) along the direction of the normal vector n, and to compare (1) and (8). Note, that the normal component of the near field velocity profile v decays like 1/S 2 towards infinity and combines different behaviour close to and away from the wall (see Fig. 2(c) and (d)). This behaviour can be rigorously justified with similar techniques as in [5,6].
For the usual definition of the Rayleigh conductivity K R it is not evident where the difference of the pressure -as it varies locally -and the volume flux -as in the original acoustic equations the fluid is compressible -shall be evaluated. The quantity k R is, however, clearly defined by (6) and (8) as the near field pressure tends to constant values for |X| → ∞ and as the near field velocity is incompressible. This results from the separation of the effects at the different length scales, namely viscous incompressible behaviour in the vicinity of the holes versus inviscid, compressible behaviour away from them, due to the asymptotic ansatz. As the near field profiles are defined in local coordinates X it has the dimensions of one over length and we denote it as effective Rayleigh conductivity of the liner.
The definition of the effective Rayleigh conductivity k R can be used for inviscid fluids as well for which ν 0 = 0 if the no-slip boundary conditions (6c) are replaced by v · n = 0.
Mesoscopic scale: the hole pattern
Pursuing with the scale of one period of the hole array and in the vicinity of one hole that tends to x Γ , we use the local coordinate X := (R, Y, Z) = ((r − R d )/δ, rθ/δ, z/δ). We consider for fixed δ > 0 the infinite periodicity cell
B δ = B δ + ∪ B δ − ∪ δ Ω hole (9) where B δ ± = (R, Y, Z) ∈ R 3 such that |Y −bZ| < √ a 2 , |Z| < 1 2 √ a , ±R > R δ ± with R δ − = 0, R δ + = h 0 δ
are two semi-infinite parallelepipeds whose opposite lateral faces |Z| = 1 2 √ a and |Y −bZ| = √ a 2 are considered to be identified with each other such that B δ ± and so B δ are topologically equivalent to a torus. With the cross-section of the periodicity cell
Γ(S) = {(R, Y, Z) ∈ R 3 such that |Y − bZ| < √ a 2 , |Z| < 1 2 √ a , R = S} e Z e Y √ a b 1/ √ a d 0 δ (a) view in (Y, Z) coordinates d 0 δ 1/ √ a h 0 δ e Z e R (b) view in (R, Z) coordinatesv δ , p δ ) of (3) as v δ = V δ 0 (x Γ , X) + O(δ) , p δ = P δ 0 (x Γ , X) + δP δ 1 (x Γ , X) + O(δ 2 )(10)
with X ∈ B δ . Inserting expansion (10) in problem (3) and identifying formally the terms of same powers of δ gives that P δ 0 (x Γ , X) is constant in X and a separation of variables for the mesoscopic corrector as
V δ 0 (x Γ , X), P δ 1 (x Γ , X) = c(x Γ ) V δ (X), P δ (X) with the mesoscopic profile (V δ , P δ ) satisfying the Darcy-type problem −ıωV δ + 1 ρ0 ∇P δ = 0, in B δ , div V δ = 0, in B δ , V δ · n = 0, on ∂B δ .(11)
Here, ∇ and div are the gradient and divergence in X. The formal identification of terms of same power in δ can be justified despite the fact that the size of the hole depends on δ as well. For this an additional scale η for the size of one hole has to be introduced that is first considered to be independent of δ due to its different meaning and later fixed to δ 2 . The expansion (10) is then in δ, where the terms of the expansion depend on η. For the brevity of the article we have chosen directly η = δ 2 . Note that (11) is equivalent to an homogeneous Laplace problem with Neumann boundary conditions for the pressure profile P δ , where the velocity profile V δ can be computed from. Following [7, Proposition 2.2], we can therefore state the following Proposition 2.2. For any fixed δ > 0, the kernel of problem (11) is of dimension 2 and spanned by the
functions (V δ N , P δ D ) = (0, 1) and (V δ D , P δ D ) such that D δ v is constant as R → ±∞. Moreover, there exists D δ ∞ ∈ C such that (V δ D , P δ D )
admits the following limit behaviour:
V δ D = 1 ıω e R + o(1), R → ±∞, P δ D = ρ 0 R ± D δ ∞ + o(1), R → ±∞.(12)
It remains to determine the constant D δ ∞ , where we are in particular interested in its asymptotic behaviour for δ → 0. To obtain this behaviour we will match the mesoscopic functions V δ D and P δ D with the near field profilesṽ andp at half-spheres Γ ± (s δ ) of radius s δ for √ δ < s δ < 2 √ δ centered at the aperture 0. First we note that due to the incompressibility and the limit behaviour of V δ D for its volume flux over the half-spheres it holds ıω 2
Γ+(s δ ) V δ D · n − Γ−(s δ ) V δ D · n = ıω 2 lim S→∞ Γ(S) V δ D · e R + Γ(−S) V δ D · e R = 1 .
Using this equality, definition (8) of the effective Rayleigh conductivity k R , the mesoscopic to microscopic variable change X = X/δ, and matching of the mesoscopic velocity V δ D and the near field velocity profilẽ v we find that
V δ D (X) ∼ ρ 0 k R δ 2ṽ ( X δ ) for √ δ < |X| < 2 √ δ and δ → 0 .
By linearity and using definition of problems (6) and (11), the gradient of the mesoscopic pressure P δ D can be matched with the gradient of the near field pressure profile as well. Integrating these gradients, using limit (6d) and Proposition 2.2 leads to
P δ D (X) ∼ ρ 0 k R δp ( X δ ) ∼ ± ρ 0 2k R δ for √ δ < |X| < 2 √ δ, ±R > 0 and δ → 0 . As for δ → 0 the mesoscopic pressure P δ D tends to ρ 0 R ± D δ ∞ if δ = o(|X|) we conclude that D δ ∞ = ρ 0 2k R δ + o(δ −1 ) .(13)
This blow up of the coefficient D δ ∞ as δ → 0 in accordance with its numerical computations based on an asymptotic analysis of (3) with only two scales [8], where the hole size is considered not as a scale but as a parameter.
Macroscopic scale and impedance conditions
Finally, away from a vicinity of the layer, the solution (v δ , p δ ) of (3) is represented by
v δ (x) = v 0 (x) + o(1), p δ (x) = p 0 (x) + o(1).(14)
Inserting this expansion in problem (3) and making a formal identification in terms of powers of δ gives that (v 0 , p 0 ) is solution of the classical Helmholtz problem
−ıωv 0 + 1 ρ0 ∇p 0 = f , in Ω \ Γ liner ,(15a)−ıωp 0 + ρ 0 c 2 div v 0 = 0, in Ω \ Γ liner ,(15b)
and a multiscale analysis [9] for rigid walls leads to the boundary conditions v 0 · n = 0, on ∂Ω .
The limit condition (3d) becomes
lim z→±∞ v 0 · (±1, 0, 0) − T 1 ± p 0 = 0,(15d)
where T 1 ± is the Dirichlet-to-Neumann operator based on the projection on the outgoing propagative modes, see [10, Eq. (2.7)] and [11]. This problem is completed by jump conditions across the interface Γ liner . To obtain the conditions we match the macroscopic pressure p 0 and flux v 0 · n in a matching zone at distance √ δ to the interface Γ liner to the mesoscopic pressure and velocity functions. For the pressure we find
p 0 (x) = C N (x Γ ) P δ N ( x−xΓ δ ) + δC D (x Γ ) P δ D ( x−xΓ δ ) for √ δ ≤ |x − x Γ | ≤ 2 √ δ and δ → 0. (16)
with two functions C N , C D that allow for slow variation along the perforated wall. With the factor δ the limit δP δ D ( x−xΓ δ ) for δ → 0 remains bounded. Subtracting the two limits of (16) for δ → 0 we obtain
p 0 (x Γ ) := lim δ→0 p 0 (x Γ + √ δn) − p 0 (x Γ − √ δn) = C D (x Γ ) ρ 0 k R .(17)
Taking the gradient in x on both sides of (16) and using (15a), the assumption that f = 0 close to the perforated wall and (11) we find
v 0 (x) · n = ρ 0 ıω ∇p 0 (x) · n = δC D (x Γ ) ρ 0 ıω ∇P δ D ( x−xΓ δ ) = C D (x Γ ) V δ D ( x−xΓ δ ) for √ δ ≤ |x − x Γ | ≤ 2 √ δ and δ → 0 .(18)
As the two limits for V δ D for R → ±∞ coincide we obtain v 0 · n (x Γ ) = 0, on Γ liner .
Finally, taking the average of (18) and the limit δ → 0 gives in view of (17) the impedance conditions
p 0 (x Γ ) = ıωρ 0 k R v 0 · n (x Γ ), on Γ liner .(19b)
Note, that the impedance conditions do not depend on the pattern of the holes, more precisely on the values a and b (see Fig. 1), but only on their area A δ , namely through ν 0 = ν/A 2 δ in the computation of the effective Rayleigh conductivity k R .
Distinguished limit Note, that the nature of the impedance condition (19b) is due to the choice of asymptotic scales. It represents a distinguished limit meaning that different choice would lead to one of the trivial conditions p 0 (x Γ ) = 0 (transparent wall) or v 0 ·n (x Γ ) = 0 (rigid wall). If we would scale the diameter of each hole like ε(δ) as well as the thickness of the perforated wall such that δ 2 = o(ε(δ)) then we would obtain transparent wall conditions in the limit δ → 0. A contrario, the impedance conditions become rigid wall conditions if we would use the scaling ε(δ) = o(δ 2 ). The choice of asymptotic scales was already stated in [12] for infinitely thin perforated wall and the Stokes flows.
Acoustic Impedance The nature of the impedance conditions is known in the literature: the notion of impedance can be found in the works of Webster in the 1910s [13]. More precisely, he defines the normalized specified acoustic impedance ζ by (note there is a complex conjugate and a different sign due to the different choice of the time-dependency convention)
ζ := − p 0 cρ 0 v 0 · n .(20)
For the derived impedance conditions (19b) and by identification, the normalized specified acoustic impedance for perforated walls is given by
ζ = ıω ck R = ıωk R c|k R | 2 .(21)
The resistance Re(ζ) and the reactance Im(ζ) are positive quantities when k R has a positive real part and a negative imaginary part. Moreover in the inviscid case k R is a positive real number, so that the normalized specified acoustic impedance ζ is purely a reactance.
Formulation in pressure only One can also remark that Problem (15) can be formulated in terms of pressure only: equations (15a)-(15d) give
∆p 0 + ω 2 c 2 p 0 = div f , in Ω \ Γ liner , ∇p 0 · n = 0, on ∂Ω, lim z→±∞ ±∂ z p 0 − ıωρ 0 T 1 ± p 0 = 0,(22a)
and impedance conditions (19a)-(19b) are written in terms of the pressure as
∇p 0 · n (x Γ ) = 0 and ∇p 0 · n (x Γ ) = k R p 0 (x Γ ) .(22b)
This kind of conditions were proposed for the inviscid case [14], where k R turns out to be the effective plate compliance.
Results and discussion
In this section, we are interested by the numerical computation of the effective Rayleigh conductivity k R , the computation of dissipation losses in acoustic ducts with the impedance conditions and comparison with data from experimental measurements.
Numerical computation of k R
The effective Rayleigh conductivity k R is defined through the solution of the near field velocity and pressure profiles in the unbounded domain Ω around a single hole. To compute k R numerically we truncate the unbounded domain, on which we use the finite element method for discretization and propose an extrapolation procedure to increase the accuracy.
of Ω for a given truncation radius S > 0.5 d 2 0 + h 2 0 (see Fig. 2(a)). It has two artificial boundaries Γ ± (S) that are no boundaries of Ω. We restrict the problem (6) to Ω(S) and ∂ Ω(S) ∩ ∂ Ω, and we approximate the conditions (6d) by settingp |Γ±(S) = ± 1 2 . From the resolution of the truncated problem we compute the approximated Rayleigh conductivity k R (S) taking as well an approximation of (8), namely
k R (S) := ıωρ 0 2 Γ+(S)ṽ · n − Γ−(S)ṽ · n ,(24)
Its approximated value k R (S) tends to the Rayleigh conductivity k R as 1/S as illustrated in Fig. 4. This first-order convergence can be explained with a rigorous analysis of the solution of problem (6) towards infinity using the Mellin transform [15] and showing that the solution of this problem on Γ ± (S) is a superposition of a radial expansion with respect to 1/S and of a cartesian expansion with terms decaying exponentially with respect to the distance to the boundary. Similar analyses were performed for the Poisson and Helmholtz problems in conical domains with a rough periodic boundary [5] or perforated wall [6]. As, more precisely, the Rayleigh conductivity k R can be expanded in powers of 1/S we use an extrapolation in 1/S of first order approximations k R (S) for different truncation radia S to obtain a second or higher order approximation of the limit value k R .
For the particular case of a straight cylindrical hole that is without loss of generality centered at y = z = 0, the domain Ω(S) is invariant under rotation around the r axis as well as the solution of the problem (6) for the near field profiles. Hence, the finite element method in two dimensions can be used Figure 4: Convergence of the real and imaginary parts of the approximated Rayleigh conductivity k R (S) to its limit value k R = 4.513 − 1.210ı in dependence of the truncation radius S for the liner DC006 at frequency f = 306 Hz (see Table 1). for the numerical resolution in a 2D axis-symmetry setting. To resolve the boundary layer of size ν 0 /ω on the wall boundary (cf. [4, Sec. 3.1]) we use the hp-adaptive strategy of Schwab and Suri [16] (see the mesh shown in Fig. 2(b)). For four liner configurations, see Table 1, from experimental studies [1,17] we have computed the near field velocity and pressure profiles and so the effective Rayleigh conductivity. The relative kinematic viscosity ν 0 is computed as quotient of the kinematic viscosity ν = 1.4660 × 10 −5 m 2 /s of air at 15 • C divided by the period δ to the power of four. In Fig. 2(b) and Fig. 2(c) we illustrate the near field pressure and velocity profilesp andṽ for the liner DC006 at frequency 306 Hz using a truncation radius S = 40. It is visible that the pressure decays almost linearly inside the cylindrical hole, but also the behaviour at distance to the hole. Moreover, the pressure shows close to the rim of the cylinder an edge singularity (i.e., a corner singularity for the 2D axis-symmetric problem) that is resolved numerically by the hp-adaptive refinement strategy. The near velocity profile shows a flux from all sides to and through the hole. It appears that the outward flux of the imaginary part ofṽ over Γ + (S) is negative (resp. positive over Γ − (S)) corresponding to a positive real part of the approximate Rayleigh conductivity k R (S) (see (24)) and so of the Rayleigh conductivity k R . This is in line with the inviscid case, where k R is real and positive. Moreover, we see the higher velocity amplitude inside the hole that decays towards its boundaries. This boundary layer phenomena is more visible for lower frequencies (see Fig. 2(d)), where one also see a local change of the velocity direction on the wall boundary.
S k R − k R (S) Re(k R − k R (S)) −Im(k R − k R (S))
In Fig. 5, we plot the effective Rayleigh conductivity k R as a function of the frequency f := ω 2π for different liner configurations given in Table 1. As expected, following the remark on the normalized specified acoustic impedance ζ, the real part of k R is positive and its imaginary part is negative. One can also remark that for liner configurations DC006 and DC007, that have a close value of the porosity σ but quite different hole repartition and hole diameter, their Rayleigh conductivities differ significantly in both their real and imaginary part. In Fig. 6, we show the computed normalized specific acoustic impedance ζ for the liner configuration DC006 in comparison with the Melling model (see [18] and [17,Eq. (12)]), that is given an analytic formula. For the latter an effective kinematic viscosityν(δ) := 2.179ν(δ) is used that shall incorporate also thermal conductivity losses near a highly conducting wall, see [19, p. 239] and [1, p. 62]. We plotted the Rayleigh conductivities obtained from our model with this effective kinematic viscosity. The reactance predicted by the two models are very close, where the resistence differs by up to 20%. The importance of taking the thermal conductivity losses into account will be seen in comparison with the measurements and be discussed later in Sec. 3.3.
Dissipation losses in acoustic ducts
Experimental Setup and Analysis
The experimental study is performed in the duct acoustic test rig with a circular cross-section (DUCT-C) at DLR Berlin at ambient conditions. The setup of the test rig is illustrated in Fig. 7. It allows high precision acoustic measurements of the damping performance of various liner configurations, including grazing and bias flow.
The test duct consists of two symmetric measurement sections (section 1 and section 2 in Fig. 7) of 1200 mm length each. They have a circular cross-section with a radius R d of 70 mm. In order to minimize the reflection of sound at the end of the duct back into the measuring section the test duct is equipped with anechoic terminations at both ends (not shown in Fig. 7). Their specifications follow the ISO 5136 standard. The damping module is a chamber of 60 mm. It has a circular cross-section with a radius of 120 mm. A total of 12 microphones are mounted flush with the wall of the test duct. They are installed at different axial positions upstream and downstream of the damping module and are distributed exponentially with a higher density towards the damping module. Two microphones are installed opposite of each other at the same axial position close to the signal source. As evanescent modes become more prominent in the vicinity of the source, their influence is reduced significantly by using the average value of these two microphones for the analysis. This technique helps to reduce the errors for frequencies approaching the cut-on frequency of the first higher order mode and thus, extends the frequency range for accurate results.
At the end of each section a loudspeaker is mounted at the circumference of the duct (A and B in Fig. 7). They deliver the test signal for the damping measurements. The signal used here is a multi-tone sine signal. All tonal components of the signal are in the plane wave range. The signal has been calibrated in a way that the amplitude of each tonal component inside the duct is about 102 dB.
The microphones used in these measurements are 1/4" G.R.A.S. type 40BP condenser microphones. Their signals are recorded with a 16 track OROS OR36 data acquisition system with a sampling frequency of 8192 Hz. The source signals for the loudspeakers are recorded on the remaining tracks. The test signal is produced by an Agilent 33220A function generator. The signals are fed through a Dynacord L300 amplifier before they power the Monacor KU-516 speakers.
For each configuration two different sound fields are excited consecutively in two separate measurements (index a and b). Speaker A is used in the first measurement and in the second measurement the same signal is fed into speaker B. Then, the data of section 1 and section 2 (index 1 and 2) are analyzed separately. This results in four equations for the complex sound pressure amplitudes for each section and measurement for j = a, b:p
1j (z) =p + 1j e ıωz/c +p − 1j e −ıωz/c (25a) p 2j (z) =p + 2j e ıω(z−L)/c +p − 2j e −ıω(z−L)/c (25b)
p + andp − are the complex amplitudes of the downstream and upstream traveling waves. The recorded microphone signals are transformed into the frequency domain using the method presented by Chung [20]. This method rejects uncorrelated noise, e.g., turbulent flow noise, from the coherent sound pressure signals. Therefore, the sound pressure spectrum of one microphone is determined by calculating the cross-spectral densities between three signals, where one signal serves as a phase reference. In our case the phase reference signal is the source signal of the active loudspeaker. As a result we obtain a phase-correlated complex sound pressure spectrum for each microphone signal.
According to Eqs. (25a)-(25b) the measured acoustic signal is a superposition of two plane waves traveling in opposite direction. In order to determine the downstream and upstream propagating portions of the wave in each section, a mathematical model is fitted to the acoustic microphone data. This model Figure 8: Illustration of the sound filed in the duct for measurements A and B by means of the sound pressure amplitudesp, the reflection coefficient r, the transmission coefficient t, and the end reflection r e considers viscous and thermal conductivity losses at the duct wall. They are included in the wave number with the following attenuation factor α as proposed by Kirchhoff [21]:
Section 1 Section 2 p + 1â p − 1ap + 2â p − 2a r + r − t + t − r + e r − e A (a) Measurement A, upstream excitation Section 1 Section 2 p + 1b p − 1bp + 2b p − 2b r + r − t + t − r + e r − e B (b) Measurement B, downstream excitationα = 1 cR d νω 2 1 + γ − 1 √ P r(26)
with the duct radius r, the speed of sound c, the kinematic viscosity ν, the angular frequency ω (as in Eq. (3)), the heat capacity ratio γ, and the Prandtl number P r. As a result of this least-mean-square fit, the four complex sound pressure amplitudesp + 1 ,p − 1 ,p + 2 andp − 2 are identified at position z = 0 for both measurements. These sound pressure amplitudes are related to each other via the reflection and transmission coefficients of the test object. This is illustrated in Fig. 8 for the two different measurements A and B. In order to calculate the reflection and transmission coefficients r + , r − , t + , and t − from the sound pressure amplitudes the following four relations can be derived for j = a, b:
p − 1j = r +p+ 1j + t −p− 2j (27a) p + 2j = r −p− 2j + t +p+ 1j (27b)
The equations from both measurements are combined and solved for the reflection
r + =p − 1ap − 2b −p − 1bp − 2â p + 1ap − 2b −p + 1bp − 2a r − =p + 2bp + 1a −p + 2ap + 1b p + 1ap − 2b −p + 1bp − 2a(28)
and transmission coefficients
t + =p + 2ap − 2b −p + 2bp − 2â p + 1ap − 2b −p + 1bp − 2a t − =p + 1ap − 1b −p + 1bp − 1â p + 1ap − 2b −p + 1bp − 2a(29)
in downstream and upstream direction, respectively. The advantage of combining the two measurements is that the resulting coefficients are independent from the reflection of sound at the duct terminations. These end-reflections are contained in the sound pressure amplitudes, but do not need to be calculated explicitly. Moreover in the case of a uniform and stagnant flow these coefficients do not depend on the direction we consider, i.e., r − = r + and t − = t + . The dissipation of acoustic energy is expressed by the dissipation coefficient. The dissipation coefficient ∆ can be calculated directly from the reflection coefficient R and the transmission coefficient T via an energy balance R ± + T ± + ∆ ± = 1.
To compute these coefficients, the integration of the acoustic energy flux in a uniform and stagnant flow yields a relation between the acoustic pressure p and acoustic power P quantities(see Blokhintsev [22] and Morfey [23]) :
P ± = πR 2 d 2ρ 0 c p ± 2(31)
Then, the energy coefficients can be given relative to the pressure coefficients as:
R + = P − 1 P + 1 = r + 2 (32a) R − = P + 2 P − 2 = r − 2 (32b) T + = P + 2 P + 1 = t + 2 (32c) T − = P − 1 P − 2 = t − 2 (32d)
where the indices 1 and 2 refer to section 1 and section 2 of the duct as illustrated in Fig. 8. With the energy balance (30) follows the definition of the energy dissipation coefficient
∆ = 1 − r ± 2 + t ± 2(33)
This is an integral value of the acoustic energy that is absorbed while a sound wave is passing the damping module. The dissipation coefficient is used to evaluate the damping performance of the test object.
Numerical simulation of dissipation losses
This setup is also simulated numerically using the equivalent problem (22a)-(22b) for the pressure with a source term corresponding to an incoming field p inc (r, θ, z) = exp(ıωz/c) from the left. The scattered field is computed numerically using the mode matching procedure with N = 5 modes [24]: we seek for the scattered field p 0 under the form (see Fig. 9(b)) p 0 (r, θ, z) = p inc (r, θ, z) +
N −1 j=0 α − j ψ j (r) exp(−ı β j z), z < 0,(34a)p 0 (r, θ, z) = N −1 j=0 α + j ψ j (r) exp(ı β j z), z > L,(34b)
inside the waveguide part, and under the form
p 0 (r, θ, z) = 2N −1 j=0 ψ j (r) α j + exp(ı β j z) + α j − exp(ı β j (L − z)) , 0 < z < L,(34c)
inside the duct part. The pairs (β j , ψ j ) and (β j , ψ j ) are solution of a "2D" transverse eigenvalue problem in the wave-guide and liner parts, using the fact that the source term p inc and the geometry are independent of the angle θ. From the mode matching and assuming that there is only one propagative mode inside the waveguide, i.e., β j ∈ ıR for j = 0, the energy dissipation coefficient is computed as
∆ := 1 − |α + 0 | 2 + |α − 0 | 2 ,(35)
and corresponds to the energy dissipation coefficient D ± (Eq. (33)) if both grazing and bias flows are absent.
(a) Figure 9: Split of the domain Ω into the two semi-infinite waveguides Ω ± and the multi-perforated liner section Ω c . Impedance transmission conditions on the interface Γ liner approximate the behaviour of the many perforations.
Ω − ∂Ω wall − ∂Ω wall − Ω + ∂Ω wall + ∂Ω wall + Ω c ∂Ω wall c ∂Ω wall c Γ liner Γ liner Γ − Γ + Γ wall + Γ wall − Γ wall + Γ wall − (b)
Numerical results and comparison with experimental data
Conclusions
It has been shown that impedance conditions with one numerically computed parameter -the effective Rayleigh conductivity -can predict well the dissipation losses of acoustic liners. The effective Rayleigh conductivity can be obtained by solving numerically an instationary Stokes problem in frequency domain of one hole with a scaled viscosity in an characteristic infinite domain with prescribed pressure at infinity. For the computation the infinite domain is truncated, where we propose approximative boundary conditions on the artificial boundaries and an extrapolation procedure to save computation time. We decoupled in a systematic way the effects at different scales and derived impedance conditions for the macroscopic pressure or velocity based on a proper matching of pressure and velocity at the different scales. In difference to a direct numerical solution for the acoustic liner the overall computation effort is separated into a precomputation of the effective Rayleigh conductivity and a computation of the Helmholtz equation for the pressure with impedance conditions, where no holes have to be resolved anymore by a finite element mesh. The comparison with measurements in the duct acoustic test rig with a circular cross-section at DLR Berlin show that the dissipation losses based on the impedance conditions with effective Rayleigh conductivity are well predicted. The derivation of the impedance conditions do not depend on the cylindrical shape of the liner and can be used for others shapes like rectangular profiles. The procedure for the computation of the effective Rayleigh conductivity can not only be extented to include thermic effects that are currently only heuristically incorporated, but also nonlinear effects inside the hole that lead to an interaction of frequencies.
Figure 1 :
1(a) Simplified geometry of a combustion liner for acoustic studies. (b) flattened liner. (c) view along a cross-section.
Figure 2 :
2(a) Computational domain for the near field problem around a single hole. (b) The near field pressure (real part) for the liner configuration DC006 (see
[ 4 ,
4Sec. 2.1.6] in timedomain). The near field velocity profile is incompressible on the scale of one hole and fulfills together with the near field pressure profile the Stokes equations with an at the scale of one hole significant viscosity ν 0 and the additional term −iωṽ that reflects a time shift between excitation and excited fields. These equations are completed by Dirichlet jump conditions at infinity lim S→∞p |Γ±(S) = ± 1 2 ,
Figure 3 :
3Representation of the periodicity cell B δ associated to the intermediate scale the symmetric difference A B := (A ∪ B) \ (A ∩ B) the boundary of the periodicity cell is given as ∂B δ = (Γ(h 0 δ) ∪ Γ(0)) δ∂ Ω hole . It consists of the wall boundary and the boundary of the hole. The periodicity cell B δ degenerates as δ → 0 and tends to the union B 0 of two semi-infinite parallelepipeds B 0 ± connected by the point 0, an infinitely small hole. Inside the periodic array of holes, we represent the solution (
First, we define the truncated domain Ω(S) = Ω ∩ max(|X − (0, 0, 0)|, |X − (h 0 , 0, 0)|) < S
1 :
1Liner configurations. The length of the liner is L = 60 mm. The value of the viscosity is ν(δ) = 1.4660 × 10 −5 m 2 /s. For all these configurations b = 0.5 √ a.
Figure 5 :
5Real and imaginary parts of k R in dependence of the frequency f =
Figure 6 :
6Comparison of the impedance with the normalized specific resistance (left) and the normalized specific reactance (right) computed by our model and Melling model as function of the frequency f = ω 2π for the liner configuration DC006. As Melling model uses an effective viscosity taking into account thermal conductivity losses we show the impedance for our model with the effective viscosity as well.
Figure 7 :
7Schematic setup of the Duct Acoustic Testrig (DUCT-C) with speakers A and B, and microphones 1-12. The anechoic terminations at both ends are not shown.
Figure 10
10shows the average dissipation of the different liner configurations (see table 1) in the DUCT-C setup (seefigure 8) as a function of the frequency. The average dissipation represents a mean value of the dissipation results for the upstream and downstream acoustic incidence (see section 3.2.1). In a symmetric setup and without grazing flow this is, of course, equal to the dissipation from either side of excitation. The graphs compare the experimental values (symbols), the former theoretical model from Melling[18] (dashed lines) and the here introduced asymptotic model (solid lines). In result, the asymptotic model indicates a better comparison to the experimental values especially for the configurations DC006 (figure 10 (a)) and DC008 (figure 10 (c)) where the Melling model slightly underestimates the dissipation in the frequency range above approximately 400 Hz. For the configuration DC007 with a porosity of 1.0 % and a hole diameter of 2.5 mm both models (Melling and asymptotic) underestimate the maximum dissipation of approximately 0.4 around 400 Hz revealed in the experimental studies.
Figure 10 :
10Average dissipation from experiments and numerical modelling plotted over the frequency comparing models for DC006, DC007, DC008, DC009.
Table
AcknowledgementsThe authors would like to thank Claus Lahiri (Rolls-Royce) for fruitful discussions.The research was supported by Einstein Center for Mathematics Berlin via the research center MATH-EON, Mathematics for Key Technologies, in Berlin as well as the Brandenburgische Technische Universität Cottbus-Senftenberg through the Early Career Fellowship of the second author.The research was partly conducted during the stay of the first and second author at the TU Berlin and the first author at BTU Cottbus-Senftenberg.
Acoustic performance of bias flow liners in gas turbine combustors. C Lahiri, Berlin, GermanyTechnische Universität BerlinPhD thesisLahiri, C.: Acoustic performance of bias flow liners in gas turbine combustors. PhD thesis, Technische Universität Berlin, Berlin, Germany (2014). https://depositonce.tu-berlin.de/handle/11303/ 4567
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On the homogenization of the Helmholtz problem with thin perforated walls of finite length. A Semin, B Delourme, K Schmidt, ESAIM: Math. Model. Numer. Anal. Accepted for publicationSemin, A., Delourme, B., Schmidt, K.: On the homogenization of the Helmholtz problem with thin perforated walls of finite length. ESAIM: Math. Model. Numer. Anal. (2017). Accepted for publication.
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On the homogenization of the acoustic wave propagation in perforated ducts of finite length for an inviscid and a viscous model. A Semin, K Schmidt, SubmittedSemin, A., Schmidt, K.: On the homogenization of the acoustic wave propagation in perforated ducts of finite length for an inviscid and a viscous model (Submitted)
Asymptotic analysis for acoustics in viscous gases close to rigid walls. K Schmidt, A Thöns-Zueva, P Joly, Math. Models Meth. Appl. Sci. 249Schmidt, K., Thöns-Zueva, A., Joly, P.: Asymptotic analysis for acoustics in viscous gases close to rigid walls. Math. Models Meth. Appl. Sci. 24(9), 1823-1855 (2014)
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Acoustic fluid flow through holes and permeability of perforated walls. J Sanchez-Hubert, E Sánchez-Palencia, J. Math. Anal. Appl. 872Sanchez-Hubert, J., Sánchez-Palencia, E.: Acoustic fluid flow through holes and permeability of perforated walls. J. Math. Anal. Appl. 87(2), 427-453 (1982)
Acoustical impedance and the theory of horns and of the phonograph. A G Webster, Proc. Nat. Acad. Sci. 57Webster, A.G.: Acoustical impedance and the theory of horns and of the phonograph. Proc. Nat. Acad. Sci. 5(7), 275-282 (1919)
Mathematical justification of the rayleigh conductivity model for perforated plates in acoustics. A Bendali, M Fares, E Piot, S Tordeux, SIAM J. Numer. Anal. 731Bendali, A., Fares, M., Piot, E., Tordeux, S.: Mathematical justification of the rayleigh conductivity model for perforated plates in acoustics. SIAM J. Numer. Anal. 73(1), 438-459 (2013)
Elliptic Boundary Value Problems in Domains with Point Singularities. V A Kozlov, V G Maz Ya, J Rossmann, Mathematical Surveys and Monographs. 52American Mathematical SocietyKozlov, V.A., Maz ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. Mathematical Surveys and Monographs, vol. 52. American Mathematical Society, Providence, RI (1997)
The p and hp versions of the finite element method for problems with boundary layers. C Schwab, M Suri, Math. Comp. 65216Schwab, C., Suri, M.: The p and hp versions of the finite element method for problems with boundary layers. Math. Comp. 65(216), 1403-1430 (1996)
A review of bias flow liners for acoustic damping in gas turbine combustors. C Lahiri, F Bake, J. Sound Vib. 400Lahiri, C., Bake, F.: A review of bias flow liners for acoustic damping in gas turbine combustors. J. Sound Vib. 400, 564-605 (2017)
The acoustic impendance of perforates at medium and high sound pressure levels. T H Melling, J. Sound Vib. 291Melling, T.H.: The acoustic impendance of perforates at medium and high sound pressure levels. J. Sound Vib. 29(1), 1-65 (1973)
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Über den Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung. G Kirchhoff, Ann. Phys. Chem. 2106Kirchhoff, G.:Über den Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung. Ann. Phys. Chem. 210(6), 177-193 (1868)
D I Blokhintsev, Acoustics of a Nonhomogeneous Moving Medium. NACA Technical Memorandum 1399. Washington, DCOriginally published 1946 in russian languageBlokhintsev, D.I.: Acoustics of a Nonhomogeneous Moving Medium. NACA Technical Memorandum 1399, Washington, DC (1956). Originally published 1946 in russian language
Acoustic energy in non-uniform flows. C L Morfey, J. Sound Vib. 142Morfey, C.L.: Acoustic energy in non-uniform flows. J. Sound Vib. 14(2), 159-170 (1971)
Simulation of reflection and tranmission properties of multiperforated acoustic liners. A Semin, A Thöns-Zueva, K Schmidt, Prog. Ind. Math. ECMI. Accepted for publicationSemin, A., Thöns-Zueva, A., Schmidt, K.: Simulation of reflection and tranmission properties of multiperforated acoustic liners. In: Prog. Ind. Math. ECMI 2016 (2017). Accepted for publication.
|
[] |
[
"Metallic state in bosonic systems with continuously degenerate minima",
"Metallic state in bosonic systems with continuously degenerate minima"
] |
[
"Shouvik Sur \nNational High Magnetic Field Laboratory\nFlorida State University\n32306TallahasseeFloridaUSA\n",
"Kun Yang \nNational High Magnetic Field Laboratory\nFlorida State University\n32306TallahasseeFloridaUSA\n"
] |
[
"National High Magnetic Field Laboratory\nFlorida State University\n32306TallahasseeFloridaUSA",
"National High Magnetic Field Laboratory\nFlorida State University\n32306TallahasseeFloridaUSA"
] |
[] |
A continuously degenerate minima of the single particle dispersion is realized in the presence of an isotropic spin-orbit coupling above one dimension. The unit codimension of the dispersion-minima leads to a divergent density of states which enhances the effects of interactions, and may lead to novel states of matter as exemplified by Luttinger liquids in one dimensional bosonic systems. Here we show that in dilute homogeneous bosonic systems above one dimension, weak inter-particle interaction stabilizes an analogue of Luttinger liquids in the presence of isotropic spin-orbit coupling at zero temperature. In this metallic phase the system possesses a quasi long-range order with non-universal scaling exponents. At fixed mean density, strengthening of either the spin-orbit coupling or the interaction destabilizes the metallic state towards charge density wave states. The magnitude of the wavevector of the dominant charge density wave state is controlled by the ratio of the mean density of bosons and the strength of spin-orbit coupling. We obtain the zero temperature phase diagram, and identify the phase boundary. arXiv:1803.05839v1 [cond-mat.quant-gas]
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10.1103/physrevb.100.024519
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[
"https://arxiv.org/pdf/1803.05839v1.pdf"
] | 55,576,966 |
1803.05839
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601f0249b35f478f07ef43d0cff3c8c28735a518
|
Metallic state in bosonic systems with continuously degenerate minima
Shouvik Sur
National High Magnetic Field Laboratory
Florida State University
32306TallahasseeFloridaUSA
Kun Yang
National High Magnetic Field Laboratory
Florida State University
32306TallahasseeFloridaUSA
Metallic state in bosonic systems with continuously degenerate minima
(Dated: March 16, 2018)
A continuously degenerate minima of the single particle dispersion is realized in the presence of an isotropic spin-orbit coupling above one dimension. The unit codimension of the dispersion-minima leads to a divergent density of states which enhances the effects of interactions, and may lead to novel states of matter as exemplified by Luttinger liquids in one dimensional bosonic systems. Here we show that in dilute homogeneous bosonic systems above one dimension, weak inter-particle interaction stabilizes an analogue of Luttinger liquids in the presence of isotropic spin-orbit coupling at zero temperature. In this metallic phase the system possesses a quasi long-range order with non-universal scaling exponents. At fixed mean density, strengthening of either the spin-orbit coupling or the interaction destabilizes the metallic state towards charge density wave states. The magnitude of the wavevector of the dominant charge density wave state is controlled by the ratio of the mean density of bosons and the strength of spin-orbit coupling. We obtain the zero temperature phase diagram, and identify the phase boundary. arXiv:1803.05839v1 [cond-mat.quant-gas]
I. INTRODUCTION
In the absence of disorder, weakly interacting bosons have a strong tendency of Bose condensing, resulting in a superfluid ground state that spontaneously breaks the global U(1) symmetry associated with particle number conservation [1]. Stronger interaction can result in a number of different phases. The simplest is a trivial Mott insulator phase that does not break any symmetry, and is possible only when bosons are loaded in a preexisting lattice with integer lattice filling [2,3]. More generally insulating states of bosons result from spontaneously breaking either continuous or discrete translation symmetry. A tantalizing possibility, namely co-existing spontaneously broken translation and U(1) symmetries that result in a supersolid phase, has been discussed extensively theoretically and explored experimentally [4][5][6]. Thus other than the trivial Mott insulator phase, all known phases formed by interacting bosons break one or more symmetries spontaneously above one-dimension (1D).
The situation is quite different in 1D. Due to enhanced fluctuations, spontaneously broken continuous symmetry is forbidden, and consequently neither superfluid nor crystalline states are stable, and the generic state is a critical metallic phase, known as the Luttinger liquid, with power-law decay of both superfluid and crystalline order parameters [7]. A long-standing question in condensed matter physics is if such a metallic phase is possible above 1D. The purpose of the present paper is to show that the answer is in the affirmative for weakly interacting dilute bosonic systems in the presence of a specific type of spin-orbit coupling (SOC).
The role of SOC in determining the properties of matter has been extensively investigated in solid state systems [8]. Owing to a dearth of naturally occurring SOC coupled bosonic systems, the effect of SOC in determining properties of interacting bosons received a significant impetus only after the advent of ultracold atom systems where synthetic SOC in bosonic systems became accessible [9,10]. Indeed a recent surge of theoretical investigations into SO coupled bosonic systems has predicted multiple novel many-body states in both trapped and homogeneous systems, including manybody 'cat'
FIG. 1: The single particle dispersion in the absence (dashed curve) and presence (solid curves) of spin-orbit coupling. In the absence of spin-orbit coupling the spectrum is doubly degenerate. The spin-orbit coupling lifts the degeneracy, and leads to an upper and a lower [given by E(K) in Eq. (2)] branch. Here we focus on the lower branch by considering energies that are smaller than the spin-orbit energy scale, E κ (dotted line). For a weakly interacting dilute system perturbation theory breaks down below a scale E λ (dot-dashed line). We utilize a bosonization based method to access the physics below E λ .
states [11], states with stripe order [12], composite fermion liquid [13], and various vortex states [14][15][16].
In this paper we focus on a homogeneous, interacting system of Rashba SO coupled pseudospin- 1 2 bosons. The pseudospin degree of freedom is associated with internal levels of an atom. An SOC between these pseudospin states is generated by dressing them with photons through the Raman effect [17]. In these experimental setups an anisotropic SOC, which may be interpreted as an equal mixture of Rashba and Dresselhaus SOC, is more readily generated, and it leads to a doubly degenerate dispersion minima along the direction of the counter-propagating laser beams. Therefore, for their immediate experimental relevance, systems with such extremely anisotropic SOC have been extensively investigated [18]. It is comparatively more complicated to realize an isotropic SOC with only Rashba or Dresselhaus terms due to the higher sym-FIG. 2: The momentum scales in the low energy model. The solid (blue) circle represents the ring shaped dispersion minima of radius κ (see Fig. 1). The effective theory in Eq. (8) is defined below an energy cutoff, E λ , which is represented by the momentum scale λ.
metry of the SOC [19][20][21][22][23][24]. In the presence of isotropic SOC, however, a qualitatively novel situation arises with the dispersion obtaining a continuously degenerate minima along a ring in d = 2 (Rashba SOC), and a spherical shell in d = 3 (Weyl SOC) [25]. For concreteness let us consider a non-relativistic Hamiltonian of N bosons in two space dimensions, in the presence of Rashba SOC (we choose units in which = 1),
H = N n=1 − ∇ 2 n 2m + E κ σ 0 − i κ m σ · ∇ n + V 0 N n>m=1 δ(r n − r m ),(1)
where r n denotes position of the n-th boson, m is the mass of a boson, κ m is the SOC strength, E κ ≡ κ 2 2m is the energy scale associated with SOC, σ 0 is the 2 × 2 identity matrix, and σ ≡ (σ x , σ y ) are Pauli matrices. The σ µ matrices act on the pseudospin degree of freedom. Here we have shifted the single particle energies by E κ to make the energy eigenvalues positive semi-definite. The SOC removes the twofold degeneracy of the single-particle spectrum (dashed curve in Fig. 1), and leads to two distinct branches (solid curves in Fig. 1). The lower branch disperses as,
E(K) = 1 2m (|K| − κ) 2 ,(2)
where K is two dimensional momentum. We note that the parameter κ corresponds to the radius of the ring over which the energy is minimized (see Figs. 1 and 2). In order to explore the low energy properties of the system, we deduce an appropriate effective theory that governs the long wavelength behavior from Eq. (1) in Section II. The quadratically dispersing lower branch and a single codimension to the zero-energy manifold results in the single particle density of states to diverge as 1/ E(K) on approaching the ring-minima. This is reminiscent of onedimensional bosonic systems, and the origin of the enhanced fluctuations that suppress superfluid order. Therefore, the interaction among bosons is expected to play a key role in determining the fate of the system. It is, however, challenging to systematically account for the effects of interaction, since a perturbative expansion near the non-interacting limit of a bosonic system is ill-defined. Thus, non-perturbative methods such as variational wave functions [11,14,26], meanfield theory, and transfer-matrix [27] have been employed in exploring the low-energy phase diagram. A general conclusion from these calculations is that the system spontaneously breaks a symmetry. Since methods based on variational wavefunctions and mean field theories are a priori biased towards specific states, usually with fixed patterns of symmetry breaking, it is unclear whether the ground state must necessarily break a symmetry everywhere on the phase diagram.
In this work, we utilize the analogy with one dimensional bosonic systems to develop a high-dimensional generalization of (1D) bosonization (as discussed in Section III), which is a non-perturbative method that does not assume a broken symmetry state. Using this multidimensional bosonization, we show that weak repulsive interactions can take advantage of the divergent single-particle density of states to stabilize a metallic state that resembles a Luttinger liquid. As the interaction strength increases the metal becomes unstable against various charge density wave (CDW) states. Our method allows for an unbiased analysis of these competing instabilities based on scaling analysis, and finds the leading instability is toward a CDW state with a wavevector of magnitude 2πρ 1D , where ρ 1D ≡ρ/κ withρ being the mean density of the bosons. We obtain the phase diagram in Fig. 3, and provide details of its determination in Sections IV and V.
II. LOW ENERGY EFFECTIVE THEORY
In this section we introduce a low energy effective model, that is appropriate for a system of dilute, weakly interacting bosons in two space dimensions in the presence of Rashba SOC. In order to construct the effective model we consider a homogeneous system in the thermodynamic limit from the perspective of effective field theory, and focus on universal features. Since the ring-minima plays a progressively important role as the energy of the system, E, decreases with respect to E κ , we project the dynamics to the lower branch by integrating out all modes that carry energy above an effective ultraviolet (UV) energy scale, E λ ≡ λ 2 2m with λ being a momentum scale. These high energy modes can be separated into two regions, E > E κ and E κ > E > E λ , with the latter corresponding to momenta lying in the region, Fig. 4). We assume that the bare interaction is weak enough to allow us to (i) completely ignore the renormalizations produced by the modes at energies E > E κ ; (ii) find a small enough λ to enable modes at energies E κ > E > E λ to produce pertubatively small but finite renormalizations.
R λ = {K | |K| < κ − λ} ∪ {K | 2κ > |K| > κ + λ} (see
In order for the effective theory with cutoff λ to be weakly renormalized compared to the bare theory, the quantum corrections generated by integrating out high-energy modes must be small compared to the bare parameters. As shown in Appendix A the quantum corrections produced by the one-loop processes in Fig. 5 are on the order of mκV 2 0 λ −1 . Therefore, the requirement of weak renormalization implies, λ mκV 0 which leads to,
λ = B λ mκV 0 ,(3)
with B λ 1 being a positive constant. Moreover, for the effective theory to be a description of the dynamics of modes lying close (compared to κ) to the ring, the two scales λ and κ must be well separated which requires the bare interaction to be weak enough to satisfy mV 0 B −1 λ . Since B λ ∼ 1 this leads to the condition for weak coupling,
mV 0 1.(4)
The dilute limit is enforced by setting the chemical potential to be the smallest energy scale,
µ E λ E κ .(5)
Since the chemical potential is related to the mean density as,
µ ∼ρV 0 ,(6)
Eqs. (4) and (5) are combined to set the condition for the dilute limit,ρ
λ 2 κ 2 .(7)
In principle, the coarse-graining both renormalizes the overall magnitude of the bare parameters, and generates momentum dependences of the effective parameters. For parameters in vertices that are 'local' in momentum space, the effective momentum dependence can be ignored within a weakcoupling expansion as they are irrelevant in a renormalization group sense. In the presence of degeneracy in the singleparticle spectrum, however, the effective momentum dependences of the parameters in 'non-local' vertices are non-trivial as they are sensitive to the degeneracy. Indeed the g-ology in one-dimensional metals, and the Landau parameters in Fermi liquids follow from such considerations. Since the interaction among modes in R λ is weak and we coarse-grain towards the ring-minima, we ignore the renormalizations to the overall magnitude of all parameters, but retain the momentum dependence of the coupling function to obtain the effective action for the low energy dynamics of interacting charge-neutral bosons at zero temperature with E ≤ E λ ,
S λ = dK Ξ λ (K) [ik 0 − µ + E(K)] |Φ(K)| 2 + 1 2 4 n=1 dK n Ξ λ (K n ) δ (3) (K 1 − K 2 + K 3 − K 4 ) V(K 1 , K 2 , K 3 , K 4 ) Φ * (K 1 )Φ(K 2 )Φ * (K 3 )Φ(K 4 ). (8)
Here k 0 is the Euclidean frequency, dK ≡ dk0dK (2π) 3 , Ξ λ (K) is a cutoff function that suppresses modes with ||K|−κ| > λ, Φ represents the low energy bosonic modes, and V({K n }) is the effective interaction potential for scattering among the bosons. In Appendix A we sketch a derivation of the functional form of V({K n }) for the bare interaction potential, V 0 ({K n }) = V 0 . We note that Eq. (8), however, serves as a low energy effective theory for more general bare interactions.
Lowering the UV cutoff below λ generates quantum corrections that are comparable or larger than the bare coupling. It signals a breakdown of conventional perturbation theory that was used to obtain Eq. (8). In general, accounting for nonperturbative effects of scatterings that do not break any symmetry requires a reorganization of perturbation theory. Here we use higher dimensional bosonization to absorb all forward scatterings into an effective Gaussian theory in analogy to 1D bosonization. In the following sections we develop and apply this method to uncover a novel critical state.
III. MULTIDIMENSIONAL BOSONIZATION
In this section we introduce a bosonization method that is analogous to multidimensional bosonization developed in the context of Fermi liquid theory [28][29][30][31][32][33]. An apparent similarity between our system and a Fermi liquid is that the low-energy modes reside along a ring (in 2D) in both cases. It is important to note that the ring-minima of single-boson dispersion, although superficially similar to a Fermi surface, differs from it in a crucial way -the lower branch curves away parabolically from the ring. This leads to the low energy dynamics in the neighborhood of the ring to be non-chiral. Thus our method differs from those applied in the study of Fermi liquid theory through the usage of non-chiral hydrodynamic modes which leads to fundamentally new physics. In subsequent sections we apply the formalism developed here to identify the low energy behavior of Eq. (8).
A. Patch approximation
Here we introduce an approximation which involves decomposing the ring into a collection of flat, linear segments of equal length (patches). This operation is equivalent to approximating the ring by a polygon.
We assume that at sufficiently low energy we can decompose the annular cutoff of width 2λ around the ring into a collection of 2N rectangular patches of length 2Λ along the ring and width 2λ (shown in Fig. 6). The value of Λ = πκ 2N is fixed by the requirement that the area of the patches must add up to yield the area of annulus, such that the total number of low energy modes is preserved. At each patch we define a local, orthonormal coordinate system with respect to its center, v α = cos π N (α + 1/2) x + sin π N (α + 1/2) y,
u α = − sin π N (α + 1/2) x + cos π N (α + 1/2) y,(9)
where α is an integer that labels the patch and −N ≥ α ≥ N − 1, ( x, y) represents the global reference frame defined with respect to the center of the disk enclosed by the ring, v α ( u α ) points along the normal (tangent) to the ring at the center of the α-th patch. In the local coordinate system a two dimensional momentum that lies closer to the center of the αth patch than any other patch is decomposed as
K = K α + k, where K α ≡ κ v α and k ≡ k ⊥ v α + k u α .
The dispersion at the α-th patch takes the form,
ε α (k) ≡ E(K α + k) = 1 2m k 2 ⊥ + O k ⊥ κ k 2 , k 2 κ 2 k 2 .(10)
The truncation above amounts to approximating the patch to in the presence of a UV cutoff λ, the phase space available for scattering from a state with momentum P to another state with momentum P is weakly dependent on P · P as long as |P − P | λ (represented by the shaded region around P ). If |P − P | > λ (i.e. P lies on the dashed curve), however, the phase space is suppressed by a factor of δθ = 2 sin −1 (λ/|P − P |) due to a restriction on P · P arising from the finiteness of λ. (b) The same constraint applies to processes in the neighborhood of the interface of adjacent patches. When Λ λ the contribution from scatterings that involve both patches is subdominant to purely intrapatch scatterings. be locally flat. It is valid under the assumption that typically k ⊥ k 2 /κ, which is true when the UV cutoff λ Λ 2 /κ. Therefore, the weakness of the local curvature of patches bounds the number of patches, N , from below.
Since at low energies the bosonic modes carry momenta that are centered around K α for some α ∈ [−N, N − 1], we define patch fields, φ α , through the mode decomposition,
Φ(τ, r) ≈ N −1 α=−N e ir·Kα φ α (τ, r).(11)
In terms of the patch fields the non-interacting part of the action in coordinate space takes the form,
S 0 = N −1 α=−N dτ dr × φ * α (τ, r) ∂ τ − 1 2m ( v α · ∇) 2 − µ φ α (τ, r),(12)
where r is conjugate to k. We note that the dynamics at each patch is effectively one dimensional which we utilize in Section III B to bosonize the action. The flat-patch approximation bounds N from below, but allows for an arbitrarily large N . Indeed in the extreme limit, N → ∞, each patch correspond to a point on the ring, and the patch is trivially flat as Λ → 0 [29]. In this limit, however, the typical momentum exchange in scattering processes greatly exceeds Λ in magnitude, and neighboring patches mix substantially. Thus, formulation of the interacting theory in terms of flat patches becomes highly non-linear owing to the importance of multi-particle processes involving adjacent patches as illustrated in Fig. 7. Therefore, in order to avoid complications arising from such non-linearities at leading order, it is necessary to introduce an upper bound on N . This is achieved by restricting λ Λ which leads to a suppression of interpatch mixing by factor(s) of λ/Λ [28]. The momentum scales that we have introduced so far are constrained as, 1 λ/Λ Λ/κ ∼ N −1 , which implies that the number of patches is constrained by,
κ λ N κ λ .(13)
The separation of scales in Eq. (13) is satisfied by the choice,
N = b c (mV 0 ) −c with 1/2 ≤ c ≤ 1. Here b c is a c-dependent positive number that is constrained by (mV 0 ) c−1/2 b c (mV 0 ) −(1−c)
. Since interpatch mixing (controlled by λ/Λ) is minimized by choosing the smallest possible value of c, we set c = 1/2 which implies,
N = B N √ mV 0 .(14)
Here B N ≡ b 1/2 and 1 B N (mV 0 ) −1/2 . We note that the precise values of the effective parameters, B λ and B N , cannot be fixed from within our formalism.
In the patch-representation the interaction term in Eq. (8) takes the form,
S I ≈ 1 2 N −1 α1,...,α4=−N 4 n=1 dk n δ(k 10 − k 20 + k 30 − k 40 ) × δ(K α1 − K α2 + K α3 − K α4 + k 1 − k 2 + k 3 − k 4 ) × V(K α1 , K α2 , K α3 , K α4 )φ * α1 (k 1 )φ α2 (k 2 )φ * α3 (k 3 )φ α4 (k 4 ),(15)
where we have expanded the effective coupling function about the ring, and retained only the most relevant pieces. The resolution of the second δ-function in the presence of the ring leads to strong kinematic constrains which select three classes of scatterings as dominant interaction channels in the low energy limit (λ κ) [34,35],
• Direct scattering (DS): K α1 = K α2 and K α3 = K α4 ,
• Exchange scattering (ES): K α1 = K α4 and K α3 = K α2 ,
• BCS scattering (BCS): K α1 = −K α3 and K α2 = −K α4 .
While the DS and ES channels conserve particle number at each patch, the BCS channel does not. We note that the nomenclature above is defined with respect to momentum transfers on the order of κ; in an isolated patch, where κ does not play any role, it is possible to have intrapatchbackscatterings because the quadratically curved dispersion at each point on the ring admits a change of sign of the velocity, ∇E(K). The intrapatch-backscatterings are not kinematically suppressed in the low energy limit as they constitute a subset of the non-BCS scatterings defined above. In subsequent parts of the paper we will explore the importance of these scattering processes.
In order to construct a minimal theory that captures the most important physics, we include only interactions in the DS and ES channels and define the dimensionless interaction matrix,
Γ α,β = Γ (DS) α,β + Γ (ES) α,β(16)
with
Γ (DS) α,β ≡ V −1 0 V(K α , K α , K β , K β ) Γ (ES) α,β ≡ V −1 0 V(K α , K β , K β , K α ).(17)
Thus in coordinate-space representation the minimal interaction takes the form,
S I | minimal = V 0 2 α,β dτ dr Γ α,β |φ α (τ, r)| 2 |φ β (τ, r)| 2 .(18)
We take advantage of the rotational symmetry along the ring to characterize the interaction matrix by N + 1 parameters, {g n }, that are generally independent,
Γ α,β = g 0 δ α,β + N n=0 δ |α−β|,n g n .(19)
We note that g n are dimensionless by construction.
B. One-patch theory
Since the minimal interaction conserves particle number at each patch, the intrapatch dynamics plays a key role in determining the physical properties of a weakly interacting theory. Here we focus on the physical properties of the fundamental entity of the patched theory: an isolated patch.
The dynamics at the α-th patch is governed by the action,
S α = dτ dr φ * α (τ, r) ∂ τ − 1 2m ∂ 2 x − µ φ α (τ, r) + V 0 g 0 dτ dr |φ α (τ, r)| 4 ,(20)
where we have chosen a local orthogonal coordinate system such thatv α ·x = 1. By expressing the non-interacting part of S α in momentum space, S α;0 = dk [ik 0 + k 2 x /(2m) − µ]|φ α (k)| 2 we note that the single-particle dynamics is one dimensional, which allows us to interpret the y-component of position in Eq. (20) as a flavor index, and S α as a the-FIG. 8: Analogy between the dynamics at an isolated patch and a stack of wires. The vertical solid (blue) line on the left represents a patch in momentum space, while the horizontal lines on the right represent the wires in coordinate space. In the absence of interwire hoppings the wire model admits a one-dimensional manifold of single-particle energy minima that is analogous to a patch. ory of multi-flavored one dimensional bosons without flavormixing. Since the momentum component that is conjugate to the y-coordinate is transverse to the patch, it is bound as k y ∈ [−Λ, Λ]. Therefore, the y-axis is analogous to a onedimensional lattice with a lattice spacing, πΛ −1 [36]. Thus Eq. (20) is analogous to a theory of isolated wires stacked alongû α (i.e. the y-axis) as shown in Fig. 8.
Assuming a uniform mode-occupancy along the ringminima, the mean density at each patch isρ/(2N ). Invoking the analogy with a stack of wires with interwire spacing,
πΛ −1 = 2N/κ, we obtainρ/(2N ) = ρ 1D κ/(2N ) where ρ 1D
is a one dimensional density that is analogous to the mean density of each wire. Thus the one dimensional density is related to the two dimensional density through,
ρ 1D =ρ κ .(21)
Further utilizing the analogy we bosonize the patch field by introducing non-chiral hydrodynamic modes associated with the fluctuations of φ α [37],
φ α (τ, r) = A ρ 1D + v α · ∇ϕ α (τ, r) e iϑα(τ,r) ,(22)
where A is a dimensionful parameter, and ϕ α and ϑ α are patchwise density and phase fluctuations, respectively. Since in the static limit |φ α | 2 =ρ/(2N ),
A = Λ/π = κ/(2N ).(23)
Thus, in terms of the hydrodynamic modes the one-patch theory takes the form,
S α A 2 2 dy dτ dx 2i(∂ τ ϕ α )(∂ x ϑ α ) + ρ 1D m (∂ x ∇ϑ α ) 2 + 2A 2 V 0 g 0 (∂ x ϕ α ) 2 ,(24)
where we have suppressed the functional dependences of ϕ α and ϑ α for notational convenience. The two hydrodynamic fields are conjugate to each other, and an effective description in terms of ϕ α (ϑ α ) may be obtained by integrating out ϑ α (ϕ α ). Thus S α describes a set of decoupled Luttinger liquids that are parameterized by the y-coordinate. Eq. (24) does not include intrapatch-backscatterings which are associated with momentum transfers on the order of ρ 1D . These backscatterings can potentially destabilize the Luttinger liquid phase governed by S α . We postpone further discussion of these destabilizing effects to Section IV B 2.
As noted below Eq. (15), the intrapatch interaction was obtained by ignoring momentum dependence of the coupling function on the order of λ. If such dependencies are retained, then the interaction takes a more general form, V 0 dτ dr dr g 0 (r − r )|φ α (τ, r)| 2 |φ α (τ, r )| 2 , which can lead to a sliding Luttinger liquid state as long as "interwire" hoppings are irrelevant [38][39][40] and g 0 (r −r ) is short-ranged [41]. In our case the single particle degeneracy along the patch guarantees the absence of "interwire" hoppings, which would otherwise lift this degeneracy.
C. Rashba-Luttinger liquid
We use the analogy between the dynamics at individual patches and the coupled-wire system discussed above to formulate a low energy effective description in terms of the hydrodynamic modes introduced in Eq. (22). Adding the contribution from the forward scattering channels to the noninteracting part in Eq. (12) we obtain the minimal action in terms of the hydrodynamic modes,
S = A 2 2 N −1 α,β=−N dτ dr δ α,β 2i(∂ τ ϕ α )( v α · ∇ϑ α ) + ρ 1D m ( v α · ∇ϑ α ) 2 + A 2 V 0 Γ α,β ( v α · ∇ϕ α )( v β · ∇ϕ β ) ,(25)
where we have suppressed the coordinate dependence of the fields for notational convenience. The minimal action is a two dimensional analogue of Luttinger liquid, which we call a Rashba-Luttinger liquid (RLL) to underscore its origin in Rashba SOC, and to distinguish it from other types of possible higher dimensional Luttinger liquids.
In order to arrive at Eq. (25), we have excluded contributions from scatterings in the BCS channel, and higher harmonics of the patch density which modulate with wavevectors 2nπρ 1D with n = 0 being an integer. The impact of these approximations can be partially elucidated by contrasting the symmetries of the actions in Eq. (8) and Eq. (25). In addition to translational, rotational, and time-reversal invariances, the action in Eq. (8) is invariant under a global U(1) symmetry, Φ → e iθ0 Φ with θ 0 a real number, which corresponds to particle number conservation. The action in Eq. α } are distinct corresponds to an emergent U(1) 2N symmetry which is associated with particle number conservation at each patch. We note that the U(1) 2N symmetry is a subgroup of the U(1) ∞ symmetry associated with particle number conservation at each point on the ring. This U(1) ∞ symmetry is identical to the one that emerges at the Fermi liquid fixed point. The invariance under a shift of ϕ α is reminiscent of the emergent 'sliding symmetry' in sliding Luttinger liquids [38,42]. Analogously it is associated with translation invariance along v α . These symmetries guarantee the presence of the RLL state. The stability of the RLL state, however, is contingent on its robustness against interaction vertices that break the emergent symmetries of Eq.
(25) is in- variant under (ϑ α , ϕ α ) → (ϑ α , ϕ α ) + (ϑ (0) α , ϕ (0) α ), where, in general, {ϑ (0) α , ϕ
(25), but are allowed by the symmetries of Eq. (8). In particular, the BCS vertex breaks the U(1) 2N symmetry, while the density wave vertices resulting from backscatterings break the sliding symmetry. In the rest of the paper we deduce the properties of the RLL state, and its stability against the excluded interaction vertices.
IV. SPECIAL CASES OF THE INTERACTING MODEL
Since the mathematical results in the presence of the most general interaction potential turn out to be rather complicated, in this section we consider two limiting cases of the interaction matrix that allow for a simpler analysis. In spite of their simplicity, these special cases elucidate key qualitative properties of the more general interacting model. In particular, the peculiarities of the spin-orbit coupled bosonic system that distinguish it from conventional higher dimensional bosonic and fermionic systems are already apparent at these simplified limits.
A. Decoupled patches
The simplest example of the interaction matrix occurs when it is diagonal, i.e. g n =0 = 0 in Eq. (19) which implies
Γ α,β → Γ α,β = 2g 0 δ α,β .(26)
Since the diagonal components of Γ generate a stiffness for intrapatch density fluctuations which leads to a well-defined interacting limit, Γ is the simplest interaction in patch-space that stabilizes the system. We note that in this limit, the inter-acting problem reduces to a set of decoupled flat-patches with non-parallel normals. Since individual patches in the presence of interactions host a type of sliding Luttinger liquid, we expect the resultant state to be a higher dimensional Luttinger liquid as well.
On including only intrapatch interactions, the action becomes diagonal in patch-space,
S = A 2 2 N −1 α=−N dk 2ik 0 ( v α · k) ϕ α (−k)ϑ α (k) + ρ 1D m ( v α · k) 2 ϑ α (−k)ϑ α (k) + 2A 2 V 0 g 0 ( v α · k) 2 ϕ α (−k)ϕ α (k) ,(27)
where dk ≡ dk0dk (2π) 3 . We note that compared to fermions at finite density, the bosonic theory becomes well defined only after the inclusion of intrapatch interactions; the non-interacting limit is ill-defined due to the divergent density of states as the ring is approached. It is straightforward to obtain the effective action for the density (phase) fluctuations by integrating out the phase (density) field,
S ϕ = A 2 2 α dk f −1 α (k) g (ϕ) α (k, 2V 0 g 0 ) ϕ α (−k)ϕ α (k), S ϑ = A 2 2 α dk f −1 α (k) g (ϑ) α (k, 2V 0 g 0 ) ϑ α (−k)ϑ α (k),(28)
where
g (ϕ) α (k, g) = k 2 0 (ρ 1D /m) + A 2 g ( v α · k) 2 , g (ϑ) α (k, g) = k 2 0 A 2 g + ρ 1D m ( v α · k) 2 ,(29)
and we have introduced the cutoff function f α (k) to enforce | v α · k| ≤ λ and | u α · k| ≤ Λ [43]. Since ϕ α is conjugate to ϑ α , S ϕ is dual to S ϑ . Depending on the correlation function of interest, it is usually convenient to use either the S ϕ or S ϑ representation of S . Thus, the propagators of ϕ α and ϑ α are
G (ϑ) α,β = A −2 f α (k)δ α,β g (ϑ) α (k, 2V 0 g 0 ) , G (ϕ) α,β = A −2 f α (k)δ α,β g (ϕ) α (k, 2V 0 g 0 ) .(30)
Since there are no off-diagonal (i.e. interpatch) terms, it is easy to derive the propagator of the microscopic bosons,
Φ(0, r)Φ † (0, 0) ∼ρ √ mV 0 (λ|r|) 2η Φ N −1 α=−N δ |vα·r|,1 e iKα·r ,(31)
with Individual patches (blue horizonal line) may be considered as the dispersion of a lattice of wires (set of parallel black lines) that lie along the normal, v α , of the patch in the absence of interwire hoppings (see Fig. 8). (a) The patch whose normal is parallel (or anti-parallel) to the spatial separation, r (red vertical arrow), between two operators in an autocorrelation function contribute to it. (b) If r makes a finite angle with v α , i.e. r · u α = 0, then the correlation function vanishes due to a symmetry that is analogous to the combination of particle number conservation and translation invariance on each wire in the wire-lattice picture.
η Φ = 1 4π g 0 (mV 0 ) 3/2 B N (ρ/κ 2 ) ,(32)
where we have expressed A, ρ 1D and N in terms of the microscopic parameters. The power-law decay of the boson propagator suggests an absence of a condensate or BEC. Instead we obtain a critical state that closely resembles a Luttinger liquid. It is characterized by a set of (anomalous) scaling exponents, and supports gapless excitations. Unlike in a Fermi liquid, here the RLL exponents are non-universal, and depend on both single particle and interaction parameters. From the term in the round bracket in Eq. (31) we conclude that only those patches whose normals are either parallel or anti-parallel to r contribute. This restriction results from an additional symmetry within each patch which arises from the absence of local curvature at each patch. The symmetry is a consequence of the combination of particle number conservation and translation invariance along each wire in the wire-lattice picture discussed in Section III B as illustrated in Fig. 9. We emphasize that this 'selection rule' does not imply an absence of rotational symmetry because the choice of the direction ofv α=0 is arbitrary, and it can be always chosen to point along r. In this sense multidimensional bosonization may be interpreted as a method for extracting the leading scaling behavior of correlation functions of the original theory, instead of a method for approximating it.
In contrast to the scaling dimension obtained in the patchdiagonal theory of Fermi liquids [30][31][32][33], here Luttinger liquid-like scaling exponents are already present in the correlation functions of an isolated patch. This is a consequence of the non-chiral dynamics at each patch, which naturally gives rise to non-trivial anomalous dimensions of various operators.
Although we considered only a subset of forward scatterings to obtain the results in this section, as we shall show in subsequent sections, the scaling behavior of the RLL obtained after the inclusion of all forward scatterings is qualitatively similar to those obtained from Eq. (28) due to a suppression of the contributions from inter-patch interactions. The dominance of the scaling exponents obtained in the limit of decoupled patches is analogous to that in Fermi liquids [30][31][32][33]. In the present case, however, the scaling exponents are more sensitive to interactions than those in Fermi liquids, since in the latter the chiral dynamics at individual patches provides additional protection against scatterings. We note that such protections due to chiral dynamics is more generic, and applies to chiral metallic states in one [44] and two dimensions [45,46]. While no such protection exists in the present case, the global curvature of the ring-minima greatly reduces the effect of interpatch interaction in the forward scattering channels.
In the absence of interpatch interactions, scatterings in the BCS channel are absent. The higher harmonics of the intrapatch density operator, however, leads to intrapatchbackscatterings which can potentially destabilize the critical state. We defer a discussion of such intrapatch-backscattering induced instabilities to the next subsection where a wider set of such scatterings will be analyzed.
B. Coupled patches: Quasi-long range effective interaction
In this subsection we consider a simple extension of the decoupled-patch model to include interpatch interactions. In order to motivate the model we consider a specific form of the effective potential, V({K n }), where it is assumed to be a function of momentum transfer only. In coordinate space representation the effective interaction vertex takes a simple form,
S I = 1 2 dτ dR drV(r)|Φ(R − r)| 2 |Φ(R + r)| 2 ,(33)
where R and r are the center-of-mass and relative coordinate, respectively. We further assumeV(r) to be isotropic with a range, a,V
(r; a) = V 0 exp −(|r|/a) 2 (a √ π) 2 .(34)
This potential has the property, lim a→0V (r; a) → V 0 δ(r).
We assume that the effective range, a, is generated by integrating out high-energy modes as discussed in Section II, and a ∼ λ −1 . In the present case, the DS and ES channels take the forms,
Γ (DS) α,β = 1 V 0 drV(r; a) = 1; Γ (ES) α,β = 1 V 0 dr e ir·(Kα−K β )V (r; a) = e −(aκ) 2 ζ 2 α,β ,(35)
where ζ α,β = sin
|θα−θ β | 2
. WhileΓ (DS) α,β contributes equally to the interaction between all pairs of patches,Γ (ES) α,β contributes dominantly to intrapatch interactions. We note that the net interaction in the forward scattering channel,Γ (DS) α,β + Γ (ES) α,β , decays as the separation between patches increases, which is consistent with the behavior of the net interaction potential discussed in Section II.
The interactions in the forward scattering channels simplify substantially in the limit, aκ 1, since lim aκ→∞Γ
(ES)
α,β → π(aκ) −2 δ α,β . Here we will consider aκ to be finite but large [47]. Since aκ ∼ κ λ N 1, the interpatch contributions of Γ (ES) α,β are suppressed by at least a factor of exp −(aκ/N ) 2 compared to its intrapatch contribution. Thus we focus on the limiting case where the interaction potential in Eq. (34) is long-ranged enough to ignore contributions from the ES channel to interpatch scatterings. For this case the minimal interaction is constituted by scatterings in the DS channel and the intrapatch component of the ES channel. With the help of Eq. (35) we obtain the interaction matrix,
Γ α,β = δ α,β + 1.(36)
We note that in terms of Eq. (19) the interaction matrix here corresponds to setting all g n = 1. Moreover, ignoring the contribution of the DS channel reduces Eq. (36) to the interaction matrix in Section IV A.
The matrixΓ α,β in Eq. (36) is readily invertible, and leads to the effective actions for the hydrodynamic modes,
S ϕ = A 2 2 α,β dk δ α,β g (ϕ) α (k, V 0 ) f −1 α (k) + A 2 V 0 ( v α · k)( v β · k) ϕ α (−k)ϕ β (k), S ϑ = A 2 2 α,β dk δ α,β g (ϑ) α (k, V 0 ) f −1 α (k) − 1 2N + 1 k 2 0 A 2 V 0 ϑ α (−k)ϑ β (k).(37)
In addition to generating interpatch interactions, the DS channel also modifies the intrapatch terms beyond those obtained in Eq. (28). By comparing the intrapatch terms, we note that the DS channel strongly renormalizes the density fluc-tuations, but leads to a perturbative correction (recall that N 1) to the dynamics of phase fluctuations. The propagators of the two hydrodynamic modes are derived in Appendix B, and they are given by,
G (ϕ) α,β (k) = δ α,β f α (k) A 2 g (ϕ) α (k, V 0 ) − V 0 (v α · k)(v β · k)f α (k)f β (k) 1 +Υ ϕ (k) g (ϕ) α (k, V 0 )g (ϕ) β (k, V 0 ) , G (ϑ) α,β (k) = δ α,β f α (k) A 2 g (ϑ) α (k, V 0 ) + k 2 0 A 4 V 0 (2N + 1) f α (k)f β (k) 1 −Υ ϑ (k) g (ϑ) α (k, V 0 )g (ϑ) β (k, V 0 ) ,(38)
whereΥ
ϕ (k) = A 2 V 0 µ (v µ · k) 2 f µ (k) g (ϕ) µ (k, V 0 ) , Υ ϑ (k) = k 2 0 A 2 V 0 (2N + 1) µ f µ (k) g (ϑ) µ (k, V 0 ) .(39)
The first term in each propagator is independent of the momentum transverse to a given patch, and survives as interpatch interactions vanish. In contrast, the second term explicitly arises from interpatch interactions, and produces a dependence on the transverse component of momentum at each patch. Both terms in each propagator contribute to the scaling exponents of correlation functions. The contribution of the first term is identical to that obtained in Section IV A.
A similar straightforward analysis of the second term is hindered by the presence of theΥ-factors, which contain contributions from all patches that satisfy |v µ · k| ≤ λ for a fixed k. Their presence, however, does not qualitatively alter the small frequency-momentum behavior of the respective propagators, and the propensity of the second term in the propagators to contribute to anomalous dimensions of various operators is not controlled by theΥ-factors. Nevertheless we will retain theΥ dependence of the propagators, since it determines the relative magnitude of contributions from the first and second terms in the propagators as shown in Appendix C. For β = α orᾱ (the labelᾱ is such that Kᾱ = −K α ) the second term does not contribute to anomalous dimensions owing to two dimensional dynamics which is non-singular. For the special values of β = α orᾱ, however, the second term contains a dynamically one dimensional part which contributes to anoma-lous dimensions. We demonstrate both properties of the second term explicitly for a 4-patch model in Appendix D. In the rest of the subsection we investigate key physical properties and instabilities of the fixed point governed by Eq. (37).
Physical properties
Here we characterize the long wavelength properties of the fixed point described by the set of actions in Eq. (37). We begin with the computation of the equal-time propagator of the 'microscopic' boson field Φ(τ, r) by utilizing Eqs. (11) and (22),
Φ(τ, r 1 )Φ † (τ, r 2 ) ≈ A 2 ρ 1D α,β e iKα·r1 e −iK β ·r2 × e − 1 2 (ϑα(τ,r1)−ϑ β (τ,r2)) 2 .(40)
Due to particle number conservation at each patch, (ϑ α (τ, r 1 ) − ϑ β (τ, r 2 )) 2 ∝ δ α,β , which implies that only the diagonal terms of the phase-propagator contributes to the propagator of Φ. Thus,
Φ(τ, r 1 )Φ † (τ, r 2 ) ∼ρ √ mV 0 B N cos(κ|r 1 − r 2 |) (λ|r 1 − r 2 |) 2ηΦ ,(41)
where the scaling dimension of Φ,
η Φ = 1 4π (mV 0 ) 3/2 2B N (ρ/κ 2 ) 1 + √ mV 0 4B N + O (mV 0 ) .(42)
While the first term results from the intrapatch dynamics and is proportional to η Φ , the second term is a result of interpatch couplings resulting from the DS channel and it is parametrically smaller than the intrapatch contribution. Presence of interpatch coupling enhances the scaling dimension, resulting in a faster decay of the propagator of Φ, which pushes the system away from a phase-coherent state. Owing to the algebraic decay of the propagator, the system exhibits a Luttinger liquid-like behavior, and it does not support a superfluid state. This is similar to one dimensional interacting bosons without spin-orbit coupling [48]. The density-density response function carries information about the two intrinsic momentum scales present in the system, κ and ρ 1D . We express the density operator as
ρ(τ, r) = ρ diag (τ, r) + ρ (τ, r),(43)
where ρ diag (τ, r) = α ρ α (τ, r) and ρ (τ, r) = α =β e i(K β −Kα)·r φ * α (τ, r)φ β (τ, r). While the long wavelength fluctuations of ρ diag (τ, r) are intrapatch density fluctuations, those of ρ (τ, r) are a combination of density and phase fluctuations. Therefore, the autocorrelation of ρ diag (τ, r) [ρ (τ, r)] modulate with wavevectors of magnitude 2nπρ 1D [2κ] with n ≥ 0.
In order to explicitly compute the autocorrelation of ρ diag (τ, r) we use the full expression of patch density operator [37],
ρ α (r) = A 2 [ρ 1D +v α · ∇ϕ α (r)] × ∞ n=−∞ exp{2in (πρ 1Dvα · r + ϕ α (r))}.(44)
The n = 0 mode of ρ α (r) was used for bosonizing the patch fields in Eq. (22). The autocorrelation function has a uniform part that is obtained from the n = 0 mode of Eq. (44), and an oscillatory part resulting from n = 0 modes. In contrast, the autocorrelation of ρ (τ, r) lacks a uniform part, and receives strongest contributions from terms with β =ᾱ. Thus in the limit λ|r| 1 we obtain
ρ(0, 0)ρ(0, r) ≈ρ 2 − c 1 2B N (ρ/κ 2 ) (mV 0 ) 3/2 κ 2 mV 0 B 2 N |r| 2 +ρ 2 mV 0 B 2 N cos (2κ|r|) (λ|r|) 4ηΦ +ρ 2 mV 0 B 2 N n≥1 cos (2nπρ 1D |r|) (λ|r|) 2η diag (n) ,(45)
where c 1 > 0 is a real constant,
η diag (n) = n 2 π 1 − π 2 32B λ B 2 N − O (mV 0 ) 2B N (ρ/κ 2 ) (mV 0 ) 3/2 ,(46)
andη Φ was defined in Eq. (32). Here we have utilized the small curvature limit which dictates B N 1. In the limit (mV 0 ) 3/2 B N (ρ/κ 2 ), the 2κ component of density modulation has the slowest decay, but it decays faster than the propagator in Eq. (41). Thus in the RLL state both the phase and density fluctuations show quasi-long range order, and the respective correlation functions spatially oscillate over a period controlled by κ −1 . At sufficiently strong coupling, however, the density fluctuation is dominated by the component that modulate over (2πρ 1D ) −1 . In the next subsection we will see that this crossover of the characteristic momentum scale from κ to ρ 1D , in fact, signals an instability of the metallic state towards a CDW state. We note that, in general, κ is not an integer multiple of ρ 1D , i.e. the corresponding length scales are incommensurate. Moreover, Eq. (7) implies κ ρ 1D . In the bosonized theory the dynamical critical exponent equals 1. The presence of the ring-minima implies that only momentum deviations perpendicular to the ring changes energy, which implies that the free energy density scales as F ∼ T 2 , where T is temperature. This is in contrast to two dimensional superfluids and crystalline states where F ∼ T 3 due to the presence of Goldstone modes. The discrepancy is an example of hyperscaling violation with a unity 'hyperscaling violation exponent' [49,50], and it arises from the presence of the ring-minima. Therefore, in the RLL state the specific heat and entropy density scale as ∂ T F ∼ T . Thus the RLL is a first example of a bosonic system above 1D which exhibits T -linear specific heat. We note that Fermi liquids are characterized by a T -linear specific heat as well, and it originates from the presence of unity codimension Fermi surface.
Instabilities
In this section we investigate the stability of the RLL state described by the minimal action, Eq. (25). In particular, we consider the effects of the BCS channel that was not included in Eq. (25), as well as density-density backscattering interactions that may drive density wave instabilities. Unlike fermionic systems, attractive interactions in a bosonic system lead to a trivial state where all bosons condense at a single point in coordinate space. Moreover, in the presence of repulsive interactions bound states cannot form, and, thus, the BCS channel does not lead to a non-trivial symmetry broken state. Therefore, we focus only on the effects of the backscattering interactions which are expected to lead to charge density wave states.
Due to the presence of two momentum scales, κ and ρ 1D , in principle, the RLL can become unstable towards the formation of density wave states carrying momenta of magnitudes, 2πnρ 1D , 2κ, and 2κ ± 2πnρ 1D with n = 0 being a positive integer. The corresponding vertices arise from the backscattering components of local interactions with lagrangian densities, α ρ 2 α (r) and α ρ α (r)ρᾱ(r), where r ≡ (τ, r). We decompose the patch-density operator as shown in Eq. (44) and consider contribution to the lagrangian densities from n = 0 modes. Thus we obtain three interaction vertices,
S (|n|) ρ 1D = g (|n|) ρ 1D N −1 α=−N dr cos {2nϕ α (r) + 2πρ 1D nv α · r} ;(47)S (|n|) κ = 1 2 g (|n|) κ N −1 α=−N dr cos {2n(ϕ α (r) + ϕᾱ(r))} ; (48) S (n1,n2) κ±ρ 1D = g (n1,n2) κ+ρ 1D N −1 α=−N dr cos {2(n 1 ϕ α (r) + n 2 ϕᾱ(r)) + 2πρ 1D (n 1 − n 2 )v α · r} ,(49)
where, by definition, n = 0 in S
C (n) ρ 1D (r) ∼ cos (2πρ 1D n|r|) (λ|r|) 2ηρ 1D (n) ,(50)C (n) κ (r) ∼ (λ|r|) −2ηκ(n) C (n1,n2) κ±ρ 1D (r) ∼ cos (2πρ 1D (n 1 − n 2 )|r|) (λ|r|) 2ηκ±ρ 1D (n1,n2)(51)
where
η ρ 1D (n) = n 2 π 1 − π 2 32B λ B 2 N 2B N (ρ/κ 2 ) (mV 0 ) 3/2 η κ (n) = 2n 2 π 2B N (ρ/κ 2 ) (mV 0 ) 3/2 η κ±ρ 1D (n 1 , n 2 ) = n 2 1 + n 2 2 π − π(n 1 − n 2 ) 2 32B λ B 2 N × 2B N (ρ/κ 2 ) (mV 0 ) 3/2 .(52)
Since the leading instability is driven by the operator with the smallest scaling dimension, we compare the relative magnitudes of the scaling exponents obtained above. With the help of the tree-level scaling dimension of the couplings, [g X ] = 2 − η X , we find that the operator corresponding to |n| = 1 in S (n) ρ 1D drives the dominant instability. The resultant CDW state arise entirely from intrapatch backscatterings, and modulates with a wavevector of magnitude 2πρ 1D . In addition to translational invariance it also breaks rotational symmetry as a specific direction for the modulation is spontaneously chosen. Although the simplest such state displays a stripe ordering pattern, more complex ordering patterns resulting from superposition of CDWs with distinct directions of wavevectors may be realized as well. We note that a finite interaction strength is necessary for driving the instability, since at arbitrarily weak coupling g (1) ρ 1D is irrelevant with η ρ 1D (±1) > 2. Therefore, at weak coupling the RLL state is stable as illustrated in the phase diagram in Fig. 3. An asymptotic expres-FIG. 10: An example of a potential CDW instability driven by backscatterings resulting from interaction vertices of the form ρ α ρ β with α = β orβ. The circle (arrow) represents the ring-minima (wavevector of the CDW state). Such CDW instabilities are expected to be suppressed by a lack of phase space.
sion for the phase boundary is obtained from the condition, η ρ 1D (±1) = 2, which leads tō
ρ κ 2 = 2π 2 B N 1 − U 1 2N g 0 − π 2 32B λ B 2 N −2 (mV 0 ) 3/2 .(53)
The phase boundary is represented by the solid curve in Fig. 3.
In principle, higher harmonics of ρ α ρ β with β = α orᾱ can drive finite coupling CDW instabilities as shown in Fig. 10. The autocorrelation functions of such backscattering operators, however, vanish identically due to the emergent symmetry associated with the flat-patch approximation (see Fig. 9). This is analogous to the fate of the BCS vertex in the bosonized description of Fermi liquids. Therefore, the vanishing of the autocorrelation function does not necessarily imply an absence of the instability. In particular, within a Wilsonian RG scheme these vertices can obtain finite quantum corrections which may lead to non-trivial RG flow. Although a detailed RG analysis of this class of CDW operators lies beyond the scope of the present work, we emphasize that, within the bosonization framework developed here, these vertices are strongly irrelevant at weak coupling. Consequently, we do not expect them to affect the low energy behavior of the system at the leading order in the weak coupling limit.
V. GENERAL SHORT-RANGE INTERACTION
In this section we explore a general interpatch interaction in the forward scattering channel. For computational convenience we express the interaction matrix in Eq. (19) as
Γ α,β = g 0 δ α,β + U α,β ,(54)
where
U α,β ≡ N n=0 δ |α−β|,n g n .(55)
We note that the models discussed in sections IV A and IV B correspond to the special cases of Eq. (54), where U α,β = g 0 δ α,β and U α,β = g 0 = 1, respectively. Here U α,β is treated as the interaction potential that couple distinct patches, along with enhancing the intrapatch interaction. Unlike the case where U α,β = 1, we cannot obtain a general expression of the matrix elements of U −1 . Thus we resort to the angular harmonics of U α,β ,
U l = α cos(θ α0 l) U α,0 ,(56)
where θ αβ = θ α − θ β = π N (α − β), and we have used the fact that U α,β depends on {α, β} only through |α − β|. In the minimal action, Eq. (25), we replace Γ α,β by Eq. (54) with
U α,β = 1 2N N −1 l=−N cos(θ αβ l) U l ,(57)
to obtain
S = A 2 2 α dk 2ik 0 ( v α · k) ϕ α (−k)ϑ α (k) + ρ 1D m ( v α · k) 2 ϑ α (−k)ϑ α (k) + A 2 V 0 g 0 ( v α · k) 2 ϕ α (−k)ϕ α (k) + A 4 V 0 2 α,β l dk U l 2N cos(θ αβ l) ( v α · k)( v β · k) ϕ α (−k)ϕ β (k).(58)
The propagators of the density and phase fluctuations are derived in Appendix E, and they are as follows,
G (ϕ) α,β (k) = f α (k) δ α,β A 2 g (ϕ) α (k, V 0 g 0 ) − V 0 f α (k)f β (k)(v α · k)(v β · k) g (ϕ) α (k, V 0 g 0 )g (ϕ) β (k, V 0 g 0 ) l,l U l U l 2N Ω (ϕ,c)
l,l (k) cos(θ α l) cos(θ β l ) + Ω (ϕ,s) l,l (k) sin(θ α l) sin(θ β l ) ,
G (ϑ) α,β (k) = f α (k) δ α,β A 2 g (ϑ) α (k, V 0 g 0 ) + k 2 0 A 2 V 0 g 0 f α (k)f β (k) A 2 g (ϑ) α (k, V 0 g 0 )g (ϑ) β (k, V 0 g 0 ) l,l U l U l 2N g 0 Ω (ϑ,c) l,l (k) cos(θ α l) cos(θ β l ) + Ω (ϑ,s) l,l (k) sin(θ α l) sin(θ β l ) ,(59)
where the inverse of the Ω-matrices are
[Ω (ϕ,c) (k)] −1 l,l = U l δ l,l + U l U l 2N µ A 2 V 0 (v µ · k) 2 f µ (k) g (ϕ) µ (k, V 0 g 0 ) cos(θ µ l) cos(θ µ l ) [Ω (ϕ,s) (k)] −1 l,l = U l δ l,l + U l U l 2N µ A 2 V 0 (v µ · k) 2 f µ (k) g (ϕ) µ (k, V 0 g 0 ) sin(θ µ l) sin(θ µ l ) ,(60)
[Ω (ϑ,c) (k)] −1 l,l = U l 1 +
U l 2g 0 δ l,l + U l U l 2g 0 δ l,−l − k 2 0 A 2 V 0 g 0 U l U l 2N g 0 µ f µ (k) cos(θ µ l) cos(θ µ l ) g (ϑ) µ (k, V 0 g 0 ) ,
[Ω (ϑ,s) (k)] −1 l,l = U l 1 +
U l 2g 0 δ l,l − U l U l 2g 0 δ l,−l − k 2 0 A 2 V 0 g 0 U l U l 2N g 0 µ f µ (k) sin(θ µ l) sin(θ µ l ) g (ϑ) µ (k, V 0 g 0 ) .(61)
Here, by construction, Ω (ϕ,c) (k) and Ω (ϑ,c) (k) are 2N × 2N matrices, while Ω (ϕ,s) (k) and Ω (ϑ,s) (k) are (2N −1)×(2N − 1) matrices due to the absence of the l, l = 0 components in the latter set of matrices. Consequently, Ω (ϕ,s) (k) and Ω (ϑ,s) (k) do not depend on the s-wave component of U .
A. S-wave only model
In order to connect the general model with the discussion in Section IV B we focus on the simplest case where only the s-wave component of the interaction U is non-vanishing. The propagators simplify to,
G (ϕ) α,β (k) = δ α,β f α (k) A 2 g (ϕ) α (k, V 0 g 0 ) − V 0 U 0 [1 + Υ ϕ (k)] −1 2N (v α · k)(v β · k)f α (k)f β (k) g (ϕ) α (k, V 0 g 0 )g (ϕ) β (k, V 0 g 0 ) G (ϑ) α,β (k) = δ α,β f α (k) A 2 g (ϑ) α (k, V 0 g 0 ) + U 0 [1 − Υ ϑ (k)] −1 2N (1 + U 0 )A 4 V 0 k 2 0 f α (k)f β (k) g (ϑ) α (k, V 0 g 0 )g (ϑ) β (k, V 0 g 0 ) , (62) where Υ ϕ (k) = A 2 V 0 U 0 2N µ (v µ · k) 2 f µ (k) g (ϕ) µ (k, V 0 g 0 ) , Υ ϑ (k) = U 0 2N (1 + U 0 ) k 2 0 A 2 V 0 µ f µ (k) g (ϑ) µ (k, V 0 g 0 ) .(63)
Since U 0 = g 0 + g N + 2
N −1 n=1 g n , the model discussed in Section IV B is a special case of the s-wave only model with all g n = 1. The scaling behavior of the s-wave only model is qualitatively similar to those discussed in Section IV B. In particular, the phase boundary is given by,
ρ κ 2 = F 0 U 0 g 0 , Λ λ , N (mV 0 ) 3/2 B N ,(64)
where
F 0 (x 0 , y, z) = 2π 2 1 − x 0 4zy(1 + x 0 ) − π 8zy y 1 dt 2/(πx 0 ) + sin −1 (1/t) .(65)
In the limit ( U 0 /g 0 ) (Λ/λ) 1 it is identical to Eq. (53) at the leading order.
B. Robustness of leading scaling behavior
Here we show that the inclusion of harmonics of U beyond s-wave does not alter the leading order scaling behavior obtained above as long as the effect of U l =0 is perturbative, i.e. U l =0 U 0 . In the interest of brevity and to directly connect the results in this section to the phase diagram, we focus only on the dynamics of density fluctuations where the equal-time correlation function, [ϕ α (r) − ϕ α (0)] 2 , plays a central role.
Since the propagator depends on U l through the second term, we focus on the Ω-factors in Eq. (60). Let us define,
Π (ϕ,c) l,l (k) = 1 2N µ A 2 V 0 (v µ · k) 2 f µ (k) g (ϕ) µ (k, V 0 g 0 ) cos(θ µ l) cos(θ µ l ) Π (ϕ,s) l,l (k) = 1 2N µ A 2 V 0 (v µ · k) 2 f µ (k) g (ϕ) µ (k, V 0 g 0 ) sin(θ µ l) sin(θ µ l ).(66)
In order to extract the coefficients of the ln(λ|r|) term in [ϕ α (r) − ϕ α (0)] 2 , we set k 0 = 0 in the Ω-factors. In the zero frequency limit Eq. (66) simplifies to
Π (ϕ,c) l,l (k) = 1 2N µ f µ (k) cos(θ µ l) cos(θ µ l ), Π (ϕ,s) l,l (k) = 1 2N µ f µ (k) sin(θ µ l) sin(θ µ l ).(67)
Due to the oscillatory factors in the summands, we conclude that |Π vanish if |l+l | is an odd integer. Therefore, the s-wave propagators in Eq. (62) dominate over contributions from higher angular harmonics, and in the limit of negligible patch curvature the leading order contribution to the coefficient of ln(λ|r|) is obtained from the first term in the respective propagators. Consequently the phase diagram remains qualitatively identical to Fig. 3 with the phase boundary satisfyinḡ
ρ κ 2 = F U 0 g 0 , U 1 g 0 , . . . , U N g 0 , Λ λ , N (mV 0 ) 3/2 B N ,(68)
where F (x 0 , x 2 , . . . , x N , y, z) ∼ 1 is a dimensionless function.
We verify the arguments presented above by explicitly computing the expression of the phase boundary by including the p-wave component of U with the condition U 0 U ±1 . Since the scaling behavior is controlled by the diagonal elements of the propagator, we obtain
G (ϕ) α,α (k) f α (k) A 4 g 0 V 0 |k| 2 1 G α (χ(k, V 0 g 0 )) − 1 N U 0 /2 g 0 + U 0 Π (ϕ,c) 0,0 (λ/|k|) + U 1 g 0 − O ( U 1 /g 0 ) 2 × cos 2 (θ α ) (G α (χ(k, V 0 g 0 ))) 2 (69)
up to singular terms. Here χ(k, g) = mk 2 0 A 2 gρ 1D |k| 2 , and G α (χ) = χ + cos 2 (θ α ). We check that in the limit U 1 = 0, Eq. (69) reduces to the corresponding term in Eq. (62). The leading behavior of the scaling dimension of backscattering operators is proportional to that obtained in Section IV B, and the phase boundary is given by,
(ρ/κ 2 ) = 2π 2 B N 1 − U 1 2N g 0 − π 2 32B λ B 2 N −2 (mV 0 ) 3/2 .(70)
VI. CONCLUSION
In this paper we deduced the phase diagram of a dilute, homogeneous, pseudospin-1 2 bosonic system in the presence of weak, short-range, repulsive interaction, and Rashba spin-orbit coupling (SOC). We take advantage of the onedimensional dynamics along the radial direction to develop a multidimensional bosonization scheme, which allows for an unbiased non-perturbative analysis of the low energy behavior. We show that at weak coupling a symmetric critical state (the Rashba-Luttinger liquid or RLL) is realized through a combined effect of SOC and interaction. The RLL phase is characterized by quasi long-range order with non-universal scaling exponents, and a T -linear specific heat. Within a treelevel scaling analysis the RLL is found to be stable at weak coupling. Strengthening of the SOC or the interaction at fixed density enhances various charge density wave (CDW) fluctuations which eventually destabilizes the RLL. The dominant instability drives the system to a CDW state with a wavevector whose magnitude is controlled by the ratio of the mean density of bosons and the spin-orbit coupling strength, ρ 1D ≡ρ/κ. We summarize our main results through the phase diagram in Fig. 3, and deduce the asymptotic form of the phase boundary. We note that more conventional CDWs with wavevectors of magnitude ∼ κ [9,12] are subdominant instabilities, and become relevant further away from the phase boundary on the symmetry broken side of the phase diagram.
The RLL is similar to 'Bose metals' that are conjectured to exist in various solid state systems, viz. near the boundary of superconductor-insulator transitions [51,52], quantum spin liquids [53][54][55], and frustrated lattices [56] (however, see [57]). Besides not breaking any symmetries, the Bose metal also hosts an extended zero-energy manifold in mo-mentum space (the 'Bose surface'), which is analogous to the ring-minima studied in this paper. In contrast to the ringminima whose origin is single particle dispersion, a Bose surface emerges at low energies only in the presence of interactions. Despite their dissimilar origin, the ring-minima and the Bose surface lead to similar physical properties like linear-T specific heat, and entanglement entropy scaling. In particular, as pointed out in Ref. [58], the presence of Bose surface(s) leads to logarithmic violation of entanglement entropy area law, similar to what happens in a free Fermi gas [59][60][61] or Fermi liquid [62]. Such violation offers a diagnostic of the RLL phase in numerical studies. Even the shape of the Bose surface can be determined by detailed studies of the entanglement entropy [63].
A generalized version of our analysis can be utilized to access the low energy behavior of three dimensional bosonic systems with symmetric (or Weyl) SOC. In the presence of Weyl SOC the single particle energy is minimized on a spherical shell of radius κ. In order to bosonize the interacting model the 'shell'-minima is approximated by a polyhedron. The faces of the polyhedron correspond to two dimensional flat patches of area ∼ Λ 2 . Due to the unit codimension of the the shell-minima, each flat patch supports one dimensional dynamics which is similar to a 2D lattice of decoupled quantum wires. Thus a suitable generalization of Eqs. (11) and (22) leads to the bosonization of the effective theory. Consequently, a three dimensional analogue of the RLL state is expected to be stabilized at weak coupling and low density, with various competing CDW instabilities arising at stronger interaction.
Although we established the phase diagram through a treelevel scaling analysis, the RLL phase and the region close to the phase boundary on the symmetry broken side are expected to be robust against quantum corrections. Deeper into the symmetry broken side of the phase diagram other CDW operators become relevant at the RLL fixed point. Generally, in the presence of multiple relevant operators the dominant instability is determined by a combination of the bare interaction strength of these operators and their scaling dimensions [41]. Various relevant operators, however, may mix under renormalization group (RG) flow to give rise to novel features that are unanticipated in our tree-level analysis. Therefore, a systematic RG analysis is required to fully characterize the phase diagram on the symmetry broken side of the phase boundary. An obvious choice of RG scheme within the hydrodynamic framework would be a coordinate-space based method, analogous to that applied to the sine-Gordon model. It is, however, non-trivial to use such a scheme in the presence of the constraint in Eq. (13). A method based on simultaneous modeelimination and increasing the number of patches may lead to a consistent scheme [64,65]. We leave such considerations to future work. Finally, we note that the phase diagram may also be modified in the region where the bare parameters no longer satisfy the contraints under which the effective theory was bosonized. In this appendix we derive the effective action in Eq. (8), along with the estimate for the UV cutoff, λ. We begin with the single-particle Hamiltonian, Eq. (1), which we quote here in terms of κ,
H 0 = |K| 2 2m + κ 2 2m σ 0 + κ m σ · K. (A.1)
With the change of coordinates, (K x , K y ) → (K cos θ k , K sin θ k ), we obtain the two branches of the spectrum,
E ± (K) = 1 2m (K ± κ) 2 , (A.2)
with respective eigenvectors, 0) and b ↓ = (0, 1) be the two pseudospin basis states. Thus the two branches can be expressed as a linear combination of the pseudospin fields,
Φ ± (θ k ) = 1 √ 2 ±e −iθ k 1 . (A.3) Let b ↑ = (1,Φ ± (θ k ) = b ↓ ± e −iθ k b ↑ , (A.4)
and they are of opposite helicities. In order to construct the effective action in Eq. (8) we proceed in two steps. In the first step we integrate out modes for which E > E κ (see Fig. 1). This includes the entire upper branch, Φ + , and modes on the lower branch with momenta, |K| > 2κ. This cannot be done exactly owing to the presence of quartic vertices, |Φ ± | 4 . For a sufficiently weak bare interaction strength, V 0 , however, the renormalizations can be ignored in comparison to the bare parameters. Therefore the effective theory that describes the dynamics for E < E κ is expressed in terms of the lower branch,
S κ ≈ dK Ξ κ (K) (ik 0 + E(K) − µ)|Φ(K)| 2 + V 0 4 n=1 dK n Ξ κ (K n ) δ(K 1 − K 2 + K 3 − K 4 ) Φ † (K 1 )Φ(K 2 )Φ † (K 3 )Φ(K 4 ). (A.5)
where κ implies ||K| − κ| < κ, dK ≡ dk0dK (2π) 3 , Ξ κ (K) is a cutoff function that suppresses modes with ||K| − κ| > κ, and Φ denotes the low energy modes.
In the second step we integrate out modes that lie in the region R λ and carry energy E κ > E > E λ as shown in Figs. 1 and 4. The quantum corrections to the interaction vertex at quadratic order in V 0 is obtained from the interaction part of Eq. (A.5),
δS int = − 1 2 S int R λ . (A.6)
The 4 scattering processes in Fig. 5 contribute to δS int ,
δS int = − V 2 0 2 dK dK dQ Ξ λ (K)Ξ λ (K )Ξ λ (K + Q)Ξ λ (K + Q) Φ * (K + Q)Φ(K)Φ * (K )Φ(K + Q) × [2C PP (K + K + Q) + 4C PH (K − K) + 4C B (Q) + 4C P (Q)] , (A.7)
where we have used the momentum independence of the coupling in Eq. (A.5). The vertex corrections C PP , C PH , C B , C P result from Figs. 5a, 5b, 5c, and 5d, respectively, and they are given by,
C PP (P ) = R λ dK G(−k 0 , −K)G(k 0 , K + P ) C PH (P ) = C B (P ) = C P (P ) = R λ dK G(k 0 , K)G(k 0 , K + P ). (A.8)
Here R λ implies that K ∈ R λ while k 0 ∈ (−∞, ∞). Since λ acts as an infrared (IR) cutoff and λ √ 2mµ, E(K) = Here the solid (blue) circle is the ring-minima of radius κ, the long-dashed circles mark the UV cutoff, κ ± λ, for the low energy effective theory, and the short-dashed circles represent the scales κ ± √ 2mµ. The (red) cross marks a generic point, K, in R λ , and the (green) circle around it represents possible values of K + P with a fixed |P |. Within the area enclosed by the short-dashed circles (shaded region) the energy E(K + P ) < µ. Thus, when a part of the (green) circle intersects the shaded region, the particle-hole diagrams become non-vanishing. The smallest such circle corresponds to |K| = κ ± λ and |P | = λ − √ 2mλ. (b) Functional dependence of C PH (P ) [measured in units of m] on P [measured in units of κ]. The filled circles are numerically evaluated values of C PH (P ), while the solid curve is a guide for the eye. For the numerical integration we used λ = 10 −2 κ, µ = 10 −5 E κ , and approximated Θ(x) → ( 1 π tan −1 (x/a) + 1 2 ) with a = 10 −13 . We note that C PH (P ) becomes appreciable only when |P | λ, as anticipated in (a). E(−K) > µ for all K ∈ R λ . Thus the frequency integral in C PH (P ) is non-vanishing only if |P | > (λ − √ 2mµ) for which E(K + P ) < µ. We show a schematic of the determination of minimum |P | in Fig. 11a.
In order to determine the order of magnitude of the quantum corrections we evaluate C PP at the BCS configuration (P = 0),
C PP (0) = dk 0 2π R λ dK 1 (ik 0 + E(K) − µ)(−ik 0 + E(−K) − µ) (A.9) = m π κ λ − 1 ≈ mκ πλ . (A.10)
After integrating over frequency the particle-hole diagrams take the form,
C PH (P ) = − R λ dK Θ(µ − E(K + P )) E(K) − E(K + P ) ≈ − R λ dK Θ(µ − E(K + P )) E(K) , (A.11)
where we utilized the fact, E(K + P ) E(K) to approximate C PH (P ). We note that C PH (P ) depends on P through |P |. We evaluate the dependence numerically (using the Cuba library for numerical integration in Mathematica) and plot the result in Fig. 11b. The scale at which C PH (P ) becomes finite is controlled by λ, and C PH (P ) decays with increasing magnitude of P . Thus the effective vertex,
V(K, K , Q) = V 0 − V 2 0 [C PP (K + K + Q) + 2C PH (K − K) + 2C B (Q) + 2C P (Q)] , (A.12)
is dependent on all three external momenta. For V 0 > 0, up to 2nd order in perturbation theory, it is enhanced (suppressed) by scatterings in the particle-hole (particle-particle) channel. We note that, unlike the UV interaction potential, V 0 (Q) = V 0 , which mediates only contact itneractions, the effective potential, V(K, K , Q), has a finite range in coordinate space and leads to more general scatterings among the bosons at low energies.
1 2 α dk [ξ α (−k)ϑ α (k) + ϑ α (−k)ξ α (k)], (B.1)
where ξ α is a source for ϑ α , and W −1 = A 2 V 0 (2N + 1). Let us define
χ(k) = k 0 √ W α ϑ α (k), (B.2) such that S χ ≡ A 2 2 dk χ(−k)χ(k) = − A 2 2 W dk k 2 0 α,β ϑ α (−k)ϑ β (k). (B.3)
We introduce auxillary fields to decompose the χ 2 term as
e −Sχ da exp − A 2 2 dk [a(−k) + iχ(−k)][a(k) + iχ(k)] − A 2 2 dk χ(−k)χ(k) (B.4) = da exp − A 2 2 dk [a(−k)a(k) + i(a(−k)χ(k) + χ(−k)a(k))] (B.5)
Thus, using Eq. (B.2), we obtain
S ϑ [ξ α ] = A 2 2 α dk g (ϑ) α (k) f α (k) ϑ α (−k)ϑ α (k) + A 2 2 dk a(−k)a(k) + A 2 2 α dk [L α (−k)ϑ α (k) + ϑ α (−k)L α (k)] . (B.6) where L α (k) = ξ α (k) + i √ W k 0 a(k). (B.7)
We integrate out ϑ α (k) for each α to obtain,
A −2 S ϑ [ξ α ] = − 1 2 α dk f α (k) g (ϑ) α (k) L α (−k)L α (k) + 1 2 dk a(−k)a(k) (B.8) = 1 2 dk 1 − α Wf α (k)k 2 0 g (ϑ) α (k) a(−k)a(k) + i α f α (k) √ Wk 0 g (ϑ) α (k) (a(−k)ξ α (k) − ξ α (−k)a(k)) − 1 2 α dk f α (k) g (ϑ) α (k) ξ α (−k)ξ α (k). (B.9) Since N > 1, g (ϑ)
α (k) > Wk 2 0 for all (k 0 , k), which implies that the coefficient of a 2 is positive definite for generic frequency and momentum. Integrating out a(k) leads to
A −2 S ϑ [ξ α ] = − 1 2 α,β dk δ α,β f α (k) g (ϑ) α (k) + W k 2 0 f α (k)f β (k) 1 − µ Wfµ(k) k 2 0 g (ϑ) µ (k) g (ϑ) α (k)g (ϑ) β (k) ξ α (−k)ξ β (k).
(B.10) Therefore, the propagator of ϑ α is
A 2 G (ϑ) α,β = δ α,β f α (k) g (ϑ) α (k) + W k 2 0 f α (k)f β (k) 1 − µ Wfµ(k) k 2 0 g (ϑ) µ (k) g (ϑ) α (k)g (ϑ) β (k)
.
(B.11)
The derivation of the propagator of ϕ α proceeds in analogy to Appendix E 2.
Appendix C: Kinematic constraints due to the curvature of the ring-minima
The sums over patches in theΥ-factors depend on the magnitude of the momentum, k. In particular, for large enough |k|, the cutoff function, f µ (k), suppresses contributions from patches with normals almost parallel to k. As a limiting case let us assume that there exists a patch, α, such that v α · k = 0. Thus k is entirely transverse at the α-th patch (i.e. k = |k| u α ), which implies that its maximum allowed magnitude is |k| ∼ Λ. Given this choice of the orientation of k, it can be carried by a boson at the β-th patch only if | v β · k| ≤ λ. Assuming the maximum possible magnitude of k, this implies a constrain on the angular separation between the α-th and β-th patches, | v β · u α | ≤ λ Λ 1, for both patches to contribute to the sum. Since | v β · u α | = | sin(θ α − θ β )|, we deduce that for |k| ∼ Λ, |α − β| ≈ 0 (mod N ) which allows for either nearly parallel or nearly anti-parallel pairs of patches. As the magnitude of k decreases, patches at progressively larger angular distance from α contribute to the sum, with all patches contributing when |k| ≤ λ. In this appendix we explicitly derive these results, and identify the most singular parts of the propagators that contribute to the scaling exponents.
For its simplicity we demonstrate the procedure with the help of the model in Section IV B. We start with the derivation of the leading behavior (in an expansion in 1/N ) of theΥ-terms in Eq. (39),
Υ ϑ (k) = k 2 0 A 2 V 0 (2N + 1) N −1 µ=−N f µ (k) g (ϑ) µ (k, V 0 ) ; Υ ϕ (k) = A 2 V 0 N −1 µ=−N (v µ · k) 2 f µ (k) g (ϕ) µ (k, V 0 ) . (C.1)
Here we choose
f µ (k) = Θ (λ − |v µ · k|) Θ (Λ − |û µ · k|) . (C.2)
Although max |k| = √ Λ 2 + λ 2 , we can set Θ (Λ − |û µ · k|) = 1 while extracting the coefficient of the ln (λ|r|) term in correlation functions because the (v µ · k) = 0 mode does not contribute to the coefficient. Since N 1 we replace the sum over µ by an integral with the choicev µ=0 ·k = 1,
Υ ϑ (k) ≈ 2N k 2 0 A 2 V 0 (2N + 1) π −π dθ 2π Θ (λ − |k|| cos θ|) k 2 0 /(A 2 V 0 ) + (ρ 1D /m)|k| 2 cos 2 θ , Υ ϕ (k) ≈ 2N A 2 V 0 π −π dθ 2π Θ (λ − |k|| cos θ|) |k| 2 cos 2 θ k 2 0 /(ρ 1D /m) + (A 2 V 0 )|k| 2 cos 2 θ . (C.3)
Therefore, as the magnitude of k increases the contribution from those patches withv α ·v 0 ≈ 1 are suppressed. In order to evaluate the integrals, it is convenient to define the ratio,
χ(k, g) = mk 2 0 A 2 gρ 1D |k| 2 , (C.4)
such thatΥ
ϑ (k) = 2N 2N + 1 χ(k, V 0 ) π −π dθ 2π
Θ (λ/|k| − | cos θ|) cos 2 θ + χ(k, V 0 )
= 1 2N + 1 4N π Θ λ |k| − 1 f ϑ (χ(k, V 0 ), 1) + Θ 1 − λ |k| f ϑ (χ(k, V 0 ), λ/|k|) (C.5) Υ ϕ (k) = 2N π −π dθ 2π Θ (λ/|k| − | cos θ|) cos 2 θ cos 2 θ + χ(k, V 0 ) = 4N π Θ λ |k| − 1 f ϕ (χ(k, V 0 ), 1) + Θ 1 − λ |k| f ϕ (χ(k, V 0 ), λ/|k|) , (C.6) where f ϑ (a, b) = b 0 dy 1 − y 2 a y 2 + a , f ϕ (a, b) = b 0 dy 1 − y 2 y 2 y 2 + a . (C.7)
It is easy to check that as b → 0 both f -functions are suppressed, which embodies the kinematic suppression due to the curvature of the ring-minima. In order to isolate the parts of the propagators that contribute to the scaling exponents, we identify the asymptotic behavior of the f -functions as a function of a,
lim a→0 f ϑ (a, b) = π √ a 2 − O (a) , lim a→∞ f ϑ (a, b) = sin −1 (b) + O a −1 (C.8) lim a→0 f ϕ (a, b) = sin −1 (b) − π √ a 2 + O (a) , lim a→∞ f ϕ (a, b) = sin −1 (b) + O a −1 . (C.9)
Since both propagators at most ∼ k −2 0 as |k 0 | → ∞ at fixed k, the frequency integrations are UV finite irrespective of the magnitude of |k|. The finiteness of |k|, however, is important for the IR finiteness of the frequency integrations. Therefore, the singular dependence of the result of the frequency integrations on |k| arises from the k 0 ≈ 0 sector. Thus we isolate the most singular terms (in the above sense) in the propagator,
G (ϕ) α,β (k) = δ α,β f α (k) A 2 g (ϕ) α (k, V 0 ) − V 0 (v α · k)(v β · k)f α (k)f β (k) 1 +Υ (0) ϕ (k) g (ϕ) α (k, V 0 )g (ϕ) β (k, V 0 ) +Υ ϕ (k) −Υ (0) ϕ (k) 1 +Υ (0) ϕ (k) 1 +Υ ϕ (k) V 0 (v α · k)(v β · k)f α (k)f β (k) g (ϕ) α (k, V 0 )g (ϕ) β (k, V 0 ) , (C.10) G (ϑ) α,β (k) = δ α,β f α (k) A 2 g (ϑ) α (k, V 0 ) + k 2 0 A 4 V 0 (2N + 1) f α (k)f β (k) 1 −Υ (0) ϑ (k) g (ϑ) α (k, V 0 )g (ϑ) β (k, V 0 ) +Υ ϑ (k) −Υ (0) ϑ (k) 1 −Υ (0) ϑ (k) 1 −Υ ϑ (k) k 2 0 A 4 V 0 (2N + 1) f α (k)f β (k) g (ϑ) α (k, V 0 )g (ϑ) β (k, V 0 ) , (C.11) whereΥ (0) ϕ (k) ≡Υ ϕ (k 0 = 0, k) = 4N π Θ λ |k| − 1 π 2 + Θ 1 − λ |k| sin −1 (λ/|k|) Υ (0) ϑ (k) ≡Υ ϑ (k 0 = 0, k) = 0. (C.12)
While the terms in the first line of each propagator contribute to the coefficient of ln(λ|r|), the term in the second line does not because the numerator produces additional suppression in the k 0 → 0 limit.
Appendix D: 4-patch theory
In this appendix we analyze the singularity structure of the propagators for the case where N = 2, i.e. four patches. This is the simplest two dimensional approximation to the Bose ring, and elucidates certain key features of two dimensional scattering processes which aids the simplification of the general-N case as discussed in the main text.
For computational convenience we define the centers of the four patches to lie at angular positions θ = −π, −π/2, 0, π/2. Considering each patch to be dynamically identical, the scattering matrix Γ is characterized by 3 parameters (couplings) corresponding to intra-patch scattering (g 0 ), scattering between antipodal patches (g 2 ), and other inter-patch scatterings (g 1 ), such that Γ α,β = g 0 δ α,β + 2 n=0 δ |α−β|,n g n .
(D.1)
It is straightforward to integrate out the phase fields (the quadratic term is diagonal in patch index) to obtain the effective action in terms of density fluctuations,
S ϕ = A 2 2 1 α,β=−2 d 3 k (2π) 3 δ α,β k 2 0 (ρ 1D /m) + A 2 V 0 Γ α,β ( v α · k)( v β · k) ϕ α (−k)ϕ β (k). (D.2)
A similar operation leads to the effective action for the phase,
S ϑ = A 2 2 1 α,β=−2 d 3 k (2π) 3 (A 2 V 0 ) −1 Γ −1 α,β k 2 0 + δ α,β m −1 ρ 1D ( v α · k) 2 ϑ α (−k)ϑ β (k). (D.3)
As a representative case we focus on the dynamics of ϑ α . The propagator is
G (ϑ) α,β (k) = A −2 δ α,β g α (k) + (1 − δ α,β − δᾱ ,β ) W 1 k 2 0 D(k) + (δ α,β + δᾱ ,β ) W 2 k 0 2 D(k) + 2g α (k)(W 1 k 2 0 ) 2 D(k)g α (k)(g α (k) − 2W 2 k 0 2 ) , (D.4)
whereᾱ-th patch is antipodal to α-th patch, and
g α (k) = k 2 0 /(A 2 V 0 ) 2g 0 − g 2 + m −1 ρ 1D (v α · k) 2 ; W 0 = 1 A 2 V 0 (2g 0 + g 2 )2g 0 − 2g 1 2 (2g 0 − g 2 )((2g 0 + g 2 ) 2 − 4g 1 2 ) , W 1 = 1 A 2 V 0 g 1 (2g 0 + g 2 ) 2 − 4g 1 2 , W 2 = 1 A 2 V 0 (2g 0 + g 2 )g 2 − 2g 1 2 (2g 0 − g 2 )((2g 0 + g 2 ) 2 − 4g 1 2 ) ; D(k) = (W 0 − W 2 )k 2 0 + m −1 ρ 1D k 2 x (W 0 − W 2 )k 2 0 + m −1 ρ 1D k 2 y − (2W 1 k 2 0 ) 2 . (D.5)
The first term in Eq. (D.4) is the renormalized intra-patch correlation, which is purely one-dimensional. The other two terms introduces two dimensional dynamics through D(k). We note that on setting g 1 = 0, W 1 vanishes which results in an effective one dimensional dynamics. Thus g 1 = 0 is crucial for retaining the two-dimensional dynamics of the boson. D(k) and g α (k) are positive definite away from the origin of the frequency-momentum space in the parametric region, 2g 0 > 0, 2g 0 > g 2 and 2g 0 + g 2 > 2g 1 . This restriction is important for the determinant of the propagator to not vanish, which is crucial for the absence of non-propagating modes. Interaction potentials that lead to a dominant intrapatch forward scattering naturally satisfy these constraints.
Correlation functions of vertex operators obtain anomalous dimensions through those components of the propagator which logarithmically diverge in the IR, viz. d 3 k G α,β (k) ∝ ln(ΛL), where L −1 (Λ) is a IR (UV) cutoff. The first term in Eq. (D.4) is IR divergent in the above sense, while the second is not. The third term possesses a hidden one-dimensionality. In order to isolate this hidden divergence, we simplify the third term to obtain
G (ϑ) α,β (k) A 2 = δ α,β g α (k) + δ α,β + δᾱ ,β 2 1 g α (k) − 2W 2 k 2 0 − 1 g α (k) + (δ α,β + δᾱ ,β ) 2(W 1 k 2 0 ) 2 D(k){g α (k) − 2W 2 k 2 0 } + (1 − δ α,β − δᾱ ,β ) W 1 k 2 0 D ϑ (k) . (D.6)
The terms in the first line diverge in the IR when integrated over (k 0 , k), the rest of the terms are IR finite (this is similar to the situation in crossed sliding Luttinger liquids discussed in Ref. [39]). Thus we conclude that, (a) the only sources of IR divergence are intra-patch correlation, and correlation between anti-podal patches; (b) the anomalous exponents are independent of W 1 . Further, we note that, had we set g 1 = 0, we would have obtained IR divergences from the same sources, since the processes relevant for generating the divergences are one dimensional. Therefore, the effect of g 1 is parametric in nature as it does not generate new IR divergences.
Appendix E: Propagators for the general U α,β model
In this appendix we derive the propagators for the phase and density fluctuations described by Eq. (58). The key method is a generalization of the one used in Appendix B which is based on the one developed in ref. [33]. While the phase field can be integrated out easily to obtain the effective action for the density fluctuations, the opposite is more complicated since it is generically hard to determine the structure of the elements of U −1 . Changing to the angular momentum basis, however, simplifies the procedure and the effective action for phase fluctuations is obtained without any approximations. We will first derive the propagator of the phase, and then move on to the derivation of the propagator of density fluctuations.
Propagator of the phase
We proceed in two steps: first we obtain the effective action for the phase field, then we obtain the propagator of the phase.
a. Integrating out density fluctuations
Recall that the action is given by
A −2 S = 1 2 α dk 2ik 0 ( v α · k) ϕ α (−k)ϑ α (k) + ρ 1D m ( v α · k) 2 ϑ α (−k)ϑ α (k) + A 2 V 0 g 0 ( v α · k) 2 ϕ α (−k)ϕ α (k) + 1 2 α,β l dk A 2 V 0 U l 2N cos(θ αβ l) ( v α · k)( v β · k) ϕ α (−k)ϕ β (k). (E.1)
We use the identity cos{(θ α − θ β )l} = cos(θ α l) cos(θ β l) + sin(θ α l) sin(θ β l) to express the ϕ α dependent terms in Eq. (E.1) as
A −2 S 1 ≡ 1 2 α dk 2ik 0 ( v α · k) ϕ α (−k)ϑ α (k) + A 2 V 0 g 0 ( v α · k) 2 ϕ α (−k)ϕ α (k) + 1 2 l α,β dk A 2 V 0 U l 2N c l α c l β + s l α s l β ( v α · k)( v β · k) ϕ α (−k)ϕ β (k), (E.2)
where we have introduced {c l α , s l α } ≡ {cos(θ α l), sin(θ α l)} for notational convenience. Let us introduce χ (c) l (k) = α c l α ( v α · k)ϕ α (k), χ ξ α (k) = ik 0 ( v α · k)ϑ α (k).
(E.4)
Thus S 1 takes the form
A −2 S 1 = 1 2 N −1 α=−N dk ξ α (k)ϕ α (−k) + ξ α (−k)ϕ α (k) + A 2 V 0 g 0 ( v α · k) 2 ϕ α (−k)ϕ α (k) − 1 2 N −1 l=−N dk A 2 V 0 U l 2N χ (c) l (−k)χ (c) l (k) + χ (s) l (−k)χ (s) l (k) . (E.5)
We note that ξ α (k) acts as a source for ϕ α (k). By introducing auxiliary fields A α=−N dk A 2 V 0 g 0 ( v α · k) 2 ϕ α (−k)ϕ α (k)
+ 1 2 N −1 α=−N dk ξ α (k) − (v α · k) 2N l − 1 2 l,l dk k 2 0
FIG. 3 :
3The T = 0 phase diagram as a function of bare interaction strength V 0 (measured in units of m −1 ), and two dimensional mean densityρ (measured in units of κ 2 ). The solid curve is the phase boundary which has the asymptotic form obtained in Eq.(53).
FIG. 4 :FIG. 5 :
45Schematic representation of the region R λ (shaded region) which contains the high energy modes that are integrate out to generate the effective action in Eq. Quantum fluctuations at one-loop order that contribute to the effective interaction vertex. The solid (dotted) lines represent boson propagators (bare interaction vertex).
FIG. 6 :
6The flat-patch approximation to the ring. The dotted circle is the ring-minima inFig. 2. Here it is approximated by a 2N -sided polygon with each side of length 2Λ. The shaded rectangles represent the restriction on the scatterings at each patch.
FIG. 7 :
7Suppression of corner processes. (a) Within a patch,
are patch-dependent real constants. The special case where all ϑ (0) α are equal corresponds to the global U(1) symmetry, while the case where {ϑ
FIG. 9 :
9A consequence of flat-patch approximation.
1D . The asymptotic behavior of the respective equal-time autocorrelation function of the lagrangian densities in Eqs. (47) -(49) are
l (k)| for |l| + |l | = 0.Thus U 0 U l Π (ϕ,c) l,0 (k) are the dominant elements in the matrix ([Ω (ϕ,c) (k)] −1 − l U l E (l) ), where E (l) is a 2N × 2N matrix with E (l)i,j = δ i,l δ j,l . The summand in Π
11: (a) Determination of the minimum magnitude of the external momentum, P , for which the particle-hole diagrams (PH, B, and P) do not vanish.
v α · k)ϕ α (k), and (E.3)
l
ξ α (k)ϕ α (−k) + ξ α (−k)ϕ α (k) + A 2 V 0 g 0 ( v α · k) 2 ϕ α (−k)ϕ α (k) (−k) + (c) → (s)
Appendix A: Derivation of effective action
We note that we have introduce the UV regulator, f α (k) (defined in the main text), to introduce the finiteness of the patches.
ACKNOWLEDGMENTSThis work was supported by the National Science Foundation No. DMR-1442366, and performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreements No. DMR-1157490 and No. DMR-1644779, and the State of Florida.Appendix B: Phase propagator of U α,β = 1 model In this appendix we derive the expressions of the propagators of the model Section IV B. Let us consider the actionWith the definition8)we note that the lagrangian density for ϕ α isIntegrating out ϕ α leads to,where we have used the identities,We note that M (c,s) are real and symmetric matrices.Integrating out A (c,s) l fields leads toAdding to the ϑ α dependent term of Eq. (E.1) we obtain the effective action for phase fluctuations,In order to obtain the propagator of the phase fields we need to invert the matrix G −1 ϑ (k) defined in Eq. (E.21). Let us first introduce sources, J (ϑ) α , for the phase fields,In the rest of the section we will derive the second equation from the first, and in the process compute the exact expression of the propagator, G ϑ (k), for any N .Let us introducewhere a (c,s) l are auxiliary fields, and X represents a column vector, while Y represents a matrix. Integrating out the phase yields,Owing to the factor of g (ϑ)α (k, V 0 g 0 ) in the denominator, we cannot simply sum over α in the 3rd line of Eq. (E.29). We could, however, use parity under α → α + N to eliminate cross-terms, c l α s l α . Let us define,such that after integrating out a (c,s) we obtain,l,l c l α c l β +Ω (ϑ,s) l,l s l α s l β .(E.33) Therefore, G (ϑ) (k) is the propagator for the phase fluctuations.Propagator of density fluctuationsHere we derive the propagator for density fluctuations. We start with the action in Eq. (E.1), and integrate out the ϑ α fields. Since the action is diagonal in ϑ α , this is straightforward, and in the presene of sources, J (ϕ) α , we obtainIn order to decouple the off-diagonal terms in ϕ α , we introduce auxiliary fields, a (c,s) l , which act as sources for,Integrating out ϕ α leads to.By symmetry under α → α + N the cross terms, containing c l α s l α , vanish when summed over α. Let us define,and integrate out the auxiliary fields to obtain,
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[
"Multiple Positive Solutions for Nonlocal Elliptic Problems Involving the Hardy Potential and Concave-Convex Nonlinearities",
"Multiple Positive Solutions for Nonlocal Elliptic Problems Involving the Hardy Potential and Concave-Convex Nonlinearities"
] |
[
"Shaya Shakerian [email protected] \nDepartment of Mathematics\nUniversity of British Columbia Vancouver BC\nV6T 1Z2Canada\n"
] |
[
"Department of Mathematics\nUniversity of British Columbia Vancouver BC\nV6T 1Z2Canada"
] |
[] |
In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities:where Ω ⊂ R n is a smooth bounded domain in R n containing 0 in its interior, and f, g ∈ C(Ω) with f + , g + ≡ 0 which may change sign in Ω. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for λ sufficiently small. The variational approach requires that 0 < α < 2, 0 < s < α < n, 1 < q < 2 < p ≤ 2 * α (s) := 2(n−s) n−α , and γ < γH (α), the latter being the best fractional Hardy constant on R n . 4 ) Γ 2 ( n−α 4 ) is the best constant in the above fractional Hardy inequality. Note that γ H (α) converges to the best classical Hardy constant (n−2) 2 4 when α → 2.
|
10.1142/s021919972050008x
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[
"https://arxiv.org/pdf/1708.01369v1.pdf"
] | 119,334,447 |
1708.01369
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aafffd997ffd57fbb651a68bdd1c67dcc0b8acce
|
Multiple Positive Solutions for Nonlocal Elliptic Problems Involving the Hardy Potential and Concave-Convex Nonlinearities
4 Aug 2017 August 7, 2017
Shaya Shakerian [email protected]
Department of Mathematics
University of British Columbia Vancouver BC
V6T 1Z2Canada
Multiple Positive Solutions for Nonlocal Elliptic Problems Involving the Hardy Potential and Concave-Convex Nonlinearities
4 Aug 2017 August 7, 2017
In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities:where Ω ⊂ R n is a smooth bounded domain in R n containing 0 in its interior, and f, g ∈ C(Ω) with f + , g + ≡ 0 which may change sign in Ω. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for λ sufficiently small. The variational approach requires that 0 < α < 2, 0 < s < α < n, 1 < q < 2 < p ≤ 2 * α (s) := 2(n−s) n−α , and γ < γH (α), the latter being the best fractional Hardy constant on R n . 4 ) Γ 2 ( n−α 4 ) is the best constant in the above fractional Hardy inequality. Note that γ H (α) converges to the best classical Hardy constant (n−2) 2 4 when α → 2.
Introduction
In this paper, we investigate the multiplicity results of positive solutions for the following fractional elliptic problem involving the Hardy potential and concave-convex non-linearities:
(−∆) α 2 u − γ u |x| α = λf (x)|u| q−2 u + g(x) |u| p−2 u |x| s in Ω u = 0 in R n \ Ω,(1)
where Ω ⊂ R n is a smooth bounded domain in R n containing 0 in its interior, 0 < α < 2, 0 < s < α < n, 1 < q < 2 < p ≤ 2 * α (s) := 2(n−s) n−α , γ < γ H (α) = 2 α Γ 2 ( n+α 4 ) Γ 2 ( n−α 4 ) , the later being the best fractional Hardy constant on R n , and f, g ∈ C(Ω) with f + , g + ≡ 0 (they are possibly change sign in Ω).
Addressing the questions regarding the effect of concave-convex non-linearities on the number of positive solutions for non-local elliptic problems has been the subject of several studies; see [1][2][3][4] and [17]. studied the sub-critical case of (1), and proved that there exists Λ > 0 such that the problem has at least two solutions for all 0 < λ < Λ when f (x) = g(x) ≡ 1, s = 0, and γ < γ H (α). The same results have been obtained by Barrios et al. in [2] in the critical case, but in the absence of the Hardy and singularity terms, i.e., when γ = s = 0. Recently, Zhang-Liu-Jiao [25] extended the results of [2] to the problem involving the sign-changing weight function f (x) ∈ C(Ω) with f + ≡ 0, and g(x) ≡ 1. When α = 2, i.e., in the case of the standard Laplacian, problem (1) has been studied extensively in the last decade; see for example [18], [19], [20], [24] and references therein. We point out that the non-local problems are still much less understood than their local counterpart. The aim of this paper is to consider the remaining cases and generalize the results of [4] and [25] to the problem involving the Hardy potential, Hardy-Sobolev singularity term, and also signchanging functions f (x) and g(x). More pricesly, we study problem (1) in the sub-critical (i.e., when 2 < p < 2 * α (s)) and critical (i.e., when p = 2 * α (s)) case, separately. Using the decomposition of the Nehari manifold as λ varies, introduced by Tarantello [24], we will prove that the problem has at least two positive solutions for λ sufficiently small. We first prove the following theorem: Theorem 1.1. Let 0 < α < 2 and 0 ≤ s < α < n. Suppose 1 < q < 2 < p < 2 * α (s) and γ < γ H (α). Then, there exits Λ > 0 such that problem (1) has at least two positive solutions for any λ ∈ (0, Λ).
The critical case is more challenging and requires information about the asymptotic behaviour of solutions of the following limiting problem at zero and infinity:
(−∆) α 2 u − γ u |x| α = u 2 * α (s)−1 |x| s in R n u ≥ 0 in R n ,(2)
where 0 < α < 2, 0 ≤ s < α, 2 * α (s) = 2(n−s) n−α , 0 ≤ γ < γ H (α) = 2 α Γ 2 ( n+α 4 ) Γ 2 ( n−α 4 ) . We get around the difficulty by working with certain asymptotic estimates for solutions of (2) recently obtained by the author and et al. in [14]; see Theorem 5.1. In order to use the results of [14], we may assume g(x) ≡ 1. Problem (1) therefore can be written as follows:
(−∆) α 2 u − γ u |x| α = λf (x)|u| q−2 u + |u| 2 * α (s)−2 u |x| s in Ω u = 0 in R n \ Ω.(3)
We then establish the following:
Theorem 1.2. Let 0 < α < 2 and 0 ≤ s < α < n. Suppose 1 < q < 2 < p = 2 * α (s) and 0 ≤ γ < γ H (α). Then, there exits Λ * > 0 such that problem (3) has at least two positive solutions for any λ ∈ (0, Λ * ).
Functional Setting
We start by recalling and introducing suitable function spaces for the variational principles that will be needed in the sequel. We first recall that the non-local operator (−∆) α 2 is defined as
(−∆) α 2 u(x) = c(n, α)P.V. R n u(x) − u(y) |x − y| n+α dy for x ∈ R n , where c(n, α) = 2 α−1 Γ n+α 2 π n 2 Γ − α 2 .
We denote by H α 2 (R n ) the classical fractional Sobolev space endowed with the so-called Gagliardo norm
u H α 2 (R n ) = u L 2 (R n ) + R n R n |u(x) − u(y)| 2 |x − y| n+α dxdy 1 2 .
For α ∈ (0, 2), the fractional Sobolev space H
α 2 0 (R n ) is defined as the completion of C ∞ c (R n ) under the norm u 2 H α 2 0 (R n ) = R n |2πξ| α |F u(ξ)| 2 dξ = R n |(−∆) α 4 u| 2 dx.
Let now Ω ⊂ R n be a smooth bounded domain. We define the space X α 2 0 (Ω) as
X α 2 0 (Ω) = u ∈ H α 2 (R n ) : u = 0 a.e. in R n \ Ω ,
and consider the following norm in X
α 2 0 (Ω) : u X α 2 0 (Ω) = R n R n |u(x) − u(y)| 2 |x − y| n+α dxdy 1 2 .
We recall that (X
α 2 0 (Ω), . X α 2 0 (Ω)
) is a Hilbert space with the scalar product
u, v X α 2 0 (Ω) = R n R n (u(x) − u(y))(v(x) − v(y)) |x − y| n+α dxdy.
Remark 2.1. It was shown in [5] that the sub-space C ∞ 0 (Ω) is dense in X α 2 0 (Ω). So, we can consider X α 2 0 (Ω) as the completion of C ∞ 0 (Ω) with with respect to the norm . .
Definition 2.2. We say u ∈ X α 2 0 (Ω) is a weak solution of (1), if for every φ ∈ X α 2 0 (Ω), we have c(n, α) u, φ X α 2 0 (Ω) − γ Ω uφ |x| α dx = λ Ω f (x)|u| q−1 φdx − Ω g(x) |u| p−1 φ |x| s dx.
The energy functional corresponding to (1) is
I λ,p (u) = c(n, α) 2 u 2 X α 0 (Ω) − γ 2 Ω |u| 2 |x| α dx − λ q Ω f (x)|u| q dx − 1 p Ω g(x) |u| p |x| s dx.(4)
Recall that any critical point u of I λ,p (u) is a weak solution for (1). The starting point of the study of existence of weak solutions to problem (1) is therefore the following fractional inequalities which will guarantee that the above functional is well defined and bounded below on the right function spaces. We start with the fractional Sobolev inequality [10], which asserts that for n > α and 0 < α < 2, there exists a constant S(n, α) > 0 such that
u 2 L 2 * α (R n ) ≤ S(n, α) R n R n |u(x)−u(y)| 2 |x−y| n+α dxdy for u ∈ H α 2 (R n ),
where 2 * α = 2n n−α . Another important inequality is the fractional Hardy inequality [16], which states that under the same conditions on n and α, there exists a constant γ(n, α) > 0 such that
γ H (α) R n |u| 2 |x| α dx ≤ R n |ξ| α |F u(ξ)| 2 dξ for u ∈ C ∞ 0 (R n ),(5)
where F u(x) = 1 (2π) n 2 R n e −iξx u(x)dx is the Fourier transform of u. It was shown in [16] that γ H (α) := 2 α Γ 2 ( n+α By Proposition 3.6 in [11], for any u ∈ H α 2 (R n ), we have the following relation between the fractional Laplacian operator (−∆) α 2 and the fractional Sobolev space H α 2 (R n ) :
R n |ξ| α |F u(ξ)| 2 dξ = c(n, α) R n R n |u(x) − u(y)| 2 |x − y| n+α dxdy.(6)
The fractional Hard inequality then can be written as
γ H (α) R n |u| 2 |x| α dx ≤ c(n, α) R n R n |u(x) − u(y)| 2 |x − y| n+α dxdy for u ∈ C ∞ 0 (R n ).(7)
By interpolating these inequalities via Hölder's inequalities, one gets the following fractional Hardy-Sobolev inequality.
Lemma 2.3 (Lemma 2.1 in [13]). Assume that 0 < α < 2, 0 ≤ s ≤ α < n and 2 < p ≤ 2 * α (s) = 2(n−s) n−α . Then, there exists a positive constant C such that
C( Ω |u| p |x| s dx) 2 p ≤ c(n, α) R n R n |u(x) − u(y)| 2 |x − y| n+α dxdy − γ Ω |u| 2 |x| α dx for u ∈ X α 2 0 (Ω),(8)
as long as γ < γ H (α) :
= 2 α Γ 2 ( n+α 4 ) Γ 2 ( n−α 4 ) .
Finally, we can define the general best Hardy-sobolev constant in the above inequality as
S p := inf u∈X α 2 0 (Ω)\{0} c(n, α) R n R n |u(x)−u(y)| 2 |x−y| n+α dxdy − γ Ω |u| 2 |x| α dx ( Ω |u| p |x| s dx) 2 p ,(9)
where 2 < p ≤ 2 * α (s) = 2(n−s) n−α , and γ < γ H (α). Note that the frational Hardy inequality (7) asserts that X α 2 0 (Ω) is embedded in the weighted space L 2 (Ω, |x| −α ) and that this embeding is continuous. If γ < γ H (α), it follows from the fractional Hardy inequality (7) that u := c(n, α)
R n R n |u(x) − u(y)| 2 |x − y| n+α dxdy − γ Ω |u| 2 |x| α dx 1 2
is well-defined on X α 2 0 (Ω). It also is equivalent to the norm . . Thus, we can rewrite the functional I λ,p as
I λ,p (u) = 1 2 u 2 − λ q Ω f (x)|u| q dx − 1 p Ω g(x) |u| p |x| s dx.
Preliminary Results
For λ > 0, we will consider the following Nehari minimization problem:
M = inf{I λ,p (u) : u ∈ N }, where N = {u ∈ X α 2 0 (Ω) : I ′ λ,p (u), u = 0}. Define φ λ,p (u) := I ′ λ,p (u), u = u 2 − λ Ω f (x)|u| q dx − Ω g(x) |u| p |x| s dx. So, φ ′ λ,p (u), u = 2 u 2 − λ q Ω f (x)|u| q dx − p Ω g(x) |u| p |x| s dx.(10)
Thus, for all u ∈ N , we have the following identities which will be used frequently in this paper.
u 2 = λ Ω f (x)|u| q dx + Ω g(x) |u| p |x| s dx,(11)I λ,p (u) = ( 1 2 − 1 q ) u 2 − ( 1 p − 1 q ) Ω g(x) |u| p |x| s dx = ( 1 2 − 1 p ) u 2 + ( 1 p − 1 q ) λ Ω f (x)|u| q dx,(12)
and
φ ′ λ,p (u), u = (2 − q) u 2 − (p − q) Ω g(x) |u| p |x| s dx = (2 − p) u 2 − λ (q − p) Ω f (x)|u| q dx.(13)
Now we split N into three parts:
N + = {u ∈ N : φ ′ λ,p (u), u > 0}, N 0 = {u ∈ N : φ ′ λ,p (u), u = 0}, N − = {u ∈ N : φ ′ λ,p (u), u < 0}.
We first show that for λ small enough, N 0 is an empty set.
Lemma 3.1. There exists a constant Λ 1 := Λ 1 (p) > 0 such that for any λ ∈ (0, Λ 1 ), we have N 0 = ∅.
Proof. We deduce by contradiction. Suppose that there exists u ∈ X α 2 0 (Ω) \ {0} such that u ∈ N 0 , that is, φ ′ (u), u = 0. We will consider the two following cases:
Case 1: Ω f (x)|u| q dx = 0. Using (13) and the fact that Ω f (x)|u| q dx = 0, we get
0 = φ ′ λ,p (u), u = (2 − p) u 2 .
On the other hand, the assumption p > 2 implies that (2 − p) u 2 < 0, which contradicts the last equality.
Case 2: Ω f (x)|u| q dx = 0. It follows from (13) that
u 2 = λ( p − q p − 2 ) Ω f (x)|u| q dx.(14)
By the definition of S p and the Hölder inequality, we get that
u 2 ≤ λ( p − q p − 2 )A Ω |u| p |x| s dx q p ≤ λ( p − q p − 2 )AS − q 2 p u q ,(15)
which yields
u ≤ λ( p − q p − 2 )AS − q 2 p 1 2−q ,(16)where A := Ω |x| sq p−q |f (x)| p p−q dx p−q p
. Define the functional J λ,p : N −→ R as
J λ,p (u) = p − 2 2 − q 2 − q p − q p−1 p−2 u 2(p−1) Ω g(x) |u| p |x| s dx 1 p−2 − λ Ω f (x)|u| q dx.
We claim that J λ (u) = 0 for all u ∈ N 0 . Indeed, by (13), we have
Ω g(x) |u| p |x| s dx = 2−q p−q u 2 and λ Ω f (x)|u| q dx = p−2 2−q u 2 for all u ∈ N 0 .(17)
Thus,
J λ,p (u) = p − 2 2 − q 2 − q p − q p−1 p−2 u 2(p−1) 2−q p−q u 2 1 p−2 − λ Ω f (x)|u| q dx = p − 2 2 − q u 2 − λ Ω f (x)|u| q dx = 0 for all u ∈ N 0 .(18)Let C(p, q) := p−2 2−q 2−q p−q p−1 p−2 .
By Hölder's inequality and the definition of S p , we obtain
J λ,p (u) ≥ C(p, q) u 2(p−1) Ω g(x) |u| p |x| s dx 1 p−2 − λA Ω |u| p |x| s dx q p ≥ Ω |u| p |x| s dx q p C(p, q)S p−1 p−2 p g 1 2−p ∞ Ω |u| p |x| s dx 1−q p − λA ≥ Ω |u| p |x| s dx q p C(p, q)S p−1 p−2 + q−1 2−q p λA(p − q) p − 2 1−q 2−q g 1 2−p ∞ − λA .(19)
Thus, we get that J λ,p (u) > 0 for λ sufficiently small. Therefore, there exists Λ 1 := Λ 1 (p) > 0 such that J λ,p (u) > 0 for all λ ∈ (0, Λ 1 ) and u ∈ N 0 . This contradicts (18) and completes the proof.
Lemma 3.2. If u ∈ N + \ {0}, then Ω f (x)|u| q dx > 0. Proof. Since u ∈ N + \ {0}, we have φ ′ λ,p (u), u > 0. It then follows from (13) that 2 − q p − q u 2 > Ω g(x) |u| p |x| s dx.
By (11) and the last inequality, we get that
λ Ω f (x)|u| q dx = u 2 − Ω g(x) |u| p |x| s dx > u 2 − 2 − q p − q u 2 > p − 2 p − q u 2 > 0.
From Lemma 3.1, we deduce that N = N + ∪ N − for any λ ∈ (0, Λ 1 ). Define
M + := inf N + I λ,p (u) and M − := inf N − I λ,p (u).
Lemma 3.3. For any λ ∈ (0, Λ 1 ), the minimizers on N are critical points for I λ,p in (X
α 2 0 (Ω)) ′ , where (X α 2 0 (Ω)) ′ is dual space of X α 2 0 (Ω).
Proof. Suppose thatū is a local minimum for I λ,p . Thus, it satisfies the following minimization problem:
min u∈X α 2 0 (Ω) I λ,p (u) : φ λ,p = I ′ λ,p (u), u = 0 ,
which gives
I λ,p (ū) = min u∈X α 2 0 (Ω) I λ,p (u) and φ λ,p (ū) = I ′ λ,p (ū),ū = 0.
It follows from the theorem of Lagrange multiplies that there exists θ such that
I ′ λ,p (ū) = θφ ′ λ,p (ū) in (X α 2 0 (Ω)) ′ . So, we have 0 = I ′ λ,p (ū),ū = θφ ′ λ,p (ū), u = θ φ ′ λ,p (ū),ū . Thus, either θ = 0 or φ ′ λ,p (ū),ū = 0. By Lemma 3.1, we get that φ ′ λ,p (u), u = 0 for u = 0. Therefore θ = 0. Thus, we obtain I ′ λ,p (ū) = θφ ′ λ,p (ū) = 0 in (X α 2 0 (Ω)) ′ , that is,ū is a critical point for I λ,p in (X α 2 0 (Ω)) ′ . Lemma 3.4. Let Λ 2 := Λ 2 (p) = p−2 p−q 2−q p−q 2−q p−2 S p−q p−2 p A −1 g q−2 p−2 ∞ .
Then, for all u ∈ X α 2 0 (Ω) \ {0} and λ ∈ (0, Λ 2 ), there exist unique t + (u) and t − (u) such that
1. 0 ≤ t + (u) < t max < t − (u). 2. t − (u)u ∈ N − and t + (u)u ∈ N + . 3. I λ,p (t − (u)u) = max t>tmax I λ,p (tu) and I λ,p (t + (u)u) = min 0≤t≤t − (u) I λ,p (tu). 4. N − = u ∈ X α 2 0 (Ω) \ {0} : t − ( u u ) = u , where t max := 2−q p−q u 2 Ω g(x) |u| p |x| s dx 1 p−2 . Moreover, t + (u) > 0 if and only if Ω f (x)|u| q dx > 0. Proof. For t ≥ 0, define h(t) = t 2−q u 2 − t p−q Ω g(x) |u| p |x| s dx. Straightforuard computations yield that h(0) = 0, lim t→∞ h(t) = −∞, h ′ (t max ) = 0, and h(t) is attained its maximum at t max . In addition, h(t) is increasing for t ∈ [0, t max ) and decreasing for t ∈ (t max , ∞). So, we have h(t max ) = p − 2 p − q 2 − q p − q 2−q p−2 u q u p Ω g(x) |u| p |x| s dx 2−q p−2 .
By Hölder's inequality and the definition of S p , we obtain
h(t max ) ≥ p − 2 p − q 2 − q p − q 2−q p−2 u q g q−2 p−2 ∞ S p(2−q) 2(p−2) p .(20)
We will now consider the two following cases:
Case 1: Ω f (x)|u| q dx ≤ 0. In this case, there exists a unique t − := t − (u) > t max such that h(t − ) = λ Ω f (x)|u| q dx and h ′ (t − ) < 0.(21)
We claim that t − u ∈ N − . Indeed, clearly t − u ∈ X α 2 0 (Ω), and (21) implies that
I ′ λ,p (t − u), t − u = t − u 2 − λ Ω f (x)|t − u| q dx − Ω g(x) |t − u| p |x| s dx = (t − ) q (t − ) 2−q u 2 − (t − ) p−q Ω g(x) |u| p |x| s dx − λ Ω f (x)|u| q dx = (t − ) q h(t − ) − λ Ω f (x)|u| q dx = 0, and φ ′ λ,p (t − u), t − u = 2 t − u 2 − λq Ω f (x)|t − u| q dx − p Ω g(x) |t − u| p |x| s dx = (2 − q) t − u 2 − (p − q) Ω g(x) |t − u| p |x| s dx = (t − ) q+1 (2 − q)(t − ) 2−q−1 u 2 − (p − q)(t − ) p−q−1 Ω g(x) |u| p |x| s dx = (t − ) q+1 h ′ (t − ) < 0.
This proves the claim, and we have that t − u ∈ N − . In order to prove that I λ,
p (t − u) = max t≥tmax I λ,p (tu),
we need to show that
d dt I λ,p (t − u) = 0 and d 2 dt 2 I λ,p (tu) < 0 for t > t max .(22)
It follows from (21) that
d dt I λ,p (t − u) = t − u 2 − λ(t − ) q−1 Ω f (x)|u| q dx − (t − ) p−1 Ω g(x) |u| p |x| s dx = (t − ) q−1 h(t − ) − λ Ω f (x)|u| q dx = 0.
We also have
t 2 d 2 dt 2 I λ,p (tu) = t − u 2 − λ(q − 1) Ω f (x)|t − u| q dx − (p − 1) Ω g(x) |t − u| p |x| s dx = (t − ) q+1 (2 − q)(t − ) 2−q−1 u 2 − (p − q)(t − ) p−q−1 Ω g(x) |u| p |x| s dx = (t − ) q+1 h ′ (t − ) < 0 for all t > t max .
Case 2: Ω f (x)|u| q dx > 0. Using Hölder's inequality and (20), we have
0 = h(0) < Ω f (x)|u| q dx ≤ λAS − q 2 p u q ≤ p − 2 p − q 2 − q p − q 2−q p−2 g q−2 p−2 ∞ S p(2−q) 2(p−2) p u q ≤ h(t max ) for 0 < λ < Λ 2 .
Using the assumption Ω f (x)|u| q dx > 0 and the fact that h(t max ) > 0, we get that there exist unique t + := t + (u) and t − := t − (u) such that t + < t max < t − , and
h(t − ) = λ Ω f (x)|u| q dx = h(t + ) and h ′ (t − ) < 0 < h ′ (t + ).
Lemma 3.5. The following hold.
1. M ≤ M + < 0. 2. Let Λ 3 := Λ 3 (p) = p−2 p−q .
Then, the functional I λ,p is coercive and bounded below on N for any λ ∈ (0, Λ 3 ].
Proof. By (12), for any u ∈ N , we have
I λ,p (u) = ( 1 2 − 1 p ) u 2 + ( 1 p − 1 q ) λ Ω f (x)|u| q dx.
1. Suppose that u ∈ N + . It follows from (13) that
I λ,p (u) < p − 2 2p p − q p − 2 λ Ω f (x)|u| q dx + q − p pq λ Ω f (x)|u| q dx = − (p − q)(2 − q) 2pq λ Ω f (x)|u| q dx. By Lemma 3.2, we have Ω f (x)|u| q dx > 0. Thus, I λ,p (u) < − (p − q)(2 − q) 2pq λ Ω f (x)|u| q dx < 0, which yields M ≤ M + < 0.
2. Using Hölder and Young's inequality, we get that
I λ,p (u) ≥ 1 2p ((p − 2) − λ(p − q)) u 2 − λ p − q pq 2 − q 2 AS − q 2 p 2 2−q .
Since 0 < λ < p−2 p−q , the functional I λ,p is coercive and bounded below on N , and we have
I λ,p (u) ≥ −λ p − q pq 2 − q 2 AS − q 2 p 2 2−q .
Lemma 3.6. For each u ∈ N \ {0}, there exist ǫ > 0 and a differentiable function σ :
B(0, ǫ) ⊂ X α 2 0 (Ω) −→ R + such that σ(0) = 1, σ(v)(u − v) ∈ N , and σ ′ (0), v = 2c(n, α) u, v X α 2 0 (Ω) − 2γ Ω uv |x| α dx − qλ Ω f (x)|u| q−1 uvdx − p Ω g(x) |u| p−2 uv |x| s dx (2 − q) c(n, α) R n R n |u(x)−u(y)| 2 |x−y| n+α dxdy − γ Ω |u| 2 |x| α dx − (p − q) Ω g(x) |u| p |x| s dx ,(23)for all v ∈ X α 2 0 (Ω). Here B(0, ǫ) := {u ∈ X α 2 0 (Ω) : u < ǫ}. Proof. For u ∈ N , define G : R × X α 2 0 (Ω) −→ R as G(t, v) = I ′ λ,p (t(u − v)), t(u − v) . So, we have G(t, v) = t 2 c(n, α) R n R n |u(x) − u(y)| 2 |x − y| n+α dxdy − t 2 γ Ω |u| 2 |x| α dx − λt q Ω f (x)|u − v| q dx − t p Ω g(x) |u| p |x| s dx,
and G(1, 0) = I ′ λ,p (u), u . By Lemma 3.1, we obtain d dt G(1, 0) = 0, that is,
0 = d dt G(1, 0) = 2tc(n, α) R n R n |u(x) − u(y)| 2 |x − y| n+α dxdy − 2tγ Ω |u| 2 |x| α dx − qλt q−1 Ω f (x)|u| q dx − pt p−1 Ω g(x) |u| p |x| s dx t=1 = (2 − q) u 2 − (p − q) Ω g(x) |u| p |x| s dx.
According to the implicit function theorem, there exist ǫ > 0 and a differentiable function σ :
B(0, ǫ) −→ R such that σ(0) = 1, and σ ′ (0), v = 2c(n, α) u, v X α 2 0 (Ω) − 2γ Ω uv |x| α dx − qλ Ω f (x)|u| q−1 uvdx − p Ω g(x) |u| p−2 uv |x| s dx (2 − q) c(n, α) R n R n |u(x)−u(y)| 2 |x−y| n+α dxdy − γ Ω |u| 2 |x| α dx − (p − q) Ω g(x) |u| p |x| s dx . Moreover, we have G(σ(v), v) = 0, for all v ∈ B(0, ǫ), which implies that I ′ λ,p (σ(v)(u − v)), σ(v)(u − v)) = 0, that is, σ(v)(u − v) ∈ N .0 (Ω) −→ R + such that σ − (0) = 1, σ − (v)(u − v) ∈ N − , and σ − ′ (0), v = 2c(n, α) u, v X α 2 0 (Ω) − 2γ Ω uv |x| α dx − qλ Ω f (x)|u| q−1 uvdx − p Ω g(x) |u| p−2 uv |x| s dx (2 − q) c(n, α) R n R n |u(x)−u(y)| 2 |x−y| n+α dxdy − γ Ω |u| 2 |x| α dx − (p − q) Ω g(x) |u| p |x| s dx ,(24)
for all v ∈ X α 2 0 (Ω). Proof. Following the proof of Lemma 3.6, we get that there exist ǫ > 0 and a differentiable function
σ − : B(0, ǫ) −→ R such that σ − (0) = 1 and σ − (v)(u − v) ∈ N , for all v ∈ B(0, ǫ). Since u ∈ N − , we have φ ′ (u), u = (2 − q) u 2 − (p − q) Ω g(x)
|u| p |x| s dx < 0.
It then follows from the continuity of φ ′ and σ − that
φ ′ (σ − (v)(u − v)), σ − (v)(u − v) = (2 − q) σ − (v)(u − v) 2 − (p − q) Ω g(x) |σ − (v)(u − v)| p |x| s dx.
Therefore, for ǫ > 0 small enough, we get that
σ − (v)(u − v) ∈ N − . Proposition 3.8. Let Λ = Λ(p) := min{Λ 1 , Λ 2 , Λ 3 }.
Then, for any λ ∈ (0, Λ), the following hold.
1. There exists a minimizing sequence (u k ) k∈N ⊂ N for I λ,p (u) such that
• I λ,p (u k ) = M + o(1). • I ′ λ,p (u k ) = o(1) in (X α 2 0 (Ω)) ′ .
2. There exists a minimizing sequence (u k ) k∈N ⊂ N − for I λ,p (u) such that
• I λ,p (u k ) = M − + o(1). • I ′ λ,p (u k ) = o(1) in (X α 2 0 (Ω)) ′ .
Proof. It follows from Lemma 3.5 that I λ,p (u) is coercive and bounded below. Then, 1. The Ekeland variational principle implies that there exists a minimizing sequence (u k ) k∈N such that
I λ,p (u k ) < inf N I λ,p (u) + 1 k = M + 1 k and I λ,p (u k ) < I λ,p (u) + 1 k u − u k for all u ∈ N .(25)
For k large enough, we use Lemma 3.5, (12) and (25) to get
( 1 p − 1 q ) λ Ω f (x)|u k | q dx ≤ I λ,p (u k ) = ( 1 2 − 1 p ) u k 2 + ( 1 p − 1 q ) λ Ω f (x)|u k | q dx < M + 1 n < M < M 2 < 0. (26) Therefore, − M 2 pq p − q < λ Ω f (x)|u k | q dx ≤ AS − q 2 p u k q ,
which yields u k = 0, for all k ∈ N. On the other hand, from (26) and Hölder's inequality, we deduce that
u k 2 < 2λ(p − q) q(p − 2) AS − q 2 p w k q .
Hence,
− M 2 pq p − q A −1 S q 2 p 1 q < u k < 2λ(p − q) q(p − 2) AS − q 2 p 1 2−q .(27)
In order to finalize the proof, it is sufficient to show that
I ′ λ,p (u k ) (X α 2 0 (Ω)) ′ → 0 as k → ∞.(28)
Indeed, it follows from Lemma 3.6 that there exists a differentiable function σ k :
B k (0, ǫ k ) −→ R + , for some ǫ k , such that σ k (u k − u) ∈ N for all k ∈ N,
where B k (0, ǫ k ) := {u ∈ X α 2 0 (Ω) : u < ǫ k }. Choose 0 < ρ < ǫ k , and for any u ∈ X α 2 0 (Ω) \ {0}, define u ρ := ρu u and η ρ := σ k (ρ k )(u k − u). Using the fact that η ρ ∈ N , and also (25) 2 , we get that
I λ,p (u k ) < I λ,p (η ρ ) + 1 k η ρ − u k , which means I λ,p (η ρ ) − I λ,p (u k ) > − 1 k η ρ − u k .(29)
Now we apply the mean value theorem to the left hand-side of the last inequality to deduce
I λ,p (η ρ ) − I λ,p (u k ) = I ′ λ,p (u k ), η ρ − u k + o( η ρ − u k ). Thus, I ′ λ,p (u k ), η ρ − u k + o( η k − u k ) ≥ − 1 k η ρ − u k .(30)
Regarding the first term in (30), we have that
I ′ λ,p (u k ), η ρ − u k = I ′ λ,p (u k ), σ k (u ρ )(u k − u ρ ) − u k = I ′ λ,p (u k ), σ k (u ρ )(u k − u ρ ) + (u ρ − u k ) − u ρ = I ′ λ,p (u k ), −u ρ + I ′ λ,p (u k ), (σ k (u ρ ) − 1)(u k − u ρ ) .
Therefore,
I ′ λ,p (u k ), −u ρ + I ′ λ,p (u k ), (σ k (u ρ ) − 1)(u k − u ρ ) ≥ − 1 k η ρ − u k .(31)
By the definition of u ρ and η ρ , we obtain
−ρ I ′ λ,p (u k ), u u + (σ k (u ρ ) − 1) I ′ λ,p (u k ) − I ′ λ,p (η ρ ), (u k − u ρ ) ≥ − 1 k η ρ − u k + o( η ρ − u k ).
The last inequality implies that
I ′ λ,p (u k ), u u ≤ 1 ρk η ρ − u k + o( η ρ − u k ) ρ + (σ k (u ρ ) − 1) ρ I ′ λ,p (u k ) − I ′ λ,p (η ρ ), (u k − u ρ ) . (32)
Note that from Lemma 3.6, it follows
lim ρ→0 |(σ k (u ρ ) − 1)| ρ = | σ ′ (0), u ρ | ρ ≤ σ ′ (0) ,
and also simple computations yield
η ρ − u k = σ k (u ρ )(u k − u ρ )u k = (σ k (u ρ ) − 1)u k − u ρ σ k (u ρ ) ≤ (σ k (u ρ ) − 1)u k + u ρ σ k (u ρ ) = |σ k (u ρ ) − 1| u k + |σ k (u ρ )|ρ.
Using the last two identities, and (32), we then get that
I ′ λ,p (u k ), u u ≤ 1 k |σ k (u ρ )| + 1 k |σ k (u ρ ) − 1| ρ u k + (σ k (u ρ ) − 1) ρ I ′ λ,p (u k ) − I ′ λ,p (η ρ ), (u k − u ρ ) .
Taking σ → 0 in the last inequality for a fixed k, and using (27), we obtain that there exists a constant C > 0 (independent of ρ) such that
I ′ λ,p (u k ), u u ≤ C k (1 + σ ′ (0) ) as σ → 0.
In order to complete the proof of (28), we only need to show that σ ′ (0) is uniformly bounded in k. It follows from (23) and Hölder's inequality that there exists a constant c > 0 such that
σ ′ (0), v ≤ c (2 − q) u k 2 − (p − q) Ω g(x) |u k | p |x| s dx .
It remains to prove that there exists a constantc > 0 such that
(2 − q) u k 2 − (p − q) Ω g(x)
|u k | p |x| s dx >c for n large enough.
We deduce by contradiction. Suppose that there exists a sub-sequence (u k ) k∈N such that
(2 − q) u k 2 − (p − q) Ω g(x) |u k | p |x| s dx = o(1) as k → ∞.(34)
Then, (27) and (34) yield
(p − q) Ω g(x) |u k | p |x| s dx = (2 − q) u k 2 + o(1) ≥ (2 − q) − M 2 pq p − q A −1 S q 2 p 1 q + o(1) as k → ∞,
which implies that there exists a constant C 1 > 0 such that
Ω g(x) |u k | p |x| s dx ≥ C 1 > 0.(35)
In addition, by (34) and the fact that (u k ) k∈N ∈ N , we have
λ Ω f (x)|u k | q dx = u k 2 − Ω g(x) |u k | p |x| s dx = u k 2 − 2 − q p − q u k 2 + o(1) = p − 2 p − q u k 2 + o(1) as k → ∞.
Hence,
u k = λ( p − q p − 2 ) Ω f (x)|u k | q dx 1 2 + o(1) ≤ λ( p − q p − 2 )AS − q 2 p 1 2−q + o(1) as k → ∞.
Following the last part of the proof of Lemma 3.1, we get that J λ,p (u k ) = o(1) as k → ∞. On the other hand, we use (35) and the fact that λ ∈ (0, Λ) to get
J λ,p (u k ) ≥ C(p, q) u k 2(p−1) Ω g(x) |u k | p |x| s dx 1 p−2 − λA Ω |u k | p |x| s dx q p ≥ Ω |u k | p |x| s dx q p C(p, q)S p−1 p−2 + q−1 2−q p λA(p − q) p − 2 1−q 2−q g 1 2−p ∞ − λA . > 0,(36)
which contradicts J λ,p (u k ) = o(1) as k → ∞. Therefore, (33) holds, and there exists a constant b > 0 such that
I ′ λ,p (u k ), u u ≤ b k .
This implies (28), and completes the proof.
2. The proof goes exactly as the first part using Lemma 3.7.
Proof of Theorem 1.1
In this section, we use the results in section 3 to prove the existence of a positive solution on N + , as well as on N + . This coupled with the fact that N − ∩ N + = ∅ yield Theorem 1.1. 2. u + is positive solution of (1).
3. I λ,p (u + ) → 0 as λ → 0.
Proof. Let (u k ) k∈N ⊂ N be a minimizing sequence for I λ,p such that I λ,p (u) = M + o(1) and I ′ λ,p (u) = o(1) in (X α 2 0 (Ω)) ′ , given in the first part of Proposition 3.8. It then follows from Lemma 3.5 and the fractional Sobolev embedding that there exists a sub-sequence (u k ) k∈N -still denote by u k -and u + ∈ X α 2 0 (Ω) such that
u k ⇀ u + weakly in X α 2 0 (Ω) u k → u + strongly in L r (Ω) for every 1 ≤ r < 2 * α .(37)
We first show that Ω f (x)|u + | q dx = 0. Indeed, suppose Ω f (x)|u + | q dx = 0. Then, by (37) 2 , and the fact that 1 < q < 2 < 2 * α , we obtain
Ω f (x)|u k | q dx → Ω f (x)|u + | q dx = 0 as k → ∞,
which means
Ω f (x)|u k | q dx = o(1) as k → ∞.
Thus,
u k 2 = λ Ω f (x)|u k | q dx + Ω g(x) |u k | p |x| s dx = Ω g(x) |u k | p |x| s dx + o(1) as k → ∞, and I λ,p (u k ) = 1 2 u k 2 − λ q Ω f (x)|u k | q dx − 1 p Ω g(x) |u k | p |x| s dx = 1 2 − 1 p Ω g(x) |u k | p |x| s dx + o(1) as k → ∞.
On the other hand,
I λ,p (u) = M + o(1) < 0 as k → ∞.
This leads us to the following contradiction:
0 ≤ p − 2 2p u k 2 + o(1) = I λ,p (u k ) = M = o(1) < 0.
Hence,
Ω f (x)|u + | q dx = 0.
We now prove that u k → u + strongly in X
I λ,p (u) ≤ I λ,p (u + ) = 1 2 u + 2 − λ q Ω f (x)|u + | q dx − 1 p Ω g(x) |u + | p |x| s dx = 1 2 − 1 p Ω g(x) |u + | p |x| s dx + 1 p − 1 q λ Ω f (x)|u + | q dx ≤ 1 2 − 1 p Ω g(x) |u k | p |x| s dx + 1 p − 1 q λ Ω f (x)|u k | q dx = M.
This yields I λ,p (u + ) = M, and u k → u + strongly in X α 2 0 (Ω). The next step is to prove that u + ∈ N + . Assume that u + ∈ N − . It then follows from Lemma 3.4 that there exist t − and t + such that t − u + ∈ N − , t + u + ∈ N + and t + < t − = 1. Following the proof of Lemma 3.4, we have that d dt I λ,p (t + u) = 0 and d 2 dt 2 I λ,p (t + u) > 0. Thus, there exists at such that t + <t < t − = 1 and I λ,p (t + u + ) < I λ,p (tu + ). We again use Lemma 3.4 to get
I λ,p (t + u + ) < I λ,p (tu + ) ≤ I λ,p (t − u + ) = I λ,p (u + ),
which is in contradiction with I λ,p (u + ) = M. Therefore, u + ∈ N + , and I λ,p (u + ) = M = M + . Since I λ,p (u + ) = I λ,p (|u + |), and |u + | ∈ N + is a solution for (1), without loss of generality, we may assume that u + is a non-negative solution of (1), and strong maximum principle [23, Proposition 2.2.8] implies that u + > 0 in Ω. To complete the proof of Theorem 4.1, we need to show that I λ,p (u + ) → 0 as λ → 0. From Lemma 3.5, it follows
−λ p − q pq 2 − q 2 AS − q 2 p 2 2−q ≤ I λ,p (u + ) < M < 0.
Thus, I λ,p (u + ) → 0 as λ → 0. 2. u − is a positive solution of (1).
Proof. Let (u k ) k∈N ⊂ N − be a minimizing sequence for I λ,p such that I λ,p (u) = M − + o(1) and I ′ λ,p (u) = o(1) in (X α 2 0 (Ω)) ′ , given in the second part of Proposition 3.8. It then follows from Lemma 3.5 and the fractional Sobolev embedding that there exists a sub-sequence (u k ) k∈N -still denote by u k -and u − ∈ X α 2 0 (Ω) such that
u k ⇀ u − weakly in X α 2 0 (Ω) u k → u − strongly in L r (Ω) for every 1 ≤ r < 2 * α .(38)
We prove that u k → u − in X α 2 0 (Ω). Indeed, if not, then we have u − < lim inf k→∞ u k . Therefore,
I ′ λ,p (u − ), u − = u − 2 − λ Ω f (x)|u − | q dx − Ω g(x) |u − | p |x| s dx < lim inf k→∞ u k 2 − λ Ω f (x)|u k | q dx − Ω g(x) |u k | p |x| s dx = 0,
which contradicts u − ∈ N − . This implies that u k → u − in X α 2 0 (Ω), and therefore I λ,p (u − ) = M − . Since I λ,p (u − ) = I λ,p (|u − |), and |u − | ∈ N − is a solution for (1), without loss of generality, we may assume that u − is a non-negative solution of (1), and the maximum principle [23,Proposition 2.2.8] implies that u − > 0 in Ω.
Proof of Theorem 1.1. It follows from Theorems 4.1 and 4.2 that there exist two positive solutions u + and u − such that u + ∈ N + and u − ∈ N − . In addition, by Lemma 3.1, N + ∩ N − = ∅. Thus, u + and u − are two distinct positive solutions for (1).
Proof of Theorem 1.2
Throughout this section, we shall assume that p = 2 * α (s) and g(x) ≡ 1.
We also use the following notations for simplicity: Λ * := Λ(2 * α (s)) and I λ (u) := I λ,2 * α (s) (u). We point out that all results (i.e., Lemmas, Propositions and Theorems) stated in the previous sections hold under condition (39). The first step to prove Theorem 1.2 is to study the existence and asymptotic behavior of the weak solutions to the following borderline problem associated with the fractional Hardy-Schrödinger operator (−∆)
α 2 − γ |x| α on R n : (−∆) α 2 u − γ u |x| α = u 2 * α (s)−1 |x| s in R n u > 0 in R n ,(40)
where 0 < α < 2, 0 ≤ s < α, 2 * α (s) = 2(n−s) n−α , 0 ≤ γ < γ H (α) = 2 α Γ 2 ( n+α 4 ) Γ 2 ( n−α 4 ) . The existence of the weak solutions to (40) was proved in [13]. Recently, the author and et al. in [14] have proved the following results regarding the asymptotic behavior of such solutions which play a crucial role in this section: Theorem 5.1 (Theorem 1.2 in [14]). Assume 0 ≤ s < α < 2, n > α and 0 ≤ γ < γ H (α). Then, any positive solution u ∈ H α 2 0 (R n ) of (40) satisfies u ∈ C 1 (R n \ {0}) and
lim x→0 |x| β−(γ) u(x) = λ 0 and lim |x|→∞ |x| β+(γ) u(x) = λ ∞ ,(41)
where λ 0 , λ ∞ > 0 and β − (γ) (resp., β + (γ)) is the unique solution in 0, n−α 2 (resp., in n−α 2 , n − α ) of the equation
Ψ n,α (t) := 2 α Γ t+α 2 Γ n−t 2 Γ n−t−α 2 Γ t 2 = γ,
with β − (0) = 0, and β + (0) = n − α.
We refer the readers to Section 2 in [14] for the definition and properties of β + (γ) and β − (γ) in detail Let u * (x) be a positive weak solution of (3). For any ǫ > 0, we define u ǫ (x) = ǫ α−n 2 u * ( x ǫ ) in R n . It is easy to show that u ǫ (x) is also a solution of (3). From the assumption on f , we know that f is a continuous function, and also f + (x) = max{f (x), 0} ≡ 0. Let Σ := {x ∈ Ω : f (x) > 0} be an open set of positive measure. Now we need to define appropriate cut-off function. Let η ∈ C ∞ 0 (Σ) be a positive cut-off function satisfying 0 ≤ η ≤ 1 in Σ. In addition, we choose ρ > 0 small enough such that B c 2ρ ⊂ Σ, η ≡ 1 in B ρ , and η ≡ 0 in B c 2ρ . One can check that ηu ǫ (x) is in X α 2 0 (Ω). For any ǫ > 0, we define
U ǫ (x) = η(x)u ǫ (x) for x ∈ R n .(42)
The following lemma is a direct consequence of the computations in Section 6.1 in [14]:
Lemma 5.2.
Assume that U ǫ defined as (42), and that u 1 be a positive solution of (3). Then, for every ǫ > 0 small enough, we have
(i) U ǫ 2 ≤ u ǫ 2 + O(ǫ β+(γ)−β−(γ) ). (ii) Ω |Uǫ| 2 * α (s) |x| s dx = Ω |uǫ| 2 * α (s) |x| s dx + o(ǫ β+(γ)−β−(γ) ).
We need the following two lemmas in order to prove Theorem 1.2.
Lemma 5.3. Assume that U ǫ defined as (42), and that u 1 be the local minimum in Theorem 5.4. Then, for every ǫ > 0 small enough, we have
Ω |u 1 + tU ǫ | 2 * α (s) |x| s dx = Ω |u 1 | 2 * α (s) |x| s dx + Ω |tU ǫ | 2 * α (s) |x| s dx + 2 * α (s)t Ω |u 1 | 2 * α (s)−2 |x| s U ǫ u 1 dx + 2 * α (s)t 2 * α (s)−1 Ω |U ǫ | 2 * α (s)−2 |x| s U ǫ u 1 dx + o(ǫ β + (γ)−β − (γ) 2
).
(43)
Proof. The proof goes exactly as (17) in [7,Theorem 1] with only minor modifications. We omit it here.
The existence of a minimizer on N +
In the following theorem, we prove the existence of a positive solution of (3) on N + .
Theorem 5.4. For any λ ∈ (0, Λ * ), there exists a minimizer u 1 ∈ N + for the functional I λ which verifies
1. I λ (u 1 ) = M = M + .
2. u 1 is positive solution of (3).
3. I λ (u 1 ) → 0 as λ → 0.
Proof. The proof is a straightforward consequence of Theorem 4.1 with p = 2 * α (s).
The existence of a minimizer on N −
In obtaining the existence result on N − , it is crucial to have the (P.S) conditions for all level σ < M + α−s 2(n−s) S n−s α−s p , which will be shown in the next two lemmas. Proof. We first note that
I λ (u 1 + tU ǫ ) = 1 2 u 1 + tU ǫ 2 − λ q Ω f (x)|u 1 + tU ǫ | q dx − 1 2 * α (s) Ω |u 1 + tU ǫ | 2 * α (s) |x| s dx.
On the other hand, simple computations yield
u 1 + tU ǫ 2 = u 1 2 + t 2 U ǫ 2 + 2t u 1 , U ǫ X α 2 0 (Ω) − 2γ Ω u 1 U ǫ |x| α dx.
Thus,
I λ (u 1 + tU ǫ ) = 1 2 u 1 2 + t 2 2 U ǫ 2 + t u 1 , U ǫ X α 2 0 (Ω) − γ Ω u 1 U ǫ |x| α dx − λ q Ω f (x)|u 1 + tU ǫ | q dx − 1 2 * α (s) Ω |u 1 + tU ǫ | 2 * α (s) |x| s dx.(44)
Now we deal with each terms separately: Regarding the first term, since u 1 is a minimizer for I λ , we have
1 2 u 1 2 = I λ (u 1 ) + λ q Ω f (x)|u 1 | q dx + 1 2 * α (s) Ω |u 1 | 2 * α (s) |x| s dx.
For the third one, we substitute test function
ηu 1 into I ′ λ (u) = 0 in X α 2 0 (Ω) to get t u 1 , U ǫ X α 2 0 (Ω) − γ Ω u 1 U ǫ |x| α dx = tλ Ω f (x)|u 1 | q−1 U ǫ dx + t Ω |u 1 | 2 * α (s)−1 |x| s U ǫ dx.
Plugging the last two inequalities and (43) into (44), we obtain
I λ (u 1 + tU ǫ ) = I λ (u 1 ) − λ q Σ f (x)|u 1 + tU ǫ | q dx − f (x)|u 1 | q dx − tqf (x)|u 1 | q−1 U ǫ dx + t 2 2 U ǫ 2 − t 2 * α (s) 2 * α (s) Ω |U ǫ | 2 * α (s) |x| s dx − t 2 * α (s)−1 Ω |U ǫ | 2 * α (s)−1 |x| s u 1 dx + o(ǫ β + (γ)−β − (γ) 2
).
(45) We also have
Σ f (x)|u 1 + tU ǫ | q dx − f (x)|u 1 | q dx − tqf (x)|u 1 | q−1 U ǫ dx = q Σ f (x) tUǫ 0 |u 1 + τ | q−1 − |u 1 | q−1 dτ dx ≥ q Σ f + (x) tUǫ 0 |u 1 + τ | q−1 − |u 1 | q−1 dτ dx ≥ 0.
In addition, we know that u 1 is a positive solution of (3). Following the iterative scheme used to prove Proposition 3.3 in [14], one can show that
u 1 (x) ≤ C|x| −β−(γ) for all x ∈ Ω.
Thus,
Ω |U ǫ | 2 * α (s)−1 |x| s u 1 dx ≤ C Ω |U ǫ | 2 * α (s)−1 |x| s |x| −β−(γ) dx = C B δ |U ǫ | 2 * α (s)−1 |x| s |x| −β−(γ) dx + C Ω\B δ |U ǫ | 2 * α (s)−1 |x| s |x| −β−(γ) dx = Cǫ n+ α−n 2 l−s−β−(γ) B ǫ −1 δ |u * | 2 * α (s)−1 |x| s |x| −β−(γ) dx + o(ǫ β + (γ)−β − (γ) 2 ) = Cǫ n+ α−n 2 l−s−β−(γ) R n |u * | 2 * α (s)−1 |x| s |x| −β−(γ) dx + o(ǫ β + (γ)−β − (γ) 2 ) = Cǫ β + (γ)−β − (γ) 2 R n |u * | 2 * α (s)−1 |x| s |x| −β−(γ) dx + o(ǫ β + (γ)−β − (γ) 2 ) = Kǫ β + (γ)−β − (γ) 2 + o(ǫ β + (γ)−β − (γ) 2
) for some K > 0, as ǫ → 0.
Note that one can use the asymptotic (41) in Theorem 5.1 to show that the last integral is finite. Therefore, there exist c > 0 such that
I λ (u 1 + tU ǫ ) ≤ I λ (u 1 ) + α − s 2(n − s) S n−s α−s p − c ǫ β + (γ)−β − (γ) 2 + o(ǫ β + (γ)−β − (γ) 2 ). < M + α − s 2(n − s) S n−s α−s p for all t > 0.
Lemma 5.6. Suppose that a sequence (u k ) k∈N satisfies the following:
1. I λ (u k ) = σ + o(1) with σ < M + α−s 2(n−s) S n−s α−s p 2. I ′ λ (u k ) = o(1) in (X α 2 0 (Ω)) ′
Then, there exists a sub-sequence of (u k ) k∈N which is strongly convergence in X α 2 0 (Ω).
Proof. It follows from Lemma 3.5 that (u k ) k∈N is bounded in X α 2 0 (Ω). Then, there exists a subsequence -still donote by u k -and u such that u k ⇀ u weakly in X α 2 0 (Ω) u k → u strongly in L r (Ω) for every 1 ≤ r < 2 * α .
Consequently from the second assumption, we obtain I ′ λ (u), w = 0 ∀w ∈ X α 2 0 (Ω). Then, u is a solution in X α 2 0 (Ω) for (3) with I λ (u) ≥ M. We first prove that u ≡ 0. Indeed, suppose u ≡ 0. Then, by (46) 2 , and the fact that 1 < q < 2 < 2 * α , we obtain Ω f (x)|u k | q dx → Ω f (x)|u| q dx = 0, which implies Ω f (x)|u k | q dx = o(1) as k → ∞.
Thus, the second assumption yields
u k 2 = λ Ω f (x)|u k | q dx + Ω |u k | 2 * α (s) |x| s dx = Ω |u k | 2 * α (s) |x| s dx + o(1) as k → ∞,(47)
and the first assumption then implies that
I λ,p (u k ) = 1 2 u k 2 − λ q Ω f (x)|u k | q dx − 1 2 * α (s) Ω |u k | 2 * α (s) |x| s dx = 1 2 − 1 2 * α (s) Ω |u k | 2 * α (s) |x| s dx + o(1) = α − s 2(n − s) Ω |u k | 2 * α (s) |x| s dx + o(1) = σ + o(1)
as k → ∞. Therefore, up to a sub-sequence, v k → 0 strongly in X α 2 0 (Ω). This implies that u k → u strongly in X α 2 0 (Ω).
We are now ready to prove the existence results on N + . Thus, N − disconnects X α 2 0 (Ω) in two connected components W 1 and W 2 , and X α 2 0 (Ω) \ N − = W 1 ∪ W 2 . By Lemma 3.4, for any u ∈ N + , there exists a unique t − ( u u ) > 0 such that 1 < t max < t − (u). Since t − (u) = 1 u t − ( u u ). Then, t − ( u u ) > u , and N + ⊂ W 1 . In particular, u 1 ∈ W 1 . Next step is to show that there exists n 0 > 0 such that u 1 + n 0 U ǫ ∈ W 2 . To prove this, we first note that there exists C > 0 such that
0 < t − ( u 1 + n 0 U ǫ u 1 + n 0 U ǫ ) < C for all n 0 > 0.(51)
Indeed, if not, there exists a sub-sequence (n k ) k∈N such that n k → ∞ and t − ( u 1 + n k U ǫ u 1 + n k U ǫ ) → 0 as k → ∞.
For all k ∈ N, let v k = u1+n k Uǫ u1+n k Uǫ . So, Lemma 3.4 implies that t − (v k )v k ∈ N − ⊂ N for all k ∈ N . Then, a straightforward computation and the Lebesgue dominated convergence theorem yield We use Lemma 5.6 and (50) to get that there exist a sub-sequence (u k ) k∈N and u 2 such that u k → u 2 strongly in X α 2 0 (Ω). So, we have that u 2 ∈ N − and I λ (u k ) → I λ (u 2 ) = M − as k → ∞. Since I λ (u 2 ) = I λ (|u 2 |), and |u 2 | ∈ N − is a solution for (3), without loss of generality, we may assume that u 2 is a non-negative solution for (3), and the maximum principle [23,Proposition 2.2.8] implies that u 2 > 0 in Ω.
Proof of Theorem 1.2. It follows from Theorem 5.4 and Proposition 5.7 that there exist two positive solutions u 1 and u 2 such that u 1 ∈ N + and u 2 ∈ N − . In addition, we have N + ∩ N − = ∅. Thus, u 1 and u 2 are two distinct positive solutions for (3).
Lemma 3. 7 .
7For each u ∈ N − \ {0}, there exist ǫ > o and a differentiable function σ − :
Theorem 4. 1 .
1Let Λ = Λ(p) := min{Λ 1 , Λ 2 , Λ 3 }. Then, for any λ ∈ (0, Λ), there exists a minimizer u + ∈ N + for the functional I λ,p which verifies 1. I λ,p (u + ) = M = M + .
Theorem 4. 2 .
2Let Λ = Λ(p) := min{Λ 1 , Λ 2 , Λ 3 }. Then, for any λ ∈ (0, Λ), the functional I λ,p has a minimizer u − ∈ N − which verifies 1. I λ,p (u − ) = M − .
Lemma 5. 5 .
5Let u 1 be the local minimum in Theorem 5.4 . Then, for ǫ > 0 small enough, we have sup t≥0 I λ (u 1 + tU ǫ ) < M + α − s 2(n − s) S
Proposition 5 . 7 .
57For any λ ∈ (0, Λ * ), there exists a minimizer u 2 ∈ N − for the functional I λ which verifies1. I λ (u 2 ) = M − < M +
).We now define|x| s dx for t > 0.By straightforward computations, we get that m attained its maximum att =Thus, for all t > 0,On the other hand, since u ǫ is an extremal for (8), we haveHence,On the other hand, it follows from(8)and(47)thatThis gives us a contradiction which implies that u can not be identically zero, and thus u ≡ 0 withWe may verify as Brzis-Lieb lemma in[6]that (see also[15,Lemma 4.2])Then,On the other hand, from the second assumption, we know that (u k ) k∈N is uniformly bounded and u is solution of (3). So,Hence,On the other hand, as k → ∞, we haveThis contradicts the fact that I λ is bounded below. Thus, (51) holds.which givesThis proves that u 1 + n 0 U ǫ ∈ W 2 . Now define I λ (τ (ξ)) and γ ⋆ (ξ) = u 1 + ξn 0 U ǫ for ξ ∈ [0, 1].We have γ ⋆ (0) ∈ W 1 and γ ⋆ (1) ∈ W 2 . So, there exists ξ 0 ∈ (0, 1) such that γ ⋆ (ξ 0 ) ∈ N − and c ⋆ ≥ M − . It also follows from Lemma 5.
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[
"Mechanical properties and electronic structure of the incompressible rhenium carbides and nitrides: A first-principles study",
"Mechanical properties and electronic structure of the incompressible rhenium carbides and nitrides: A first-principles study"
] |
[
"Naihua Miao ",
"Baisheng Sa ",
"Jian Zhou ",
"Zhimei Sun [email protected] ",
"Rajeev Ahuja \nDepartment of Physics and Materials Science\nCondensed Matter Theory Group\nUppsala University\nBox 53075121UppsalaSweden\n",
"\nDepartment of Materials Science and Engineering\nCollege of Materials\nXiamen University\n361005XiamenPeople's Republic of China\n"
] |
[
"Department of Physics and Materials Science\nCondensed Matter Theory Group\nUppsala University\nBox 53075121UppsalaSweden",
"Department of Materials Science and Engineering\nCollege of Materials\nXiamen University\n361005XiamenPeople's Republic of China"
] |
[] |
By means of first-principles calculations, the structural stability, mechanical properties and electronic structure of the newly synthesized incompressible Re 2 C, Re 2 N, Re 3 N and an analogous compound Re 3 C have been investigated. Our results agree well with the available experimental and theoretical data. The proposed Re 3 C is shown to be energetically, mechanically and dynamically stable and also incompressible. Furthermore, it is suggested that the incompressibility of these compounds is originated from the strong covalent bonding character with the hybridization of 5d orbital of Re and the 2p orbital of C or N, and a zigzag topology of interconnected bonds, e.g., Re-Re, Re-C or Re-N bonding.
|
10.1016/j.ssc.2011.08.011
|
[
"https://arxiv.org/pdf/1503.02048v1.pdf"
] | 119,200,198 |
1503.02048
|
cae0e70d4eb044885c3a6c805c8f23bbea4a7415
|
Mechanical properties and electronic structure of the incompressible rhenium carbides and nitrides: A first-principles study
Naihua Miao
Baisheng Sa
Jian Zhou
Zhimei Sun [email protected]
Rajeev Ahuja
Department of Physics and Materials Science
Condensed Matter Theory Group
Uppsala University
Box 53075121UppsalaSweden
Department of Materials Science and Engineering
College of Materials
Xiamen University
361005XiamenPeople's Republic of China
Mechanical properties and electronic structure of the incompressible rhenium carbides and nitrides: A first-principles study
1 *Author to whom correspondence should be addressed. Tel and Fax: +86-592-2186664. Email addresses:A Hard materialsD Mechanical propertiesE First-principles 2
By means of first-principles calculations, the structural stability, mechanical properties and electronic structure of the newly synthesized incompressible Re 2 C, Re 2 N, Re 3 N and an analogous compound Re 3 C have been investigated. Our results agree well with the available experimental and theoretical data. The proposed Re 3 C is shown to be energetically, mechanically and dynamically stable and also incompressible. Furthermore, it is suggested that the incompressibility of these compounds is originated from the strong covalent bonding character with the hybridization of 5d orbital of Re and the 2p orbital of C or N, and a zigzag topology of interconnected bonds, e.g., Re-Re, Re-C or Re-N bonding.
Introduction
Transition-metal (TM) carbides, nitrides and borides are of great interest and importance for their useful mechanical and electrical properties. Among the TMs, rhenium receives considerable attentions due to its high bulk modulus and shear modulus. Rhenium diborides have been synthesized and broadly studied for their incompressibility and great hardness [1]. Recently, Re 2 C (P63/mmc) [2][3][4], Re 2 N (P63/mmc) and Re 3 N (P-6m2) [5] have been prepared and shown to be incompressible, which are placed for potential technological applications. Rather great bulk modulus of B=405±30 GPa for Re 2 C, B=401±10 GPa for Re 2 N and B=395±7 GPa for Re 3 N have been reported [2,3,5], respectively, which is comparable to that of the well-known incompressible materials c-BN. The unique property of these carbides and nitrides opens an interesting topic demanding of further exploration. Among these compounds, Re 2 C and Re 3 N have been studied theoretically and experimentally [2-4, 6, 7]. However, an extended study on the physical properties of Re 2 N and Re 3 C has not been performed and that why these compounds are so incompressible is still unknown. This provides us the motivation to undertake a detail investigation on their mechanical properties and electronic structure and to understand the origin of their incompressibility.
First-principles computational technique provides us a powerful tool to explore the phase stability and physical and mechanical properties of materials. In the present work, by means of first-principles calculations, we focused on rhenium carbides and nitrides (Re n X, n=2 or 3, X=C or N). Among them, the proposed Re 3 C (P-6m2) with an isostructure of Re 3 N has not yet been reported experimentally and its structural stability is still unknown. Hence, we began with our study on the thermodynamic stabilities of Re n X, and then systematically explored their mechanical properties and 3 electronic structure. Our results will provide a fundamental understanding on these incompressible materials and will be helpful for the further experimental and theoretical investigations on this class of materials.
Calculation Methods
The present first-principles calculations are based on the density functional theory (DFT). We use the Vienna ab initio simulation package (VASP) code [8] as implemented to solve the Kohn-Sham equations employing the projector augmented wave (PAW) method [9][10][11][12]. The semi-core s and p states and possibly the semi-core d states were treated as valence states. PAW-GGA-PBE [11,13,14] pseudo-potentials with electronic configurations of C 2s 2 2p 2 , N 2s 2 2p 3 , and Re 5p 6 6s 2 5d 5 were employed.
The cutoff energy for plane wave basis set was 800 eV. 13×13×4 k-points for Re 2 C and Re 2 N and 14×14×6 for Re 3 C and Re 3 N with Monkhorst-Pack (MP) scheme [15] were adopted for Brillouin zone sampling. The relaxation convergence for ions and electrons were 1×10 -5 eV and 1×10 -6 eV, respectively. The lattice parameters, force constants, elastic constants and density of states and electron localization function (ELF) [16,17] analyzed by VESTA [18] were calculated for the equilibrium structures. The elastic constants were calculated by using the stress-strain methods.
The stress tensors were calculated by VASP and then the elastic coefficients were extracted according to the Hooke's law. The phonon calculations for Re 3 C were performed through the supercell approach [19]. Force constants of supercells were obtained by using the VASP, and the PHONOPY code [20,21] was performed to calculate the phonon frequencies and phonon density of states.
Results and Discussion
The calculated lattice parameters of Re 2 C, Re 2 N, Re 3 N and Re 3 C are given in Table 1. It is noted that the lattice parameters predicted by VASP are in good agreement with the available experimental data for Re 2 C, Re 2 N and Re 3 N [3,5] and the theoretical results for Re 2 C [6]. The calculated values of a and c are around 1% larger than the experimental ones which is within the GGA overestimated error. Moreover, the predicted z constants also coincide well with the experiments [3,5]. For Re 3 C, there is no available experimental or theoretical data, hence our results can serve as a prediction for future investigations.
To investigate the thermodynamic stability of these rhenium carbides and nitrides (Re n X, n=2 or 3, X=C or N), we have calculated the formation energy E form by Eq. (1) according to the reaction (2).
E form = E total (Re n X) − nE total (Re) − E total (X)(1)
nRe + X →Re n X (2) The total energy E total of Re, C and N were calculated for stable crystalline rhenium (P63/mmc), diamond (Fd-3m), and alpha dinitrogen (Pa-3), respectively. The results are also given in Table 1. For all the studied compounds, including the proposed compound Re 3 C, small formation energies are observed, suggesting that all of them are thermodynamically stable under certain conditions (e.g., high temperature and high pressure [2,3,5]). It is worth noting that the positive E form value 43.14 meV/atom for Re 2 N points out that the synthesis of Re 2 N through reaction (2) is not a self-driven process. Nevertheless, such a tiny positive value of formation energy indicates that the reaction can be easily driven by a certain temperature or pressure.
Among these compounds, Re 2 C, Re 2 N and Re 3 N have been prepared by experiments [2,3,5], as the negative E form value -74.24meV/atom for Re 3 C is calculated, it is expected that Re 3 C could be also obtained by experiments. Moreover, a phonon dispersion calculation along the high symmetry directions has been performed to further explore the dynamical stability of the new predicted compounds Re 3 C, which 5 is illustrated in Fig. 1. It is known that imaginary frequencies indicate the dynamical instability of crystals. As seen from the phonon dispersion curves in Fig. 1, no negative frequency have been found for Re 3 C, suggesting it is dynamically stable.
For hexagonal crystals, there are five independent elastic stiffness constants. The shear modulus (G) and the bulk modulus (B) according to Voigt approximations [22] are defined as: G = (2c 11 Table. 2. The Born mechanical stability criteria for hexagonal crystal is given as: c 44 >0, c 11 −c 12 >0, (c 11 +c 12 )c 33 >2c 13 2 . It is obvious that all of the studied rhenium carbides and nitrides satisfy the Born criteria, hence they are all mechanically stable. As seen from Table 2, our calculated bulk modulus are in excellent agreement with the experimental values [3,5] (where the difference are less than 0.5%), and also coincide with the theoretical values [3][4][5][6][7].
Moreover, the bulk modulus of these studied rhenium carbides and nitrides are greater than 390 GPa which are greater than that of c-BN (376 GPa) [23], hence all of them are considered to be incompressible materials. It is also noted in Table 2 that, for all the compounds studied here, their Poisson's ratio (v) are small, indicating all of them are relatively stable against shear.
To study the ductility and brittleness of Re n X, we refer to the Cauchy pressure (c 13c 44 for hexagonal crystals) and the ratio of bulk modulus to shear modulus (B/G).
Generally, a positive Cauchy pressure, or a B/G value larger than 1.75, reveals damage tolerance and ductility of a crystal, while a negative Cauchy pressure, or a B/G value smaller than 1.75, demonstrates brittleness [24]. The Cauchy pressure for Fig. 3 (a) and -N-Re-Re-Re-N-for Re 3 N in Fig. 3 (b), respectively.
Hence, it is clear that along the c-axis of Re n N, the interconnected bonding strengths are mainly attributing to the hybridization of 5d-2p (Re-N bonding) and 5d-5d (Re-Re bonding). Moreover, it can be seen in Fig. 3 (a) Therefore, it is evident that these strong directional bonding chains with a zigzag topology account for the incompressibility of Re n N (Re n C), which is quite similar to that in the incompressibility of 5d transition-metal diborides [26].
Conclusions
In summary, the structural stability, mechanical properties and electronic structure of rhenium carbides and nitrides have been systematically studied by means of first-principles calculations. The calculated lattice parameters and elastic constants for them are in good agreement with the available experimental data. All the studied compounds are shown to be incompressible with bulk modulus greater than 390 GPa.
The calculated density of states of Re n X indicates that all these borides display a metallic conductivity. Furthermore, our analysis on their electronic structure and electron localization function suggests that the incompressibility of these compounds mainly attributes to strong covalent bonding character with the hybridization of 5d
orbital of Re and the 2p orbital of C or N, and a zigzag topology of interconnected bonds, e.g., Re-Re, Re-C or Re-N bonding. For the proposed compound Re 3 C which has been demonstrated to be energetically, mechanically and dynamically stable, future experimental works are recommended for further confirmation. [3]. b Ref. [5]. c Ref. [6]. [3]. b Ref. [5]. c Ref. [4]. d Ref. [6]. e Ref. [7].
+c 33 −c 12 −2c 13 +6c 44 +3c 66 )/15; B = (2c 11 +2c 12 +c 33 +4c 13 )/9. Then the Young's modulus (E), and Poisson's ratio (v) are calculated by: E = 9BG/(3B+G); v = (3B−2G)/2(3B+G). Based on these equations, we derived the corresponding values which are presented in
Re 2 C
2, Re 3 C, Re 2 N and Re 3 N are -32 GPa, -42 GPa, 98 GPa and 107 GPa, respectively. , the results of B/G and Cauchy pressure indicate the ductile nature of Re n C and the brittle nature Re n N. To quantify the elastic anisotropy of Re n X, we have calculated the shear anisotropic factor A = 4c 44 / (c 11 +c 33 -2c 13 ) for the {1 0 0} shear planes between the <0 1 1> and <0 1 0> directions, which is identical to the shear anisotropy factor for the {0 1 0} shear planes between <1 0 1> and <0 0 1> directions. For an isotropic crystal, A is equal to 1. The magnitude of the deviation from 1 is a measure of the degree of elastic anisotropy possessed by the crystal[25].The calculated shear anisotropic factor A for Re 2 C, Re 3 C, Re 2 N and Re 3 N are 0.79, 0.83, 0.90 and 0.97, respectively. Hence, Re n C are relatively more anisotropic than Re n N.To explore the origin of incompressibility of Re n X and gain an understanding of their electronic structure and chemical bonding, we have calculated the electronic density of states (DOS) and electron localization function (ELF). Either the DOS or the ELF of Re n C and Re n N show similar characters, hence we took the DOS plots of Re 2 N and the ELF plots of Re 2 N and Re 3 N for further discussion, which have been presented inFig. 2 and Fig. 3. Note that there are finite values at the Fermi levels of the DOS inFig. 2, indicating all of them display a metallic conductivity. The typical feature of the DOS inFig. 2is the presence of the so called pseudo-gap, which is the borderline of the bonding and anti-bonding states, suggesting that a strong covalent bonding character in these compounds. From the partial density of states (PDOS) inFig. 2, it is observed that the hybridization of 5d orbital of Re and the 2p orbital of N (or C) contribute to the Re-Re and Re-N (or Re-C) bonding. As seen from Fig. 3, along the c-axis, there are zigzag topology of interconnected bonds of -Re-N-Re-Re-N-Re-for Re 2 N in
and (b) that the ELF value of Re-N bonding are greater than 0.75, suggesting a strong covalent bonding character, while the ELF value of Re-Re bonding are around 0.50, indicating a typical metallic Re-Re interatomic bonding. The similar character can be observed in the ELF of Re n C.
Figure caption : Fig. 1 .
:1The calculated phonon dispersion curve for Re 3 C.
Fig. 2 .
2(Color online) The calculated density of states for (a) Re 2 C, (b) Re 3 C, (c) Re 2 N, (d) Re 3 N. The Fermi levels have been set to 0 eV and marked by short dash lines.
Fig. 3 .
3(Color online) Structures and contour plots of ELF on the (110) plane of Re n N for (a) Re 2 N, (b) Re 3 N. The color scale for the ELF value is given at the bottom of the figure, where all the mappings are under the same saturation levels and the interval between two nearest contour lines is 0.13.
Table 1 .
1The calculated lattice parameters (a, c and z) and formation energy E form (meV/atom) for Re n X. Calc., Exp. and Theo. represent the calculated values in the present work, the experimental and theoretical results, respectively. Calc. Exp. Theo. Calc. Exp. Re 2 C 2.866 2.836 a 2.862 c 9.931 9.86 a 9.903 c 0.109 0.111 a -92.482 Re 2 N 2.864 2.844 b 2.837 b 9.892 9.796 b 9.799 b 0.106 0.106 b 43.139Re n X
a(Å)
c(Å)
z
E form
Calc. Exp.
Theo.
Re 3 C 2.849
-
-
7.181
-
-
0.195
-
-74.237
Re 3 N 2.835 2.811 b 2.825 b 7.202 7.154 b 7.159 b 0.198 0.198 b -32.483
a Ref.
Table 2 .
2The calculated elastic stiffness constants c ij (in GPa) bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (v) for polycrystalline Re n X aggregates.Re n X c 11 c 12 c 13 c 33 c 44B(GPa)
G(GPa) E(GPa)
v
Calc. Exp.
Theo.
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[] |
[
"CONSERVATION LAWS AND LINE SOLITON SOLUTIONS OF A FAMILY OF MODIFIED KP EQUATIONS",
"CONSERVATION LAWS AND LINE SOLITON SOLUTIONS OF A FAMILY OF MODIFIED KP EQUATIONS"
] |
[
"Stephen C Anco ",
"M L Gandarias ",
"Elena Recio "
] |
[] |
[] |
1 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 2 department of mathematics faculty of sciences, university of cádiz puerto real, cádiz, spain, 11510 Abstract. A family of modified Kadomtsev-Petviashvili equations which includes the integrable case is studied. The explicit line soliton solution and all conservation laws of low order are derived and compared to their counterparts in the integrable case.
| null |
[
"https://arxiv.org/pdf/1902.06365v1.pdf"
] | 119,315,111 |
1902.06365
|
a467e8c081485785023e9727e4fce351ba5aad38
|
CONSERVATION LAWS AND LINE SOLITON SOLUTIONS OF A FAMILY OF MODIFIED KP EQUATIONS
18 Feb 2019
Stephen C Anco
M L Gandarias
Elena Recio
CONSERVATION LAWS AND LINE SOLITON SOLUTIONS OF A FAMILY OF MODIFIED KP EQUATIONS
18 Feb 2019arXiv:1902.06365v1 [math-ph]
1 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 2 department of mathematics faculty of sciences, university of cádiz puerto real, cádiz, spain, 11510 Abstract. A family of modified Kadomtsev-Petviashvili equations which includes the integrable case is studied. The explicit line soliton solution and all conservation laws of low order are derived and compared to their counterparts in the integrable case.
Introduction
An integrable generalization of the modified Korteveg-de Vries (mKdV) equation in 2+1 dimensions is the modified Kadomtsev-Petviashvili (mKP) equation [10] (u t − αu 2 u x ± 2αγu x ∂ −1
x u y + βu xxx ) x + γu yy = 0 (1) where α, β, γ are non-zero constants. This equation arises in several physical applications [6,7,15] pertaining to dispersive nonlinear wave phenomena. Unlike the better known Kadomtsev-Petviashvili (KP) equation [9], the mKP equation contains a nonlocal term and has no obvious Lagrangian structure. Its line soliton solutions and some conservation laws can be found in Refs. [8,11,12,17].
In the present paper, we consider a family of mKP equations
(u t − αu 2 u x + κu x ∂ −1 x u y + βu xxx ) x + γu yy = 0(2)
with arbitrary non-zero constant coefficients α, β, γ, κ. The integrable mKP equation is given by the case
κ 2 = 2αγ > 0(3)
The main goals will be to determine the conservation laws and line soliton solutions of the mKP family (2) and compare the results to the integrable mKP case. First, in section 2, the mKP family (2) is formulated as a local PDE by use of the potential w, with u = w x .
Next, in section 3, all low-order conservation laws of the mKP family in potential form are derived. The admitted conservation laws are found to consist of a topological charge and a generalized momentum for arbitrary κ, plus an energy in the integrable case of the mKP equation. Computational aspects are summarized in an appendix. 1 In section 4, all line solitons u = U(x + µy − νt) are derived, where the parameters µ and ν determine the direction and the speed of the line soliton. The basic kinematical properties of these solutions are discussed and compared to the mKP line solitons.
Finally, a few concluding remarks are made in section 5.
Potential form
The mKP family (2) is equivalent to a local PDE system
u t − αu 2 u x + κu x v + βu xxx + γv y = 0, v x = u y(4)
This system can be expressed as a single PDE by the introduction of a potential w given by
u = w x , v = w y ,(5)
which yields 0 = w tx − αw 2 x w xx + κw xx w y + βw xxxx + γw yy (6) By applying a general scaling transformation t → λ 1 t, x → λ 2 x, y → λ 3 y, w → λ 4 w, where λ 1 , λ 2 , λ 3 , λ 4 = 0, we can fix three of the four coefficients α, β, γ, κ: |α| = β = |γ| = 1 and also we can fix the sign κ > 0, without loss of generality.
Hence, we will consider the mKP family in the scaled potential form
0 = w tx + (σ 1 w 2 x + κw y )w xx + w xxxx + σ 2 w yy , σ 1 , σ 2 = ±1, κ > 0 (7)
which is a one-parameter family where κ (rescaled) is an arbitrary positive constant. We will refer to σ 1 = 1 as the focussing case, and σ 1 = −1 as the defocussing case; this distinction will be significant when line soliton solutions are considered. The corresponding scaled mKP family has the form
(u t + σ 1 u 2 u x + κu x ∂ −1 x u y + u xxx ) x + σ 2 u yy = 0, σ 1 , σ 2 = ±1, κ > 0(8)
in which the scaled mKP equation is the case
κ 2 = 2, σ 1 σ 2 = −1(9)
namely,
(u t + σ 1 u 2 u x + √ 2u x ∂ −1 x u y + u xxx ) x − σ 1 u yy = 0, σ 1 = ±1(10)
Conservation laws
Conservation laws are of basic importance for nonlinear evolution equations because they provide physical, conserved quantities as well as conserved norms. A general treatment of conservation laws is given in Refs. [13,4,5,2].
For the mKP family in potential form (7), a local conservation law is a continuity equation
D t T + D x X + D y Y = 0(11)
holding for all solutions w(x, y, t) of equation (7), where T is the conserved density, and (X, Y ) is the spatial flux, which are functions of t, x, y, w, and derivatives of w. When solutions w(x, y, t) are considered in a given spatial domain Ω ⊆ R 2 , every local conservation law yields a corresponding conserved integral
C[w] = Ω T dx dy(12)
satisfying the global balance equation
d dt C[w] = − ∂Ω (X, Y ) ·n ds(13)
wheren is the unit outward normal vector of the domain boundary curve ∂Ω, and where ds is the arclength on this curve with clockwise orientation. This global equation (13) has the physical meaning that the rate of change of the quantity (12) on the spatial domain is balanced by the net outward flux through the boundary of the domain. A conservation law is locally trivial [13,5,2] if, for all solutions w(x, y, t) in Ω, the conserved density T reduces to a spatial divergence D x Ψ x + D y Ψ y and the spatial flux (X, Y ) reduces to a time derivative −D t (Ψ x , Ψ y ), since then the global balance equation (13) becomes an identity. Likewise, two conservation laws are locally equivalent [13,5,2] if they differ by a locally trivial conservation law, for all solutions w(x, y, t) in Ω. We will be interested only in locally non-trivial conservation laws.
Any non-trivial conservation law (11) can be expressed in an equivalent characteristic form [13,5,2] which is given by a divergence identity holding off of the space of solutions w(x, y, t).
For the mKP family in potential form (7), conservation laws have the characteristic form
D tT + D xX + D yỸ = (w tx + (σ 1 w 2 x + κw y )w xx + w xxxx + σ 2 w yy )Q(14)
whereT ,X,Ỹ are functions of t, x, y, w, and derivatives of w, and where the conserved densityT and the spatial flux (X,Ỹ ) reduce to T and (X, Y ) when restricted to all solutions w(x, y, t) of equation (7). This divergence identity is called the characteristic equation for the conservation law, and the function Q is called the conservation law multiplier. Note that, when a conservation law is non-trivial, Q will be non-singular when it is evaluated on any solution w(x, y, t).
All multipliers Q are determined by
E w (w tx + (σ 1 w 2 x + κw y )w xx + w xxxx + σ 2 w yy )Q = 0(15)
holding off of solutions of equation (7), where E w is the Euler operator [13,5,2] with respect to w. For Q having any specified form, this determining equation (15) splits with respect to all variables that do not appear in Q, yielding an overdetermined system to be solved for Q.
A variety of methods [16,3,5,2] can be used to derive the conserved densityT and spatial flux (X,Ỹ ) arising from any given multiplier Q.
Here we will explicitly find all low-order conservation laws of the mKP family in potential form (7) by determining all multipliers having the general form
Q(t, x, y, w, ∂w, ∂ 2 w, ∂ 3 w)(16)
where ∂ k w denotes the set of all partial derivatives of order k ≥ 0 of w. Some remarks on the computations are provided in the appendix.
Proposition 3.1. All low-order multipliers (16) admitted by the potential form of the mKP family (7) with κ = 0, σ 2 1 = 1, σ 2 2 = 1 are given by (i) κ arbitrary:
Q 1 = f 1 (t),(17)Q 2 = κw x f 2 (t) + yf ′ 2 (t); (18) 3 (ii) κ 2 = 2, σ 2 = −σ 1 : Q 3 = (4yw x − 2κσ 2 x)f 3 (t) + κy 2 f ′ 3 (t),(19)Q 4 = − (κσ 1 w xxx + 1 3 κw 3 x − σ 1 w x w y − 1 4 κσ 1 w t )f 4 (t) + ( 1 4 κσ 1 xw x )f ′ 4 (t) + ( 1 8 κy 2 w x + 1 4 σ 1 xy)f ′′ 4 (t) + 1 24 y 3 f ′′′ 4 (t);
(20)
where f 1 (t), f 2 (t), f 3 (t), f 4 (t) are arbitrary functions.
These multipliers yield all non-trivial conservation laws of low order, summarized as follows.
Theorem 3.1. (i) The low-order conservation laws admitted by the mKP family in potential form (7) for arbitrary κ are given by (up to equivalence)
T 1 =0,(21a)X 1 = w xxx + κw x w y + 1 3 σ 1 w 3 x + w t f 1 (t),(21b)Y 1 = σ 2 w y − 1 2 κw 2 x f 1 (t); (21c) T 2 = 1 2 κf 2 (t)w 2 x ,(22a)X 2 = κw x w xxx − 1 2 κw 2 xx + 1 4 κσ 1 w 4 x + 1 2 κ 2 w 2 x w y − 1 2 κσ 2 w 2 y f 2 (t) + w xxx + 1 3 σ 1 w 3 x + κw x w y + w t yf ′ 2 (t),(22b)Y 2 = κσ 2 w x w y − 1 6 κ 2 w 3 x f 2 (t) + (σ 2 w y − 1 2 κw 2 x )y − σ 2 w f ′ 2 (t); (22c)
where f 1 (t), f 2 (t) are arbitrary functions.
(ii) Additional low-order conservation laws are admitted only when κ 2 = 2, σ 2 σ 1 = −1.
These conservation laws consist of (up to equivalence):
T 3 = 2yw 2 x − 2κσ 1 w f 3 (t),(23a)X 3 = (2κσ 1 w xxx + 4σ 1 w x w y + 2 3 κw 3 x + 2κσ 1 w t )x + (σ 1 w 4 x + 2κw 2 x w y + 2σ 1 w 2 y − 2w 2 xx + 4w x w xxx )y − 2κσ 1 w xx f 3 (t) + κw xxx + 1 3 κσ 1 w 3 x + 2w x w y + κw t y 2 f ′ 3 (t),(23b)Y 3 = − (2σ 1 w 2 x + 2κw y )x + (4σ 1 w x w y + 2 3 κw 3 x )y f 3 (t) + (κσ 1 w y + w 2 x )y 2 − 2κσ 1 yw f ′ 3 (t);
(23c)
T 4 = 3 8 κσ 1 w 2 xx − 1 4 σ 1 w 2 x w y − 1 16 κw 4 x − 1 8 κw 2 x f 4 (t) + 1 8 κσ 1 xw 2 x f ′ 4 (t) + 1 16 κy 2 w 2 x − 1 4 σ 1 yw f ′′ 4 (t),(24a)4 X 4 = − 1 2 κσ 1 w 2 xxx + (σ 1 w x w y + 1 4 κσ 1 w t + 1 3 κw 3 x )w xxx + 1 2 σ 1 w 2 xx w y − 1 2 κw 2 xy + 1 8 κσ 1 w 2 t − (σ 1 w x w xy − 3 4 κσ 1 w tx + κw yy )w xx + ( 1 12 κw 3 x + 1 2 σ 1 w x w y )w t + 5 12 w 4 x w y + 1 18 κσ 1 w 6 x + 1 6 w 3 y + 1 2 κσ 1 w 2 x w 2 y f 4 (t) + ( 1 4 κσ 1 w x w xxx − 1 8 κσ 1 w 2 xx + 1 16 κw 4 x + 1 8 κw 2 y + 1 4 σ 1 w 2 x w y )x − 1 4 κσ 1 w x w xx f ′ 4 (t) + ( 1 8 κw x w xxx + 1 16 κσ 1 w 2 y + 1 8 w 2 x w y − 1 16 κw 2 xx + 1 32 κσ 1 w 4 x )y 2 + ( 1 4 κσ 1 w x w y + 1 4 σ 1 w xxx + 1 4 σ 1 w t + 1 12 w 3 x )xy − 1 4 σ 1 yw xx f ′′ 4 (t) + 1 24 w xxx + 1 24 w t + 1 72 σ 1 w 3 x + 1 24 κw x w y y 3 f ′′′ 4 (t),(24b)Y 4 = − 1 2 σ 1 w x w 2 xx − κw xx w xy + ( 1 4 κw y + 1 4 σ 1 w 2 x )w t + 1 3 κσ 1 w 3 x w y + 1 2 w x w 2 y + 1 12 w 5 x f 4 (t) − 1 4 κw x w y + 1 12 σ 1 w 3 x xf ′ 4 (t) − ( 1 4 yw y + 1 8 κσ 1 yw 2 x − 1 4 w)x + ( 1 8 κσ 1 w x w y + 1 24 w 3 x )y 2 f ′′ 4 (t) − ( 1 24 σ 1 w y + 1 48 κw 2 x )y 3 − 1 8 σ 1 y 2 w f ′′′ 4 (t); (24c)
where f 3 (t), f 4 (t) are arbitrary functions.
Conserved quantities. Conservation law (22) yields
P[w, f ] = 1 2 κ R 2 w 2 x f (t) dx dy = 1 2 f (t) Ω κu 2 dx dy (25)
which is a generalized momentum quantity, in analogy with the same conserved integral known to hold for the potential form of the mKdV equation. Conservation law (21) in contrast yields a spatial flux quantity which describes a conserved topological charge
F [w, f ] = ∂Ω w xxx + κw x w y + 1 3 σ 1 w 3 x + w t , σ 2 w y − 1 2 κw 2 x ·n ds = 0 (26)
holding for all closed curves ∂Ω in R 2 , without any boundary conditions on w.
The two additional conservation laws (23)-(24) arise in the case of the mKP equation. Conservation law (23) yields an additional momentum quantity
Q[w, f ] = Ω (2yw 2 x − 2κσ 1 w)f (t) dx dy = 2f (t) Ω yu 2 − κσ 1 w dx dy(27)
while conservation law (24) yields an energy quantity
E[w, f ] = Ω 3 8 κσ 1 w 2 xx − 1 4 σ 1 w 2 x w y − 1 16 κw 4 x − 1 8 κw 2 y f (t) + 1 8 κσ 1 xw 2 x f ′ (t) + 1 16 κy 2 w 2 x − 1 4 σ 1 yw f ′′ (t) dx dy =f (t)
Line soliton solutions
A line soliton is a solitary wave in two dimensions,
u = U(x + µy − νt) (29) with U, U ′ , U ′′ , etc. → 0 as |x|, |y| → ∞,(30)
where the parameters µ and ν determine the direction and the speed of the wave. A more geometrical form for a line soliton is given by writing x + µy = (x, y) · k with k = (1, µ) being a constant vector in the (x, y)-plane. The travelling wave variable can then be expressed as
ξ = x + µy − νt = |k|(k · (x, y) − ct)(31)
where the unit vectork = (cos θ, sin θ), tan θ = µ
gives the direction of propagation of the line soliton, and the constant c = ν/|k|, |k| 2 = 1 + µ 2
gives the speed of the line soliton.
We will now derive the explicit line soliton solutions (29) for the scaled mKP family (8). It will be convenient to use the coordinate form of the travelling wave variable ξ = x + µy − νt for this derivation. Thus, we have u x = U ′ , u y = µU ′ , u t = −νU ′ , and so on, while ∂ −1
x u y = µ∂ −1 ξ U ′ = µU by the solitary wave conditions (30). Substitution of the line soliton expression (29) into equation (8) yields a nonlinear fourth-order ODE
(σ 2 µ 2 − ν)U ′′ + σ 1 (U 2 U ′ ) ′ + κµ(UU ′ ) ′ + U ′′′′ = 0(34)
We can straightforwardly reduce this ODE to a separable form
U ′2 = (ν − σ 2 µ 2 ) − 1 3 κµU − 1 6 σ 1 U 2 U 2(35)
after use of conditions (30).
u = 6(ν − σ 2 µ 2 ) 6σ 1 ν + (κ 2 − 6σ 1 σ 2 )µ 2 cosh( ν − σ 2 µ 2 (x + µy − νt)) + κµ (36a) where ν − σ 2 µ 2 > 0 if σ 1 = 1; 0 < ν − σ 2 µ 2 < 1 6 κ 2 µ 2 and µ > 0 if σ 1 = −1 (36b)
With respect to the x axis, the angle θ of the direction of motion of the line soliton is given by arctan(µ), while the speed of the line soliton is given by ν/ (1 + µ 2 ). These two parameters obey the kinematic condition (36b) which depends crucially on the signs of σ 1 and σ 2 .
In the case (9) representing the scaled mKP equation (10), the general line soliton solution (36) becomes u = 3 √ 2(ν + σ 1 µ 2 ) 3σ 1 ν + 4µ 2 cosh( ν + σ 1 µ 2 (x + µy − νt)) + µ (37a) 6 with the kinematic condition
ν > −µ 2 if σ 1 = −σ 2 = 1; µ 2 < ν < 4 3 µ 2 and µ > 0 if σ 1 = −σ 2 = −1 (37b)
We will next discuss a few properties of the mKP family of line solitons (36) in comparison to the mKP line solitons (37).
4.1.
Subfamily containing the mKP equation. To begin, we examine the case σ 1 σ 2 = −1, where the mKP family constitutes a one-parameter (κ) extension of the mKP equation. The line soliton (36) in this case is given by
u = 6(ν + σ 1 µ 2 ) 6σ 1 ν + (κ 2 + 6µ 2 cosh( ν + σ 1 µ 2 (x + µy − νt)) + κµ , σ 1 = ±1, κ > 0 (38)
with the kinematic conditions
ν > −µ 2 if σ 1 = 1 (39) µ 2 < ν < ( 1 6 κ 2 + 1)µ 2 and µ > 0 if σ 1 = −1(40)
In the focussing case, σ 1 = 1, there is a minimum negative speed c > −µ 2 / 1 + µ 2 which is determined by the angle θ = arctan(µ), while there is no maximum speed. In the defocussing case, σ 1 = −1, the speed has both a positive minimum and maximum, µ 2 / 1 + µ 2 < c < (1 + 1 6 κ 2 )µ 2 / 1 + µ 2 . Note κ 2 = 2 recovers the mKP equation in both cases. In the focussing case, plots of the mKP line soliton for different speeds and angles are shown in Fig. 1. Comparison plots of the mKP family line soliton with κ 2 = 2 are shown in Fig. 2 and Fig. 3.
In the focussing case, σ 1 = 1, here the speed has a positive minimum c > µ 2 / 1 + µ 2 and no maximum. In the defocussing case, σ 1 = −1, there is a minimum negative speed c > −µ 2 / 1 + µ 2 , while the maximum speed is either positive if κ 2 > 6 or negative if κ 2 < 6. These conditions are qualitatively different compared to the conditions (39) in the mKP-like case.
Plots of the mKP family line soliton (41) for different speeds and angles are shown in Fig. 6 for the focussing case, and in Fig. 7 and Fig. 8 for the defocussing case. Our results can be used as a starting point to investigate the stability of the line soliton solutions and to determine how their stability may depend on the coefficient of the nonlocal term.
1 u 2 x − 1 4 σ 1 u 2 w y − 1 16 κu 4 − 1 8 κw 2 y dx dy + 1 8 κσ 1 f ′ (t) Ω xu 2 dx dy + f ′′ (t)Ω 1 16 κy 2 u 2 − 1 4 σ 1 yw dx dy(28) 5
Proposition 4. 1 .
1The general line soliton solution of the scaled mKP family (8) is given by
Figure 1 .Figure 2 . 2 Figure 3 .
1223mKP line soliton (37) in the focussing case σ 1 = 1 with fixed speed c and different angles θ A similar comparison in the defocussing case is shown in Fig. 4 and Fig. 5. mKP family line soliton (38) in the focussing case σ 1 = 1 with c = 1 and different angles (a) κ 2 = 10 (b) κ 2 = 1 mKP family line soliton (38) in the focussing case σ 1 = 1 with c = −1 and different angles 4.2. Subfamily excluding the mKP equation. Last, we examine the case σ 1 σ 2 = 1, where the mKP family is a strict generalization of the mKP equation. The line soliton (36) in this case is given by u = 6(ν − σ 1 µ 2 ) 6σ 1 ν + (κ 2 − 6µ 2 cosh( ν + σ 1 µ 2 (x + µy − νt)) + κµ , σ 1 = ±1, κ > 0 (41) with the kinematic conditions ν > µ 2 if σ 1 = 1 (42) 8 (a) c = 1
Figure 4 . 2 Figure 5 .
425mKP line soliton (37) in the defocussing case σ 1 = −1 with fixed speed and different angles (a) κ 2 = 10 (b) κ 2 = 1 mKP family line soliton (38) in the focussing case σ 1 = 1 with c = 1 and different angles − µ 2 < ν < ( 1 6 κ 2 − 1)µ 2 and µ > 0 if σ 1 = −1
Figure 6 .
6mKP family line soliton (41) in the focussing case σ 1 = 1 with c = 2 and different angles (a) κ 2 = 10
Figure 7 .
7mKP family line soliton (41) in the defocussing case σ 1 = −1 with c = 2 and different angles 5. Concluding remarks We have obtained all line soliton solutions in an explicit form and all low-order conservation laws for the family (2) of mKP equations which includes the well-known integrable case of the mKP equation (1). Depending on the coefficient of the nonlocal term in this equation, the line solitons can have a qualitatively different kinematic behaviour compared to the mKP line solitons.
Figure 8 .
8mKP family line soliton (41) in the defocussing case σ 1 = −1 with c = −1 and different angles
Acknowledgements S.C.A. is supported by an NSERC research grant and thanks the University of Cádiz for additional support during the period when this work was initiated.AppendixThe determining equation(15)for multipliers(16)with differential order less than four splits with respect to the set of variables {∂ 4 w, ∂ 5 w, ∂ 6 w}. We have carried out the setting up and splitting of the determining equation by using Maple. This yields an overdetermined system consisting of 3356 equations to be solved for Q as well as for κ = 0, with σ 2 1 = σ 2 2 = 1. Solving the system is a nonlinear problem because Q appears linearly in products with κ. We use the Maple package 'rifsimp' to find the complete case tree of solutions. For each solution case in the tree, we solve the system of equations by using Maple 'pdsolve' and 'dsolve', and we check that the solution has the correct number of free constants/functions and satisfies the original overdetermined system. Finally, we merge overlapping cases by following the method explained in Ref.[14].
On the evolution of packets of water waves. M J Ablowitz, H Segur, J. Fluid Mech. 92M.J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech. 92 (1979), 691-715.
Generalization of Noether's theorem in modern form to non-variational partial differential equations. S C Anco, Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. 79S.C. Anco, Generalization of Noether's theorem in modern form to non-variational partial differential equations. In: Recent progress and Modern Challenges in Applied Mathematics, Modeling and Compu- tational Science, 119-182, Fields Institute Communications, Volume 79 (2017).
Conservation laws of scaling-invariant field equations. S C Anco, J. Phys. A: Math. and Gen. 36S.C. Anco, Conservation laws of scaling-invariant field equations, J. Phys. A: Math. and Gen. 36 (2003), 8623-8638.
Direct construction method for conservation laws of partial differential equations Part II: General treatment. S C Anco, G Bluman, Euro. J. Appl. Math. 41S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations Part II: General treatment, Euro. J. Appl. Math. 41 (2002), 567-585.
G W Bluman, S C Cheviakov, Anco, Applications of Symmetry Methods to Partial Differential Equations. New YorkSpringerG.W. Bluman, A Cheviakov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations. New York: Springer (2009).
A generalized modified Kadomtsev-Petviashvili equation for interfacial wave propagation near the critical depth level. Y Chen, P L , .-F Liu, Wave Motion. 27Y. Chen, P. L.-F. Liu, A generalized modified Kadomtsev-Petviashvili equation for interfacial wave propagation near the critical depth level, Wave Motion 27 (1998), 321-339.
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Explicit construction of solutions of the modified Kadomtsev-Petviashvili equation. F Gesztesy, H Holden, E Saab, B Simon, J. Funct. Anal. 981F. Gesztesy, H. Holden, E. Saab, B. Simon, Explicit construction of solutions of the modified Kadomtsev- Petviashvili equation, J. Funct. Anal. 98(1) (1991), 211-228.
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Some new integrable nonlinear evolution equations in 2+1 dimensions. B Konopelchenko, V Dubrovsky, Phys. Lett. A. 102B. Konopelchenko, V. Dubrovsky, Some new integrable nonlinear evolution equations in 2+1 dimensions, Phys. Lett. A 102 (1984), 15-17.
Inverse spectral transform for the modified Kadomtsev-Petviashvili equation. B G Konopelchenko, V G Dubrovsky, Studies in Applied Math. 863B.G. Konopelchenko and V.G. Dubrovsky, Inverse spectral transform for the modified Kadomtsev- Petviashvili equation, Studies in Applied Math. 86(3) (1992), 219-268.
Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem. R Naz, Z Ali, I Naeem, Abstract and Applied Analysis. R. Naz, Z. Ali, and I. Naeem, Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem. Abstract and Applied Analysis (2013), 340564- 340575.
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Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations. E Recio, S C Anco, J. Math. Anal. Appl. 452E. Recio, S.C. Anco, Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations, J. Math. Anal. Appl. 452 (2017) 1229-1261.
Modified Kadomtsev-Petviashvili (MKP) equation and electromagnetic soliton. V Veerakumar, M Daniel, Math. Comput. Simulat. 62V. Veerakumar and M. Daniel, Modified Kadomtsev-Petviashvili (MKP) equation and electromagnetic soliton, Math. Comput. Simulat. 62 (2003), 163-169.
A comparison of four approaches to the calculation of conservation laws. T Wolf, Euro. J. Appl. Math. 13T. Wolf, A comparison of four approaches to the calculation of conservation laws, Euro. J. Appl. Math. 13 (2002), 129-152.
Solitary wave solutions for the modified Kadomtsev-Petviashvili equation. X Zhao, W Xu, H Jia, H Zhou, Chaos, Solitons and Fractals. 342X. Zhao, W. Xu, H. Jia, and H. Zhou, Solitary wave solutions for the modified Kadomtsev-Petviashvili equation, Chaos, Solitons and Fractals 34(2) (2007), 465-475.
|
[] |
[
"Transport in organic semiconductors in large electric fields: From thermal activation to field emission",
"Transport in organic semiconductors in large electric fields: From thermal activation to field emission"
] |
[
"J H Worne \nDepartment of Electrical and Computer Engineering\nRice University\n6100 Main St77005HoustonTX\n",
"J E Anthony \nDepartment of Chemistry\nUniversity of Kentucky\n40506-0055LexingtonKY\n",
"D Natelson \nDepartment of Electrical and Computer Engineering\nRice University\n6100 Main St77005HoustonTX\n\nDepartment of Chemistry\nUniversity of Kentucky\n40506-0055LexingtonKY\n\nDepartment of Physics and Astronomy\nRice University\n6100 Main St77005HoustonTX\n"
] |
[
"Department of Electrical and Computer Engineering\nRice University\n6100 Main St77005HoustonTX",
"Department of Chemistry\nUniversity of Kentucky\n40506-0055LexingtonKY",
"Department of Electrical and Computer Engineering\nRice University\n6100 Main St77005HoustonTX",
"Department of Chemistry\nUniversity of Kentucky\n40506-0055LexingtonKY",
"Department of Physics and Astronomy\nRice University\n6100 Main St77005HoustonTX"
] |
[] |
Understanding charge transport in organic semiconductors in the presence of large electric fields is relevant to many potential applications. Here we present transport measurements in organic field-effect transistors based on poly(3-hexylthiophene) (P3HT) and 6,13-bis(triisopropyl-silylethynyl) pentacene (TIPS-pentacene) with short channels, from room temperature down to 4.2 K. Near 300 K transport in both systems is well described by a model of thermally assisted hopping with Poole-Frenkel-like electric field enhancement of the mobility. At low temperatures and large gate voltages, transport in both materials becomes nearly temperature independent, crossing over into a regime described by field-driven tunneling. These data, particularly in TIPS-pentacene, show that great caution must be exercised when considering more exotic models of low-T transport in these materials.Charge transport in organic semiconductors in the presence of large electric fields is important in organic light-emitting diodes (OLEDs), the channels of organic field-effect transistors (OFETs), and organic photovoltaic devices. Much recent work has focused on the effects of large electric fields on the injection process 1-3 as well as on transport within the semiconductor bulk 2,4-8 . In general, this is a complex, nonequilibrium problem, involving charge carriers with moderately strong couplings to vibrational degrees of freedom, and a disordered environment resulting in localization and a strongly energy dependent density of states. As a result, transport characteristics, usually parametrized by a mobility, µ, depend nontrivially on temperature, electric field, and carrier density 6,7,9,10 .Near room temperature, the field dependence of the mobility is often reasonably described (over a limited temperature and field range) by an effective Poole-Frenkel (PF) model of mobility 11 . In this model 11 , mobility µ PF ∝ µ 0 (T ) exp(γ √ E), where µ 0 is the zero-field mobility and E is the electric field. Both µ 0 and γ vary like 1/T in this limited range 4,5,8 . In the Poole-Frenkel regime, one can think of the field dependence arising from field-induced distortion of the disorder potential of the localized carrier states.At lower temperatures and strong source-drain fields, it has been noted 5 that conduction in OFETs is highly nonlinear and approaches a temperature-independent regime as T → 0. The physical mechanism for this temperature independence is in dispute. Recent work done by Dhoot, et al. 12 argued that the crossover to temperature-independent nonlinear conduction at low temperatures and large gate voltages was a signature of a voltage-driven insulator-to-metal transition. Prigodin and Epstein 13 argue instead that what is observed is a field-induced crossover from thermal activation to a field emission hopping regime 14 previously examined by Shklovskii 15 . This point of view is ex-tended by Wei et al.16, who find that a multistep tunneling model explains the observed strong dependences on gate and source-drain voltage. More recently, Yuen et al.17have looked at the evolution of carrier transport from 300K to 4K and below, and claim further evidence for an insulator-to-metal transition in devices based on the and poly(2,5-bis(3-tetradecylthiophen-2yl)thieno[3,2-b]thiophene) (PBTTT). Specifically, they contend that their data are well described by a Tomonaga-Luttinger Liquid (TLL) model of transport originally developed for truly one-dimensional metals.To address this controversial issue, we performed transport experiments from room temperature down to cryogenic temperatures in short-channel bottomcontact OFET devices based on two different molecules, P3HT 18 and 6,13-bis(triisopropyl-silylethynyl) (TIPS)pentacene 19 . P3HT is a polymer semiconductor that tends toward glassy or nanocrystalline structure, while TIPS-pentacene is a solution-processable small molecule that forms van der Waals bonded molecular crystals. As the temperature is decreased, we observe that charge carrier behavior evolves from Poole-Frenkel-like activated hopping at high temperatures to temperatureindependent hopping consistent with field emission at low temperatures in both P3HT and TIPS-pentacene systems. While the TLL analysis approach 17,20 produces compelling plots, we argue that this is fortuitous, particularly since there is no reason to expect TLL physics to be relevant in the TIPS-pentacene case.We used degenerately doped p-type silicon with 200 nm of thermally grown oxide, which serves both as the substrate and the gate in our experiments. Platinum electrodes were fabricated using standard electron beam lithography, electron beam evaporation and liftoff processing with channel widths, W , of 50µm and channel length, L, of 300nm (device A) and W = 200µm and L = 1µm (device B). On both substrates other electrodes of fixed width and varying channel lengths were prepared for transmission line estimates of the contact
|
10.1063/1.3309704
|
[
"https://arxiv.org/pdf/1001.3160v3.pdf"
] | 118,886,077 |
1001.3160
|
b5b7b4471fd5696ed426db84727f14924c3257a2
|
Transport in organic semiconductors in large electric fields: From thermal activation to field emission
7 Mar 2010
J H Worne
Department of Electrical and Computer Engineering
Rice University
6100 Main St77005HoustonTX
J E Anthony
Department of Chemistry
University of Kentucky
40506-0055LexingtonKY
D Natelson
Department of Electrical and Computer Engineering
Rice University
6100 Main St77005HoustonTX
Department of Chemistry
University of Kentucky
40506-0055LexingtonKY
Department of Physics and Astronomy
Rice University
6100 Main St77005HoustonTX
Transport in organic semiconductors in large electric fields: From thermal activation to field emission
7 Mar 2010
Understanding charge transport in organic semiconductors in the presence of large electric fields is relevant to many potential applications. Here we present transport measurements in organic field-effect transistors based on poly(3-hexylthiophene) (P3HT) and 6,13-bis(triisopropyl-silylethynyl) pentacene (TIPS-pentacene) with short channels, from room temperature down to 4.2 K. Near 300 K transport in both systems is well described by a model of thermally assisted hopping with Poole-Frenkel-like electric field enhancement of the mobility. At low temperatures and large gate voltages, transport in both materials becomes nearly temperature independent, crossing over into a regime described by field-driven tunneling. These data, particularly in TIPS-pentacene, show that great caution must be exercised when considering more exotic models of low-T transport in these materials.Charge transport in organic semiconductors in the presence of large electric fields is important in organic light-emitting diodes (OLEDs), the channels of organic field-effect transistors (OFETs), and organic photovoltaic devices. Much recent work has focused on the effects of large electric fields on the injection process 1-3 as well as on transport within the semiconductor bulk 2,4-8 . In general, this is a complex, nonequilibrium problem, involving charge carriers with moderately strong couplings to vibrational degrees of freedom, and a disordered environment resulting in localization and a strongly energy dependent density of states. As a result, transport characteristics, usually parametrized by a mobility, µ, depend nontrivially on temperature, electric field, and carrier density 6,7,9,10 .Near room temperature, the field dependence of the mobility is often reasonably described (over a limited temperature and field range) by an effective Poole-Frenkel (PF) model of mobility 11 . In this model 11 , mobility µ PF ∝ µ 0 (T ) exp(γ √ E), where µ 0 is the zero-field mobility and E is the electric field. Both µ 0 and γ vary like 1/T in this limited range 4,5,8 . In the Poole-Frenkel regime, one can think of the field dependence arising from field-induced distortion of the disorder potential of the localized carrier states.At lower temperatures and strong source-drain fields, it has been noted 5 that conduction in OFETs is highly nonlinear and approaches a temperature-independent regime as T → 0. The physical mechanism for this temperature independence is in dispute. Recent work done by Dhoot, et al. 12 argued that the crossover to temperature-independent nonlinear conduction at low temperatures and large gate voltages was a signature of a voltage-driven insulator-to-metal transition. Prigodin and Epstein 13 argue instead that what is observed is a field-induced crossover from thermal activation to a field emission hopping regime 14 previously examined by Shklovskii 15 . This point of view is ex-tended by Wei et al.16, who find that a multistep tunneling model explains the observed strong dependences on gate and source-drain voltage. More recently, Yuen et al.17have looked at the evolution of carrier transport from 300K to 4K and below, and claim further evidence for an insulator-to-metal transition in devices based on the and poly(2,5-bis(3-tetradecylthiophen-2yl)thieno[3,2-b]thiophene) (PBTTT). Specifically, they contend that their data are well described by a Tomonaga-Luttinger Liquid (TLL) model of transport originally developed for truly one-dimensional metals.To address this controversial issue, we performed transport experiments from room temperature down to cryogenic temperatures in short-channel bottomcontact OFET devices based on two different molecules, P3HT 18 and 6,13-bis(triisopropyl-silylethynyl) (TIPS)pentacene 19 . P3HT is a polymer semiconductor that tends toward glassy or nanocrystalline structure, while TIPS-pentacene is a solution-processable small molecule that forms van der Waals bonded molecular crystals. As the temperature is decreased, we observe that charge carrier behavior evolves from Poole-Frenkel-like activated hopping at high temperatures to temperatureindependent hopping consistent with field emission at low temperatures in both P3HT and TIPS-pentacene systems. While the TLL analysis approach 17,20 produces compelling plots, we argue that this is fortuitous, particularly since there is no reason to expect TLL physics to be relevant in the TIPS-pentacene case.We used degenerately doped p-type silicon with 200 nm of thermally grown oxide, which serves both as the substrate and the gate in our experiments. Platinum electrodes were fabricated using standard electron beam lithography, electron beam evaporation and liftoff processing with channel widths, W , of 50µm and channel length, L, of 300nm (device A) and W = 200µm and L = 1µm (device B). On both substrates other electrodes of fixed width and varying channel lengths were prepared for transmission line estimates of the contact
Understanding charge transport in organic semiconductors in the presence of large electric fields is relevant to many potential applications. Here we present transport measurements in organic field-effect transistors based on poly(3-hexylthiophene) (P3HT) and 6,13-bis(triisopropyl-silylethynyl) pentacene (TIPS-pentacene) with short channels, from room temperature down to 4.2 K. Near 300 K transport in both systems is well described by a model of thermally assisted hopping with Poole-Frenkel-like electric field enhancement of the mobility. At low temperatures and large gate voltages, transport in both materials becomes nearly temperature independent, crossing over into a regime described by field-driven tunneling. These data, particularly in TIPS-pentacene, show that great caution must be exercised when considering more exotic models of low-T transport in these materials.
Charge transport in organic semiconductors in the presence of large electric fields is important in organic light-emitting diodes (OLEDs), the channels of organic field-effect transistors (OFETs), and organic photovoltaic devices. Much recent work has focused on the effects of large electric fields on the injection process 1-3 as well as on transport within the semiconductor bulk 2,4-8 . In general, this is a complex, nonequilibrium problem, involving charge carriers with moderately strong couplings to vibrational degrees of freedom, and a disordered environment resulting in localization and a strongly energy dependent density of states. As a result, transport characteristics, usually parametrized by a mobility, µ, depend nontrivially on temperature, electric field, and carrier density 6,7,9,10 .
Near room temperature, the field dependence of the mobility is often reasonably described (over a limited temperature and field range) by an effective Poole-Frenkel (PF) model of mobility 11 . In this model 11 , mobility µ PF ∝ µ 0 (T ) exp(γ √ E), where µ 0 is the zero-field mobility and E is the electric field. Both µ 0 and γ vary like 1/T in this limited range 4,5,8 . In the Poole-Frenkel regime, one can think of the field dependence arising from field-induced distortion of the disorder potential of the localized carrier states.
At lower temperatures and strong source-drain fields, it has been noted 5 that conduction in OFETs is highly nonlinear and approaches a temperature-independent regime as T → 0. The physical mechanism for this temperature independence is in dispute. Recent work done by Dhoot, et al. 12 argued that the crossover to temperature-independent nonlinear conduction at low temperatures and large gate voltages was a signature of a voltage-driven insulator-to-metal transition. Prigodin and Epstein 13 argue instead that what is observed is a field-induced crossover from thermal activation to a field emission hopping regime 14 previously examined by Shklovskii 15 . This point of view is ex-tended by Wei et al. 16 , who find that a multistep tunneling model explains the observed strong dependences on gate and source-drain voltage. More recently, Yuen et al. 17 have looked at the evolution of carrier transport from 300K to 4K and below, and claim further evidence for an insulator-to-metal transition in devices based on the and poly(2,5-bis(3-tetradecylthiophen-2yl)thieno[3,2-b]thiophene) (PBTTT). Specifically, they contend that their data are well described by a Tomonaga-Luttinger Liquid (TLL) model of transport originally developed for truly one-dimensional metals.
To address this controversial issue, we performed transport experiments from room temperature down to cryogenic temperatures in short-channel bottomcontact OFET devices based on two different molecules, P3HT 18 and 6,13-bis(triisopropyl-silylethynyl) (TIPS)pentacene 19 . P3HT is a polymer semiconductor that tends toward glassy or nanocrystalline structure, while TIPS-pentacene is a solution-processable small molecule that forms van der Waals bonded molecular crystals. As the temperature is decreased, we observe that charge carrier behavior evolves from Poole-Frenkel-like activated hopping at high temperatures to temperatureindependent hopping consistent with field emission at low temperatures in both P3HT and TIPS-pentacene systems. While the TLL analysis approach 17,20 produces compelling plots, we argue that this is fortuitous, particularly since there is no reason to expect TLL physics to be relevant in the TIPS-pentacene case.
We used degenerately doped p-type silicon with 200 nm of thermally grown oxide, which serves both as the substrate and the gate in our experiments. Platinum electrodes were fabricated using standard electron beam lithography, electron beam evaporation and liftoff processing with channel widths, W , of 50µm and channel length, L, of 300nm (device A) and W = 200µm and L = 1µm (device B). On both substrates other electrodes of fixed width and varying channel lengths were prepared for transmission line estimates of the contact resistance. Samples were rinsed using isopropanol and acetone followed by an oxygen plasma cleaning for two minutes. Samples were then spin-coated with hexamethyldisilazane (HMDS) at 3000 RPM for 30 seconds, followed by a bake at 130 C for 20 minutes. P3HT was spin-cast from chloroform at a 0.1% by weight concentration onto device A and TIPS-pentacene was drop cast from toluene at a 1% by weight concentration onto device B. Samples were measured in a variable temperature probe station with base pressure of 1x10 −6 Torr. Transmission line measurements on the cofabricated device arrays were made on chip to measure device mobilities, which were 4.6x10 −2 cm 2 /Vs (P3HT) and 1.1x10 −4 cm 2 /Vs (TIPS-pentacene), respectively. We then extracted contact resistances for our devices and determined that they are much smaller than the resistance within the device channel, indicating these devices are bulk dominated. As shown in previous work 21 , as the temperature is reduced, bulk channel resistance increases more rapidly than the contact resistance, so that our measurements remain bulk limited down to low temperatures. Measurement source-drain voltages V DS were set so that the average source-drain electric fields in the channels of both device A and B were similar, with a maximum electric field of 20 MV/m in device A and 10 MV/m in device B.
We confirmed that our P3HT sample follows PF behavior at high T , as shown in figure 1. The inset of figure 1 is a cartoon describing our device geometry. The data shown here are at one particular gate voltage, V G , at a variety of temperatures. At each temperature data for all gate voltages are analyzed 8 using the form
I D = µ 0 wC i L exp(γ V DS /L) (V G − V T )V DS − V 2 DS 2 ,
(1) where µ 0 is the (gate-and temperature-dependent) zerofield mobility, C i is the capacitance per area of the gate oxide, γ is a prefactor that is found to vary like 1/T , and V T is the threshold voltage. As T decreases, we note a deviation of the data from the PF model. This will be addressed below. Analysis of the TIPS-pentacene data is qualitatively identical.
We tried plotting our data in the manner suggested by the TLL analysis 17,20 . The expression for the current in the TLL picture is 17,22,23
I = I 0 T α+1 sinh(γ ′ eV /k B T )|Γ((1 + β)/2 + iV /πk B T )| 2 ,
(2) where Γ is the Gamma function, α and β are phenomenological exponents estimated from plots of conductance vs. T and I(V ), respectively, and γ ′ is a phenomenological parameter thought to be related to the amount of disorder (tunneling barriers) along the 1d structure. Based on this equation, the idea is (at fixed V G ) to plot I D /T α+1 vs. (eV DS /k B T ), where α is a fit parameter. If a particular value of α collapses all of the data over the whole temperature range onto a single curve, it is tempting to conclude that the TLL picture is valid. Critical to the TLL theory validity is that the system in question be one-dimensional. In polymers such as PBTTT and P3HT, carrier mobility along the polymer chain is generally much higher than between chains, implying that they may be relevant.
We present the same style of analysis in Figure 2, with the P3HT data shown in A and the TIPS-pentacene data shown in B. We are able to collapse our data onto a single line as T is decreased, with choices for α (5.43 for P3HT, 7.1 for TIPS-pentacene) that are not wildly different from those reported 17 for PBTTT (5.4, 4.3) or polyaniline fibers 20 (5.5). Recall that TIPS-pentacene is a short chain molecule, without the mobility anisotropy found in PBTTT or P3HT. It seems extremely unlikely that TIPS-pentacene can be described by TLL theory for a one-dimensional metal, despite the apparent collapse of its I D − V DS curves onto a single master line.
With the freedom to adjust α, plotting scaled data as in Fig. 2 becomes unwise. As T decreases, both P3HT and TIPS-pentacene I D −V DS curves become increasingly non-linear and temperature independent. Plotting the data of Figure 2 turns roughly power law trends over a limited voltage range into a linear segment on such a log-log plot. The freedom to choose α while plotting allows fine-tuning of the subsequent temperature curves to lie on the same line. Because current and voltage are plotted as I D /T α+1 and eV DS /k B T , decreasing T moves subsequent temperature data sets up and to the right on the graph, even if the data themselves do not change with temperature at all. Data collapse with this plotting for both fits, we used the theoretical expectation β = α + 1. As explained in the text, the apparent scaling collapse is fortuitous, rather than the result of Tomonaga-Luttinger Liquid physics. procedure is not sufficient to demonstrate TLL physics.
Instead, as T decreases, we propose that carrier transport evolves from activation hopping into field emission hopping 13,16 , where µ P F becomes µ F E ∝ µ 0 exp(− E 0 /E) in an equation analogous to Eq. (1). Here E = V DS /L is the average electric field within the channel, while E 0 (V G ) is temperature independent and is expected 13 to depend on the disorder of the sample. We plot the data for our lowest temperature I D − V DS curves in figure 3 and note that near this temperature our data are well fit by a temperature-independent µ 0 . A detailed comparison to the multiple tunneling model 16 will require further extensive data, particularly examining the effects of gate voltage on E 0 and attempting to analyze the crossover regime of temperature and voltage.
In conclusion, we present data that illustrates both the high and low temperature transport behavior of two chemically unique organic semiconductors. The data are consistent with a crossover from Poole-Frenkellike activated hopping near room temperature to a temperature-independent field emission hopping process (∼ exp(− E 0 /E)) at cryogenic temperatures. We further find that a scaling approach is insufficient to test for Tomonaga-Luttinger Liquid physics, as it shows apparent TLL consistency even in TIPS-pentacene, a material that has no microscopic basis for TLL physics. Further detailed investigations should be able to test sophisticated models for the full temperature and voltage dependence of transport in these rich material systems.
FIG. 1 .
1ID−VDS curves for device A over a 100 K temperature range at Vg = −80V . Fit lines are generated from a Poole-Frenkel-like field dependence of the mobility, as explained in the text. The deviation from theory as T decreases indicates the beginning of the crossover from activated hopping into field emission.
FIG. 2 .
2Plotting ID vs. VDS in scaled coordinates as suggested by Eq. (2). Top data is on device A (P3HT, VG = −80 V), while bottom data is on device B (TIPS-pentacene, VG = −70 V). Solid lines are fits to Eq. (2). For device A, α = 5.43, γ ′ = 4 × 10 −3 ; for device B, α = 7.1, γ ′ = 3 × 10 −3 ;
FIG. 3 .
3Top is P3HT (device A) measured at various gate voltages at 4.2 K; bottom is TIPS-pentacene (device B) measured at 5K. Black lines are fits using the field emission hopping model expression for mobility, µ ∝ exp(−(E0/E) 1/2 ).
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|
[] |
[
"An agent-based model for modal shift in public transport",
"An agent-based model for modal shift in public transport"
] |
[
"Thibaut Barbet \nEcole des Ponts ParisTech\n\n",
"Amine Nacer-Weill \nEcole des Ponts ParisTech\n\n",
"Changtao Yang \nEcole des Ponts ParisTech\n\n",
"Juste Raimbault *[email protected] \nCenter for Advanced Spatial Analysis\nUniversity College London\n\n"
] |
[
"Ecole des Ponts ParisTech\n",
"Ecole des Ponts ParisTech\n",
"Ecole des Ponts ParisTech\n",
"Center for Advanced Spatial Analysis\nUniversity College London\n"
] |
[] |
Modal shift in public transport as a consequence of a disruption on a line has in some cases unforeseen consequences such as an increase in congestion in the rest of the network. How information is provided to users and their behavior plays a central role in such configurations. We introduce here a simple and stylised agent-based model aimed at understanding the impact of behavioural parameters on modal shift. The model is applied on a case study based on a stated preference survey for a segment of Paris suburban train network. We systematically explore the parameter space and show non-trivial patterns of congestion for some values of discrete choice parameters linked to perceived wait time and congestion. We also apply a genetic optimisation algorithm to the model to search for optimal compromises between congestion in different modes.
|
10.1016/j.trpro.2022.02.088
|
[
"https://arxiv.org/pdf/2107.11399v1.pdf"
] | 236,428,304 |
2107.11399
|
ba1e52bdfdfc26d11dd07ad2ecd7b289c066c94c
|
An agent-based model for modal shift in public transport
Thibaut Barbet
Ecole des Ponts ParisTech
Amine Nacer-Weill
Ecole des Ponts ParisTech
Changtao Yang
Ecole des Ponts ParisTech
Juste Raimbault *[email protected]
Center for Advanced Spatial Analysis
University College London
An agent-based model for modal shift in public transport
Modal shiftPublic transport disruptionAgent-based modelingModel exploration
Modal shift in public transport as a consequence of a disruption on a line has in some cases unforeseen consequences such as an increase in congestion in the rest of the network. How information is provided to users and their behavior plays a central role in such configurations. We introduce here a simple and stylised agent-based model aimed at understanding the impact of behavioural parameters on modal shift. The model is applied on a case study based on a stated preference survey for a segment of Paris suburban train network. We systematically explore the parameter space and show non-trivial patterns of congestion for some values of discrete choice parameters linked to perceived wait time and congestion. We also apply a genetic optimisation algorithm to the model to search for optimal compromises between congestion in different modes.
Introduction
Disruptions in public transport networks are not rare events, often worsened by an increased complexity of these networks and their management (Dekker et al., 2018). Network resilience is then tightly linked to patterns of modal shift (Stamos et al., 2015), and a better understanding of these is crucial both from a theoretical and operational viewpoint. In that context, users have to make decisions in a limited time and under partial information (Lyons, 2006). The study of modal shift under disruption is therefore improved when taking into account users behavior and a detailed representation of users cognition (Brisbois, 2010). More particularly, the role of information provided in real time can in some cases become crucial, as rerouting may in fact increase the congestion in other parts of the network and ultimately increase the average travel time (Chatterjee et al., 2002;Chorus et al., 2006).
To study mechanisms linking information given to users with modal shift, and more generally the evolution of multi-modal network flows under disruptions, agent-based modeling has been highlighted as a relevant approach (Leng and Corman, 2020b). For example, Leng and Corman (2020a) show that the issue time of information has a significant impact on the total congestion. Agent-based models are used in similar studies of modal share, such as by Baindur and Viegas (2011) for freight. Ambra et al. (2019) include the reaction to disruptions in a multimodal agent-based model. Raney et al. (2003) describe an application of the MATSim model, which is a data-driven agent-based and activity-based transport model, to a large sample of Switzerland transport network. The MATSim model can be applied at large scales in a reproducible manner, such as in the case of Ile-de-France illustrated by Hörl and Balac (2020).
Large transport agent-based models such as MATSim however require an extensive parametrisation on real data, are difficult to systematically validate given their runtime and large parameter space, and despite their high modularity can be tuned only to some extent regarding a precise description of users behavior in public transport and their interactions with a network disruption. This paper therefore proposes to introduce a simple and stylised agent-based model to understand the role of information and user behavior in modal shift under disruptions in public transport. With a reduced computational complexity but also parameter space to explore, under limited requirements for data, such a model can be used to systematically explore if some qualitative stylised facts are robustly found under different scenario, and possibly extended into a decision-making tool. Our contribution is threefold: (i) we contextualise the construction of the model with a reduced stated preferences survey, applied on a case study in the Parisian public transport network on a specific segment often subject to disruption; (ii) we introduce the simple agent-based model as an open-source tool, which can be extended or modified to test concurrent hypotheses on user behavior; (iii) we proceed to a systematic exploration of the model parameter space, unveiling 1 arXiv:2107.11399v1 [cs.MA] 23 Jul 2021 some non-linear patterns and a counter-productive reaction of users to the disruption in terms of congestion for some scenarios.
The rest of this paper is organised as follows: we first describe an exploratory stated preference survey; we then describe the agent-based model and results of numerical experiments. We finally discuss potential applications and developments.
Stated preferences survey
We first proceeded to a small size stated preferences survey (Kroes and Sheldon, 1988), in order to have a qualitative overview of processes needed in the model and to have a case study for model application. Martin et al. (2016) have indeed shown that there exists a high heterogeneity in user reaction to disruptions. We choose to study the line A of the RER suburban train in Paris, which has the highest load in the region and has a non-negligible frequency of disruption. We focus on a subpart of the network, in order to sample users which have a higher chance of realising a given origin-destination pattern, and for which several modal alternatives exist. Therefore, the segment Etoile-La Défense was chosen, as it features alternative trips with the Metro Line 1 and several bus lines.
We surveyed a total of N = 48 users, among which a sub-sample of N = 27 were regular users for which the full questionnaire was given. Surveys were realised on peak hours (between 7am and 9am) of weekdays in January 2021, on the platform of the Etoile station. The exact survey which was used (in French) is available on the open repository of the project (see model implementation below).
Main results which can inform our model context are shown in Fig. 1. We first confirm that disruptions are indeed frequent on the line, as around 50% of users experience a disruption at least twice a month. We then find that under no information, user behavior is heterogeneous and a large part shifts to alternative itineraries. Finally, when users are informed of a 10 minutes disruption, on the contrary more than 75% decide to wait on place. Therefore, the model must include various modal shifts, but also take into account how perceived wait time will influence user choice.
This survey was designed in an exploratory manner as a pilot study, and could not be continued on a larger scale for various practical reasons. Our sample size thus remains relatively small and this part of the study must be considered as qualitative. Significantly larger sample sizes are needed to obtain robust inferential results, for example to forecast travel demand (Horváth and Horváth, 2015) or to estimate discrete choice parameters (Bliemer and Rose, 2009). In our case, we consider the simulation model as a tool to overcome such data limitations and still study stylised behavioural processes using qualitative insights from our survey data.
Model description
The agent-based model is based on two types of agents: users and RER trains. The simulated segment of the network is included for the main mode (RER train) and alternative modes (metro, bus, taxi, bike, walking). Main mode is simulated in full granularity, i.e. including train boarding, train capacity and train alighting. Other modes are simulated as queues with a given capacity. One simulation covers peak hours, which we take as a duration of 4 hours, with one-minutes time steps. The model is therefore stopped at t f = 240. Arrival of users for each mode are simulated using stationary Poisson random draws.
More precisely, the model simulates sequentially the following processes at each time step.
1. New users enters the network: for each mode, a fixed arrival rate λ i is used for a random Poisson law draw giving the number of new users entering the segment on this mode. Users entering the main RER mode are set on the platform as waiting, while users on other modes are queued at the start of the segment.
2. Users waiting for a RER train on the platform evaluate a discrete choice utility difference (between waiting and shifting to an other mode) given by
∆U = β c · c + β τ τ(1)
where c is perceived congestion given by the current number of user waiting normalised by platform capacity (which corresponds in our case only to a parameter rescaling as we consider a single station); τ is perceived time that we assume proportional to user current travel time (time since departure of the user trip) and current waiting time between trains. β c and β τ are thus behavioural parameters capturing the influence of perceived time and congestion on user behaviour. Note that a model generalisation with a broader scope (more stations for example) will have to introduce more accurate proxies for this processes or other behavioural aspects. 3. These users can then switch mode with the corresponding discrete choice probability probability given by
p = 1/(1 + exp ∆U )(2)
and with fixed nested probabilities for the choice of the other mode once a shift has been decided;
4. Next train possibly enter station (once the time interval I between trains has elapsed) and users board at a given speed and given train capacity. Train capacity C captures congestion upstream this segment, and is given as a maximal number of users which can board a train. In the meantime, users alight at the end of the segment if a train has arrived at the terminus station.
5. Train queue on the segment is simulated, i.e. each train is advanced at its maximal speed, if the next slot is not occupied by a preceding train.
6. Other modes are simulated as queues with a given user capacity: users enter the queue if it contains less users than its capacity; users in the queue are advanced according to the mode speed; arrived users are removed from the queue.
Model results are assessed with indicators giving the average travel time and average congestion over the peak hour and all users. Congestion indicators can be computed for each mode separately.
The model has several parameters which can be set with realistic values (see setup below), while others remain variables of interest which influence on model behavior will be studied in numerical experiments. In particular, varying parameters are the behavioural parameters β c and β τ , time interval between train I and train capacity C.
The discrete choice function does not include control variables such as differences between modes in ticket price, level of service, quality of information. These are implicitly taken into account in our model through the empirical values of modal shift observed in the survey, and could be included in a refined version of the model.
Results
Implementation and model setup
The model is implemented in NetLogo which is a platform and programming language specifically suited for such agent-based simulations (Tisue and Wilensky, 2004). We show a visualisation of model interface in Fig. 2. The open-source code of the model and results is available as a git repository at https://github.com/JusteRaimbault/ ReportMasse. We systematically explored model parameter space by using the OpenMOLE model exploration software (Reuillon et al., 2013), which allows embedding any type of model, provides a seamless access to high performance computing infrastructures, and integrates state-of-the-art model validation and exploration methods.
Several model parameters are set following real world values. Respective speed for each mode for example correspond to travel times for the segment in an uncongested setting. Mode capacities also have realistic values (for example for Metro 1, 700 users per train with 5 trains on the segment). We take a boarding speed for the RER of 1000 users per minute (MI09 trains have a total capacity of 2600 users and stop for not more than 1min30sec during peak hours). A train takes 4min for the segment. We assume a maximal waiting time in station of 2 minutes. We fix the user arrival rate λ RER = 100 in experiments, to study borderline scenarios in terms of congestion. Finally, user transfer time between modes is taken as 5 minutes.
Parameter space exploration
We run 10 stochastic replications of the model, for parameters β c , β τ , train capacity C, and train arrival interval I varying within a grid. This corresponds to 24, 000 runs of the model. We show in Fig. 3 the variation of average travel time and congestion as a function of β c . We find an interesting non-linear behavior of travel time (Fig. 3, Right panel) for low values of β τ (dark blue, right column), where travel time is maximal around a neutral position of users to congestion: in that context of a low tolerance to waiting, either a low or high tolerance to congestion are better than being indifferent. Larger values of β τ induce a decreasing travel time as a function of β c (the more users are not tolerant to congestion, the lower the travel time), and similarly travel time is decreasing as a function of β τ . Regarding the congestion indicator (Left panel of Fig. 3), it increases with both discrete choice parameters but differently in the various disruption scenarios.
Model optimisation
We then propose to apply the model in an optimisation context. We use a genetic algorithm for multi-objective optimisation (more precisely the NSGA2 algorithm implemented into the OpenMOLE platform) to search for compromises between congestion in the train and congestion in alternative modes. We run the algorithm in a congested context (arrival rate λ RER = 100, train capacity C = 500 and train interval I = 5), with free parameters β c and β τ . Optimisation objectives are the average congestion in the RER and the average congestion in other modes.
Simulation results obtained with an algorithm population of µ = 200 after 2000 generations are shown in Fig. 4. We find that acting on different dimensions of user behavior can mitigate the different dimensions of congestion. More precisely, a Pareto front is obtained between congestion in the RER and congestion in the other modes. Congestion in the RER can be minimised, at the detriment of other modes. This occurs when congestion and perceived time parameters have both high values. On the contrary, a low congestion in other modes implies a high tolerance of users and a high congestion in the RER. The shape of the Pareto front is interesting, witnessing two linear regimes. A good compromise is thus found at the breakpoint. At this point, users are rather tolerant to perceived time (negative β τ values) but can have various behaviours regarding congestion. This shows that mitigating congestion can be achieved by acting on user behaviour, and that a compromise for the multimodal network can be found.
Discussion
Our simulation results show how this model can be applied to explore complex congestion patterns in the stylised network following interactions between different components of user behavior. Our model can be applied to the simulation of disruption events, and possible scenarios to improve network resilience. It can also be applied to test interventions to mitigate congestion in targeted modes. Its advantage in comparison to more complicated models relies on its simplicity, which allows systematic explorations of the parameter space and more advanced numerical experiments as we illustrated above.
Several developments can furthermore be considered. The behavioural model is rather simple, and as the implementation is generic, other functions or more complicated processes can be tested. The variables we used in the discrete choice utility are only proxies of perceived information, and how to account for the information effectively received by the users depending on the context remains an open question (Gao et al., 2018). The disutility due to perceived travel time is in our case simply the sum of previous travel time and announced waiting time: studying other functions and aspects is an other interesting development of the model. For example, including congestion within the trains may be crucial regarding recent health concerns linked to crowding in closed spaces (Raimbault and Batty, 2021).
Besides, the possible mitigation of congestion through interventions acting on the provided information may raise ethical issues -how to ensure an overall optimisation while remaining fair to all users: it was shown in other modes such as car travel that inequalities in congestion may rise from path shift due to live information (Cabannes et al., 2017). We also assumed on this issue of access to information that users did not necessarily had access to some synthetic information on all alternatives (with a web application for example) and that there was no unified mobility service. Exploring the impact of such aspects on behavioural and congestion patterns is also a potential refinement of the model.
Finally, a more data-driven version of the model, including estimated discrete choice parameters, would be an interesting path to explore, assuming that a further survey and data collection is possible. This may also imply to include control variables in discrete choices utilities to account for differences between modes for example in terms of price or level of service.
Conclusion
The study of modal shift following a disruption in public transport networks is an important aspect to ensure resilience and a high level of service. We introduced in this paper the basis for a simple agent-based model focused on user decision based on perceived congestion and waiting time. We showed how the model can be used to explore the interaction between different behavioural components of users, and to optimise the congestion in different modes. Our model is open source and aimed at being extended and applied to other case studies. Systematic model exploration and validation should provide both practical results in terms of transport management, but also more general results on generic and specific processes in modal shift.
Figure 1 :
1Results of the stated preference survey. (Top) Number of times a month users experience a disruption; (Middle) Behavior of users when no information is given; (Bottom) Behavior of users when a 10 minutes traffic stop is announced.3
Figure 2 :
2Model interface. Main varying parameters are included as sliders, while mode queues are visualised in a spatialised manner and as a plot showing the load for each mode in time. Various numerical indicators are provided.
Figure 3 :
3Model exploration results. (Left) Average congestion as a function of β c , for varying β τ (colour), train interval (columns) and train capacity (rows); (Right) Average travel time for same parameter values.
Figure 4 :
4Model optimisation. We plot the Pareto front of compromise points between the two optimisation objectives. Point colour level gives β τ while point size gives β c .
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We apply the Lefschetz thimble formulation of field theories to a couple of different problems. We first address the solution of a complex 0-dimensional φ 4 theory. Although very simple, this toy-model makes us appreciate a few key issues of the method. In particular, we will solve the model by a correct accounting of all the thimbles giving a contribution to the partition function and we will discuss a number of algorithmic solutions to simulate this (simple) model. We will then move to a chiral random matrix (CRM) theory. This is a somehow more realistic setting, giving us once again the chance to tackle the same couple of fundamental questions: how many thimbles contribute to the solution? how can we make sure that we correctly sample configurations on the thimble? Since the exact result is known for the observable we study (a condensate), we can verify that, in the region of parameters we studied, only one thimble contributes and that the algorithmic solution that we set up works well, despite its very crude nature. The deviation of results from phase quenched ones highlights that in a certain region of parameter space there is a quite important sign problem. In view of this, the success of our thimble approach is quite a significant one. arXiv:1507.03858v3 [hep-lat]
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10.1103/physrevd.92.085030
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"https://arxiv.org/pdf/1507.03858v3.pdf"
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1507.03858
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b11c96c1c54433d9d74a5be1ff05adc974d16395
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Thimble regularization at work: from toy models to chiral random matrix theories
September 28, 2015
F Di Renzo
Dipartimento di Fisica e Scienze della Terra
Università di Parma
INFN
Gruppo Collegato di ParmaI-43100ParmaItaly
G Eruzzi
Dipartimento di Fisica e Scienze della Terra
Università di Parma
INFN
Gruppo Collegato di ParmaI-43100ParmaItaly
Thimble regularization at work: from toy models to chiral random matrix theories
September 28, 2015
We apply the Lefschetz thimble formulation of field theories to a couple of different problems. We first address the solution of a complex 0-dimensional φ 4 theory. Although very simple, this toy-model makes us appreciate a few key issues of the method. In particular, we will solve the model by a correct accounting of all the thimbles giving a contribution to the partition function and we will discuss a number of algorithmic solutions to simulate this (simple) model. We will then move to a chiral random matrix (CRM) theory. This is a somehow more realistic setting, giving us once again the chance to tackle the same couple of fundamental questions: how many thimbles contribute to the solution? how can we make sure that we correctly sample configurations on the thimble? Since the exact result is known for the observable we study (a condensate), we can verify that, in the region of parameters we studied, only one thimble contributes and that the algorithmic solution that we set up works well, despite its very crude nature. The deviation of results from phase quenched ones highlights that in a certain region of parameter space there is a quite important sign problem. In view of this, the success of our thimble approach is quite a significant one. arXiv:1507.03858v3 [hep-lat]
Introduction
The so-called sign problem is one of the current big challenges for lattice field theories. It is in fact the major obstacle to tackling a non-perturbative study of the QCD phase diagram. Following pioneering work by Witten [1], Lefschetz thimble regularization has been proposed as a possible solution [2,3] (for more recent contributions see also [4,5,6,7]): the functional integral is defined in terms of fields taking values on non-trivial manifolds on which the imaginary part of the action stays piecewise (i.e. on the distinct thimbles attached to different critical points) constant. It is an elegant, although with many respects non-trivial alternative to the standard formulation of field theories. It has intriguing connections with resurgence theory, a few results of which motivate the conjecture that the semi-classical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles 1 [8]. Morse theory [9] is the natural framework for discussing the thimble regularization, even though it could be that it does not necessarily have the last word on the subject. In this work we will discuss the thimble solution of two different models, having in mind two big issues. First of all, since there is a thimble attached to every critical point of the (complexified) theory one is considering, we need to understand how many thimbles do give contribution to the solution of a given theory. A second relevant problem is that of devising Monte Carlo algorithms to correctly sample thimbles, which are manifolds for which we lack a local description.
Toy models can be a precious tool to approach hard problems in theoretical physics, with the hope that a simplified model can nevertheless capture the relevant issues. Being the sign problem both relevant and hard, it does not come as a surprise that toy models have been around for a while. Quite interestingly, some of these have been resisting the efforts to solve them in much the same way the real problem is still far for being fully solved. The first application we discuss is the solution of a toy model that dates back to almost thirty years ago [10]. It can be regarded as a 0-dim version of a φ 4 field theory. This simple model became a benchmark for the complex Langevin treatment of the sign problem, and quite interestingly only partial success has been claimed over the years. We will show that the thimble regularization completely solves the model. In this case we will show how in different regimes one or more thimbles give contribution to the solution. Moreover, it is possible to perform numerical simulations on the thimble(s): in this simple case we will have a number of algorithmic solutions at work. A partial account of these results has already been given in [11].
We will then address the solution of a chiral random matrix theory. This is a somehow more realistic problem, for which once again the application of the complex Langevin method has been shown to be non-trivial: in a given parametrization it fails [12], while in a different one it works [13]. Actually one can show that the sign problem can be quite severe for this model. The theory has an adjustable parameter (the dimension N of the matrices) which controls the dimensionality of the problem one has to solve. Since the analytical solution for the observable we will study (a mass condensate) is known, it appeared to us a perfect setting for testing a conjecture: it could well be that more than one thimble contribute (just like in the 0-dim φ 4 theory) in low dimensions, but as N grows it could be that a single thimble dominates in the thermodynamic limit. While we were ready for a richer scenario, in the region of parameters we studied we actually did not find any other thimble but the one attached to the global minimum. We did not find any problem related to the parametrization of the theory; in particular, the parametrization that was failing in the case of complex Langevin was absolutely fine for the thimble treatment of the theory. In this case algorithmic problems are non-trivial. We will show how a natural parametrization of the thimble can be the starting point for an algorithmic solution that in this case works in its simplest version (admittedly a very crude one).
The paper is organized as follows. In section 2 we give a brief account of the Lefschetz thimble approach to field theories: this is a short review of results that are collected to facilitate the reader. Section 3 is dedicated to the 0-dim φ 4 toy model, showing that thimbles provide a complete solution; in particular we show that we can effectively numerically simulate the model, making use of different algorithms. In section 4 we address the chiral random matrix theory, showing that we can numerically solve it by thimble regularization: all the analytical results are correctly reconstructed. In the final section we draw a few conclusions and mention natural steps forward.
Thimble regularization in a nutshell
In the following we collect the basics of thimble regularization: the interested reader is referred to [1,2,3] for further details and references.
A conceptual starting point to approach the thimble regularization is that of generalizing saddle point integration. The latter displays a couple of features which appear as good candidates to tackle the sign problem: stationary phase and localization. The full generalization of saddle point techniques is formulated in the framework of Morse theory [9].
• One starts with an integral on a real domain of the form C d n x g(x 1 , . . . , x n ) e −S(x 1 ,...,xn) (1) in which C is a real domain of real dimension n 2 and both S(x 1 , . . . , x n ) = S R (x 1 , . . . , x n )+ iS I (x 1 , . . . , x n ) and g(x 1 , . . . , x n ) are holomorphic functions. The notation for the exponential makes it clear that we have already in mind a functional integral (even if the normalizing factor Z −1 is missing). For such an integral the following decomposition holds C d n x g(x 1 , . . . , x n ) e −S(x 1 ,...,xn) = σ n σ Jσ d n z g(z 1 , . . . , z n ) e −S(z 1 ,...,zn) (2) in which an extension from a real domain to a complex one has been performed (see the complex variables z i = x i + iy i as opposed to the real ones x i ).
(2) holds in the homological sense, i.e. C = σ n σ J σ .
• The index σ labels the critical points of the complex(ified) S(z 1 , . . . , z n ) and to each critical point p σ a stable thimble J σ is attached. Each J σ is defined as the union of all the Steepest Ascent (in the following SA; we will also write SD for Steepest Descent) paths falling into p σ at (minus) infinite time, i.e. the union of the solutions of
dx i dτ = ∂S R (x, y) ∂x i dy i dτ = ∂S R (x, y) ∂y i(3)satisfying z(τ = −∞) = x(τ = −∞) + iy(τ = −∞) = p σ .
The real dimension of each thimble J σ is n. It is quite natural to regard thimbles as manifolds embedded in C n (which is instead of real dimension 2n).
• For each critical point one also defines unstable thimbles K σ as the union of all flows satisfying equation (3) and going to the critical point in the opposite time limit, i.e. such that z(τ = ∞) = p σ . The coefficients n σ count the intersections of the K σ with the original domain of integration n σ = C, K σ .
• The imaginary part S I stays constant on thimbles, i.e. there is a phase associated to each thimble.
Note that in the framework of field theories a natural picture of universality emerges. A single thimble can give us a formulation of a field theory with the same degrees of freedom, the same symmetries 3 and symmetry representations, the same Perturbation Theory and naive continuum limit of the original formulation (see [2] for details). In force of universality we expect that these properties essentially determine the behavior of physical quantities in the continuum limit. Moreover a simple argument suggests that in the thermodynamic limit only thimbles attached to global minima can survive, as it is easily seen (we now call φ σ the critical points, having in mind field configurations, and consider the partition function of the field theory)
Z = σ n σ e −iS I (φσ) Jσ d n z e −S R (z) = σ n σ e −S(φσ) Jσ d n z e −(S R (z)−S R (φσ))
In the end, it could well be that a full resolution in terms of all the thimbles could turn out to be with many respect overkilling the original problem. Having said this, we nevertheless stress that both the universality argument and the thermodynamic argument can not be regarded as conclusive. It is worth noting that resurgence theory [17] with many respect even asks for more than one thimble in view of the interpretation of the semi-classical approximations as trans-series 4 . All in all, the fact that in certain cases a single thimble dominance can take place has to be regarded as a conjecture: as we will see, this was in a sense one of the motivation of this work on a Random Matrix Model. The subject deserves deeper investigation and we think it will certainly receive it.
It is good to have a somehow more constructive approach to the thimble formulation. We therefore now sketch a few more technical details that the reader will see at work in the following sections. The integral we have in mind will be the functional integral of a field theory. Let us first of all parametrize the field in the vicinity of a critical point as Φ i = φ i − φ σ,i . Here and in the following i is a multi-index; in particular it can refer to a 3 Since we have discussed the case of a non-degenerate hessian, one could wonder how the method can be applied in case where Spontaneous Symmetry Breaking (SSB) is in place. In [14] a solution has been described and shown to be effective: one introduces an explicit Symmetry Breaking term h and studies the limit h → 0. This is not the only way to proceed: for a different thimble approach to SSB the reader is referred to [3]. Symmetries are dealt in yet another way in the case of Gauge Theories: the construction of thimbles was discussed in [15] and reviewed in [1]; see also the discussion in [2,16]. 4 A nice account of many issues connected to resurgence has been recently provided in [18].
real or imaginary part. The real part of the action can be expressed as
S R (φ) = S R (φ σ ) + 1 2 Φ T HΦ + O φ 3(4)
where the 2n × 2n matrix H is the hessian evaluated at the critical point
H ij = ∂ 2 S R ∂φ i ∂φ j φ=φσ H can be put in diagonal form Λ = diag (λ 1 , · · · , λ n , −λ 1 , · · · , −λ n )
by a transformation H = W ΛW T defined by the orthogonal matrix W whose columns are given by the normalized eigenvectors of H, that is {v (i) } i=1···2n . Half of the eigenvalues of H are positive; the corresponding eigenvectors span the tangent space to the thimble at the critical point. Any combination of these vectors is a direction along which the real part of the action grows. If we leave the critical point along these directions integrating the SA equations we span the thimble. On the other side, the other directions (which are attached to the negative eigenvalues) would take us along the unstable thimble. At a generic point Z ∈ J σ we miss a priori the knowledge of the tangent space T Z J σ ; in general we expect that the latter is not parallel to the canonical basis of C n whose duals appear in d n z = dz 1 ∧ · · · ∧ dz n . We thus want to perform the relevant change of coordinates from the canonical ones (of C n ) to the basis of T Z J σ , given by the (complex) vectors {U (i) } i=1···n (these are orthonormal with respect to the standard hermitian metric of C n ). Let ϕ : N ⊂ J σ → R n be a local chart in a neighbourhood N ⊂ J σ of Z
ϕ Z + n i=1 U (i) y i = Y + O y 2 ∈ R n
If we denote U the n × n complex unitary matrix whose columns are the vectors {U (i) }, we can express the integral of a generic function f (Z) on the thimble as
N d n z f (Z) = ϕ(N ) n i=1 dy i f ϕ −1 (Y ) det U ϕ −1 (Y )(5)
In this expression the quantity det U = e iω (U is unitary) has appeared; this is what has been termed the residual phase [2] (see [19] for further details). This could in principle reintroduce a sign problem in the thimble formulation, but it is expected that this is not the case. Not any phase gives raise to a serious sign problem, and in particular one expects that a phase changing rather smoothly can be safely taken into account by reweighting. This expectation could appear optimistic, but has been till now confirmed (see [3]) and will be confirmed also in this work.
We end this brief introduction to the thimble formulation by going back to the constructive point of view: we can span the thimble by integrating the SA equations for the field
φ 5 dφ i dt = ∂S R ∂φ i i = 1 · · · 2n
This has a counterpart in parallel-transport equations for the n basis vectors which defines the tangent space to the thimble (see [2,3])
dV (i) j dt = 2n k=1 ∂ 2 S R ∂φ k ∂φ j V (i) k i = 1 · · · n j = 1 · · · 2n.(6)
We can set up a similar equation for any other vector with an initial condition on the tangent space at the critical point; (6) expresses the parallel transport of a vector along the gradient flow. In the vicinity of the critical point one knows the asymptotic (t → −∞) solutions
t 1 φ j (t) ≈ φ σ,j + n i=1 v (i) j e λ i t n i j = 1 · · · 2n | n| 2 = 1 V (i) j (t) ≈ v (i) j e λ i t j = 1 · · · 2n i = 1 · · · n(7)
Note that from a practical point of view the former parametrization is viable only provided one introduces a reference time t 0 1 at which the former asymptotic solution holds. We will make estensive use of the former equations, which in particular can be regarded as initial conditions for a given flow on the thimble, e.g. for eq. (6). It thus emerges a natural picture in which a generic point Φ ∈ J σ is unambiguously defined by a choice ofn 6 and the time t: Φ = Φ (n, t) (this has been very effectively discussed in [3]). Note also that one could insist on regarding the (7) as valid all over; this would in turn mean one is considering a purely quadratic action (i.e. the free field approximation).
3 The 0-dim φ 4 toy model
We will now put at work what we have just seen, applying the thimble regularization to the study of the action
S (φ) = 1 2 σφ 2 + 1 4 λφ 4
with φ ∈ R, λ ∈ R + and σ = σ R + i σ I ∈ C. This is obviously a toy model, and one regards as correlators plain one-dimensional integrals such as
φ n = 1 Z R dφ φ n e −S(φ)(8)
with the partition function given by:
Z = R dφ e −S(φ)
The solution is given in terms of a modified Bessel function, i.e. Z = σ 2λ e σ 2 8λ K − 1 4 ( σ 2 8λ ), differentiating appropriately which one can get any of the (8). The choice of a complex σ is a prototypal case of the sign problem: with a complex action, we miss a positive semi-definite measure and hence a probability distribution to start with; in particular, a direct access to Monte Carlo methods is ruled out.
It was realized long time ago that a solution to the sign problem could be searched in the context of Stochastic Quantization: the Langevin equation admits a formal solution also for complex actions, in particular via the Fokker-Planck formulation [20,21]. Turning the formal arguments into a rigorous proof eventually turned out to be hard and numerical instabilities (suggesting problems) were in particular discussed in the context of the theory at hand [10]. Much experience has been gained over the years and much progress has been done [22]. The question of convergence of complex Langevin equation remains a subtle one, and quite interestingly even the simple model at hand displays delicate issues. For a recent and thorough study of complex Langevin dynamics of this model, the reader can refer to [23]. One peculiar feature of this model is that complex Langevin simulations display divergences for φ n with n > 4 in a certain region of parameters. The relation between complex Langevin and thimbles has been investigated in [24,25].
We can complexify the field by setting φ = x + i y. As a result, real and imaginary part of the action read
S R = 1 2 σ R x 2 − y 2 − 2σ I xy + 1 4 λ x 4 + y 4 − 6x 2 y 2 S I = 1 2 σ I x 2 − y 2 + 2σ R xy + λ x 3 y − xy 3
The hessian is built from the second derivatives of S R and takes the form:
H (x, y) = σ R + 3λx 2 − 3λy 2 −σ I − 6λxy −σ I − 6λxy −σ R − 3λx 2 + 3λy 2(9)
There are 3 critical points: φ 0 = 0 and φ ± = ± − σ λ (which are the two, complex valued, "Higgs vacua"). The question is now which thimbles do give a contribution to the integrals we want to compute, and the answer is quite different in the 3 cases σ R > 0, σ R < 0 and σ R = 0: in each case we computed the stable and unstable thimbles associated to each critical point. This can be done putting at work the constructive definition of thimbles we discussed in the previous section.
In practice, we want to integrate the equations of SA starting in the vicinity of the critical point φ σ for an arbitrarily long flow time t. We can do this provided that the initial condition is chosen correctly: for the stable thimble this means we leave the critical point along the direction (in the xy plane) which is given by the eigenvector of positive eigenvalue of the hessian (9) computed at the critical point. Once we have singled out the relevant direction, we can ascend in two ways (namely, increasing or decreasing x), both of which we have to take to cover all the thimble. By holomorphicity the hessian has two eigenvalues opposite in sign. Since S R always increases along the flow, exp (−S R ) goes to 0 as t → +∞, thus ensuring convergence of the integrals along the thimble. To obtain the unstable thimble K σ , we can repeat the same procedure described above, but picking up the eigenvector of the hessian of S R with negative eigenvalue. Note that the unstable thimble is needed because the coefficient n σ in our master equation (2) counts the intersection of such thimbles with the original domain of integration, which in our case is the real axis (the sign ambiguity is not resolved just by this definition, but it can be deduced by means of other considerations). Figures 1 (left panel) and 2 show the results for the three cases σ R > 0, σ R < 0 and σ R = 0 (see also [26]). From figure 1, we see that when σ R > 0 the unstable thimbles related to the Higgs vacua do not intersect the real axis. Therefore these points do not contribute to the integrals, that is n ± = 0 and n 0 = 1. By integrating along the stable thimble attached to φ 0 , we recover the correct results for, say, Z = e −S (the integration can be easily carried on along the real axis both analytically and numerically). The case σ R < 0 depicted in the left panel of figure 2 is a totally different matter, as we cross the Stokes ray σ R = 0 while changing sign to σ R . Now we see that the unstable thimbles connected to the Higgs vacua do intersect the real axis and therefore n ± = 0, as well as n 0 = 0. The correct combination which recovers the expected results for the integrals turns out to be n 0 = −1 and n ± = +1. What is the origin of this discontinuity? and, above all, if we hadn't known the correct result from the beginning, how would have we calculated the n σ ? The answer lies in considering the case σ R = 0, showed in the right panel of figure 2. The stable thimble connected to 0 exhibits the Stokes phenomenon: in fact it "collapses" into the Higgs vacua, from which it does not "move" any more; the unstable thimble continues to say that n 0 = 0. The stable thimbles connected to the Higgs vacua display the same shape, but their unstable counterparts collapse into 0 (by overlapping its stable thimble) and therefore there is intersection with the real axis; so, n ± = 0. However, there is no integer-valued combination of n σ that recovers the correct results for σ R = 0. This is quite expected, as the Morse decomposition along thimbles is not legitimate when we are on a Stokes ray, on which we clearly are (the imaginary axis in the complex σ plane is a "Stokes ray") 7 . Now, the original integral is continuous (in fact, it is holomorphic) in σ and therefore there cannot be any discontinuity in the computation of the partition function Z in σ R = 0. Thus, we must have
Z [σ R → 0 + ] = Z [σ R → 0 − ] = Z [σ R = 0]
. By examining the integration along the thimble connected to 0, we find that it is discontinuous in σ R = 0, and again, this is not surprising as the thimble shape undergoes a radical change between the two cases. The change in sign of the n σ is precisely the only one which keeps the original integral continuous while crossing the Stokes ray.
A variety of algorithmic solutions
Within the thimble regularization we were able to perform numerical simulations of the quartic toy model, making use of different algorithms. In particular, we were able to numerical compute all the possible moments (8).
It was observed in [2] that the Langevin algorithm is the obvious candidate for sampling configurations on the thimble. In
dφ i dt = − ∂S R ∂φ i + η i i = 1 · · · 2n(10)
the drift term constrains the field on the thimble by definition, so that the problem boils down to extracting a convenient noise, i.e. a noise tangent to the thimble. We do not discuss here the original solution which was put forward in [2] (the Aurora algorithm); there will be a convenient time for such a discussion when we later approach the CRM model. Here it suffices to say that, being the thimble 1-dimensional, at every point the tangent space reduces to the direction singled out by the drift term itself. As a matter of fact, Langevin works pretty well; in the right panel of figure 1 one can see how the simulation correctly samples configurations on the thimble. Here parameters are the same of the left panel, so one thimble is relevant, i.e. the one attached to the origin (which in the notation of (2) we denote p 0 ). The (Aurora) Langevin algorithm samples points according to the measure normalized by 8
Z (0) ≡ J 0 dτ e −S R(11)
We now denote
O 0 ≡ J 0 dτ O e −S R Z (0)(12)
and stress that this is not what we have to compute. Properly including the residual phase, the correct result was computed as
O = e iω O 0 e iω 0(13)
When σ R < 0 the thimbles associated to all the three critical points 9 contribute and we have to compute
O = 2 i=0 n i e −i S I (p i ) J i dτ e −S R O e iω 2 i=0 n i e −i S I (p i ) J i dτ e −S R e iω(14)
which can be written
O = e iω O 0 + α 1 e iω O 1 + α 2 e iω O 2 e iω 0 + α 1 e iω 1 + α 2 e iω 2(15)
with
α i = n i e −i S I (p i ) Z (i) n 0 e −i S I (p 0 ) Z (0) i = 1, 2(16)
On each thimble J i (i = 0, 1, 2) the quantities e iω O i and e iω i can be computed via (Aurora) Langevin simulations. The (complex) unknown coefficients α i can then be fixed by relations which can be regarded as renormalization conditions in a physical scheme, i.e.
e iω O i 0 + α 1 e iω O i 1 + α 2 e iω O i 2 e iω 0 + α 1 e iω 1 + α 2 e iω 2 = X i i = 1, 2(17)
where the X i are known values of given observables O i (e.g. , in the case of moments (8), two of them). As always in such an approach, one gives up predicting everything, but after normalizing results to a (minimum) number of external inputs, one has full predictive power for (all the) other quantities. Of course computing the moments (8) for the toy model at hand is not such a big numerical success; nevertheless the outline of the method is quite general. In particular, we will refer to it in section 4.3.
Another algorithmic solution for this simple setting is provided by the Metropolis algorithm which is described in [27]. The method relies on a correspondence between the full model one has to simulate and a gaussian approximation associated to it. The latter is obtained by diagonalizing the hessian at a critical point and truncating the expansion of the action around it, i.e.
S R (η) = S R (φ σ ) + 1 2 2n k=1 λ k η 2 k ≡ S R (φ σ ) + S G (η)(18)
where the η k are the Φ k = φ k − φ σ,k of equation (4) expressed in the basis provided by the eigenvectors of the hessian, with a convenient ordering in which λ k > 0 for k = 1 . . . n and λ k < 0 for k = n + 1 . . . 2n. For the gaussian action (18) it is very simple to construct the associated stable thimble. It is a flat thimble in which the tangent space is known once and for all, i.e. the span of the eigenvectors associated to {λ k |k = 1 . . . n}: we term it a gaussian thimble. For the gaussian thimble the solution in the right hand side of (7) is valid all over the manifold.
The simulation is run as a quite standard Metropolis algorithm controlled by an accept/reject test, with a mechanism for proposing configurations which is dictated by the correspondence between the thimble one has to sample and its gaussian approximation. We sketch the method in the case of more than one thimble contributing to the final result, to stress how also in this case we were able to run numerical simulations on thimbles, for both σ R > 0 and σ R < 0.
The method always handles a couple of configurations, i.e. one φ field on the thimble we have to sample and one auxiliary η field on the associated gaussian thimble. In order to extract a new φ field one proceeds as follows:
• One proposes a thimble σ (i.e. a critical point) with a probability |n σ | σ |n σ | • One extracts a configuration η on the gaussian thimble associated to that critical point according to the weight e −S G . This is trivial, given the gaussian form.
• One starts a SD on the gaussian thimble with η as initial condition. The integration is carried on over a time extentτ such that one ends up close enough to the critical point, namely at a point where the gaussian thimble and the thimble one has to sample effectively sit on top of each other (this means that the solution (7) holds for both thimbles). We callη the configuration that has been obtained in this way.
• Takingη as the initial condition, one integrates the SA equations for the complete theory over the same time extentτ . This generates the new configuration φ .
• φ is accepted with probability
P acc = min 1, e −[S R (φ )−S R (φ)]+[S G (η )−S G (η)]
The result for a given observable O is obtained as
O = 1 T T t=1 e iω(φt) O(φ t ) 1 T T t=1 e iω(φt)
where the index t runs over all the configurations sampled by Metropolis.
Notice that the previous accounting of the Metropolis algorithm is technically different from the proposal of [27] 10 . The latter relies on an exponential mapping for the integration time (i.e. r = e −t ) and an adjustable parameter is introduced to control convergence properties (the interested reader can refer to figure 4 of [11]). For a given, effective choice of this parameter figure 3 displays how the three thimbles giving contribution in the region σ R < 0 are sampled in a Metropolis simulation.
We stress that also in this case one could think of situations in which the weights n σ are unknown. However, here the situation is different from that of Langevin, since we only need to know a few integers values. In other terms, given the knowledge of the set of relevant integers {n σ }, the problem is solved on an entire region of parameter space: in the case at hand, for each σ R < 0 (technically, over the entire region which ends up in a point where a Stokes phenomenon shows up). Notice that in principle there can be different ways of finding the relevant set of integers (e.g. known asymptotic solutions in a convenient region) 11 . Both (Aurora) Langevin and Metropolis could correctly compute the moments (8), in both regions σ R > 0 and σ R < 0. For example, figure 3 (right panel) displays the computed values of φ 8 over a range of both σ R > 0 (one thimble being relevant) and σ R < 0 (three thimbles to be taken into account) values.
Note that there is another natural way of computing on a thimble, and this takes advantage of the fact that on a thimble there is a one-to-one correspondence in between configurations and values of S R . To make things simple, let us consider the case in which only one thimble is relevant and let us write the partition function Z = Z up + Z down : these are 10 We decided to enlighten the rationale of the algorithm, leaving out the technicalities. 11 The α i introduced for the (Aurora) Langevin algorithm entail instead the values of partition functions and are given at a given point of parameter space; in the case at hand, for a given value of σ R < 0. the two contributions resulting from the two pieces of the thimble we have already referred to. Namely, they are associated to leaving the critical point along the direction dictated by the eigenvector of the hessian in one of the two possible ways (i.e. increasing or decreasing x values). Each Z up/down has the global phase e −i S I (pσ) as a factor and features an integrand which is the residual phase times a monotonic function of S R . It thus can be written taking the action as the integration variable, e.g.
Z up = e −i S I (pσ) ∞ Sp σ dS e −S R |∇S R | −1 e i tan −1 (∂yS R /∂xS R )(19)
We could have written the integral by taking the flow time as the integration variable (also in this case there is a one-to-one correspondence with the configurations along the thimble, each reached at a given flow time). Note that in computing (19) one proceeds by integrating the SA. We illustrated the issue by taking into account the Z, but we showed that all the moments (8) can be successfully computed in this way. In a sense, (19) is the prototype of a parametrization we will see at work for the CRM model.
All in all, we think that the simple toy model we discussed is a perfect playground to see thimble regularization at work: it is instructive both from the point of view of inspecting the structure of relevant thimbles and from the algorithmic point of view (we can compute on thimbles).
Chiral random matrix model
We now address the chiral random matrix model defined by the partition function
Z N f N (m) = dΦdΨ det N f (D(µ) + m) exp −N · Tr[Ψ † Ψ + Φ † Φ] ,(20)
where
D(µ) + m = m i cosh(µ)Φ + sinh(µ)Ψ i cosh(µ)Φ † + sinh(µ)Ψ † m .(21)
The degrees of freedom of the model are N × N general complex matrices Ψ and Φ. Since its introduction it has attracted attention due to the many features which it shares with QCD [28,29,30]: they both have in their functional integral the determinant of a Dirac operator and the flavor symmetries and explicit breaking hereof are identical. Chiral perturbation theory at leading order in the -domain is the relevant low energy theory in the microscopic limit for both theories, which resulted in a lot of interesting insights into QCD coming from the (much simpler) random matrix theory. The microscopic limit in which contact is made with -regime of chiral perturbation theory is that of N → ∞ withm ≡ N m andμ ≡ √ N µ kept constant. A sign problem is there for this theory as it is for QCD. This sign problem can be a severe one, as it is made manifest by considering the observable we will be concerned with, i.e. the mass dependent chiral condensate Our interest in the model was triggered by [12,13]: the nature of the sign problem (which is due to the determinant) has a counterpart in a non-trivial success of the complex Langevin method (which needs to take the logarithm of the determinant to define the standard effective action dictating the drift term of Langevin equation). While the application of complex Langevin in the most direct parametrization of the theory fails [12], a different parametrization (resulting in a different complexification) reproduces the right results [13].
1 N ηη = 1 N ∂ m log (Z) .(22)
How many thimbles should we take into account?
We take the most direct path to complexification, i.e. for each field (we directly deal with the matrix elements) Φ ij = a ij + ib ij and Ψ ij = α ij + iβ ij , each real component gets complexified (e.g. β ij = β
(R) ij + iβ (I)
ij ). We adhere to the notation of [12] and denote the action as with
S (a, b, α, β) = N i,j a 2 ij + b 2 ij + α 2 ij + β 2 ij − N f Tr log m 2 1 N ×N − XYX ij = i cosh µ (a ij + ib ij ) + sinh µ (α ij + iβ ij ) Y ij = i cosh µ (a ji − ib ji ) + sinh µ (α ji − iβ ji )
Once we have complexified the degrees of freedom, the first step for the thimble approach is the identification of critical points of the resulting action. First candidate is the absolute minimum which is already there for the real formulation, i.e. Ψ = Φ = 0. All the relevant formulae for the spectral analysis of the hessian of S R are collected in appendix A. Here we simply state that the hessian in 0 has the expected number of positive eigenvalues, i.e. the real dimension of the thimble attached to 0 is 4N 2 . Note that there is a huge degeneracy: we have only two different eigenvalues, with the two eigenspaces having the same dimension. As the (rescaled) massm gets smaller, the gap between the two eigenvalues gets larger. Some insight can now be gained from equation (7): in first approximation, the closer the eigenvalues, the more isotropic we expect the thimble to be. This expectation turned out to be correct in view of the results of our simulations.
We tried to identify other critical points. In our study we explored different values ofm (at different values of N ) while keeping fixed N f = 2 andμ = 2. One approach was solving ∇S = 0 via Newton-Raphson method. We cross-checked results by applying the Nelder-Mead simplex method to minimize ∇S 2 . We found two classes of extrema, both outside the original domain and featuring an action smaller than S R (Ψ = Φ = 0), which turns out to be the absolute minimum in the original domain. Under such conditions, since the unstable thimbles attached to the extrema we found can not intersect the original domain of integration, we expect no contribution from their stable thimbles (n σ = 0; see section II.B.3 of [2] for a more extensive discussion).
Algorithmic issues for the CRM model
While for the 0-dim toy model the original algorithmic solution proposed in [2] is trivial, this is not the case for the CRM model. However, previous experience with the Bose gas [14] taught us that there can be lucky cases. Let us remind the reader of the Aurora algorithm and of its gaussian approximation (which successfully deals with the lucky cases we were referring to). We want to extract a proper noise vector for the Langevin dynamics
dφ i dt = − ∂S R ∂φ i + η i i = 1 · · · 2n
We can proceed as follows [2]:
• We extract a gaussian noise η
i (0), where the superscript qualifies this quantity as an initial proposal and the argument has to be thought as a flow time in a sense that will be clear soon.
• We evolve it following the flow (6) downwards (i.e. with a change of sign with respect to (6)), aiming at getting close enough to the critical point in order to make contact with the regime of (7). This will hold at a given descent time τ * .
• We then project with
η i = P ij η (0) j (τ * ) P ≡ 1 2 H √ H 2 + 1(23)
and normalize the result
η(τ * ) = r η η(24)
r being extracted according to the n-dimensional χ distribution.
• We then ascend along the flow, covering again a time interval of length τ * . The result is the noise η i we will put in our Langevin equation.
Extracting the noise vector is not yet the end of the story, since any finite order approximation to Langevin equation, e.g. the Euler scheme
φ i = φ i − δt ∂S R ∂φ i + √
δt η i will introduce systematic effects; since the manifold is not flat, the final point φ i will be moved away from the thimble. The obvious remedy for this effect is to repeat just the same we did for the noise vector (move the configuration along the flow downward, close to the critical point, project it onto the tangent space, move it upward along the flow for the same time length). Note that it is expected that, in all the descent/ascent mechanisms we have just described, the downward flow, i.e. the SD will be numerically delicate. It is thus much better to formulate the descent as a boundary value problem (BVP) rather than as an initial value problem, as it was observed in [14].
A much more appealing observation was also made in [14]: there are lucky cases in which a quite rough approximation holds; with a slight linguistic abuse we call it a gaussian approximation 13 . Roughly speaking, this means taking the minimum value for the τ * technical parameter, i.e. τ * = 0. This formally relies on the assumption that integrating the system on the vector space defined by the tangent space at the critical point actually takes into account the relevant configurations giving the most important contribution to the functional integral. This was actually holding in the case of [14].
Does the gaussian approximation hold true also for the CRM model? Figure 5 reveals that there is actually a regime in which it can do pretty well. Not surprisingly, it is a regime in which results are not that far away from the phase quenched approximation; we know that this is a regime in which the two different eigenvalues of the hessian at the critical point are quite close to each other and the problem appears all in all quite symmetric and not that far away from the regime of (7). Note that the value of the (rescaled) mass at which the solution provided by the gaussian approximation departs from the correct one varies with N .
Next step was to leave the gaussian approximation aside and try to implement the full Aurora algorithm. There are a couple of issues one should be aware of: we need a solid estimate for τ * ; also, within a time length of order τ * we have to make sure we have under good numerical control both the SA and the SD. The latter is the critical one, for which we have already made clear that a BVP formulation is the choice to go for. Our implementation was along the same lines of the code available at [31]. All in all, our experience with the complete Aurora algorithm for the CRM model was at first somehow inconclusive: a clearcut indication of values of τ * at the same time safe and manageable was missing. We will come back to this observation later, in the framework of the other numerical approach that we chose to implement.
A different numerical approach
We now want to take advantage of the parametrization Φ ∈ J σ ↔ (n, t) (a few of the formulae we will need to implement our strategy were clearly stated in [3], while the strategy itself we will see at work in the following was first described in [32]).
In (25) we have taken into account that S I is constant on J σ . Moreover, there could be more than one relevant chart and on a given chart we have to take into account the residual phase. For the sake of simplicity in notation, we now take a few shortcuts. First of all, we discard the overall phase e −iS I ; it will be easy to account for it when we come back to the actual computation of an observable. We also discard the fine detail of more than one chart, since in practice this is not a issue 14 . Finally, we leave the residual phase aside, having in mind that we can take it into account a posteriori by reweighting. We thus write a new quantity, which is the one we will further manipulate in order to single out the contributions from the single ascents. We define 15
Z (σ) ≡ n i=1 dy i e −S R(26)
Roughly speaking, this is the quantity that can have a probabilistic interpretation. The key point is now to write an expression for 1
1 = ∆n (t) n k=1 dn k δ | n| 2 − 1 dt n i=1 δ (y i − y i (n, t))(27)
where {y i (n, t)} are the coordinates of the field as expressed in the local (orthonormal) basis {U
∆n (t) = det δ(| n | 2 −1) δt δ(| n | 2 −1) δn 1 · · · δ(| n | 2 −1) δnn δ(y 1 −y 1 (n,t)) δt δ(y 1 −y 1 (n,t)) δn 1 · · · δ(y 1 −y 1 (n,t)) δnn . . . . . . . . . . . . δ(yn−yn(n,t)) δt δ(yn−yn(n,t)) δn 1 · · · δ(yn−yn(n,t)) δnn or ∆n (t) = det 0 2n 1 · · · 2n n δy 1 (n,t) δt δy 1 (n,t) δn 1 · · · δy 1 (n,t) δnn . . . . . . . . . . . .
δyn(n,t) δt δyn(n,t) δn 1 · · · δyn(n,t)
δnn (28)
The first column of this determinant can be easily related to the gradient of the action. It turns out that to compute the generic matrix element we need to do the following:
• We need to evolve not only the field, but the entire basis by integrating (6).
• We construct the 2n × n matrix V whose columns are the {V (i) (t)}.
• We construct the 2n × n matrix u whose columns are the vectors {u (i) } i=1···n which are obtained from the {V (i) (t)} by means of Gram-Schmidt orthonormalization procedure.
• The relation V = uE holds, with
E ij = V (j) · u (i) j ≥ i 0 j < i
• The entries of the determinant we are looking for are now given by
δy i δt = n k=1 λ k n k E ik δy i δn j = E ij(29)
15 For the sake of notational simplicity we also omit the explicit indication that the integration is on the thimble, as it is easy to recognize we are assuming. Notice that (26) is the generalization of (11) of section 3.1.
Not surprisingly, there is a lot of information in the (tremendous amount of) computations we have just sketched. In particular, if we now introduce the n × 2n complex space projector P P = 1 n×n i 1 n×n (30) then the n × n complex matrix U = P u is unitary; this is precisely the matrix of eq. (5), whose determinant is the residual phase e iω .
The details of the previous computation of ∆n (t) are given in appendix B. We now proceed to make use of the expression for the identity encoded in (27). Inserting it in (26) we get
Z (σ) = n i=1 dy i e −S R = n i=1 dy i e −S R ∆n (t) n k=1 dn k δ | n| 2 − 1 dt n i=1 δ (y i − y i (n, t)) = n k=1 dn k δ | n| 2 − 1 dt n i=1 dy i δ (y i − y i (n, t)) ∆n (t) e −S R = n k=1 dn k δ | n| 2 − 1 dt ∆n (t) e −S R (n,t)
which has a possible interpretation in terms of
Z (σ) = n k=1 dn k δ | n| 2 − 1 Z (σ) n(31)
i.e. there is a contribution to the partition function for each SA path
Z (σ) n = +∞ −∞ dt ∆n (t) e −S R (n,t)(32)
Note that the procedure naturally defines a probability, i.e. that for a point reached at time t on the SA defined byn
Pn (t) = ∆n (t) e −S R (n,t) Z (σ) n(33)
One can also naturally define the cumulative distribution function (it is manifestly nondecreasing, positive definite and has the correct normalization)
Fn (t) = 1 Z (σ) n t −∞ dt ∆n (t ) e −S R (n,t )(34)
Since we can easily invert this function numerically, we have a tool to ideally sample configurations on a single SA. Namely, we extract a random number ξ ∈ [0, 1] and then get the point on the SA (rather, the time at which the point is reached) by t = F −1 n (ξ). Actually this is not that useful. The fact that we ascend all the way along a given SA in order to compute Z (σ) n suggests that it is rather convenient to compute the entire contribution which is attached to that given ascent. On the other hand, the relative weight of a given SA (within the complete partition function Z (σ) ) is given by Z (σ) n /Z (σ) .
We now want to take advantage of the parametrization Φ ∈ J σ ↔ (n, t) in the computation of an observable. In the following, we will assume we are in a case in which only one single thimble is relevant. This is not the general case, but for what we want to obtain it is not a limitation. In the cases in which more than one thimble contribute, we can address the problem using the same strategy described in 3.1 in the context of the quartic toy model: it will be easy for the reader to generalize eq. (14) and the discussion following it. With this caveat in mind, we first of all write
O = Jσ dz 1 ∧ . . . ∧ dz n O e −S Z (σ) = n i=1 dy i O e iω e −S R n i=1 dy i e iω e −S R = e iω O σ e iω σ where . . . σ ≡ Z (σ)−1 n i=1 dy i . . . e −S R .
We have till now simply generalized eq. (13). We can go further by making use of the new parametrization we introduced 16
< O > = Dn dt ∆n(t) e −S R (n,t) e iω(
Eq. (35) is in a sense a new average. Namely, the different directionsn can now be regarded as the new degrees of freedom of the overall integral, the quantities to be measured are the O e iω n and e iω n (i.e. partial averages attached to single SA), and the weights are given by the Z (σ) n . It is rather obvious that • The basic building block are complete ascents. This is good, since we can have their computation under good numerical control. In other words, sampling on the thimble is not a problem: we stay on the thimble by definition.
• The way to importance sampling now appears tricky. This is easy to understand, since picking up a contribution means picking up an, whose weight Z (σ) n is not known a priori, but only after the SA path associated ton has been obtained.
• The crudest approach one can think of is of course a uniform sampling of then-space; this is a static, crude Monte Carlo, which can easily become inefficient (in particular for large systems). 16 We denote Dn ≡ n k=1 dn k δ | n| 2 − 1 .
In the following we will just be satisfied with the last approach of static, crude Monte Carlo: this will be enough to show that we can reproduce the correct results for the model at hand (even in regions where the sign problem is quite severe) and this holds true taking into account the contribution of the single critical point we found. From this very basic approach there will be something to be learnt also with respect to the Aurora algorithm (and on the computation of density of states as well). We will finally report on a few speculations on smarter algorithms which we are trying to devise. Figure 6 displays the results we obtained from simulations performed in the static Monte Carlo approach we have just discussed. All these results come from the contribution of one single thimble. As we have already pointed out, one original motivation of ours turned out to be not relevant: we were ready for looking for dominance of one thimble in some asymptotic regime (in the thermodynamic limit) and, in the region of parameters we studied, we actually found no other thimble but the trivial one. Also, complexifying the theory in the parametrization that was shown to be problematic for complex Langevin [12] did not result in any problem. Results are shown for N = 1, 2, 3, 4 and fixed N f = 2,μ = 2. It is interesting to regard the parametrization we have employed from another point of view. In the upper panel of figure 7 we plot the quantity ∆n(t) e −S R (n,t) /Z (σ) n as a function of S R along a given ascent (remember that on each ascent one single value of S R is only read once). This is the real weight of the functional integral for the configurations which lie on that given ascent: it can be thought of as a different way of looking at the density of states (namely, this is the contribution attached to a given ascent). In the lower panel of figure 7 we plot the same quantity as a function of the flow time. A first remark is due for the long initial flat region. Consider eq. (7). When we want to compute the contribution from a single ascent (that is from a singlen) we need an initial condition, i.e. an initial value t 0 at which the asymptotic regime holds. In principle, the more back in time we take t 0 , the better initial condition we prepare. There are of course accuracy issues one has to live with. The flat initial region reflects the fact that we do our best to ensure we stay on the thimble. On the other side, one would like to know till what value of the flow time the asymptotic regime holds to a reasonable confidence. We can think of more than one indicator for the latter condition, e.g. the gaussian approximation of the action is very close to the actual value of S, or the factor ∆n(t) is very close to its gaussian approximation (see appendix B.1). We mark with a (red) star a value of flow time which can be assumed to be the boundary of the region we have just described. Now, an efficient dynamic Monte Carlo is supposed to sample configurations in regions where the weight is concentrated. In this sense we can say that the distance (in flow time) from the region around the maximum of ∆n(t) e −S R (n,t) /Z Both plots show that reweighting for the residual phase is essential to get the correct results; this happens to be the case in particular for low dimensions. 17 Note that in this case we take the residual phase into account in the numerator.
Results for the CRM model
We admittedly made use of the crudest possible application of the parametrization contained in eq. (35), i.e. a static, crude Monte Carlo sampling of the integral. It is static, i.e. it is not based on a stochastic process. It is crude, because it does not implement importance sampling 18 . In general, importance sampling computes an integral I = dxf (x) as an av-
erage I = dx ρ(x) f (x)
ρ(x) and all the point is being able to extract configurations distributed according to ρ(x). The natural importance sampling for (35) would try to sample the space of ascents (i.e. the differentn) according to the weights Z (σ) n /Z (σ) and we have already made the point that this is tricky. Crude Monte Carlo simply extractsn with constant probability. As it is well known, the more non-trivial the profile of Z (σ) n /Z (σ) is, the more inefficient one expects the crude Monte Carlo to be. This can be clearly seen at low masses, where the problem is less symmetric with respect to different choices of then: we have already made this point while commenting on the failure of the gaussian approximation for the Aurora algorithm. The interested reader is referred to appendix B.1 to get more insight on this in the case of the gaussian action. Here we want to stress that the problem with crude Monte Carlo does not necessarily relate to the dimension N (of the matrices): going low enough in mass at any fixed N can already result in a difficult computation 19 . As the mass gets lower, one indeed clearly sees larger error bars in figure 6. Needless to say, this is a region in which we had to collect many ascents: hundreds of thousands, actually, by definition all statistically independent. This is a huge numerical effort. On the other side, even accepting that at some point our crude Monte Carlo has to give up in front of a non-trivial profile of the weights Z (σ) n /Z (σ) , it is reassuring to see that we nevertheless solved a non-trivial problem: figure 6 clearly shows that we were able to solve the problem also in regions in which the sign problem shows up as quite severe.
All in all, we saw that implementing importance sampling is hard, since the relative weights Z (σ) n /Z (σ) are only known after the complete SA associated ton has been computed. This could sound a very pessimistic conclusion. On the other hand, the gaussian approximation of the Z (σ) n can be easily computed (see appendix B.1), which suggests the idea of making use of them to formulate proposals for then. This is something we are currently investigating.
Conclusions and prospects
We discussed the solution of a simple toy model via thimble regularization. Quite interestingly, this model, which dates back to some thirty years ago and was proposed as a sort of benchmark for complex Langevin, was still missing a full solution in the context of the latter. In thimble regularization the solution is clear and can be implemented numerically by a number of simulation algorithms. 18 We recall that the Monte Carlo methods we are mostly familiar with are dynamic Monte Carlo in which importance sampling is obtained via convergence to the equilibrium distribution of a stochastic process, which in virtually all the cases is a Markov chain. 19 Of course going to higher values of N would result in extra computational effort, but what is really crucial is to see at each value of N what is the threshold in mass below which we have a too much non-trivial profile of the Z (σ)
n /Z (σ) .
We then investigated the Chiral Random Matrix model and showed that thimble regularization can successfully deal with the sign problem that the system displays. In the region of parameters we studied, a single thimble accounts for the results. We made use of a parametrization in terms of contributions attached to SA (which are the basic building blocks to define the thimble). This was done by crude Monte Carlo, leaving open the problem of devising a smarter algorithm (importance sampling) to take advantage of the parametrization we made use of. This is the subject we are currently investigating in view of other applications.
A The Hessian for the CRM model
We want to compute the hessian at the critical point a ij = b ij = α ij = β ij = 0. We need the following second derivatives (the fields are complexified by setting e.g. a ij = a R ij + i a I ij etc.) ) and the coefficients are given by
∂ 2 S R ∂a R mn ∂a R ij 0 = − ∂ 2 S R ∂a I mn ∂a I ij 0 = A − δ mi δ nj ∂ 2 S R ∂a R mn ∂a I ij 0 = ∂ 2 S R ∂a I mn ∂a R ij 0 = 0 ∂ 2 S R ∂b R mn ∂b R ij 0 = − ∂ 2 S R ∂b I mn ∂b I ij 0 = A − δ mi δ nj ∂ 2 S R ∂b R mn ∂b I ij 0 = ∂ 2 S R ∂b I mn ∂b R ij 0 = 0 ∂ 2 S R ∂α R mn ∂α R ij 0 = − ∂ 2 S R ∂α I mn ∂α I ij 0 = A + δ mi δ nj ∂ 2 S R ∂α R mn ∂α I ij 0 = ∂ 2 S R ∂α I mn ∂α R ij 0 = 0 ∂ 2 S R ∂β R mn ∂β R ij 0 = − ∂ 2 S R ∂β I mn ∂β I ij 0 = A + δ mi δ nj ∂ 2 S R ∂β R mn ∂β I ij 0 = ∂ 2 S R ∂β I mn ∂β R ij 0 = 0 ∂ 2 S R ∂a R mn ∂b R ij 0 = ∂ 2 S R ∂a I mn ∂b I ij 0 = ∂ 2 S R ∂a R mn ∂b I ij 0 = ∂ 2 S R ∂a I mn ∂b R ij 0 = 0 ∂ 2 S R ∂a R mn ∂α R ij 0 = ∂ 2 S R ∂a I mn ∂α I ij 0 = 0 ∂ 2 S R ∂a R mn ∂α I ij 0 = ∂ 2 S R ∂a I mn ∂α R ij 0 = Bδ mi δ nj ∂ 2 S R ∂a R mn ∂β R ij 0 = ∂ 2 S R ∂a I mn ∂β I ij 0 = ∂ 2 S R ∂a R mn ∂β I ij 0 = ∂ 2 S R ∂a I mn ∂β R ij 0 = 0 ∂ 2 S R ∂b R mn ∂α R ij 0 = ∂ 2 S R ∂b I mn ∂α I ij 0 = ∂ 2 S R ∂b R mn ∂α I ij 0 = ∂ 2 S R ∂b I mn ∂α R ij 0 = 0 ∂ 2 S R ∂b R mn ∂β R ij 0 = ∂ 2 S R ∂b I mn ∂β I ij 0 = 0 ∂ 2 S R ∂b R mn ∂β I ij 0 = ∂ 2 S R ∂b I mn ∂β R ij 0 = Bδ mi δ nj ∂ 2 S R ∂α R mn ∂β R ij 0 = ∂ 2 SA − = 2 N − N f cosh 2 µ m 2 A + = 2 N + N f sinh 2 µ m 2 B = −2N f cosh µ sinh µ m 2
The hessian for N = 1 is (with the conventional choice of ordering: a R , b R , α R , β R , a I , b I , α I , β I )
H (1) = A − 0 0 0 0 0 B 0 0 A − 0 0 0 0 0 B 0 0 A + 0 B 0 0 0 0 0 0 A + 0 B 0 0 0 0 B 0 −A − 0 0 0 0 0 0 B 0 −A − 0 0 B 0 0 0 0 0 −A + 0 0 B 0 0 0 0 0 −A +
As the second derivatives are manifestly diagonal with respect to the indices i, j, m, n, the hessian for a generic N is block-diagonal
H (N ) = N H (1)
The model thus features a huge degeneracy of eigenvalues, as the spectrum of N = 1 is repeated N times. The spectrum for N = 1 features 4 positive eigenvalues and 4 eigenvalues opposite in sign (as expected by holomorphicity). We are interested in the positive part of the spectrum. An explicit computation shows that the distinct positive eigenvalues of H (1) are actually 2 and they are
λ ± = 1 2m 2 2N f cosh (2µ) ± √ 2 8m 4 − 8N f m 2 + N 2 f + N 2 f cosh (4µ)
We note in passing that (here and in many other places) we could have written formulas in the complex notation of Takagi factorization theorem (see [3]), which we decided not to employ here and in all the paper to stick to a completely real notation.
B Computing ∆n(t)
We want to compute ∆n(t), which is defined in (27). The main point is that we are ascending along a given flow, and while doing that we are transporting along the flow also the basis vectors, i.e. we are integrating (6) as well. Near the critical point the (7) hold, in which the parametrization Φ ∈ J σ ↔ (n, t) is manifest. Given a reference point t 0 1, the (7) can be regarded as initial conditions for the flow associated ton. Near a generic point, under infinitesimal variations of t andn, the variation of the point δΦ is given by
δΦ = δΦ (n, t) = n i=1 V (i) (t) δc (i)
This is so because δΦ is itself a vector belonging to the tangent space T Φ J σ . The (constant) coefficients δc (i) can be worked out from the asymptotic form of Φ (t) near the critical point
δΦ ≈ δ φ σ + n i=1 v (i) e λ i t n i = n i=1 v (i) n j=1 δn j ∂ ∂n j + δt ∂ ∂t e λ i t n i = = n i=1 v (i) e λ i t (δn i + λ i n i δt) ≈ n i=1 V (i) (t) (δn i + λ i n i δt)
from which it follows
δc (i) = δn i + λ i n i δt
Being δΦ a vector of T Φ J σ , we can write it as a decomposition on the (orthonormal) u-basis δΦ = n i=1 u (i) δy i and from this we have
δy i = 2n j=1 u (i) j δφ j
Let us now consider the terms δy i δ appearing in ∆n (t), where is either t or n j . For these we have Now we make use of the explicit form of δc (k) , which gives δc (k) δt = λ k n k and δc (k) δn j = δ kj , from which one can easily derive the (29).
B.1 ∆n(t) in the gaussian approximation
We can compute ∆n (t) for the gaussian case (purely quadratic action), where the asymptotic form for the SA and parallel-transport equations is correct arbitrarily far away from the critical point. In that case the entries of ∆n (t) are (E = 1 n×n ) δy i δt = λ i n i e λ i t δy i δn j = e λ i t δ ij and the determinant is
∆n (t) = 2 n i=1 λ i n 2 i e n i=1 λ i t
For the gaussian action S R (n, t) = S R (φ σ ) + 1 2 n k=1 λ k n 2 k e 2λ k t , so that collecting everything we can write an expression for the weight itself From this expression it is easy to understand that the more the eigenvalues differ from each other (which in our case happens for low values of the mass parameter), the more various Z (σ) n can differ.
B.2 A useful consistency relation
Since in the end we are performing quite a lot of computations, it is useful to have a consistency relation to be checked while ascending along the flow. The gradient of the action is yet another vector belonging to the tangent space T Φ J σ . Let us write the decomposition
∇ Φ S R = n i=1 V (i) (t) g (i)
the coefficients g (i) can be found with the aid of the asymptotic form of the action
∇ Φ S R ≈ ∇ Φ S R (φ σ ) + 1 2 Φ T HΦ = HΦ ≈ H n i=1 v (i) n i e λ i t = = n i=1 v (i) e λ i t n i λ i ≈ n i=1 V (i) (t) n i λ i
where we have used the symmetry of the hessian and the fact that Hv (i) = λ i v (i) . We have found that g (i) = n i λ i , so while integrating the flow equations, we can keep checked the norm
∇ Φ S R − n i=1
V (i) (t) n i λ i and make sure that it is small with respect to the size of the system.
Figure 1 :
1Thimbles structure for σ = 0.5 + i0.75, λ = 2 (left panel). In this case only the unstable thimble attached to z = 0 intersects the real axis and thus only one critical point contributes. On the right we can see how the Langevin simulation correctly covers the relevant thimble.
Figure 2 :
2Thimbles structure for σ = −0.5 + i0.75, (left panel) and σ = i0.75 (right); in both cases λ = 2. For σ R < 0 (left) all the three critical points contribute. σ R = 0 (right) is an example of a Stokes phenomenon.
Figure 3 :
3Left panel: the three thimbles associated to σ = −0.5 + i0.75 correctly sampled by a Metropolis simulation. Right panel: for the same choice of parameters, the computed values of φ 8 over a range of both σ R > 0 and σ R < 0.
Figure 4
4displays both the exact (solid red line) and the phase quenched (dashed blue line) results for 1 N η η as a function ofm for N = 1, 2, 3, 4 and fixed N f = 2,μ = 2. As it can be seen, the sign problem is indeed severe in certain regimes of small (rescaled) masses 12 .
Figure 4 :
4Exact (solid red line) and phase quenched (dashed blue line) results for the condensate, at fixed N f = 2,μ = 2.
Figure 5 :
5Exact (solid red line), phase quenched (dashed blue line) and gaussian approximation results for the condensate, at fixed N f = 2,μ = 2.Basically one describes a generic point by locating it on the SA curve it lies on. This means providingn (the direction one is taking while leaving the critical point) and the time t at which one reaches Φ while integrating the SA equations. The first goal is now to rewrite the contribution to the partition function which is attached to one thimble. In full detail this reads Z (σ) = Jσ dz 1 ∧ . . . ∧ dz n e iω e −S R
n
(t)} parallel-transported along the SA defined byn until time t. The solution for ∆n (t) is in terms of (the module of) a determinant
n,t) O(n, t) Dn dt ∆n(t) e −S R (n,t) e iω(n,t) ∆n(t) e −S R (n,t) e iω(n,t)
Figure 6 :
6Exact (solid red line), phase quenched (dashed blue line) and thimble simulations results for the condensate, at fixed N f = 2,μ = 2.
Figure 7 :
7Real weight of the functional integral for the configurations which lie on the SA defined by a particularn as a function of S R (upper panel) and t (lower panel) for N = 2, m = 7, N f = 2,μ = 2.
the flow time marked with the star is a reasonable indicator of the τ * parameter of the Aurora algorithm.A comment is due concerning the residual phase: we encountered no problem in taking it into account by reweighting. As it was expected, it is a smooth function on the ascents, so that O e iω n and e iω n can be safely computed and wild cancellations are never there at any stage of our computations. The fact that the residual phase is smooth does not of course mean it has no net effect, as can be seen infigure 8. In the upper panel we show the effect of neglecting the contribution of the denominator of eq. (35): this amounts to computing Z (σ)−1 Dn Z (σ) n e iω O n 17 . In the lower panel we show yet another phase quenched computation, namely what we could term a residual phase quenched result. In this case we simply omit the contribution of the residual phase and compute Z (σ)
Figure 8 :
8The effect of not accounting for the residual phase. In the upper panel its contribution is not accounted in the denominator of eq. (35), i.e. we compute Z (σ)−1 Dn Z (σ) n e iω O n . In the lower panel we omit the residual phase completely and compute Z (σ)−1 Dn Z (σ) n O n .
δc (k) δ
where use of the Cauchy-Riemann equations has been made (the other derivatives are trivially related to these by Schwarz theorem, e.g. ∂ 2 S RR
∂α I
mn ∂β I
ij 0
=
∂ 2 S R
∂α R
mn ∂β I
ij 0
=
∂ 2 S R
∂α I
mn ∂β R
ij 0
= 0
∂a R
mn ∂a R
ij
= ∂ 2 S R
∂a R
ij ∂a R
mn
This is a literal citation from[8]; we will have more to quote later on the subject of resurgence.
In what follows we will be a little bit sloppy in our notation: whenever there is not subscript attached to a symbol (e.g. x), that will denote a n-dimensional coordinate.
It is worth to recall here that the subscript i is a multi-index, in which real and imaginary parts are on the same footing.6 The hat notation remembers us of the normalization condition, i.e.n singles out a direction in the tangent space at the critical point.
see[1] for a detailed explanation of the Stokes phenomenon with respect to the Airy integral
τ is the real coordinate on the (one-dimensional) thimble.9 In the notation of (2) we now denote φ + = p 1 and φ − = p 2 .
The value of the condensate is real. This has to be understood later when we will compare to our results; we will always plot only the real part, the imaginary one having been correctly verified to be zero within errors.
One should not confuse this with the gaussian approximation described in 3.1. In that case the action is approximated with its leading (gaussian) term and all the thimble analysis is performed consequently. In this (algorithmic) gaussian approximation one pretends that the thimble manifold we are interested in and the thimble associated to the gaussian approximation of the action sit on top of each other also away from the critical point.
We will be concerned with single ascents, for which we will revert to a different, smooth parametrization.
AcknowledgmentsWe warmly thank Luigi Scorzato for many valuable discussions and for all the common work on the subject in recent years. We are indebted to Kim Splittorff for very fruitful conversations and for having introduced us to the subject of the CRM model. We also acknowledge useful discussions with Marco Cristoforetti and Michele Brambilla and we are grateful to Christian Torrero who has collaborated with us at an early stage of this work. We finally thank A. Alexandru for a useful conversation which made us consider a more extended comment on the effects of the residual phase. This
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|
[] |
[
"Zipf's laws of meaning in Catalan",
"Zipf's laws of meaning in Catalan"
] |
[
"Neus Català \nComputer Science Departament\nTALP Research Center\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain\n",
"Jaume Baixeriesid \nComputer Science Departament\nLARCA Research Group, Complexity and Quantitative Linguistics Laboratory\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain\n",
"‡ ",
"Ramon Ferrer-I-Cancho \nComputer Science Departament\nLARCA Research Group, Complexity and Quantitative Linguistics Laboratory\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain\n",
"Lluís 2☯ ",
"Padró \nComputer Science Departament\nTALP Research Center\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain\n",
"‡ ",
"Antoni Herná Ndez-Ferná Ndezid \nComputer Science Departament\nLARCA Research Group, Complexity and Quantitative Linguistics Laboratory\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain\n\nSocietat Catalana de Tecnologia\nSecció de Ciències i Tecnologia\nInstitut d'Estudis Catalans -Catalan Studies Institute\nBarcelonaCataloniaSpain\n"
] |
[
"Computer Science Departament\nTALP Research Center\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain",
"Computer Science Departament\nLARCA Research Group, Complexity and Quantitative Linguistics Laboratory\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain",
"Computer Science Departament\nLARCA Research Group, Complexity and Quantitative Linguistics Laboratory\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain",
"Computer Science Departament\nTALP Research Center\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain",
"Computer Science Departament\nLARCA Research Group, Complexity and Quantitative Linguistics Laboratory\nUniversitat Politècnica de Catalunya\nBarcelonaCataloniaSpain",
"Societat Catalana de Tecnologia\nSecció de Ciències i Tecnologia\nInstitut d'Estudis Catalans -Catalan Studies Institute\nBarcelonaCataloniaSpain"
] |
[] |
In his pioneering research, G. K. Zipf formulated a couple of statistical laws on the relationship between the frequency of a word with its number of meanings: the law of meaning distribution, relating the frequency of a word and its frequency rank, and the meaning-frequency law, relating the frequency of a word with its number of meanings. Although these laws were formulated more than half a century ago, they have been only investigated in a few languages. Here we present the first study of these laws in Catalan. We verify these laws in Catalan via the relationship among their exponents and that of the rank-frequency law. We present a new protocol for the analysis of these Zipfian laws that can be extended to other languages. We report the first evidence of two marked regimes for these laws in written language and speech, paralleling the two regimes in Zipf's rank-frequency law in large multiauthor corpora discovered in early 2000s. Finally, the implications of these two regimes will be discussed.
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10.1371/journal.pone.0260849
| null | 235,694,124 |
2107.00042
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36591b8db8ad9c1f1227d25ee50f0299d88664af
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Zipf's laws of meaning in Catalan
Neus Català
Computer Science Departament
TALP Research Center
Universitat Politècnica de Catalunya
BarcelonaCataloniaSpain
Jaume Baixeriesid
Computer Science Departament
LARCA Research Group, Complexity and Quantitative Linguistics Laboratory
Universitat Politècnica de Catalunya
BarcelonaCataloniaSpain
‡
Ramon Ferrer-I-Cancho
Computer Science Departament
LARCA Research Group, Complexity and Quantitative Linguistics Laboratory
Universitat Politècnica de Catalunya
BarcelonaCataloniaSpain
Lluís 2☯
Padró
Computer Science Departament
TALP Research Center
Universitat Politècnica de Catalunya
BarcelonaCataloniaSpain
‡
Antoni Herná Ndez-Ferná Ndezid
Computer Science Departament
LARCA Research Group, Complexity and Quantitative Linguistics Laboratory
Universitat Politècnica de Catalunya
BarcelonaCataloniaSpain
Societat Catalana de Tecnologia
Secció de Ciències i Tecnologia
Institut d'Estudis Catalans -Catalan Studies Institute
BarcelonaCataloniaSpain
Zipf's laws of meaning in Catalan
RESEARCH ARTICLE ☯ These authors contributed equally to this work. ‡ These authors also contributed equally to this work. * [email protected]
In his pioneering research, G. K. Zipf formulated a couple of statistical laws on the relationship between the frequency of a word with its number of meanings: the law of meaning distribution, relating the frequency of a word and its frequency rank, and the meaning-frequency law, relating the frequency of a word with its number of meanings. Although these laws were formulated more than half a century ago, they have been only investigated in a few languages. Here we present the first study of these laws in Catalan. We verify these laws in Catalan via the relationship among their exponents and that of the rank-frequency law. We present a new protocol for the analysis of these Zipfian laws that can be extended to other languages. We report the first evidence of two marked regimes for these laws in written language and speech, paralleling the two regimes in Zipf's rank-frequency law in large multiauthor corpora discovered in early 2000s. Finally, the implications of these two regimes will be discussed.
Introduction
During the 1st half of the last century, G. K. Zipf carried out a vast investigation of statistical regularities of languages [1][2][3], that lead to the formulation of linguistic laws [4]. Among them, a subset has received very little attention: laws that relate the frequency of a word with its number of meanings in two ways. On the one hand, the law of meaning distribution, that relates the frequency rank of a word with its number of meanings (the most frequent word has rank 1, the 2nd most frequent word has rank 2,. . .). On the other hand, the meaning-frequency law, that relates the frequency of words to their number of meanings.
Both Zipfian laws of meaning indicate the more frequent words tend to have more meanings and their mathematical definition takes the form of a power law [3,5]. The relationship between the number of meanings of a word, μ, and word frequency, f, Zipf where δ � 1/2 [5][6][7]. The law of meaning distribution that relates the meanings μ of a word with its frequency rank i as
m / i À g ;ð2Þ
where γ � 1/2 [6,8]. Interestingly, G. K. Zipf never investigated the meaning-frequency law empirically but deduced it from the law of meaning distribution [5,9] and the popular rank-frequency law, relating f (the frequency of a word) with i (its frequency rank) approximately as [1,10] f / i À a :
Zipf deduced Eq 1 with δ = 1/2 from α = 1 (for Zipf's rank-frequency law), and γ = 1/2 (for the law of meaning distribution) [3,5]. Recently, it has been shown that the three exponents of the power laws are related by [6,8]
d ¼ g a :ð4Þ
It should be noted that a weak version of Zipf's meaning-frequency law simply indicates that there is a positive correlation between word frequency (f) and the number of meanings (μ) what has been connected with a family of Zipfian optimization models of communication [11]. To date, experimental evidence has been accumulating for Zipf's law of meaning distribution, either fitting a power law function or computing the correlation between word frequency and meaning [7,[12][13][14][15].
Thus, there is new empirical evidence for the weak version of Zipf's law of meaning distribution on eight languages from different language families (Indo-European, Japonic, Sino-Tibetan and Austronesian), also retrieving exponents of Zipf's law of meaning distribution (γ) between 0.21 and 0.51 depending on the language (see [12] for details). The results of Bond et al (2019) [12] are consistent with previous works that they review and that have verified the correlation between the frequency of words and their meanings [7,15] even in child language and language-directed speech [13,14]. In fact, Bond et al (2019) already noted the influence of both binning size and Zipf's law deviations on the predictive power of the Zipfian laws for the meaning although without considering Eq 4. Eq 4 actually involves assuming the validity of Zipf's rank-frequency law, which has previously been seen not always happen, either due to the appearance of more than one regime in the distribution of words [16][17][18] or because the data fits better with other mathematical functions [17,19,20].
This work is the first empirical study of Zipf's laws of meaning in Catalan and also the first one that considers two sources of different modality for word frequency: an speech corpus and a written one. Our study consists of investigating the exponents of Zipf's meaning-frequency law (Eq 1) and meaning distribution (Eq 2) and then to test the validity of the relationship between the exponents (Eq 4) previously proposed [6,8]. As far as we know this is the first time that the three Zipfian laws have been analyzed together empirically in one language: Catalan.
On Catalan
Catalan is a Romance language spoken in the Western Mediterranean by more than ten million people. Catalan is considered a language between the Ibero-Romance (Spanish, Portuguese, Galician) and Gallo-Romance (French, Occitan, Franco-Provençal) languages [21].
Catalan can be considered a language of intermediate complexity from the quantitative perspective of information theory [22]: Catalan has an intermediate level of entropy rate of 5.84, with languages mean around 5.97±0.91 [23] and morphological complexity (in terms of word complexity, with Catalan ranked in 202 position over 520 languages [24]). As it happens with other Romance languages Catalan does have a great inflectional variability [21], especially in verbs but also in nouns and adjectives [25] with some lexical peculiarities [26]. Besides, derivation is a very productive procedure in the formation of new words in Catalan, suffixation being the most important (above the prefixation and infixation) [27], as it is also usual in other Romance languages [21]. Despite the limited geographical extension of Catalan, there are local variations in suffixation processes that have been reviewed in detail (see [27] and references therein).
Catalan is also a language that has recently been studied in depth under the paradigm of quantitative linguistics [28], recovering the best-known linguistic laws in which meaning does not intervene (as is the case of Zipf's rank-frequency law, Herdan-Heaps' law, the brevity law or the Menzerath-Altmann law) in this speech corpus (Glissando) and in its transcripts [29]. In addition, although the issue is still debated [30], it has also been found the lognormal distribution of words, lately proposed as a new linguistic law [4,29] but, nevertheless, the statistical patterns of meaning in Catalan has not been addressed until the present study.
Since Zipf's pioneering research, one of the most remarkable discoveries on the rankfrequency law in large multi-author textual corpus is that the power law put forward by Zipf (Eq 3) has to be generalized, on a first approximation, as a double power law of the form [16][17][18][31][32][33],
f � ( i À a 1 for i � i � i À a 2 for i � i � ;ð5Þ
where α 1 is the exponent for the low rank (high frequency) power-law regime corresponding to Eq 3, α 2 is that of the high rank (low frequency) regime, and i � is the breakpoint rank. In the British National Corpus it was found that α 1 = 1 and α 2 = 2 [16]. Precisely, these two scaling regimes in Zipf's rank-frequency law were referred to as the kernel lexicon, for the most frequent words usually shared by the majority of speakers of the language, while the rarer words would be part of the so-called unlimited lexicon, formed by less common, more specialized or technical words or, in the case of more extensive diachronic corpus, that have fallen into disuse [16,31].
Here we will provide evidence, in a novel way, that such a double power-law regime also applies to Zipf's laws of meaning. First, the law of meaning distribution becomes m �
( i À g 1 for i � i � i À g 2 for i � i � ;ð6Þ
where γ 1 is the exponent for high frequency regime, corresponding to δ in Eq 1, and γ 2 is the exponent for the low frequency regime. Second, the meaning-frequency law becomes
m � ( f d 2 for f � f ði � Þ f d 1 for f � f ði � Þ;ð7Þ
where f(i � ) is the frequency of the word of rank i � , δ 2 is the exponent for the low frequency regime and δ 1 is the exponent for the high frequency regime (Eq 1). Notice that, according to our convention, the subindex 1 (in α 1 , γ 1 and δ 1 ) is used to refer to high frequencies (low ranks) while the subindex 2 (in α 2 , γ 2 and δ 2 ) is used to refer to low frequencies (high ranks). The remainder of the article is organized as follows. In Section On Materials and methods, we introduce the materials used, i.e. two Catalan corpora, one based on written texts (CTILC) and the other based on transcribed speech (Glissando), as well as the DIEC2 normative dictionary [34], from which the number of meanings (or polysemy of words) were obtained. We also present the lemmatization process with FreeLing [35] and the binning method used, following Zipf [5]. In Section Result theoretic model selection, we first present an empirical exploration of the three Zipfian laws outlined above assuming a single regime followed by and analysis assuming two regimes. We will show that Eq 4 offers a poor prediction of δ when a single power-law regime as in Zipf's classic work for the multi-author corpora described above. In contrast, the assumption of two regimes, namely
d 1 ¼ g 1 a 1ð8Þd 2 ¼ g 2 a 2 ;ð9Þ
improves the quality of predictions. Finally, in Section Discussion, we discuss these results for Catalan in the context of previous studies [12][13][14][15] and revisit the hypothesis of the existence of a core vocabulary and a peripheral vocabulary [16,18,31].
Materials and methods
Materials
One of the most important institutional contributions to Catalan corpus linguistics is the Corpus Textual Informatitzat de la Llengua Catalana (CTILC), a corpus that covers Catalan written language in texts from 1833 to 1988 (available at https://ctilc.iec.cat/). Based on this corpus, of which an expanded and updated version is in progress, the eminent linguist Joaquim Rafel i Fontanals was able to create a frequency dictionary of Catalan words [36]. By way of example, this implies that a trendy word at present (unfortunately) such as coronavirus does not appear in CTILC corpus and, however, there are words that have fallen into disuse from the 19th century such as coquessa (a cooker who is hired to make meals on holidays). This frequency dictionary contains more than 160,000 distinct lemmas, with about 52 million word tokens in total: about 29 million tokens from non-literary texts (56%) and 23 million tokens from literary texts (the remaining 44%) [36]. A more recent corpus is the Glissando corpus recorded in 2010, that contains more than 12 hours of read speech-news-and 12 more hours of studio recordings of dialogues which have been transcribed and aligned to the voice signal [37]. In fact, Glissando is an annotated speech corpus, that is bilingual (Spanish and Catalan) and consisting of the recordings of twenty eight speakers per language, with about 93,000 word tokens and more than 5,000 types in Catalan [29,37].
Methods
Preprocessing. The empirical study of meaning, from the experimental perspective of quantitative and corpus linguistics, has several methodological challenges that were already pointed out by Zipf in his seminal study [5]. In fact, Zipf devoted the first pages of [5] to the reflection and justification of the corpus he chose in his analysis. Zipf's pioneering work assumed the concept of corpus lemmatization, although without explicitly citing it [3,5]. Zipf refers to a "dictionary forms"_ and concludes, interestingly, that ". . .we have no reason to suppose that any 'law of meanings' would be seriously distorted if we concentrated our attention upon lexical units and simply ignored variations in number, case, or tense." (see in [3], p. 29]).
In general, this type of approach usually starts from a study corpus (either written or transcribed from orality) that is not lemmatized. Lemmatization is the process of grouping together the different forms of a word (inflectional forms or derivationally related) in a single linguistic element, identified by the word's lemma or dictionary form, since it is the form that is usually found in dictionaries [38].
This process of lemmatization is necessary to compare the study corpus with the dictionary corpus in order to establish which subset of words from the study corpus are present in the dictionary. In general, not all the words from a study corpus will be in the dictionary although the dictionary will be responsible for indicating the number of meanings of each word. The situation is depicted in Fig 1, where the corpora (the spoken corpus and written corpus) and the dictionary used in this work are displayed as a Venn diagram. Venn diagram where the sources used in the present study are displayed as a sets: the normative dictionary of the Catalan language (DIEC2), providing the number of meanings of each word; the written corpus CTILC, providing the basis for the descriptive dictionary and the dictionary of word frequencies of Catalan; and the speech corpus Glissando, that is not initially lemmatized, but that has been lemmatized here with FreeLing [35].
https://doi.org/10.1371/journal.pone.0260849.g001
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Our analysis of the relationship between the number of meanings of a lemma and its frequency consists in combining information about the number of meanings from the official dictionary of the Catalan language (Diccionari de la llengua catalana, DIEC2 [34]) with two sources for the frequency: written language (Corpus Textual Informatitzat de la Llengua Catalana, CTILC) and speech (Glissando Corpus). As shown in Fig 1, we work with a subset of each corpus, that is defined by the intersection of the study corpus (spoken or written) with the dictionary, which implies necessarily a decrease in the number of lemmas with respect to the initial corpus. Finally, the total number of different lemmas of each corpus is specified in Table 1. The intersection of the CTILC corpus with DIEC2 yields 52, 578 lemmas, while the intersection of the Glissando corpus with DIEC2 yields 3, 083 lemmas.
CTILC is manually lemmatized and annotated with Parts-of-Speech (PoS) tags and Glissando contains the direct transcriptions of spoken dialogues. In order to be able to perform the same analysis in both corpora, we resorted to FreeLing [35] to lemmatize the Glissando corpus.
FreeLing is an open-source library offering a variety of linguistic analysis functionalities for more than 12 languages, including Catalan. Details can be found at http://nlp.cs.upc. edu/freeling.
More specifically, the natural language processing layers used in this work were:
Tokenization & sentence splitting: Given a text, split the basic lexical terms (words, punctuation signs, numbers, etc.), and group these tokens into sentences.
Morphological analysis: Find out all possible Parts-of-Speech (PoS) for each token.
PoS-Tagging and Lemmatization: Determine the right PoS for each word in its context. Determining the right PoS allows inferring the right lemma in almost all cases.
Named Entity Recognition:
Detect proper nouns in the text, which may be formed by one or more tokens. We used only pattern-matching based detection relying mainly on capitalization.
We used FreeLing to perform PoS-tagging, lemmatization, and proper noun detection on Glissando corpus. We then filtered out all tokens marked as punctuation, number, or proper noun, before proceeding to count occurrences of each lemma and cross them with DIEC2. FreeLing was configured deactivating date, number, and multiword detection. In addition, proper noun configuration was changed from the default (gluing toghether proper nouns as multiwords) to keep proper noun tokens separated. Finally dictionary entries for contractions and related retokenization options were set to avoid contraction splitting. The link to the configuration file and execution command used to process the files is in Data availability section.
Of the 93, 069 tokens remaining after the filtering, 82, 838 (89%) were lemmatized by FreeLing, producing 4, 510 different lemmas. The remaining 11% tokens correspond to Table 1. Summary of the data. Number of tokens, different lemmas and availability details of the different sources used in the present study. The lemmas of the Glissando corpus were obtained after lemmatization with Freeling [35]. For the DIEC2 dictionary, and its intersection with the respective corpora, the number of tokens is not applicable because the dictionary only includes lemmas.
Corpus
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interjections, hesitations, half-words, foreign words (Spanish or English), colloquial expressions, non-capitalized proper nouns, or transcription errors (where the transcription was made phonetically, and not with the right form of the intended word) and were left out of the study. Fitting method. After intersecting each corpus with DIEC2 (Fig 1), we analized the resulting data sets and fitted the power law functions of the three Zipfian laws, using linear Least Squares (LS) on a logarithmic transoformation of both axes: the rank-frequency law, the law of meaning distribution and the meaning-frequency law, as defined in Eqs 1-3, respectively.
Following Baayen's method [39], we calculated the most likely breakpoint, i � (the data point where the two regimes cross) in both corpora. The method consists of scanning all possible breakpoints and compute, for each, the deviation (sum of squared errors) or deviance, as Baayen's refers to it, between the real points and Eq 5. This breakpoint is the rank that minimizes the deviance. Some care is required in this method because there might be more than one local optimum, as we will see.
Since we are studying three interrelated Zipfian laws (rank-frequency law, law of meaning distribution and meaning-frequency law), we decided, for simplicity, to transfer the breakpoint obtained for the rank-frequency law to the other double regime laws. The translation of the breakpoint to the double regime law of meaning distribution (Eq 6) is immediate: the breakpoint, i � , is the same as in the rank-frequency law. The translation of the i � to the breakpoint of the double regime meaning-frequency law (Eq 7) requires computing f(i � ). The value of f(i � ) is calculated from the mean of the frequencies of the data in the bin where i � is located. i � and f(i � ) are used to complete the fitting and retrieve the exponents of the double regime laws. Accordingly, for the law of meaning distribution, we fit Eq 6, where the relevant parameters are γ 1 and γ 2 . For the meaning frequency law, we fit Eq 7 where the relevant parameters are δ 1 and δ 2 .
Curve smoothing. The fitting method described above is applied to two kinds of data: the raw data and the smoothed data. In the raw data approach, every point of the curve corresponds to a "word". No bucketing or binning was applied. The smoothed data approach follows from Zipf, who applied a linear binning technique to reduce noise in his analysis of the law of meaning distribution [3,5]. In previous work [13], Zipf's analysis of the law in English [3] was revisited and values for exponents very close to those already obtained by Zipf were retrieved but, surprisingly, with sources of data differing from Zipf's work: in the case of the law of meaning distribution, γ achieving a value of 0.5 as in Zipf's pioneering research [5].
Given the robustness of this method, the data smoothing approach consisted of estimating the exponents of the three Zipfian laws after applying a linear binning on the data sets. In particular, we applied equal-size binning (K bins, each with n/K data points), in the sense of having the same number of data points in each bin, considering n the total number of lemmas of each subcorpus studied (Table 1). In equal-size binning, resulting bins have an equal number of observations in each group. Bin sizes have been chosen from divisors of the number of data points (lemmas) in each corpus to warrant that every bin has the same number of points and that no data point is lost. CTILC \ DIEC2 contains 53, 578 lemmas and Glissando \ DIEC2 contains 3, 083 lemmas.
Information theoretic model selection
We applied information theoretic model selection [40] to assess if a two-regime model approach provides a better description of the data than a single-regime model. In particular, we employed the corrected Akaike Information Criterion (AICc) and the Bayesian Information Criterion (BIC) [41]. Both scores favour models that are more likely but penalizing for their complexity, measured with k, the number of parameters. Both scores take into consideration n, the number of observations.
AICc is a variant of the plain AIC that incorporates a correction for n. BIC and the plain AIC are defined as
AIC ¼ 2k À 2 lnðL Þ BIC ¼ k lnðnÞ À 2 lnðL Þ;
whereL is the likelihood of the model. In turn, AICc is defined as [40] AICc
¼ AIC þ 2k 2 þ 2k n À k À 1ð10Þ
AIC and BIC values were computed using the R library stats. AICc was computed applying the values delivered by R to Eq 10.
For each Zipfian law, k = 3 for the one-regime models (intercept, slope and error variance). The number of parameters of the two-regime models varies. For Zipf's rank-frequency law, k = 5 (intercept, 2 slopes, breakpoint and error variance). For Zipf's law of meaning distribution and Zipf's meaning-frequency law, k = 4 (intercept, 2 slopes and error variance). Notice that the breakpoint here is not a free parameter because it is set by Zipf's rank-frequency law. Table 2, both in the case of equal-size binning and also when no smoothing is performed. Interestingly, γ approaches 0.5 in CTILC as the bin size increases and the difference between δ and δ 0 (the predicted value of δ from α and γ in Eq 4), is slightly reduced when binning is used in both corpora.
Results
One regime analysis
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Two regime analysis
After visual inspection of the rank-frequency law (Figs 2 and 3), the appearance of at least two regimes, i.e. straight lines in log-log scale with different slopes, is suggested in the CTILC written corpus (Fig 2). That is a well-known feature arising in large multi-author corpora [16][17][18][31][32][33]. However, interestingly, these two regimes are apparently not visually observed in the Glissando speech corpus (Fig 3) but they might be hidden in the greater dispersion of the points in meaning distribution, despite the linear binning, that can be seen in the speech corpus (Fig 5) with respect to CTILC (Fig 4).
To confirm the presence of two regimes in the CTILC corpus and shed light on the possible presence of two regimes in the Glissando corpus, we perform a careful analysis using Baayen's method of breakpoint detection [39]. As can be seen in Fig 8, the deviance has only a single minimum in the case of CTILC corpus, which allows to determine a breakpoint easily. Besides, a global minimum and a and and additional local minimum are found in the Glissando corpus (Fig 9): in the first binning (23 words per bin), the global minimum does correspond to a meaningful two-regime breakpoint but, in the other two binnings (46 and 67 words per bin), the global mininum corresponds to a spurious result due to the abundance of hapax legomena in the tail of the rank distribution, a phenomenon that can also be observed in the first binning (Fig 9). This spurious minimum in deviance can be explained by an artifact of the deviance minimization procedure for breakpoint detection, that becomes too sensitive to the concentration of points in the tail and the scarcity of points in other parts of the curve when bin size increases [39]. As a result it is observed that when bin size increases, the local minima in deviance decreases (see right panels in Fig 9).
The value of the breakpoint increases with the size of the bin. This value is approximately 20, 606 (127 words per bin), 22, 149 (414 words per bin) and 23, 241 (762 words per bin) in the CTILC corpus (Fig 8). In the Glissando corpus (Fig 9), the non-spurious breakpoint is found at 333, 5 (23 words per bin), 437, 0 (46 words per bin) and 502, 5 (67 words per bin). Evidently, the size of the corpus influences the the values of these breakpoints. Fig 10 shows the meaning-frequency law with this two regime analysis in CTILC and Glissando corpora, and Fig 11 shows the two-regime in law of meaning distribution. In both cases, by varying the bin size, the fitting curve for the CTILC corpus has a clear breakpoint that divides the two regimes, while for the Glissando corpus this breakpoint is not so clear visually,
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as there is greater dispersion of data in the speech corpus and, consequently, a weaker correlation. Table 3 summarizes the exponents of the Zipfian laws obtained in each of the two regimes for the CTILC and Glissando corpora. The difference between δ 1 and its predicted value d 0 1 from Eq 8, and the difference between δ 2 and its predicted value d 0 2 from Eq 9, are, in general, smaller than when a single regime was assumed in Table 2, providing support for the reality of two-regime structure.
Model selection
Now we turn to the question of which of the two approaches, a two-regime model or a oneregime model, actually provides a better description of the data. To address the problem of the trade-off between the goodness of fit of the model and its number of parameters, we recur to information theoretic model selection [40]. Tables 4, 5 and 6 show AICc and BIC values for each Zipfian law, corpus and kind of binning. All values favour two-regime models over one-regime models, except for the law of meaning distribution and the meaning-frequency law on the Glissando corpus with no binning, where the scores for each model are very close to each other. https://doi.org/10.1371/journal.pone.0260849.g007 Table 2. One regime analysis (CTILC and Glissando corpora). The exponents of the Zipfian laws: the rank-frequency law (α, Eq 3), the law of meaning distribution (γ, Eq 2) and meaning-frequency law (δ, Eq 1). δ 0 is the exponent δ predicted by Eq 4, obtained from α and γ. For estimating the values of the parameters, we have used Least Squares (LS) on a logarithmic transformation of both axes. Concerning equal-size binning, see the Methods section for the rationale behind the choice of the bin sizes.
Binning
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Discussion
Zipf's laws of meaning and robustness of the equation between exponents
Zipf's laws of meaning have been verified in Catalan. Assuming a single regime and applying equal-size binning to the CTILC corpus, the γ exponent of the law of meaning distribution that Zipf found for the word meaning distribution law in English [3,5] was recovered approximately, γ � 1/2, (Table 2), consistently with previous work [6,8]. This is not the case of the Glissando speech corpus where γ � 0.304 − 0.312. Nonetheless, considering the results for the fitting of a single slope (γ) for the law of meaning distribution, the γ values obtained here are consistent with previous works [13,15] that extended the fitting of this law to non-Indo-European languages [12]. As shown in previous research, verified here again for Catalan, there is a certain variability in the exponents of the Zipfian laws according to the size of the binning [12,13,15]. However, there were clear deviations in Zipf's rank-frequency law exponent, i.e. α, with respect to Catalan (α = 1.42 in [29] using different methods) and other languages in normal conditions [42,43]. Consequently, this fact affected the exponent δ 0 obtained indirectly with Eq 4 (see Table 2). These considerations notwithstanding, δ 0 gives approximately a similar Table 3. Breakpoints (i � ) are depicted as dashed blue lines.
https://doi.org/10.1371/journal.pone.0260849.g011 Table 3. Two regime analysis (CTILC and Glissando corpora). The exponents of each regime of the Zipfian laws: the rank-frequency law (α 1 and α 2 ), the law of meaning distribution (γ 1 and γ 2 ) and meaning-frequency law (δ 1 and δ 2 ). d 0 1 is the exponent δ 1 predicted by Eq 9 while d 0 2 is the exponent δ 2 predicted by Eq 8. To obtain the exponents, we used LS following Baayen's method [39]. Concerning equal-size binning, see the Methods section for the rationale behind the choice of the bin sizes. Subindexes of the exponents correspond to the regimes according to Eqs 5-7.
CTILC
Binning bin size
α 1 α 2 γ 1 γ 2 δ 1 d 0 1 δ 2 d 0 2
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value as the δ obtained directly from the meaning-frequency law (δ 0 = 0.187 − 0.21 (CTILC corpus) and δ 0 = 0.20 − 0.21 (Glissando corpus)) but deviates in both cases from the previously estimated for English [5,6]. The well-known deviations in Zipf's rank-frequency law (Eq 3) imply that it cannot be assumed that α = 1 in general [42,43], therefore, according to Eq 4, it cannot be expected that γ = δ as in Zipf's early work [5,12]. Nevertheless, the Eq 4 to obtain δ 0 indirectly is even robust in the case of the analysis of one regime without binning in both corpus ( Table 2). This is especially interesting given that, in this case, both in the CTILC corpus (α � 2.20, γ � 0.39) and in Glissando (α � 1.46, γ � 0.26) the Zipfian exponents are far from the usual ones but δ � δ 0 . On the other hand, the minimal differences found in α with respect to the previous study of the Glissando corpus [29] (where α � 1.42) may be due to the fact that here we have worked with a subcorpus of Glissando (Glissando \ DIEC2) and with different methods.
On the other hand, after verifying the existence of two regimes in Zipf's rank-frequency law in CTILC corpus (Figs 2 and 8) and in Glissando corpus (Figs 3 and 9), as previous works pointed out in big corpora [16,18,31], we have seen that this affects the meaning-frequency law (Fig 10). Then, the analysis of the two regimes in the CTILC corpus allowed us to obtain in the first regime γ 1 � 1/2 and δ 1 � 0.30 (with d 1 � d 0 1 ), and in the second regime γ 2 = 0.37 − 0.40 and δ 2 � 0.06 − 0.07 (and again d 0 2 � 0:08). In the case of the Glissando corpus in the Table 3.
Finally, our experimental results show that the relationship (Eq 4) between the three Zipfian exponents [6,8] is specially robust when we two regimes are assumed as expected from the higher precision in the estimation of the exponents. Again, δ and δ 0 deviate from previously reported for English, δ � 0.5 [5,6,8]. δ 0 may deviate from the expected value because of the value of α retrieved here for Catalan. Besides, δ may deviate from the expected value because of the two regime structure of the data. Therefore, as a hypothesis to corroborate in future research employing more languages, the variations in γ or δ could be explained by deviations in Zipf's rank-frequency law or the underlying two regime structure reported here.
Our comparison of the error of the theoretical predictions of Eq 4 for the two-regime model against those of the single-regime model, can be seen as a form of model selection based on the mathematical theory of Zipfian power laws [6,8]. As explained above, that model selection approach has provided indirect support for the existence of these two regimes and, in particular, support for the double power-law model over the single power-law model for each Zipfian law. Alternatively, we have obtained direct support for the two regimes following a formal approach using information theoretic model selection. We have found that the double power-law model fits the data better than the single power-law model in terms of the trade-off between parsimony and goodness of fit (Tables 4-6).
Two regimes in Zipfian laws: Core vocabulary?
In sum, although we have verified Zipf's meaning-frequency law (Eq 1) between the number of meanings and the frequency of words in Catalan employing Zipf's binning technique [5], data is better-described when two regimes are assumed. These two scaling regimes could be explained simply as the outcome of aggregating texts: previous work indicate that scaling breaks in rank-frequency distributions are a consequence of the mixing and composition of texts and corpora [18].
Previous works showed that some variability was found in the breakpoints in the case of corpus of English, Spanish and Portuguese, depending on the size of the corpus [18]. Here, for both corpora, we have seen that the breakpoint tends to increase with the size of the corpus and also with the size of the bin, and in the case of CTILC corpus (167, 079 lemmas) the breakpoint varies from 20, 606 to 23, 241, and in Glissando corpus (4510 lemmas) from 333.5 to 502.5. Therefore, in the case of Catalan we have also seen this dependence with the size of the corpus.
In our opinion, as seen, the effects of corpus size, composition and heterogeneity previously suggested [17,18,42,44] are not incompatible with the existence of a core and a peripheral vocabulary or "unlimited lexicon" [16,33], but this dichotomy is not necessarily be an observable property of Zipfian distributions by means of the breakpoint as [18] pointed out. Besides, if we understand the core vocabulary as the real basic vocabulary of a linguistic community at a given time, then Glissando turns out to be a better source than CTILC to capture that subset of the vocabulary, because CTILC corpus mixed sources from different time periods.
One the one hand, the CTILC written corpus includes from literary and journalistic texts to scientific ones, in a time interval of more than a century, with the diachronic variations that this implies. The sum of the size effect and the greater linguistic variability of combining heterogeneous texts in the CTILC corpus could explain the appearance of two regimes in Zipf's rank-frequency law [17,18,32] and, as we have seen, consequently, of two regimes in the Zipf's meaning-frequency law. On the other hand, Glissando is a synchronous speech corpus in which the interlocutors cooperate and circumscribe themselves to a single communicative context, as is often promoted in the systematic design of speech corpora [37]. That is, to the usual reduction in the use of rare words that occurs in oral communication, in a pragmatic context with broad Gricean implications (see [45,46] for a review), it must be added that in the construction of speech corpus like Glissando, literally communicative scenarios are 'designed' [37]. This fact causes that we are not really facing a spontaneous speech corpus, causing a tendency to unify the vocabulary used by the different informants involved in or, at least, reducing the use of infrequent words. Infrequent words are typical of the peripheral vocabulary of multi-author corpus that deal with diverse topics. However, in the speech corpus a greater dispersion of the points in the meaning-frequency distribution is appreciated, but the two regimes are still found.
Lexical diversity is defined as the variety of vocabulary deployed in a text or transcript by either a speaker or a writer [47]. In speech one expects to find a smaller lexical diversity and core vocabulary given the stronger cognitive constraints of spontaneous oral language (Glissando) with respect to written language (CTILC), to which the effects of the lemmatization are added here (see next subsection). Size effects have been shown to influence lexical diversity, even in small corpus [48].
Therefore, other speech corpora of different size and spontaneous speech should be analysed in the future to corroborate these two regimes observed, exploring the size limits that had previously been considered for the appearance of a kernel vocabulary [16]. It also remains as future work to verify if these effects are present in the comparison of spoken and written corpora of other languages and, in addition, if these two regimes appear in other languages, analogously to Catalan, in multi-author texts or whether they appear just as a consequence of text mixing [18,31], given the influence of semantics on Zipfian distributions [11,49].
Lemmatization and binning effects
Glissando is a corpus smaller in size than CTILC and with less thematic variability. However, the effect of binning seems to be added to the effect of corpus size. As Table 1 shows, only the lemmas for which we had their meanings have been analyzed (intersection of each corpus with the DIEC2 dictionary). Thus, there is eventually an order of magnitude difference between the number of lemmas in both corpora (the intersection of CTILC corpus with DIEC2 has 52, 578 lemmas and the intersection of Glissando corpus with DIEC2 only 3, 083 lemmas). The quantitative effects of corpus size have been related to variations of Zipf's law and other linguistic laws in such a way that a larger size in a corpus implies increasing the probability of rare words, so that word frequency distributions are Large Number of Rare Events (LNRE) as [44] explain clearly. Thus, in the case of the speech corpus such as Glissando, smaller in size, also there is a smaller number of rare words, since fewer technical words and jargon are used in orality than in written corpus [50], but the two regimes are nevertheless observed. The lemmatization process implies a decrease in the LNRE by including under the same lemma inflected words, of special importance in Catalan [36], as in other Romance languages [21,26].
Regarding the deviations observed in the exponents of Zipf's rank-frequency law (Eq 3) also found in other languages [51], notice that by lemmatizing the corpus the morphological complexity and, by extension, the diversity of the vocabulary, is reduced. As explained in the Introduction, Catalan is a Romance language with a rich inflection and derivation [21,25,27], that, however, it does not stand out typologically compared to other languages in terms of indicators of entropy rate or phonological and morphological richness [22][23][24].
The robustness of Zipf's law under lemmatization for single-author written texts was already checked in previous work studying Spanish, French and English with different methods [52]. The exponent of lemmas and word forms may vary but both are correlated in texts [52] and the same could happen in speech corpora. However, one could consider an opposite hypotheses that should be confirmed in future works, including more languages: that the lemmatization, as a source of reduction of morphological richness, should vary the exponent of Zipf's rank-frequency law [52], and this variation would depend on the inflectional and derivational complexity of the language (in the sense of Bentz's works [22][23][24]), that is, it will affect the languages with more morphological variability to a greater extent.
Following [53] the α exponent of Zipf's rank-frequency law "reflects changes in morphological marking" [53], so that more inflection is correlated to a higher α and longer tail of hapax legomena [22,53]. Comparing modalities, in our case orality shows a lower α than writing, contrary to [53] but consider that here we follow different methods. Besides, it has been shown that there is variability in α related to both the text genre and the linguistic typology for languages of different linguistic families [51]. Future research should control for both effects (modality and morphological complexity) with corpora that are closer and employing uniform methods.
Future work should clarify how factors such as corpus size, binning and linguistic variability, influence Zipf's meaning-frequency law. The relationship between the exponents of the laws confirmed here for Catalan should be investigated for other languages. In any case, other aspects in the formation of words that affect the lemmatization could be considered in future work: the work carried out here for the first time can serve then as a protocol to replicate or refine in future studies with other languages. Our findings call for a revision of previous research of these laws assuming one regime.
Fig 1 .
1Graphical representation of the different sources used.
Figs 2 and 3
3show Zipf's rank-frequency law for the CTILC corpus and the Glissando corpus, respectively. Similarly, Figs 4 and 5 show the meaning distribution law for CTILC and Glissando, respectively, and Figs 6 and 7 show the meaning-frequency law for CTILC and Glissando, respectively. The values of the exponents of each law are summarized in
Fig 2 .
2Zipf's rank-frequency law in CTILC corpus. Average frequency (f) as a function of rank (i) after applying equal-size binning (blue). The best fit of a power law is also shown (red). Left: bin size of 127 words. Center: bins size of 414 words. Right: bin size of 762 words. https://doi.org/10.1371/journal.pone.0260849.g002
Fig 3 .
3Zipf's rank-frequency law in Glissando corpus. Average frequency (f) as a function of rank (i) after applying equal-size binning (blue). The best fit of a power law is also shown (red). Left: bin size of 23 words. Center: bins size of 46 words. Right: bin size of 67 words. https://doi.org/10.1371/journal.pone.0260849.g003
Fig 4 .
4Zipf's law of meaning distribution in CTILC corpus. Average number of meanings (μ) as a function of frequency rank (i) after applying equalsize binning (blue). The best fit of a power law is also shown (red). Left: bin size of 127 words. Center: bin size of 414 words. Right: bin size of 762 words. Sources: Catalan words in CTILC, using DIEC2 meanings and CTILC frequencies.https://doi.org/10.1371/journal.pone.0260849.g004
Fig 5 .
5Zipf's law of meaning distribution in Glissando corpus. Average number of meanings (μ) as a function of frequency rank (i) after applying equal-size binning (blue). The best fit of a power law is also shown (red). Left: bin size of 23 words. Center: bin-size of 46 words. Right: bin-size of 67 words. Sources: Catalan words in Glissando, using DIEC2 meanings and Glissando frequencies.https://doi.org/10.1371/journal.pone.0260849.g005
Fig 6 .
6Zipf's meaning-frequency law in CTILC corpus. Average number of meanings (μ) as a function of frequency (f) after applying equal-size binning (blue). The best fit of a power law is also shown (red). Left: bin size of 127 words. Center: bin size of 414 words. Right: bin size of 762 words. Sources: Catalan words in CTILC, using DIEC2 meanings and CTILC frequencies.https://doi.org/10.1371/journal.pone.0260849.g006
Fig 7 .
7Zipf's meaning-frequency law in Glissando corpus. Average number of meanings (μ) as a function of frequency (f) after applying equal-size binning (blue). The best fit of a power law is also shown (red). Left: bin size of 23 words. Center: bin-size of 46 words. Right: bin-size of 67 words. Sources: Catalan words in Glissando, using DIEC2 meanings and Glissando frequencies.
Fig 8 .
8Breakpoint analysis for Zipf's rank-frequency law in CTILC corpus. Breakpoints (i � ) are depicted as blue dashed lines. Left: frequency (f) as a function of rank (i). The best fit of a two-regime power law is also shown (red). Right: Baayen's deviance as a function of rank (i). The choice of the bin sizes is the same as inTable 3and influences the breakpoint. Row A) 127 words per bin and breakpoint 20, 606, row B) 414 words per bin and breakpoint 22, 149, and row C) 762 words per bin and breakpoint 23, 241. https://doi.org/10.1371/journal.pone.0260849.g008 PLOS ONE Fig 9. Breakpoint analysis for Zipf's rank-frequency law in Glissando corpus. Blue dashed lines and gray dashed lines are used to indicate, respectively, the first and the second local minimum of deviance. The 1st local minimum is taken as the meaningful breakpoint (i � ). Left: frequency (f) as a function of rank (i). The best fit of a two-regime power law is also shown (red). Right: Baayen's deviance as a function of rank (i). The choice of the bin sizes is the same as in Table 3 and influences the first non-spurious breakpoint. Row A) 23 words per bin and breakpoint 333.5, row B) 46 words per bin and breakpoint 437.0, and row C) 67 words per bin and breakpoint 502.5. https://doi.org/10.1371/journal.pone.0260849.g009 Fig 10. The meaning-frequency law in the CTILC and Glissando corpora. Row A) CTILC corpus and row B) Glissando corpus. The choice of the bin sizes is the same as in Table 3. Breakpoints (i � ) are depicted as dashed blue lines. https://doi.org/10.1371/journal.pone.0260849.g010 PLOS ONE Fig 11. The law of meaning distribution in the CTILC and Glissando corpora. Row A) CTILC corpus and row B) Glissando corpus. The choice of the bin sizes is the same as in
Table 5 .
5AICc and BIC values for the one-regime model (1r) and the two-regime model (2r) for Zipf's law of meaning distribution.n is the number of points.CTILC
Binning
bin size
n
AICc
BIC
1r
2r
1r
2r
No binning
-
52,578
-7021.764
-7185.055
-6995.154
-7149.576
Equal-size
127
414
-1639.696
-1670.549
-1627.677
-1654.543
414
127
-620.296
-659.527
-611.958
-648.479
762
69
-352.745
-393.797
-346.412
-385.486
Glissando
No binning
-
3,083
3234.723
3228.888
3252.816
3253.01
Equal-size
23
134
-155.601
-163.972
-147.093
-152.691
46
67
-111.902
-121.156
-105.669
-112.982
67
46
-83.455
-91.648
-78.541
-85.309
https://doi.org/10.1371/journal.pone.0260849.t005
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PLOS ONE | https://doi.org/10.1371/journal.pone.0260849December 16, 2021
AcknowledgmentsWe are grateful to C. Bentz, J. M. Garrido, C. Santamaria, J. Rafel and the technicians of Oficines Lexicogràfiques de l'Institut d'Estudis Catalans (Institute of Catalan Studies) for providing us with the data and helpful comments.Author Contributions
Funding acquisition: Antoni Hernández-Fernández. Investigation: Neus Català, Ramon Ferrer-i-Cancho, Antoni Hernández-Fernández. Methodology: Neus Català, Ramon Ferrer-i-Cancho, Lluís Padró, Antoni Hernández-Fernández. Neus Català, Ramon Ferrer-i-Cancho, Antoni Hernández-FernándezFormal analysis: Neus Català, Ramon Ferrer-i-Cancho, Antoni Hernández-Fernández. Funding acquisition: Antoni Hernández-Fernández. Investigation: Neus Català, Ramon Ferrer-i-Cancho, Antoni Hernández-Fernández. Methodology: Neus Català, Ramon Ferrer-i-Cancho, Lluís Padró, Antoni Hernández- Fernández.
Supervision: Neus Català, Ramon Ferrer-i-Cancho, Antoni Hernández-Fernández. Validation: Neus Català, Ramon Ferrer-i-Cancho. Jaume Baixeries, Antoni Hernández-Fernández. Software: Neus Català, Jaume Baixeries, Lluís Padró. Antoni Hernández-FernándezProject administration: Jaume Baixeries, Antoni Hernández-Fernández. Software: Neus Català, Jaume Baixeries, Lluís Padró. Supervision: Neus Català, Ramon Ferrer-i-Cancho, Antoni Hernández-Fernández. Validation: Neus Català, Ramon Ferrer-i-Cancho, Lluís Padró, Antoni Hernández- Fernández.
Visualization: Neus Català, Ramon Ferrer-i-Cancho. Antoni Hernández-FernándezVisualization: Neus Català, Ramon Ferrer-i-Cancho, Antoni Hernández-Fernández.
Writing -original draft: Neus Català, Ramon Ferrer-i-Cancho. Antoni Hernández-FernándezWriting -original draft: Neus Català, Ramon Ferrer-i-Cancho, Antoni Hernández- Fernández.
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[] |
[
"Low-complexity End-to-End Performance Optimization in MIMO Full-Duplex Relay Systems",
"Low-complexity End-to-End Performance Optimization in MIMO Full-Duplex Relay Systems"
] |
[
"Himal A Suraweera [email protected] ",
"Ioannis Krikidis ",
"Gan Zheng ",
"Chau Yuen [email protected] ",
"Peter J Smith ",
"H A Suraweera ",
"C Yuen ",
"\nDepartment of Electrical and Electronic Engineering\nUniversity of Peradeniya\n20400PeradeniyaSri Lanka\n",
"\nDepartment of Electrical & Computer Engineering\nSingapore University of Technology and Design\n20 Dover Drive138682Singapore\n",
"\nis with the Interdisciplinary Centre for Security, Reliability and Trust (SnT)\nUniversity of Cyprus\n1678NicosiaCyprus\n",
"\nDepartment of Electrical and Computer Engineering\nUniversity of Luxembourg\n4 rue Alphonse WeickerL-2721Luxembourg\n",
"\nThe University of Canterbury\nPrivate Bag 4800ChristchurchNew Zealand\n"
] |
[
"Department of Electrical and Electronic Engineering\nUniversity of Peradeniya\n20400PeradeniyaSri Lanka",
"Department of Electrical & Computer Engineering\nSingapore University of Technology and Design\n20 Dover Drive138682Singapore",
"is with the Interdisciplinary Centre for Security, Reliability and Trust (SnT)\nUniversity of Cyprus\n1678NicosiaCyprus",
"Department of Electrical and Computer Engineering\nUniversity of Luxembourg\n4 rue Alphonse WeickerL-2721Luxembourg",
"The University of Canterbury\nPrivate Bag 4800ChristchurchNew Zealand"
] |
[] |
In this paper, we deal with the deployment of full-duplex relaying in amplify-and-forward (AF) cooperative networks with multiple-antenna terminals. In contrast to previous studies, which focus on the spatial mitigation of the loopback interference (LI) at the relay node, a joint precoding/decoding design that maximizes the end-to-end (e2e) performance is investigated. The proposed precoding incorporates rank-1 zero-forcing (ZF) LI suppression at the relay node and is derived in closed-form by solving appropriate optimization problems. In order to further reduce system complexity, the antenna selection (AS) problem for full-duplex AF cooperative systems is discussed. We investigate different AS schemes to select a single transmit antenna at both the source and the relay, as well as a single receive antenna at both the relay and the destination. To facilitate comparison, exact outage probability expressions and asymptotic approximations of the proposed AS schemes are provided. In order to overcome zerodiversity effects associated with the AS operation, a simple power allocation scheme at the relay node is also investigated and its optimal value is analytically derived. Numerical and simulation results show that the joint ZF-based precoding significantly improves e2e performance, while AS schemes are efficient solutions for scenarios with strict computational constraints.Index TermsMIMO relay networks, full-duplex relaying, precoding, antenna selection, outage probability.H. A. Suraweera is with the
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10.1109/twc.2013.122313.130608
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[
"https://arxiv.org/pdf/1311.3428v1.pdf"
] | 267,425 |
1311.3428
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a8ff5a03865a572effe753f8cf9ea955607513d9
|
Low-complexity End-to-End Performance Optimization in MIMO Full-Duplex Relay Systems
14 Nov 2013
Himal A Suraweera [email protected]
Ioannis Krikidis
Gan Zheng
Chau Yuen [email protected]
Peter J Smith
H A Suraweera
C Yuen
Department of Electrical and Electronic Engineering
University of Peradeniya
20400PeradeniyaSri Lanka
Department of Electrical & Computer Engineering
Singapore University of Technology and Design
20 Dover Drive138682Singapore
is with the Interdisciplinary Centre for Security, Reliability and Trust (SnT)
University of Cyprus
1678NicosiaCyprus
Department of Electrical and Computer Engineering
University of Luxembourg
4 rue Alphonse WeickerL-2721Luxembourg
The University of Canterbury
Private Bag 4800ChristchurchNew Zealand
Low-complexity End-to-End Performance Optimization in MIMO Full-Duplex Relay Systems
14 Nov 2013
In this paper, we deal with the deployment of full-duplex relaying in amplify-and-forward (AF) cooperative networks with multiple-antenna terminals. In contrast to previous studies, which focus on the spatial mitigation of the loopback interference (LI) at the relay node, a joint precoding/decoding design that maximizes the end-to-end (e2e) performance is investigated. The proposed precoding incorporates rank-1 zero-forcing (ZF) LI suppression at the relay node and is derived in closed-form by solving appropriate optimization problems. In order to further reduce system complexity, the antenna selection (AS) problem for full-duplex AF cooperative systems is discussed. We investigate different AS schemes to select a single transmit antenna at both the source and the relay, as well as a single receive antenna at both the relay and the destination. To facilitate comparison, exact outage probability expressions and asymptotic approximations of the proposed AS schemes are provided. In order to overcome zerodiversity effects associated with the AS operation, a simple power allocation scheme at the relay node is also investigated and its optimal value is analytically derived. Numerical and simulation results show that the joint ZF-based precoding significantly improves e2e performance, while AS schemes are efficient solutions for scenarios with strict computational constraints.Index TermsMIMO relay networks, full-duplex relaying, precoding, antenna selection, outage probability.H. A. Suraweera is with the
I. INTRODUCTION
Cooperative communications with relaying is a promising solution to extend the network coverage and ensure higher throughputs and quality-of-service (QoS). Relaying techniques can be classified as either half-duplex or full-duplex [1]. In order to complete the relaying operation, half-duplex relaying requires two orthogonal channels and the associated bandwidth loss recovery has been an active research area for several years. With full-duplex relaying, the relay node receives and transmits simultaneously on the same channel and therefore utilizes the spectrum resources more efficiently [2], [3]. However, the main limitation in full-duplex operation is the loopback interference (LI) (also known in the literature as the loopback self-interference) due to signal leakage from the relay's output to the input at the reception side [4]- [7]. Specifically, the main drawback of full-duplex operation is the large power differential between the LI generated by the full-duplex terminal and the received signal of interest coming from a distant source. The large LI spans most of the dynamic range of the analog-to-digital converter at the receiver side and thus its mitigation is critical for the implementation of full-duplex operation. In modern communication systems such as WiFi, Bluetooth, and Femtocells, the transmission power and the distance between communicating devices has been decreased. This important architectural modification decreases the power differential between the two received signals. This attribute, combined with the high computation capabilities of modern terminals, significantly facilitates the implementation of the full-duplex radio technology [8]- [10].
In the literature, the combination of multiple-input multiple-output (MIMO) techniques with relaying has been invoked to further enhance the communication performance [11], [12]. While most work has focused on MIMO half-duplex relaying, recent work has also considered MIMO full-duplex relaying.
MIMO provides an effective means to suppress the LI in the spatial domain [6], [13], [14]. With multiple transmit or receive antennas at the full-duplex relay, precoding at the transmitter and decoding at the receiver can be jointly optimized to mitigate the LI effects. Zero forcing (ZF) and minimum mean square error (MMSE) are two widely adopted criteria in the literature for the precoding and decoding design [15]. ZF aims to completely null out undesired interference and provides an interference-free channel.
Although ZF normally results in sub-optimal solutions, its performance is nearly optimal in the high signal-to-noise ratio (SNR) regime. MMSE is an improved precoder/decoder design compared to ZF, which takes into account the background noise. The MMSE-based precoder has a more complicated structure but it can improve the achievable QoS. Due to the implementation simplicity and the efficiency in the high SNR regime, ZF becomes a useful design criterion to completely cancel the LI and break the closed-loop between the relay input and output.
Assuming there is no closed-loop processing delay, the optimal precoding matrix for a full-duplex amplify-and-forward (AF) relay that maximizes the mutual information under an average power constraint is studied in [16]. In this case, the design approach and the resulting precoding solution are similar to the half-duplex case. The joint precoding and decoding design for a full-duplex relay is studied in [6], [17], where both ZF solutions and MMSE solutions are discussed. Notice that the ZF solution used in [6], [17] and most early works uses a conventional approach based on the singular value decomposition of the loopback self-interference channel. The main drawback of this approach is that the ZF solution only exists given that the numbers of antennas at the source, full-duplex relay and the receiver satisfy a certain condition. In order to overcome this limitation, [13] adopts an alternative criterion and proposes to maximize the signal-to-interference ratios between the power of the useful signal to the power of LI at the relay input and output, respectively. Conventional ZF precoding and decoding are chosen via the singular vectors of the LI channels, however, this design does not take into account the other channels and the end-to-end (e2e) performance. In [18], a joint design of ZF precoding and decoding is proposed to fully null out the LI at the relay, taking into account the source-relay and relay-destination channels.
A simple approach is studied in [19], where an iterative algorithm that jointly optimizes the precoding and decoding vectors in respect of the e2e performance, is investigated.
Most of the work in the literature does not deal with the joint optimization of the precoding and decoding process, even for scenarios with multiple antennas at the terminals. Hence, the focus has been restricted to full duplex relay processing which has led to strictly suboptimal e2e performance.
Furthermore, the available ZF-based solutions which do aim to optimize e2e performance are not given in closed-form. Hence, in this paper, we consider a general case where each terminal can have arbitrary multiple antennas and we jointly design precoding and decoding at the source, the relay and the destination in order to maximize the achievable rate. For simplicity, a single data stream is transmitted and ZF criteria are used by the full-duplex relay to handle the LI. We give the closed-form precoder/decoder solutions for transmit and receive ZF schemes. Furthermore, the diversity orders are derived for the different schemes.
In addition, we also propose several low-complexity antenna selection (AS) schemes 1 for MIMO fullduplex relaying and analyze the outage probability of each scheme. The complexity of implementing MIMO systems can be significantly decreased with AS, which employs fewer radio frequency chains than antenna elements and then connects the chains to the best available antenna element [20]. Some 1 We follow the footsteps of recent work such as [6] and investigate the performance of AS since both precoding/decoding and AS schemes belong to the general category of MIMO spatial suppression techniques. ZF precoding/decoding designs and different AS schemes studied in this paper eliminate/mitigate the effect of LI respectively, and hence offer different performance/complexity tradeoff choices to a system designer. Moreover, AS can be viewed as a special case of precoding where the beamforming vector only contains a single non zero unit element whose entry depends on the selected transmit/receive antenna. limited work on AS in full-duplex relay systems can be found in [6], [14]. In [6], several spatial LI suppression techniques based on antenna sub-set selection and joint transmit/receive beam selection have been investigated. In [14], several low complexity antenna sub-set selection schemes have been proposed with the objective to suppress LI at the relay's transmit side. However, a basic limitation of the current work is that AS is used only to achieve LI suppression. On the other hand, from a system performance standpoint, it is important to deploy MIMO AS techniques such that the e2e signal-to-interference noise ratio (SINR) at the destination is maximized.
The performance of AS in half-duplex relay systems is a mature topic and well studied, see for e.g., [21]- [24]. On the other hand, to the best of authors' knowledge, the current paper is the first to analytically investigate the AS performance for full-duplex relay systems. Moreover, our analysis presents new results in addition to earlier work such as [25], [26] where the outage probability of single antenna full-duplex systems have been studied. Specifically, we select single transmit antennas at the source and the relay, respectively, and single receive antennas at the relay and the destination, respectively. The performance of the aforementioned system set-up with different AS schemes is quantified by deriving exact, and asymptotic outage probability expressions. The asymptotic expressions illuminate the network performance by revealing the comparative performances of the AS schemes in terms of the system and channel parameters. Furthermore, in order to eliminate the zero-diversity behavior of the full-duplex relaying due to the LI, we propose a new simple power allocation scheme 2 at the relay, which only involves a single parameter optimization. We also present optimal values for this parameter to minimize the outage probability from a diversity perspective. These closed-form expressions are in the form of fractions of the number of source/relay/destination antennas and reveal the spatial degrees of freedom offered by each AS scheme. Moreover, these values can be calculated directly once a particular system configuration is decided.
The main contributions of this paper are twofold.
• A low complexity joint precoding/decoding design for e2e SNR maximization is proposed. Specifically, based on ZF loopback self-interference suppression, receive/transmit beamforming vectors at the relay are designed. Closed-form solutions for the scheme's outage probability as well as high SNR simple expressions are derived. Our analysis clearly reveals insights on system performance and shows the impact on the achieved diversity order.
• Several AS schemes are proposed including the optimal AS scheme that maximizes the e2e SNR at the destination and various sub-optimal AS schemes. In order to eliminate the zero diversity behavior 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 in such full-duplex MIMO systems, we propose a simple power allocation method at the relay. The outage performance of the AS schemes are analytically investigated. Using the derived high SINR outage approximations, we also investigate the optimal power allocation coefficient values.
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The rest of the paper is organized as follows. Section II presents the overall MIMO system model.
Sections III and IV present the joint precoding/decoding designs and AS schemes, respectively. The outage probability of the precoding and AS schemes is analyzed in Section V and numerical results are given in Section VI. Finally, Section VII concludes the paper and summarizes several key findings.
Notation: The lowercase and uppercase boldface letters (e.g., x and X) indicate column vectors and matrices, respectively. I is the identity matrix and diag (a 1 , a 2 , . . . , a n ) denotes a diagonal matrix with elements l = {a 1 , a 2 , . . . , a n }. We use (·) † to denote the conjugate transpose, · is the Frobenius norm and Tr(·) is the trace operation. λ max (X) denotes the maximum eigenvalue of a matrix X and u max (X)
represents the eigenvector associated with λ max (X). The expectation operator is denoted by E(·) and
Pr{·} is probability. K ν (z) is the modified Bessel function of the second kind of order ν.
II. SYSTEM MODEL
We consider a basic three-node MIMO relay network consisting of one source S, one relay R, and one destination, D as shown in Fig. 1. We use N T and N R to denote the number of transmit and receive antennas at S and D, respectively. The relay is equipped with two groups of antennas; M R receive and M T transmit antennas for full-duplex operation. S has no direct link to D, which may result from heavy path loss and high shadowing between S and D.
A. Channel Model
All wireless links in the network are subject to non-selective independent Rayleigh block fading and additive white Gaussian noise (AWGN); H SR and H RD denote the S − R and R − D channels, respectively, while H RR denotes the LI channel. In order to reduce the effects of self-interference on system performance, an imperfect interference cancellation scheme (i.e. analog/digital cancellation) is used at R and we model the residual LI channel as a fading feedback channel [4], [27], [28]. Moreover, the noise at the nodes is modeled as complex AWGN with zero mean and normalized variance. In addition, the single-input single-output channel corresponding to the i-th receive and the j-th transmit
antenna from terminal X to terminal Y , is denoted by h i,j XY where X ∈ {S, R} and Y ∈ {R, D}. As for the average S − R and R − D channel statistics; we assume E{|h i,j SR | 2 } = c SR and E{|h i,j RD | 2 } = c RD .
The experimental-based study in [8] has demonstrated that the amount of LI suppression achieved by an analog/digital cancellation technique is influenced by several system and hardware parameters. Since each implementation of a particular analog/digital LI cancellation scheme can be characterized by a specific residual power, a parameterization by H RR with elements satisfying E{|h i,j RR | 2 } = c RR allows these effects to be studied in a generic way [5]. We assume that the channel coefficients remain approximately stationary for a long observation time (time slot), but change independently from one slot to another according to a Rayleigh distribution. The channel coherence time is equal to one time slot. This assumption applies to networks with a low mobility and corresponds to slow fading (block) channels where coding is performed over one block.
B. System Model
This work studies full-duplex operation at a system level using some well-known models for the characterization of the residual loop interference [6]. We note that the developed schemes do not refer to a specific analogue or baseband implementation and can be applied to both by taking into account related practical aspects (i.e., training sequence, antenna impedance mismatch, dynamic range etc). Further implementation issues as well as more realistic radio environments (i.e., frequency selectivity) are beyond the scope of this paper.
In order to keep the complexity low, we assume that a single data stream 3 is transmitted and each node employs only linear processing, i.e., S applies a precoding vector t on the data stream, while D uses a linear receive vector r to decode the signal, where t = r = 1. With the recent trend to increase the number of antennas at the terminals (e.g., massive MIMO), linear processing solutions offer an attractive solution for low complexity implementation. In contrast, the complexity of the optimal non linear signal detection approach grows exponentially with the number of transmit antennas [29]. The relaying operation is based on the AF policy with an amplification matrix W that keeps the transmitted power at the relay node below the threshold P R . We jointly optimize t, r and W to maximize the e2e system performance.
1) Joint Precoding/Decoding
Design: ZF is chosen as the design criterion for the relay amplification matrix W, such that there is no loopback self-interference from the relay output to relay input. To simplify the problem, we further decouple W as W = w t w † r , where w r is the receive beamforming vector and w t is the transmit beamforming vector both at the relay node. By fixing w r (or w t ), w t (or w r ) can be jointly optimized with t at S and r at R to realize the overall zero loopback self-interference at R and maximize the e2e SNR.
2) Antenna Selection: AS schemes can be considered as a special case of our system model with one element of r and t being unity and the rest zero. Hence, only one element of W is non-zero and this entry depends on the selected transmit and receive antennas at the relay. Specifically, in the case of AS, we assume that at each terminal, a single antenna is selected either to maximize the e2e SINR at D (with optimal AS) or to maximize SNRs/SINRs associated with S − R, R − R and R − D links (with sub-optimal AS). The search complexity of the optimal scheme is high especially with a large number of antennas at each terminal, therefore, the sub-optimal schemes provide a better trade-off between implementation complexity and e2e system performance. Moreover, if S transmits with a power P S , we model the transmit power at R, as P α S where 0 < α ≤ 1. The parameter α provides a dB scaling of the relay transmit power which is necessary in the presence of residual LI. Hence, α captures the effects of power control on the achieved performance as it allows the analysis of different relative power gains between the SINR and the SNR of the S − R and R − D hops, respectively. Although the proposed ZF precoding design operation is optimal with the use of full power at the relay (α = 1), as we show later (in Section V), when AS schemes are implemented, an appropriate α can protect the MIMO relay system from error floor effects and thus a zeroth-order diversity.
III. JOINT PRECODING/DECODING DESIGN Based on the above system model, the equivalent S − R and R − D channels become
h SR H SR t, and h RD = H † RD r.(1)
We first assume t, r are fixed and study their optimal design together with w r and w † t according to different criteria.
By assuming a processing delay at R, given by τ [5], [6], the input and the output at R can be written as
r[n] = h SR x S [n] + H RR x R [n] + n R [n],(2)
and
x R [n] = Wr[n − τ ],(3)
respectively, where x S [n] is the transmitted symbol at S with zero mean, average power P S and n R is the M R × 1 AWGN vector with zero mean and identity covariance matrix.
Using (2) and (3) recursively, the relay output can be rewritten as
x R [n] = Wr[n − τ ] = Wh SR x S [n − τ ] + WH RR x R [n − τ ] + Wn R [n] = W ∞ j=0 (H RR W) j (h SR x S [n − jτ − τ ] + n R [n − jτ ]) .(4)
Note that we aim to maximize the e2e SNR, and the optimal W should possess a minimum mean square error (MMSE) structure, which is nontrivial to solve. To simplify the signal model, and find lowcomplexity closed-form rather than optimal solutions, we add the additional ZF constraint that the design of W ensures no loopback self-interference for the full-duplex operation. To realize this, it is easy to check from (4) that the following condition is sufficient,
WH RR W = 0.(5)
As a result, (4) becomes
x R [n] = W (h SR x S [n − τ ] + n R [n]) ,(6)
with the covariance matrix
E[x R x † R ] = P S Wh SR h † SR W † + WW † .(7)
The relay output power is
P R = Tr(E[x R x † R ]) = Wh SR 2 P S + W 2 .(8)
The received signal at D can be written as
r D [n] = h † RD x R [n] + n D [n] = h † RD Wh SR x S [n − τ ] + h † RD Wn R [n] + n D [n].(9)
The e2e SINR, denoted as γ, is expressed as
γ = P S |h † RD Wh SR | 2 h † RD W 2 + 1 .(10)
We aim to optimize the relay processing matrices W in order to maximize the e2e SINR. Mathematically, the optimization problem is formulated as
max W γ (in Eq. 10) (11) s.t. P S Wh SR 2 + W 2 ≤ P R , WHW = 0.
To further simply the problem, we assume W = w t w † r , where w r is the receive beamforming vector and w t is the transmit beamforming vector. It is noted that W is of rank-1 and this is reasonable since there is only a single data stream. Then the ZF condition is simplified to w † r H RR w t = 0. To achieve this requirement, we can design w r or w t jointly with t and r, as described below.
A. Receive ZF with M R > 1
We assume maximum ratio transmission (MRT) with w t = h RD and optimize w r based on the ZF criterion. Consequently, problem (11) reduces to
max wr P S h RD 4 |w † r h SR | 2 h RD 4 w r 2 + 1 (12) s.t. P S h RD 2 |w † r h SR | 2 + h RD 2 w r 2 ≤ P R , w † r H RR h RD = 0.
Note that the first power constraint needs to be satisfied with equality, otherwise, w r can be increased without violating any constraint and this leads to a higher objective value. Hence, the objective function (12) can be written as
P S h RD 4 |w † r h SR | 2 h RD 2 (P R − P S h RD 2 |w † r h SR | 2 ) + 1 ,
which is monotonically increasing in |w † r h SR |. As a result, (12) is equivalent to
max wr |w † r h SR | 2 (13) s.t. P S h RD 2 |w † r h SR | 2 + h RD 2 w r 2 ≤ P R , w † r H RR h RD = 0. Let E I + P S h SR h † SR and E 1/2 w r = v r .
With this definition, we can formulate a simple optimization problem for v r as follows:
max vr |v † r E −1/2 h SR | 2 (14) s.t. v r 2 ≤ P R h RD 2 , v † r E −1/2 H RR h RD = 0.
From the ZF constraint, we know that v r lies in the null
space of E −1/2 H RR h RD . Hence, v r = Du r , where D I− E −1/2 HRRhRDh † RD H † RR E −1/2 E −1/2 HRRhRD 2 .
The objective function in (14) then becomes |u † r DE −1/2 h SR | 2 and the optimal u r should align with DE −1/2 h SR . Using the facts that the first power constraint should be met with equality and D is idempotent, we can express the optimal solutions of (14) and (13) as
v r = DE −1/2 h SR DE −1/2 h SR P R h RD 2 , and(15)w r = E −1/2 DE −1/2 h SR DE −1/2 h SR P R h RD 2 .
The objective value in (12) involves |w † r h SR | 2 and w r 2 which, from (15), are given by
|w † r h SR | 2 = h † SR E −1/2 DE −1/2 h SR DE −1/2 h SR 2 P R h RD 2 = P R h RD 2 h † SR E −1/2 DE −1/2 h SR = P R h RD 2 h † SR E −1/2 2 − h † SR E −1 H RR h RD 2 E −1/2 H RR h RD 2 = P R h RD 2 H RR h RD 2 h SR 2 − |h † SR H RR h RD | 2 H RR h RD 2 + P S ( H RR h RD 2 h SR 2 − |h † SR H RR h RD | 2 ) ,(16)
and
w r 2 = P R h RD 2 − P S |w † r h SR | 2 (17) = P R h RD 2 − P S P R h RD 2 H RR h RD 2 h SR 2 − |h † SR H RR h RD | 2 H RR h RD 2 + P S ( H RR h RD 2 h SR 2 − |h † SR H RR h RD | 2 ) (18) = P R h RD 2 H RR h RD 2 H RR h RD 2 + P S ( H RR h RD 2 h SR 2 − |h † SR H RR h RD | 2 ) .(19)
Using (16) and (17) in (12), the achievable e2e SNR can be derived as
γ = P S Dh SR 2 P R h RD 2 P S Dh SR 2 + P R h RD 2 + 1 ,(20)
where D I −
HRRhRDh † RD H † RR HRRhRD 2 .
Next, we can address the design of t and r. Notice from (1) that t and r are embedded in Dh SR 2 and h RD 2 , respectively, so we propose the following solution to separately optimize t and r:
t * = arg max t =1 Dh SR 2 = arg max t =1 DH SR t 2 (21) = u max (H † SR DH SR ), and r * = arg max r =1 h RD 2 = arg max r =1 H † RD r 2 (22) = u max (H † RD H RD ),
where we have used the fact that D is idempotent. Note that D also depends on r via h RD , so the above solutions may not be optimal. Nevertheless, the choice of r * in (22) uniquely maximizes h RD 2 and given this choice of r, t * in (21) uniquely maximizes Dh SR 2 . Hence, this approach is very appealing and these simple closed-form solutions facilitate both the precoder/receive vector design and performance analysis.
Substituting t * and r * back into (20), the e2e SNR can be expressed as
γ = P S DH SR 2 2 P R H RD 2 2 P S DH SR 2 2 + P R H RD 2 2 + 1 ,(23)
where X 2 2 = λ max (XX † ).
B. Transmit ZF with M T > 1
We assume that w r = h SR , i.e., the relay employs a maximal-ratio combining (MRC) receive beamforming vector, and optimizes the transmit ZF vector w t . In this case, we can simplify problem (11) as:
max wt P S |h † RD w t | 2 h SR 4 |h † RD w t | 2 h SR 2 + 1 (24) s.t. w t 2 ≤ P R h SR 4 P S + h SR 2 , h † SR H RR w t = 0,
or equivalently using monotonicity,
max wt |h † RD w t | 2 (25) s.t. w t 2 ≤ P R h SR 4 P S + h SR 2 , h † SR H RR w t = 0.
Following the same procedure employed to obtain (15), the solution of (25) is given by
w * t = P R h SR 4 P S + h SR 2 Bh RD Bh RD ,(26)
where we have defined B I −
H † RR hSRh † SR HRR h † SR HRR 2 .
With w * t , the optimized e2e SNR can be expressed as
γ = P S h SR 2 P R Bh RD 2 P S h SR 2 + P R Bh RD 2 + 1 .(27)
Similar to the receive ZF scheme, we propose the following solutions for t and r (which may not be optimal)
t * = arg max t =1 h SR 2 = arg max t =1 H SR t 2 (28) = u max (H † SR H SR ), and r * = arg max r =1 Bh RD 2 = arg max r =1 BH RD r 2 (29) = u max (H † RD BH RD ),
respectively. Finally, substituting t * and r * into (27), the e2e SNR can be expressed as The AF process at R employs the conventional amplification factor [5, Eq. (4)] which guarantees the stability of the relay and prevents oscillation. This particular choice of amplification process is also simple to use since R can adaptively adjust its transmit power to a constant level. In this case, the instantaneous e2e SINR is expressed as [4], [5] γ i,j,k,l =
γ = P S H SR 2 2 P R BH † RD 2 2 P S H SR 2 2 + P R BH † RD 2 2 + 1 .(30)γ i,j SR γ i,l RR +1 γ k,l RD γ i,j SR γ i,l RR +1 + γ k,l RD + 1 ,(31)
where γ i,j SR = P S |h i,j SR | 2 , and γ k,l RD = P α S |h k,l RD | 2 are the instantaneous SNRs of the S − R and the R − D links while γ i,l RR = P α S |h i,l RR | 2 is the instantaneous interference-to-noise ratio (INR) of the R − R link. In order to facilitate the analysis of the outage probability in Section V-B, we also restate the average SNRs of the S − R and the R − D links asγ SR P S c SR andγ RD P α S c RD , respectively. Moreover, γ RR P α S c RR is the average INR of the R − R link.
A. Optimal Antenna Selection
Denote the selected receive and transmit antenna indexes at R and S, and the receive and transmit antenna indexes at D and R are by I, J, K, L, respectively. The optimal AS (OP AS) scheme can be expressed as
{I, J, K, L} = argmax 1≤i≤MR,1≤j≤NT 1≤k≤NR,1≤l≤MT γ i,j,k,l .(32)
The OP AS scheme maximizes the e2e SINR, however it has a high computation and implementation complexity. In a centralized architecture, a central unit requires the knowledge of all links (S − R, R − R and R − D) in order to decide on the selected antennas.
B. max-max Antenna Selection
The max − max AS (MM AS) scheme selects the best S − R and R − D links without considering the LI and can be expressed as
{I, J} = argmax 1≤i≤MR,1≤j≤NT γ i,j SR , {K, L} = argmax 1≤k≤NR,1≤l≤MT γ k,l RD .(33)
Note that the MM AS scheme, which is SNR optimal in conventional half-duplex relaying [21], becomes strictly sub-optimal in full-duplex relaying since it does not take into account the effect of LI. However, the MM AS scheme can be easily implemented by estimating the S − R channels at R and using channel feedback (on the R − D link) from D to R, related to the selected antenna index K.
C. Partial Antenna Selection
The partial AS (PR AS) scheme 4 simplifies the selection problem by decoupling the two relaying hops according to the following rule
{I, J, L} = argmax 1≤i≤MR,1≤j≤NT ,1≤l≤MT γ i,j SR γ i,l RR + 1 , {K} = argmax 1≤k≤NR γ k,L RD .(34)
The PR AS scheme provides a good performance/implementation complexity trade-off since it reduces the searching set of the optimal solution while it also takes into account the LI. It is worth noting that channel feedback from D to R is not required since the relay transmit antenna is selected independently of the second hop.
D. Loop Interference Antenna Selection
The loop interference AS (LI AS) scheme selects the receive/transmit antennas in order to minimize the effects of LI according to
{I, L} = argmin 1≤i≤MR,1≤l≤MT γ i,l RR , {J} = argmax 1≤j≤NT γ I,j SR , {K} = argmax 1≤k≤NR γ k,L RD .(35)
This scheme is analogous to the LI suppression policies with relay precoders proposed in [6], [14]. The LI AS aims to minimize the deleterious effects of LI, while some improvement in the S − R, R − D channels is also extracted by selecting antennas at S and D.
V. OUTAGE PROBABILITY ANALYSIS
In this section, we study the outage probability of the precoding/decoding designs as well as the AS schemes presented in Sections III and IV, respectively. We derive exact expressions for the outage probability and based on these results, the asymptotic behavior is also studied to reveal important insights such as the diversity order.
A. Joint Precoding/Decoding Designs
The rate outage probability, P out , is defined as the probability that the instantaneous mutual information, I = log 2 (1 + γ), falls below a target rate of R 0 bits per channel use (BPCU). Hence,
P out = Pr (log 2 (1 + γ) ≤ R 0 ) = F γ (γ T ) ,(36)
where γ T = 2 R0 − 1 and F γ (·) is the cumulative distribution function (cdf) of the e2e SNR. (23), we can now derive the outage probability of the system. To this end, we first note that DH SR 2 2 = λ max H † SR D † DH SR can be written as
1) Receive ZF: From
DH SR 2 2 = λ max H † SR I − H RR h RD h † RD H † RR H RR h RD 2 H SR = λ max H † SR Φ † (I − diag (1, 0, . . . , 0)) ΦH SR = λ max H † SR diag (0, 1, . . . , 1) H SR = λ max H † SRH SR ,(37)
where Φ is a unitary matrix, H SR = ΦH SR andH SR is a (M R − 1) × N T matrix. In (37), the first equality follows from the fact that D = D † D. The second equality is due to the eigen decomposition ( HRRhRD HRRhRD is a M R × 1 normalized column vector and has rank 1). Hence, DH SR 2 2 is the maximum eigenvalue of a Wishart matrix H † SRH SR with dimensions (M R − 1) × N T . We now derive the exact outage probability with receive ZF using the result for DH SR is the probability density function (pdf) of γ SR , with γ SR = P S DH SR 2 2 and γ RD = P R H SR 2 2 . By using [32,Eq. (23)], we can obtain the pdf of γ SR and the cdf of γ RD as
f γSR (x) = min(NT ,MR−1) a=1 (NT +MR−1)a−2a 2 b=|NT −MR+1| a b+1 d 1 (a, b) (b)!γ b+1 SR x b e − ax γ SR , and F γRD (x) = 1 − min(MT ,NR) k=1 (MT +NR)k−2k 2 l=|MT −NR| l m=0 k m d 2 (k, l) (m)!γ m RD x m e − kx γ RD ,
respectively, where the average SNR of the S − R and R − D links are given byγ SR = P S c SR and γ RD = P R c RD . The coefficients, d l (i, j), l = 1, 2 are given in [32] for some system configurations and can be efficiently computed using the algorithm in [34]. We now substitute the above pdf and cdf into the integral representation of F γ (γ T ) and solve it in closed-form using [35, Eq. (3.471.9)] to yield
F γ (γ T ) = 1 − s1 a=1 (NT +MR−1)a−2a 2 b=|NT −MR+1| s2 k=1 (MT +NR)k−2k 2 l=|MT −NR| l m=0 (38) × m u=0 b v=0 2 m u b v d 1 (a, b)d 2 (k, l)k u+v+m+1 2 γ m+2b+u−v+1 2 T (1 + γ T ) m−u+v+1 2 b!m!a u+v−m−2b−1 2γ 2b−u−v+m+1 2 SRγ u+v+m+1 2 RD × e − ā γ SR + k γ RD γT K u+v−m+1 2 ak (1 + γ T ) γ T γ SRγRD ,
where s 1 = min (N T , M R − 1) and s 2 = min (M T , N R ).
In order to further obtain insights, such as diversity order, we now present a simplified asymptotic outage probability. Specifically, we adopt the upper bound, γ ≤ min (γ SR , γ RD ), to γ. This bound is tight for medium-to-high SNR values and in [36] it was shown that it is also asymptotically-exact in the high SNR regime [36]. Therefore, using simple order statistics we can express the asymptotic cdf of the
e2e SNR as F ∞ γ (x) = F γ ∞ SR (x) + F γ ∞ RD (x) − F γ ∞ SR (x)F γ ∞ RD (x).
It can be easily shown that at high SNRs, F ∞ γ (x) can be approximated by a single term polynomial approximation. To see this, we first need polynomial approximations for γ SR and γ RD . These results can be borrowed from [37,Eq. (7)] and with the aid of F ∞ γ (x) we can show that
P ∞ out = s 1 −1 k=0 k! s 1 −1 k=0 (t1+k)! γT γSR NT (MR−1) N T (M R − 1) < M T N R , s 1 −1 k=0 k! s 1 −1 k=0 (t1+k)! γT γSR NE + s 2 −1 k=0 k! s 2 −1 k=0 (t2+k)! γT γRD NE N T (M R − 1) = M T N R = N E , s 2 −1 k=0 k! s 2 −1 k=0 (t2+k)! γT γRD MT NR N T (M R − 1) > M T N R ,(39)
where t 1 = max (N T , M R − 1) and t 2 = max (M T , N R ). By inspecting (39), we see that our full-duplex receive ZF design achieves a diversity order of min (N T (M R − 1), M T N R ).
2) Transmit ZF:
Using an equivalent approach to that used for the receive ZF scheme and omitting details for conciseness, the exact outage probability can be expressed as
F γ (γ T ) = 1 − s3 a=1 (NT +MR)a−2a 2 b=|NT −MR| s4 k=1 (MT +NR−1)k−2k 2 l=|MT −NR−1| l m=0 (40) × m u=0 b v=0 2 m u b v d 1 (a, b)d 2 (k, l)k u+v+m+1 2 γ m+2b+u−v+1 2 T (1 + γ T ) m−u+v+1 2 b!m!a u+v−m−2b−1 2γ 2b−u−v+m+1 2 SRγ u+v+m+1 2 RD × e − ā γ SR + k γ RD γT K u+v−m+1 2 ak (1 + γ T ) γ T γ SRγRD , where s 3 = min (N T , M R ) and s 4 = min (M T − 1, N R ).
Furthermore, we can express the asymptotic outage probability of transmit ZF as
P ∞ out = s 3 −1 k=0 k! s 3 −1 k=0 (t3+k)! γT γSR NT MR N T M R < (M T − 1)N R , s 3 −1 k=0 k! s 3 −1 k=0 (t3+k)! γT γSR ME + s 4 −1 k=0 k! s 4 −1 k=0 (t4+k)! γT γRD ME N T M R = (M T − 1)N R = M E , s 4 −1 k=0 k! s 4 −1 k=0 (t4+k)! γT γRD (MT −1)NR N T M R > (M T − 1)N R ,(41)
B. Antenna Selection
In this subsection, we investigate the outage probability of the proposed full-duplex based AS schemes.
We derive exact as well as approximate outage expressions when P S → ∞ for comparison of the proposed AS schemes. By considering the definition of the outage probability, we can write 5
P ⋆ = Pr log 2 1 + γ I,J SR γ I,L RR +1 γ K,L RD γ I,J SR γ I,L RR +1 + γ K,L RD + 1 < R 0 .
(42) 5 In the following subsections, the statistical distributions of γ I,J SR , γ I,L RR and γ K,L RD may differ depending on the AS scheme. Any remark concerning the distributions of these RVs is strictly limited to the particular AS scheme.
For "optimal", "max − max", "partial" and "loop interference" AS schemes, the subscript ⋆ in (42) refers to OP, MM, PR and LI, respectively.
1) Optimal Antenna Selection:
Let γ OP denote the e2e SINR at D for the OP AS scheme. The outage probability of the OP AS scheme can be written as
P OP = F γOP (γ T ) ,(43)
where F X (·) denotes the cdf of the random variable (RV), X. Obtaining an analytical expression for P OP appears to be a cumbersome problem due to the dependencies between the SINR variables being maximized. Therefore, we have performed simulations to evaluate the outage performance of the OP AS scheme in Section V. Further, under some special antenna configurations, for example with M R = M T = 1, the OP AS scheme is equivalent to the MM AS scheme for which an analytical expression is presented below.
We now state the asymptotic behavior of the OP AS scheme in Proposition 1.
Proposition 1:
The outage probability of the OP AS scheme as P S → ∞ can be approximated by
P OP ≈ C 1 γ RR γ T γ SR NT MR + C 2 γ T γ RD MT NR ,(44)
where C 1 > 0 and C 2 > 0 are two positive constants.
Proof: We first lower bound γ OP by γ MM , where γ MM is the SINR of the suboptimal MM AS scheme.
In the following subsection, we show that as P S tends to infinity, the corresponding upper bound, P OP ≤ . Next we upper bound γ OP by γ UB , 6 defined as γ UB
X 1 Y X2 X 1 Y +X2+1
where X 1 and X 2 are the maximum of N T M R and M T N R exponential RVs with parameters,γ SR andγ RD , respectively, while Y is a RV chosen as the minimum of M T M R exponential RVs with parameterγ RR . As P S tends to infinity, we can show that the corresponding lower bound, P OP > P LB can be approximated by
P LB ≈ (NT MR)! (MT MR) N T M R γRRγT γSR NT MR + γT γRD MT NR .
Since the upper and lower bounds of P OP have the same diversity order, (44) follows and the proof is completed.
Using the above asymptotic result, we now derive the optimal α to yield the power allocation solution at the relay. Following the respective definitions and expressingγ SR ,γ RR andγ RD explicitly in terms of P S , we see that the first term in (44) decays as P −(1−α)NT MR S while the second term decays as P −αMT NR S . 6 The SINR upper bound, γUB corresponds to a "virtual" system in which transmit/receive AS is decoupled to consider the best S − R and R − D links and the weakest LI (R − R) link, respectively, since such a strategy will maximize the e2e SINR in (31). However, clearly such a AS scheme is not possible in our system, since selecting a particular transmit/receive antenna pair at R will automatically fix the LI link, i.e., AS for the links can not be performed independently. Therefore, depending on the value of α, the first or the second term in (44) becomes dominant and determines the total asymptotic outage probability. Outage minimization from a diversity perspective occurs when (1 − α)N T M R = αM T N R and we have
α OP opt = N T M R N T M R + M T N R ,(45)
with P α OS opt S as the optimal power allocation solution at the relay. Moreover, the highest diversity order, d max,OP , achieved with the OP AS scheme is given by
d max,OP = 1 (M T N R ) −1 + (N T M R ) −1 .(46)
2) max − max Antenna Selection: With this scheme, γ I,J SR is simply the largest of N T M R exponential RVs with parameterγ SR , γ K,L RD is simply the largest of M T N R exponential RVs with parameterγ RD , and, since the R − R link is ignored, γ I,L R,R is an exponential RV with parameterγ RR . The outage probability of MM AS can be written as
P MM = 1 − ∞ 0 F X (y + γ T + 1)γ T y f Y (y + γ T )dy,(47)
where X = γ I,J SR γ I,L RR +1 , Y = γ K,L RD and F X (·) denotes the complementary cdf of the RV, X. Clearly, in order to evaluate (47) we first need to find the cdf and the pdf of X and Y , respectively. The cdf of X can be expressed as
F X (x) = 1 γ RR ∞ 0 F γ I,J SR ((y + 1)x) e − ȳ γ RR dy (48) = 1 − N T M R NT MR−1 p=0 (−1) p NT MR−1 p e − (p+1)x γ SR (p + 1) 1 + (p+1)γRRx γSR .
The second equality in (48) follows since the binomial expansion F γ I,J
SR (x) = 1 − e − x γ SR
NT MR
can be written as
F γ I,J SR (x) = 1 − N T M R NT MR−1 p=0 (−1) p ( N T M R −1 p ) p+1 e − (p+1)x γ SR .
We can now write (47) as
P MM = 1 − N T M R M T N R NT MR−1 p=0 (−1) p NT MR−1 p p + 1 MT NR−1 q=0 (−1) q MT NR−1 q γ RD (49) × ∞ 0 e − (p+1)(y+γ T +1)γ T γ SR y − (q+1)(y+γ T ) γ RD 1 + (p+1)(y+γT +1)γRRγT γSRy dy.
Eq. (49) does not admit a closed-form solution. However, it can be easily evaluated numerically using standard mathematical software tools.
In order to derive an accurate closed-form outage expression applicable in the asymptotic regime (P S → ∞), we consider
P MM ≥ Pr min γ I,J SR γ I,K RR + 1 , γ K,L RD < γ T (50) → 1 − N T M R M T N R NT MR−1 p=0 (−1) p NT MR−1 p (p + 1) 1 + (p+1)γRRγT γSR MT NR−1 q=0 (−1) q MT NR−1 q q + 1 e − p+1 γ SR + q+1 γ RD γT , where F min(X,Y ) (·) = 1−(1 − F X (·)) (1 − F Y (·)
) has been used. Ignoring the product term F X (·) F Y (·)
as it gives higher order terms, we observe that the asymptotic behavior of P MM can be further approximated
as P MM ≈ F X (γ T ) + F Y (γ T ). Consider F X (γ T ) as P S → ∞; for small x = γT γSR we can simplify F X (x) = e − x γ SR γRR ∞ 0 1 − e − x γ SR NT MR e − ȳ γ RR dy as F X (x) ≈ x NT MR γ RR ∞ 0 y NT MR e − ȳ γ RR dy (51) = (N T M R )! (γ RR x) NT MR .
Similarly, we can show that as
P S → ∞, F Y (γ T ) ≈ γT γRD MT NR
. Therefore, (50) can be simplified for 0 < α < 1 as
P MM ≈ (N T M R )! γ RR γ T γ SR NT MR + γ T γ RD MT NR .(52)
As an immediate observation, from (44) and (52) we see that the OP AS scheme and the MM AS scheme achieve the same diversity performance. As a result, we have α MM opt = α OP opt with P α MM opt S as the optimal power allocation solution at the relay and the highest diversity order, achieved with the MM AS scheme is also d max,MM = 1 (MT NR) −1 +(NT MR) −1 . However, compared to the MM AS scheme, the OP AS scheme has a higher array gain as verified in Section VI.
3) Partial Antenna Selection:
The outage probability of this scheme can be evaluated from Substituting the required cdf and the pdf into F X (x) with simplifications yields
P PR = 1 − ∞ 0 F X (y + γ T + 1)γ T y f Y (y + γ T )dy,(53)F X (x) = 1 − N T NT −1 p=0 (−1) p NT −1 p e − (p+1)x γ SR (p + 1) 1 + (p+1)γRRx MTγSR MR .(54)
Furthermore, we notice that the RV, Y = γ K,L RD , is simply the largest among N R exponential RVs with parameterγ RD . Therefore, the pdf of Y can be written as f Y (y) = NR γRD NR−1
q=0 (−1) q NR−1 q e − (q+1)ȳ γ RD .
Combining these results, the exact outage probability of the PR AS scheme can be written as
P PR = 1 − N R γ RD NR−1 q=0 (−1) q N R − 1 q I q ,(55)
where the integral I q is defined as
I q = ∞ 0 1 − 1 − N T NT −1 p=0 (−1) p NT −1 p e − (p+1)(y+γ T +1)γ T γ SR y (p + 1) 1 + (p+1)γRR(y+γT +1)γT MTγSRy MR e − (q+1)(y+γ T ) γ RD dy.(56)
In order to derive an accurate closed-form expression for the outage probability with P S → ∞ we consider P PR ≥ Pr min γ I,J SR γ I,K RR + 1
, γ K,L RD < γ T (57) ≈ N T ! M NT T MR γ RR γ T γ SR NT MR + γ T γ RD NR ,
for 0 < α < 1. We see that the first term decays as P −(1−α)NT MR S while the second term decays as
P −αNR S
. Therefore, outage minimization occurs when (1 − α)N T M R = αN R and we have
α PR opt = N T M R N T M R + N R ,(58)
with P α PR opt S as the optimal power allocation solution at the relay. Therefore, the highest diversity order, d max,PR , achieved with the PR AS scheme can be expressed as
d max,PR = 1 N −1 R + (N T M R ) −1 .(59)
4) Loop Interference Antenna Selection:
In the case of the LI AS scheme, the outage probability can be evaluated from
P LI = 1 − ∞ 0 F X (y + γ T + 1)γ T y f Y (y + γ T )dy,(60)
where X = γ I,J SR γ I,L RR +1 and Y = γ K,L RD . Since receive/transmit antennas at R are selected to minimize the LI, with this scheme γ I,L RR is the minimum of M R M T exponential RVs with parameterγ RR , while γ I,J SR and γ K,L RD are the largest of N T and N R exponential RVs with parametersγ SR andγ RD , respectively. Therefore, the required cdf of X can be found using
F X (x) = 1 − N T M R M T γ RR NT −1 p=0 (−1) p NT −1 p p + 1 ∞ 0 e − (p+1)(y+1)x γ SR e − M R M T ȳ γ RR dy.(61)
Simplifying the integral in (61) yields
F X (x) = 1 − N T NT −1 p=0 (−1) p NT −1 p e − (p+1)x γ SR (p + 1) 1 + (p+1)γRRx MRMTγSR .(62)
Now, combining the pdf of Y and (62) we can express the exact outage probability as
P LI = 1 − N T N R γ RD NT −1 p=0 (−1) p NT −1 p p + 1 NR−1 q=0 (−1) q N R − 1 q ∞ 0 e − (p+1)(y+γ T +1)γ T γ SR y e − (q+1)(y+γ T ) γ RD 1 + (p+1)γRRγT (y+γT +1) MRMTγSRy dy.(63)
We now present an asymptotic approximation for the outage probability of the LI AS scheme. The outage probability as P S → ∞ can be approximated by
P LI ≥ Pr min γ I,J SR γ I,K RR + 1 , γ K,L RD < γ T (64) → 1 − N T N R NT −1 p=0 (−1) p NT −1 p e − (p+1)x γ SR (p + 1) 1 + (p+1)γRRx MRMTγSR NR−1 q=0 (−1) q NR−1 q q + 1 e − (q+1)ȳ γ RD .
Eq. (64) can be simplified as
P LI ≈ N T ! (M R M T ) NT γ RR γ T γ SR NT + γ T γ RD NR ,(65)
for 0 < α < 1. We see that the first term decays as P
−(1−α)NT S
while the second term decays as P −αNR S .
As for the previous AS schemes, the optimum α value can be found from (1 − α)N T = αN R and is given by
α LI opt = N T N T + N R ,(66)
to yield P α LI opt S as the optimal power allocation solution at the relay.
Scheme Diversity Order Complexity
Receive ZF min (N T (M R − 1), M T N R ) high Transmit ZF min (N T M R , (M T − 1) N R ) high OP AS 1 (MT NR) −1 +(NT MR) −1 N T M R M T N R MM AS 1 (MT NR) −1 +(NT MR) −1 N T M R + M T N R PR AS 1 N −1 R +(NT MR) −1 N T M R M T + N R LI AS 1 N −1 T +N −1 R N T + M R M T + N R
Further, the highest diversity order, d max,LI , achieved with the LI AS scheme can be expressed as
d max,LI = 1 N −1 T + N −1 R .
(67)
C. Comparisons of the Schemes
d precoding > d OP = d MM > d PR > d LI .(68)
As for the complexity, the precoding schemes utilize all the antennas and require a radio frequency chain for each antenna element. In addition, the computation of the beamforming vectors involves demanding mathematical operations such as matrix multiplication, matrix inversion and eigen-decomposition giving a general complexity of O(n 3 ). Therefore, although ZF precoding designs achieve higher diversity performance, they are characterized by a higher complexity in comparison to AS schemes. The proposed AS schemes also correspond to different complexities and are appropriate for networks with different computational capabilities. In order to provide a simple comparison of their complexity, we use as a metric the number of channels that should be examined in order to apply each AS scheme. It is worth noting that each channel in most of the cases is associated with a feedback channel (and a training process) in a centralized implementation. The OP AS examines all the possible combinations and therefore corresponds to a high complexity equal to N T M R M T N R channels. The MM AS scheme decouples the AS selection into two independent groups and therefore has a complexity of (N used at D, the performance can be improved by selecting appropriate design parameters at S and R. This attribute of the system is useful under different conditions; e.g., when fixed infrastructure based relays are employed, they can be equipped with many antennas while user terminals that act as relays have space constraints, and here the source can be equipped with many antennas. We also observe that although (2, 2, 2, 1) and (2, 3, 2, 1) enjoys a diversity order of two, the latter has a superior performance as a result of higher array gain. The same observation can be seen when (3, 2, 3, 1) and (2, 3, 3, 1) are compared.
In the first case, additional performance gain is obtained via increasing M R (also (2, 3, 2, 1) has one more total number of antennas compared to (2, 2, 2, 1)). However, in the second case, while (3, 2, 3, 1)
and (2, 3, 3, 1) have the same number of total antennas, swapping N S with M R improves the outage probability. For comparison, we have included results for half-duplex hop-by-hop beamforming [33] with two configurations, namely (2, 2, 1, 1) and (2, 3, 3, 1) and γ T = 2 2R0 − 1. These results can be compared for example with (2, 2, 1, 1) full-duplex operation and refer to the so called "RF chain preserved" condition and the "number of antenna preserved" (at the relay) condition. We show results for transmit ZF based precoding design with different antenna configurations in Fig. 3.
The achievable diversity orders of the considered configurations, given by min (N T M R , (M T − 1)N R ), are again 1, 2 and 3, respectively. We also compare the performance of the (2, 3, 2, 3) configuration under receive and transmit ZF designs, and the achievable diversity order of the former design given by min (N T (M R − 1), M T N R ) is four. Interestingly, receive ZF design exhibits a superior performance to transmit ZF since the former enjoys fourth order diversity order while the latter only has a diversity order of three. Clearly, this observation demonstrates that while under some configurations (M T = 1) or (M R = 1) only one form (receive or transmit) of precoding design can be deployed, in other configurations, when both designs can be applied, the system designer has to carefully decide on the configuration as well as the precoding design.
B. Antenna Selection
In Figs. 4-6, we have set N T = M R = M T = N R = 2. Fig. 4 shows the outage probability as a function of P S for the considered AS schemes. No power control at R is adopted and thus we adopt α = 1.
Clearly, we see that all full-duplex schemes suffer from a zero-diversity order. Among the full-duplex AS schemes, the OP AS scheme provides the best performance. The PR AS scheme exhibits the next best performance and converges to the same error floor as the OP AS scheme. With low P S , the MM AS performs better than both PR AS and LI AS schemes. Furthermore, for comparison with full-duplex, we have also plotted results for half-duplex operation with two cases; namely, the total number of antennas at the relay (n V ) is 2 and 4, respectively. With half-duplex transmission, the AS principle is simple; i.e.,
antennas are selected at each node to maximize the SNRs of the S − R and R − D links, respectively.
The half-duplex results were plotted using [21,Eq. (9)] with γ T = 2 2R0 − 1 due to the two time slot operation. The full-duplex AS schemes shows a favorable outage performance at a low-to-medium range of P S , while the superiority of half-duplex transmission at high P S is clearly evident since it avoids LI and enjoys the benefits of diversity. Fig. 5 shows the outage probability of the AS schemes with optimal α. In contrast to the results in Fig. 4, where outage probability exhibits a saturated behavior at high P S (zero diversity), all AS schemes are now able to provide some diversity and outage decays as P S increases. For the considered system set up, α OS opt = 0.5, α MM opt = 0.5, α PR opt = 0.667 and α LI opt = 0.5, and the achieved diversity orders of the OP, MM, PR and LI AS schemes are respectively, 2, 2, 1.33 and 1. Moreover, as expected, the OP AS scheme is able to provide the best performance among all the considered AS schemes in the work. When c RR is high (0.5), a performance gap between OP AS and MM AS is observed (although both OP AS and MM AS provides the same diversity, the former has a higher array gain). However, we see that the performance difference between MM AS and OP AS schemes are almost negligible at c RR = 0.1. The usefulness of our asymptotic results can also be appreciated from Fig. 5. With increasing P S , we see that the asymptotic plots match the exact results very well. In Fig. 6, the outage behavior of the PR AS scheme with several values of α is illustrated. For α values close to one, the outage begins to suffer from low diversity (e.g., the curve corresponding to α = 0.99 almost converge to an error floor and exhibit a near zero diversity behavior). Clearly, the value of α PR opt = 0.667 yields the best performance in the asymptotic regime. Interestingly, for P S < 30 dB, α = 0.99 and 0.9 are able to provide a better performance than the optimal case before they begin to experience the decremental effects of low diversity. Therefore, depending on the operating region, an appropriate value for α can be selected. In the cases of OP AS, MM AS and LI AS, similar outage behavior with different α values can be observed as well.
VII. CONCLUSION
In this paper, we considered full-duplex MIMO relaying with multi-antenna source and destination nodes. We introduced joint precoding/decoding designs which incorporate rank-1 zero-forcing selfinterference suppression at the relay node. Our analysis delivered closed-form results which were further analyzed to reveal several interesting observations. Exact as well as asymptotic expressions for the outage probability were derived to explicitly reveal insights such as the achievable diversity order and the array gain. These results were also verified from simulations to confirm their correctness. The outage probability is influenced by the number of antennas deployed at each node as well as the adopted precoding (receive ZF or transmit ZF) design. In order to further reduce system complexity, we also presented several AS schemes. The investigated AS schemes have been analyzed in terms of the outage probability and exact expressions as well as asymptotic approximations have been derived. A simple power allocation scheme at the relay was proposed to overcome the zero-diversity limitation. A single parameter in the power allocation scheme can be set to obtain the desired outage performance while optimum values of this parameter were presented for diversity maximization of the investigated AS schemes.
Fig. 1 .
1Full-duplex MIMO relaying with multi-antenna source and destination nodes. The dashed line denotes the loopback self-interference.
deals with the problem of AS for the full-duplex MIMO relay channel considered. AS is proposed as an alternative to the e2e optimization and is particularly relevant to systems with stricter computational/energy constraints. Full-duplex relay AS introduces new design challenges due to the presence of LI and differs from the existing body of AS literature in several ways. As explained below, with full-duplex operation, several AS choices that provide different performance/complexity tradeoff exist while a straightforward AS strategy (see for e.g.[21]) can be used to maximize the performance in half-duplex AS systems. Moreover, power allocation is an important issue with different full duplex AS schemes while half-duplex AS schemes can use full power at the relay (in the absence of LI).
The required cdf of the e2e SNR can be derived by adopting a similar approach as in[33, Appendix I]. Specifically, we can express the cdf of γ as F γ (γ T ) = PrγSRγRD γSR+γRD+1 < γ T = 1− ∞ 0F γRD (γT +y+1)γT y f γSR (γ T +y)dy, whereF γRD (x) is the complementary cdf of γ RD , and f γSR (x)
where t 3
3= max (N T , M R ) and t 4 = max (M T − 1, N R ). From Eq. (41) we see that with transmit ZF, a diversity order of min (N T M R , (M T − 1)N R ) can be achieved. On the other hand, half-duplex MIMO hop-by-hop (MRT/MRC) beamforming exhibits a diversity order of min (N T M R , M T N R ). As a result, although half-duplex hop-by-hop beamforming delivers a superior diversity performance in general, in certain antenna configurations, half-duplex hop-by-hop beamforming and full-duplex ZF designs offer the same diversity.
P
MM , can be approximated by P MM ≈ (N T M R )!
RR
+1 and Y = γ K,L RD . The required distributions of X and Y are different to the previous case of max − max AS and in order to calculate P PR we need to evaluate them. For any i-th relay receive antenna, the ratio γ i,j SR γ i,l RR +1 is maximized when the strongest S − R channel and the weakest R − R channel from the ith antenna (i = 1, . . . , M R ) are selected. Since there are M R antennas, the cdf of X can be evaluated as F X (x) = ∞ 0 F A ((y + 1)x) f B (y)dy MR , where A is a RV defined as the largest among N T exponentially distributed RVs, while B is the smallest out of M T exponentially distributed RVs.
T M R + M T N R ) channels. The PR AS scheme decouples the R − D link in the selection process and gives a complexity of (N T M R M T + N R ) channels. Finally, the LI AS scheme is based on the LI channel and thus has a complexity of (N T + M R M T + N R ) channels. VI. NUMERICAL RESULTS In this section, we give numerical examples for the outage probability of the proposed precoding and AS schemes. The simulation set-up follows the system model of Section II with R 0 = 2 BPCU, and c SR = c RD = 1. Although we have considered a symmetric setup, i.e., c SR = c RD , the main observations shown for AS schemes in Figs. 4-6 are also valid for asymmetric setups, where c SR = c RD .A. Joint Precoding/Decoding DesignsFig. 2 shows the results for the receive ZF based precoding design with different antenna configurations. The specific values of N T , M R , M T , N R for each antenna configuration are shown inside the figure labels as (N T , M R , M T , N R ) respectively. These results reveal several interesting observations useful for system designers. The achievable diversity orders of the considered configurations, given by min (N T (M R − 1), M T N R ), are 1, 2 and 3, respectively. Therefore, although only one receive antenna is
Fig. 2 .
2Outage probability versus per hop average SNR for the receive ZF based precoding design with different antenna configurations.
Fig. 3 .
3Outage probability versus per hop average SNR of precoding designs with different antenna configurations.
Fig. 4 .
4Outage probability versus PS; cRR = 0.05 and α = 1. The results for OP, MM, PR and LI AS schemes are computed via simulations and (49), (55), (63) respectively.
Fig. 5 .
5Outage probability versus PS with optimum α. The results for OP AS scheme are computed from simulations while asymptotic results for MM, PR, and LI AS schemes are due to (52), (57), (65) respectively.
Fig. 6 .
6Outage probability versus PS for the PR AS scheme and different α; cRR = 0.1.
TABLE I DIVERSITY
IORDER AND COMPLEXITY FOR THE PRECODING DESIGNS AND AS SCHEMES.
Table 1
1summarizes the diversity order achieved from the investigated schemes as well as their associated complexity. The first main observation is that the precoding designs outperform the AS schemes in terms of diversity gain. The utilization of all antenna elements mitigates the LI effects and ensures a diversity order that is dominated by the weakest relaying branch. We note that due to the received/transmitted ZF operation, one antenna element is reserved for spatial cancellation at the relay's input/output, respectively. On the other hand, OP AS and MM AS schemes achieve similar diversity performance and significantly outperform the PR AS and LI AS schemes. Another interesting observation is that the diversity order of the PR AS scheme does not depend on M T . Similarly, in the case of LI AS the diversity order is independent of the number of relay antennas. By comparing the results in Column 2 ofTable 1, it is easy to see that with M R , M T > 1
November 15, 2013 DRAFT
It should be noted that due to the influence of LI, power adaptation (or "gain control"[26]) is an important issue for full-duplex AF relaying.November 15, 2013 DRAFT
Single stream beamforming delivers maximum diversity/array gains and is suitable in a slow fading environment. Also, with multiple antennas at all three nodes, multiple independent data streams can be simultaneously sent, namely, multistream beamforming systems. In such systems, although full-duplex operation mode aimed at utilizing spectrum resources more efficiently can promise rate gains due to spatial multiplexing[30], in general they experience poor error performance[31].November 15, 2013 DRAFT
The name for this AS scheme was adopted in the same spirit where selection schemes based on the first-hop CSI are identified as partial relay selection in the literature[28].November 15, 2013 DRAFT
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Asymptotic analysis in MIMO MRT/MRC systems. Q Zhou, H Dai, EURASIP J Wireless Commun. and Networking. Q. Zhou and H. Dai, "Asymptotic analysis in MIMO MRT/MRC systems," EURASIP J Wireless Commun. and Networking, pp. 1-8, 2006. [Online]. Available: http://jwcn.eurasipjournals.com/content/pdf/1687-1499-2006-045831.pdf
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[] |
[
"A Census of Protostellar Outflows in Nearby Molecular Clouds",
"A Census of Protostellar Outflows in Nearby Molecular Clouds"
] |
[
"Duo Xu \nDepartment of Astronomy\nThe University of Texas at Austin\n78712AustinTXUSA\n\nDepartment of Astronomy\nUniversity of Virginia\n22904-4235CharlottesvilleVAUSA\n",
"Stella S R Offner \nDepartment of Astronomy\nThe University of Texas at Austin\n78712AustinTXUSA\n",
"Robert Gutermuth \nDepartment of Astronomy\nUniversity of Massachusetts\n01003AmherstMAUSA\n",
"Shuo Kong \nSteward Observatory\nUniversity of Arizona\n85719TucsonAZUSA\n\nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n",
"Hector G Arce \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n"
] |
[
"Department of Astronomy\nThe University of Texas at Austin\n78712AustinTXUSA",
"Department of Astronomy\nUniversity of Virginia\n22904-4235CharlottesvilleVAUSA",
"Department of Astronomy\nThe University of Texas at Austin\n78712AustinTXUSA",
"Department of Astronomy\nUniversity of Massachusetts\n01003AmherstMAUSA",
"Steward Observatory\nUniversity of Arizona\n85719TucsonAZUSA",
"Department of Astronomy\nYale University\n06511New HavenCTUSA",
"Department of Astronomy\nYale University\n06511New HavenCTUSA"
] |
[] |
We adopt the deep learning method casi-3d (Convolutional Approach to Structure Identification-3D) to systemically identify protostellar outflows in 12 CO and 13 CO observations of the nearby molecular clouds, Ophiuchus, Taurus, Perseus and Orion. The total outflow masses are 267 M , 795 M , 1305 M and 6332 M for Ophiuchus, Taurus, Perseus and Orion, respectively. We show the outflow mass in each cloud is linearly proportional to the total number of young stellar objects. The estimated total 3D deprojected outflow energies are 9 × 10 45 ergs, 6 × 10 46 ergs, 1.2 × 10 47 ergs and 6 × 10 47 ergs for Ophiuchus, Taurus, Perseus and Orion, respectively. The energy associated with outflows is sufficient to offset turbulent dissipation at the current epoch for all four clouds. All clouds also exhibit a break point in the spatial power spectrum of the outflow prediction map, which likely corresponds to the typical outflow mass and energy injection scale.
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10.3847/1538-4357/ac39a0
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[
"https://arxiv.org/pdf/2111.07995v1.pdf"
] | 244,117,564 |
2111.07995
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93e437a181467b14ce0166708550cb47751bae36
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A Census of Protostellar Outflows in Nearby Molecular Clouds
November 16, 2021
Duo Xu
Department of Astronomy
The University of Texas at Austin
78712AustinTXUSA
Department of Astronomy
University of Virginia
22904-4235CharlottesvilleVAUSA
Stella S R Offner
Department of Astronomy
The University of Texas at Austin
78712AustinTXUSA
Robert Gutermuth
Department of Astronomy
University of Massachusetts
01003AmherstMAUSA
Shuo Kong
Steward Observatory
University of Arizona
85719TucsonAZUSA
Department of Astronomy
Yale University
06511New HavenCTUSA
Hector G Arce
Department of Astronomy
Yale University
06511New HavenCTUSA
A Census of Protostellar Outflows in Nearby Molecular Clouds
November 16, 2021Draft version Typeset using L A T E X preprint2 style in AASTeX63ISM: outflows -ISM: clouds -methods: data analysis -stars: formation
We adopt the deep learning method casi-3d (Convolutional Approach to Structure Identification-3D) to systemically identify protostellar outflows in 12 CO and 13 CO observations of the nearby molecular clouds, Ophiuchus, Taurus, Perseus and Orion. The total outflow masses are 267 M , 795 M , 1305 M and 6332 M for Ophiuchus, Taurus, Perseus and Orion, respectively. We show the outflow mass in each cloud is linearly proportional to the total number of young stellar objects. The estimated total 3D deprojected outflow energies are 9 × 10 45 ergs, 6 × 10 46 ergs, 1.2 × 10 47 ergs and 6 × 10 47 ergs for Ophiuchus, Taurus, Perseus and Orion, respectively. The energy associated with outflows is sufficient to offset turbulent dissipation at the current epoch for all four clouds. All clouds also exhibit a break point in the spatial power spectrum of the outflow prediction map, which likely corresponds to the typical outflow mass and energy injection scale.
INTRODUCTION
Protostars launch energetic collimated bipolar outflows during the star formation process. This outflow gas entrains and accelerates the surrounding material to higher velocities than that of the ambient cloud gas, injecting a substantial amount of energy into the host molecular cloud (Frank et al. 2014;Bally 2016). Numerical simulations suggest protostellar [email protected] [email protected] flows not only reduce the core and stellar mass (e.g., Offner & Chaban 2017) but also depress the star formation rate and star formation efficiency in molecular clouds (e.g., Federrath et al. 2014;Federrath 2015;Cunningham et al. 2018). Meanwhile, protostellar outflows likely control the shape of the stellar initial mass function (Cunningham et al. 2018;Guszejnov et al. 2021). Theoretical and numerical studies indicate that protostellar outflows are highly efficient at driving turbulent motions (Matzner 2007;Nakamura & Li 2007;Moraghan et al. 2013). In small to interme-diate size clouds, energy injected by protostellar outflows can compensate for rapid turbulent dissipation over at least several dynamical time scales (Cunningham et al. 2006;Li & Nakamura 2006;Matzner 2007;Carroll et al. 2009). Consequently, protostellar outflows may stall the local gravitational collapse of molecular clouds, thereby extending the cloud lifetime (Wang et al. 2010). The role of protostellar feedback on the fate of molecular clouds is still under debate (Bally 2016).
Unfortunately, the mechanism by which protostellar outflows convert their kinetic energy into turbulent energy remains poorly understood observationally (Frank et al. 2014). In practice, astronomers usually evaluate outflow impact by comparing the total kinetic energy from protostellar outflows with the turbulent energy of their host cloud. For example, Arce et al. (2010) identified 60 outflow candidates in the Perseus molecular cloud and concluded the total outflow energy is sufficient to replenish the dissipation of turbulence. A similar result was also found in Taurus (Li et al. 2015), in Ophiuchus (Nakamura et al. 2011), and in Orion (Feddersen et al. 2020).
Accurately determining the impact of protostellar outflows, especially quantifying their effect on the cloud energy budget, requires a complete census of outflows. However, it is challenging to identify most of the outflows that are deeply embedded in dense clouds (Arce et al. 2010;Dunham et al. 2014a). The morphology of these embedded outflows are not as distinct as isolated ones in low velcity channels and the outflowing gas is difficult to separate from the ambient cloud. Protostars are often clustered and their outflows interact, making visual identification challenging. One solution is to calculate the outflow gas whose velocity is significantly above the cloud velocity dispersion and extrapolate the total mass (Arce et al. 2010;Li et al. 2015;Feddersen et al. 2020). However, this leads to an order of magnitude uncertainty (Arce et al. 2010).
Recent developments in machine learning approaches have enabled automated detection, which can separate outflows from cloud emission and conduct systematic identification of outflow features (Zhang et al. 2020;Xu et al. 2020b). Zhang et al. (2020) employed Support Vector Machines (SVM) to identify molecular outflows in a dark cloud complex from molecular line emission. Although SVM performs robustly in classification tasks, it requires preprocessing of the raw data cubes, i.e., manually extracting feature vectors that represent the raw data. The choice of input features are determined subjectively by visual inspection, and extracting these features from data cubes is nontrivial. Moreover, the manually extracted features discard part of the information embedded in the raw spectral cubes, which introduces uncertainty and affects the performance of classification. Convolutional neural networks (CNNs) are a powerful new approach being applied to identify structures or objects in astronomical data, such as exoplanets (Shallue & Vanderburg 2018), stellar feedback bubbles (Van Oort et al. 2019;Xu et al. 2020a) and protostellar outflows (Xu et al. 2020b). Given a labeled training set of images or spectral cube data, CNNs can be applied to efficiently identify features in large surveys. Van Oort et al. (2019) developed a Convolutional Approach to Shell/Structure Identification, casi, to identify stellar feedback bubbles in 2D density slices and 12 CO integrated intensity maps. Xu et al. (2020a) and Xu et al. (2020b) successfully extended casi to casi-3d, which is able to identify stellar feedback bubbles and protostellar outflows in position-position-velocity (PPV) molecular line spectra cubes. Xu et al. (2020b) applied casi-3d to 12 CO observations of the Perseus molecular cloud and identified all 60 previously visually identified outflows. Additionally, casi-3d found 20 new high-confidence outflows. Apart from structure identification, casi-3d successfully predicts hidden information in the data cube, such as the fraction of mass associated with feedback, which provides a more accurate feedback mass estimation. Xu et al. (2020a) showed that the actual mass associated with stellar feedback in Taurus is an order of magnitude smaller than the previous visual estimates. These results illustrate the capability of CNNs to identify complex structures and infer embedded information.
In this paper, we adopt the deep learning method casi-3d to systemically identify protostellar outflows in the nearby molecular clouds, Ophiuchus, Taurus, Perseus and Orion. We describe casi-3d and the CO observations of these nearby molecular clouds in Section 2. In Section 3, we present the performance of our CNN models in identifying protostellar outflows in the observational data, calculate the physical properties of outflows and discuss their impact on the clouds. We discuss the broader impact of outflows and compare them with stellar wind driven bubbles in Section 4 and summarize our results and conclusions in Section 5.
DATA AND METHOD
COMPLETE Survey
The 12 CO J=1-0 (115.271 GHz) and 13 CO J=1-0 (110.201 GHz) lines were observed simultaneously in surveys of Ophiuchus, Taurus and Perseus between 2002 and 2005 using the 13.7 m Five College Radio Astronomy Observatory (FCRAO) Telescope (Ridge et al. 2006;Narayanan et al. 2008). The main beam of the antenna pattern has a FWHM of 45 for 12 CO and 47 for 13 CO. The data are obtained on the fly (OTF), but they are resampled onto a uniform 23 grid (Ridge et al. 2006).
The Ophiuchus data has root-mean-square (RMS) antenna temperatures of 0.98 K and 0.33 K for 12 CO and 13 CO, respectively. We resam-pled the spectra with a lower velocity resolution of 0.125 km/s to ensure a uniform velocity resolution for all regions and to match the resolution of the training set. If the velocity resolution is too low (> 0.3 km s −1 ), the structure generated by stellar feedback is not distinguishable across multiple velocity channels. On the other hand, if the velocity resolution is too high (< 0.1 km s −1 ), the training data will hit the limitation of GPU memory. The noise levels for the new Ophiuchus 12 CO and 13 CO spectra are reduced by a factor of square root of 2 to 0.69 K and 0.23 K, respectively. The final Ophiuchus data cube has a velocity range between -0.8 and 7.5 km s −1 with 67 channels.
The Taurus data has a RMS antenna temperature of 0.28 K for 12 CO and 0.125 K for 13 CO. There are 80 and 76 channels with 0.26 and 0.27 km s −1 spacing for 12 CO and 13 CO, respectively. The velocity range of the Taurus data spans -5.1 to 14.9 km s −1 .
The Perseus 12 CO and 13 CO data have a RMS antenna temperature of 0.25 K and 0.2 K, respectively. We resampled the spectra with a lower velocity resolution of 0.125 km/s to ensure a uniform velocity resolution for all regions and to match the resolution of the training set. The noise levels for the new 12 CO and 13 CO spectra is reduced by a factor of square root of 2 to 0.17 K and 0.14 K, respectively. The final Perseus data cube has a velocity range between -2.0 and 15.0 km s −1 with 137 channels.
NRO45 Orion Survey
The 12 CO J=1-0 and 13 CO J=1-0 observations of Orion A were carried out from 2007 to 2017 by the Nobeyama Radio Observatory 45 m telescope (NRO 45m), using two different receivers (Shimajiri et al. 2015a,b;Ishii et al. 2019;Nakamura et al. 2019). The two maps were calibrated to the same intensity scale and combined on a common grid with a pixel scale of 7 .5, which corresponds to an effective angular resolution of 22 . Ishii et al. (2019) smoothed the combined map to a velocity resolution of 0.22 km/s, and further converted the intensity to the main beam temperature scale that is benchmarked with data from FCRAO and IRAM 30m. The final sensitivity for the 12 CO map is 0.35 K, while the sensitivity for the 13 CO map is 0.40 K. The velocity range of the Orion data spans -2.9 to 19.3 km s −1 . More detailed description about the data can be found in Kong et al. (2018) and Ishii et al. (2019).
YSO Catalogue
To validate our outflow identifications, we compare our feedback maps with the observed distributions of YSOs. We use the YSO catalog for Ophiuchus and Perseus from SESNA (Spitzer Extended Solar Neighborhood Archive, Gutermuth et al. in prep) used by Pokhrel et al. (2020). SESNA uses an updated implementation of the data treatment, source catalog construction, and YSO identification and classification processes on Spitzer surveys (Gutermuth et al. 2009). SESNA classified YSOs into four groups: deeply embedded protostars, Class I YSOs, Class II YSOs, and transition disks. For further analysis, we combine the former two groups as "Younger YSOs" and merge the latter two groups as "Older YSOs".
We adopt the YSO catalog from Rebull et al. (2010) for Taurus, which contains 215 previously identified YSOs and 148 newly identified YSOs. Rebull et al. (2010) adopted observations from IRAC and MIPS between 2005 and 2007. Rebull et al. (2010) applied colormagnitude criteria to select the YSO candidates and classified these YSO candidates into four different classes: Class I, II, III and Flat. For further analysis, we combine YSOs in Class I and Flat into a single category of "Younger YSOs" and merge YSOs in Class II and III into a single category of "Older YSOs".
We adopt the YSO catalog from Megeath et al. (2012) for Orion, which contains a total of 3479 dusty YSOs from IRAC and MIPS obser-vations. Megeath et al. (2012) classified these YSOs into three categories, which includes 2991 young stars with disks, 428 protostars, and 50 faint candidate protostars. For further analysis, we designate protostars as "Younger YSOs" and young stars with disks as "Older YSOs".
YSOs in the group of "Younger YSOs" have an average age smaller than 1 Myr, while YSOs in the group of "Older YSOs" have an average age around 1-3 Myrs (Dunham et al. 2014b). However, it is worth noting that truly younger YSOs and older YSOs may be misclassified based on their orientation with respect to the line of sight (Offner et al. 2012), which means the actual age of an individual YSO in the group of "Younger YSOs" is not necessarily smaller than that of an object in the group of "Older YSOs".
Finally, we note that not all the surveys completely cover the area of observed molecular emission. We focus our analysis on the areas with YSO coverage and indicate the boundaries of the surveys on the maps.
Method
casi-3d
We adopt the previously trained casi-3d model from Xu et al. (2020b) to identify protostellar outflows in 12 CO. casi-3d is an encoderdecoder based convolutional neural network combining both residual networks (He et al. 2016) and a "U-net" (Ronneberger et al. 2015). casi-3d is trained on synthetic 12 CO data, which models forming stars that launch protostellar outflows. We adopt the same training set and ranges of magnetohydrodynamic (MHD) model properties as described in Xu et al. (2020b, Table 1). We adopt three different 12 CO abundances and two different cloud kinetic temperatures when conducting synthetic observations as described in Xu et al. (2020b, Table 2). casi-3d takes 12 CO data cubes as input and predicts the position of outflows on the voxel level. Xu et al. (2020b) trained two casi-3d models to identify protostellar outflows. One model, model ME1, is trained to predict the 12 CO emission that is associated with outflows. The other model, model MF, is trained to predict the fraction of the mass that comes from stellar feedback. A more detailed description of how we generate training data for these two models can be found in Xu et al. (2020b).
Data Preprocessing
Before we apply our casi-3d models to the observational data, we apply the same preprocessing steps that we adopt for the training data. Due to the relatively large dynamic range of the 12 CO emission, we take the logarithm of the emission and then normalize the values by subtracting the mean and dividing by the standard deviation of the full map. Figure 1 illustrates the cumulative distribution of normalized emission for the four clouds. Although the 12 CO emission varies significantly between clouds, after preprocessing, the normalized emission of the four clouds is similar. This allows casi-3d to perform stably across a variety of conditions. Xu et al. (2020b) examined the performance of casi-3d on conditions that are not included in the training set, such as different kinetic temperatures, 12 CO abundances, beam sizes and noise levels. casi-3d is able to successfully identify outflows across a variety of physical and observational conditions. See Appendix C in Xu et al. (2020b) for more details. These tests suggest that the casi-3d identifications are relatively robust across the range of conditions expected to be characteristic of the observed clouds.
After preprocessing the observational data, we adopt the same strategy as Xu et al. (2020a) to carry out the full 12 CO map prediction. We crop the full 12 CO map into a stack of threedimensional chunks. Each chunk has a size of 64 × 64 × 32 (position-position-velocity). To reduce the bias due to the position of outflows, each chunk has at least 84% volume overlap with adjacent chunks. To construct the fullmap prediction, we combine the predictions for the individual chunks and adopt the maximum predicted value for each voxel.
Mass Calculations
We follow the same strategy as Arce et al. (2010) to calculate the outflow mass by combining both 12 CO and 13 CO data. If there is distinct 13 CO emission at the corresponding position, we use 13 CO to calculate the outflow mass. We assume the 13 CO emission line is optically thin and the 12 CO emission line is optical thick. We adopt an excitation temperature of 25 K or the 12 CO peak temperature, whichever is higher, to calculate the mass (Arce et al. 2010;Narayanan et al. 2012;Li et al. 2015;Feddersen et al. 2020). If there is no distinct 13 CO emission at the corresponding position, we assume the 12 CO emission line is optically thin to derive the mass. Under the assumption of local thermodynamic equillibrium (LTE), the mass estimation scales linearly with the excitation temperature. From previous feedback mass estimates, the choice of excitation temperature ranges from 10 K to 50 K. This could potentially introduce a factor of two uncertainty in the mass estimation. We take 62 as the abundance ratio between 12 CO and 13 CO and 10 −4 as the abundance ratio between 12 CO and H 2 (Arce et al. 2010;Feddersen et al. 2020). We verify that the 13 CO emission is generally optically thin for all four clouds. For example, Orion has the largest cloud mass and highest column density among the four clouds. The optical depth of 13 CO in Orion is less than 1 for 99.4% of the pixels (see also Kong et al. 2018). Only 0.6% of the pixels have τ 13 > 1, and these are mostly in the OMC-2/3 and L1641-N regions. The maximum value of τ 13 is 7.8.
RESULTS
Outflows Identified in the Full Map
In this section, we present the prediction by models ME1 and MF on the four star-forming clouds (Ophiuchus, Taurus, Perseus and Orion). Most molecular clouds have a global velocity gradient, which indicates there is no one unique central velocity of a cloud. To better visualize the blue-shifted and red-shifted lobes of outflows, we remove the large scale gradient to make local outflow motions clearer, i.e., shifting the velocity zero point to the flux-weighted central velocity of the spectrum for each pixel. We calculate the flux-weighted central velocity from the 1 st moment maps. In order to reduce the velocity fluctuations on small scales, we apply a Gaussian kernel with a FWHM that corresponds to 0.1 pc to convolve the 1 st moment maps. For each pixel, we show the integrated prediction over the channels that have absolute velocities greater than the central velocity by the specified thresholds.
3.1.1. Ophiuchus Figure 2 shows ME1 and MF model predictions for Ophiuchus. Both models identify a large amount of outflow activity in the region of the large, central star cluster. In contrast, the models identify much less outflow activity just north of the cluster, which has only a few YSOs. Figure 3 shows the predicted outflow activity by models ME1 and MF toward the young star cluster region Lynds 1688 in Ophiuchus. The L1688 region has over 120 young stars and a large number of interacting outflows. Model ME1, which performs similarly to human visual identification, identifies almost everything as feedback in this region. In contrast, Model MF provides an approach to disentangle the outflow emission from the ambient cloud emission. Quantitively, model ME1 predicts 81% of the mass in this region as outflows. However, model MF predicts that only 12% of the total mass is associated with outflows. Figure 3 shows that both models ME1 and MF are able to identify coherent high velocity structures in the position-velocity diagram.
It is challenging to separate outflow emission from the cloud emission in 12 CO (1-0) "by eye," since the cloud emission dominates the 12 CO (1-0) emission. Consequently, even an expert astronomer can only separate the high-velocity component from the host cloud emission but cannot identify the outflow morphology near the rest frame velocity of the cloud. Figure 4 and 5 show the ME1 and MF model predictions for Taurus. The structures in the outflow prediction maps are more discrete than those in Ophiuchus. One possible reason is that YSOs in Taurus are more sparsely distributed than those in Ophiuchus. Although the total number of YSOs in Taurus is slightly larger, they are less clustered. In Ophiuchus, the outflows driven by YSOs are more likely to interact and overlap with each other. While in Taurus, the outflows are more isolated. Figure 6 shows the outflow activity predicted by models ME1 and MF toward a previously identified outflow, TMO 06 (Li et al. 2015 Red contours indicate the integrated prediction over the channels that have velocities greater than V cen + 0.4 km s −1 for model ME1 and V cen + 0.2 km s −1 for model MF. Blue contours indicate the integrated prediction over the channels that have velocities smaller than V cen − 0.4 km s −1 for model ME1 and V cen − 0.2 km s −1 for model MF. "Y" and "O" indicates the location of YSOs, as described in Section 2.3. The contours start at 25th percentile of the sorted pixel values and end at 99.7th percentile of the sorted pixel values of the data, with 6 levels evenly spaced. It is worth noting that the absolute values of the contour levels for models ME1 and MF are different. The contour levels for the model ME1 prediction start at 2.8 K·km/s, for the blue-shifted lobes and 2.2 K·km/s, for the red-shifted lobes, and end at 19.3 K·km/s, and 15.3 K·km/s, for the blue-and red-shifted lobes, respectively. The contour levels for the model MF prediction start at 0.80 K·km/s, (blue) and 0.96 K·km/s, (red), and end at 7.2 K·km/s, (blue) and 8.6 K·km/s, (red). The method of plotting contours is the same for Figure 4 outflow. However, the model MF prediction is more extended in the position-velocity diagram.
Taurus
-23°00' 30' -24°00' 30' -25°00' 30' RA Dec O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O OO O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O OO O O O O O O OO O O O O OO O O O O OO O O O O O O O O OO OO O O O O O O O O O O O O O O O O O O O O O OOO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY YY YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y-23°00' 30' -24°00' 30' -25°00' 30' RA Dec O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O OO O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O OO O O O O O O OO O O O O OO O O O O OO O O O O O O O O OO OO O O O O O O O O O O O O O O O O O O O O O OOO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY YY YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YO O O O O OO O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y16
In high velocity channels, the 12 CO emission produced by outflows is usually faint while the fraction of outflow mass is high. Model ME1 only identifies the location of outflows and gives stronger emission a higher weight. In this subregion, model ME1 predicts that 43% of the mass in this region is associated with outflows. However, model MF predicts that only 8% of the total mass is outflows gas.
Perseus
Xu et al. (2020b) followed up previous visually identified outflow targets and validated the model performance. Here we extend that analysis by carrying out a "blind search" where we analyze the full cloud. Figure 7 which indicates outflows exist everywhere. This is consistent with the extremely dense distribution of YSOs in Orion. Figure 9 shows the predicted outflow activity by models ME1 and MF toward a previously identified outflow, Outflow No. 7 (Tanabe et al. 2019), in Orion. Both models similarly highlight the coherent high-velocity structures. There are several young YSOs located around this outflow, which could be the driving source. For reference, we illustrate the high- velocity components in blue and red contours, which are considered to be outflows by Tanabe et al. (2019). The mass of these high-velocity components is 268 M . Model MF predicts the outflow mass to be 308 M , which is similar to previous estimates. In this subregion, model ME1 predicts 61% of the mass in this region is associated with outflows. However, model MF predicts that only 13% of the total mass is outflow gas.
O O O O O O O O O O O O O O OO O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO OO O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O OO O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Blue: < V4 h 48 m 36 m 24 m 12 m 30°2 8°2 6°2 4°2 2°R A Dec O O O O O O O O O O O O O O OO O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO OO O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O OO O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Blue: < V
Physical Properties of Outflows
In this section, we study the physical properties of the outflows we identified and analyze the correlation between the physical quantities of outflows and the number of YSOs in the clouds. This analysis help us evaluate the robustness of the outflow identification by casi-3d. If the outflows are accurately identified by casi-3d, we should expect a linear correlation between the outflow mass and the number of YSOs. This is exactly what we find in this section and Section 3.3. Because our analysis aims to correlate the physical properties of outflows with the number of YSOs, we only calculate mass, energy and momentum of outflows in the source catalog covered area.
We follow the method in Section 2.4.3 to calculate the mass of outflows. We adopt the model MF estimates for the fiducial values. Figure 10 shows the mass estimates for the outflows in the four molecular clouds. When there are more YSOs in the region, the predicted outflow mass is higher. In Ophiuchus, on average, each YSO (including both young and old) contributes 0.43 M outflow mass to the host cloud.
While the values are 1.7 M , 2.8 M , and 3.4 M for Taurus, Perseus and Orion, respectively. On average, in each of the four clouds, the total mass associated with feedback is about 10% of the mass of the host cloud within the area studied.
We define the 1D (line-of-sight, LOS) momentum as the sum of the gas mass in each channel multiplied by the channel velocity, where we have shifted the mean cloud velocity to zero. To better quantify LOS momentum, we subtract the central velocity along each sightline to re- (Gutermuth et al. in prep). The contour levels for the model ME1 prediction start at 2.5 K·km/s (blue) and 2.1 K·km/s (red), and end at 17 K·km/s (blue) and 15 K·km/s (red). The contour levels for the model MF prediction start at 0.96 K·km/s (blue) and 0.90 K·km/s (red), and end at 8.6 K·km/s (blue) and 8.0 K·km/s (red).
3 h 48 m 42 m 36 m 30 m 24 m 33°3 2°3 1°3 0°R A Dec O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O OO OO OO O O O O O O OO O O O O O O O O OO O O O O O O O O O O O O OO O O O OO O O O O O O O O O O O O O O O O O O O O O O O O OOO OO O OO O O O O O O OO O O OOO O OO O O O O O O O O O OO O O O O O O OOO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OOO O O O O O O O O O OO O O O O O O O O O O O OO O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O OO O O O O O O O O O O OO O O O O OO OO O O OO O O O OO OO O O O O OO O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y YY YY Y Y Y Y Y Y33°3 2°3 1°3 0°R A Dec O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O OO OO OO O O O O O O OO O O O O O O O O OO O O O O O O O O O O O O OO O O O OO O O O O O O O O O O O O O O O O O O O O O O O O OOO OO O OO O O O O O O OO O O OOO O OO O O O O O O O O O OO O O O O O O OOO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OOO O O O O O O O O O OO O O O O O O O O O O O OO O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O OO O O O O O O O O O O OO O O O O OO OO O O OO O O O OO OO O O O O OO O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y YY YY Y Y Y Y Y Y
duce the effect of large velocity gradients across the entire cloud, as described in Section 3.1.
Subtracting the velocity gradient provides a better estimate of LOS momentum, which re- duces the impact of large-scale motions on our estimates. For comparison, we also calculate the LOS moment without subtracting the velocity gradient. The LOS momenta of the outflows and the host clouds are calculated by equations:
5 h 38 m 36 m 34 m -5°00' 30' -6°00' 30' -7°00' RA Dec Y YY Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y YY Y Y Y Y Y Y YY Y Y YY Y Y YY Y Y Y YY YYY Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y YY Y Y Y YY Y YY Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y YY Y Y YY Y YY Y Y Y Y Y Y Y Blue: < V-6°00' 30' -7°00' RA Dec Y YY Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y Y YY Y Y Y Y Y Y YY Y Y YY Y Y YY Y Y Y YY YYY Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y YY Y Y Y YY Y YY Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y YY Y Y YY Y YY Y Y Y Y Y Y Y Blue: < VP (V Gsub) = i,j,k M CO,i,j,k (v i,j,k − v i,j ), (1) P (V Gnonsub) = i,j,k M CO,i,j,k (v i,j,k − v global ).
(2) Similarly, we define the LOS kinetic energy as
E(V Gsub) = i,j,k 1 2 M CO,i,j,k (v i,j,k − v i,j ) 2 ,(O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y 5E(V Gnonsub) = i,j,k 1 2 M CO,i,j,k (v i,j,k − v global ) 2 .
(4) Figure 11 shows the LOS outflow momentum estimates for the four molecular clouds. We list two sets of momentum estimates, one that subtracts the velocity gradient and a second that does not subtract the velocity gradient. We adopt the momentum estimate by model MF, which subtracts the velocity gradients as the fiducial estimate. The trend between the momentum driven by outflows and the number of YSOs is similar to that of the mass estimates for four clouds. In Ophiuchus, the 1D momentum driven by outflows is around 10% of the total 1D momentum of the host cloud. In Taurus, this momentum ratio is 8%. In Perseus and Orion, the 1D momentum driven by outflows is around 16% of the total 1D momentum, which indicates there are likely more high veloc-ity structures identified as outflows. This is consistent with the star formation history of these clouds. Taurus has a relatively low star formation rate and mainly forms low mass stars. On the other hand, Ophiuchus and Orion are home to clusters forming higher-mass stars and have more active star formation. Perseus also contains some intermediate mass stars, including a couple of B-type stars (Arce et al. 2011).
Ophiuchus, Perseus and Orion have larger velocity gradients than Taurus, which indicates these clouds contain more complex cloud kinematic structure. These velocity gradients are likely caused by large scale processes, such as gas accretion (Klessen & Hennebelle 2010) or converging atomic gas flows caused by spiral density waves (Wolfire et al. 1995;Heitsch et al. 2006). Given the magnitude of these bulk motions, we expect that the momentum estimate that includes the velocity gradient overestimates the contribution of outflows, while the momentum that subtracts the velocity gradient provides a better estimate. Figure 12 shows the LOS outflow kinetic energy for the four molecular clouds. We adopt the kinetic energy estimates by model MF that subtracts the velocity gradient for the comparisons. In Ophiuchus, the LOS kinetic energy injected by outflows is around 6% of the total LOS kinetic energy of the host cloud. In Taurus, this energy ratio is 4%. In Perseus and Orion, the LOS kinetic energy injected by outflows is around 14% of the total LOS kinetic energy. Since kinetic energy is proportional to v 2 , one caveat here is that a small amount of mass located at high velocities likely dominates the kinetic energy estimates. However, due to observational limits, some high velocity gas emission vanishes into background noise, such that our kinetic energy calculation is likely an underestimate. As discussed in Xu et al. (2020b), this could potentially introduce a factor of two uncertainty.
Correlation Between the Outflow Properties and the Number of YSOs
In this section, we study the correlation between the total mass, momentum and energy associated with outflows and the number of YSOs. Due to the extended and overlapping nature of the outflow emission it is not possible to definitively associate particular sources with particular outflows. Instead, we consider an approach that independently counts the number of YSOs and the outflow impact in a region. We define a "window" to scan through the entire cloud region. We adopt window sizes of 0.5 pc×0.5 pc, 1 pc×1 pc and 2 pc×2 pc to scan Ophiuchus, and 2 pc×2 pc, 3 pc×3 pc and 5 pc×5 pc to scan Taurus and Orion. Since Perseus has the poorest physical resolution, and its full map is narrow, we only adopt window sizes of 2 pc×2 pc and 3 pc×3 pc to prevent the window exceeding the region. We examine the effect of our choice of window sizes in Appendix A. We set a scanning step size such that each box overlaps with its neighbors by at least 80%. We calculate the mass, momentum and energy inside the window and count the number of YSOs inside. We consider the outflow relationship between the number of both young YSOs and total YSOs. Figure 13 shows the correlation between the mass associated with feedback and the number of YSOs in the four clouds. The outflow mass grows linearly with the number of young YSOs for all four clouds. However, the clouds have different offsets, indicating that the outflow mass per young YSO varies. Orion has the most mass associated with outflows per young YSO. However, Orion also has a very large population of older YSOs, which may also drive outflows. The right panel of Figure 13 shows that the correlation between the outflow mass and the number of total YSOs is tighter, and most of the separation between different clouds disappears. There is a linear relation between the outflow mass and the number of all YSOs, where on average, each YSO is associated with one M of outflow material. The outflow mass directly launched by the protostar is expected to be 10-30% of the accreted gas, i.e., 10-30% of the star mass (e.g., Shu et al. 1988;Pelletier & Pudritz 1992). However, here our outflow estimate also counts entrained mass, which is much higher (e.g., Offner & Chaban 2017). Our results suggest one solar mass of outflow material per source. Assuming a one solar mass star for example, theoretical models for outflow launching predict that the outflow mass directly launched would be about 0.1-0.3 M (e.g., Shu et al. 1994;Bontemps et al. 1996). This implies that 0.7-0.9 M of the outflow gas we identify is entrained. This gives a mass-loading factor of 2.3-9. This is comparable to the mass-loading factor estimated from simulations of individual protostars and the av-erage core-mass-function to stellar IMF offset (Machida & Hosokawa 2013;Offner & Arce 2014;Offner & Chaban 2017). It is worth noting that our outflow estimate is larger than previous observational estimates. The main reason is that casi-3d is able to capture the outflow material that has relatively low velocities. The low-velocity outflowing gas entrains a significant amount of mass that also contributes to the momentum and kinetic energy of the host cloud, but it is not possible to separate this gas from the ambient gas by eye. Previous work mainly considered the high-velocity outflow mass only.
Younger sources power stronger outflows (Bally 2016), so we might expect a tighter relation between the outflow mass and the number of young YSOs. However, we see better correlation between outflow mass and the total YSO number. There are several possible explanations. First, low-mass sources, with age up to 3 to 4 Myr (Bally 2016) continue to launch outflows, albeit weaker ones. We discuss an example in Section 4.1, where an outflow is likely driven by old evolutionary stage YSOs. Second, although young YSOs have a higher massloss rate, the outflow mass is a combination of both mass-loss rate and the launching duration. The older YSOs likely have ejected a significant amount of gas by the time they are observed. In addition, our method may include outflow "relics." The dissipation timescale of outflow features produced by young stars is much longer than the lifetime of the driving source. These fossil outflows might remain even as their driving sources evolve to older evolutionary stages. Cunningham et al. (2006) found a similar scenario in numerical simulations, where fossil outflows retain speeds above the turbulent velocity for a timescale that is 10 times the duration of their driving source. Similarly, Offner & Chaban (2017) found in simulations that although the high accretion and active outflow phase lasts for ∼ 0.05 Myr, their impact on the velocity dispersion of the host clouds remains for several 0.1 Myr.
In our outflow mass estimate, a significant amount of the total outflow mass derives from emission in the cloud velocity channels (|v| 1 km s −1 ). This raises the concern that our results may be contaminated by dense, non-outflow material. In Appendix E we evaluate the effect of dense gas contamination on the derived outflow masses and YSO relations. We show that our outflow mass estimate is not significantly contaminated by dense gas.
We show the correlation between the outflow momentum and the number of YSOs in Figure 13. The trend is similar to that of outflow mass. On average, one YSO injects 1 M km/s. This indicates the mass weighted LOS velocity is 1 km/s, which is even smaller than the cloud turbulent velocity derived from 12 CO. This is because the 13 CO emitting region dominates the mass of the cloud and the outflows, which usually has a velocity dispersion only half that of 12 CO. For example, after subtracting the velocity gradients, the 12 CO velocity dispersion in Orion is ∼1.5 km/s, while that of 13 CO is only 0.8 km/s. The mass-weighted LOS velocity dispersion in Orion is 0.98 km/s. Model MF predicted outflow gas has a LOS velocity dispersion of 1.02 km/s in Orion. All these numbers are similar for the four clouds. Although the outflow 12 CO emission spans 2-3 km/s across the velocity channels, most of the mass is associated with gas within 0.8 km/s of the cloud mean velocity, i.e., the 13 CO emitting regions. Near the rest-frame velocity channels, the gas mass increases significantly but the fraction of gas associated with feedback drops. Consequently, there is a competition between these two effects. Xu et al. (2020b) examined the outflow mass located near the rest-frame velocity, which was ignored in previous outflow surveys. As shown in Figure 16 and 17 in Xu et al. (2020b), the restframe gas that is associated with the outflows accounts for almost 75% of the total outflow mass. Thus, the mass-weighted LOS velocity is relatively low, because high-velocity gas contributes only a small portion of the total outflow mass.
In addition, the resolution and low-signal-to-noise of the data makes detecting high-velocity outflow emission difficult. Consequently, the mass weighted LOS velocity of outflows is only 1 km/s for all four clouds. We show a position-velocity diagram of an outflow in 13 CO in Appendix D.
Our mass-weighted LOS outflow velocity of 1 km/s is low compared to typical outflow velocities estimated in previous works of 1-4 km/s (e.g. Arce et al. 2010). This is caused by the mass-weighting. In most cases, observers discard the rest-frame gas emission, whose outflow morphology is difficult to identify. This leads to a higher LOS velocity estimate. However, casi-3d is able to pick out the outflow emission from the confused ambient gas emission near the restframe velocity. Although the fraction of mass associated with feedback is low, perhaps a few percent, the total mass near rest-frame, especially in 13 CO emission channels, is significantly higher than that located at high-velocity channels. After correcting for the contamination, the outflow gas near the rest frame still dominates the total mass, which is consistent with the conclusion in Xu et al. (2020b). On the other hand, high-velocity gas is less dense and may be lost in the background noise. Thus, we likely underestimate the total outflow momentum by at least 10% Xu et al. (2020b).
The correlation between the outflow kinetic energy and the number of YSOs in the four clouds is similar to that of mass, as shown in Figure 13.
Quantifying the Impact of Feedback with Turbulent Statistics
Feedback, including stellar winds and outflows, injects kinetic energy into the host cloud. The input energy influences the shape of spatial power spectrum (SPS) of the integrated intensity map of 12 CO. Xu et al. (2020a) found that stellar wind-generated bubbles flatten the SPS of the 13 CO integrated intensity map, which indicates mass and energy are injected at small scales. In this section, we investigate how protostellar outflows affect the SPS of the 12 CO integrated intensity maps.
The SPS is defined as the square of the 2D Fourier transform of an image:
P(k) = | k|=k |M 0 ( k)| 2 = | ∞ −∞ ∞ −∞ M 0 ( x)e −2πj k x d x| 2 ,(5)
where M 0 is the 0 th moment (integrated intensity) of 12 CO. Figure 14 shows the SPS of Taurus where the emission is above 0.2 K (i.e., excluding noise) and the SPS of the model ME1 and MF predicted feedback regions. Figure 14 shows that the SPS of the feedback predictions breaks into two power laws. The break point corresponds to a physical scale of ∼ 0.5 pc, which might indicate the typical outflow mass and energy injection scale. For reference, we show the scale bar of this injection length in a subregion of Taurus in Figure 14. This outflow injection scale is comparable to the typical outflow size in Taurus. in the four clouds. The window sizes are 0.5 pc×0.5 pc, 1 pc×1 pc and 2 pc×2 pc for Ophiuchus, and 2 pc×2 pc, 3 pc×3 pc and 5 pc×5 pc for Taurus and Orion. The window sizes for Perseus are 2 pc×2 pc and 3 pc×3 pc.
We find similar broken power laws in Ophiuchus, Perseus and Orion. We list the SPS fits for all four regions in Table 3.4. For a given cloud, we find the model ME1 break point scale to be similar and usually slightly higher, than that of model MF. Table 1 also lists the median of the distance between young YSOs and their four nearest young companions.
There are several competing effects that likely influence the location of the break point. For isolated outflows, the break point size is close to the outflow physical scale. However, the typical outflow size is not necessarily the same in all regions. It is influenced by the gas density and how readily the outflow can expand, as well as the source mass and age. In very clustered regions, outflows interact with each other, since the separation between sources is smaller. This causes the outflow emission to blend together, which increases the break point.
The physical scale of the break point is small for Orion and Ophiuchus, and Taurus has a intermediate break point scale, while Perseus has the largest. Among the four clouds, Taurus has the most distributed and least clustered YSOs. Taurus also has the lowest average gas density. The mass/energy injection scale in Taurus reflects the physical size of individual molecular outflows. While in Orion and Ophiuchus, the average distance between stars is small, which increases the probability that outflows from different sources interact. The mean gas densities are also higher, and high external pressure from the surrounding gas may also act to limit the propagation of outflows. For example, Kirk et al. (2017) found all dense cores in Orion are pressure confined. All of these effects help to explain the relatively small mass/energy injection scale.
The physical scale of the break point in Perseus is significantly larger than that of the other regions. This might be due to both its typical density, which is lower than that of Orion and Ophiuchus, and the presence of multiple clusters in Perseus. Perseus hosts several intermediate mass star-forming clusters, such as NGC 1333 and IC348. Outflows in these clusters interact and blend together, such that mass/energy injection occurs on a relatively larger scale compared to individual stars. The average young YSO separation is slightly larger in Perseus than that in Orion and Ophiuchus, in part because the two main clusters contain few young YSOs. Consequently, the combination of these effects may explain the large mass/energy injection scale in Perseus.
To investigate how YSO clusters affect the SPS break point, we conduct a SPS analysis towards two star clusters, NGC1333 in Perseus and L1688 in Ophiuchus. We adopt different window sizes to explore the location of the break point in the cluster when viewed on different size scales. When the window size is smaller than the cluster size, we find there is no break point in the SPS for both clustered region and the slope is steeper. When the window size is two or more times larger than the cluster size, the break point appears. However, the break point scale is not neatly correlated with the size of the cluster nor the window size. NGC1333 has a break point between 0.6 pc and 0.7 pc, while L1688 has a break point scale between 0.3 pc and 0.4 pc. These scales are similar to those of the full predictions, which suggests that the outflows in the clustered regions are influencing the location of the overall break point.
We also explore the effect of different observation resolutions on the results. The observations of Ophiuchus, Taurus and Orion have a similar physical scale per pixel, 0.013 − 0.015 pc/pixel. However, the physical scale per pixel of Perseus is twice that of the others. We convolve the Ophiuchus and Orion observations to a similar effective resolution as Perseus, and apply the casi-3d models to these convolved data cubes and conduct the same SPS analysis on the convolved data. However, we find only minor changes in the break point. For example, the break point remains the same in the convolved Ophiuchus model MF prediction and increases by 0.1 pc in the convolved Ophiuchus model ME1 prediction. However, it decreases by 0.08 pc in the convolved Orion ME1 prediction, while increasing by 0.08 pc in the Orion MF prediction. We find no significant trend when convolving with a larger beam and conclude the analysis is relatively insensitive to the resolution. However, this test suggests that the break point has an uncertainty of ±0.1 pc, which is smaller than the difference between the break point of Perseus and those of the other clouds.
Finally, to test whether (non-feedback) gas structures influence the break point location, we compute the SPS for the dense regions of the clouds that are traced by 13 CO. We mask out the low intensity pixels to isolate only the dense gas. We compare the results for two clouds, Orion and Taurus. When all pixels with intensities below 5 K are removed, there is no break point in the Orion SPS. For the densest regions, i.e., as defined using an intensity cutoff of 10 K, we find a break point at 0.6 pc. This is two times larger than that of the break point of the outflow maps. We find a similar result in Taurus. When applying a high intensity cut-off, we find a break point around 1 pc, which is a sizescale twice that of the break point identified by the Taurus SPS outflow analysis. We conclude that our outflow predictions are not correlated with dense cloud structures.
DISCUSSION
Outflows without Driving Sources and False Detections
Confirming outflow identifications requires ancillary data, e.g., YSO catalogs. With the help of YSO locations, we are able to increase confidence in the casi-3d predictions, which are based on 12 CO data only. In this section, we discuss two cases predicted by casi-3d: a region in Taurus with no detected YSOs, and a region in Orion with no young YSOs. Figure 15 presents the channel by channel prediction by model MF on a region with no YSOs in Taurus. This region is considered to be a trans-Alfvenic flow regulated by magnetic fields (Heyer & Brunt 2012;Heyer et al. 2016), which has low surface brightness 12 CO emission. There are no known YSOs observed by GAIA or Herschel (Roccatagliata et al. 2020). Consequently, we believe there is no outflow activity. However, model MF predicts outflow activity in this region. Inspection of the channel map shows that a coherent jet-like structure exists across several velocity channels. These structures actually visually resemble outflows, which might be reason for the model failure. Figure 16 shows the model MF prediction along a position-velocity cut of the fake outflow in Figure 15. In the position-velocity diagram, we identify two faint coherent high velocity blobs between 4 and 5 km/s, Their morphology is similar to some of the faint outflows (e.g., TMO22 in Li et al. 2015). Since these 12 CO structures are indistinguishable from true outflows, casi-3d is not able to recognize these false detections based only on 12 CO morphology. In terms of morphology, casi-3d performs robustly in identifying coherent high velocity features that are similar to outflows. But many mechanisms may cause coherent high velocity features, including but not limited to cloud formation, cloud-cloud collision, gas phase transition near the cloud boundary or gas flows regulated by MHD waves (Heyer & Brunt 2012;Motte et al. 2014;Nakamura et al. 2014;Heyer et al. 2016). Figure 17 shows an example of the performance of models ME1 and MF toward a region where there are no young YSOs in Orion. In the position-velocity diagram, we identify a clear outflow-like feature above 10 km/s, which is also identified by the two models. We notice that there are several old evolutionary stage YSOs nearby, which might act as driving sources. One reason might be that the boundary between young and old is not so well defined, which is discussed in Section 3.3. Or these outflow structures might exist for a much longer timescale than that of their driving sources. We recognize that there are two velocity components in the molecular cloud at this position. A narrow gas bridge located around 8 km/s connects these two gas components, which is likely an outflow. We cannot confirm the origin of this high velocity component. It might be caused by the fossil outflows from YSOs or by two converging gas flows. All these mechanisms cause similar high-velocity features, which cannot be easily distinguished either visually or using casi-3d.
O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Blue: < V
In order to validate that our casi-3d models are sensitive to gas velocity structures and morphology but not only the dense regions, we examine the casi-3d performance on several previously identified filaments in Taurus. Panopoulou et al. (2014) identified 10 filametary structures in 13 CO emission in Taurus, and illustrated the PPV diagrams of the filaments in Figure 15 in their paper. The filaments have very coherent motions and exhibit small velocity dispersions. Their filaments 3, 4, 5, 6, and 7 do not have significant high-velocity components. When comparing with our outflow predictions (Figure 4 and 5 in this work) and bubble predictions (Figure 17 and 18 in Xu et al. 2020a), we find that both models pre- dict little feedback along these filaments. However, when we look at Filament 2 (also known as L1495/B213), we see a clear high-velocity structure with a "U" shape. This morphology is consistent with the theoretical bubble morphology shown in Figure 5 in Arce et al. (2011). Unsurprisingly, both casi-3d models that identify bubbles and outflows predict the presence of feedback at the location of filament 2. casi-3d may return a false detection if the morphology of a structure is similar to that of outflows or bubbles. In this case, visual inspection suggests that filament 2 also contains a bubble, so it is likely that the feedback identified by our models at this location is real. We emphasize that machine learning models are not perfect tools. They must be applied with caution and checked thoroughly.
Outflows versus Bubbles
In this section we investigate the overlap between a model identifying outflow feedback and one identifying stellar wind feedback. We present the prediction by two sets of models: those trained to identify protostellar outflows and those trained to identify stellar wind driven bubbles.
The morphology of bubbles is more symmetric and arc-like compared to that of outflows. However, when the line of sight is parallel to an outflow launching axis, i.e., looking through the cavity of the outflow, the morphology of the outflow can resemble that of a bubble. Figure 18 illustrates an example of a synthetic outflow whose launching axis is parallel to the line of sight. Consequently, models trained to identify outflows are likely to also identify some bubble structures. Figure 19 shows the prediction by models trained to identify stellar wind driven bubbles towards Ophiuchus L1688 Region. Comparison with Figure 3 shows that the prediction by models trained to identify protostellar outflows covers most of the regions that are identified as bubbles. Quantitively, 68% of the voxels are identified as feedback by both MF models, 21% of the voxels are identified as only outflow feedback, while 11% of the voxels are identi-5.3-5.5km/s 5.5-5.8km/s 5.8-6.0km/s 6.0-6.3km/s 6.3-6.5km/s 6.5-6.8km/s fied as only belonging to bubbles. Statistically, we might neglect 13% of the feedback gas associated with stellar winds when adopting the mass estimates predicted by models trained to identify protostellar outflows. On average, the outflow mass predicted by models can serve as a decent estimate for feedback mass.
To better investigate the difference between outflows and bubbles, we employ an unsu-pervised learning method t-SNE (t-distributed Stochastic Neighbor Embedding) to study the similarities between the spectra of outflows and that of bubbles. We discuss t-SNE and the results in detail in Appendix B. Our main finding is that there is a variation in spectra shape for outflows and bubbles. And even the spectra that are dominated by outflow gas show various spectral shapes. Meanwhile, there are some "transitional spectra" with a shape similar to both. We note that the t-SNE representation encodes only 1D information about the spectra and does not represent any correlations in the spatial dimension. Consequently, this analysis provides intuition for understanding the features that the models select when they identify feedback, and it illustrates the challenge of distinguishing outflows and bubbles based only on the LOS spectrum.
Impact of Outflows on Molecular Clouds
In this section, we place our energy estimations (Section 3.2) in the context of prior work and discuss the broader implications of the global impact of outflows. Most prior work derives the total outflow energy by assuming a moderate inclination angle and multiplying the 1D energy estimate by a factor of 3. For example, Dunham et al. (2014a) adopted an inclination angle of 57 • .3 which yields a energy correction factor of 3.4. Here we make the same assumption and report the 3D total, where we multiply our 1D estimate by a factor of 3.
To investigate the impact of outflows on turbulence, a typical approach is to compare the turbulence dissipation rate and the outflow energy injection rate (e.g., Arce et al. 2010; Feddersen et al. 2020). However, the outflow energy injection timescale is highly uncertain. As discussed in Section 3.3, we find the outflow mass is linearly correlated with the total number of YSOs instead of only the younger sources, which indicates that the driving timescale of outflows is even less clear. Instead, we compute the total outflow kinetic energy -a quantity estimated as part of previous analyses in the determination of the outflow injection rate -and compare our new values to those in the prior published studies. This allows us to conclude whether the impact of outflows is relatively similar to, more or less than that in prior analyses. Nakamura et al. (2011) conducted observations of 12 CO (3-2) and 12 CO (1-0) towards an active cluster-forming clump in Ophiuchus and identified six molecular outflows in both data. Nakamura et al. (2011) derived the outflow mass from 12 CO (3-2) and 12 CO (1-0) emission located in the high-velocity line wings. They derived a total outflow kinetic energy of 6 × 10 44 ergs. After comparing the total outflow energy injection rate with the dissipation rate of the supersonic turbulence, Nakamura et al. (2011) concluded that outflows inject signifi-0.5 pc 12 CO Slice Tracer Integrated 12 CO Integrated Tracer Figure 18. A synthetic outflow whose launching axis is parallel to the line of sight.
cantly more energy than needed to offset the dissipation of turbulence. In our work, we find that the total outflow kinetic energy is 9 × 10 45 ergs, which is an order of magnitude larger than the value in Nakamura et al. (2011). A larger area coverage in our work and using the optically thin tracer 13 CO in mass estimates might explain this difference. 12 CO (3-2) and 12 CO (1-0) are likely optically thick, which could cause their outflow mass, momentum and energy to be underestimated. More importantly, our method does not discard the outflow material ejected perpendicular to the line of sight that is located near the cloud central velocity. Since our energy calculation leads to a higher energy estimate, our result confirms the conclusion that outflow kinetic energy is sufficient to compensate for the dissipation of turbulence in Ophiuchus. Li et al. (2015) conducted a feedback census in Taurus and identified 55 outflows and 37 bubbles. They derived a total outflow energy of 3.9 × 10 45 ergs, which is 1% of the Taurus cloud turbulent energy. However, Li et al. (2015) found that bubbles dominate the feedback en-ergy. When also combining the kinetic energy from bubbles, they found the total kinetic energy from stellar feedback in Taurus is 30% of the cloud turbulent energy. Since the energy injection rate from stellar feedback is comparable to the dissipation rate of the cloud turbulence, they concluded that stellar feedback, mainly in the form of bubbles, is sufficient to maintain turbulence in the current epoch. However, Xu et al. (2020a) showed that the Li et al. (2015) bubble energy estimate was substantially overestimated due to LOS contamination, which is caused by gas emission that is not associated with feedback being included in the outflow/bubble estimate. After correction, the kinetic energy from bubbles decreased by a factor of four. In our current work, we find the total outflow kinetic energy is 5.4 × 10 46 ergs, which is an order of magnitude larger than the value in Li et al. (2015). This difference is likely caused by a similar reason to that in Ophiuchus, i.e., our method includes the outflow material located around the cloud central velocity that is excluded in traditional approaches. Our calculation indicates that the kinetic energy from outflows is 14% of the cloud turbulent energy. If we combine the contribution from bubbles (Xu et al. 2020a), the total kinetic energy from stellar feedback is 22% of the cloud turbulent energy. Li et al. (2015) concluded that the outflow energy injection rate is marginally comparable to the dissipation rate of turbulence in Taurus, although outflow energy is only 1% of the Taurus cloud turbulent energy. With our updated feedback energy, which is 22% of the cloud turbulent energy, we conclude that the feedback energy injection rate can compensate for the turbulent dissipation rate (Li et al. 2015;Xu et al. 2020a). Arce et al. (2010) identified 60 outflow candidates in Perseus and derived a total outflow energy of 2×10 46 ergs, which is 12.5% of the total cloud turbulent energy. However, it is worth noting that the method of calculating cloud turbulent energy used in Arce et al. (2010) is different from our method. Arce et al. (2010) adopted an average line width of 2 km/s to calculate the turbulent energy. While in this work, we calculate the turbulent energy channel by channel followed by Equation 3. The cloud turbulent energy derived by Arce et al. (2010) is 1.6 × 10 4 7 ergs, while our approach indicates an estimate of 8×10 4 7 ergs, which is a factor of 5 larger than the estimate by Arce et al. (2010). To make a fair comparison, we adopt our turbulent energy estimate to discuss the impact of outflows in Perseus. Under these circumstances, the kinetic energy of the previously identified outflows are only 2.4% of the total cloud turbulent energy. Xu et al. (2020b) applied casi-3d to the same outflow catalog and predicted a total outflow energy of 7.8×10 45 ergs, which is one fourth of the value in Arce et al. (2010). This is caused by the lower mass-weighted LOS velocity as discussed in Section 3.3. In our present work, we derive a total outflow energy of 1.2 × 10 47 ergs, which is 6 times larger than that in Arce et al. (2010). There are several possible reasons for this higher value. First, there are newly identified isolated outflows as discussed in Xu et al. (2020b). It should be noted that Xu et al. (2020b) just followed up the 60 previous outflows and identified a few new outflows in the vicinity of these, whereas our current work make a prediction using the full CO map, which includes the entire cloud and a couple of clusters. This likely increase the outflow kinetic energy by a factor of a few. Second, casi-3d is able to identify all feedback structures in clustered regions, which are not fully identified in previous work. This also may enlarge the outflow energy by a factor of a few. Moreover, outflows identified by Arce et al. (2010) are all distinct outflows with obvious coherent high-velocity structures that are likely powered by strong driving sources. However, in our work, casi-3d also identifies less distinct outflows whose velocity is small compared to previously identified ones. These weaker, less distinct outflows might be driven by more evolved or lower mass sources. This indicates that casi-3d provides a more inclusive outflow sample. Arce et al. (2010) concluded that the 60 outflow candidates they identified play an important role in maintaining turbulence at the current epoch in Perseus, although the kinetic energy of outflows is 12.5% of the total cloud turbulent energy in their study, or 2.4% of the total cloud turbulent energy in our turbulence calculation. However, our updated outflow kinetic energy estimate is 15% of the cloud turbulent energy. This indicates that outflows still play an important role in maintaining turbulence in Perseus. Feddersen et al. (2020) identified 45 outflows near Herschel Orion Protostar Survey (HOPS) protostars (Furlan et al. 2016) in Orion but skipped the OMC-1 region. They derived a total outflow energy of 0.7 − 1.7 × 10 46 ergs. They found that the total outflow energy injection rate is comparable to the dissipation rate of turbulence. In our work, we derive a total outflow energy of 2.1 × 10 47 ergs, which is an order of magnitude larger than previous estimates. The main reason for the difference is that our survey covers a larger area and does not only focus on gas near the position of YSOs. In particular, we include the most active star forming region OMC-1. casi-3d predicts the kinetic energy from outflows in OMC-1 is 7.7 × 10 46 ergs, which is 37% of the entire outflow energy in Orion. Thus, we confirm the prior conclusion that outflows are sufficient to maintain turbulence in Orion.
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In conclusion, our outflow kinetic energy estimates for all four clouds are larger than previous estimates. Consequently, we confirm the previous conclusion that outflows are an important agent to maintain turbulence at the current epoch.
Uncertainties of Outflow Estimates
In Section 3.2, we calculate the masses and dynamical properties of the outflows for the four clouds. In this section, we attempt to quantify the uncertainty of these outflow estimates.
First, we assume a moderate excitation temperature of 25 K or the 12 CO peak temperature, whichever is higher, to calculate the outflow mass. Most studies adopt values of excitation temperature in the range of 10-50 K (Dunham et al. 2014a;Feddersen et al. 2020). Since the mass scales linearly with the excitation temperature, the choice of excitation temperature introduces a factor of two uncertainty.
Meanwhile, Xu et al. (2020b) found that model MF predicts the outflow mass within a scatter of 0.41. The most extreme offset case over-/under-estimates the mass by a factor of two. We did not find any offset that would cause a systematic over or under-estimation. Meanwhile, we carry out one additional test to constrain the degree of possible contamination from cloud emission at rest-frame gas velocities. Appendix F shows an analysis of a more complex synthetic observation, which contains a number of interacting outflows. We conclude our casi-3d model is generally able to exclude contamination from the ambient gas near the rest frame velocity and may even underestimate the outflow contribution from low velocities in complex star-forming areas.
Furthermore, high-velocity, low-density gas emission is easily missed in high-velocity channels due to the low signal-to-noise. Xu et al. (2020b) used the simulations to estimate that 10% of the outflow gas is lost. The correction factors for LOS momentum and LOS energy are 1.3 and 1.8 (Xu et al. 2020b), respectively.
In addition, the inclination angle of the outflow plays an important role in converting LOS momentum/energy to 3D momentum/energy. In this work, we adopt correction factors of √ 3 and 3 to convert LOS momentum and LOS energy to 3D estimates. These correspond to an inclination angle of 55 • . If the inclination angle is between 20 • and 70 • , this leads to a factor of 2 uncertainty for 3D momentum and a factor of 4 uncertainty for 3D energy.
Finally, chemical conditions, i.e., 12 CO and 13 CO abundances, have an influence on the mass estimate. In our mass calculation, we adopt constant abundance ratios, [ 12 CO]/[H 2 ]=10 −4 and [ 12 CO]/[ 13 CO]=62. However, these abundance ratios vary across diffident clouds. We cannot evaluate this uncertainty without sophisticated chemical modeling for each cloud.
Consequently, by combining these uncertainties and using error propagation, we find a factor of 3 uncertainty for mass, a factor of 3.4 uncertainty for 3D momentum, and a factor of 5 uncertainty for 3D energy.
CONCLUSIONS
We apply the deep learning method casi-3d to four nearby molecular clouds, Ophiuchus, Taurus, Perseus and Orion, to systemically identify protostellar outflows and study the impact of outflows on their host clouds. Our main findings are the following: In this section, we examine the effect of our choice of "window" sizes when studying the correlation between the total mass associated with outflows and the number of YSOs. We define windows with physical sizes of 0.5 pc×0.5 pc, 1 pc×1 pc, 2 pc×2 pc, 3 pc×3 pc and 5 pc×5 pc to scan through the entire cloud region. We set a step size such that each box overlaps with its neighbors by at least 80%. Then we calculate the mass inside the window and count the number of YSOs enclosed. We consider the outflow relationship between the number of both young YSOs and total YSOs. Figure 20 shows the correlation between the model MF outflow prediction and the number of young YSOs for the five window sizes applied to the four clouds. For Ophiuchus, Perseus and Orion, when the window size is above certain threshold, the correlations between outflow mass and the number of YSOs become similar, which indicates the choice of window size does not significantly change the results. Taurus is more sensitive to the window size than the other regions. This is likely caused by the large separations between YSOs. However, it still shows a clear trend of smaller separation between different correlations when the window size gets larger.
The trends are similar when considering total YSOs. Consequently, we conclude that the study of the correlation between the outflow properties and the number of YSOs is robust.
B. SPECTRA OF OUTFLOWS AND BUBBLES
In this section, we employ an unsupervised learning algorithm t-SNE (t-distributed Stochastic Neighbor Embedding) to compare the spectra identified as outflow feedback to those identified as belonging to bubbles. t-SNE is a tool to visualize high-dimensional data by converting the similarities between data points into a low-dimensional manifold. It has been used for classification and outlier detection in a variety of astronomy data (Jofré et al. 2017;Reis et al. 2018;Lochner & Bassett 2020;Fluke & Jacobs 2020). For example, Reis et al. (2018) successfully applied t-SNE to cluster stars based on their spectra and identify outliers based on the t-SNE map.
We take the L1688 star cluster region in Ophiuchus as an example. Figures 3 and 19 show the prediction of this star cluster region by models trained to identify only outflows and only bubbles, respectively. In some cases the models clearly identify only one or the other type of feedback, however, there is also significant overlap. In order to classify the spectra, we calculate outflow mass and bubble mass of each spectra. If the outflow mass or bubble mass for a given line of sight is above 5σ compared with the background noise, we consider that spectrum to contain feedback. If the outflow mass is larger than the bubble mass, we label it an outflow spectrum. If the bubble mass is larger than the outflow mass, we label it a bubble spectrum. If there is emission but neither outflow mass nor bubble mass exceeds the 5σ threshold, or the spectrum has less than three channels that contain feedback, we label this spectrum as non-feedback. We neglect any spectra that have no emission above 3σ sigma. We use the spectra of the L1688 region at each line of sight as input to t-SNE. t-SNE projects the higher dimensional data into a lower dimensional (2D) space and organizes it based on similarities (e.g., the line shape and intensity of a spectrum) in the high-dimension space. Figure 21 shows the t-SNE clusters. We color the data points based on the casi-3d predictions, where spectra predicted to be primarily associated with outflows are in red, those primarily associated with bubbles are in blue and non-feedback in black. It is clear that the spectral shape for bubbles and outflows are different. The spectra of bubbles show two Gaussian components with two peaks of different intensities. While the spectra of outflows have various shapes, e.g., broader line wings, two Gaussian components but with almost equal peaks. It's worth noting that broad line wings are often used to visually identify outflows (Li et al. 2015;Tanabe et al. 2019). t-SNE separates some bubble spectra from the outflow spectra. Meanwhile, we notice that there are some blended/transiting regions, where outflow spectra and bubble spectra are clustered together. In these cases, we find that bubble and outflow identifications have similar spectral shapes. This demonstrates that we cannot distinguish outflows and bubbles based only on single spectra. There is also a mixed region where non-feedback spectra are blended with outflow spectra. The spectral shape of this region indeed show less outflow features, where there is a faint broad line-wing on the blue shifted side. This analysis explores the similarity between outflow and bubble feedback using an unsupervised machine learning approach. In some cases, the casi-3d predictions are clearly distinct. In other cases, where both models identify feedback emission, it is clear that the spectra are truly similar. This may be because both types of feedback can produce such velocity features or because such regions genuinely contain both types of feedback. In the current analysis it is not possible to distinguish between these two possibilities. However, this t-SNE clustering gives confidence that feedback is shaping these spectra and that casi-3d is correctly identifying feedback signatures. Figure 21. t-SNE map of the 12 CO spectra in the L1688 star cluster region in Ophiuchus. Red dots indicate outflow spectra. Blue dots indicate bubble spectra. Black dots indicate non-feedback spectra. The sub-plots show the average spectra of each subregion.
Bubble Outflow
C. YSOS IN ORION
Orion has an extensive population of older YSOs, so in Figure 8 we plot only the young YSOs for readability. Here, Figure 22 shows the location of both young and old YSOs in Orion. While we expect younger YSOs to contribute more feedback per source, the large number of older YSOs indicates that they likely contribute feedback throughout the cloud, and this is consistent with the prediction in Section 3.1.4.
D. OUTFLOWS IN 13 CO
In some regions 12 CO is optically thick and we combine both 12 CO and 13 CO to derive the kinematic estimates rather than using 12 CO only (e.g., Arce et al. 2010 Figure 22. Intensity of 12 CO integrated over all velocity channels for Orion. Letters "Y" and "O" mark YSO positions, as described in Section 2.3.
-7°00' RA Dec Y O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OOO O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O OO O O O O O O O O OO O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O OO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
where there is 13 CO emission dominate the mass in each spectrum. Here we compare the identification from the 12 CO map to the 13 CO emission. Figure 23 shows the 13 CO emission of the outflow in Figure 9. We can clearly see the velocity range of 13 CO is substantially smaller than that of 12 CO. However, the mass in the 13 CO emitting region dominates the total mass of the cloud. This also explains why the mass weighted LOS velocity is smaller than the velocity calculated based on 12 CO only. Figure 9 but where the background colorscale indicates the 13 CO integrated intensity. White contours show the prediction by models ME1 and MF on the 12 CO data.
O O O O O O O O O O O O O O O O O O O O Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y5
E. CORRELATION BETWEEN THE GAS MASS AND THE NUMBER OF YSOS
In this appendix, we examine the relationship between the number of YSOs, the total gas mass, and the dense gas mass. Prior work has shown that the number of YSOs is well-correlated with the gas column density, which is Σ * ∝ Σ 2 gas , or M gas ∝ N 0.5 * (e.g., Gutermuth et al. 2011;Pokhrel et al. 2020Pokhrel et al. , 2021, where the gas mass was estimated from dust emission. Pokhrel et al. (2020Pokhrel et al. ( , 2021 found that the surface density of protostars (i.e., younger YSOs) are tightly correlated with gas column following the relation above. Gutermuth et al. (2011) found a similar but less tight relation when considering all YSOs. Consequently, we expect YSOs and dense gas to be correlated. Figure 24 shows a nearly linear relation between the total mass and the number of all YSOs. We find the slope is 0.93, which is slightly smaller than the slope of the outflow mass-YSO relation, 1.03.
To check whether the outflow mass estimate is correlated with the dense gas due to contamination, we examine the relation excluding the rest-frame velocity gas, which is located within the turbulent velocity range (±1 km/s). Figure 25 shows the correlation between the total high velocity gas (|v − v cen | > 1 km/s) mass and the number of all YSOs, and the correlation between the high velocity outflow mass (|v − v cen | > 1 km/s) and the number of all YSOs. The slopes are similar to that of the total gas/outflow gas relation including the rest-frame velocity gas. Therefore, we conclude the correlation between gas mass and YSOs exists because outflows are linearly proportional to the number of YSOs launching them. YSOs form from dense gas and thus by nature are more likely to be located in denser regions. By focusing only on the high-velocity material, which traces the outflows most directly, we show that the trend arises directly from the expected correlation between YSOs and outflows rather than indirectly from contamination by the dense gas. This result provides further evidence that our method performs well in dense regions where contamination from the cloud would otherwise be a serious problem. In this appendix, we examine a prediction by the two casi-3d models on a synthetic observation of a simulation snapshot with a number of interacting outflows. This synthetic observation and the simulation it derives from is not included in our previous training or testing, so it constitutes a more challenging test of the model performance. This simulation is run using a different initial turbulent seed and with twice the inital gas density as the other simulations in the training set. The mean magnetic field strength is 0.8 µG. The evolutionary time of the snapshot is 0.7 t ff . There are 13 stars launching outflows in the simulation box, whereas our training set is constructed from snapshots with 1 to 5 stars. Consequently, this output provides an independent check on the casi-3dmodel performance. Figure 26 shows the integrated CO emission, tracer fractions and predictions. We find the casi-3d model does not actually over-predict the amount of emission in the cloud velocity channels but instead slightly under-predicts the location and mass of the outflows. More quantitatively, model MF identifies 76% of the outflow emission at high-velocity channels (above 2km s −1 ). However, model MF only identifies 40% outflow emissions near the rest-frame velocity (within 2 km s −1 ). This implies casi-3d might underestimate the outflow mass by a factor of 2. This is consistent with the model uncertainty discussed in Section 4.4. This analysis gives further confidence that casi-3d correctly excludes contamination from the ambient gas near the rest frame velocity, even in relatively clustered and complex regions.
Figure 1 .
1Cumulative distribution function (CDF) of the normalized emission for the four clouds.
Figure 2 .
2Intensity of 12 CO (1-0) integrated over all velocity channels for Ophiuchus, overlaid with the model ME1 prediction (upper panel in red and blue contours) and with the model MF prediction (lower panel in red and blue contours).
, 5, 7 and 8. this outflow. Both models are able to iden-tify the blueshifted and redshifted lobes of this
Figure 3 .
3Position-velocity diagram of 12 CO emission toward the Ophiuchus L1688 region. Left panel: integrated intensity of 12 CO over the the full velocity range (from -0.8 km/s to 7.5 km/s) overlaid with the model ME1 and MF predictions in white contours. Letters "Y" and "O" mark YSO positions, as described in Section 2.3. The purple line illustrates the cut direction of the position-velocity diagram. Middle and right panel: position-velocity diagram of 12 CO emission overlaid with the model ME1 and MF predictions in white contours.
Figure 4 .
4Intensity of 12 CO (1-0) integrated over all velocity channels for Taurus, overlaid with the model ME1 prediction (red and blue contours). Letters "Y" and "O" mark YSO positions, as described in Section 2.3. The grey line encloses the Spitzer coverage, where YSOs are identified(Rebull et al. 2010). The contours start at the 25th percentile of the sorted pixel values (0.47 K·km/s, for blue contours and 0.49 K·km/s, for red contours) and end at the 99.7th percentile of the sorted pixel values of the data (2.9 K·km/s, for blue contours and 3.4 for red contours), with 6 levels evenly spaced. Note that the absolute values of the contour levels for models ME1 and MF are different. much of this emission is cloud contamination and excludes it.3.1.4. OrionFigure 8 shows the model ME1 and MF predictions for Orion. Due to the large number of old evolutionary stage YSOs in Orion, we only show the younger YSOs to reduce confusion. We show the position of all Orion YSOs in Figure 22 in Appendix C. The prediction by both models covers most of the emission in Orion,
Figure 5 .
5Intensity of 12 CO (1-0) integrated over all velocity channels for Taurus, overlaid with the model MF prediction (red and blue contours). Letters "Y" and "O" mark YSO positions, as described in Section 2.3. The grey line encloses the Spitzer coverage, where YSOs are identified(Rebull et al. 2010). The contours start at the 25th percentile of the sorted pixel values (0.27 K·km/s, for blue contours and 0.27 K·km/s, for red contours) and end at the 99.7th percentile of the sorted pixel values of the data (1.7 K·km/s, for blue contours and 1.5 for red contours), with 6 levels evenly spaced.
Figure 6 .
6Position-velocity diagram of 12 CO emission toward a previously identified outflow, TMO 06(Li et al. 2015), in Taurus. Left panel: integrated intensity of 12 CO over the the full velocity range (from -1.5 km/s to 13.4 km/s) overlaid with the model ME1 and MF predictions in white contours. Letters "Y" and "O" mark YSO positions, as described in Section 2.3. The purple line illustrates the cut direction of the position-velocity diagram. Middle and right panel: position-velocity diagram of 12 CO emission overlaid with the model ME1 and MF predictions in white contours.
Figure 7 .
7Intensity of 12 CO (1-0) integrated over all velocity channels for Perseus, overlaid with the model ME1 prediction (upper panel in red and blue contours) and with the model MF prediction (lower panel in red and blue contours). Letters "Y" and "O" mark YSO positions, as described in Section 2.3. The grey line encloses the Spitzer coverage, where YSOs are identified
Figure 9 .
9Position-velocity diagram of 12 CO emission toward a previously identified outflow, Outflow No. 7(Tanabe et al. 2019), in Orion. Left panel: integrated intensity of 12 CO over the the full velocity range (from 2.3 km/s to 17.1 km/s) overlaid with the model ME1 and MF predictions in white contours. The blue and red contours indicate the high velocity components, integrated over the raw emission with velocity ranges of 2.3-9.5 km s −1 and 12.3-17.0 km s −1 , respectively. Letters "Y" and "O" mark YSO positions, as described in Section 2.3. The purple line illustrates the cut direction of the position-velocity diagram. Middle and right panel: position-velocity diagram of 12 CO emission overlaid with the model ME1 and MF predictions in white contours.
Figure 10 .
10Outflow mass (solid green colors) and the number of YSOs (hatched blue and red) in the four molecular clouds. Numbers in the hatched bars indicate the number of YSOs. Numbers in the unhatched bars indicate the gas mass in units of 10 3 M . The total mass indicates the mass of the host cloud within the area studied.
Figure 11 .
11Outflow line-of-sight momentum (solid colors) and the number of YSOs (hatched colors ) in the four molecular clouds. Numbers in the hatched bars indicate the number of YSOs. Numbers in the unhatched bars indicate the gas momentum in units of 10 3 M km s −1 . The total momentum indicates the momentum of the host cloud within the area studied.
Figure 12 .
12Outflow line-of-sight kinetic energy (solid colors) and the number of YSOs (hatched colors) in the four molecular clouds. Numbers in the hatched bars indicate the number of YSOs. Numbers in the unhatched bars indicate the kinetic energy in units of 10 46 ergs. The total energy indicates the energy of the host cloud within the area studied.
Figure 13 .
13Correlation between the mass (top row)/momentum (middle row)/energy (bottom row) associated with feedback and the number of young YSOs (left panel) or the number of all YSOs (right panel) in the four clouds. The filled circles indicate the total outflow mass (top row)/momentum (middle row)/energy (bottom row) and the total number of YSOs (young YSOs in the left panel, all YSOs in the right panel)
Figure 14 .
14Upper panel: intensity of 12 CO (1-0) integrated over all velocity channels for a subregion in Taurus, overlaid with the model MF prediction (red and blue contours). The physical scale of the break point in the SPS is also shown for reference. Lower panels: the spatial power spectrum (SPS) of the regions in Taurus where the emission is above 0.2 K (excluding noise) and the SPS of the model ME1 and MF predicted feedback regions. The top right number in each panel indicates the slope of the fit. The middle bottom number in each panel indicates the physical scale of the broken point.
Figure 16 .
16Position-velocity diagram of 12 CO emission toward a region with no YSOs in Taurus. Left panel: integrated intensity of 12 CO over the the full velocity range (from -1.5 km/s to 13.4 km/s) overlaid with the model ME1 and MF predictions in white contours. The purple line illustrates the cut direction of the position-velocity diagram. Middle and right panel: position-velocity diagram of 12 CO emission overlaid with the model ME1 and MF predictions in white contours.
Figure 17 .
17Position-velocity diagram of 12 CO emission toward a region without young YSOs in Orion. Left panel: integrated intensity of 12 CO over the full velocity range (from 2.3 km/s to 17.1 km/s) overlaid with the model ME1 and MF predictions in white contours. Letters "O" mark old YSO positions. The purple line illustrates the cut direction of the position-velocity diagram. Middle and right panel: position-velocity diagram of 12 CO emission overlaid with the model ME1 and MF predictions in white contours. et al. 2015;
Figure 19 .
19Position-velocity diagram of 12 CO emission towards Ophiuchus L1688 Region, similar toFigure 3but predicted by models trained to identify stellar wind driven bubbles.
Figure 20 .
20Correlation between the mass associated with feedback and the number of young YSOs. The different colors indicate different window sizes for the estimation. The filled circles indicate the total outflow mass and the total number of young YSOs in the four clouds.
Figure 23 .
23Same as
Figure 24 .
24Left: correlation between the total mass and the number of all YSOs. Right: correlation between the outflow mass and the number of all YSOs. F. casi-3d PREDICTION ON A COMPLEX SYNTHETIC OBSERVATION
Figure 25 .
25Left: correlation between the total high velocity gas (|v − v cen | > 1 km/s) mass and the number of all YSOs. Right: correlation between the high velocity outflow mass (|v − v cen | > 1 km/s) and the number of all YSOs.
Figure 26 .
26Synthetic 12 CO observations and casi-3d prediction in different velocity channels.
shows the full map predictions for both models. Both model ME1 and MF predictions are more concentrated towards star clusters. The model ME1 prediction is more spatially extended than that by model MF. This might be caused by a strong emission region with a low outflow mass fraction, where model ME1 counts the entire voxel as feedback while model MF recognizes that4 h 48 m
36 m
24 m
12 m
30°2
8°2
6°2
4°2
2°R
A
Dec
cen 0.3 km/s Red: > V cen + 0.3 km/s MFFigure 8. Intensity of 12 CO integrated over all velocity channels for Orion, overlaid with the model ME1 prediction (left panel in red and blue contours) and with the model MF prediction (right panel in red and blue contours). Letters "Y" mark young YSO positions, as described in Section 2.3. The contour levels for the model ME1 prediction start at 2.1 K·km/s (blue) and 2.6 K·km/s (red), and end at 14 K·km/s (blue) and 18 K·km/s (red). The contours levels for the model MF prediction start at 0.48 K·km/s (blue) and 0.55 K·km/s (red), and end at 4.3 K·km/s (blue) and 4.9 K·km/s (red).
Table 1 .
1Fitting Results of the Spatial Power Spectrum Single power-law fit results for the spatial power spectra for the emission regions (excluding noise). b Broken power-law fit results for the spatial power spectra applied to the ME1 and MF feedback maps. c Median of the separation between YSOs and their four nearest companions.Cloud
Full Map a
ME1 b
MF b
Young YSO
Slope
Slope
Break (pc)
Slope
Break (pc) Seperation c (pc)
Ophiuchus -3.17±0.02
-2.47±0.05
0.36
-1.82±0.05
0.38
0.24
-3.71±0.07
-3.78±0.08
Taurus
-2.92±0.02
-1.59±0.02
0.56
-1.49±0.02
0.49
0.85
-3.26±0.04
-3.47±0.04
Perseus
-3.07±0.02
-2.18±0.04
1.03
-2.21±0.03
0.65
0.31
-3.40±0.06
-3.47±0.05
Orion
-2.95±0.02
-2.29±0.03
0.36
-2.06±0.03
0.27
0.22
-3.75±0.05
-4.83±0.05
Notes:
a
Figure 15. 12 CO channel map of a region with no YSOs in Taurus overlaid with the prediction by model MF in white contours.6.8-7.0km/s
7.0-7.3km/s
7.3-7.6km/s
4 h 57 m 56 m 55 m 54 m
27°45'
30'
15'
00'
RA
Dec
6.8-7.0km/s
NSF grant AST-1812747. D.X. acknowledges support from David Alan Benfield Memorial Scholarship in Astronomy and from the Virginia Initiative on Cosmic Origins (VICO). S.S.R.O. acknowledges support from NSF Career grant AST-1748571. R.A.G. acknowledges support from NASA ADAP grant NNX17AF24G. HGA acknowledges support from NSF grant AST-1714710. The Texas Ad-vanced Computing Center (TACC) at the University of Texas at Austin provided HPC resources that have contributed to the research results reported within this paper. APPENDIX A. DISCUSSION ON THE EFFECT OF BOX SIZES ON THE YSO-MASS RELATION1. The total outflow masses are 267 M , 795
M , 1305 M and 6332 M for Ophi-
uchus, Taurus, Perseus and Orion, respec-
tively. On average, the mass associated
with feedback is around 10% of the host
cloud mass for all four clouds.
2. The total 3D outflow energies are 9 × 10 45
ergs, 6 × 10 46 ergs, 1.2 × 10 47 ergs and 6 ×
10 47 ergs for Ophiuchus, Taurus, Perseus
and Orion, respectively.
3. The outflow mass is linearly proportional
to the total number of YSOs for all four
clouds. On average, each YSO is associ-
ated with one solar mass of outflow mate-
rial.
4. The outflow momentum is linearly pro-
portional to the total number of YSOs
for all four clouds. On average, each out-
flow has a mass-weighted LOS velocity of
1 km/s. The relatively low value is be-
cause more mass is located near the cloud
central velocity.
5. We compute the spatial power spectrum
of the outflow prediction map. We find
all four clouds exhibit a break point,
which ranges from 0.27 pc (Orion) to 0.65
(Perseus) pc. The break point likely indi-
cates the typical outflow mass and energy
injection scale.
6. Models trained to identify outflows are
likely to also identify bubble structures,
which may be produced by main sequence
massive stars.
7. We compare the energy associated with
outflows to the rate of turbulent dissipa-
tion and conclude that feedback is suffi-
cient to maintain turbulent dissipation at
the current epoch for all four clouds.
D.X., S.S.R.O., and R.A.G. acknowledge sup-
port by
). In most cases, the velocity channels5 h 38 m
36 m
34 m
-5°00'
30'
-6°00'
30'
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. D Xu, S S R Offner, R Gutermuth, C Van Oort, 10.3847/1538-4357/abc7bfApJ. 905172Xu, D., Offner, S. S. R., Gutermuth, R., & Van Oort, C. 2020b, ApJ, 905, 172, doi: 10.3847/1538-4357/abc7bf
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|
[] |
[
"Casimir edge effects",
"Casimir edge effects"
] |
[
"Holger Gies \nInstitut für Theoretische Physik\nPhilosophenweg 1669120HeidelbergGermany\n",
"Klaus Klingmüller \nInstitut für Theoretische Physik\nPhilosophenweg 1669120HeidelbergGermany\n"
] |
[
"Institut für Theoretische Physik\nPhilosophenweg 1669120HeidelbergGermany",
"Institut für Theoretische Physik\nPhilosophenweg 1669120HeidelbergGermany"
] |
[] |
We compute Casimir forces in open geometries with edges, involving parallel as well as perpendicular semi-infinite plates. We focus on Casimir configurations which are governed by a unique dimensional scaling law with a universal coefficient. With the aid of worldline numerics, we determine this coefficient for various geometries for the case of scalar-field fluctuations with Dirichlet boundary conditions. Our results facilitate an estimate of the systematic error induced by the edges of finite plates, for instance, in a standard parallel-plate experiment. The Casimir edge effects for this case can be reformulated as an increase of the effective area of the configuration.
|
10.1103/physrevlett.97.220405
|
[
"https://export.arxiv.org/pdf/quant-ph/0606235v1.pdf"
] | 29,780,244 |
quant-ph/0606235
|
3ac0d863e1a6954a21b00ef6d769bb1d1a6a48e2
|
Casimir edge effects
28 Jun 2006
Holger Gies
Institut für Theoretische Physik
Philosophenweg 1669120HeidelbergGermany
Klaus Klingmüller
Institut für Theoretische Physik
Philosophenweg 1669120HeidelbergGermany
Casimir edge effects
28 Jun 2006arXiv:quant-ph/0606235v1numbers: 4250Lc0370+k1110-z
We compute Casimir forces in open geometries with edges, involving parallel as well as perpendicular semi-infinite plates. We focus on Casimir configurations which are governed by a unique dimensional scaling law with a universal coefficient. With the aid of worldline numerics, we determine this coefficient for various geometries for the case of scalar-field fluctuations with Dirichlet boundary conditions. Our results facilitate an estimate of the systematic error induced by the edges of finite plates, for instance, in a standard parallel-plate experiment. The Casimir edge effects for this case can be reformulated as an increase of the effective area of the configuration.
I. INTRODUCTION
Casimir's prediction for the force F per unit area A between two perfectly conducting infinite parallel plates at a distance a [1],
F A = −2γ c a 4 , γ = π 2 480 ≃ 2.056 × 10 −2 ,(1)
has a remarkable property: a straightforward dimensional analysis already fixes the powers of , c, and a uniquely. In absence of any other dimensionful quantity, the effects of quantum fluctuations in this geometry can be summarized by a simple number: 2γ . This coefficient is universal in the sense that it does not depend on the microscopic details of the interactions between the fluctuating field and the constituents of the surfaces. It is completely fixed by specifying the geometry, the nature of the fluctuating field and the type of boundary conditions. For instance, for a fluctuating real scalar field with Dirichlet boundary conditions, the parallel-plate coefficient reduces exactly to γ ; the factor of 2 in Eq. (1) can be traced back to the two polarization modes of the electromagnetic field. Away from the ideal Casimir limit, corrections to Eq. (1) arise from finite conductivity, surface roughness, thermal fluctuations and deviations from the ideal geometry. All these come with additional dimensionful scales, such as plasma frequency, length scales of roughness variation, temperature or surface-curvature radii. The corrections generically cannot be predicted from dimensional analysis, but its functional dependence on the further parameters has to be computed [2,3,4,5,6,7,8].
The present work is devoted to an investigation of the Casimir force between disconnected rigid surfaces, which exhibits properties similar to Casimir's classic parallelplate configuration: unique dimensional scale dependencies and universal coefficients. The first property implies that the geometry is characterized by only one length scale, such as the distance parameter a. New Casimir configurations therefore necessarily involve edges, whose influence on the Casimir effect is an interesting and difficult question in itself. In view of the rapid progress in the fabrication and use of micro-and nano-scale mechanical devices accompanied by precision measurements of the Casimir forces in these systems [9,10,11,12,13,14,15], a detailed understanding of Casimir edge effects is indispensable.
Straightforward computations of Casimir edge effects are conceptually complicated, since the fluctuation spectrum carries the relevant information in a subtle manner. A technique that facilitates Casimir computations from first field-theoretic principles is given by worldline numerics [16], combining the string-inspired approach to quantum field theory [17] with Monte Carlo methods. As a main advantage, the worldline algorithm can be formulated for arbitrary Casimir geometries, resulting in a numerical estimate of the exact answer [18]. Since the approach is based on Feynman path-integral techniques, the problem of determining the Casimir fluctuation spectrum is circumvented [19]. The resulting algorithms are trivially scalable, and computational efforts increase only linearly with the parameters of the numerics.
Recent results obtained by worldline numerics [20] go hand in hand with those obtained by new analytical methods [21,22,23] which are based on advanced scattering-theory techniques; excellent agreement has been found for the experimentally important sphere-plate and cylinder-plate Casimir configurations.
In the present work, we use worldline numerics to examine Casimir edge effects induced by a fluctuating scalar field, obeying Dirichlet boundary conditions ("Dirichlet scalar"). We compute Casimir interaction energies and forces between rigid surfaces. Our results can directly be applied to Casimir configurations in ultracoldgas systems [24] where massless scalar fluctuations exist near the phase transition. For Casimir configurations probing the electromagnetic fluctuation field, the results for the universal coefficients may quantitatively differ, but our values can be used for an order-of-magnitude estimate of the error induced by edges of a finite configuration, thus providing an important ingredient for the data analysis of future experiments.
In addition to being a simple and reliable quantitative method, the worldline formalism also offers an intuitive picture of quantum-fluctuation phenomena. The fluctuations are mapped onto closed Gaußian random paths (worldlines) which represent the spacetime trajectories of virtual loop processes. The Casimir interaction energy between two surfaces can thus be obtained by identifying all worldlines that intersect both surfaces. These worldlines correspond to fluctuations that would violate the boundary conditions; their removal from the ensemble of all possible fluctuations thereby contributes to the (negative) Casimir interaction energy. The latter measures only that part of the energy that contributes to the force between rigid surfaces; possibly divergent self-energies of the single surfaces [25] are already removed.
For a massless Dirichlet scalar, the worldline representation of the Casimir interaction energy reads [18,19]
E = − 1 2 1 (4π) 2 ∞ 0 dT T 3 Θ Σ [x] x .(2)
The expectation value in (2) has to be taken with respect to an ensemble of closed worldlines,
. . . x := x(T )=x(0) Dx . . . e − 1 4 T 0 dτẋ 2 ,(3)
with implicit normalization 1 x = 1. In Eq. (2),
Θ Σ [x] = 1 if a worldline x intersects both surfaces Σ = Σ 1 + Σ 2 , and Θ Σ [x] = 0 otherwise.
The worldline integral can also be evaluated locally, e.g., with the restriction to worldlines with a common center of mass, x CM , resulting in the interaction energy density ε(x CM ),
E = d 3 xε(x CM )
. The interaction energy serves as a potential for the Casimir force between rigid surfaces; the force is thus obtained by simple differentiation with respect to the distance parameters. The worldline numerical algorithm corresponds to a Monte Carlo evaluation of the path integral of Eq. (3) with a discretized propertime τ . In this work, we exploit the recent algorithmic developments detailed in [26].
II. CASIMIR EDGE CONFIGURATIONS
A. Perpendicular Plates
Let us first analyze a semi-infinite plate perpendicularly above an infinite plate at a minimal distance a, as first proposed in [19]. This configuration is illustrated in Fig. 1 together with a worldline which contributes to the Casimir interaction energy, since it intersects both plates. This configuration is translationally invariant only in the direction pointing along the edge with a being the only dimensionful length scale. The Casimir force per unit length L along the edge direction is thus unambiguously fixed by dimensional analysis,
F ⊥ L = −γ ⊥ c a 3 .(4)
Evaluating the worldline integral as outlined above, we obtain an estimate for the corresponding Casimir interaction energy density ε(x), a contour plot of which is given
The error is below the 1% level for a path ensemble of 40 000 loops with 200 000 points per loop (ppl) each. This coefficient is in agreement with the Casimir interaction energy computed in [19].
B. Semi-infinite plate parallel to an infinite plate
Next we consider a first variant of the parallel-plate configuration, where one of the plates is only semi-infinite with an edge on one side; see Fig. 3. This configuration can be viewed as an idealized limit of a real experimental situation where a smaller controllable finite plate is kept parallel above a larger fixed substrate. In this case, the dominant contribution to the force is given by the universal classic parallel-plate result of Eq. (1) with A being the surface area of the smaller plate.
In the ideal limit of A as well as the edge length L going to infinity, the sub-leading Casimir edge effect is also universal. Dimensional analysis requires the exact force to be of the form where F denotes the parallel-plate force for the Dirichlet scalar, i.e., without the factor 2 in Eq. (1). A priori, the universal coefficient γ 1si can be positive or negative. The sign can easily be guessed within the worldline picture: owing to their spatial extent, a sizable fraction of worldlines can intersect both plates even if their center of mass is located outside the plates. This can quantitatively be verified by the energy density, the peak of which indeed extends into the outside region; see Fig. 4. This peak in the outside region contributes to the total interaction energy, implying an increase of the Casimir force compared to the pure parallel-plate formula. Therefore, the Casimir edge effect leads to further attraction, and the sign of the universal coefficient γ 1si must be positive. Quantitatively, we find
F = F − γ 1si c a 3 L,(6)γ 1si = 5.23(2) × 10 −3 ,(7)
again with 40 000 loops, 200 000 ppl.
C. Parallel semi-infinite plates
Another variant of the parallel-plate configuration is given by two parallel semi-infinite plates with parallel edges; see an idealized parallel-plate experiment where both plates have the same area size A. In the ideal limit of infinite A as well as infinite edge length L, the exact form of the force is again given by dimensional analysis,
F = F − γ 2si c a 3 L,(8)
equivalent to Eq. (6). Qualitatively, the situation is similar to the preceding one with one semi-infinite plate. Quantitatively, fewer worldlines in the outside as well as the inside region near the edge intersect both plates. Both aspects are visible in the plot of the interaction energy density in Fig. 6: the peak height and width is reduced near the edge both inside and outside the plates. We still observe a positive universal coefficient,
γ 2si = 2.30(1) × 10 −3(9)
(93 000 loops, 500 000 ppl), which is a bit less than half as big as the preceding case with one semi-infinite plate. Again, the Casimir edge effect increases the force in comparison with the pure parallel-plate estimate F .
III. EDGE-CONFIGURATION ESTIMATES
The universal results for the idealized configurations presented above can immediately be used to derive estimated predictions for further Casimir configurations.
A. Casimir comb
Replicating the perpendicular-plate configuration in the horizontal direction of Figs. 1 and 2, we obtain a stack of semi-infinite plates (a "Casimir comb") perpendicularly above an infinite plate. Let d be the distance between two neighboring semi-infinite plates, i.e., the distance between two teeth of the comb. In the limit d ≫ a, we obtain the Casimir force between the Casimir comb and the infinite plate by simply adding the forces for the individual perpendicular plates. The reliability of this approximation is obvious from Fig. 2, which shows that the dominant contribution to the energy is peaked inside a region with length scale ∼ a. The resulting force is
F comb = −γ ⊥ c a 3 d A,(10)
with A = Lnd being the total area of a comb with n teeth. For a fixed comb, i.e., fixed d, the short-distance Casimir force thus has a weaker dependence on a than for the parallel-plate case. In the opposite limit d ≪ a, we expect the force between the comb and the plate to rapidly approach that of the parallel-plate case (1). This is because a generic worldline contributing to the force will have a spatial extent of order a, such that the finer comb scale d ≪ a will not be resolved by the worldline ensemble to first approximation. A similar observation has been made in studies of periodic corrugations [28].
B. Finite parallel-plate configurations
In a real parallel-plate experiment, the finite extent of the plates induces edge effects. If the typical length scale L of a plate (such as the edge length of a square plate or the radius of a circular disc) is much larger than the plate distance a, our results for the idealized limits studied above can be used within a good approximation. The force law can then be summarized as
F = −γ c a 4 A eff ,(11)
where the effective area A eff also carries the information about the edge effects. For the case of a smaller plate with area A and circumference C above a much larger substrate, the effective area is given by
A eff = A + γ 1si γ aC,(12)
e.g., C = 4L for a square plate with edge length L. For the case of two parallel plates of equal size and shape with area A and circumference C, Eq. (12) holds with γ 1si replaced by γ 2si . Obviously, the effective area A eff is larger than the physical area in either case. Consider, for instance, a square plate of edge length L above a larger substrate: the Casimir edge effects induce a correction on the 1% level if a 1% of L. In the experiment of reference [27], the edge length is L = 1.2mm and the distance goes up to a = 3µm. One of the edges faces an edge of the substrate, similar to Fig. 5, whereas the other three correspond to Fig. 3. For the Dirichlet scalar this results in a correction of 0.2%, which is much smaller than the 15% precision level of the experiment.
IV. CONCLUSIONS
We have performed a detailed quantitative study of Casimir edge effects induced by a fluctuating scalar field obeying Dirichlet boundary conditions. All of our results exhibit a uniquely fixed dependence on dimensionful scales, as for Casimir's classic result. The effect of quantum fluctuations is quantitatively encoded in a universal dimensionless coefficient, which only depends on the geometry, the nature of the fluctuating field and the boundary conditions. From the perspective of a scattering-theory approach, Casimir edge effects are dominated by diffractive contributions to the correlation functions which are difficult to handle for direct approximation techniques [29,30]; hence, our results give an important first insight into the properties of diffractive contributions to Casimir forces. For Casimir measurements involving electromagnetic fluctuations, our results serve as a first order-of-magnitude estimate of the error induced by edges of finite configurations -an error that any parallel-plate experiment has to deal with.
FIG. 1 :
1Sketch of the perpendicular-plates configuration. The minimal distance a between the edge of the upper semi-infinite plate (thick solid line) and the lower infinite plate represents the only dimensionful length scale in the problem.
FIG. 2 :
2Contour plot of the Casimir interaction energy density ε for the perpendicular-plate configuration. The white lines mark the position of the plates to guide the eye. Ensemble parameters: 2000 loops with 10 000 ppl. in Fig. 2. For the universal coefficient, we obtain γ ⊥ = 1.200(4) × 10 −2 .
FIG. 3 :
3Sketch of the configuration of a semi-infinite plate parallel to an infinite plate at a distance a. A worldline can intersect both plates even if its center of mass is located outside the two plates.
FIG. 4 :
4Contour plot of the Casimir interaction energy density ε for a semi-infinite plate parallel to an infinite plate. The white lines mark the position of the plates to guide the eye. The energy-density peak extends into the outside region, since worldlines can intersect both plates even if their center of mass is in the outside region. Ensemble parameters: 1000 loops, 10 000 ppl.
Fig. 5 .FIG. 6 :
56This Contour plot of the Casimir interaction energy density ε for two parallel semi-infinite plates. The energy-density peak extends into the outside region, since worldlines can intersect both plates even if their center of mass is in the outside region. Ensemble parameters: 2000 loops, 10 000 ppl.
The authors acknowledge support by the DFG Gi 328/1-3 (Emmy-Noether program) and Gi 328/3-2.
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|
[] |
[
"Radial geodesics as a microscopic origin of black hole entropy. III: Exercise with the Kerr-Newman black hole",
"Radial geodesics as a microscopic origin of black hole entropy. III: Exercise with the Kerr-Newman black hole"
] |
[
"V V Kiselev [email protected] \nRussian State Research Center \"Institute for High Energy Physics\"\nPobeda 1Protvino\n\nMoscow Region\n142281Russia\n\nInstitutskii per. 9\nMoscow Institute of Physics and Technology\nDolgoprudnyi Moscow Region\n141700Russia\n"
] |
[
"Russian State Research Center \"Institute for High Energy Physics\"\nPobeda 1Protvino",
"Moscow Region\n142281Russia",
"Institutskii per. 9\nMoscow Institute of Physics and Technology\nDolgoprudnyi Moscow Region\n141700Russia"
] |
[] |
We specify an angular motion on geodesics to reduce the problem to the case of radial motion elaborated in previous chapters. An appropriate value of entropy for a charged and rotating black hole is obtained by calculating the partition function on thermal geodesics confined under horizons. The quantum aggregation is classified in a similar way to the Reissner-Nordstrøm black hole.
| null |
[
"https://arxiv.org/pdf/gr-qc/0412110v3.pdf"
] | 119,038,391 |
gr-qc/0412110
|
eabfea221e4b8b51675e4391cdbbfd58a49d9afe
|
Radial geodesics as a microscopic origin of black hole entropy. III: Exercise with the Kerr-Newman black hole
V V Kiselev [email protected]
Russian State Research Center "Institute for High Energy Physics"
Pobeda 1Protvino
Moscow Region
142281Russia
Institutskii per. 9
Moscow Institute of Physics and Technology
Dolgoprudnyi Moscow Region
141700Russia
Radial geodesics as a microscopic origin of black hole entropy. III: Exercise with the Kerr-Newman black hole
arXiv:gr-qc/0412110v3 4 Jan 2005
We specify an angular motion on geodesics to reduce the problem to the case of radial motion elaborated in previous chapters. An appropriate value of entropy for a charged and rotating black hole is obtained by calculating the partition function on thermal geodesics confined under horizons. The quantum aggregation is classified in a similar way to the Reissner-Nordstrøm black hole.
Continuation of Preface
A rotation of Kerr-Newman black hole involves a new feature in the description of geodesics responsible for the entropy of black hole: a nonzero projection of angular momentum on the axis of rotation is permitted by the symmetry of the problem. Therefore, we start with a specification of angular motion on geodesics of conserved orbital momentum in Section 2. Then, the procedure has a little to differ from the Reissner-Nordstrøm black hole. The difference is reduced to a particular dependence of mass sum on the polar angle, that allows us to evaluate the partition function and entropy for a cool state of aggregation in agreement with the Bekenstein-Hawking formula [1][2][3] in Section 3. A short summary is situated in Section 4.
Angular motion and reduced geodesics
The Kerr-Newman metric of charged and rotating black hole can be written in the form
ds 2 KN = ∆ Σ dω 2 t − Σ ∆ dr 2 − Σ dθ 2 − sin 2 θ Σ dω 2 φ ,(1)with dω t = dt − a sin 2 θ dφ, dω φ = (r 2 + a 2 ) dφ − a dt, Σ = r 2 + a 2 cos 2 θ, ∆ = (r − r + )(r − r − ),(2)
where the black hole parameters: mass M, charge Q, and angular momentum J, are given by
M = 1 2 (r + + r − ), Q 2 + a 2 = r + r − , J = a M.(3)
In order to proceed with the Hamilton-Jacobi equation for a test particle with a mass m
g µν ∂ µ S HJ ∂ ν S HJ − m 2 = 0,(4)
one has to invert the metric in terms of {t, φ} given by the following elements:
g tt = 1 Σ (∆ − a 2 sin 2 θ), g tφ = − a sin 2 θ Σ (∆ − r 2 − a 2 ), g φφ = sin 2 θ Σ (a 2 sin 2 θ − r 2 − a 2 ).(5)
For the corresponding 2 × 2-matrixĝ we get the determinant
detĝ = ∆ sin 2 θ Σ 2 [−a 4 sin 4 θ − (r 2 + a 2 ) 2 + 2(r 2 + a 2 )a 2 sin 2 θ],(6)
which enters the inverse matrix elements
g tt = 1 detĝ g φφ , g tφ = − 1 detĝ g tφ , g φφ = 1 detĝ g tt .(7)
Then, we introduce two integrals of motion: an energy E and an angular momentum µ, which determine the action in the form
S HJ = −E t + µ φ + S HJ (r),(8)atθ ≡ 0. From (4) we deduce ∂S HJ ∂r 2 = ∆ Σ E 2 g tt + 2Eµ g tφ + µ 2 g φφ − m 2 ≡ ∆ Σ h.(9)
which results in
S HJ (r) = r(t) r 0 dr h ∆ Σ .(10)
The trajectory is implicitly determined by equations
∂S HJ ∂E = t 0 = −t + r(t) r 0 dr ∆ h Σ [E g tt + µ g tφ ],(11)∂S HJ ∂µ = φ 0 = φ + r(t) r 0 dr ∆ h Σ [E g tφ + µ g φφ ].(12)
Taking the derivative of (11), (12) with respect to the time ‡, we get
1 =ṙ ∆ h Σ [E g tt + µ g tφ ],(13)φ = −ṙ ∆ h Σ [E g tφ + µ g φφ ],(14)
determining the angular motion bẏ
φ = − E g tφ + µ g φφ E g tt + µ g tφ .(15)
Further, we use a relation specifying the angular motion by
µ = E a sin 2 θ,(16)that givesφ = a r 2 + a 2 .(17)
Note, for such the angular velocity dω φ = 0, and the interval takes the form
ds 2 = Σ ∆ (r 2 + a 2 ) 2 (dt 2 − dr 2 * ),(18)
with
dr * = r 2 + a 2 ∆ dr,(19)
yielding
r * = r + r 2 + + a 2 r + − r − ln r r + − 1 − r 2 − + a 2 r + − r − ln r r − − 1 .(20)
Then, we can repeat the Hamilton-Jacobi formalism for the interval of reduced motion in (18) and find
1 m 2 ∂S HJ ∂r * 2 = E A − U(r),(21)with E A = 1/A, U(r) = Σ · ∆ (r 2 + a 2 ) 2 ,(22)and dt dr * = E A E A − U(r) .(23)
Therefore, on the geodesics we get the causal interval
ds 2 A = U 2 (r) E A − U(r) (r 2 + a 2 ) 2 ∆ 2 dr 2 .(24)
For the ground state we will consider further, E A → −0 and
ds 2 = r 2 + a 2 cos 2 θ (r + − r)(r − r − ) dr 2 , r − < r < r + ,(25)‡ As usual ∂ t f (t) =ḟ .
while the increment of time per cycle is given by
∆ c t E = 2π (r + + r − ).(26)
Further, due to (18) and (20) we can introduce two consistent maps under the horizons in terms of Kruskal isotropic variables in a manner of our treatment in Chapter II, with inverse temperatures
β + = 4π r 2 + + a 2 r + − r − , β − = 4π r 2 − + a 2 r + − r − ,(27)
and a quantum ratio of horizon areas
A + A − = 4π(r 2 + + a 2 ) 4π(r 2 − + a 2 ) = l ∈ N.(28)
The winding numbers on geodesics are equal to the same values as in Chapter II,
n + = 2l l − 1 , n − = 2 l − 1 .(29)
The increment of interval per cycle ∆ c s(cos θ) follows from (25), while at θ = π/2 we easily get ∆ c s(0) = π(r + + r − ).
(30)
Thus, we can follow the analogy with the Reissner-Nordstrøm black hole.
Entropy
Introducing a sum of particles moving at a fixed value of angle θ in the maps ±,
σ ± (cos θ) = ± mc θ ,(31)
for pure "ice" state of aggregation we get the partition function ln Z + = −n + ∆ c s(cos θ) σ + (cos θ).
In order to get the most probable configuration the product ∆ c s(cos θ) σ + (cos θ) should be invariant under the variation of θ and take its minimal value, which is reached at cos θ = 0, so that
ln Z + = −n + ∆ c s(0) σ + (0) = − β + 2 σ + (0).(33)
Thus, in the thermal equilibrium, the sum of masses σ(cos θ) adjusts its value in order to compensate the dependence of ∆ c s on cos θ. An example of such the adjustment is shown in Fig. 1. From (33) and Chapter II we deduce the expressions
σ + (0) = 2M − 1 2β + A + ,(34)
and the entropy
S + = 1 4 A + ,(35)
valid due to the standard relation between the temperature and 'surface gravity' (see discussion in review [4]), A discrimination of two phases of aggregation is the same as for the charged black hole.
T + = 4 ∂M ∂A + , at dQ 2 ≡ 0, dJ ≡ 0.(36)
Conclusion
In present chapter we have shown how the consideration of rotating black hole is reduced to the charged one of Reissner and Nordstrøm. The only actual difference is the adjustment of mass sum versus the polar angle.
Finally, let us make a short note on the extremal black hole with r + = r − , that corresponds to solitonic BPS states in superstrings [5]. We have found that the extremum takes place, when the sphere of euclidian time and radius degenerates to a torus with an infinitely small radius (two poles of sphere are glued, but the radius tends to zero). The corresponding winding number tends to infinity for the ground state. However, coming from such the singular ground state to an excited state, one could adjust the level of winding number (the value of x) and tune the sum of particle masses in order to preserve the final value of entropy, which is independent of particular value of temperature at the given area of external horizon. Due to the thermal quantization of ratio of horizon areas, the limit of extremal black hole cannot be reached continuously: one has got a quantum jump of r − .
Thus, we finalize considering the basics of method for the evaluation of black hole entropy by calculating the microscopic partition function on classical geodesics confined under the horizons by their thermal quantization. We hope that the tool provides a new insight to the entropy of black hole, not excluding some new particular problems and questions.
This work is partially supported by the grant of the president of Russian Federation for scientific schools NSc-1303NSc- .2003, and the Russian Foundation for Basic Research, grants 04-02-17530, 05-02-16315.
Figure 1 .
1The variation of σ + versus cos θ at a = 1, r − = 2, r + = 3.
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[] |
[
"WEIGHTED EMPIRICAL LIKELIHOOD IN SOME TWO-SAMPLE SEMIPARAMETRIC MODELS WITH VARIOUS TYPES OF CENSORED DATA",
"WEIGHTED EMPIRICAL LIKELIHOOD IN SOME TWO-SAMPLE SEMIPARAMETRIC MODELS WITH VARIOUS TYPES OF CENSORED DATA",
"WEIGHTED EMPIRICAL LIKELIHOOD IN SOME TWO-SAMPLE SEMIPARAMETRIC MODELS WITH VARIOUS TYPES OF CENSORED DATA",
"WEIGHTED EMPIRICAL LIKELIHOOD IN SOME TWO-SAMPLE SEMIPARAMETRIC MODELS WITH VARIOUS TYPES OF CENSORED DATA"
] |
[
"Jian-Jian Ren \nUniversity of Central Florida\n\n",
"Jian-Jian Ren \nUniversity of Central Florida\n\n"
] |
[
"University of Central Florida\n",
"University of Central Florida\n"
] |
[
"The Annals of Statistics",
"The Annals of Statistics"
] |
In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator (θn,Fn) for the underlying parameter θ0 and distribution F0 is derived, and the strong consistency of (θn,Fn) and the asymptotic normality ofθn are established. Under biased sampling models, the weighted empirical log-likelihood ratio is shown to have an asymptotic scaled chi-squared distribution for censored data aforementioned. For right censored data, doubly censored data and partly interval-censored data, it is shown that √ n(Fn − F0) weakly converges to a centered Gaussian process, which leads to a consistent goodness-of-fit test for the case-control logistic regression models.
|
10.1214/009053607000000695
|
[
"https://arxiv.org/pdf/0803.1752v1.pdf"
] | 7,391,465 |
0803.1752
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2ea4eb02ec35b47daaefb5149ab8154295537c57
|
WEIGHTED EMPIRICAL LIKELIHOOD IN SOME TWO-SAMPLE SEMIPARAMETRIC MODELS WITH VARIOUS TYPES OF CENSORED DATA
2008
Jian-Jian Ren
University of Central Florida
WEIGHTED EMPIRICAL LIKELIHOOD IN SOME TWO-SAMPLE SEMIPARAMETRIC MODELS WITH VARIOUS TYPES OF CENSORED DATA
The Annals of Statistics
361200810.1214/009053607000000695
In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator (θn,Fn) for the underlying parameter θ0 and distribution F0 is derived, and the strong consistency of (θn,Fn) and the asymptotic normality ofθn are established. Under biased sampling models, the weighted empirical log-likelihood ratio is shown to have an asymptotic scaled chi-squared distribution for censored data aforementioned. For right censored data, doubly censored data and partly interval-censored data, it is shown that √ n(Fn − F0) weakly converges to a centered Gaussian process, which leads to a consistent goodness-of-fit test for the case-control logistic regression models.
1. Introduction. Consider the following two-sample semiparametric model: X 1 , . . . , X n 0 is a random sample with density f 0 (x), (1.1) Y 1 , . . . , Y n 1 is a random sample with density g 0 (x) = ϕ(x; θ 0 )f 0 (x),
where the two samples are independent, and ϕ(x; θ 0 ) is a known function with x ∈ R and a unique unknown parameter θ 0 ∈ R q , while f 0 and g F 0 and G 0 , respectively. This model (1.1) includes biased sampling models (Vardi [32]) and case-control logistic regression models (Prentice and Pyke [22]) as special cases, for which there has not been any published work dealing with censored data. In this article, we study model (1.1) when at least one of the two samples is not completely observable due to censoring. As follows, we use random sample X 1 , . . . , X n 0 to illustrate the censoring models under consideration here, while Examples 1 and 2 discuss biased sampling models and case-control logistic regression models, respectively.
Right censored sample. The observed data are O i = (V i , δ i ), 1 ≤ i ≤ n 0 , with
V i = X i , if X i ≤ C i , δ i = 1, C i , if X i > C i , δ i = 0, (1.2)
where C i is the right censoring variable and is independent of X i . This type of censoring has been extensively studied in the literature in the past few decades.
Doubly censored sample. The observed data are O i = (V i , δ i ), 1 ≤ i ≤ n 0 , with
V i = X i , if D i < X i ≤ C i , δ i = 1, C i , if X i > C i , δ i = 2, D i , if X i ≤ D i , δ i = 3,(1.3)
where C i and D i are right and left censoring variables, respectively, and they are independent of X i with P {D i < C i } = 1. This type of censoring has been considered by Turnbull [31], Chang and Yang [4], Gu and Zhang [11] and Mykland and Ren [17], among others. One recent example of doubly censored data was encountered in a study of primary breast cancer (Ren and Peer [28]).
Interval censored sample.
Case 1. The observed data are O i = (C i , δ i ), 1 ≤ i ≤ n 0 , with δ i = I{X i ≤ C i }. (1.4) Case 2. The observed data are O i = (C i , D i , δ i ), 1 ≤ i ≤ n 0 , with δ i = 1, if D i < X i ≤ C i , 2, if X i > C i , 3, if X i ≤ D i ,(1.5)
where C i and D i are independent of X i and satisfy P {D i < C i } = 1 for Case 2. These two types of interval censoring were considered by Groeneboom and Wellner [10], among others. In practice, interval censored Case 2 data were encountered in AIDS research (Kim, De Gruttola and Lagakos [16]; see discussion in Ren [26]).
Partly interval-censored sample.
"Case 1" partly interval-censored data. The observed data are
O i = X i , if 1 ≤ i ≤ k 0 , (C i , δ i ), if k 0 + 1 ≤ i ≤ n 0 , (1.6) where δ i = I{X i ≤ C i } and C i is independent of X i .
General partly interval-censored data. The observed data are
O i = X i , if 1 ≤ i ≤ k 0 , (C, δ i ), if k 0 + 1 ≤ i ≤ n 0 , (1.7)
where for N potential examination times C 1 < · · · < C N , letting C 0 = 0 and C N +1 = ∞, we have C = (C 1 , . . . , C N ) and δ i = (δ
(1) i , . . . , δ (N +1) i ) with δ (j) i = 1, if C j−1 < X i ≤ C j ; 0,
elsewhere. This means that for intervals (0, C 1 ], (C 1 , C 2 ], . . . , (C N , ∞), we know in which one of them X i falls. These two types of partly interval-censoring were considered by Huang [12], among others. As pointed out by Huang [12], in practice the general partly intervalcensored data were encountered in Framingham Heart Disease Study (Odell, Anderson and D'Agostino [18]), and in the study on incidence of proteinuria in insulin-dependent diabetic patients (Enevoldsen et al. [5]).
Example 1 (Biased sampling model). In (1.1), let
ϕ(x; θ 0 ) = θ 0 w(x), θ 0 ∈ R, (1.8)
where w(x) is a weight function with positive value on the support of F 0 , and θ 0 = 1/w 0 is the weight parameter satisfying w 0 = ∞ 0 w(x) dF 0 (x). Then, (1.1) is a two-sample biased sampling problem, for which the case with lengthbiased distribution G 0 , that is, w(x) = x in (1.8), was considered by Vardi [32], and the empirical log-likelihood ratio for the mean of F 0 was shown to have an asymptotic chi-squared distribution by Qin [23]. More general biased sampling models were considered by Vardi [33], Gill, Vardi and Wellner [9], who discussed various application examples, and showed that the maximum likelihood estimator for F 0 is asymptotically Gaussian and efficient. For right censored samples in (1.1), Vardi [33] gave an estimator for F 0 based on the EM algorithm, but the asymptotic properties of the estimator were not studied. Below, we discuss practical examples of biased sampling problem with censored data.
In Patil and Rao [20], the biased sampling problem is discussed in the context of efficiency of early screening for disease. Using our notations in (1.1), if F 0 is the d.f. of the duration of the preclinical state of certain chronic disease, then the first sample in (1.1) is taken from those whose clinical state is detected by the usual medical care. If at a certain point in 4 J.-J. REN time some individuals in the preclinical state begin participating in an early detection program, then such a program identifies them by a length-biased sampling. In other words, the second sample in (1.1) is taken from those who participated in the early detection program, and G 0 is a length-biased distribution. However, in reality a usual screening program for "disease" is conducted by examining an individual periodically with a fixed length of time between two consecutive check-ups. The data encountered in such a screening program is typically a doubly censored sample (1.3); that is, the actually observed data for the second sample in (1.1) is doubly censored. In statistical literature, examples of doubly censored data encountered in screening programs have been given by Turnbull [31] and Ren and Gu [27], among others.
Example 2 (Case-control logistic regression model). In (1.1), let ϕ(x; θ 0 ) = e α 0 +β 0 x , (1.9) F 0 (x) = P {T ≤ x|Z = 0}, G 0 (x) = P {T ≤ x|Z = 1};
then under reparameterization by Qin and Zhang [24], model (1.1) is equivalent to the following case-control logistic regression model (Prentice and Pyke [22]):
P {Z = 1|T = x} = exp(α * + β 0 x) 1 + exp(α * + β 0 x) , (1.10)
where θ 0 = (α 0 , β 0 ) ∈ R 2 , Z is the binary response variable (with value 1 or 0 to indicate presence or absence of a disease or occurrence of an event of interest), T is the covariate variable, and (α * , β 0 ) is the regression parameter satisfying α 0 = α * + ln[(1 − π)/π] for π = P {Z = 1}. Qin and Zhang [24] established asymptotic normality of the semiparametric maximum likelihood estimators (SPMLE) for (θ 0 , F 0 ) in (1.9) with two complete samples in (1.1), and provided a goodness-of-fit test for (1.10). Below, we discuss an example to illustrate the situation with censored covariate variable T .
In the example of early detection of breast cancer considered by Ren and Gu [27], T is the age at which the tumor could be detected when screening mammogram is the only detection method, and based on series screening mammograms the observed data on T are doubly censored. This example is part of a study on the effectiveness of screening mammograms; see Ren and Peer [28] for precise description of left and right censored observations. Here, to study the effects of screening mammograms on survival, we consider those individuals who had breast cancer, and let Z = 1 represent death due to breast cancer within 5 years of diagnosis; Z = 0, otherwise. Then under (1.9), for those "dead" (i.e., Z = 1) the second sample in (1.1) is taken from the available screening mammogram records; thus the actually observed data from G 0 (x) = P {T ≤ x|Z = 1} is a doubly censored sample. Similarly, for those "survived" (i.e., Z = 0) the first sample in (1.1) is also taken from screening mammogram records; thus also a doubly censored sample. Fitting the logistic regression model (1.10) with these two doubly censored casecontrol samples, we obtain P {Z = 1|T = x 0 }, which is the probability of "death" for an individual whose tumor was detected by screening mammogram at age x 0 .
In this article, we apply weighted empirical likelihood (Ren [25]) to model (1.1) with the following two independent samples for n = n 0 + n 1 :
O X 1 , . . . , O X n 0 is the observed sample for sample X 1 , . . . , X n 0 , (1.11) O Y 1 , . . . , O Y n 1 is the observed sample for sample Y 1 , . . . , Y n 1 ,
where O X i 's or O Y j 's is possibly one of those censored samples described above, and we denoteF andĜ as the nonparametric maximum likelihood estimators (NPMLE) for F 0 and G 0 based on O X i 's and O Y j 's, respectively. Section 2 provides a heuristic explanation of the concept of weighted empirical likelihood. For censored data (1.2)-(1.7) aforementioned, Section 3 derives the weighted empirical likelihood-based SPMLE (θ n ,F n ) for (θ 0 , F 0 ), and establishes the strong consistency of (θ n ,F n ) and the asymptotic normality ofθ n , while Section 4 further discusses Example 1 on biased sampling models, and shows that the weighted empirical log-likelihood ratio has an asymptotic scaled chi-squared distribution. For right censored data, doubly censored data and partly interval-censored data, Section 3 also shows that √ n(F n − F 0 ) weakly converges to a centered Gaussian process, while Section 5 further discusses Example 2 on case-control logistic regression models, and provides a consistent goodness-of-fit test.
We note that the weighted empirical likelihood approach used in this article can be adapted to deal with more general biased sampling models. Also note that based on Ren and Gu [27], our results here on the case-control logistic regression models can be extended to k-dimensional (k > 1) covariate T , where T contains one component that is subject to right censoring or doubly censoring.
For interval censored data (1.4)-(1.5), the weighted empirical likelihood approach enables us to obtain the strong consistency of the SPMLE (θ n ,F n ), the asymptotic normality ofθ n , and the limiting distribution of the loglikelihood ratio via the asymptotic results on the NPMLEF orĜ for interval censored data by Groeneboom and Wellner [10] and Geskus and Groeneboom [6], among others. However, the techniques used in our proofs show that the weak convergence ofF n for interval censored data relies on that of F orĜ for interval censored data, which is now unknown. 2. Weighted empirical likelihood. For random sample X 1 , . . . , X n 0 from d.f. F 0 , the empirical likelihood function (Owen [19]) is given by
L(F ) = n 0 i=1 [F (X i ) − F (X i −)]
, where F is any d.f. The weighted empirical likelihood function in Ren [25] may be understood as follows.
For each type of censored data aforementioned, the likelihood function has been given in literature, and the NPMLEF for F 0 is the solution which maximizes the likelihood function. Moreover, it is shown that from observed
censored data {O X i ; 1 ≤ i ≤ n 0 }, there exist m 0 distinct points W X 1 < W X 2 < · · · < W X m 0 along withp X j > 0, 1 ≤ j ≤ m 0 , such thatF can be expressed asF (x) = m 0 i=1p X
i I{W X i ≤ x} for above right censored data (Kaplan and Meier [15]), doubly censored data (Mykland and Ren [17]), interval censored data Case 1 and Case 2 (Groeneboom and Wellner [10]) and partly intervalcensored data (Huang [12]). Since in all these casesF is shown to be a strong uniform consistent estimator for F 0 under some suitable conditions, we may expect a random sample X * 1 , . . . , X * n 0 taken fromF to behave asymptotically the same as X 1 , . . . ,
X n 0 . If F * n 0 denotes the empirical d.f. of X * 1 , . . . , X * n 0 , then fromF ≈ F * n 0 we have n 0 i=1 P {X i = x i } ≈ n 0 i=1 P {X * i = x * i } = m 0 j=1 (P {X * 1 = W X j }) k j ≈ m 0 j=1 (P {X * 1 = W X j }) n 0 [F (W X j )−F (W X j −)] = m 0 j=1 (P {X * 1 = W X j }) n 0p X j , where k j = n 0 [F * n 0 (W X j ) − F * n 0 (W X j −)]
. Thus, the weighted empirical likelihood function (Ren [25])
L(F ) = m 0 i=1 [F (W X i ) − F (W X i −)] n 0p X i (2.1)
may be viewed as the asymptotic version of the empirical likelihood function L(F ) for censored data. When there is no censoring,L(F ) coincides with L(F ).
3. SPMLE and asymptotic results. This section derives the semiparametric maximum likelihood estimator for (θ 0 , F 0 ) in (1.1) using censored data (1.11), and studies related asymptotic properties.
As general notations throughout this paper, letF andĜ be the NPMLE for F 0 and G 0 in (1.1) based on observed censored data O X 1 , . . . , O X n 0 and O Y 1 , . . . , O Y n 1 in (1.11), respectively. From Section 2, we know that there
exist distinct points W X 1 < · · · < W X m 0 and W Y 1 < · · · < W Y m 1 withp X i > 0 andp Y i > 0 such thatF andĜ can be expressed aŝ F (x) = m 0 i=1p X i I{W X i ≤ x} andĜ(x) = m 1 i=1p Y i I{W Y i ≤ x} (3.1)
respectively, for those censored data aforementioned. We also let
(W 1 , . . . , W m ) = (W X 1 , . . . , W X m 0 , W Y 1 , . . . , W Y m 1 ), (p 1 , . . . ,p m ) = (p X 1 , . . . ,p X m 0 ,p Y 1 , . . . ,p Y m 1 ), (3.2) (ω 1 , . . . , ω m ) = (ρ 0p X 1 , . . . , ρ 0p X m 0 , ρ 1p Y 1 , . . . , ρ 1p Y m 1 ), where m = m 0 + m 1 , ρ 0 = n 0 /n and ρ 1 = n 1 /n.
To derive an estimator for (θ 0 , F 0 ) using both samples in (1.11), we apply weighted empirical likelihood function (2.1) to model (1.1), and obtain
m 0 i=1 [F (W X i ) − F (W X i −)] n 0p X i m 1 j=1 [G(W Y j ) − G(W Y j −)] n 1p Y j = m 0 i=1 [F (W X i ) − F (W X i −)] n 0p X i × m 1 j=1 {ϕ(W Y j ; θ 0 )[F (W Y j ) − F (W Y j −)]} n 1p Y j .
Thus, from (3.2) the weighted empirical likelihood function for model (1.1) is given by
L(θ, F ) = m i=1 p nω i i m j=m 0 +1 [ϕ(W j ; θ)] nω j (3.3) for p i = F (W i ) − F (W i −),
and the SPMLE (θ n ,F n ) for (θ 0 , F 0 ) is the solution that maximizes L(θ, F ). One may note that the use of weighted empirical likelihood function (2.1) here provides a simple and direct way to incorporate the model assumption of (1.1) in the derivation of likelihood function (3.3) for censored data. Also note that using the usual likelihood functions for specific types of censored data would result in a much more complicated likelihood function which is very difficult to handle. To find (θ n ,F n ), we need to solve the following optimization problem:
max L(θ, p) = m i=1 p nω i i m j=m 0 +1 [ϕ(W j ; θ)] nω j 8 J.-J. REN (3.4) subject to p i ≥ 0, m i=1 p i = 1, m i=1 p i ϕ(W i ; θ) = 1,
where the last constraint reflects the fact that ϕ(
x; θ)[F (x) − F (x−)]i=1p X i = m 1 i=1p Y i = 1 in (3.1)
, which will not be needed later on for our main results of the paper. To solve (3.4), we first maximize L(θ, p) with respect to p = (p 1 , . . . , p m ) for fixed θ, then maximize l(θ) = ln L(θ,p) = max p ln L(θ, p) over θ to findθ n . Noting that for
U i (θ) = ϕ(W i ; θ), constraints in (3.4) imply m i=1 p i [U i (θ) − 1] = 0, we know that θ must satisfy [U (1) (θ) − 1] < 0 < [U (m) (θ) − 1]. (3.5)
Using the Lagrange multiplier method, it can be shown that for any fixed θ satisfying (3.5), the convexity of ln L(θ, p) ensures that L(θ, p) is uniquely maximized by L(θ,p) (see pages 90-91 and 164 of Bazaraa, Sherali and Shetty [1]), wherẽ
p i = ω i 1 + λ(θ)[U i (θ) − 1] , i = 1, . . . , m, (3.6) with λ(θ) as the unique solution on interval (−[U (m) (θ) − 1] −1 , −[U (1) (θ) − 1] −1 ) for 0 = ψ(λ; θ) ≡ m i=1 ω i [U i (θ) − 1] 1 + λ[U i (θ) − 1] . (3.7)
Thus, we have l(θ) = n m i=1 ω i lnp i + n m j=m 0 +1 ω j ln ϕ(W j ; θ). For our examples, we have θ 0 ∈ R or θ 0 ∈ R 2 in (1.1), and that for some functions h 1 (θ) and h 2 (x), the following assumption holds for ϕ(x; θ) with θ ∈ R or θ ∈ R 2 :
(AS0) ∇ϕ(x; θ) = ϕ(x; θ)h 1 (θ)(1, h 2 (x)) ⊤ for ∇ = (∂/∂θ 1 , ∂/∂θ 2 ) ⊤ , where 0 < h 1 (θ) ∈ R is twice differentiable for θ ∈ Θ; 0 ≤ h 2 (x) ∈ R is mono- tone for x ≥ 0;
in the case θ ∈ R, we have degenerating h 2 (x) ≡ 0; in the case θ ∈ R 2 , we always have strictly monotone h 2 (x) on the support of F 0 . Throughout this paper, our notations mean that for the case θ ∈ R, only the nondegenerating component in equations, vectors and matrices is meaningful. To minimize l(θ), from (3.2), (3.6)-(3.7), ψ(λ(θ); θ) = 0 and constraints in (3.4), we obtain that under assumption (AS0):
∂l ∂θ 1 = −nλ(θ)h 1 (θ) m i=1p i ϕ(W i ; θ) + nh 1 (θ) m j=m 0 +1 ω j WEIGHTED EMPIRICAL LIKELIHOOD 9 = nh 1 (θ)[ρ 1 − λ(θ)], (3.8) ∂l ∂θ 2 = nh 1 (θ) ρ 1 m j=m 0 +1p j h 2 (W j ) − λ(θ) m i=1p i ϕ(W i ; θ)h 2 (W i ) ,
where the use of ∇λ(θ) in deriving (3.8) can easily be justified by the theorems on implicit functions in mathematical analysis. Ifθ n is a solution of ∇l(θ) = 0, then λ(θ n ) = ρ 1 and
m j=m 0 +1p j h 2 (W j ) − m i=1p i ϕ(W i ;θ n )h 2 (W i ) = 0. (3.9)
In the Appendix, we show thatθ n is equivalently given by the solution of equation(s):
0 = g 1 (θ) ≡ ∞ 0 ϕ(x; θ) ρ 0 + ρ 1 ϕ(x; θ) dF (x) − ∞ 0 1 ρ 0 + ρ 1 ϕ(x; θ) dĜ(x), 0 = g 2 (θ) ≡ ∞ 0 ϕ(x; θ)h 2 (x) ρ 0 + ρ 1 ϕ(x; θ) dF (x) − ∞ 0 h 2 (x) ρ 0 + ρ 1 ϕ(x; θ) dĜ(x),(3.10)
by which we always mean thatθ n ∈ R is the solution of
g 1 (θ) = 0 if h 2 (x) ≡ 0.
For our examples, the unique existence of solutionθ n for (3.10) is shown in Sections 4 and 5, respectively, and it can be shown thatθ n maximizes l(θ) over those θ satisfying (3.5) (the proofs are omitted). Thus,θ n is the SPMLE for θ 0 in (1.1). Consequently, replacing θ byθ n in (3.6), we obtain the following SPMLEF n for F 0 :
F n (t) = m i=1p i I{W i ≤ t} = t 0 1 ρ 0 + ρ 1 ϕ(x;θ n ) d[ρ 0F (x) + ρ 1Ĝ (x)]. (3.11)
Since the equations in (3.10) only depend on the NPMLEF andĜ, thus for the rest of the paper,θ n denotes the solution of (3.10) without
assumption m 0 i=1p X i = m 1 i=1p Y i = 1 in (3.1)
, and is used to computeF n in (3.11). In the following theorems, some asymptotic results on (θ n ,F n ) are established under some of the assumptions listed below, while the proofs are deferred to the Appendix.
(AS1) (a) ϕ(x; θ) is monotone in x for any fixed θ ∈ Θ, where Θ = {θ 1 |a 1 < θ 1 < ∞} if θ ∈ R; Θ = {(θ 1 , θ 2 )|a i < θ i < ∞, i = 1, 2} if θ ∈ R 2 ; (b) ϕ(x; θ)
is increasing in θ 1 (and in θ 2 if θ ∈ R 2 ) for any fixed x > 0; (c) for fixed x > 0 (and fixed θ 2 if θ ∈ R 2 ), ϕ(x; θ) → ∞(0), as θ 1 → ∞(a 1 ); (d) for θ = (θ 1 , θ 2 ) ∈ R 2 and fixed x > 0, when −θ 1 /θ 2 → γ with 0 ≤ γ ≤ ∞: ϕ(x; θ) → 0(∞) if x < γ(x > γ), as θ 2 → ∞; ϕ(x; θ) → 0(∞) if x > γ(x < γ), as θ 2 → a 2 ; (AS2) ρ 0 = n 0 n and ρ 1 = n 1 n remain the same as n → ∞;
10 J.-J. REN (AS3) √ n 0 ∞ 0 [h 2 (x)] k ϕ(x;θ 0 ) ρ 0 +ρ 1 ϕ(x;θ 0 ) d[F (x) − F 0 (x)] D → N(0, σ 2 F,k ), as n → ∞, √ n 1 × ∞ 0 [h 2 (x)] k ρ 0 +ρ 1 ϕ(x;θ 0 ) d[Ĝ(x) − G 0 (x)] D → N(0, σ 2 G,k ), as n → ∞, where k = 0, 1, and [h 2 (x)] 0 ≡ 1; (AS4) F − F 0 a.s. → 0, Ĝ − G 0 a.s. → 0, as n → ∞; (AS5) ∞ 0 [h 2 (x)] k d[F (x) − F 0 (x)] a.s. → 0, ∞ 0 [h 2 (x)] k d[Ĝ(x) − G 0 (x)] a.s. → 0, as n → ∞, with finite ∞ 0 [h 2 (x)] k dF 0 (x) and ∞ 0 [h 2 (x)] k dG 0 (x), where k = 1, 2, 3; (AS6) √ n 0 (F − F 0 ) w ⇒ G F , √ n 1 (Ĝ − G 0 ) w ⇒ G G , as n → ∞,
where G F and G G are centered Gaussian processes. → θ 0 , as n → ∞;
(ii) √ n(θ n − θ 0 ) D → N(0, Σ 0 ), as n → ∞; (iii) F n − F 0 a.s. → 0, as n → ∞.
Remark 1 (Assumptions of theorems).
For our examples, (AS0)-(AS1) hold, which will be discussed in Sections 4 and 5, respectively. From Gill [7], Gu and Zhang [11], Huang [12], Huang and Wellner [13] and Geskus and Groeneboom [6], we know that under some suitable conditions, (AS3) holds for censored data (1.2)-(1.7) aforementioned. We also know that for these types of censored data, (AS4) holds under some suitable conditions; see Stute and Wang [30], Gu and Zhang [11], Huang [12] and Groeneboom and Wellner [10]. For right censored data, (AS5) holds under some regularity conditions (Stute and Wang [30]). For other types of censored data, (AS5) is implied by (AS4) if the support of F 0 is finite. On the other hand, if weaker consistency result is desired in Theorem 1(i), assumption (AS5) can be weakened. Moreover, from Gill [7], Gu and Zhang [11] and Huang [12], we know that (AS6) holds under some suitable conditions for right censored data, doubly censored data and partly interval-censored data. The techniques used in our proofs show that the weak convergence ofF n for interval censored data relies on that of NPMLEF orĜ for interval censored data, which is now unknown.
Biased sampling models.
For the biased sampling problem in Example 1, this section discusses assumptions (AS0)-(AS1), shows the unique existence of SPMLEθ n for θ 0 ∈ R in (1.8), and studies the weighted empirical log-likelihood ratio for w 0 .
Under (1.8), we have that in (AS0), h 1 (θ) = 1/θ for θ ∈ Θ = {θ|a 1 = 0 < θ < ∞} and h 2 (x) ≡ 0, and that (AS1)(a)-(c) obviously hold for any monotone weight function w(x), while (AS1)(d) does not apply. Since h 2 (x) ≡ 0, θ n ∈ R is determined by the first equation of (3.10). Note that (AS1)(c) and the Dominated Convergence Theorem (DCT) imply: lim θ→0 g 1 (θ) = −Ĝ(∞)/ρ 0 < 0 and lim θ→∞ g 1 (θ) =F (∞)/ρ 1 > 0. Thus, the solutionθ n of equation g 1 (θ) = 0 uniquely exists because g ′ 1 (θ) > 0 for θ > 0.
Weighted empirical log-likelihood ratio. From (3.3) and (3.6), we know that under (1.8), the weighted empirical likelihood ratio is given byR
(F ) = L(θ, F )/L(θ n ,F n ) = (θ/θ n ) nρ 1 m i=1 (p i /p i ) nω i , where F (x) = m i=1 p i I{W i ≤ x}, θ = 1/[ m i=1 p i w(W i )] andp i = ω i /[ρ 0 + ρ 1θn w(W i )]. Then, set S = { w(x) dF (x)|R(F ) ≥ c} may be used as confidence interval for w 0 , where 0 < c < 1 is a constant. Let r(θ 0 ) = sup (θ 0 /θ n ) nρ 1 m i=1 (p i /p i ) nω i |p i ≥ 0, (4.1) m i=1 p i = 1, m i=1 p i w(W i ) = 1 θ 0 .
It is easy to show that S is an interval expressed by S = [X L , X U ], and that
X L ≤ w 0 ≤ X U if and only if r(θ 0 ) ≥ c, where X L = inf{ ∞ 0 w(x) dF (x)|F ∈ F} and X U = sup{ ∞ 0 w(x) dF (x)|F ∈ F} for F = {F |R(F ) ≥ c, p i ≥ 0, m i=1 p i = 1}.
We call [X L , X U ] the weighted empirical likelihood ratio confidence interval for w 0 , and the limiting distribution of weighted empirical log-likelihood ratio for those censored data (1.2)-(1.7) is given in the following theorem with a proof sketched in the Appendix. 5. Case-control logistic regression models. For the case-control logistic regression model in Example 2, this section discusses assumptions (AS0)-(AS1), shows the unique existence of SPMLEθ n for θ 0 ∈ R 2 in (1.9), and provides a goodness-of-fit test for model (1.10).
Under (1.9), we have that in (AS0)-(AS1), h 1 (θ) ≡ 1 for θ ∈ Θ with a 1 = a 2 = −∞ and h 2 (x) = x, and that (AS1) holds for ϕ(x; θ) = exp(α + βx) with θ = (α, β) ∈ R 2 . In the Appendix, we show that the solutionθ n of (3.10) exists uniquely. Goodness-of-fit test. To assess the validity of logistic regression model assumption (1.10) with censored data, note that there are two ways to estimate d.f. F 0 in (1.9) using censored data (1.11). One is the NPMLEF based on the first sample, and the other is the SPMLEF n based on both samples under model assumption (1.10), that is, (1.9). Based on Theorems 1 and 2, we have the following corollary on the asymptotic properties ofF andF n with proofs deferred to the Appendix. (1.9). Then, as n → ∞: (iii) √ n(F n −F ) weakly converges to a centered Gaussian process under model (1.10) and assumption (AS6).
Corollary 1. Assume (AS2)-(AS5) for model(i) F n −F a.s.
Thus, from Remark 1 we know that for right censored data, doubly censored data and partly interval-censored data, we may use the following Kolmogorov-Smirnov-type statistic to measure the difference betweenF andF n , which gives a goodness-of-fit test statistic for case-control logistic regression model (1.10):
T n = √ n F n −F = √ n sup 0≤t<∞ |F n (t) −F (t)|. (5.1)
Bootstrap method. To compute the p-value for test statistic T n in (5.1), we suggest the following n out of n bootstrap method. Sinceθ n = (α n ,β n ) is determined by (3.10), it is a functional of the NPMLEF andĜ, denoted asθ n = θ(F ,Ĝ); in turn, (3.11) implies thatF n (t) −F (t) is a functional of F andĜ, denoted asF n −F = τ (F ,Ĝ). Note that under model (1.1), θ 0 is the unique solution of equation(s):
0 = g 01 (θ) ≡ ∞ 0 ϕ(x; θ) ρ 0 + ρ 1 ϕ(x; θ) dF 0 (x) − ∞ 0 1 ρ 0 + ρ 1 ϕ(x; θ) dG 0 (x),(5.2)0 = g 02 (θ) ≡ ∞ 0 ϕ(x; θ)h 2 (x) ρ 0 + ρ 1 ϕ(x; θ) dF 0 (x) − ∞ 0 h 2 (x) ρ 0 + ρ 1 ϕ(x; θ) dG 0 (x),
by which we always mean that θ 0 ∈ R is the solution of g 01 (θ) = 0 if h 2 (x) ≡ 0. Thus, under (1.9) we have θ 0 = (α 0 , β 0 ) = θ(F 0 , G 0 ); in turn, τ (F 0 , G 0 ) ≡ 0, which means T n = √ n τ (F ,Ĝ)− τ (F 0 , G 0 ) under model (1.10). Hence, from the formulation given in Bickel and Ren [3], the distribution of T n under model (1.10) can be estimated by that of T * n = √ n τ (F * ,Ĝ * ) − τ (F ,Ĝ) , whereF * andĜ * are calculated based on the n out of n bootstrap samples, respectively. For instance,F * is calculated based on the bootstrap sample O X * 1 , . . . , O X * n 0 taken with replacement from {O X 1 , . . . , O X n 0 }. The p-value is estimated by the percentage of T * n 's that are greater than test statistic T n . Note that the n out of n bootstrap consistency for √ n 0 (F − F 0 ) estimated by √ n 0 (F * −F ) has been established for right censored data, doubly censored data and partly interval-censored data by Bickel and Ren [2] and Huang [12].
Remark 2. The proposed test (5.1) can be used for any type of censored data as long as (AS2)-(AS6) hold. When (AS6) does not hold, such as for interval censored data, Corollary 1 shows that we may graphically check the model fitting for (1.10) by comparing curves ofF andF n . Note that when model (1.10) does not hold, statistic T * n is still asymptotically a function of a centered Gaussian process, but T n a.s. → ∞ based on Corollary 1(ii). Thus, our proposed test is consistent. In terms of computing (α n ,β n ), it can be done using the Newton-Raphson method described on page 374 of Press et al. [21] to solve (3.10); a computation routine in FORTRAN is available from the author. Although not presented here, our extensive simulation studies on (α n ,β n ) and the comparison between the distributions of T n and T * n give excellent results. i=1p Y i = 1, the first equation of (3.9) is equivalent to ψ(ρ 1 ; θ) = 0, which by (3.7) and (3.1)-(3.2), gives g 1 (θ) = 0 in (3.10). The proof follows from that (3.6) and λ(θ) = ρ 1 imply that the second equation of (3.9) is 0 = −ρ 0 g 2 (θ).
Proof of "UNIQUE EXISTENCE OFθn IN EXAMPLE 2." Let
R n (θ) = ∞ 0 1 ρ 1 ln[ρ 0 + ρ 1 ϕ(x; θ)] dF (x) (A.1) + ∞ 0 1 ρ 0 ln ρ 0 + ρ 1 ϕ(x; θ) ϕ(x; θ) dĜ(x).
SinceF andĜ are step functions with finite jumps, we know that R n (θ) is well defined on R 2 . From (A.1) and (3.10), we have ∇R n (θ) = h 1 (θ)(g 1 (θ), g 2 (θ)) ⊤ and
Σ Rn,θ = ∂ 2 R n ∂θ 2 1 ∂ 2 R n ∂θ 2 ∂θ 1 ∂ 2 R n ∂θ 1 ∂θ 2 ∂ 2 R n ∂θ 2 2 = (g 1 (θ), g 2 (θ)) ⊤ (∇h 1 (θ)) ⊤ (A.2) + h 2 1 (θ) ∞ 0 1 h 2 (x) h 2 (x) h 2 2 (x) × ϕ(x; θ) [ρ 0 + ρ 1 ϕ(x; θ)] 2 d[ρ 0F (x) + ρ 1Ĝ (x)].
Thus, ∇R n (θ) = 0 is equivalent to (3.10) because h 1 (θ) > 0 by (AS0). For Example 2, we have h 1 (θ) ≡ 1 and h 2 (x) = x, which imply that Σ Rn,θ is a positive-definite matrix. Hence, R n (θ) is strictly convex. Moreover, note that under (1.9), we have in (A.1) R n (θ) ≥ (ln ρ 0 )/ρ 1 + (ln ρ 1 )/ρ 0 for any θ = (α, β) ∈ R 2 , and that by a similar argument used in (6.5) of Ren and Gu [27], we can show: lim λ→∞ inf R n (λe 1 , λe 2 ) = ∞ for any e 2 1 + e 2 2 = 1. Hence, R n (θ) has a unique global minimum point which must be the solution of (3.10) (see pages 101-102 of Bazaraa, Sherali and Shetty [1]).
Proof of Theorem 1(i). Letμ(x) = ρ 0F (x) + ρ 1Ĝ (x); then (3.10) givesF
(∞) = ∞ 0 dμ(x) ρ 0 + ρ 1 ϕ(x;θ n ) ≤ 1 ρ 0 , (A.3)Ĝ (∞) = ∞ 0 ϕ(x;θ n ) dμ(x) ρ 0 + ρ 1 ϕ(x;θ n ) ≤ 1 ρ 1 , where (AS4) impliesF (∞) a.s. → 1,Ĝ(∞)
a.s. → 1, as n → ∞. As follows, we show θ n = O(1) almost surely for caseθ n = (θ (1) n ,θ (2) n ) ∈ R 2 (the proof for casẽ θ n ∈ R is similar). Assumeθ (2) n ≥ 0. Ifθ (1) n → ∞, then from integration by parts, the boundedness of the integrand function, (AS1)(b)-(c) and the DCT, we have that in (A.3):
1 = lim n→∞ ∞ 0 dµ 0 (x) ρ 0 + ρ 1 ϕ(x;θ n ) ≤ ∞ 0 lim n→∞ dµ 0 (x) ρ 0 + ρ 1 ϕ(x;θ(1)n , 0) = 0, (A.4) a contradiction, where µ 0 (x) = ρ 0 F 0 (x) + ρ 1 G 0 (x). Thus,θ(2)n ≥ 0 implies θ (1) n = O(1) orθ (1)
n → a 1 . Similarly, we know that 0 ≤θ
(2) n ≤ M 2 < ∞ and θ (1) n → a 1 imply 1 = lim ∞ 0 [ρ 0 +ρ 1 ϕ(x;θ n )] −1 dµ 0 (x) ≥ ∞ 0 lim[ρ 0 +ρ 1 ϕ(x;θ(1)
n , M 2 )] −1 dµ 0 (x) = 1/ρ 0 , a contradiction. Hence, ifθ (2) n ≥ 0, thenθ
(2) n = O(1) impliesθ (1) n = O(1). Assumeθ (2) n → ∞, −θ (1) n /θ (2) n → γ with 0 ≤ γ ≤ ∞. Similarly as (A.4), (AS1) gives 1 = ∞ 0 lim n→∞ dµ 0 (x) ρ 0 + ρ 1 ϕ(x;θ n ) = µ 0 (γ) ρ 0 , (A.5)
where we must have 0 < γ < ∞ to be inside the support of F 0 ; a contradiction otherwise. Also, if we let n → ∞ in the second equation of (3.10), then from (AS4)-(AS5), Hölder's inequality, the DCT and an argument similar to above, we have
1 ρ 1 ∞ γ h 2 (x) dF 0 (x) = 1 ρ 0 γ 0 h 2 (x) dG 0 (x). (A.6) However, (A.5)-(A.6) contradict [G 0 (γ) ∞ γ h 2 (x) dF 0 (x) −F 0 (γ) × γ 0 h 2 (x) dG 0 (x)] = x<γ<y [h 2 (y) − h 2 (x)
] dF 0 (y) dG 0 (x) = 0, which is implied by (AS0). Thus, ifθ (2) n ≥ 0, we must haveθ Assumeθ n → η 0 , as n → ∞. Then, from (3.10) and an argument similar to that used in (A.6), we know that η 0 is a solution of (5.2). Note that for nondegenerating h 2 (x), to obtain the second equation of (5.2) for η 0 we use (AS5) and the proof of Lemma 3 of Gill [8], noticing that h 2 (x) is monotone and [ρ 0 + ρ 1 ϕ(x; η 0 )] −1 is bounded and continuous. Hence, the proof follows from the uniqueness of the solution for (5.2).
Proof of Theorem 1(ii). Here, we only prove the caseθ n ∈ R 2 , because the proof for caseθ n ∈ R is similar. For R n (θ) in (A.1), we have that under model (1.1):
∇R n (θ 0 ) = h 1 (θ 0 )([g 1 (θ 0 ) − g 01 (θ 0 )], [g 2 (θ 0 ) − g 02 (θ 0 )]) ⊤ , (A.7)
∇R n (θ n ) = ∇R n (θ 0 ) + Σ Rn,θ 0 (θ n − θ 0 ) ⊤ + 1 2 (r 1 (θ n ), r 2 (θ n )) ⊤ , where g 1 , g 2 and g 01 , g 02 are given in (3.10) and (5.2), respectively; Σ Rn,θ is given in (A.2); and from (AS5), Theorem 1(i) and straightforward calculation based on (A.2), we have r i (θ n ) = o p (θ n − θ 0 ). From (A.7), (AS3), the independence betweenF andĜ, and page 4 of Serfling [29], we know that √ n∇R n (θ 0 ) converges in distribution to a normal random vector, while (A.2), (5.2) and a similar argument in (A.6) imply Σ Rn,θ 0 a.s. Proof of Theorem 1(iii). Here, we only prove the caseθ n ∈ R 2 , because the proof for caseθ n ∈ R is similar. For any t > 0, we letF n (t) ≡ g 3 (θ n ) in (3.11); theñ
→ Σ 1 = h 2 1 (θ 0 ) ∞ 0 1 h 2 (x) h 2 (x) h 2 2 (x) ϕ(x; θ 0 ) dµ 0 (x) [ρ 0 + ρ 1 ϕ(x; θ 0 )] 2 (A.8) as n → ∞,F n (t) = g 3 (θ n ) (A.10) = g 3 (θ 0 ) + (θ n − θ 0 )∇g 3 (θ 0 ) + 1 2 (θ n − θ 0 )Σ g 3 ,ξn (θ n − θ 0 ) ⊤ ,
where ξ n is betweenθ n and θ 0 , and → 0 for any fixed t > 0; in turn, the proof follows from (A.12) and Pólya's Theorem.
∇g 3 (θ) = −ρ 1 h 1 (θ) t 0 (1, h 2 (x)) ⊤ ϕ(x; θ) [ρ 0 + ρ 1 ϕ(x; θ)] 2 dμ(x), Σ g 3 ,θ = ∂ 2 g 3 ∂θ 2 1 ∂ 2 g 3 ∂θ 2 ∂θ 1 ∂ 2 g 3 ∂θ 1 ∂θ 2 ∂ 2 g 3 ∂θ 2 2 = [h 1 (θ)] −1 ∇g 3 (θ)[∇h 1 (θ)] ⊤ (A.11) − ρ 1 h 2 1 (θ) t 0 1 h 2 (x) h 2 (x) h 2 2 (x)
Proof of Theorem 2. Here, we only prove the caseθ n ∈ R 2 , because the proof for caseθ n ∈ R is similar. Let (v 1 ,v 2 ) ⊤ = ∇g 3 (θ 0 ) as in (A.11), and let (v 1 , v 2 ) ⊤ = −ρ 1 h 1 (θ 0 ) t 0 (1, h 2 (x)) ⊤ ϕ(x; θ 0 )[ρ 0 + ρ 1 ϕ(x; θ 0 )] −2 dµ 0 (x). From (AS4) and integration by parts, we have |v k (t) − v k (t)| a.s.
+ √ n t 0 u 0 (x) dμ(x) − F 0 (t) − (v 1 (t),v 2 (t))Σ −1 1 ∇R n (θ 0 ) = o p (1) + √ n(Û F − U F ) + √ n(Û G − U G ),
where for s 1 (t) = h 1 (θ 0 )[λ 11 v 1 (t) + λ 21 v 2 (t)] and s 2 (t) = h 1 (θ 0 )[λ 12 v 1 (t) +
λ 22 v 2 (t)], √ n[Û F (t) − U F (t)] ≡ τ 1 ( √ n 0 (F − F 0 )) = √ n ρ 0 t 0 u 0 (x) d[F (x) − F 0 (x)] (A.14) − s 1 (t) ∞ 0 u 1 (x) d[F (x) − F 0 (x)] − s 2 (t) ∞ 0 u 1 (x)h 2 (x) d[F (x) − F 0 (x)] , √ n[Û G (t) − U G (t)] ≡ τ 2 ( √ n 1 (Ĝ − G 0 )) = √ n ρ 1 t 0 u 0 (x) d[Ĝ(x) − G 0 (x)] (A.15) + s 1 (t) ∞ 0 u 0 (x) d[Ĝ(x) − G 0 (x)] + s 2 (t) ∞ 0 u 0 (x)h 2 (x) d[Ĝ(x) − G 0 (x)] .
As (A.14) is a linear functional of √ n 0 (F − F 0 ), (AS6) implies √ n(Û F − U F ) w ⇒ τ 1 (G F ), as n → ∞, where from pages 154-157 of Iranpour and Chacon [14], we know that τ 1 (G F ) is a centered Gaussian process. Similarly, √ n[Û G − U G ] in (A.15) weakly converges to a centered Gaussian process τ 2 (G G ). The proof follows from (A.13)-(A.15), and that τ 1 (G F ) and τ 2 (G G ) are two independent centered Gaussian processes.
Proof of Corollary 1. Note that part (i) follows directly from Theorem 1(iii) and (AS4), while part (iii) follows from some minor adjustments in the proof of Theorem 2. Thus, we only give the proof of part (ii) as follows.
0 are the density functions of unknown nonnegative distribution functions (d.f.) This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2008, Vol. 36, No. 1, 147-166. This reprint differs from the original in pagination and typographic detail.
Theorem 1 .
1Assume (AS0)-(AS5). Under model (1.1), we have:
Theorem 2 .
2Assume (AS0)-(AS6). Under model (1.1), we have that √ n(F n − F 0 )weakly converges to a centered Gaussian process.
Theorem 3 .
3Assume (AS2)-(AS5) for model(1.8). Then, −2 ln r(θ 0 ) D → c 0 χ 2 1 , as n → ∞, where 0 < c 0 < ∞ is a constant and χ 2 1 has a chi-squared distribution.
→ 0
0when model (1.10) does not hold [i.e., g 0 (x) a.e. = ϕ(x; θ 0 )× f 0 (x) does not hold], where F 1 = F 0 ;
APPENDIX
Proof of "θ n IS EQUIVALENTLY GIVEN BY THE SOLUTION OF (3.10)."
0. Hence, we haveθ n = O(1) almost surely.
where Σ 1
1is positive-definite. Hence, ∇R n (θ n ) = 0, (A.7)-(A.8) and Theorem 1(i) give √ n(θ n − θ 0 ) = −Σ −1 1 √ n∇R n (θ 0 ) + o p (1).
x; ξ n )h 2 (x)[(∂ 2 h 1 (ξ n )/∂θ 2 2 ) + h 2 1 (ξ n )h 2 (x)] [ρ 0 + ρ 1 ϕ(x; ξ n )] 2 dμ(x) = O a.s. (1),which also holds for other partial derivatives in (A.11). Thus, Theorem 1(ii) implies that with (θ n − θ 0 )∇g 3 (θ 0 ) = o a.s. (1), (A.10) can be written asF n (t) = g 3 (θ 0 ) + (θ n − θ 0 )∇g 3 (θ 0 ) + O a.s. (|θ n − θ 0 | 2 ). (A.12)From (AS4) and integration by parts, we have |g 3 (θ 0 ) − F 0 (t)| a.s.
fixed t > 0, where k = 1, 2. Sincev k (t) and v k (t) are continuous and monotone in t, then from (AS5) and a similar argument used in the proof of Theorem 1(i) for showing η 0 as the solution of (5.2), we have v k − v k a.s. → 0, as n → ∞. Thus, if we let u 0 (x) = 1/[ρ 0 + ρ 1 ϕ(x; θ 0 )], u 1 (x) = u 0 (x)ϕ(x; θ 0 ), and λ ij the elements of Σ −1 1 , then (A.7), (A.9), (A.12), (1.1) and Theorem 1(ii) imply √ n[F n (t) − F 0
Acknowledgments. The author is very grateful to the referees, the Associate Editor and Editor Jianqing Fan for their comments and suggestions on the earlier version of the manuscript, which led to a much improved paper.J.-J.RENHere, we have h 1 (θ) ≡ 1 and h 2 (x) = x; thus in (A.2) we have ∇h 1 (θ) ≡ 0. From the proofs of the unique existence ofθ n and Theorem 1(i), we know that when model (1.10) does not hold,θ n is still well defined, and satisfies |θ n − θ 1 | a.s. → 0, as n → ∞, where θ 1 = (α 1 , β 1 ) is the unique solution of (5.2) for ϕ(x; θ) = exp(α + βx). Applying this, (AS4) and integration by parts to (3.11), we have F n − F 1 a.s.It is easy to verify that F 1 = F 0 when (1.10) does not hold [otherwise, we have g 0 (x) a.e. = ϕ(x; θ 1 )f 0 (x) with θ 1 = θ 0 ], and that the first equation of (5.2) implies that F 1 is a distribution function.Proof of Theorem 3. For a simpler argument, we assume m 0, which can be removed with some additional work in our proof here. To get an expression of r(θ 0 ), it can be shown by using the Lagrange multiplier method that the solution of the maximization problem in (4.1)Using Taylor's expansion on φ(λ), we have that from ψ(ρ 1 ;θ n ) = 0 in (3.7),where ξ is between ρ 1 and λ 0 , and ξ i is between θ 0 andθ n . From (AS4), integration by parts and Theorem 1(i), we know that (A.17) impliesAlso using Taylor's expansion, we havewhere η i is between ρ 1 and λ 0 , while ζ i and ζ are between θ 0 andθ n . Since (A.18) and Theorem 1(ii) imply (ρ 1 − λ 0 ) = O p (n −1/2 ), then fromHence, the proof follows from Theorem 1(ii) and applying (A.18) to (A.22), where the limits of the coefficients of n(θ n − θ 0 ) 2 are handled similarly to (A.8).
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|
[] |
[
"Efficient Group Key Management Schemes for Multicast Dynamic Communication Systems",
"Efficient Group Key Management Schemes for Multicast Dynamic Communication Systems"
] |
[
"Muhammad Yasir ",
"Malik "
] |
[] |
[] |
Key management in multicast dynamic groups, where users can leave or join at their ease is one of the most crucial and essential part of secure communication. Various efficient management strategies have been proposed during last decade that aim to decrease encryption costs and transmission overheads. In this report, two different types of key management schemes are proposed. First proposed scheme is based on One-way function tree (OFT).The proposed scheme fulfills the security gaps that have been pointed out in recent years. Second proposed scheme is based on logical key hierarchy (LKH). This proposed scheme provides better performance for, rather inflexible and expensive, LKH scheme.
| null |
[
"https://arxiv.org/pdf/1211.3502v1.pdf"
] | 1,976,866 |
1211.3502
|
2020262e540d2bdf76a5fa1b6ea2669bdc86df14
|
Efficient Group Key Management Schemes for Multicast Dynamic Communication Systems
2012 i
Muhammad Yasir
Malik
Efficient Group Key Management Schemes for Multicast Dynamic Communication Systems
2012 iGroup communication, key
Key management in multicast dynamic groups, where users can leave or join at their ease is one of the most crucial and essential part of secure communication. Various efficient management strategies have been proposed during last decade that aim to decrease encryption costs and transmission overheads. In this report, two different types of key management schemes are proposed. First proposed scheme is based on One-way function tree (OFT).The proposed scheme fulfills the security gaps that have been pointed out in recent years. Second proposed scheme is based on logical key hierarchy (LKH). This proposed scheme provides better performance for, rather inflexible and expensive, LKH scheme.
Table of Contents
Internet protocol (IP) multicast, also known as multicast, is used to share contents with multiple users in a group. This form of communication tends to be efficient in terms of bandwidth as compared to unicast protocols, as it transmits information to every user in the group simultaneously. Internet group management protocol (IGMP) [9] is an example of multicast systems, in which any member can broadcast data to all n members in the group. Any user can join and receive contents in IGMP, which makes it a scalable system. Lack of access control and authentication poses security threats to IGMP, as any host can send and receive data from these systems. Conventional method to enforce restrictions on data flow is use of encryption to secure data contents, such that only the desired hosts can gather the data by utilizing cryptographic keys. Only the members with appropriate key can decrypt the data, which makes the communication secure and reliable.
Key distribution centre (KDC) or server is responsible for authentication of a user interested in joining the group. Server authenticates the user and allocates location to newly joined user. Server also provides the necessary keys to the user, which enables this user to communicate within the group.
Key management protocols are responsible for key pre-distribution and key updating in case of changes in the group. Group keys are shared among all the members and the contents to be shared are encrypted with this key and broadcasted to all group members. Such groups are supposed to be flexible enough to allow new hosts to join and present members to leave. Joining and leaving of hosts require change in the group key, so that the privacy and secrecy of the group members and their communication can be preserved. To maintain group keys, secure key management protocols are devised and employed. These protocols provide the authentication services, along with changing of the group keys with each user joining and leaving. The process of changing keys on every user join or leave is called key updating or rekeying.
Lack of presence of any key management protocol has rekeying cost of nK for n users. Logical key hierarchy (LKH) [5] and one-way function key tree (OFT) [1] are two much efficient centralized key management schemes. These both schemes differ in their functionalities; OFT follows down-up strategy as opposed to LKH which follows a static key tree structure. LKH has broadcast costs of 2hk h for n users, where k is key size in bits and h is the height of tree. OFT scheme includes users along with the server in key updating process.
This makes the scheme more efficient and it cuts the overhead cost at rekeying by 1/ 2 , i.e., hk h .
Security Requirements
Centralized key management schemes must, in all conditions, fulfill some security requirements. Their basic security requirements are forward and backward secrecy in the group. Security requirements are discussed in detail in Chapter 2.
Forward Secrecy: Evicted members of the group are unable to access new information in the group, which states they cannot compute (or access) newer group keys.
Backward Secrecy: New members of the group are unable to access previous information in the group, which states that they cannot compute older group keys.
Objectives
To fill the security gap caused by collusion attacks on OFT and to reduce comparatively higher cost of LKH scheme, several improvements in both of these key management schemes have been proposed in this thesis.
OFT scheme is found to be weak against attacks by adversaries. We propose an improved OFT scheme, which guarantees better security at minimum costs.
We also propose a simple LKH scheme for key distribution which provides the same functionalities at lower transmission cost.
Our improvements guarantee less cost in both schemes.
Key Management Schemes
This chapter describes two main centralized key management schemes in detail; logical key hierarchy and one-way function key tree. This chapter also covers collusion attacks on OFT scheme and revisited security requirements
for key management schemes.
Logical Key Hierarchy
LKH maps all members of the group as leaves of a structured tree, most commonly as a balanced binary tree. Group key is at the root of the tree, whereas the leaves represent group members. Group members have to store their individual keys, group key, and all node keys in the path from member to the group key. On each user leave or join, group key and other node keys in the path must be changed. New group key can be distributed by following the algorithms for join and leave functions, which will be defined in next subsections.
The complexity for key distribution to n users will be ) (log n O .
LKH Tree Structure
All users in the group store 2 log 1 n keys, out of which one is their individual key and all other h keys belong to the middle nodes in its path to the group node key. These h keys with the user need to be changed after each join or leave.
On User Join
Suppose user 8 u joins the group and users 1 u to 7 u are the present members.
The group key k and two node keys 78 k and 58 k are changed to ' k , 78 ', k and 58 ', k respectively.
The key distribution process for each user join can be listed as:
1. Server authenticates the interested user 8 u and allocates it an empty place in the group tree. Server also provides individual key 8 k to the new user.
2. All the keys which 8 u needs in order to communicate with group members, i.e., ' k , 78 ', k and 58 ' k are sent by unicast to 8 u encrypted with its individual key 8 k .
New group key '
k is sent by multicast to 5 6 7 ( , , ) u u u and 1 2 3 4 ( , , , ) u u u u after being encrypted with 58 ' k and 14 , k respectively.
On User Leave
On eviction of user, middle node keys must be changed to preserve communication secrecy in the group.
Suppose user 8 u leaves the group. Here, the group key k and node key 58 k must be changed. Their new values can be represented as ' k and 57 ', k respectively.
User 8
u node is deleted from the key tree at first.
2. User 8 u sibling's node, 7 , u moves to its parent's node. 3.
One-Way Function Key Tree
OFT key management scheme decreases server-level computation, as computation load is distributed between the server and group members.
Rekeying overhead for OFT is hk h . Details of these functions are as follows.
• One-way function, () g : The keys are passed through a strong oneway function to hide the contents of the original key. These "blinded" keys can be shared to corresponding users without any security concerns.
• Joining function, ( , ) f a b : This function concatenates or combines two entities, a and . b Members can calculate desired key i k by following formula
( ) ( ) ( ( ), ( )) i left i right i k f g k g k where () left i k and () right i k
denote left and right children node keys of the node key i k .
OFT Structure
In OFT, keys are dependent on each other. Group members have knowledge of certain blinded node keys, which enables them to generate new keys on every user join and leave. All members know their sibling's blinded node key as well as their ancestors' sibling's blinded node keys. For example, user 1 u also stores k Now, members can compute their node keys and more importantly, the group key by using these known values. Left subgroup with users 1 2 3 4 ( , , , ) u u u u performs the following operations to find their node keys as ( ( ), ( )).
k f g k g k k f g k g k k f g k g k
Group key k can be generated by all members of the group as 14 58
( ( ), ( )). k f g k g k
Here, server as well as all group users participates to compute the group key.
On User Join
:{ ( )} multicast u u g k k server u u g k k u g k k where { ( )} i gk denotes the encryption of value () g by key i k .
On User Leave
On every user leave, group key must be changed to preserve forward and backward secrecy. The sibling of the evicted user is assigned with new key and it moves up to their parent's node. The group key alters because of these steps.
Suppose user 8 u leaves the group. Group key ' k and node key 58 ' k are computed by members of the group by using blinded node keys given as
58 56 7 14 58 ' ( ( ), ( ' )) ' ( ( ), ( ' )). k f g k g k k f g k g k
Updated right subgroup key 58 ' k is shared with complementary left subgroup, whose members can calculate the new group key.
By using this procedure, the need of multicasting all updated keys is reduced, as the users can compute necessary keys themselves. Only few blinded node keys are sent to particular users and subgroups. OFT reduces the required broadcasts to nearly half as compared to LKH scheme.
Collusion Attacks on OFT
Horng's Attack
Horng [6] shows that OFT scheme is susceptible to collusion attacks, where leaving and joining users can collude their information of group keys to find older or newer group keys as shown in Figure 2 Here we present an example of such an attack on OFT.
Ku and Chen's Attack
Ku and Chen [10] present some other cases of collusion attacks on OFT. We present these conditions here. If Alice is evicted at state t 1 and Bob is added to the same subgroup at time t 2 , they can collude to get the value of G l subgroup key between time intervals t 1 and t 2 , as observed in Horng's attack. As both members also know the blinded node key of subgroup G r , they can easily compute group key between time intervals t 1 and t 2 . Also, consider that Alice leaves the group at time t 1 .
After this, Bob joins the group at time t 2 , whereas Candy joins the group at time t 3 at locations shown in Figure 2 Alice and Bob are associated with G l , whereas Candy belongs to G r . Here, Alice knows the blinded node key of subgroup G r between times t 1 and t 3 .
Candy knows the blinded node key of subgroup G l between time t 2 and t 3 .
Alice and Candy can share their information about blinded node keys of subgroups to compute group key between time t 2 and t 3 .
In both of these cases, successful collusion attacks occur, which compromise the security of group communication. OFT scheme has security vulnerabilities, which must be addressed.
Security Requirements
Now we list some security requirements which must be fulfilled by all key management schemes in order to ensure secure communication in dynamic multicast systems.
1. Group key secrecy: Any passive adversary is unable in any way to compute previous or existing group key. This also implies that adversary is also unable to find any changed node key in the group.
Mathematical operations and random numbers involved in rekeying must be cryptographically strong.
2.
Forward key secrecy: Passive adversaries or former members of the group, who may know any subset of older group keys, cannot find any new group key.
3. Backward key secrecy: Passive adversaries or present members of the group, who may know any subset of group keys, are unable to discover any previously used group key.
Key independence: Passive adversaries or former and present
members of the group, who may know any subset of group keys, are unable to discover any other group key.
5. Reuse of known node keys: Evicted members must not discover any new information that is flowing within the group. Sometimes evicted users can use their prior knowledge of node keys to decrypt any future transmission. All node keys known to a leaving member must be changed during rekeying process.
6. OFT group key segments: Group key in OFT scheme is combination of blinded node keys of its two children. These two children nodes represent left and right subgroup node keys. For each user leave, both segments of the group key must be changed. This step prevents occurring of any collusion attacks.
New Secure OFT Scheme
In this chapter, improvement in the OFT schemes will be introduced, that is, OFT scheme is vulnerable to collusion attacks and thus this scheme needs extra steps to make it reliable enough for proper functionalities.
We propose new OFT scheme with more security and lesser costs.
Introduction
In OFT scheme, all members are given the blinded node keys of their siblings and their ancestors' siblings. They can then calculate the desired node and group keys by using these blinded node keys. In this way, a part of computation load is transferred to the users from the server. In OFT, rekeying overhead decreases as a result of combined computations by the server and members.
First we define the levels and locations of the members in the group. siblings, to communicate within group and update the keys whenever needed.
These blinded node keys are kept with their indices, in the user's memory as shown in Figure 3-2. This helps in maintaining and updating these keys whenever there is some change in the group.
For example, the keys with the user 1 u can be written as shown in Table 3.1.
On User Leave
In case of eviction, the sibling of the removed user takes the place of their parent. Also its sibling key changes, which causes change in the node key and the group key. Figure 3-2 shows eviction of 8 u after which 7 u promotes to a higher level. Shaded nodes are the ones to be altered.
Key Requirements
Node keys are dependent in OFT scheme and users contribute in generation of node keys. Users have blinded node keys of their siblings and parent's siblings, which are used to efficiently generate new keys without much intervention from server. Group members update the keys present with them and generate new node keys and the group key. New keys can be generated as explained in the following protocol.
Algorithm on User Eviction
We present the algorithm which is followed by our scheme at user eviction.
1.
Server removes the node of evicted member from the key tree and promotes its sibling, if any, to higher level which was the level of its parent before eviction.
2.
Server provides new node key to evicted member's sibling. It also shares the changed blinded node keys with appropriate neighboring members.
All members of the affected subgroup can compute new subgroup
node key.
Server shares this new subgroup node key in blinded form with
neighboring unaffected subgroup members.
5.
For the unaffected subgroup, server generates and shares a random number with its members.
6.
Members update stored node keys by XORing them with the provided random number, after which the resulting values are passed through one-way function to generate new node keys.
All members of unaffected subgroup can calculate new subgroup
node key by utilizing one-way and combining functions on blinded node key.
8.
This new subgroup node key is then shared with the neighboring (unaffected) subgroup.
9.
All members of the group can use blinded node keys of affected 6. Users can change the blinded node keys with them, so as to alter the overall left blinded subgroup node keys. Already available blinded node keys can be changed like as shown in Table 3.2.
Simulation and Results
Results of our proposed scheme and the conventional key management schemes are shown in this section. Table 3.3 shows security properties of some of the known schemes, and shows that our proposed scheme has got better security strength.
Forward Backward Simple Y Y Y GKMP [4] N Y Y LKH [9]
Y Y Y OFT [7] N N N Ku and Chen [10] Y
Y Y Xu et al. [11] Y Y Y Proposed sol. Y Y Y
Tables 3.4 and 3.5 show performance of various schemes. They show that our scheme has less broadcast costs for user leave, as compared to scheme by Ku and Chen [10]. Our scheme performs even more efficiently than OFT for large group sizes. (log ( )) log ( ) nn Proposed sol. As seen in this case also, Alice is not able to collude with present members in order to find illegitimate group keys.
Thus, the proposed scheme prevents any collusion attacks.
Multicast Scheme Based on LKH
In this chapter, we will provide a new scheme based on LKH. Our proposed scheme is more efficient than the original LKH scheme in terms of communication overheads needed at rekeying.
System Design
Design Principles
Firstly, we outline some basic principles, which our proposed scheme will comply. We use binary key tree, which tends to balance itself in order to maintain the symmetry. Interested hosts can join the group through a process which includes sending request to the server and passing the authentication by the server.
Server provides an empty place on the group to the new users. Server also provides the new member with all the necessary keys through the key management protocol. Similarly, present members can leave the group by sending request to the server, who in return eradicates the user and its corresponding node from the key tree. Sibling of the leaving node moves to the position of their parent node.
Detailed Outline
On each join or leave, keys in the path from that location to server need to be changed. This ensures the backward and forward secrecy requirements of secure group communication. Server changes 2 log n keys for each join, and 2 log 1 n keys for each user leave.
The keys, which intend to be changed, affects 2 l users, where l is the level of the key. This calls for an efficient protocol, capable of sharing new keys among all members of the group. Our proposed scheme possesses the following distinctive features while distributing keys for right subgroup.
• Our scheme follows bottom-to-top approach, where bottom node keys are firstly distributed to the desired members, moving upwards.
• Higher level node keys are encrypted with lower level ones, and multicast to subsequent subgroups. For example, subgroup key at level-1 will be distributed by encrypting it with individual keys at level-0. After that, a subgroup key at level-2 is encrypted with level-1 key before multicast, and so on.
• Instead of wasting resources by sending all keys to only one user, our proposed scheme moves in a step-wise manner thus providing all essential keys to members.
On User Join
After the interested host is successfully authenticated, server allocates the host an empty location in the group. Group key tree is renewed, according to following protocol. Example protocol for height 3 h is described below, which will be generalized afterwards.
Key Requirements
Depending on their location in the group, members require different keys to update the essential keys. Desired keys by user can be outlined as
Protocol for User Join
To share keys among the members of the group, they are encrypted by individual or subgroup keys and sent through unicast or multicast to respective members.
{} ij kk represents encryption of the group key k by any subgroup key ij k . The same notation is used in describing the schemes.
The protocol for key management on user join is given below, where the keys are being transmitted by the server to various locations.
On User Leave
User leave makes an empty slot in the binary balanced key tree. Sibling of the leaving user gets promoted to the position of its parent's node.
Key Requirements
For each user leave, server generates and shares 2 log 1 n keys to remaining members. The demand for keys differs with the member's location. Keys needed for update can be outlined as
Protocol for User Leave
Less number of keys is shared on each user leave, contrary to the number of keys distributed for each user join.
The protocol for key management on each user leave is described below, where the keys are being transmitted by the server to various locations encrypted by either individual or subgroup keys.
Simulation and Results
Performance of our proposed scheme as compared to LKH scheme is shown in this section.
Performance Comparison
Tables 4.1 and 4.2 show performance of various schemes. They show that our scheme has less broadcast costs for user leave and join as compared to other key management schemes. Our scheme performs more efficiently than LKH and the communication overhead of our scheme is less than that of LKH. Encryption cost of our proposed scheme is also less than the original LKH scheme.
Empirical Analysis
Following table shows broadcast costs at user join and leave for LKH and our scheme. The improvement in results can be analyzed from the table. Table 4.3 shows that our scheme has less broadcast costs for user join and leave, as compared to LKH. Thus, our scheme is more efficient than LKH in terms of costs. Leave 10 30 20 20 18 12 36 24 24 22 13 39 26 26 24 14 42 28 28 26 15 45 30 30 28 16 48 32 32 30 17 51 34 34 32 18 54 36 36 34 Table 4.4 compares our proposed scheme at user join and leave with original LKH scheme. It is clear that even for large groups, our proposed scheme gives better overhead costs. The following figures also show that our proposed scheme performs better with different number of users.
Conclusion
Key management in dynamic groups, where users can leave or join at their ease is an important part of secure communication. Different strategies have been proposed during last decade that aim to either improve the security or the performance of key management schemes. Decreasing the encryption and transmission overheads has also been a major concern for such schemes.
In this thesis, we proposed two schemes based on different architectures. One of the schemes improves the security of OFT scheme. We showed the resilience of proposed scheme by analyzing different cases. The other proposed scheme improves the performance of independent key hierarchy system (LKH).
Both proposed schemes provide better broadcast and transmission costs than previously published schemes.
Figure 1 - 1
11of Contents .................................................................................. ii List of Tables ......................................................................................... v 1 Introduction ................................................................................... 1 1.1 Background ....................................................................................... 1 1.2 Security Requirements ...................................................................... 3 1.3 Objectives .........................................................................................Secure centralized multicast system. ............................................... 2 Figure 2-1 Logical key tree. .............................................................................. 6 Figure 2-2 OFT key tree of height h=3. ............................................................ 8 Figure 2-3 User u 3 is leaving the group and user u 5 joins the group. .............. 12 Figure 2-4 User u 1 leaves the group and users u 4 and u 5 join the group. ....... 13 Figure 3-1 Key tree with levels ....................................................................... 17 Figure 3-2 Key tree with location indices. ...................................................... 18 Figure 4-1 Binary key tree on a user join. ....................................................... 28 Figure 4-2 Binary key tree on a user leave. ..................................................... 31 Figure 4-3 Comparison between the proposed scheme and LKH on user join. ......................................................................................................................... 35 Figure 4-4 Comparison between the proposed scheme and LKH on user leave. ......................................................................................................................... 35 v List of Tables
Figure 1 -
11 shows structure of a secure centralized multicast system.
Figure 1 - 1
11Secure centralized multicast system.
Figure 2 -
21 shows a balanced binary LKH key tree with height 3. h If the number of users in the group is n , then height of group is given as 2 log hn .
Figure 2 - 1
21Logical key tree.
58 k
58and k are updated to new values, '
Figure 2 -
22 shows OFT key management scheme, where ( , ) f a b is a mixing function and (.) g a one-way hash function. The value of (.) g is called blinded node key.
Figure 2 - 2
22OFT key tree of height h=3.
gk along with its individual key 1 .
node key 14 k , which is left child node of group key, is shared with other subgroup. Users5 6 7 8 ( , , , ) u u u u can compute key for right child node of group key, which is then shared with neighboring
Figure 2 -
22 shows the case of use joining, where a new user has just entered the group. When a user, say8 u , joins the group, he will receive his sibling'values of the calculated keys are encrypted with their sibling keys and advertised to existing group members by multicast as follows:
- 3 .
3This weakens the security notions of forward and backward secrecy.
Figure 2 - 3
23User u 3 is leaving the group and user u 5 joins the group.Suppose the initial group key to is unsuccessful to provide forward secrecy against3 u and backward secrecy against5 . u
Figure 2 - 4
24User u 1 leaves the group and users u 4 and u 5 join the group.ConsiderFigure 2-4 for these attacks. Alice is represented by user u 1 in the group; Bob is associated with user u 4 in the key tree; Candy is related to user u 5 . Also, time intervals follow the relation t 3 > t 2 > t 1 .
- 4 .
4
Figure 3 -
31 shows how we name each user based on their level and location within level in the group. This will help us in introducing the scheme which provides better key management performance along with ensuring better security against the collusion attacks.
Figure 3 - 1
31Key tree with levels.In the above figure,• TEK: traffic encryption key (group key), used for communicating with all the users in the group (highest level node key)• KEK: key encryption keys, also called as sub-group keys. They are used for encrypting the group key for its transfer (intermediate level node keys).• IK: Individual keys of users (lowest level node keys)• Height of tree (h): Number of levels in the tree. For example, the tree shown in Figure 3-1 has height of 3.
Figure 3 - 2
32Key tree with location indices. We provide all the users with location indices which give information about their level and location in the group. Users must have 2 log n blinded node keys, which are the blinded node keys of their siblings and their ancestors'
Figure 4
4keys, namely, k , 58 , k and 78 k are affected. Change of 78 k will affect two users 7 u and 8 u . Similarly, changing subgroup key 58 k affects four users 58 uu . Each group member needs to change its group key k , on every user join and leave.
Figure 4 - 1
41Binary key tree on a user join.
Figure 4
4the group, forming a subgroup with 7 u . The shaded keys in the figure are changed to new ones by the server.
Figure 4 -
42 shows user 8 u leaving the tree, after which user7 u resides on level-1. Shaded nodes show the compromised keys that will be changed by the server.
Figure 4 - 2
42Binary key tree on a user leave.
are new group key and subgroup key, respectively.
Figure 4 - 3
43Comparison between the proposed scheme and LKH on user join.
Figure 4 - 4
44Comparison between the proposed scheme and LKH on user leave.
Table
Table 3 .
3) ............................................................ 19 Security of key management schemes ........................................... 23 Table 3.4 Performance comparison of key management schemes ................. 23 Table 3.5 Encryption costs for different schemes ........................................... 24 Table 4.1 Performance comparison of hierarchical schemes ......................... 33 Table 4.2 Encryption costs for different schemes ........................................... 33 Table 4.3 Broadcast costs of schemes ............................................................. 34 Table 4.4 Comparison of our solution ............................................................. 34 With increasing communication services, users are often grouped in various applications. They normally form centralized or decentralized structures, capable of handling entities involved in functions ranging from web and mail to sensor networks, file sharing to databases, and so on. Applications of such groups are enormous, and so are the demands for secure and reliable communication in these groups.1 The keys with user 1
u (0,1Table 3.2 Blinded node key change operations for user 1
u (0,1) .................... 22
Table 3.3
Table 3 .
31 The keys with user
1
u (0,1)
Indices
Key stored
(0,2)
2
()
gk
(1,2)
34
()
gk
(2,2)
58
()
gk
Similarly, other users have the blinded node keys stored with the same
pattern.
Our scheme performs only at user eviction. On user join, it follows the
scheme of original OFT.
Table 3 .
32 Blinded node key change operations for user 1
u (0,1)
Original blinded node keys
New blinded node keys
Table 3 .
33 Security of key management schemes
Schemes
Secrecy
Secure against
collusion attacks
Table 3 .
34 Performance comparison of key management schemes
Schemes
Message
Join
Leave
multicast
unicast
Simple
nK
K
nK
GKMP
2K
2K
-
LKH
Table 3 .
35 Encryption costs for different schemes
Schemes
Join
Leave
LKH
2
3log ( )
n
2
2log ( )
n
OFT
2
2log ( ) 2
n
2
log ( )
n
Ku and Chen
2
2log ( ) 2
n
2
22
2. For right subgroup, key distribution starts at the bottom where the bottom-most node key 78 ' k is encrypted with individual keys of both1. Server encrypts new group key '
k with subgroup key 14
k and
multicasts it to left subgroup as
1
4
14
:{ '} .
u
u
k k
users 7
u and 8
u , before being unicast to them as
7
78
7
8
78
8
:{ ' }
:{ ' } .
u
k
k
u
k
k
3. Level-2 node key 58
'
k is encrypted with level-1 node keys 56
k and
78
'
k , and shared as
56
58 56
78
58
78
and
:{ ' }
and
:{ ' } ' .
u
u
k
k
u
u k
k
4. Right subgroup gets its new group key '
k by encrypting and
multicasting it as
48
58
:{ '} ' .
u
u k k
1 .
1Server encrypts new group key ' k with subgroup key 14 k and multicasts it to left subgroup as 2. As for right subgroup, key distribution starts at the bottom. Level-21
4
14
:{ '} .
u
u
k k
node key 57
'
k is encrypted with node key 56
k and individual key of
user 7
u , 7
k , before being multicast and unicast, respectively.
56
57 56
7
57 7
and :{ ' }
:{ ' }
u
u k
k
u k
k
3. Right subgroup gets its new group key '
k by encrypting and
multicasting it as
47
57
:{ '} ' .
u u k k
Table 4 .
41 Performance comparison of hierarchical schemesSchemes
Message
Join
Leave
Multicast
Unicast
Simple
nK
K
nK
GKMP
2K
2K
-
LKH
2
2log ( )
n
2
log ( )
n
2
2log ( )
n
Our solution
2
log ( )
n
2
log ( ) 1
n
2
log ( ) 1
n
Table 4 .
42 Encryption costs for different schemesSchemes
Join
Leave
LKH
2
3log ( )
n
2
2log ( )
n
Our solution
2
2log ( )
n
2
2log ( ) 2
n
Table 4 .
43 Broadcast costs of schemesSchemes
Height
LKH
Our solution
Join
Leave
Join
Table 4 .
44 Comparison of our solution Height Our sol./LKH Following Figures 4-3 and 4-4 show the performance comparison of our proposed scheme with LKH for different heights of key tree.10
12
13
14
15
16
17
18
Join
0.67 0.67
0.67
0.67
0.67
0.67
0.67
0.67
Leave
0.9
0.917 0.923 0.928 0.933 0.9375 0.941 0.944
, ( ) k g k
and unaffected subgroup to calculate new group key.An Example of User EvictionUser8u leaves the subgroup with users5 6 7,, u u u , and we define this subgroup as right, r G . The other subgroup is defined as left subgroup, l G .Group key is combination function of blinded node keys of subgroups r G and l G . Our protocol changes both of these subgroup keys in order to prevent collusion attacks. Group keys at eviction and joining for different cases are given below.Case 1:Consider the case when both Bob and Candy join same subgroup, r G .Then, we can write group key G k at different steps as follows.Step 1: Alice evictsStep 2: Bob joinsStep 3: Candy joinsAs obvious in this case, Alice is not able to collude with any present member in order to find illegitimate group keys.Case 2:Consider the case when Bob joins subgroup l G , whereas Candy joins the subgroup r G .Then, we can write group key G k at different steps as follows.Step Step 3: Candy joins r G
Key management for large dynamic groups: One-way functions trees and amortized initialization. D Balenson, D Mcgrew, A Sherman, IETF Internet Draft. D. Balenson, D. McGrew, and A. Sherman, "Key management for large dynamic groups: One-way functions trees and amortized initialization," IETF Internet Draft, 1999.
A taxonomy of multicast security issues. R Canetti, B Pinkas, IETF Internet Draft. R. Canetti and B. Pinkas, "A taxonomy of multicast security issues," IETF Internet Draft, 1998
Multicast security: A taxonomy and efficient constructions. R Canetti, J Garey, G Itkis, D Micciancio, M Naor, B Pinkas, Proceedings of IEEE Infocomm'99. IEEE Infocomm'992R. Canetti, J. Garey, G. Itkis, D. Micciancio, M. Naor, and B. Pinkas, "Multicast security: A taxonomy and efficient constructions," Proceedings of IEEE Infocomm'99, vol. 2, pp. 708-716, Mar. 1999.
Group key management protocol architecture. H Harney, C Muckenhirn, T Rivers, IETF. H. Harney, C. Muckenhirn, and T. Rivers, "Group key management protocol architecture," IETF, RFC2093, 1997.
Logical key hierarchy protocol," draft-harneysparta-lkhp-sec-00.txt, IETF Internet Draft. H Harney, E Harder, H. Harney and E. Harder, "Logical key hierarchy protocol," draft-harney- sparta-lkhp-sec-00.txt, IETF Internet Draft, 1999.
Cryptanalysis of a key management scheme for secure multicast communications. G Horng, IEICE Trans. Commun. 5G. Horng, "Cryptanalysis of a key management scheme for secure multicast communications" IEICE Trans. Commun., vol. E85-B, no. 5, pp. 1050-1051, 2002.
Key establishment in large dynamic groups using one-way function trees. D Mcgrew, A David, T Alan, A Sherman, no.0755TIS Labs at Network Associates, IncGlenwood, MDTIS ReportD. McGrew, A. David, T. Alan, and A. Sherman, "Key establishment in large dynamic groups using one-way function trees," TIS Report no.0755, TIS Labs at Network Associates, Inc., Glenwood, MD, 1998.
Multicast over wireless networks. U Varshney, Commun. ACM. 4512U. Varshney, "Multicast over wireless networks," Commun. ACM, vol. 45, no. 12, pp. 31-37, Dec. 2002.
Key management for multicast: Issues and architectures. D Wallner, E Harder, R Agee, IETF. 2627D. Wallner, E. Harder, and R. Agee, "Key management for multicast: Issues and architectures," IETF, RFC2627, 1999.
An improved key management scheme for large dynamic groups using one-way function trees. Wei- , Chi Ku, Shuai-Min Chen, Proc. nullWei-Chi Ku and Shuai-Min Chen," An improved key management scheme for large dynamic groups using one-way function trees," Proc.
ICPPW'03. ICPPW'03, pp 391-396, Oct. 2003.
Preventing collusion attacks on the one-way function tree (OFT) scheme. Xuxin Xu, Lingyu Wang, Amr Youssef, Bo Zhu, ACNS '07Xuxin Xu, Lingyu Wang, Amr Youssef, and Bo Zhu, "Preventing collusion attacks on the one-way function tree (OFT) scheme," ACNS '07
Proceedings of the 5th International Conference on Applied Cryptography and Network Security Springer-Verlag. the 5th International Conference on Applied Cryptography and Network Security Springer-VerlagBerlin, HeidelbergProceedings of the 5th International Conference on Applied Cryptography and Network Security Springer-Verlag Berlin, Heidelberg, 2007.
A survey of key management for secure group communication. S Rafaeli, D Hutchison, ACM Computing Surveys. 35S. Rafaeli and D. Hutchison, "A survey of key management for secure group communication," ACM Computing Surveys, Volume 35 Issue 3, Sept. 2003.
|
[] |
[
"MAGNUS-TYPE INTEGRATOR FOR THE FINITE ELEMENT DISCRETIZATION OF SEMILINEAR PARABOLIC NON-AUTONOMOUS SPDES DRIVEN BY MULTIPLICATIVE NOISE",
"MAGNUS-TYPE INTEGRATOR FOR THE FINITE ELEMENT DISCRETIZATION OF SEMILINEAR PARABOLIC NON-AUTONOMOUS SPDES DRIVEN BY MULTIPLICATIVE NOISE"
] |
[
"Antoine Tambue ",
"§ , ¶Jean Daniel Mukam "
] |
[] |
[] |
This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE driven by multiplicative noise by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square L 2 norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order O h 2 1 + max(0, ln tm/h 2 + ∆t 1/2 . Numerical simulations to illustrate our theoretical finding are provided.
| null |
[
"https://arxiv.org/pdf/1809.04438v1.pdf"
] | 119,674,077 |
1809.04438
|
6a212bb3f6f84a351e34203da2caf5cbec69ddc1
|
MAGNUS-TYPE INTEGRATOR FOR THE FINITE ELEMENT DISCRETIZATION OF SEMILINEAR PARABOLIC NON-AUTONOMOUS SPDES DRIVEN BY MULTIPLICATIVE NOISE
Sep 2018
Antoine Tambue
§ , ¶Jean Daniel Mukam
MAGNUS-TYPE INTEGRATOR FOR THE FINITE ELEMENT DISCRETIZATION OF SEMILINEAR PARABOLIC NON-AUTONOMOUS SPDES DRIVEN BY MULTIPLICATIVE NOISE
Sep 2018Magnus-type integratorStochastic partial differential equationsMultiplicative noiseStrong convergenceNon-autonomous equationsFinite element method
This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE driven by multiplicative noise by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square L 2 norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order O h 2 1 + max(0, ln tm/h 2 + ∆t 1/2 . Numerical simulations to illustrate our theoretical finding are provided.
1. Introduction. We consider the numerical approximations of the following semilinear parabolic non-autonomous SPDE driven by mutiplicative noise dX = [A(t)X + F (t, X)]dt + B(t, X)dW (t), in Λ × (0, T ], X(0) = X 0 , in Λ, (1.1) in the Hilbert space L 2 (Λ), where Λ is a bounded domain of R d , d = 1, 2, 3 and T ∈ (0, ∞). The family of unbounded linear operators A(t) are not necessarily selfadjoint. Each A(t) is assumed to generate an analytic semigroup S t (s) := e A(t)s . The nonlinear functions F and B are respectively the drift and the diffusion parts. Precise assumptions on A(t), F and B to ensure the existence of the unique mild solution of (1.1) are given in the next section. The random initial data is denoted by X 0 . We denote by (Ω, F , P) a probability space with a filtration (F t ) t∈[0,T ] ⊂ F that fulfills the usual conditions (see [30,Definition 2.1.11]). The noise term W (t) is assumed to be a Q-Wiener process defined on a filtered probability space Ω, F , P, {F t } t∈[0,T ] , where the covariance operator Q : H −→ H is assumed to be linear, self adjoint and positive definite. It is well known [30] that the noise can be represented as
W (t, x) = ∞ i=0 √ q i e i (x)β i (t),(1.2)
where (q i , e i ) i∈N are the eigenvalues and eigenfunctions of the covariance operator Q, and (β i ) i∈N are independent and identically distributed standard Brownian motions. The deterministic counterpart of (1.1) finds applications in many fields such as quantum fields theory, electromagnetism, nuclear physics (see e.g. [4] and references therein). It is worth to mention that models based on SPDEs can offer a more realistic representation of the system than models based only on PDEs, due to uncertainty in the input data. In many situations it is very hard to exhibit explicit solutions of SPDEs. For instance the following non-autonomous linear Stratonovich stochastic ordinary differential equation
dy = G 0 (t)ydt + d j=1 G j (t)ydW j (t), y(0) = y 0 ∈ R m (1.3)
does not have explicit solution (see e.g. [2,18]), unless G i and G j commute for all i, j ≥ 0. Numerical algorithms are therefore excellent tools to provide good approximations. Numerical approximations of (1.1) based on implicit, explicit Euler methods and exponential integrators with A(t) = A, where A is self-adjoint are thoroughly investigated in the literature, see e.g. [16,19,20,37,38,23,36] and the references therein. If we turn our attention to the case of time independent operator A(t) = A, with A not necessary self-adjoint, the list of references become remarkably short, see e.g., [22,26]. To the best of our knowledge numerical approximations of (1.1) with time dependent linear operator A(t) are not yet investigated in the scientific literature, due to the complexity of the linear operator A(t) and its semigroup S t (s) := e A(t)s . Our aim in this paper is to fill that gap and propose an explicit numerical scheme to approximate (1.1). We use the finite element method for spatial discretization and Magnus-type integrator for temporal discretization. Magnus-type integrator is based on a truncation of Magnus expansion, which was first proposed in [25] to represent the solution of non-autonomous homogeneous differential equation in the exponential form. Magnus expansion was further studied in [2,3,4]. The first numerical method based on magnus expansion was proposed in [14] for deterministic time-dependent homogeneous Schröndinger equation. The study in [14] was extended in [10] for partial differential equation of the following form u ′ (t) = A(t)u(t) + b(t), 0 < t ≤ T, u(0) = u 0 .
(1.4)
We follow [10] and apply the Magnus-type integrator method to the semi-discrete problem (2.37) and obtain the fully discrete scheme (2.41), called stochastic Magnustype integrators (SMTI). We investigate the strong convergence of the new fully discrete scheme toward the exact solution. Due to the complexity of the linear operator and the corresponding semi discrete linear operator after space discretisation, novel technical estimates are provided to achieve convergence orders comparable of that of autonomous SPDEs [22,19,26]. The result indicates how the convergence orders in both space and time depend on the regularity of the initial data and the noise. In particular for multiplicative trace class noise, we achieve optimal convergence orders of O h β + ∆t min(β,1)/2 , where β is the regularity's parameter, defined in Assumption 2.1. The rest of this paper is organised as follows. Section 2 provides the general setting, the fully discrete scheme and the main result. In Section 3 we provide some preparatory results and we present the proof of the main result. Section 4 provides some numerical experiments to confirm our theoretical result.
2. Mathematical setting, numerical scheme and main result.
Notations and main assumptions .
Let (H, ., . H , . ) be a separable Hilbert space. For a Banach space U , we denote by L 2 (Ω, U ) the Banach space of all equivalence classes of square-integrable U -valued random variables. Let L(U, H) be the space of bounded linear mappings from U to H endowed with the usual operator norm . L(U,H) . By L 2 (U, H) := HS(U, H), we denote the space of Hilbert-Schmidt operators from U to H equipped with the norm
l 2 L2(U,H) := ∞ i=1 lψ i 2 , l ∈ L 2 (U, H), (2.1) where (ψ i ) ∞ i=1
is an orthonormal basis of U . Note that this definition is independent of the orthonormal basis of U . For simplicity, we use the notations L(U, U ) =: L(U ). and L 2 (U, U ) =: L 2 (U ). For all l ∈ L(U, H) and l 1 ∈ L 2 (U ) we have ll 1 ∈ L 2 (U, H) and
ll 1 L2(U,H) ≤ l L(U,H) l 1 L2(U) .
(2.
2)
The space of Hilbert-Schmidt operators from Q 1/2 (H) to H is denoted by L 0 2 := L 2 (Q 1/2 (H), H) = HS(Q 1/2 (H), H). As usual, L 0 2 is equipped with the norm
l L 0 2 := lQ 1/2 HS = ∞ i=1 lQ 1/2 e i 2 1/2 , l ∈ L 0 2 , (2.3) where (e i ) ∞ i=1
is an orthonormal basis of H. This definition is independent of the orthonormal basis of H. For an L 0 2 -predictable stochastic process φ : In the rest of this paper, we consider H = L 2 (Λ). To guarantee the existence of a unique mild solution of (1.1) and for the purpose of the convergence analysis, we make the following assumptions. Assumption 2.1. The initial data X 0 : Ω −→ H is assumed to be measurable and
[0, T ]×Λ −→ L 0 2 such that t 0 E φQ 1/2 2 HS ds < ∞, t ∈ [0, T ],(2.E t 0 φdW (s) 2 = t 0 E φ 2 L 0 2 ds = t 0 E φQ 1/2 2 HS ds, t ∈ [0, T ],(2.satisfies X 0 ∈ L 2 Ω, D (−A(0)) β/2 , 0 ≤ β ≤ 2.
Assumption 2.2.
(i) As in [10,11,13], we assume that D (A(t)) = D, 0 ≤ t ≤ T and the family of linear operators A(t) : D ⊂ H −→ H to be uniformly sectorial on 0 ≤ t ≤ T , i.e. there exist constants c > 0 and θ ∈ 1 2 π, π such that
(λI − A(t)) −1 L(L 2 (Λ)) ≤ c |λ| , λ ∈ S θ , (2.6)
where S θ := λ ∈ C : λ = ρe iφ , ρ > 0, 0 ≤ |φ| ≤ θ . As in [13], by a standard scaling argument, we assume −A(t) to be invertible with bounded inverse.
(ii) Similarly to [11,13,10,29], we require the following Lipschitz conditions: there exists a positive constant K 1 such that
(A(t) − A(s)) (−A(0)) −1 L(H) ≤ K 1 |t − s|, s, t ∈ [0, T ], (2.7) (−A(0)) −1 (A(t) − A(s)) L(D,H) ≤ K 1 |t − s|, s, t ∈ [0, T ]. (2.8)
(iii) Since we are dealing with non smooth data, we follow [32] and assume that
D ((−A(t)) α ) = D ((−A(0)) α ) , 0 ≤ t ≤ T, 0 ≤ α ≤ 1 (2.9)
and there exists a positive constant K 2 such that for all u ∈ D((−A(0)) α ) the following estimate holds uniformly for t ∈ [0, T ]
K −1 2 (−A(0)) α u ≤ (−A(t)) α u ≤ K 2 (−A(0)) α u .(2.(−A(t)) α e sA(t) L(H) ≤ C 1 s −α , s > 0, (2.11) (−A(t)) −δ I − e sA(t) L(H) ≤ C 1 s δ , s ≥ 0,(2.U (t, s) L(H) ≤ K 0 , 0 ≤ s ≤ t ≤ T. (2.13) (ii) U (., s) ∈ C 1 (]s, T ]; L(H)), 0 ≤ s ≤ T , ∂U ∂t (t, s) = −A(t)U (t, s), 0 ≤ s < t ≤ T,(2.
14)
A(t)U (t, s) L(H) ≤ K 0 t − s , 0 ≤ s < t ≤ T. (2.15) (iii) U (t, .)x ∈ C 1 ([0, t[; H), 0 < t ≤ T , x ∈ D(A(0)) and ∂U ∂s (t, s) = −U (t, s)A(s)x, 0 ≤ s ≤ t ≤ T,(2.
16)
A(t)U (t, s)A(s) −1 L(H) ≤ K 0 , 0 ≤ s ≤ t ≤ T. (2.17) We equip V α (t) := D (−A(t)) α/2 , α ∈ R with the norm u α,t := (−A(t)) α/2 u .
Due to (2.9)-(2.10) and for the seek of ease notations, we simply write V α and . α . We follow [32] and assume the nonlinear operator F to satisfy the following Lipschitz condition.
F (s, 0) ≤ K 3 , F (t, u) − F (s, v) ≤ K 3 |t − s| β/2 + u − v , (2.18)
for all s, t ∈ [0, T ] and u, v ∈ H. Assumption 2.6. We assume the diffusion function B : [0, T ] × H −→ L 2 0 to be β/2-Hölder continuous with respect to the first variable and Lipschitz continuous with respect to the second variable, i.e. there exists a positive constant K 4 such that
B(s, 0) L 0 2 ≤ K 4 , B(t, u) − B(s, v) L 0 2 ≤ K 4 |t − s| β/2 + u − v , (2.19)
for all s, t ∈ [0, T ] and u, v ∈ H.
The following theorem ensures the existence of a unique mild solution of (1.1). Theorem 2.7. [32, Theorem 1.3] Let Assumptions 2.1, 2.2 (i)-(ii), 2.5 and 2.6 be fulfilled. Then the non-autonomous SPDE (1.1) has a unique mild solution X(t) ∈ L 2 Ω, D (−A(0)) β/2 , which takes the following form
X(t) = U (t, 0)X 0 + t 0 U (t, s)F (s, X(s))ds + t 0 U (t, s)B(s, X(s))dW (s),(2.20)
where U (t, s) is the evolution system of Proposition 2.4. Moreover, there exists a positive constant K 5 such that
sup 0≤t≤T X(t) L 2 (Ω,D((−A(0)) β/2 )) ≤ K 5 1 + X 0 L 2 (Ω,D((−A(0)) β/2 )) . (2.21)
To achieve optimal convergence order in space for multiplicative noise when β ∈ [1, 2], we require the following further assumption, also used in [19,17,36,22,26]. Assumption 2.8. We assume that there exists a positive constant (2.22) where β comes from Assumption 2.1.
c 1 > 0, such that B s, D((−A(0)) β−1 2 ) ⊂ HS Q 1/2 (H), D (−A(0)) β−1 2 (−A(0)) β−1 2 B(s, v) L 0 2 ≤ c1 (1 + v β−1 ) , v ∈ D (−A(0)) β−1 2 , s ∈ [0, T ],
2.2.
Fully discrete scheme and main result. For the seek of simplicity, we assume the family of linear operators A(t) 1 to be of second order and has the following form
A(t)u = d i,j=1 ∂ ∂x i q ij (x, t) ∂u ∂x j − d j=1 q j (x, t) ∂u ∂x j . (2.23)
We require the coefficients q i,j and q j to be smooth functions of the variable x ∈ Λ and Hölder-continuous with respect to t ∈ [0, T ]. We further assume that there exists a positive constant c such that the following ellipticity condition holds
d i,j=1 q ij (x, t)ξ i ξ j ≥ c|ξ| 2 , (x, t) ∈ Λ × [0, T ].(F (v))(x) = f (x, v(x)), (B(v)u)(x) = b(x, v(x)).u(x),(2.25)
for all x ∈ Λ, v ∈ H and u ∈ Q 1/2 (H), where f : Λ × R −→ R and b : Λ × R −→ R are continuously differentiable functions with globally bounded derivatives. Under the above assumptions on q ij and q j , it is well known that the family of linear operators defined by (2.23) fulfills Assumption 2.2 (i)-(ii) with D = H 2 (Λ) ∩ H 1 0 (Λ), see [28,Section 7.6] or [35,Section 5.2]. The above assumptions on q ij and q j also imply that Assumption 2.2 (iii) is fulfilled, see e.g. [32, Example 6.1] or [1,31]. As in [9,22], we introduce two spaces H and V , such that H ⊂ V , depending on the boundary conditions for the domain of the operator −A(t) and the corresponding bilinear form. For Dirichlet boundary conditions we take
V = H = H 1 0 (Λ) = {v ∈ H 1 (Λ) : v = 0 on ∂Λ}. (2.26)
For Robin boundary condition and Neumann boundary condition, which is a special case of Robin boundary condition (α 0 = 0), we take V = H 1 (Λ) and
H = {v ∈ H 2 (Λ) : ∂v/∂v A + α 0 v = 0, on ∂Λ}, α 0 ∈ R. (2.27)
Using Green's formula and the boundary conditions, we obtain the corresponding bilinear form associated to −A(t)
a(t)(u, v) = Λ d i,j=1 q ij (x, t) ∂u ∂x i ∂v ∂x j + d i=1 q i (x, t) ∂u ∂x i v dx, u, v ∈ V,
for Dirichlet boundary conditions and
a(t)(u, v) = Λ d i,j=1 q ij (x, t) ∂u ∂x i ∂v ∂x j + d i=1 q i (x, t) ∂u ∂x i v dx + ∂Λ α 0 uvdx.
for Robin and Neumann boundary conditions. Using Gårding's inequality, it holds that there exist two constants λ 0 and c 0 such that
a(t)(v, v) ≥ λ 0 v 2 1 − c 0 v 2 , ∀v ∈ V, t ∈ [0, T ].
(2.28)
By adding and subtracting c 0 u on the right hand side of (1.1), we obtain a new family of linear operators that we still denote by A(t). Therefore the new corresponding bilinear form associated to −A(t) still denoted by a(t) satisfies the following coercivity property
a(t)(v, v) ≥ λ 0 v 2 1 , ∀v ∈ V, t ∈ [0, T ]. (2.29)
Note that the expression of the nonlinear term F has changed as we have included the term −c 0 u in a new nonlinear term that we still denote by F . The coercivity property (2.29) implies that A(t) is sectorial on L 2 (Λ), see e.g. [21]. Therefore A(t) generates an analytic semigroup S t (s) = e sA(t) on L 2 (Λ) such that
[12] S t (s) = e sA(t) = 1 2πi C e sλ (λI − A(t)) −1 dλ, s > 0, (2.30)
where C denotes a path that surrounds the spectrum of A(t). The coercivity property (2.29) also implies that −A(t) is a positive operator and its fractional powers are well defined and for any α > 0 we have
(−A(t)) −α = 1 Γ(α) ∞ 0 s α−1 e sA(t) ds, (−A(t)) α = ((−A(t)) −α ) −1 , (2.31)
where Γ(α) is the Gamma function (see [12]). The domain of (−A(t)) α/2 are characterized in [9,7,21] for 1 ≤ α ≤ 2 with equivalence of norms as follows.
D((−A(t)) α/2 ) = H 1 0 (Λ) ∩ H α (Λ) (for Dirichlet boundary condition) D(−A(t)) = H, D((−A(t)) 1/2 ) = H 1 (Λ) (for Robin boundary condition) v H α (Λ) ≡ ((−A(t)) α/2 v := v α , ∀v ∈ D((−A(t)) α/2 ). The characterization of D((−A(t)) α/2 ) for 0 ≤ α < 1 can be found in [27, Theorem 2.1 & Theorem 2.2].
Let us now turn our attention to the space discretization of the problem (1.1). We start by splitting the domain Λ in finite triangles. Let T h be the triangulation with maximal length h satisfying the usual regularity assumptions, and V h ⊂ V be the space of continuous functions that are piecewise linear over the triangulation T h . We consider the projection
P h from H = L 2 (Λ) to V h defined for every u ∈ H by P h u, χ H = u, χ H , φ, χ ∈ V h . (2.32) For all t ∈ [0, T ], the discrete operator A h (t) : V h −→ V h is defined by A h (t)φ, χ H = A(t)φ, χ H = −a(t)(φ, χ), φ, χ ∈ V h . (2.33)
The coercivity property (2.29) implies that there exist constants C 2 > 0 and θ ∈ ( 1 2 π, π) such that (see e.g. [21, (2.9)] or [9,12])
(λI − A h (t)) −1 L(H) ≤ C 2 |λ| , λ ∈ S θ (2.34)
holds uniformly for h > 0 and t ∈ [0, T ]. The coercivity condition (2.29) implies that for any t ∈ [0, T ], A h (t) generates an analytic semigroup S h t (s) := e sA h (t) , s ∈ [0, T ]. The coercivity property (2.29) also implies that the smooth properties (2.11) and (2.12) hold for A h uniformly for h > 0 and t ∈ [0, T ], i.e. for all α ≥ 0 and δ ∈ [0, 1], there exists a positive constant C 3 such that the following estimates hold uniformly for h > 0 and t ∈ [0, T ], see e.g. [9,12]
(−A h (t)) α e sA h (t) L(H) ≤ C 3 s −α , s > 0, (2.35) (−A h (t)) −δ I − e sA h (t) L(H) ≤ C 3 s δ , s ≥ 0. (2.36)
The semi-discrete version of (1.1) consists of finding
X h (t) ∈ V h , t ∈ [0, T ] such that X h (0) := P h X 0 and dX h (t) = [A h (t)X h (t) + P h F (t, X h (t))]dt + P h B(t, X h (t))dW (t), (2.37)
for t ∈ (0, T ]. Let us consider the following linear system of non-autonomous ordinary differential equations (ODEs) y ′ (t) = A(t)y(t), y(0) given. (2.38) It was shown by Magnus [25] that the solution of (2.38) can be represented in the following exponential form
y(t) = e Θ(t) y(0), t ≥ 0, (2.39)
where Θ(t) called Magnus expansion is given by the following series [25, (3.28)]
Θ(t) = t 0 A(τ )dτ + 1 2 t 0 A(τ ), τ 0 A(σ)dσ dτ + 1 4 t 0 τ 0 σ 0 A(µ)dµ, A(σ) dσ, A(τ ) dτ + 1 12 t 0 τ 0 A(σ)dσ, τ 0 A(µ)dµ, A(τ ) dτ + · · · . (2.40) Here the Lie-product [u, v] of u and v is given by [u, v] = uv − vu.
For deterministic problems, numerical methods based on this expansion received some attentions since one decade, see e.g. [4,10,14,15,24]. For the time-dependent Schrödinger equation [10], the Magnus expansion (2.40) was truncated after the first term and the integral was approximated by the mid-point rule. This mid-point rule approximation of Θ(t) was also used in [14] to obtain a second-order Magnus type integrator for non-autonomous deterministic parabolic partial differential equation (PDE). Note that the convergence analysis in [10,14] was only done in time.
Throughout this paper, we take [10,14], we introduce the following fully discrete scheme for (1.1), called stochastic Magnus-type integrators (SMTI)
t m = m∆t ∈ [0, T ], where T = M ∆t for m, M ∈ N, m ≤ M . Motivated byX h m+1 = e ∆tA h,m X h m + ∆tϕ 1 (∆tA h,m )P h F t m , X h m + e ∆tA h,m P h B t m , X h m ∆W m , m = 0, · · · , M, (2.41) X h 0 = P h X 0 ,
where the linear operator ϕ 1 (∆tA h,m ) is given by Note that the numerical scheme (2.41) can be written in the following integral form, useful for the error analysis
ϕ 1 (∆tA h,m ) := 1 ∆t ∆t 0 e (∆t−s)A h,m ds, A h,m := A h (t m ) ,(2.X h m+1 = e ∆tA h,m X h m + tm+1 tm e (tm+1−s)A h,m P h F t m , X h m ds + tm+1 tm e ∆tA h,m P h B t m , X h m dW (s). (2.44)
We also note that an equivalent formulation of the numerical scheme (2.41), easy for simulation is given by
X h m+1 = X h m + P h B tm, X h m ∆Wm + ∆tϕ1(∆tA h,m ) A h,m X h m + P h B tm, X h m ∆Wm + P h F tm, X h m . (2.45)
With the numerical method in hand, we can now state its strong convergence result toward the exact solution, which is in fact our main result. In the rest of this paper C is a generic constant independent of h, m, M and ∆t that may change from one place to another.
E X(t m ) − X h m 2 1/2 ≤ C h β + ∆t β/2 . (2.46)
(ii) If 1 ≤ β < 2 and moreover if Assumption 2.8 is satisfied, then the following error estimate holds
E X(t m ) − X h m 2 1/2 ≤ C h β + ∆t 1/2 . (2.47)
(iii) If β = 2 and if Assumption 2.8 is fulfilled, then the following error estimate holds
E X(t m ) − X h m 2 1/2 ≤ C h 2 1 + max(0, ln(t m /h 2 ) + ∆t 1/2 .(2.48)
3. Proof of the main result. The proof of the main result needs some preparatory results.
Preparatory results.
The following lemma will be useful in our convergence proof.
K −1 (−(A h (0)) −γ v ≤ ((−A h (t)) −γ v ≤ K ((−A h (0)) −γ v , v ∈ V h , (3.1) K −1 (−(A h (0)) γ v ≤ ((−A h (t)) γ v ≤ K ((A h (0)) γ v , v ∈ V h , (3.2)
where K is a positive constant independent of t and h. Lemma 3.2.
[33] Under Assumption 2.2, the following estimates hold
(A h (t) − A h (s))(−A h (r)) −1 u h ≤ C|t − s| u h , r, s, t ∈ [0, T ], u h ∈ V h , (3.3) (−A h (r)) −1 (A h (s) − A h (t)) u h ≤ C|s − t| u h , r, s, t ∈ [0, T ], u h ∈ V h . (3.4)U h (t, s) = S h s (t − s) + t s S h τ (t − τ )R h (τ, s)dτ, (3.5) where S h s (t) := e A h (s)t , R h (t, s) := ∞ m=1 R h m (t, s), with R h m (t,R h (t, s) = R h 1 (t, s) + t s R h 1 (t, τ )R h (τ, s)dτ. (3.7)
The mild solution of (2.37) is therefore given by
X h (t) = U h (t, 0)P h X 0 + t 0 U h (t, s)P h F s, X h (s) ds + t 0 U h (t, s)P h B s, X h (s) dW (s).∂U h ∂t (t, s) = −A h (t)U h (t, s), 0 ≤ s ≤ t ≤ T, (3.9) A h (t)U h (t, s) L(H) ≤ C t − s , 0 ≤ s < t ≤ T. (3.10) (ii) U h (t, .)u ∈ C 1 ([0, t[; H), 0 < t ≤ T , u ∈ D(A h (0)) and ∂U h ∂s (t, s)u = −U h (t, s)A h (s)u, 0 ≤ s ≤ t ≤ T, (3.11) A h (t)U h (t, s)A h (s) −1 L(H) ≤ C, 0 ≤ s ≤ t ≤ T.(3.R h 1 (t, s) L(H) ≤ C, R h m (t, s) L(H) ≤ C m! (t − s) m−1 , m ≥ 1, (3.13) R h (t, s) L(H) ≤ C, U h (t, s) L(H) ≤ C, 0 ≤ s ≤ t ≤ T.
(3.14)
(ii) For any 0 ≤ α ≤ 1, 0 ≤ γ ≤ 1 and 0 ≤ s ≤ t ≤ T , the following estimates holds
(−A h (r)) α U h (t, s) L(H) ≤ C(t − s) −α , r ∈ [0, T ], (3.15) U h (t, s)(−A h (r)) α L(H) ≤ C(t − s) −α , r ∈ [0, T ], (3.16) (−A h (r)) α U h (t, s)(−A h (s)) −γ L(H) ≤ C(t − s) γ−α , r ∈ [0, T ].(3.17)
(iii) For any 0 ≤ s ≤ t ≤ T the following useful estimates hold
(U h (t, s) − I) (−A h (s)) −γ L(H) ≤ C(t − s) γ , 0 ≤ γ ≤ 1, (3.18) −A h (r)) −γ (U h (t, s) − I L(H) ≤ C(t − s) γ , 0 ≤ γ ≤ 1. (3.19)
The following space and time regularity of the semi-discrete problem (2.37) will be useful in our convergence analysis. Lemma 3.6. Let Assumptions 2.1, 2.2 (i)-(ii), 2.5 and 2.6 be fulfilled with the corresponding 0 ≤ β < 1. Then for all γ ∈ [0, β] the following estimates hold
(−A h (r)) γ/2 X h (t) L 2 (Ω,H) ≤ C, 0 ≤ r, t ≤ T, (3.20) X h (t 2 ) − X h (t 1 ) L 2 (Ω,H) ≤ C(t 2 − t 1 ) β/2 , 0 ≤ t 1 ≤ t 2 ≤ T. (3.21)
Moreover if Assumption 2.8 is fulfilled, then (3.20) and (3.21) hold for β = 1.
Proof. We first show that sup
t∈[0,T ] X h (t) 2 L 2 (Ω,H) ≤ C.
Taking the norm in both side of (3.8) and using the inequality
(a + b + c) 2 ≤ 3a 2 + 3b 2 + 3c 2 , a, b, c ∈ R + yields X h (t) 2 L 2 (Ω,H) ≤ 3 U h (t, 0)P h X0 2 L 2 (Ω,H) + 3 t 0 U h (t, s)P h F s, X h (s) ds 2 L 2 (Ω,H) ds + 3 t 0 U h (t, s)P h B s, X h (s) dW (s) 2 L 2 (Ω,H) := I0 + I1 + I2. (3.22)
Using Lemma 3.5 (i) and the uniformly boundedness of P h , it holds that
I 0 ≤ 3 X 0 2 L 2 (Ω,H) ≤ C. (3.23)
Using again Lemma 3.5 (i), Assumption 2.5 and the uniformly boundedness of P h , it holds that
I1 ≤ 3 t 0 U h (t, s)P h F s, X h (s) L 2 (Ω,H) 2 ≤ C t 0 C + X h (s) L 2 (Ω,H) ds 2 .
Using Hölder inequality yields
I 1 ≤ C + C t 0 X h (s) 2 L 2 (Ω,H) ds. (3.24)
Applying the itô-isometry's property, using Lemma 3.5 (i) and Assumption 2.6, it holds that
I2 = 3 t 0 U h (t, s)P h B s, X h (s) 2 L 0 2 ds ≤ C + C t 0 X h (t) 2 L 2 (Ω,H) ds.X h (t) 2 L 2 (Ω,H) ≤ C + C t 0 X h (s) 2 L 2 (Ω,H) ds. (3.26)
Applying the continuous Gronwall's lemma to (3.26) yields
X h (t) 2 L 2 (Ω,H) ≤ C, t ∈ [0, T ].(3.27)
Let us now prove (3.20). Pre-multiplying (3.8) by (−A h (r)) γ/2 , taking the norm in both sides and using triangle inequality yields Inserting (−A h (0)) −γ/2 (−A h (0)) γ/2 , using Lemma 3.5 (ii) and Lemma 3.1, it holds that
(−A h (r)) γ/2 X h (t) L 2 (Ω,H) ≤ (−A h (r)) γ/2 U h (t, 0)P h X0 L 2 (Ω,H) + t 0 (−A h (r)) γ/2 U h (t, s)P h F s, X h (s) L(Ω,H) ds + t 0 (−A h (r)) γ/2 U h (t, s)P h B s, X h (s) dW (s)II0 ≤ (−A h (r)) γ/2 U h (t, 0)(−A h (0)) −γ/2 L(H) (−A h (0)) γ/2 X0 ≤ C. (3.29)
Using Lemmas 3.1, 3.5 (ii), Assumption 2.5 and (3.27) yields
II 1 ≤ C t 0 (−A h (s)) γ/2 U h (t, s) L(H) sup t∈[0,T ] F s, X h (s) ds ≤ C sup s∈[0,T ] 1 + X h (s) L 2 (Ω,H) t 0 (t − s) −γ/2 ds ≤ C. (3.30)
Applying the Itô-isometry property, using Lemmas 3.1, 3.5 (ii), Assumption 2.6 and (3.27) yields
II 2 2 = t 0 (−A h (0)) γ/2 U h (t, s)P h B s, X h (s) 2 L 0 2 ds ≤ C sup s∈[0,T ] 1 + X h (s) 2 L 2 (Ω,H) t 0 (t − s) −γ ds ≤ C.X h (t 2 ) − X h (t 1 ) L 2 (Ω,H) ≤ (U h (t 2 , 0) − U h (t 1 , 0)) P h X 0 L 2 (Ω,H) + t1 0 (U h (t 2 , s) − U h (t 1 , s)) P h F s, X h (s) L 2 (Ω,H) ds + t2 t1 U h (t 2 , s)P h F s, X h (s) L 2 (Ω,H) ds + t1 0 U h (t 2 , s) − U h (t 1 , s))P h B s, X h (s) dW (s) L 2 (Ω,H) + t2 t1 U h (t 2 , s)P h B s, X h (s) dW (s) L 2 (Ω,H) := III 0 + III 1 + III 2 + III 3 + III 4 . (3.32)
Inserting an appropriate power of −A h (t 1 ), using Lemmas 3.
(ii)-(iii) and [26, Lemma 1] yields
III0 = (U h (t2, t1) − I)U h (t1, 0)P h X0 L 2 (Ω,H) ≤ (U h (t2, t1) − I)(−A h (t1)) −β/2 L(H) × (−A h (t1)) β/2 U h (t1, 0)(−A h (t1)) −β/2 L(H) (−A h (t1)) β/2 P h X0 L 2 (Ω,H) ≤ C(t2 − t1) β/2 . (3.33)
Using Assumption 2.6, (3.20), Lemma 3.5 (ii) and (iii) yields
III1 ≤ t 1 0 (U h (t2, t1) − I)U h (t1, s) L(H) P h F s, X h (s) L 2 (Ω,H) ds ≤ C t 1 0 (U h (t2, t1) − I)(−A h (t1)) −β/2 L(H) (−A h (t1)) β/2 U h (t1, s) L(H) ds ≤ C t 1 0 (t2 − t1) β/2 (t1 − s) −β/2 ds ≤ C(t2 − t1) β/2 . (3.34)
Using Lemma 3.5 (i) and Assumption 2.5, it holds that
III 2 ≤ C t2 t1 sup s∈[0,T ] F s, X h (s) L 2 (Ω,H) ds ≤ C(t 2 − t 1 ). (3.35)
Using the Itô-isometry property, Assumption 2.8, (3.20), Lemma 3.5 (ii)-(iii) and following the same lines as the estimate of III 1 yields
III 2 3 ≤ C(t 2 − t 1 ) β . (3.36)
Using the Itô-isometry property and following the same lines as that of III 2 yields The corresponding semi-discrete problem in space is: find u h ∈ V h such that
III 2 4 ≤ C(t 2 − t 1 ).u ′ h (t) = A h (t)u h , u h (τ ) = P h v, t ∈ (τ, T ], τ ≥ 0. (3.39)
Let us define the operator
T h (t, τ ) := U (t, τ ) − U h (t, τ )P h , (3.40) so that u(t)−u h (t) = T h (t, τ )v.
The following lemma will be useful in our convergence analysis. Lemma 3.7.
[33] Let r ∈ [0, 2] and 0 ≤ γ ≤ r. Let Assumption 2.2 be fulfilled. Then the following error estimate holds for the semi-discrete approximation (3.39)
u(t) − u h (t) = T h (t, τ )v ≤ Ch r (t − τ ) −(r−γ)/2 v γ , v ∈ D (−A(0)) γX(t) − X h (t) L 2 (Ω,H) ≤ Ch β , 0 ≤ t ≤ T. (3.42)
(ii) If 1 ≤ β < 2 and moreover if Assumption 2.8 is fulfilled, then the following error estimate holds
X(t) − X h (t) L 2 (Ω,H) ≤ Ch β , 0 ≤ t ≤ T,(3.X(t) − X h (t) L 2 (Ω,H) ≤ Ch 2 1 + max 0, ln(t/h 2 ) , 0 < t ≤ T.(3.44)
Proof. Subtracting (3.8) form (2.20), taking the L 2 norm and using triangle inequality yields
X(t) − X h (t) L 2 (Ω,H) ≤ U (t, 0)X0 − U h (t, 0)P h X0 L 2 (Ω,H) + t 0 U (t, s)F (s, X(s)) − U h (t, s)P h F s, X h (s) ds L 2 (Ω,H) + t 0 U (t, s)B (s, X(s)) − U h (t, s)P h B s, X h (s) dW (s) L 2 (Ω,H) =: IV0 + IV1 + IV2. (3.45)
Using Lemma 3.7 with r = γ = β yields
IV 0 ≤ Ch β X 0 L 2 (Ω,D((−A(0)) β/2 )) ≤ Ch β . (3.46)
Using Lemma 3.7 with r = β, γ = 0, Assumption 2.5, Lemmas 3.6 and 3.5 yields
IV 1 ≤ t 0 U (t, s)F (s, X(s)) − U (t, s)F s, X h (s) L 2 (Ω,H) ds + t 0 U (t, s)F s, X h (s) − U h (t, s)P h F s, X h (s) L 2 (Ω,H) ds ≤ C t 0 X(s) − X h (s) L 2 (Ω,H) ds + Ch β t 0 (t − s) −β/2 ds ≤ Ch β + C t 0 X(s) − X h (s) L 2 (Ω,H) ds. (3.47)
Using the Itô-isometry property, Lemma 3.6, Lemma 3.7 with r = β and γ = β−1 Applying the continuous Gronwall's lemma to (3.49) yields
2 yields IV 2 2 = t 0 U (t, s)B (s, X(s)) − U h (t, s)P h B s, X h (s) 2 L 0 2 ds ≤ t 0 U (t, s)B (s, X(s)) − U (t, s)B s, X h (s) 2 L 0 2 ds + t 0 U (t, s)B s, X h (s) − U h (t, s)P h B s, X h (s) L 0 2 ds ≤ C t 0 X(s) − X h (s) 2 L 2 (Ω,H) ds + Ch 2β t 0 (t − s) −1+β ds ≤ Ch 2β + C t 0 X(s) − X h (s) L 2 (Ω,H) ds.X(t) − X h (t) L 2 (Ω,H) ≤ Ch β . (3.50)
For non commutative operators H j on a Banach space, we introduce the following notation for the composition
k j=l H j = H k H k−1 · · · H l if k ≥ l, I if k < l. (3.51)
The following lemma will be useful in our convergence proof. Lemma 3.9.
[33] Let Assumption 2.2 be fulfilled. Then the following estimate holds
m j=l e ∆tA h,j (−A h,l ) γ L(H) ≤ Ct −γ m−l , 0 ≤ l < m, 0 ≤ γ < 1,(3.52) (−A h,k ) γ 1 m j=l e ∆tA h,j (−A h,l ) −γ 2 L(H) ≤ Ct γ 2 −γ 1 m−l , 0 ≤ l < m, (3.53) 0 ≤ γ 1 ≤ 1, 0 < γ 2 ≤ 1,
where C is a positive constant independent of m, l, h and ∆t. Lemma 3.10.
(i) For all α ≥ 0, the following estimate holds
R h (t, s)(−A h (s)) α L(H) ≤ C(t − s) −α , t, s ∈ [0, T ]. (3.54)
(ii) For all α ∈ [0, 1], the following estimate holds
U h (t j , t j−1 ) − e ∆tA h,j−1 (−A h,j−1 ) −α L(H) ≤ C∆t 1+α . (3.55)
(iii) For all α ∈ [0, 1), the following estimate holds
U h (t j , t j−1 ) − e ∆tA h,j−1 (−A h,j−1 ) α L(H) ≤ C∆t 1−α . (3.56)
(iv) For all α ∈ [0, 1], the following estimate holds
(−A h,j−1 ) −α U h (t j , t j−1 ) − e ∆tA h,j−1 L(H) ≤ C∆t 1+α . (3.57)
Proof. From the integral equation (3.7), we have
R h (t, s)(−A h (s)) α = e A h (s)(t−s) (−A h (s)) α + t s R h 1 (t, τ )R h (τ, s)(−A h (s)) α dτ. (3.58)
Taking the norm in both sides of (3.58), using (2.36) and Lemma 3.5 yields
R h (t, s)(−A h (s)) α L(H) ≤ e A h (s)(t−s) (−A h (s)) α L(H) + t s R h 1 (τ, s) L(H) R h (τ, s)(−A h (s)) α L(H) dτ ≤ C(t − s) −α + C t s R h (τ, s)(−A h (s)) α L(H) dτ. (3.59)
Applying the continuous Gronwall's lemma to (3.59) yields
R h (t, s)(−A h (s)) α L(H) ≤ C(t − s) −α . (3.60)
This completes the proof of (i). From (3.5) and (3.7), we have
U h (t j , t j−1 ) − e ∆tA h,j−1 = tj tj−1 e (tj −τ )A h (τ ) R h (τ, t j−1 )dτ = tj tj−1 e (tj −τ )A h (τ ) R h 1 (τ, t j−1 )dτ + tj tj−1 e (tj −τ )A h (τ ) τ tj−1 R h 1 (τ, s)R h (s, t j−1 )ds dτ = tj tj−1 e (tj −τ )A h (τ ) (A h (τ ) − A h (t j−1 )) e A h,j−1 (τ −tj−1) dτ + tj tj−1 e (tj −τ )A h (τ ) τ tj−1 R h 1 (τ, s)R h (s, t j−1 )ds dτ. (3.61)
Therefore, from (3.61), for all α ∈ [0, 1], using (2.36) and Lemma 3.5, it holds that
U h (t j , t j−1 ) − e ∆tA h,j−1 (−A h,j−1 ) −α L(H) ≤ tj tj−1 e (tj −τ )A h (τ ) (A h (τ ) − A h (t j−1 )) (−A h,j−1 ) −1 .e A h,j−1 (τ −tj−1) (−A h,j−1 ) 1−α L(H) dτ + tj tj−1 e (tj −τ )A h (τ ) L(H) τ tj−1 R h 1 (τ, s)R h (s, t j−1 ) L(H) ds dτ ≤ tj tj−1 e (tj −τ )A h (τ ) L(H) (A h (τ ) − A h (t j−1 )) (−A h,j−1 ) −1 L(H) × e A h,j−1 (τ −tj−1) (−A h,j−1 ) 1−α L(H) dτ + C tj tj−1 τ tj−1 dsdτ ≤ C tj tj−1 (τ − t j−1 ) α dτ + C∆t 2 ≤ C∆t 1+α . (3.62)
This completes the proof of (ii). The proof of (iii) and (iv) are similar to that of (ii) using (i).
The following lemma can be found in [21] Lemma 3.11. For all α 1 , α 2 > 0 and α ∈ [0, 1), there exist two positive constants C α1,α2 and C α,α2 such that
∆t m j=1 t −1+α1 m−j+1 t −1+α2 j ≤ C α1,α2 t −1+α1+α2 m , (3.63) ∆t m j=1 t −α m−j+1 t −1+α2 j ≤ C α,α2 t −α+α2 m . (3.64)
Proof. The proof of (3.63) follows from the comparison with the integral
t 0 (t − s) −1+α1 s −1+α2 ds. (3.65)
The proof of (3.64) is a consequence of (3.63).
The following lemma is fundamental in our convergence analysis. Using the telescopic sum, (3.68) can be rewritten as follows
m j=i U h (t j , t j−1 ) − m j=i e ∆tA h,j−1 = m−i+1 k=1 m j=i+k U h (t j , t j−1 ) U h (t i+k−1 , t i+k−2 ) − e ∆tA h,i+k−2 . i+k−2 j=i e ∆tA h,j−1 . (3.69)
Writing down explicitly the first term of (3.69) gives
m j=i U h (t j , t j−1 ) − m j=i e ∆tA h,j−1 = m j=i+1 U h (t j , t j−1 ) U h (t i , t i−1 ) − e ∆tA h,i−1 + m−i+1 k=2 m j=i+k U h (t j , t j−1 ) U h (t i+k−1 , t i+k−2 ) − e ∆tA h,i+k−2 . i+k−2 j=i e ∆tA h,j−1 . (3.70)
Taking the norm in both sides of (3.70), using Lemma 3.5, Lemma 3.10 (ii) and Lemma 3.9 yields
m j=i U h (tj , tj−1) − m j=i e ∆tA h,j−1 L(H) ≤ U h (tm−i+1, ti) L(H) U h (ti, ti−1) − e ∆tA h,i−1 L(H) + m−i+1 k=2 U h (tm, t i+k−1 ) L(H) U h (t i+k−1 , t i+k−2 ) − e ∆tA h,i+k−2 (−A h,i+k−2 ) −1+ǫ L(H) × (−A h,i+k−2 ) 1−ǫ i+k−2 j=i e ∆tA h,j−1 L(H) ≤ C∆t + C m−i+1 k=2 ∆t 2−ǫ t −1+ǫ k−1 ≤ C∆t 1−ǫ . (3.71)
This completes the proof of (i). The proof of (ii) is similar to that of (i) using (3.53) and Lemma 3.11. With the above preparatory results in hand, we can now prove our main result.
3.2.
Proof of Theorem 2.9. Using triangle inequality, we split the fully discrete error in two parts as follows.
X(t m ) − X h m L 2 (Ω,H) ≤ X(t m ) − X h (t m ) L 2 (Ω,H) + X h (t m ) − X h m L 2 (Ω,H) =: V + V I. (3.72)
The space error V is estimated in Lemma 3.7. It remains to estimate the time error V I. Note that the mild solution of (2.37) can be written as follows.
X h (t m ) = U h (t m , t m−1 )X h (t m−1 ) + tm tm−1 U h (t m , s)P h F s, X h (s) ds + tm tm−1 U h (t m , s)P h B s, X h (s) dW (s). (3.73)
Iterating the mild solution (3.73) yields
X h (tm) = m j=1 U h (tj, tj−1) P h X0 + tm t m−1 U h (tm, s)P h F s, X h (s) ds + tm t m−1 U h (tm, s)P h B s, X h (s) dW (s) + m−1 k=1 t m−k t m−k−1 m j=m−k+1 U h (tj, tj−1) U h (t m−k , s)P h F s, X h (s) ds + m−1 k=1 t m−k t m−k−1 m j=m−k+1 U h (tj, tj−1) U h (t m−k , s)P h B s, X h (s) dW (s). (3.74)
Iterating the numerical scheme (2.44) by substituting X h j , j = m − 1, · · · , 1 only in the first term of (2.44) by their expressions yields
X h m = m−1 j=0 e ∆tA h,j X h 0 + tm t m−1 e (tm−s)A h,m−1 P h F tm−1, X h m−1 ds + tm t m−1 e ∆tA h,m−1 P h B tm−1, X h m−1 dW (s) + m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j e (t m−k −s)A h,m−k−1 P h F t m−k−1 , X h m−k−1 ds + m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j e ∆tA h,m−k−1 P h B t m−k−1 , X h m−k−1 dW (s). (3.75)
Substracting (3.75) from (3.74) yields Taking the norm in both sides of (3.76) yields
X h (tm) − X h m = m j=1 U h (tj, tj−1) P h X0 − m−1 j=0 e ∆tA h,j P h X0 + tm t m−1 U h (tm, s)P h F s, X h (s) ds − tm t m−1 e (tm−s)A h,m−1 P h F tm−1, X h m−1 ds + tm t m−1 U h (tm, s)P h B s, X h (s) dW (s) − tm t m−1 e ∆tA h,m−1 P h B tm−1, X h m−1 dW (s) + m−1 k=1 t m−k t m−k−1 m j=m−k+1 U h (tj, tj−1) U h (t m−k , s)P h F s, X h (s) ds − m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j e (t m−k −s)A h,m−k−1 P h F t m−k−1 , X h m−k−1 ds + m−1 k=1 t m−k t m−k−1 m j=m−k+1 U h (tj, tj−1) U h (t m−k , s)P h B s, X h (s) dW (s) − m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j e ∆tA h,m−k−1 P h B t m−k−1 , X h m−k−1 dW (s) =: V I1 + V I2 + V I3 + V I4 + V I5.X h (t m ) − X h m 2 L 2 (Ω,H) ≤ 25 5 i=1 V I i 2 L 2 (Ω,H) . (3.77)
In what follows, we estimate separately V I i L 2 (Ω,H) , i = 1, · · · , 5.
3.2.1. Estimate of V I 1 , V I 2 and V I 3 . Using Lemma 3.12, it holds that
V I 1 L 2 (Ω,H) ≤ m j=1 U h (t j , t j−1 ) − m−1 j=0 e ∆tA h,j L(H) X 0 L 2 (Ω,H) ≤ C∆t 1−ǫ .(3.V I2 L 2 (Ω,H) ≤ tm t m−1 U h (tm, s)P h F s, X h (s) L 2 (Ω,H) ds + tm t m−1 e (tm−s)A h,m−1 P h F tm−1, X h m−1 − P h F tm−1, X h (tm−1) L 2 (Ω,H) ds + tm t m−1 e (tm−s)A h,m−1 P h F tm−1, X h (tm−1) L 2 (Ω,H) ds ≤ C tm t m−1 ds + C tm t m−1 X h (tm−1) − X h m−1 L 2 (Ω,H) ds + C tm t m−1 ds ≤ C∆t + C∆t X h (tm−1) − X h m−1 L 2 (Ω,H) . (3.79)
Applying the Itô-isometry property, using Assumption 2.6, (2.35), Theorem 2.7 and Lemma 3.5 yields
V I3 2 L 2 (Ω,H) ≤ 9 tm t m−1 E U h (tm, s)P h B s, X h (s) 2 L 0 2 ds + 9 tm t m−1 E e ∆tA h,m−1 P h B tm−1, X h m−1 − P h B tm−1, X h (tm−1) 2 L 0 2 ds + 9 tm t m−1 E e ∆tA h,m−1 P h F tm−1, X h (tm−1) 2 L 0 2 ds ≤ C tm t m−1 ds + C tm t m−1 X h (tm−1) − X h m−1 2 L 2 (Ω,H) ds + C tm t m−1 ds ≤ C∆t + C∆t X h (tm−1) − X h m−1 2 L 2 (Ω,V I4 = m−1 k=1 t m−k t m−k−1 m j=m−k+1 U h (tj, tj−1) U h (t m−k , s) P h F s, X h (s) − P h F t m−k−1 , X h (t m−k−1 ) ds + m−1 k=1 t m−k t m−k−1 m j=m−k+1 U h (tj, tj−1) [U h (t m−k , s) − U h (t m−k , t m−k−1 )] P h F t m−k−1 , X h (t m−k−1 ) ds + m−1 k=1 t m−k t m−k−1 m j=m−k U h (tj, tj−1) − m−1 j=m−k−1 e ∆tA h,j P h F t m−k−1 , X h (t m−k−1 ) ds + m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j e ∆tA h,m−k−1 − e (t m−k −s)A h,m−k−1 P h F t m−k−1 , X h (t m−k−1 ) ds + m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j e (t m−k −s)A h,m−k−1 P h F t m−k−1 , X h (t m−k−1 ) − P h F t m−k−1 , X h m−k−1 ds =: V I41 + V I42 + V I43 + V I44 + V I45. (3.81)
Using Lemma 3.5, Assumption 2.5 and Lemma 3.6 yields
V I41 L 2 (Ω,H) ≤ C m−1 k=1 t m−k t m−k−1 P h F s, X h (s) − P h F t m−k−1 , X h (t m−k−1 ) L 2 (Ω,H) ds ≤ C m−1 k=1 t m−k t m−k−1 (s − t m−k−1 ) β/2 ds + C m−1 k=1 t m−k t m−k−1 X h (s) − X h (t m−k−1 ) L 2 (Ω,H) ds ≤ C∆t β/2 + m−1 k=1 t m−k t m−k−1 (s − t m−k−1 ) min(β,1)/2 ds ≤ C∆t min(β,1)/2 . (3.82)
Using Lemma 3.5, Assumption 2.5 and Theorem 2.7 gives
V I42 L 2 (Ω,H) ≤ C m−1 k=1 t m−k t m−k−1 U h (tm, t m−k )U h (t m−k , s)(I − U h (s, t m−k−1 ) L(H) × P h F t m−k−1 , X h (t m−k−1 ) L 2 (Ω,H) ds ≤ C m−1 k=1 t m−k t m−k−1 U h (tm, t m−k )(−A h,m−k ) 1−ǫ L(H) (−A h,m−k ) −1+ǫ U h (t m−k , s)(−A h,m−k ) 1−ǫ L(H) × (−A h,m−k ) −1+ǫ (I − U h (s, t m−k−1 )) L(H) ds ≤ C m−1 k=1 t m−k t m−k−1 (tm − t m−k ) −1+ǫ (s − t m−k−1 ) 1−ǫ ds ≤ C∆t 1−ǫ m−1 k=1 t m−k t m−k−1 t −1+ǫ k ds ≤ C∆t 1−ǫ m−1 k=1 ∆tt −1+ǫ k ≤ C∆t 1−ǫ .(3.≤ m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j e (s−t m−k−1 )A h,m−k−1 − I e (t m−k −s)A h,m−k−1 L(H) × P h F t m−k−1 , X h (t m−k−1 ) L 2 (Ω,H) ds ≤ C m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j (−A h,m−k−1 ) 1−ǫ L(H) × (−A h,m−k−1 ) −1+ǫ e (s−t m−k−1 )A h,m−k−1 − I L(H) e (t m−k −s)A h,m−k−1 L(H) ds ≤ C m−1 k=1 t m−k t m−k−1 t −1+ǫ k (s − t m−k−1 ) 1−ǫ ds ≤ C∆t 1−ǫ m−1 k=1 t m−k t m−k−1 t −1+ǫ k ∆t ≤ C∆t 1−ǫ .× P h F t m−k−1 , X h (t m−k−1 ) L 2 (Ω,H) ds ≤ C m−1 k=1 t m−k t m−k−1 m−1 j=m−k e ∆tA h,j (−A h,m−k ) 1−ǫ L(H) × (−A h,m−k ) −1+ǫ I − e (s−t m−k−1 )A h,m−k−1 L(H) e (t m−k −s)A h,m−k−1 L(H) ≤ C m−1 k=1 t m−k t m−k−1 t −1+ǫ k (s − t m−k−1 ) 1−ǫ ds ≤ C∆t 1−ǫ m−1 k=1 t m−k t m−k−1 t −1+ǫ k ds ≤ C∆t 1−ǫ . (3.85)
Using Lemma 3.9 and Assumption 2.5 yields
V I 45 L 2 (Ω,H) ≤ C m−1 k=1 t m−k t m−k−1 X h (t m−k−1 ) − X h m−k−1 L 2 (Ω,H) ≤ C∆t m−1 k=0 X h (t k ) − X h k L 2 (Ω,H) .(3.V I5 = m−1 k=1 t m−k t m−k−1 m j=m−k+1 U h (tj, tj−1) U h (t m−k , s) P h B s, X h (s) − P h B t m−k−1 , X h (t m−k−1 ) dW (s) + m−1 k=1 t m−k t m−k−1 m j=m−k+1 U h (tj, tj−1) [U h (t m−k , s) − U h (t m−k , t m−k−1 )] P h B t m−k−1 , X h (t m−k−1 ) dW (s) + m−1 k=1 t m−k t m−k−1 m j=m−k U h (tj, tj−1) − m−1 j=m−k−1 e ∆tA h,j P h B t m−k−1 , X h (t m−k−1 ) dW (s) + m−1 k=1 t m−k t m−k−1 m−1 j=m−k−1 e ∆tA h,j P h B t m−k−1 , X h (t m−k−1 ) − P h B t m−k−1 , X h m−k−1 dW (s) =: V I51 + V I52 + V I53 + V I54. (3.88)
Using the Itô-isometry property, Lemma 3.5, Assumption 2.6 and Lemma 3.6 yields
V I 51 2 L 2 (Ω,H) = m−1 k=1 t m−k t m−k−1 E U h (t m , s) P h B s, X h (s) − P h B t m−k−1 , X h (t m−k−1 ) 2 L 0 2 ds ≤ C m−1 k=1 t m−k t m−k−1 (s − t m−k−1 ) β ds + C m−1 k=1 t m−k t m−k−1 X h (s) − X h (t m−k−1 ) 2 L 2 (Ω,H) ds ≤ C∆t β + C m−1 k=1 t m−k t m−k−1 (s − t m−k−1 ) min(β,1) ds ≤ C∆t min(β,1) . (3.89) ds ≤ C∆t 1−ǫ .
(3.90)
Applying the Itô-isometry property, using Lemma 3.12, Assumption 2.6 and Lemma 3.6 yields Note that to achieve optimal convergence 1/2 when β ≥ 1, we only need to re-estimate V I 52 L 2 (Ω,H) and V I 53 L 2 (Ω,H) by using Assumption 2.8 and Lemma 3.12 (ii). This is straightforward. The proof of Theorem 2.9 is therefore completed. where β i,j (t) are independent and identically distributed standard Brownian motions, λ i,j , (i, j) ∈ N 2 are the eigenvalues of Q, with λ i,j = i 2 + j 2 −(β+δ) , β > 0, (4.3) in the representation (4.2) for some small δ > 0. To obtain trace class noise, it is enough to have β + δ > 1. In our simulations, we take β ∈ {1.5, 2} and δ = 0.001. In (2.25), we take b(x, u) = 4u, x ∈ Λ and u ∈ R. Therefore, from [17,Section 4] it follows that the operators B defined by (2.25) fulfills Assumption 2.6 and Assumption 2.8.
V I 53 2 L 2 (Ω,H) = m−1 k=1 t m−k t m−k−1 E m j=m−k U h (t j , t j−1 ) − m−1 j=m−k−1 e ∆tA h,j . P h B t m−k−1 , X h (t m−k−1 ) 2 L 0 2 ds ≤ C m−1 k=1 t m−k t m−k−1 ∆t 1−ǫ ds ≤ C∆t 1−ǫ .
The function F is given by F (t, v) = − e −t v 1 + |v| , t ∈ [0, T ], v ∈ H and obviously satisfies where v is the Darcy velocity. We obtain the Darcy velocity field v = (q i ) by solving the following system and −k ∇p(x, t) · n = 0 in Γ 1 N . Here, we use a constant permeabily tensor k and have obtained almost a linear presure p. Clearly D(A(t)) = D(A(0)), t ∈ [0, T ] and D((−A(t)) α ) = D((−A(0)) α ), t ∈ [0, T ], 0 ≤ α ≤ 1. The function q ij (x, t) defined in (2.23) is given by q ii (x, t) = 1 + e −t , and q ij (x, t) = 0, i = j. Since q ii (x, t) is bounded below by 1 + e −T , it follows that the ellipticity condition (2.24) holds and therefore as a consequence of Section 2.2, it follows that A(t) is sectorial. Obviously Assumption 2.2 is fulfilled. In Figure 4.1, we can observe the convergence of the the stochastic Magnus scheme for two noise's parameters. Indeed the order of convergence in time is 0.57 for β = 1 and 0.54 for β = 2. These orders are close to the theoretical orders 0.5 obtained in Theorem 2.9 for β = 1 and β = 2.
∇ · v = 0, v = −k∇p,(4.
12) see e.g. [13, (2.1)]. Proposition 2.4. [28, Theorem 6.1, Chapter 5] Let ∆(T ) := {(t, s) : 0 ≤ s ≤ t ≤ T }. Under Assumption 2.2 there exists a unique evolution system [28, Definition 5.3, Chapter 5] U : ∆(T ) −→ L(H) such that (i) There exists a positive constant K 0 such that
abstract form (1.1), the nonlinear functions F : H −→ H and B : H −→ HS(Q 1/2 (H), H) are defined by
42) and for any M ∈ N, ∆t = T /M , t m = m∆t, m = 0, 1, · · · , M and ∆W m := W (m+1)∆t − W m∆t .(2.43)
Lemma 3.1. [33] Let Assumption 2.2 be fulfilled. Then for any γ ∈ [0, 1], the following estimates hold uniformly in h > 0 and t ∈ [0, T ]
Remark 3. 3 .
3From Lemma 3.2 and the fact that D(A h (t)) = D(A h (0)), it follows from [28, Theorem 6.1, Chapter 5] that there exists a unique evolution system U h : ∆(T ) −→ L(H), satisfying [28, (6.3), Page 149]
. 4 .
4Under Assumption 2.2, the evolution system U h : ∆(T ) −→ H satisfies the following (i) U h (., s) ∈ C 1 (]s, T ]; L(H)), 0 ≤ s ≤ T and
( 3 . 25 )
325Substituting (3.25), (3.24) and (3.23) in (3.22) yields
L 2 (
2Ω,H):= II0 + II1 + II2.(3.28)
3.31),(3.30) and(3.29) in (3.28) completes the proof of (3.20). The proof of (3.21) follows from(3.8). In fact from (3.8) we have
3.37),(3.36),(3.35),(3.34) and(3.33) in(3.32) completes the proof of (3.21). Let us consider the following deterministic problem: find u ∈ V such that u ′ = A(t)u, u(τ ) = v, t ∈ (τ, T ].(3.38)
X
(t) − X h (t) 2 L 2 (Ω,H) ≤ Ch 2β + C t 0 X(s) − X h (s)2 L 2 (Ω,H) ds. (3.49)
Lemma 3 . 12 .
312Let Assumption 2.2 be fulfilled. Then for all 1 ≤ i ≤ m ≤ M .
I − e (s−t m−k−1 )A h,m−k−1 e (t m−k −s)A h,
.
P h B t m−k−1 , X h (t m−k−1 ) − P h B t m−k−1 2 (Ω,H) ≤ C∆t min(β,1) + C∆t m−1 k=0 X h (t k ) − X h k 2 L 2 (Ω,H). discrete Gronwall's lemma to (3.94) yieldsX h (t m ) − X h m L 2 (Ω,H) ≤ C∆t min(β,1−ǫ)/2 . (3.95)
4 .j
4Numerical experiments. We consider the following stochastic reactive dominated advection diffusion reaction with constant diagonal difussion tensordX = (1 + e −t ) (∆X − ∇ · (qX)) − e −t X |X| + 1 dt + XdW, X(0) = 0,(4.1) with mixed Neumann-Dirichlet boundary conditions on Λ = [0, L 1 ] × [0, L 2 ]. The Dirichlet boundary condition is X = 1 at Γ = {(x, y) : x = 0} and we use the homogeneous Neumann boundary conditions elsewhere. The eigenfunctions {e i,j } = {e } i,j≥0 of the covariance operator Q are the same as for the Laplace operator −∆ with homogeneous boundary condition, l ∈ {1, 2} , x ∈ Λ. We assume that the noise can be represented as W (x, t) = (i,j)∈N 2 λ i,j e i,j (x)β i,j (t), (4.2)
Assumption 2. 5 .
5The nonlinear operator A(t) is given byA(t) = (1 + e −t ) (∆(.) − ∇.v(.)) , t ∈ [0, T ],(4.4)
{0} × [0, L 2 ] 0 in {L 1 } × [0, L 2 ]
Fig. 4 . 1 .
41Convergence of the implicit scheme for β = 1, and β = 2 in (4.3). The order of convergence in time is 0.57 for β = 1, 0.54 for β = 2. The total number of samples used is 100.
Estimate of V I 4 . To estimate V I 4 , we split it in five terms as follows.H) .
(3.80)
3.2.2.
83)Using Lemma 3.9, Assumption 2.5, Theorem 2.7, (2.35) and (2.36) yields
V I 43 L 2 (Ω,H)
Estimate of V I 5 . To estimate V I 5 , we split it in four terms as follows86)
Substituting (3.86), (3.85), (3.84), (3.83) and (3.82) in (3.81) yields
V I 4 L 2 (Ω,H) ≤ C∆t min(β,1)/2 + C∆t
m−1
k=0
X h (t k ) − X h
k L 2 (Ω,H) .
(3.87)
3.2.3.
Indeed the operators A(t) are identified to their L 2 realizations given in (2.23) (see[9]).
k=1 t m−k t m−k−1 t −1+ǫ k (s − t m−k−1 ) 1−ǫ ds ≤ C∆t 1−ǫ m−1 k=1 t m−k t m−k−1 t −1+ǫ k
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[] |
[
"SUR L'APPLICATION DES PÉRIODES D'UNE VARIATION DE STRUCTURE DE HODGE ATTACHÉE AUX FAMILLES D'HYPERSURFACESÀ SINGULARITÉS SIMPLES",
"SUR L'APPLICATION DES PÉRIODES D'UNE VARIATION DE STRUCTURE DE HODGE ATTACHÉE AUX FAMILLES D'HYPERSURFACESÀ SINGULARITÉS SIMPLES"
] |
[
"Philippe Eyssidieux ",
"Damien Mégy "
] |
[] |
[] |
Soit n ∈ N * un entier positif pair et d un entier positif. Pour toute famille complète Z d'hypersurfaces de P n+1 de degré dà singularités isolées de type A-D-E, nous construisons d'après une idée de Carlson et Toledo reprise dans [Sim93, Mé12] un champ de Deligne-MumfordZ d'espace de modules Z auquel la représentation de monodromie de la famille se prolonge. Nouś etudions l'application de périodes associée et montrons un théorème de Torelli infinitésimal le long des strates isosingulières de Z sous des hypothèses de transversalité. Enfin, nous appliquons ce résultatà l'étude du revêtement universel deZ.Dans [Sim93] est décrite une classe de surfaces projectives algébriques S munies de Q-Variations de Structure de Hodge V S qui sont intéressantes du point de vue de la théorie de Hodge non-abélienne : (S, V S ) ne peut pas s'exprimer par tiré en arrièreà partir de systèmes locaux sur des courbes, variétés abéliennes ou espaces localement symétriques hermitiens. Ce sont des exemples particulièrement intéressants pour l'uniformisation en plusieurs variables complexes (voir [Eys11] pour un survey récent) et l'un de nous a généralisé cette construction jusqu'en dimension 6 et a entamé l'étude cohomologique de ces exemples [Mé12]. La conjecture de Toledo stipulant que H 2 (π 1 (S), Q) = 0 n'est décidée dans cette classe d'exemples que dans certains cas[Mé12]. La motivation initiale de ce travail est d'étudier pour cette classe d'exemples l'autre problème ouvert général de l'uniformisation en plusieurs variables complexes, c'està dire la conjecture de Shafarevich prédisant que le revêtement universel d'une variété projective algébrique complexe est holomorphiquement convexe (cf. [Eys11] pour la définition de la convexité holomorphe et une discussion du problème).Décrivons la construction de [Sim93, Mé12] qui reprend une idée de Carlson et Toledo. Dans ce qui suit X désigne une variété projective complexe connexe de dimension n + 1, n ≥ 1, L un faisceau inversible tel que |L| n'a pas de point base. Soient X ⊂ |L| × X l'hypersurface universelle et p 1 : X → |L| la projection sur le premier facteur.le point correspondant. Définissons l'ouvert de Zariski U (0) := U (X, L)(0) comme le lieu des [f ] ∈ |L| tels que X f est une hypersurface lisse et notons D := |L| − U (0) le lieu discriminant. On pose X (0) = p −11 (U (0)). Sur U (0), on construit le système local de monodromie R n p 1 * Q X (0)
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10.4171/cmh/369
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[
"https://arxiv.org/pdf/1305.3780v1.pdf"
] | 119,330,310 |
1305.3780
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4c2b7b2f6650afdeb5cd340215e759d31e08d02a
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SUR L'APPLICATION DES PÉRIODES D'UNE VARIATION DE STRUCTURE DE HODGE ATTACHÉE AUX FAMILLES D'HYPERSURFACESÀ SINGULARITÉS SIMPLES
16 May 2013
Philippe Eyssidieux
Damien Mégy
SUR L'APPLICATION DES PÉRIODES D'UNE VARIATION DE STRUCTURE DE HODGE ATTACHÉE AUX FAMILLES D'HYPERSURFACESÀ SINGULARITÉS SIMPLES
16 May 2013
Soit n ∈ N * un entier positif pair et d un entier positif. Pour toute famille complète Z d'hypersurfaces de P n+1 de degré dà singularités isolées de type A-D-E, nous construisons d'après une idée de Carlson et Toledo reprise dans [Sim93, Mé12] un champ de Deligne-MumfordZ d'espace de modules Z auquel la représentation de monodromie de la famille se prolonge. Nouś etudions l'application de périodes associée et montrons un théorème de Torelli infinitésimal le long des strates isosingulières de Z sous des hypothèses de transversalité. Enfin, nous appliquons ce résultatà l'étude du revêtement universel deZ.Dans [Sim93] est décrite une classe de surfaces projectives algébriques S munies de Q-Variations de Structure de Hodge V S qui sont intéressantes du point de vue de la théorie de Hodge non-abélienne : (S, V S ) ne peut pas s'exprimer par tiré en arrièreà partir de systèmes locaux sur des courbes, variétés abéliennes ou espaces localement symétriques hermitiens. Ce sont des exemples particulièrement intéressants pour l'uniformisation en plusieurs variables complexes (voir [Eys11] pour un survey récent) et l'un de nous a généralisé cette construction jusqu'en dimension 6 et a entamé l'étude cohomologique de ces exemples [Mé12]. La conjecture de Toledo stipulant que H 2 (π 1 (S), Q) = 0 n'est décidée dans cette classe d'exemples que dans certains cas[Mé12]. La motivation initiale de ce travail est d'étudier pour cette classe d'exemples l'autre problème ouvert général de l'uniformisation en plusieurs variables complexes, c'està dire la conjecture de Shafarevich prédisant que le revêtement universel d'une variété projective algébrique complexe est holomorphiquement convexe (cf. [Eys11] pour la définition de la convexité holomorphe et une discussion du problème).Décrivons la construction de [Sim93, Mé12] qui reprend une idée de Carlson et Toledo. Dans ce qui suit X désigne une variété projective complexe connexe de dimension n + 1, n ≥ 1, L un faisceau inversible tel que |L| n'a pas de point base. Soient X ⊂ |L| × X l'hypersurface universelle et p 1 : X → |L| la projection sur le premier facteur.le point correspondant. Définissons l'ouvert de Zariski U (0) := U (X, L)(0) comme le lieu des [f ] ∈ |L| tels que X f est une hypersurface lisse et notons D := |L| − U (0) le lieu discriminant. On pose X (0) = p −11 (U (0)). Sur U (0), on construit le système local de monodromie R n p 1 * Q X (0)
du morphisme H n (X, Q) ⊗ Q U(0) → R n p 1 * Q X (0) . On a, par le théorème de semisimplicité, un isomorphisme R n p 1 * Q X (0) ≃ V U(0) ⊕ H n (X, Q) ⊗ Q U(0) . Fixons une fois pour toute unélément général f gen ∈ H 0 (X, L) et prenons η := [f gen ] ∈ U (0) comme point base.
Supposons désormais que n est pair. Le système local V U(0) est le produit tensoriel par Q d'un système sous jacentà une Z-Variation de Structures de Hodge de poids n dont la polarisation est orthogonale et la fibre en η est H n ev (X η , Q) := ker(H n (X η , Q) → H n (X, Q)). On note par propre sur son espace de module U contenant U (0) comme un ouvert de Zariski de sorte que, le morphisme surjectif π 1 (U (0), η) → π 1 ( U , η) (voir [Noo04,Noo05] pour les groupes d'homotopie des champs topologiques) associéà l'inclusion de U (0) dans U a un noyau contenu dans celui de ρ. La représentation ρ descend alorsà une représentation
ρ : π 1 ( U , η) → O := O(H n ev (X η , Q), Xη − ∪ −).
apparaissant comme l'holonomie d'une Q-Variation de Structures de Hodge polarisée de poids n (V U , F , S) sur U . L'image Γ de ρ est la même que celle deρ et par un théorème classique de Beauville [Bea86], c'est un sous-groupe arithmétique du groupe orthogonal G = O(H n ev (X η , R), Xη − ∪ −). On note par D := G/U le domaine de Griffiths attachéà (V U , F, S) [Gri73], Ú etant le sous-groupe qui stabilise la structure de Hodge sur H n ev (X η , Q). On rappelle que D a une structure naturelle de variété complexe homogène naturelle et porte une distribution holomorphe horizontale G-équivariante. L'action de Γ sur D est proprement discontinue et le champ quotient [Γ\D] est un orbifold complexe. L'application des périodes de (V U , F , S) définit une application holomorphe horizontale de champs complexes analytiques p : U → [Γ\D].
Pour toute variété Z propre sur U (ce qui existe avec dim(Z) < c), on peut construire un revêtement Galoisien fini Z ′ → Z de groupe Gétale au dessus de U (1)∩Z et un morphisme [G\Z ′ ] → U ce qui fournit un système local G-équivariant V Z ′ qui est sous jacentà une Q-VSH G-équivariante sur Z ′ . Plus généralement pour tout morphisme F : Y → U avec Y projective-algébrique V Y := F * (V U , F , S) est une Q-VSH d'application de périodes p • F . Pour plus de détails sur ces exemples, diverses généralisations et uneétude cohomologique de V P ′ si P ⊂ |L| est un espace linéaire générique de dimension ≤ 6 voir [Mé12].
Soit k ∈ N. L'ensemble U (k) ⊂ |L| = PH 0 (X, L) des hypersurfaces X f := {f = 0}à singularités simples de nombre de Tjurina total τ (f ) inférieur ouégalà k est un ouvert dont le complémentaire a codimension c ≥ min(7, k) dès que L est k-ample. On a U (0) ⊂ U (k) ⊂ U et on note U (k) = U × U U (k). C'est un sous champ de U d'espace de modules U (k) Le principal résultat de cet article est un théorème de Torelli infinitésimal. Pour l'énoncer, nous avons besoin d'une définition. Si I ⊂ O X est un idéal cohérent, on note p X (I) = min{m, h 1 (X, I(m)) = 0}. D'autre part, on note H k l'ensemble des idéaux cohérents dont le cosupport est un sous schéma artinien de longueur ≤ k, et qui sont localement des idéaux jacobiens de singularités isolées simples. Alors sup I∈H k p X (I) < ∞. On note s k (X) ce supremum.
Theorème 1. Si X = P n+1 , n ≥ 2 pair, et L = O X (d) et k ∈ N vérifie d ≥ n + 3 + s k (P n+1 )
, l'action de P GL(n + 2) sur |L| se relèveà U en préservant les U (k) et la différentielle de la restriction de l'application de périodes de V Uà chaque strate U (k) − U (k − 1) a pour noyau le tangent de l'orbite de P GL(n + 2).
On a s 0 (P n+1 ) = s 1 (P n+1 ) = 0, s 2 (P n+1 ) = 2. Nos bornes ne sont pas optimales. Si les surfaces quintiques avec un noeud sont obtenues par notre théorème, ce dernier est vide pour les surfaces quartiquesà singularités simples alors que Torelli infinitésimal est bien connu dans ce cas. L'obtention de bornes optimales nécessiterait des arguments nettement plus fins non développés ici.
Corollaire 2. Sous les hypothèses précédentes
p : [P GL(n + 2)\ U (k)] → [Γ\D] est finie.
Le résultat avec k = 0 est un résultat classique de Griffiths [Gri69]. La preuve du théoreème 1 repose sur le calcul de la différentielle de l'application de périodes pour des hypersurfaces nodales issue du travail fondamental [DimSa06] et de l'étude de leur filtration de Hodge dans [DiSaWo09]. Nousétendons une partie des résultats de ces articles aux hypersurfaces a singularités simples. Cette extension effectuée, l'énoncé de type Torelli infinitésimal repose sur une variante donnée au lemme 3.2.1 du théorème de Macaulay pour des hypersurfacesà singularités isolées quasihomogènes, exactement comme dans [Voi02]. Techniquement, nos résultats sont complémentaires de ceux de [DimSt11,Dim12] qui ne considèrent pas la question de Torelli infinitésimal. 2 Une généralisation du théorème de Griffiths sur les intégrales rationnelles donnant une interprétation de la différentielle de l'application de périodes comme opérateur de multiplication pour les directions transverses aux strates isosingulières ne semble pas avoirété considérée de façon systématique dans la littérature. De même il est probable que le théorème 1 se généralise pour X quelconque pourvu que L soit assez ample mais là encore nous n'avons pas trouvé de référence dans la littérature. Nous laissons ces questions pour de futures recherches. L'applicationà la conjecture de Shafarevich est immédiate en utilisant une construction de [Eys97,Eys04] : 1.1. Complémentsà la construction de [Mé12]. Cette construction peut se reformuler et s'étendre comme suit : Proposition 1.1.1. Soient X, L et U comme ci-dessus. Il existe un champ de Deligne-Mumford U = k U (k) compactifiant U (0) et fini sur son espace de modules U , tel que la représentation de monodromie de π 1 (U (0)) se prolongeàρ :=ρ (X,L) : π 1 ( U , η) → O(H n (X η , Q)) où η ∈ U (0) est un point base arbitraire.
Corollaire
Preuve: Soit f ∈ H 0 (X, L) tel que l'hypersurface X f ⊂ X soità singularités isolées. Tout voisinage de [f ] dans PH 0 (X, L) induit une déformation de X f et donc, pour chaque singulier P ∈ |Σ|, une déformation du germe singulier (X f , P ). On en déduit un morphisme de germes
λ : (|L|, [f ]) → P ∈|Σ| Def (X f , P )
où Def (X f , P ) est la base de (( la )) déformation miniverselle de la singularité isolée d'hypersurface (X f , P ). Il est connu que dans ce cas, cette base Def (X f , P ) est lisse, et naturellement isomorpheà un voisinage de 0 dans l'espace vectoriel Def (X f , P ). En choisissant une trivialisation de L P de façon adéquate, l'application λ a pour différentielle l'application d'évaluation des jets aux points singuliers de X f :
O Σ,P . La déformation (X × |L|, P × [f ]) ⊃ (X , P × [f ]) → (|L|, [f ]) estev [f ] = P ∈|Σ| ev P : T [f ] |L| → P ∈|Σ| O Σ,P ⊗ OΣ L qui provient par quotient de l'application naturelle H 0 (X, L) → H 0 (X, L ⊗ O Σ ).
Stratification par le nombre de Tjurina.
Soit k ∈ N. Posons Z(k) = U (k) − U (k − 1) et supposons [f ] ∈ U (k). Z(k) est localement fermé dans U . Pour l ≤ k le germe (Z(k), [f ]) se laisse décrire comme l'image réciproque par λ de la réunion sur toutes les partitions de l l = P ∈|Σ| l P de P ∈|Σ| Z u (l P ) où Z u (l P ) ⊂ Def (X f , P ) est le lieu des germes de nombre de Tjurinaégalà l P .
Si
ev [f ] est surjectif, λ estéquivalentà la seconde projection du produit (Z(k), [f ])× P ∈|Σ| Def (X f , P ) [Loo84, Ch. 6] et la stratification U (k) = k l=0 Z(l) est de
Whithney en [f ] puisque c'est le cas pour la stratification par les nombres de Tjurina de la déformation miniverselle d'un germe de singularité simple. La strate Z(l) est alors de codimension l.
Revêtement Galoisien neutralisant la monodromie localeà l'infini 3
Soit B un voisinage de [f ] suffisamment petit dans |L|. Notons ρ la représentation de monodromie de R n p 1 * Q X (1) et ρ P la représentation de π 1 (Def (X, P ) − Z(1)) définie par la monodromie de la fibre de Milnor de P ∈ |Σ|. Alors, ρ| π1(B−Z(1)) se décompose comme somme directe d'un facteur trivial et du tiré en arrière par λ de ⊕ P ρ P . Puisque, en dimension paire, la monodromie locale ρ P est d'image finie, il suit que ρ(π 1 (B − Z(1)) est fini.
Par un théorème de Selberg, il existe un sous-groupe normal sans torsion d'indice fini dans ρ(π 1 (|L| − Z(1))). Le revêtementétale fini correspondant η : U ′ (0) → U − Z(1) est galoisien de groupe G et se prolonge par le théorème de Grauert-Remmert en un revêtement Galoisien normal encore noté η :
|L| ′ → |L|. On note aussi U ′ (k) = |L| ′ × |L| U (k) et η : U ′ (k) → U (k) la restrictionà cet ouvert. Orbifold U (k). Considérons le champ quotient U (k) = [U ′ (k)/G]. Si ev [f ] est surjective pour tout [f ] ∈ U (k), U ′ (k) est lisse (voir [Mé12]) donc U (k) est un champ de Deligne- Mumford lisse d'espace de modules U (k) et l'isotropie en [f ]
est le produit des groupes de monodromie locale des singularités. On note π 1 ( U (k)) le groupe fondamental du champ topologique sous-jacent [Noo04,Noo05].
Par construction, toute composante connexe V de la préimage de U − Z(1) dans U ′ (k), vérifie ρ(π 1 (V )) = {e} et que U ′ (k) soit lisse ou non, la représentation ρ • η * est induite par une représentation ρ ′ :
π 1 (U ′ (k), x) → O(H n (X g , Q)) où x ∈ U ′ (1) est un point base et [g] = η(x). Le système local correspondant est G-équivariant et descendà un système local V U(k) sur U (k) et on noteraρ : π 1 ( U (k), x) → O(H n (X g , Q)) sa monodromie. Si U (k) est lisse ce système local est sous jacentà une Q-VSH par [Gri73]. 2
Notre discussion implique aussi la précision suivante sauf pour le dernier point pour lequel on renvoieà [Mé12].
Proposition 1.1.2. Si, de plus, L est k-jet-ample, U (k) est lisse, codim |L| U (k) c ≥ max(7, k) et la stratification U (k) = ∪ k l=0 Z(l) (où l'on a posé Z(l) = U (l) − U (l + 1)) est de Whithney.
Notons que γ : Z(k) → Z(k) est une gerbe de lien un schéma en groupes fini si, pour tout [f ] ∈ Z(k), ev [f ] est surjectif. Ceci permet de définir un système local V Z(k) := γ * V Z(k) qui est sous jacentà une VSH polarisée sur Z(k).
1.2.
Une question ouverte. Dans le cas X = P 3 , L = O P 3 (4) on sait queρ est un isomorphisme sur son image grâce au théorème de Torelli pour les surfaces K3. Une conjecture de Carlson-Toledo prédit que les seuls réseaux de groupes algébriques réels semisimples apparaissant comme groupes kählériens sont ceux des groupes de type hermitiens symétriques. Cette conjecture implique queρ n'est pas injective pour d ≥ 5. Ceci motive :
Conjecture 1. Pour d ≥ 5 le noyau deρ =ρ (P 3 ,O P 3 (d)) est un groupe infini.
La méthode de [CaTo99] pour prouver le fait analogue dans le cas de ρ ne s'applique malheureusement pas ici. Nous ne voyons pas comment construire d'autres représentations linéaires de π 1 ( U (P 3 , O P 3 (d)), η). Ceci motive la : Question 2. Le groupe π 1 ( U (P 3 , O P 3 (d)), η) admet il d'autres représentations complexes que les représentations de la forme α •ρ où α est une représentation rationnelle de O(H n (X g , R)) ?
Pour d = 4 la réponseà cette question est négative par le théorème de superrigidité de Margulis.
1.3. Relèvement de l'action de P Aut(X, L). Le groupe Aut(X, L) des automorphismes du couple (X, L) agit sur |L|à travers P Aut(X, L) = Aut(X, L)/C * en préservant les U (k). Proposition 1.3.1. Si le groupe P Aut(X, L) est semisimple, son action sur U se relèveà une action sur U .
Preuve: L'algèbre de Lie paut(X, L) op ⊂ H 0 (Θ |L| ) est une algèbre de Lie de champs de vecteurs holomorphes qui sont tangentsà chaque Z(k). Or Z(1) est le lieu de ramification de |L| ′ → |L|. Donc ces champs de vecteurs se relèventà des champs de vecteurs sur |L| ′ car tout champ de vecteurs tangent au lieu de
ramification de Z → Y avec Y lisse et Z normal se relèveà Z.
En effet, le relèvement a lieu en codimension un car un germe de champ de vecteurs de la forme a(z, (w j ))z ∂ ∂z + b i (z, (w j )) ∂ ∂wi a, b ∈ C{z, w 1 , . . . , w n } se relève bienà un germe de champ de vecteurs holomorphes par un morphisme de la forme (z, (w j ) → (z e , (w j )). Ce relèvement en codimension un se prolongeà Y tout entier car le faisceau des champs de vecteurs holomorphes sur l'espace normal Y est réflexif.
On dispose donc d'un morphisme d'algèbres de Lie paut(X, L) op → H 0 (Θ U ′ ) G qui s'exponentie en un morphisme de groupes de Lie complexes du revêtement universel topologiqueP de P Aut(X, L) vers le centralisateur C(G, Aut(U ′ ) de G dans Aut(U ′ ).
Si P Aut(X, L) est semisimple, le groupe P est un groupe algébrique affine semisimple et le morphisme correspondant est un morphisme de groupes algébriques. Le noyau N du morphisme P → Aut ′ (U ) est contenu dans le noyau N ′ de P → P Aut(X, L) car l'action de P redescendà une action de P sur U factorisant via P Aut(X, L). Mais N ′ /N commuteà G et préserve η. surjectif pour tout X f ∈ Z(k). Les sous faisceaux de Θ Z(k) définis par les noyaux des Gr F (∇) pour les VSH sous-jacentes aux systèmes locaux Gr n+2 W R n+1 (p 1 ) * Q (Z(k)×X−X Z(k) ) et V Z(k) sontégaux. Remarquons que l'hypothèse de la proposition est satisfaite si par exemple L est le produit tensoriel d'au moins τ (f ) fibrés très amples.
Pour montrer la proposition, on compare la cohomologie de X fà celle de son complémentaire (2.1) ainsi qu'à la structure limite (
(2.1) H n+1 (X, Z) → H n+1 (X − X f , Z) → H n+2 X f (X, Z) → H n+2 (X, Z)
Toujours parce que X f est une variété d'homologie rationnelle, on a un isomorphisme de structure de Hodge H n (X f , Q)(1) → H n+2 X f (X, Q). Ainsi Gr n+2 W H n+1 (X− X f , Z) est une sous structure de Hodge pure de poids n + 2 de H n+2 X f (X, Z) et sa filtration de Hodge vérifie
0 = F n+2 ⊂ F n+1 ⊂ . . . ⊂ F 1 = F 0 = Gr n+2 W H n+1 (X − X f , Z)
On note que si H n+1 (X, Z) = 0, ce qui est le cas si X = P n+1 puisque n est pair, alors H n+1 (X − X f , Z) est pure de poids n + 2. La première flèche a du triangle distingué canonique
Q X f [n] → ψ t (Q X∆ ) can − − → φ t (Q X∆ ) +1 − − →isosingulière Z(k) de X f . Notons enfin g la composée X T π − → T t − → ∆. On a sur X S un triangle distingué Q XS [n] sp − → ψ g (Q XT ) can − − → φ g (Q XT ) +1 − − → .
Le dernier terme est supporté sur le lieu singulier de g qui est une union finie de multisectionsétales de X S → S qui décrivent les points singuliers des X s pour s ∈ S. Comme ces singularités sont de type ADE donc rigides, le dernier terme est donc une somme directe, de tirés en arrière de faisceaux gratte-ciel sur la déformation universelle locale de chaque singularité.
En appliquant π * , et en utilisant l' isomorphisme naturel π * ψ g ≃ ψ t π * , on obtient sur S le triangle
π * Q XS → ψ t (π * Q XT ) → π * φ g (Q XT ) +1 − − → .
Ce dernier donne en cohomologie une suite exacte sur S KoMo98,p. 135]à la famille universelle p 1 : X → U on trouve un revêtement ramifié r : U * → U et une application holomorphe propre et lisseū : X * → U * avec un morphisme π : X * → X × U U * qui soit une résolution simultanée, c'està dire que pour tout s ∈ U * π s : X * s → X s soit une résolution. De plus, ces résolutions peuventêtre supposées minimales. Le morphisme r factorise par un morphisme fini r ′ : U * → U . L'existence globale de r n'est pasévidente et il n'est pas clair pour nous que r puisseêtre choisi de façonà ce que r ′ soitétale. Toutefois, cf [BuWa74], c'est le cas si on restreint uà un petit voisinage d'un point de U . Ceci implique que V U,[f ] ≃ H 2 (X ′ f , Q) comme structures de Hodge où X ′ f → X f est la résolution minimale mais aussi que r ′ * V U ≃ R 2ū * Q X * comme variations de structures de Hodge polarisables.
0 → R n π * Q XS → V S ⊕ H n (X, Q) → x∈|Σ| R n π * φ g (Q X∆ ) x .
Notons Z * (k) = r −1 (Z(k)−Z(k+1)) et appelons X * (k), resp. X (k) la restriction de X * , resp. Xà Z * (k). X * (k) est lisse sur Z(k) et, quitteà faire un revêtement etale de Z 0 (k), on peut supposer que sur chaque composante connexe Z 0 (k) de Z * (k) l'ensemble singulier de X (k) est un produit de Z 0 (k) par un ensemble fini. L'ensemble exceptionnel de π k : X * (k) → X (k) est globalement un produit de Z 0 (k) par une réunion de configurations de courbes rationnelles du type A-D-E adéquat.
Ceci donne des suites exactes de Variation de Structure de Hodge :
0 → Q(1) ⊕k Z 0 (k) → R 2ū * Q X * (k) | Z 0 (k) ≃ (r ′ ) * V U | Z 0 (k) → R 2 (p 1 ) * Q X (k) → 0 Notons v : U * × X − X * → U * la première projection. La suite exacte (2.1) im- plique que R 2 (p 1 ) * Q X (k) (1) ≃ Gr 4 W R 3 v * Q (U * ×X−X * ) | Z 0 (k)
, puis par le théorème de semisimplicité :
(r ′ ) * V U | Z 0 (k) ≃ Q(1) ⊕k Z 0 (k) ⊕ Gr 4 W R 3 v * Q (U * ×X−X * ) | Z 0 (k)
Ceci implique que sur Z 0 (k), les VSH Gr 4
W R 3 v * Q (U * ×X−X ) | Z 0 (k) et V U | Z 0 (k) ne difH 0 (L m−(n+2)D ) ⊗ Λ n+2 G → . . . → H 0 (L m−D ) ⊗ G → H 0 (L m )
où le premier terme du complexe est par convention en degré 0. Considéronségalement, avec la même convention, sa version faisceautique
K(G) • m : O X (L m−(n+2)D ) ⊗ Λ n+2 G → . . . → O X (L m−D ) ⊗ G → O X (L m )
L'hypercohomologie de ce complexe de faisceaux est décrite par le lemme suivant.
(1) H n+2 (K(G) • m ) ≃ H n+1 (K(G) • m ) ≃ P ∈|Σ| O Σ,P .
Preuve: Le support |Σ| de Σ consiste en un nombre fini de points de X et
O Σ = P ∈|Σ| O Σ,P
est une somme de faisceaux gratte-ciel. Par abus de langage on identifie O Σ,P et l'algèbre artinienne locale de Σ en P . Ensuite, pour tout P ∈ |Σ|, désignant par m P ⊂ O X,P l'idéal maximal, on a dim C G P /m P G P = n + 1 et toute famille (g 0 , . . . g n ) dans G P induisant une base de G P /m P G P est une suite régulière dans O X,P engendrant G P voir par exemple [Kap70, Thm 129]. On peut donc choisir une base (g 1 , . . . , g n , g n+1 ) de G de sorte que (g 1 , . . . g n ) est une suite régulière engendrant G P . Dans la catégorie dérivée D b (M od(O X,P )) on a donc
K(G) • 0,P = K(g 1,P , . . . , g n+1,P ) • 0 ⊗ K(g n+2,P ) • 0 ≃ O Σ,P [−n − 1] ⊗ L K(g n+2,P ) • ≃ (O Σ,P [−n − 1] 0 → O Σ,P [−n − 2])
puis H n+1 (K(G) • 0,P ) ≃ H n+2 (K(G) • 0,P ) ≃ O Σ,P comme O X,P -modules, les autres faisceaux de cohomologieétant nuls. Par suite, les faisceaux de cohomologie non nuls
de K(G) • m sont les gratte-ciels H n+1 (K(G) • m ) ≃ H n+2 (K(G) • m ) ≃ P ∈|Σ| O Σ,P puisque K(G) •
0 est acyclique hors de |Σ|. La suite spectrale d'hypercohomologie de K(G) • m n'a donc qu'un seul terme non nul en E 1 qui est d 1 :
H 0 (H n+1 (K(G) • m )) → H 0 (H n+2 (K(G) • m )
) et d 1 = 0 car c'est le cas après localisation. Donc, pour tout m ∈ Z, H i (K(G) • m ) = 0 pour i = n + 2, n + 1 et
(2) H n+2 (K(G) • m ) ≃ H n+1 (K(G) • m ) ≃ P ∈|Σ| O Σ,P . 2
Par ailleurs, on a la filtration bête de K(G) 0 définie par
σ ≥p K(G) • = O X (L −pD ) ⊗ Λ n+1−p G → . . . → O X .
Cette filtration décroissante induit une filtration sur H * (K(G) • m ) et la suite spectrale correspondante a un terme E p,q On a
1 = H q (Gr p σ (K(G) • )). Elle dégénère en E n+3 . On remarqueégalement que (E •,0 1 , d 1 ) = K(G) • m , de sorte que E p,0 2 = H p (K(G) • m ). Notamment, E n+2,0 2 = H n+2 (K(G) • m ) = (R/(G)) mImd n+2 = ker E n+2,0 n+2 ։ E n+2,0 n+3 = ker E n+2,0 1 ։ E n+2,0 n+3 ker E n+2,0 1 ։ E n+2,0 n+2 = ker E n+2,0 1 ։ E n+2,0 n+3 ֒→ H n+2 (K(G) • m ) ker E n+2,0 1 ։ E n+2,0 n+2
Or, la flèche composée
E n+2,0 1 ։ E n+2,0 2 = E n+2,0 n+2 ։ E n+2,0 n+3 ֒→ H n+2 (K(G) • m )
On en déduit
Im d n+2 = I m Im d 1 = I m (G) m . Deuxièmeétape : la coimage de d n+2 : E 0,n+1 n+2 → E n+2,0 n+2 s'identifie par dualité de Serreà (I/(G)) ∨ −m+(n+2)(d−2) = (I/(G)) ∨ σ−m .
La suite spectrale duale ((E p,q r ) ∨ , t d r ) s'identifie par dualité de Serreà une renumérotation de la suite spectrale de la filtration bête de :
ω P n+1 (L −m ) → ω P n+1 (L −m+d−1 ) ⊗ G ∨ → . . . → ω P n+1 (L −m+(n+2)(d−1) ) ⊗ Λ n+2 G ∨
Le complexe obtenu en tensorisant par la droite complexe Λ n+2 G ≃ C s'identifieà K(G) • −m+σ en utilisant ω P n+1 ≃ O P n+1 (n + 2). La premièreétape permet alors de conclure. Preuve: Soit P unélément du noyau. On a, pour tout Q ′′′ ∈ (A/J) d−n−2 , P Q ′′′ = 0 mod J. De là P Q ′′′ Q ′′ = 0 mod J pour tout Q ′′ ∈ A/J ≥0 . Puisque A/J ≥d−(n+2) est engendré en degré d − (n + 2) > 0, il suit que P Q ′ = 0 mod J pour tout Q ′ ∈ A/J ≥d−(n+2) . Comme σ − 2d + n + 2 = (n + 2)
E 1,n+1 2 ≃ H n+1 (K(G) • σ−m ) ∨ . 2 Fixons m 0 tel que ev m0 : H 0 (O P n+1 (m 0 )) → H 0 (O Σ (m 0 )) soit surjective.(d − 1) − 2d = nd − n − 2 ≥ d − (n + 2) on a ∀Q ∈ I/J d−n−2 ∀Q ′ ∈ I/J σ−2d+n+2 < Q; P Q ′ > d−(n+2) = 0 où < −; − > d désigne l'accouplement de dualité défini par d n+1 entre I/J d et I/J σ−d au lemme 3.2.1.
Or on a, par fonctorialité de la dualité de Serre,
< Q; P Q ′ > d−(n+2) =< QQ ′ ; P > σ−d .
Puisque I/J ≥d−(n+2) est engendré en degré d − (n + 2), P est orthogonalà l'espace I/J σ−d entier. Donc P = 0 mod J. 2
Une preuve plus courte du corollaire 3.2.4 valable dans le cas d'une hypersurface avec un seul noeud nous aété communiquée par A. Otwinowska au moment où nous finissions la présente preuve.
Théorème de Torelli local sur les strates isosingulières
Soit n un entier pair strictement positif, et soit X f ⊂ P n+1 une hypersurface de degré dà singularités isolées simples. Comme dans le cas où X f est lisse, une déformation isosingulière de X f fournit une variation de structure de Hodge dont la tige en X f est H n+1 (P n+1 − X f ). Rappelons que H f := H n+1 (P n+1 − X f ) est pur de poids n + 2 par la discussion de la section 2 et que sa filtration de Hodge vérifie ] α X f ,y le plus petit zéro de la b-fonction de la singularité. Dans notre cas, la singularité est quasihomogène et on note w 1 , ..., w n+1 les poids correspondants. Alors, il est connu que α X f ,y = i w i . Par exemple, pour une singularité A 1 , on a α X f ,y = (n + 1)/2. Un examen deséquations des singularités simples montre que l'on a toujours α X f ,y > 1. Plus précisemment, si n ≥ 4, alors ⌊ α X f ,y ⌋ > 1, et si n = 2, alors ⌊ α X f ,y ⌋ = 1.
0 = F n+2 ⊂ F n+1 ⊂ . . . ⊂ F 1 = H n+1 (P n+1 − X f , C
Lorsque n = 2 on définit un idéal homogène I ′ de l'anneau des polynômes de n + 2 variables de la façon suivante. Définissons d'abord comme [DiSaWo09, (2.1.4)] le faisceau d'idéaux I ′ (1) ⊂ O P 3 cosupporté aux points singuliers y de X f par F 1 O P 3 ,y ( * X f ) = I ′ (1) O P 3 (2X f ), où le membre de gauche désigne la filtration de Hodge du D-module O P 3 ( * X f ) correspondant au Module de Hodge Mixte sous jacentà Rj * Q P 3 −X f . On pose alors I ′ k = Γ(P 3 , I ′ (1) (k)) puis I ′ = ⊕ k∈N I ′ k . Par [DiSaWo09, Theorem 2.2], on a : Preuve: Il s'agit de voir que F 1 O P 3 ,y ( * X f ) = ( ∂h ∂x1 . . . ∂h ∂xn )h −2 O P 3 ,y où h est uneéquation locale de X f près de y Or, d'après [DiSaWo09,(1.3.2)] (qui réfèreà [Saito09]), la filtration de Hodge sur O P 3 ,y ( * X f ) est dans ce cas donnée par :
4.2.
Formule de Dimca-Saito pour le premier gradué de la connexion de Gauss-Manin de P n+1 − X f . Il y aégalement une seconde filtration sur H n+1 (P n+1 − X f , C) la filtration par l'ordre du pôle notée P • et l'on a F i ⊂ P i [DeDi90].
Le long de la strate isosingulière S de X f dans l'espace projectif paramétrisant les hypersurfaces de degré d, la connexion de Gauss Manin vérifie ∇P i ⊂ P i−1 ⊗ Ω 1 S du moins si dim P i est localement constante près de [X f ] ce qui est vrai sur un ouvert dense S ′ ⊂ S et [DimSa06] donne une formule pour Gr P ∇ ξ si ξ ∈ T [X f ] S = I/(f ) d en termes de la partie libre du module de Brieskorn.
Ceci est exploité dans [DiSaWo09, Remarks 3.9] dont nous tirons la proposition suivante : Preuve: En faisant attention au fait que la notation n ici correspondà ce qui est noté n − 1 dans [DiSaWo09], cela résulte de [DimSa06] de la même façon que dans [DiSaWo09, Remarks 3.9] au moins sur l'ouvert S ′ . On conclut par passageà la limite. 2 connexes fermés maximaux de Z ρ et donc que R n'a pas de sous-espace complexe analytique compact de dimension positive.
Pour montrer que R est de Stein, on utilise la solution de Narasimhan du problème de Levi. On peut construire des fonctions C ∞ positives et exhaustives sur D dont le hessien complexe est défini positif le long de la distribution horizontale de D et il est aisé de les modifier pour construire une fonction d'exhaustion strictement plurisousharmonique sur R, voir [Eys04] pour plus de détails.
2
1 .
1On désigne par V U(0) le conoyau Date: 16 Mai 2013. 1. Pour tout espace T et tout groupe A, A T désigne le faisceau des fonctions localement constantes sur Tà valeurs dans A. Plus généralement si W T est un système local sur T et φ : T ′ → T une application continue, on note W T ′ = φ * W T ′ . De même, pour X → T une application continue, on note X T ′ = X × T T ′ .
ρ
: π 1 (U (0), η) → O(H n ev (X η , Q), Xη − ∪ −) la représentation d'holonomie du système local V U(0) . Introduisons U := U (X, L) ⊂ |L| l'ouvert de Zariski formé des hypersurfaces n'ayant que de singularités isolées et de type A-D-E.Évidemment, U (0) ⊂ U . Nous construisons un champ algébrique de Deligne-Mumford U := U (X, L) séparé et
Nous tenonsà remercier D. Barlet, N. Borne, M. Brion, A. Dimca, S. Druel, L. Gruson, C. Peters, C. Voisin, M. H. Saito et tout particulièrement A. Otwinowska pour d'utiles remarques sur les questions traitées ici. 1. Structure orbifold sur U (k) et prolongement de la représentation de monodromie Soit X une variété projective lisse complexe de dimension impaire n + 1, L un fibré en droites sans point base sur X, et f ∈ H 0 (X, L)−{0} telle que l'hypersurface {f = 0} := X f n'ait que des singularités isolées. Soit Σ ⊂ X f le sous schéma artinien de X de support X sing f défini par l'annulation du premier jet de f . La longueur de Σ, i.e. le nombre de Tjurina total τ (f ), est défini par la relation coordonnées locales et une trivialisation locale de L et notant f P la fonction qui définit f dans ces coordonnées, on voit que O Σ,P est isomorpheà l'algèbre de Tjurina O C n+1 ,0 /(f P , ∂fP ∂x1 , . . . , ∂fP ∂xn+1 ). Soit k ∈ N, et U (k) ⊂ |L| l'ouvert de Zariski constitué des [f ] ∈ L tels que X f n'ait que des singularités isolées, simples (autrement dit de type A-D-E), et telles que τ (f ) ≤ k. On a U (0) ⊂ U (1) ⊂ . . . ⊂ U = k U (k). Remarquons que U est non vide puisque l'ouvert U (0) des [f ] tels que X f soit lisse est non vide par Bertini. Dans la suite de cette section, on introduit (1.1) un champ de Deligne-Mumford muni d'une variation de structure de Hodge. Dans certains cas (1.3), ce champ est egalement muni de l'action d'un groupe algébrique, et la variation de structure de Hodge descend au champ d'Artin quotient.
Comme η est Galoisien de groupe G on déduit que N ′ /N ⊂ Z(G) et que donc, en divisant U ′ par N ′ /N , on obtient une action de P Aut(X, L) sur U ′′ := U ′ /(N ′ /N ) qui commuteà Gal(U ′′ /U ) = G/(N ′ /N ) := G ′ et descendà une action de P Aut(X, L) sur U ≃ [U ′′ /G ′ ]. 2Corollaire 1.3.2. L'application des périodes P : U → [Γ\D] attachéeàρ descend a une application définie sur le champ quotient [P Aut(X, L)\ U ] → [Γ\D]. 2. Interprétation géométrique de la représentation de monodromie prolongée Avec les notations de 1.1, soit [f ] ∈ U et X f ⊂ X l'hypersurface de Xà singularités isolées simples définie par f et ev [f ] la flèche d'évaluation des jets aux points singuliers de X f . Proposition 2.0.3. Supposons ev [f ]
2. 2 .
2Structure de Hodge de X f et structure de Hodge limite. Soit i : ∆ → U un disque analytique tel que i(0) = [f ] et i(∆ * ) ⊂ U (0). On pose X ∆ = X × U ∆. Alors t : X ∆ → ∆ est un morphisme projectif plat lisse hors de {t = 0} et (X ∆ ) 0 = X f . Si la monodromie de X ∆ → ∆ * est unipotente,c'està dire triviale puisque les groupes de monodromie locale près de [f ] sont finis, ceci permet de définir la structure de Hodge limite H n lim (X f , Q) au sens de [Ste76]. Grâceà [Saito90], on a un isomorphisme de structures de Hodge puresH n lim (X f , Q) = H 0 (X f , ψ t (Q X∆ [n])) L'objet φ t (Q X∆ [n])est concentré aux points singuliers de X f . Le Module de Hodge Mixte ψ t (Q X∆ [n]) = ψ t (IC X∆ (Q)) est un module de Hodge polarisable pur car IC X∆ (Q) est un module de Hodge polarisable et que le logarithme de la monodromie vérifie N = 0, en vertu de [Saito88, (0.7), p. 852].
est donc un morphisme de Modules de Hodge Polarisables de même poids. La catégorie des modules de Hodge Polarisables de poids donnéétant abélienne et semi-simple [Saito88, Lemme 5, p. 854] il suit que ψ t (Q X∆ ) ≃ Q X f [n] ⊕ Coker(a) où Coker(a) a même poids que Q X f [n]. Il suit que φ t (Q X∆ ) est isomorpheà Coker(a).Ceci fournit une suite exacte scindée de structures de Hodge pures :(2.2) 0 → H n (X f , Q) → H n lim (X f , Q) → x∈|Σ| H n (φ t (Q X∆ )) x →0 2.3. Comparaison avec la variation de structure de Hodge des variétés singulières. Soit avec les notations du paragraphe 1.1 un germe de disque analytique j : (∆, 0) → p∈|Σ| Def (X f , p) tel que j(∆ * ) ⊂ p∈|Σ| Def (X f , p)(0) et dont la monodromie est nulle. On définit un germe T := (|L|, [f ]) × λ,j ∆ muni de la famille d'hypersurfaces X × T ⊃ X T π → T . La projection naturelle t : T → ∆ définit une fonction holomorphe et S := {t = 0} est exactement le germe en [f ] de la strate
férent que par un système local de monodromie finie donc d'application de périodes constante. En particulier, Torelli infinitésimal pour V U | Z 0 (k)é quivautà Torelli infinitésimal pour Gr 4 W R 3 v * Q X×U * −X | Z 0 (k) ce quiéquivautà la proposition 2.0.3 dans ce cas.3. Théorème de Macaulay avec singularités modérées Claire Voisin nous a signalé qu'une variante du théorème de Macaulay autorisant un peu de singularités devrait suivre en adaptant [Voi02, pp. 427-428]. Mettons en oeuvre de cette suggestion : après quelques généralités sur le complexe de Koszul, nous obtenons la propriété d'injectivité 3.2.4 en corollaire du résultat de dualité 3.2.1. Cela est suffisant pour montrer l'injectivité de l'application des périodes de V U sur certaines strates de U . 3.1. Un lemme sur le complexe de Koszul d'une presqu'intersection complète. Soit D ∈ N * un entier positif. Soit G ⊂ H 0 (X, L D ) un sous espace de dimension n + 2. Notons G ⊂ O X le faisceau d'idéaux engendré par G, et Σ ⊂ X le sous schéma tel que O Σ = O X /G.Posons A = A(X, L) = ⊕ k∈N H 0 (X, L k ). On appelle I l'idéal gradué de A(X, L) défini en degré k par I k = H 0 (G(k)). On a (G) ⊂ I.Soit K(G) • m le complexe de Koszul en degré m :
.
Si Σ ⊂ X est artinien et localement d'intersection complète, alors, pour tout m ∈ Z, H i (K(G) • m ) = 0 pour i = n + 2, n + 1 et
et une variation légère de la preuve du lemme 3.1.1 permet de voir que G a une suite régulière de longueur n + 1 et donc que E p,02 = H p (K(G) • m ) = 0 pour p ≤ n.3.2. Dualité de Macaulay pour des hypersurfaces de P n+1à singularités isolées quasihomogènes. On se place dans le cas particulier suivant.Soit f ∈ H 0 (P n+1 , O P n+1 (d)) − {0} telle que l'hypersurface X f := {f = 0} n'aie que des singularités isolées quasi-homogènes. On considère G ⊂ H 0 (P n+1 , O P n+1 (d − 1))le sous espace vectoriel engendré par les dérivées partielles de f , et (G) = J est alors l'idéal jacobien de f . Avec les notations précédentes, D = d−1, et Σ ⊂ X f s'identifie au sous schéma artinien de X de support X sing f défini par l'annulation du premier jet de f , Ià l'idéal de A s'annulant sur Σ et G = J Σ s'identifie, après introduction de coordonnées locales,à l'idéal de Tjurina de la fonction correspondantà f . Puisque les singularités de X f sont quasi-homogènes, les idéaux de Milnor et de Tjurina coïncident donc Σ est intersection complète locale.Si l ∈ Z, on note ev l : H 0 (P n+1 , O P n+1 (l)) → H 0 (P n+1 , O Σ (l)) la fléche induite par la surjection de faisceaux O P n+1 → O Σ . On note enfin σ = (n + 2)(d − 2). Sur l'espace projectif P n+1 , les faisceaux inversibles n'ont de cohomologie qu'en degré 0 ou n + 1, on en déduit l'annulation de la plupart des termes de la suite spectrale introduite en 3.1 : on a E p,q r = 0 sauf si q = 0 ou q = n + 1, et donc les seules fléches d r non nulles sont les fléches d 1 et la fléche d n+2 : E 0,n+1 n+2 → E n+2,0 n+2 . Les deux propositions suivantes précisent son image et sa coimage.
et (I/(G)) m . Preuve: Cette fléche induit un isomorphisme entre sa coimage et son image. Il suffit de calculer ces deux espaces. Premièreétape : l'image de d n+2 dans E n+2,0 n+2 = R/(G) m est (I/(G)) m .
apparaissant au numérateur coïncide avec H 0 (O(m)) → H 0 (O/G(m)). Son noyau est donc H 0 (G(m)) = I m . D'autre part, au dénominateur,
Si la flèche d'évaluation ev m : H 0 (O P n+1 (m)) → H 0 (O Σ (m)) est surjective, H n+1 (K(G) • σ−m ) = 0. Preuve: Au vu du lemme 3.1.1, reprenant les notations de la premièreétape de la preuve du lemme 3.2.1, on a aussi surjectivité de E n+2,0 n+3 → H n+2 (K(G) • m ). De ceci suit que E 1,n+1 2 = 0. Or, par la secondeétape de la preuve du lemme 3.2.1,
Pour tout m ≤ σ − m 0 , H n+1 (K(G) • m ) = 0. Preuve: Soit m ≤ σ − m 0 . La flèche ev σ−m : H 0 (O P n+1 (σ − m)) → H 0 (O Σ (σ −m)) est surjective. On peut donc conclure avec le lemme 3.2.2.2 Plus généralement, on a h n+1 (K(G) • m ) = h 1 (J Σ (σ − m)).
Supposons que d − (n + 2) > 0. Supposons que A/J ≥d−(n+2) est engendré en degré d− (n+ 2) 4 et que I/J ≥d−(n+2) est engendré en degré d− (n+ 2). Si m 0 ≤ d − (n + 2), l'application linéaire induite par la multiplication I/J d → Hom(A/J d−(n+2) , I/J 2d−(n+2) ) est injective.
(P n+1 − X f , C) = A/J d−n−2 , Gr n F H n+1 (P n+1 − X f , C) = A/J 2d−n−2 si n ≥ 4 I ′ /J 2d−n−2 si n = 2Dans [DiSaWo09, Lemma 1.5] esténoncé que, pour les surfaces nodales, I ′ (1) est l'idéal des fonctions qui s'annulent sur les points singuliers de X f (avec structure réduite). Il est facile de généraliser :Lemme 4.1.2. L'idéal I ′ (1) coïncide avec l'idéal de Tjurina : I ′ (1) = J Σ .
F 1 O
1P 3 ,y ( * X f ) = F 1 D P 3 ,y (h −1 O),où D P 3 ,y est filtré par l'ordre de l'opérateur. En effet, suivant les notations de loc. cit. k 0 = 0 et O ≥1 P 3 ,y = O P 3 ,y . 2 Corollaire 4.1.3. Avec les notations de la section 3.2, I ′ = I.
.
Si n ≥ 4 Gr F ∇ ξ : Gr n+1 F H f → Gr n F H f s'identifie par l'isomorphisme de la proposition 4.1.1à −1 fois la multiplication I/(f ) d ⊗ A/J d−n−2 → A/J 2d−n−2 et si n = 2,à −1 fois la multiplication I/(f ) d ⊗A/J d−n−2 → I/J 2d−n−2 .
3. Hypothèses et notations comme au théorème 1. Soit Z une variété projective lisse et f : Z → U (k + 1) un morphisme fini. Alors le revêtement universel de Z est une variété de Stein.Dans le cas où f est génériquement fini, l'étude de la conjecture de Shafarevich semble beaucoup plus délicate et nous ne savons pas non plus la décider dans tous les cas.2. Alors que nous finissions de rédiger ce travail, A. Dimca nous a signalé que le lemme 3.2.1 résultait de[DimSa12] qui traite le cas plus général des singularités isolées quelconques, moyennant une traduction qui n'est pas siévidente pour nous. Notre preuve demandant moins de technologie et restant assez courte, nous avons donc préféré la conserver.
induite par la déformation miniverselle [Loo84, Ch. 6]. En particulier leséléments de |L| proches de [f ] singuliers près de P sont ceux que λ envoie dans le discriminant de
2.2), d'abord ponctuellement puis en famille (2.3), ce qui donne le résultat. Une preuve alternative et plus constructive est donnée dans le cas de familles de surfaces en (2.4). 2.1. Relation entre les structures de Hodge de X f et de X − X f . Une singularité simple a une forme d'intersection définie négative, en particulier non dégénérée, donc X f est une variété d'homologie rationnelle [Di92, prop. 4.7]. Le groupe H n (X f , Q) coïncide donc avec le groupe de cohomologie d'intersection IH n (X f , Q) et porte une structure de Hodge pure. Le groupe H n+1 (X − X f , Z) porte une structure de Hodge mixte [Del71-75] de poids n + 1 et n + 2, entrant dans une suite exacte :
La suite exacte (2.2) implique que les tiges des termes de cette suite exacte sont des structures de Hodge pures de même poids et donc que cette suite exacte est une suite exacte de Modules de Hodge polarisables purs qui est scindée toujours par [Saito88, Lemme5, p. 854]. Finalement, en tout point de S le noyau de Gr F (∇) pour la Q-Variation de Structures de Hodge R n π * Q XS et celui de V S sont les mêmes. Interprétation dans le cas n = 2. Donnons un argument alternatif permettant de démontrer la proposition 2.0.3 dans le cas n = 2. Si on applique le théorème de résolution simultanée des singularités DuVal [Art74] [Bri70], cf. [Compte tenu de la suite exacte (2.1), ceciétablit la proposition 2.0.3.
2.4.
) .
)L'IVHS correspondante aétéétudié par Dimca et Saito[DimSa06] et avec plus de détails dans le cas nodal par ces mêmes auteurs et Wotzlaw [DiSaWo09]. Dans cette section, nous extrayons de leur travail tous les renseignements dont nous aurons besoin en ajoutant quelques petits points supplémentaires. Ceci permet d'appliquer le résultat d'injectivité 3.2.4 et d'aboutirà la démonstration du théorème 1. 4.1. Formule de Dimca-Saito-Wotzlaw pour les deux premiers termes de la filtration de Hodge. Si y est un point singulier de X f , on note comme [DiSaWo09, section 1.1
. L'hypothèse n ≡ 0[2] n'est utilisée qu'à partir de ce point.
. Ce qui est garanti dès que m 0 ≤ d − (n + 2).
. Ce qui, puisque X = P n+1 avec n ≥ 2, implique que H 0 (P n+1 , O(d ′ )) → O(d ′ )/J Σ est surjectif pour d ′ ≥ d − (n + 2) et donc que les résultats de de la section 2 s'appliquent en posant d ′ = d.
Proposition 5.1.1. La variété Z ρ est holomorphiquement convexe.Preuve: Ceci résulte des résultats de[Eys04]modulo le fait que {f * ρ } est constructible absolu. Le cas présent est particulièrement simple et il est facile de décrire la réduction de Cartan-Remmert en termes de l'application des périodes.Notons que f * ρ est sous jacentà une VSH polarisable définie sur Z qui est f * V U . Notons Γ ′ = f * ρ(π 1 (Z, z)). L'application des périodes attachéeà f * V U se relèveà une application Γ ′ -équivariante P : Z ρ → D. Comme Γ ′ agit proprement discontinûment sur D puisque Γ ′ est discret, il suit que P est propre. Considérons sa factorisation de Stein Z ρ α → R β → D, où R est un espace complexe normal, α est propre surjectiveà fibres connexes, et β finie.Comme il n'existe pas d'application holomorphe horizontale M → D où M est compacte complexe ([Gri73]), les fibres de α sont les sous espaces analytiques
Ceciétablit le théorème 1 pour d ≫ k > n + 2 grâce aux résultats de la section 2. Soyons maintenant plus précis sur les conditions que d doit satisfaire pour la condition suffisante d. X (k) Z(l) )/Z(l), le noyau de Gr F ∇ ξ : Gr n+1 F H f → Gr n F H f s'identifie via la proposition 4.2.1à J d ⊂ I d qui est l'espace tangent de l'orbite de [f ] sous P GL(n + 2). engendrement de la proposition 4.2.1. Plaçons nous en f ∈ Z(k) d'idéal Jacobien J Σ . Nous devons nous assurer que H 0 (P n+1 , O(d−(n+2)Preuve du théorème 1. Le corollaire 3.2.4 signifie précisément que pour la famille isosingulière (P n+1 ×Z(l)−X (k) Z(l) )/Z(l), le noyau de Gr F ∇ ξ : Gr n+1 F H f → Gr n F H f s'identifie via la proposition 4.2.1à J d ⊂ I d qui est l'espace tangent de l'orbite de [f ] sous P GL(n + 2). Ceciétablit le théorème 1 pour d ≫ k > n + 2 grâce aux résultats de la section 2. Soyons maintenant plus précis sur les conditions que d doit satisfaire pour la condition suffisante d'engendrement de la proposition 4.2.1. Plaçons nous en f ∈ Z(k) d'idéal Jacobien J Σ . Nous devons nous assurer que H 0 (P n+1 , O(d−(n+2))) →
Une condition suffisante est que d − (n + 2) ≥ Reg(J Σ ) où Reg est la régularité de Castelnuovo-Mumford. Or, puisqu'il définit un sous-schéma artinien, J Σ est mrégulier (pour m ≥ n + 1) si et seulement si H 1 (P n+1 , J Σ (m − 1) = 0, c'està dire si et seulement si H 0 (P n+1 , O(m − 1)) → O(m − 1)/J Σ est surjectif. La condition suffisante obtenue est donc tout simplement la surjectivité. /J Σ , de : H 0O(d − (n + 2))/J Σ est surjectif 5 et surtout que I/J est engendré sur l'anneau de polynômes A en degré d −Ce dernier point est garanti dès que I est est engendré sur l'anneau de polynômes A en degré d − (n + 2), c'està dire dès que J Σ (d − (n + 2)) est engendré par ses sections globales. P n+1 , O(d − (n + 3))) → O(d − (n + 3O(d − (n + 2))/J Σ est surjectif 5 et surtout que I/J est engendré sur l'anneau de polynômes A en degré d − (n + 2). Ce dernier point est garanti dès que I est est engendré sur l'anneau de polynômes A en degré d − (n + 2), c'està dire dès que J Σ (d − (n + 2)) est engendré par ses sections globales. Une condition suffisante est que d − (n + 2) ≥ Reg(J Σ ) où Reg est la régularité de Castelnuovo-Mumford. Or, puisqu'il définit un sous-schéma artinien, J Σ est m- régulier (pour m ≥ n + 1) si et seulement si H 1 (P n+1 , J Σ (m − 1) = 0, c'està dire si et seulement si H 0 (P n+1 , O(m − 1)) → O(m − 1)/J Σ est surjectif. La condition suffisante obtenue est donc tout simplement la surjectivité de : H 0 (P n+1 , O(d − (n + 3))) → O(d − (n + 3))/J Σ .
Applicationà la conjecture de Shafarevich sur l'Uniformisation 5.1. Variétés propres sur U. Applicationà la conjecture de Shafarevich sur l'Uniformisation 5.1. Variétés propres sur U . Dans ce paragraphe on reprend les notations de
La représentationρ : π 1 ( U , x) → O(H n (X g , Q)) construiteà la proposition 1.1.1 induit une représentation f * ρ : π 1 (Z, z) → O(H n (X g , Q)). Notons Z un → Z le revêtement universel de Z. Soit Z Une Variété Connexe Projective Lisse, F : Z → U Un Morphisme Et Z ∈ Z Un Point Base, et Z ρ := kerSoit Z une variété connexe projective lisse, f : Z → U un morphisme et z ∈ Z un point base. La représentationρ : π 1 ( U , x) → O(H n (X g , Q)) construiteà la proposition 1.1.1 induit une représentation f * ρ : π 1 (Z, z) → O(H n (X g , Q)). Notons Z un → Z le revêtement universel de Z, et Z ρ := ker(
En effet Z ρ ≃ R est de Stein et tout revêtement topologique d'une variété de Stein est Stein. Preuve, Preuve: En effet Z ρ ≃ R est de Stein et tout revêtement topologique d'une variété de Stein est Stein.
La discussion peutêtre résumée ainsi : si f : Z → U est finie, le revêtement universel de Z est de Stein sauf si Z contient une courbe C telle que V C a monodromie finie. Notons m : U → U resp. m ′ : [Γ\D] → Γ\D les applications canoniques vers les espace des modules des champs considérés. Notons qu'il existe une application holomorphe p red : U → Γ\D. Il est clair que, si f : Z → U est finie, P est finie si et seulement si p red est finie sur Z r := m • f (Z). La discussion peutêtre résumée ainsi : si f : Z → U est finie, le revêtement uni- versel de Z est de Stein sauf si Z contient une courbe C telle que V C a monodromie finie. Notons m : U → U resp. m ′ : [Γ\D] → Γ\D les applications canoniques vers les espace des modules des champs considérés. Notons qu'il existe une application holomorphe p red : U → Γ\D. Il est clair que, si f : Z → U est finie, P est finie si et seulement si p red est finie sur Z r := m • f (Z).
Si f : Z → U est finie et si Z r est un espace projectif ou plus généralement a la propriété que toute application holomorphe Z r → M (avec M un espace complexe) est constante ou finie. Z un est de SteinProposition 5.1.3.Proposition 5.1.3. Si f : Z → U est finie et si Z r est un espace projectif ou plus généralement a la propriété que toute application holomorphe Z r → M (avec M un espace complexe) est constante ou finie, Z un est de Stein.
Le corollaire 3 résulte du théorème 1 car celui-ci implique que l'application des périodes p • f : Z → Γ\D est finie sur son image. X Cas, En effetCas où X = P n+1 . Le corollaire 3 résulte du théorème 1 car celui-ci implique que l'application des périodes p • f : Z → Γ\D est finie sur son image. En effet, par
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38402 Saint Martin d'Hères Cedex, France philippe.eyssidieux@ujf-grenoble. BP. Lorraine. InstitutÉlie Cartan. B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France. Damien.Megy@univ-lorraine74Philippe Eyssidieux Institut Universitaire de France. Université Grenoble I. Institut Fourier. 100 rue des MathsPhilippe Eyssidieux Institut Universitaire de France. Université Grenoble I. Institut Fourier. 100 rue des Maths, BP 74, 38402 Saint Martin d'Hères Cedex, France [email protected] http ://www-fourier.ujf-grenoble.fr/∼eyssi/ Damien Mégy Université de Lorraine. InstitutÉlie Cartan. B.P. 70239, F-54506 Vandoeuvre-lès- Nancy Cedex, France. [email protected] http ://www.iecn.u-nancy.fr/∼megy/
|
[] |
[
"HiT-DVAE: Human Motion Generation via Hierarchical Transformer Dynamical VAE",
"HiT-DVAE: Human Motion Generation via Hierarchical Transformer Dynamical VAE"
] |
[
"Xiaoyu Bie \nInria\nLJK\nUniv. Grenoble Alpes\nCNRS\nGrenoble INP\nFrance\n",
"Wen Guo \nInria\nLJK\nUniv. Grenoble Alpes\nCNRS\nGrenoble INP\nFrance\n\nInstitut de Robòtica i Informàtica Industrial\nCSIC-UPC\nSpain\n",
"Simon Leglaive \nCentraleSupélec\nIETR\nFrance\n",
"Lauren Girin \nUniv. Grenoble Alpes\nGrenoble-INP\nGIPSA-lab\nFrance\n",
"Francesc Moreno-Noguer \nInstitut de Robòtica i Informàtica Industrial\nCSIC-UPC\nSpain\n",
"Xavier Alameda-Pineda \nInria\nLJK\nUniv. Grenoble Alpes\nCNRS\nGrenoble INP\nFrance\n"
] |
[
"Inria\nLJK\nUniv. Grenoble Alpes\nCNRS\nGrenoble INP\nFrance",
"Inria\nLJK\nUniv. Grenoble Alpes\nCNRS\nGrenoble INP\nFrance",
"Institut de Robòtica i Informàtica Industrial\nCSIC-UPC\nSpain",
"CentraleSupélec\nIETR\nFrance",
"Univ. Grenoble Alpes\nGrenoble-INP\nGIPSA-lab\nFrance",
"Institut de Robòtica i Informàtica Industrial\nCSIC-UPC\nSpain",
"Inria\nLJK\nUniv. Grenoble Alpes\nCNRS\nGrenoble INP\nFrance"
] |
[] |
Studies on the automatic processing of 3D human pose data have flourished in the recent past. In this paper, we are interested in the generation of plausible and diverse future human poses following an observed 3D pose sequence. Current methods address this problem by injecting random variables from a single latent space into a deterministic motion prediction framework, which precludes the inherent multi-modality in human motion generation. In addition, previous works rarely explore the use of attention to select which frames are to be used to inform the generation process up to our knowledge. To overcome these limitations, we propose Hierarchical Transformer Dynamical Variational Autoencoder, HiT-DVAE, which implements auto-regressive generation with transformer-like attention mechanisms. HiT-DVAE simultaneously learns the evolution of data and latent space distribution with time correlated probabilistic dependencies, thus enabling the generative model to learn a more complex and time-varying latent space as well as diverse and realistic human motions. Furthermore, the auto-regressive generation brings more flexibility on observation and prediction, i.e. one can have any length of observation and predict arbitrary large sequences of poses with a single pre-trained model. We evaluate the proposed method on HumanEva-I and Human3.6M with various evaluation methods, and outperform the state-of-the-art methods on most of the metrics. of observations into a single latent embedding and learn to generate future motions by combining samples from this single latent space with past observations. Our method HiT-DVAE (right) aims to learn a sequential generative model on joint distribution of data and latent variables. Diverse motion generation is conducted by alternate use of generation on z t and x t .Human motion generation has several challenges: i) diversity: different from deterministic prediction, the generation methods should not just learn average patterns, but need to faithfully reflect the intrinsic intra-class variability. ii) dynamics: the generation process must inherently model the dynamics of the 3D human pose data, so that they can be transferred to the generated data and avoid collapsing to a stable motion. iii) smoothness: the generated data should be smooth.Some of these challenges have been partially addressed in past studies, although up to our knowledge, there is no existing methodology designed to face all three challenges. For instance, MT-VAE [58] combined the RNN-based motion prediction model with conditional variational autoencoders (VAE), where the difference between the observed and future poses was encoded into the latent variable, which was then concatenated with the RNN's hidden state to account for the dynamics. DLow [60] proposed to generate a diverse set of motion sequences, by training fifty different encoders (and a single decoder), and obtaining fifty different instances of the latent variable, and thus fifty different generated motions. GSPS [39] inherited the diversity loss from DLow, but utilized a more powerful motion prediction framework, based on graph convolutional network (GCN) rather than the RNN. Diversity was obtained by concatenating random noise to the observed sequences. Finally, ACTOR, a Transformer-VAE method, was proposed in [45] to perform sequential generation, as opposed to previous methods that were designed for a fixed output length. ACTOR[45]learned an action class informed token as input to the encoder and decoder, thus conditioning the generation with manually annotated labels. All the above methods encode the observed human poses into sequence-level embedding, meaning that the entire observed sequence is encoded into a single time-independent embedding. This motivates us to propose HiT-DVAE inspired by the recent literature on dynamical variational autoencoders (DVAE)[19]. Our method belongs to the very general family of variational autoencoders (VAE) and is therefore a probabilistic method inherently able to generate stochastic, and hence diverse output. More precisely, using DVAE, the sequence of observations is encoded into a sequence of latent variables instead of a single latent variable, thus offering larger representation power to learn and exploit the motion dynamics, seeFigure 1. Besides, we model the generative process with auto-regressive dependencies, meaning that the generation of each frame depends on the previous ones and can be done sequentially. We can then train this model to generate sequences of arbitrary length. Finally, we implement these auto-regressive dependencies with a transformer-like (attention-based) encoderdecoder architecture, thus learning to automatically select which are the best frames to inform the generation of the next human 3D pose. Overall, HiT-DVAE implements auto-regressive probabilistic dependencies with transformer-like attention mechanisms enabling the learning of pose sequence dynamics as well as stochastic motion generation. Hence, the proposed solution deals with all the three challenges mentioned above.Considering the evaluation of generated data, recent works either evaluate the generations directly on the joint location of poses[39,60], or using the feature extracted from a pretrained feature extractor[21,45]. Both protocols have clear shortcomings: the former just evaluates the best generated
|
10.48550/arxiv.2204.01565
|
[
"https://arxiv.org/pdf/2204.01565v1.pdf"
] | 247,939,428 |
2204.01565
|
8137ee3367bb3b8ffa431501d27562e3967a90f4
|
HiT-DVAE: Human Motion Generation via Hierarchical Transformer Dynamical VAE
Xiaoyu Bie
Inria
LJK
Univ. Grenoble Alpes
CNRS
Grenoble INP
France
Wen Guo
Inria
LJK
Univ. Grenoble Alpes
CNRS
Grenoble INP
France
Institut de Robòtica i Informàtica Industrial
CSIC-UPC
Spain
Simon Leglaive
CentraleSupélec
IETR
France
Lauren Girin
Univ. Grenoble Alpes
Grenoble-INP
GIPSA-lab
France
Francesc Moreno-Noguer
Institut de Robòtica i Informàtica Industrial
CSIC-UPC
Spain
Xavier Alameda-Pineda
Inria
LJK
Univ. Grenoble Alpes
CNRS
Grenoble INP
France
HiT-DVAE: Human Motion Generation via Hierarchical Transformer Dynamical VAE
human motion generationdynamical variational autoencodersprobabilistic trans- formers
Studies on the automatic processing of 3D human pose data have flourished in the recent past. In this paper, we are interested in the generation of plausible and diverse future human poses following an observed 3D pose sequence. Current methods address this problem by injecting random variables from a single latent space into a deterministic motion prediction framework, which precludes the inherent multi-modality in human motion generation. In addition, previous works rarely explore the use of attention to select which frames are to be used to inform the generation process up to our knowledge. To overcome these limitations, we propose Hierarchical Transformer Dynamical Variational Autoencoder, HiT-DVAE, which implements auto-regressive generation with transformer-like attention mechanisms. HiT-DVAE simultaneously learns the evolution of data and latent space distribution with time correlated probabilistic dependencies, thus enabling the generative model to learn a more complex and time-varying latent space as well as diverse and realistic human motions. Furthermore, the auto-regressive generation brings more flexibility on observation and prediction, i.e. one can have any length of observation and predict arbitrary large sequences of poses with a single pre-trained model. We evaluate the proposed method on HumanEva-I and Human3.6M with various evaluation methods, and outperform the state-of-the-art methods on most of the metrics. of observations into a single latent embedding and learn to generate future motions by combining samples from this single latent space with past observations. Our method HiT-DVAE (right) aims to learn a sequential generative model on joint distribution of data and latent variables. Diverse motion generation is conducted by alternate use of generation on z t and x t .Human motion generation has several challenges: i) diversity: different from deterministic prediction, the generation methods should not just learn average patterns, but need to faithfully reflect the intrinsic intra-class variability. ii) dynamics: the generation process must inherently model the dynamics of the 3D human pose data, so that they can be transferred to the generated data and avoid collapsing to a stable motion. iii) smoothness: the generated data should be smooth.Some of these challenges have been partially addressed in past studies, although up to our knowledge, there is no existing methodology designed to face all three challenges. For instance, MT-VAE [58] combined the RNN-based motion prediction model with conditional variational autoencoders (VAE), where the difference between the observed and future poses was encoded into the latent variable, which was then concatenated with the RNN's hidden state to account for the dynamics. DLow [60] proposed to generate a diverse set of motion sequences, by training fifty different encoders (and a single decoder), and obtaining fifty different instances of the latent variable, and thus fifty different generated motions. GSPS [39] inherited the diversity loss from DLow, but utilized a more powerful motion prediction framework, based on graph convolutional network (GCN) rather than the RNN. Diversity was obtained by concatenating random noise to the observed sequences. Finally, ACTOR, a Transformer-VAE method, was proposed in [45] to perform sequential generation, as opposed to previous methods that were designed for a fixed output length. ACTOR[45]learned an action class informed token as input to the encoder and decoder, thus conditioning the generation with manually annotated labels. All the above methods encode the observed human poses into sequence-level embedding, meaning that the entire observed sequence is encoded into a single time-independent embedding. This motivates us to propose HiT-DVAE inspired by the recent literature on dynamical variational autoencoders (DVAE)[19]. Our method belongs to the very general family of variational autoencoders (VAE) and is therefore a probabilistic method inherently able to generate stochastic, and hence diverse output. More precisely, using DVAE, the sequence of observations is encoded into a sequence of latent variables instead of a single latent variable, thus offering larger representation power to learn and exploit the motion dynamics, seeFigure 1. Besides, we model the generative process with auto-regressive dependencies, meaning that the generation of each frame depends on the previous ones and can be done sequentially. We can then train this model to generate sequences of arbitrary length. Finally, we implement these auto-regressive dependencies with a transformer-like (attention-based) encoderdecoder architecture, thus learning to automatically select which are the best frames to inform the generation of the next human 3D pose. Overall, HiT-DVAE implements auto-regressive probabilistic dependencies with transformer-like attention mechanisms enabling the learning of pose sequence dynamics as well as stochastic motion generation. Hence, the proposed solution deals with all the three challenges mentioned above.Considering the evaluation of generated data, recent works either evaluate the generations directly on the joint location of poses[39,60], or using the feature extracted from a pretrained feature extractor[21,45]. Both protocols have clear shortcomings: the former just evaluates the best generated
Introduction
Modeling 3D human pose and motion is a fundamental problem towards understanding human behaviour, with a wide range of applications in medical prognosis, 3D content production, autonomous driving and human-robot interaction. One of the most studied computer vision tasks is pose estimation either from single images [23,42,43,53], monocular videos [44,46,46], or from multi-camera settings [5,49,54]. This development is certainly due to the availability of large-scale datasets such as Human3.6M [28]. Beyond extracting 3D human pose information from various types of visual data, a number of works have been proposed around the development of machine learning methods allowing to forecast future motion [1,2,9,10,12,14,18,20,22,25,26,29,30,32,33,37,38,40,41,51,56] and more recently on generating plausible future sequences of realistic human 3D pose data [3,4,6,11,21,31,36,39,40,45,57,58,60]. sample and the diversity of all generations, ignoring the performance of the generations except the best one; while the latter depends on the quality of the feature extractor. To thoroughly evaluate the generation quality, we use both evaluation methods, and broaden the first one to take performance stability into consideration.
We thoroughly evaluate HiT-DVAE on HumanEva-I and Human3.6M datasets, using both explicit and implicit metrics to measure the quality of generated data. Experimental results show that our method achieves state-of-the-art on most of the metrics for both datasets, proving that the generation of HiT-DVAE has high quality (smaller errors, better features, correct action) with better performance stability.
2 Related Work
Modeling Future Human motion
The forecasting of human motion has been addressed under two paradigms so far: deterministic motion prediction and stochastic human motion generation. The former aims at using deterministic approaches to regress a single future motion from the past observation which is the most likely to the ground truth; while the latter focuses at generating various possibilities of the future to model the multi-modal nature of human motions.
Deterministic human motion prediction
Due to the inherent sequential structure of human motion, 3D human motion prediction has been mostly addressed with recurrent models [18,29,41]. However, although RNNs can achieve great success in motion prediction, they represent the entire past motion history with a fixed-size hidden state and tend to converge to a static pose. Some works alleviate this problem by using RNN variants [12,37], sliding windows [9,10], convolutional models [25,26,32] or adversarial training [20]. Since human body pose data are structured, directly encoding the whole body into a compact latent embedding neglects the spatial connectivity of human joints. To this end, recent work tend to leverage the forward graph convolutional network (GCN) [30,56] with a predefined or learnable adjacency matrix [1,2,14,33,38,40,51]. While deterministic methods have achieved promising results on accurate predictions, they exhibit strong limitations when it comes to model the diversity of plausible human motion forecasts. Stochastic methods are promising tools to overcome these limitations.
Stochastic human motion generation
To generate multiple future outcomes given a sequence of past observations, two types of approaches have been studied in the recent past: (i) the enhancement of deterministic methods with stochastic variations, e.g., incorporating noise, and (ii) leveraging conditional variational architectures that learn a probability distribution. In the first category, early works include combining random noise with hidden states either by concatenation [31,36] or addition [6]. More recently, Mao et al. [39] further investigated this paradigm with a GCN based motion prediction model [40], and showed promising results with dedicated designed losses. In the second category, past observations are encoded to learn a posterior latent space, then a random variable will be sampled and then combined with observations to predict the future [3,4,11,57,58]. Rencently, DLow [60] proposed to explicitly generate a large number of samples during training, then to use a energy function to promote the diverse generation. ACTOR [45] first introduce a Transformer-VAE to obtain long term attention. Rather than modeling the whole observation into a single embedding, Action2Motion [21] and HuMor [47] exploit a autoregressive generative model that the current generation will depend on the past prediction. However, they do not model the entire sequence, but only on the last frame, thus resulting in non-smooth motion generation.
Deep generative modeling
The family of stochastic human motion generation methods are mostly based in the general paradigm of variational inference, and of variational autoencoders. VAEs model the joint distribution of an observation x and a latent variable z. In stochastic human motion generation, the observations x often corresponds to a sequence of poses, rather than a single pose. However, up to our knowledge, most of the previous methods use a single latent variable z to encode the entire observed sequence. Alternatively, one could consider a sequence of latent variables and of observations, and use a VAE to model the relationship between between x t and z t without any time dependencies. However, the dynamics and any temporal relationships cannot be modeled in this case, which is obviously not desirable. Dynamical variational autoencoders (DVAEs) [19] offer the possibility to model data sequences within the general paradigm of variational inference. DVAE is a general class of models and different models are obtained when considering various dependencies between the variables, e.g., variational recurrent neural networks [13] or stochastic recurrent neural networks [17]. However, current DVAE models have a major limitation: the probabilistic dependencies between variables are always implemented with recurrent neural networks (or variants), thus avoiding the possibility to select which past frames are used to inform the generation of the current frame. In addition and up to our knowledge, the use of DVAEs for human motion forecasting has not been investigated so far. This motivates us to explore the use of attention mechanisms within the DVAE paradigm with applications to human motion forecasting, as explained in the following.
Method
We address the problem of 3D human motion generation that we formalise as follows. Given a sequence of O observed 3D poses of a person x 1:O = [x 1 , . . . , x O ], our aim is to generate a sequence of G 3D poses x O+1:O+G = [x O+1 , . . . x O+G ], that follow the observations x 1:O . Each pose vector x t ∈ R J×3 encodes the location of the J joints of a person at time t in Cartesian coordinates. Different from deterministic human motion prediction, we intend to generate multiple plausible future motion sequences with arbitrary length. To this end, we propose a new method named Hierarchical Transformer Dynamical Variational AutoEncoder or HiT-DVAE. Our method is based on the recently reviewed family of dynamical variational autoencoders [19], which formulates the generative process of time series in an autoregressive and time-dependent perspective. Up to our knowledge, this general methodology has never been combined with attention-based mechanisms. On the one hand, existing variants of DVAEs are always implemented with recurrent networks [19] (or standard variants such as LSTM and GRU). On the other hand, even if self-attention has been proven useful when combined with a Conditional VAE [45], the architectures proposed so far encode the entire sequence into a single latent variable z, therefore potentially limiting the representation capabilities of temporal dynamics. We propose HiT-DVAE to get the best of both worlds, enabling stochastic motion generation together with dynamic sequence modeling via transformer-like attention mechanisms.
HiT-DVAE
The proposed method is based on the very general DVAE methodology (see [19] for an exhaustive presentation on the topic). The basic principle of DVAEs is that for every observation x t there is a corresponding latent variable z t , as opposed to VAEs which would encode the entire observed sequence x 1:O into a single latent variable z. The sequence of observations and corresponding latent variables will be denoted by x 1:T = [x t ] T t=1 and z 1:T = [z t ] T t=1 , respectively. For the time being, we will assume that T = O + G, as if all 3D poses were observed even if this is not our setting. We will discuss the impact of having hybrid half-observed half-generated sequences later on.
In addition to the time-dependent latent variable z 1:T , and inspired by [34,45], we add a timeindependent latent variable w. Very differently from [45], w will be learned in an unsupervised manner within the DVAE methodology, see [34], thus without requiring action class labels. Formally, the proposed generative model writes: meaning that the generative processes of both the observed and latent variables are auto-regressive, with cross-dependencies. We set θ = θ w ∪θ z ∪θ x . In order to learn this generative model, we introduce
p θ (x 1:T , z 1:T , w) = T t=1 p θ (x t , z t , w|x 1:t−1 , z 1:t−1 ) (1) = p θw (w) T t=1 p θx (x t |x 1:t−1 , z t , w)p θz (z t |x t−1 , z 1:t−1 , w),(2)an inference model (φ = φ w ∪ φ z ): q φ (z 1:T , w|x 1:T ) = q φ w (w|x 1:T ) T t=1 q φ z (z t |x 1:T , w).(3)
The training objective is to maximize the evidence lower bound (ELBO):
L(θ, φ; x 1:T ) = E q φ (z 1:T ,w|x 1:T ) [ln p θ (x 1:T , z 1:T , w) − ln q φ (z 1:T , w|x 1:T )] .(4)
Although the above equations define the probabilistic dependencies between the different random variables, there are plenty of ways of implementing these dependencies. In this paper, we propose to use a hierarchical transformer-based architecture. In our ablation study, we discuss other -perhaps more conventional-ways of implementing such dependencies, that exhibit lower performance and demonstrate the interest of having both attention and a hierarchical structure, and thus justify the proposed HiT-DVAE. Both the encoder and the decoder of the proposed method exploit a spatial graph convolutional network (SGCN) to extract pose features from the raw poses x t . We will denote this pose feature extraction operation as f , and we will let the encoder and decoder fine-tune their pose extractor leading to f E and f d , see below for more details. Figure 2 shows an overview of our proposed model. Specifically, we employ the transformer architecture [55] jointly with GCN-based feature extractors to formulate the inference and generation on the sequential human motion data.
Generative Model (HiT-DVAE Decoder)
The generation of both x 1:T and z 1:T is performed via the attention mechanisms proposed in the original transformer architecture [55]. Specifically, the generative model will use multi-head crossattention, after the GCN-based feature extractor f d . The generative processes of x and z differ on what variables are used as queries, keys and values in the attention mechanism. The output of the two decoders will be the parameters of the respective probability distributions, defined in (2). Both distributions are considered to be Gaussian. While we learn both the mean and covariance matrix of the generative distribution of the latent variable z t , the covariance matrix of the observations x t is considered to be the identity as in previous works [3,21,45,47,60]. In particular we have
p θx (x t |x 1:t−1 , z t , w) = N (x t ; µ θx,t , I) and p θz (z t |x t−1 , z 1:t−1 , w) = N (z t ; µ θz,t , Σ θz,t ) with µ θx,t = MaskedMultiHead (Q θx,t , K θx , V θx ) ,(5)Q θx,t = z t w , K θx = V θx = [f d (x 1 ), . . . , f d (x T )],(6)µ θz,t Σ θz,t = MaskedMultiHead (Q θz,t , K θz , V θz ) ,(7)Q θz,t = f d (x t−1 ) w , K θz = V θz = [z 1 , . . . , z T ],(8)
where a mask is used to prevent z t and x t from being generated from future latent and observed variables [55]. Finally, we have p θw (w) = N (w; 0, I), where 0 and I are the zero vector and the identity matrix of appropriate dimensions.
Inference Model (HiT-DVAE Encoder)
The inference of the latent variables w and z 1:T from x 1:T is performed via a temporal GCN and the multi-head self-attention mechanism of the transformer encoder (TE). The extracted pose features are fed into the temporal GCN with T nodes, where each node indicates a time frame, and then into a fully connected (FC) layer to output the mean and variance of w, namely µ φ w and Σ φ w . Samples are drawn from the corresponding posterior q φ w (w|x 1:
T ) = N (w; µ φ w , Σ φ w )
, concatenated to all pose features and then fed into the transformer encoder such that q φ z (z t |x 1:
T , w) = N (z t ; µ φ z ,t , Σ φ z ,t ) with µ φ z ,t Σ φ z ,t = MultiHead Q φ z ,t , K φ z , V φ z ,(9)Q φ z ,t = f E (x t ) w, , K φ z = V φ z = f E (x 1 ), . . . , f E (x T ) w . . . , w .(10)
Feature Extractor on Human Poses
As mentioned above, we extract pose features via a spatial GCN f . This strategy was suggested in [13].
In HiT-DVAE, the spatial GCN is composed of J nodes in the graph, each of them representing a joint in the pose skeleton. While the architecture of the spatial GCN is the same as that of the generative and inference models, these two spatial GCNs are trained separately (i.e., the weights are not shared).
Training losses
3.2.1 ELBO.
Optimising the evidence lower bound (4) in the case of the proposed HiT-DVAE, boils down to (i) minimising the L 2 loss on the reconstructed poses while (ii) minimising the KL divergence between the posterior and prior distributions over the latent variables. Because directly optimising the ELBO would not encourage diversity in our generative model, we inspire from [39], and we explicitly generate K motion sequences {x k 1:T } K k=1 and compute the ground-truth reconstruction loss and multi-modal reconstruction loss:
L r = min k ||x k 1:T − x 1:T || 2 , L mm = 1 M M m=1 min k ||x k 1:T − x m 1:T || 2 ,(11)
where x 1:T is the ground-truth, and x m 1:T are the pseudo-ground truth sequences, which are selected from the training set following the same protocol as in [39]. In addition to the reconstruction losses, we need to minimise the KL divergence:
L kl-z = 1 T T t=1 D KL (q φ z (z t |x 1:T , w)||p θz (z t |x t−1 , z 1:t−1 , w))(12)L kl-w = D KL (q φ w (w|x 1:T )||p θw (w)).(13)
The final evidence lower bound (ELBO) writes:
L elbo = λ r L r + λ mm L mm + λ kl-z L kl-z + λ kl-w L kl-w .(14)
3.2.2 Diversity loss.
As suggested by [39,60], we add diversity promoting losses on upper body and lower body:
L div = p∈{l,u} λ div-p 2 K(K − 1) K k=1 K k =k+1 exp − ||x k,p 1:T −x k ,p 1:T || 1 α p ,(15)
where l (u) indicates the lower (upper) body part and α p is a normalizing factor.
Realistic pose loss.
Follow [39], we employ three extra losses to penalize for unrealistic poses, L l for shifting limb length, L a for aberrant joint angles and L nf for negative prior pose probability from a pre-trained pose prior model based on normalizing flow [16,48]. Altogether, our final training loss writes:
L = L elbo + L div + λ l L l + λ a L a + λ nf L nf .(16)
HiT-DVAE for diverse human motion generation
The losses above allow to train the proposed HiT-DVAE to reconstruct full sequences. In practice, as stated in the problem formulation, HiT-DVAE must input O observed frames x 1:O and generate the following G frames x O+1:O+G . However, if we use the model trained with ground-truth input over the entire sequence x 1:T (T = O + G) we encounter severe difficulties since after O frames the input distribution changes from the ground-truth to the generated one, and the generation fails. One alternative could be training with ground-truth input until O x 1:O and then complete with generated datax O+1:O+G . Unfortunately, at the beginning of the training the generated data is pattern-less, and the training diverges. In order to overcome this issue, we use scheduled sampling [7]: we start training only with ground-truth data, and we progressively add more generated frames (randomly) with a proportion starting at 0% and up to 100%. Once our model is trained, we can use it to generate various future motion sequences in arbitrary length. Given O observations in our setting, we can get the posterior of z 1:O and w from the inference model. Then, we can generate the next G framesx O+1:O+G by recursively applying the generative function on p θx (x t |x 1:t−1 , z t , w) and p θz (z t |x t−1 , z 1:t−1 , w). The diversity comes simply from the different samples of z O+1:O+G and w. datasets. For each case we show 'Start', which is the last observed frame and the subsequent frames correspond the the last frames of 10 different generated sequences. GT is the ground truth last frame. Note that gsps and DLow highly diverge from that last GT frame, while our approach, while generating different alternatives, they all keep the essence of the particular action. Table 1: Results on HumanEva-I. "Real data" means real motion in testing set, shown the theoretical upper bounds of accuracy on generation methods. The suffix 'b' or 'm' represents the best/medium metrics. ↑ (↓) means higher (lower) is better. † indicates results taken from DLow, indicates results obtained by using the official code repository.
Experiments
Evaluation Protocols
Acc (%) ↑ FID ↓ APD (m) ↑ ADEb (m) ↓ FDEb (m) ↓ MMADEb (m) ↓ MMFDEb (m) ↓ ADEm (m) ↓ FDEm (m) ↓ MMADEm (m) ↓ MMFDEm (m) ↓ Real data 88.3 - - - - - - - - - - ERD
Explicit evaluation metrics
Following [39,60], we evaluate error and diversity of our results with the following metrics, calculating directly on the joint locations of poses: i) Average Pairwise Distance (APD): average L2 distance of all pairs among the generated sequences:
1 K(K−1) K i=1 K j =i x i O+1:O+G −x j O+1:O+G 2 .
APD measures the capacity of the model to generate diverse samples without considering their quality. ii) Average Displacement Error (ADE): L2 distance between the ground truth and the 'best' generated sample among all, taking the average over all frames of the sequence: 1 G min k x k O+1:O+G − x O+1:O+G . Here 'best' means the one which is closest to the ground truth. ADE evaluates the upper bound of the generation quality of the model among all the generation results. iii) Final Displacement Error (FDE): Similar to ADE, FDE evaluates the distance between the ground truth and the best sample, but just on the final frame instead of the whole sequence: min k x k O+G − x O+G . iv) Multi-Modal ADE (MMADE) and Multi-Modal FDE (MMFDE): multi-modal version of ADE and FDE.
The MPJPE based metrics are widely used for evaluating the quality of generated motion, considering the diversity of the generated data and accurancy of the best generated one. While the shortcoming of them is obvious. On one hand, when we use the generator to generate actions, we could not always have the groundtruth to judge which generation is the best one; on the other hand, as described in Sec 2.1, generating one best example is what deterministic motion prediction aims at, while stochastic methods should generate various good samples. For example, a batch of generated motions where only one sample is exact while the others are all super crazy will result in a large APD and tiny errors which seems perfect numerically, but this is certainly not a good generation we want. So just considering the above metrics is not comprehensively and proper.
Thus, we take two solutions: 1) instead of just evaluating ii-iv) on the best generated sample, we also evaluate these criteria on the 'medium' example, which holds the medium distance to the groundtruth among all the generated samples. 2) beside of this explicit measurement based on poses, we also consider implicit measurements based on a pretrained action classifier, as described below. Table 2: Results on Human3.6M. "Real data" means real motion in testing set, shown the theoretical upper bounds of accuracy on generation methods. The suffix 'b' or 'm' represents the best/medium metrics. ↑ (↓) means higher (lower) is better. † indicates results taken from DLow, indicates results obtained by using the official code repository.
Acc (%) ↑ FID ↓ APD (m) ↑ ADEb (m) ↓ FDEb (m) ↓ MMADEb (m) ↓ MMFDEb (m) ↓ ADEm (m) ↓ FDEm (m) ↓ MMADEm (m) ↓ MMFDEm (m) ↓ Real data 85.5 - - - - - - - - - - ERD † [18] - - 0 0.
Implicit evaluation metrics
Following [21,45], we use a GRU-based action classifier pre-trained on real data to evaluate the quality of generated data by: i) calculating Recognition Accuracy (Acc) of the classifier on generated data to evaluate if the generated data could be recognized as the correct action class. ii) extracting features of the generated data and real data respectively by the action classifier, and calculate the Frechet Inception Distance (FID) of these two distributions to evaluate the overall quality of generated data. For the two datasets, we trained a classifier for each of them on their training splits.
Implementation details
We set the dimension of z t to 16 and w to 32, and employ the same GCN architecture described in [40]. We use 1 GCN block with hidden size of 8 for spatial GCN and 4 GCN blocks with hidden size of 64 for temporal GCN. For the Transformer encoder and the decoder for generating z t , we set the input feature dimension to 64, with 4 multi-head, followed by a FC layer with dimension of 256, whereas for the Transformer decoder to generate x t , we set those parameters to 256, 4, 1024 respectively. We generate K = 50 samples for each observation. We train the model for 500 epochs with 1000 training samples per epoch, using Adam optimizer, and set learning rate to 0.001, batch size to 64 for HumanEva and 32 for H3.6M. We applied a linear KL annealing [52] for the first 20 epochs to warm-up the latent space, then we take 80 epochs to increase the probability of schedule sampling from 0 to 1. For HumanEva, we train with a sequence length of 75, where the inference of w only takes 15 frames with a random start point. The weights of different loss terms (λ r , λ mm , λ d,l , λ d,u , λ l , λ a , λ nf , λ z , λ w ) and the normalizing factors (α l , α u ) are set to (10, 5, 0.1, 0.2, 100, 1, 0.001, 0.5, 0.1) and (15,50). For H3.6M, we train with a sequence length of 125, where w is inferred from 25 frames. The weights of different loss terms and the normalizing factors are set to (20, 10, 0.1, 0.2, 100, 1, 0.01, 0.5, 0.1) and (100, 300) respectively.
Quantitative results
We evaluate the generated motions on HumanEva-I and Human3.6m dataset by the explicit and implicit metrics described in Sec 4.1, and found that our methods outperforms the state of the art methods on most of the evaluation metrics.
HumanEva-I
As shown in Tab 1, our method achieves comparable results with state of the art on explicit evaluation of diversity (APD) and errors of 'best' sample (ADE b , FDE b , MMADE b , MMFDE b ). As discussed in Table 3: Ablation Study on HumanEva-I and Human3.6M. 'w/o SS' means without scheduled sampling, 'w/o Att.' means using an LSTM instead of transformer, 'w/o w' means without using the time-independent latent variable w. 'w/o Att. & w ' means without attention and no use of w. Sec 4.1, we know that just considering these errors and diversity is not reliable, because these erros just evaluate the best sample among all generations and APD just evaluates diversity without considering the generation quality. And we should note that APD is not always better for larger values, because although we want the generated data to have diversity, crazy large diversity represents that some of the generated samples might totally fail and the quality of generation is not guarantied. So in order to comprehensively measure the performance of generated data, we test the error of 'medium' samples, action-recognition based accuracy ACC and feature-based FID scores on our data, and also on other code-released state of the art methods [39,60]. We find that our method achieves significantly better results than other state of the art methods on the implicit metrics ACC and FID, which means that our generated data is better in feature distribution, and could generate more reasonable results that could be recognized as the right action. And our method is also clearly better on errors of medium sample (ADE m , FDE m , MMADE m , MMFDE m ), which indicates higher stability of our overall generation quality.
Architecture ACC (%) ↑ FID ↓ APD (m) ↑ ADEb (m) ↓ FDEb (m) ↓ MMADEb (m) ↓ MMFDEb (m) ↓ ADEm (m) ↓ FDEm (m) ↓ MMADEm (m) ↓ MMFDEm (m) ↓
Human3.6M
Similar conclusion could be drawn for Human3.6M dataset, as shown in Tab 2. When training the action classifier for Human3.6M dataset, instead of training on all the 15 action labels, we group the 15 actions into 5 groups. This is because that some actions in Human3.6M dataset are not with much difference, so it is is not suitable for training the action classier. For example, we could not see the difference between 'eating' and 'smoking' with just the skeletons of the person. With this grouping, the average classification accuracy on real data of our classifier increases from 48.1% to 85.5%. Note that even on the 15-action classifier with low real-data-accuracy, our method still has higher Acc and lower FID comparing with other state of the art methods. While we report the 5-group classifier here because we believe that a better classifier is more reliable for calculating accuracy and extracting features for FID. More details about the action classifier could be found in supplementary material.
Ablation study
Tab 3 shows ablation studies on our method with different architecture design. We bold the best results and underlined the second best ones. We could find that, without schedule sampling, our method tends to generate more diverse results, but with worse quality either on explicit metrics and implicit metrics. Looking in the results of Human3.6M, the use of attention mechanism brings higher generation quality on explicit measurement. The global time-independent variable w bring more diversity on both two datasets (see the results w/o w). When we consider a vanilla DVAE model (w/o Att. & w), without the hierarchical transformer (HIT) architecture, it is very likely to collapse to a static state on the sequential latent space, which leads to moderate generation quality, and much worse diversity. The final setting of HiT-DVAE we used performs good on almost all the metrics and is a balance of different evaluations.
Qualitative results
To qualitatively evaluate our generated results, we visualize various generating samples of our methods in Fig 3 comparing with other state of the art methods. we could see that other methods either generates very similar samples for all the generations, either result in some weird actions, while our method performs well on all the generations, with diverse but always reasonable results. More visualisations could be found in supplementary material.
Conclusions
In this paper we have investigated the use of attention combined with temporal probabilistic models for human motion generation. In particular, we proposed HiT-DVAE, a variational method modeling temporal dependencies between the observations and the latent variables, and exploiting attention to select which observations will be used to inform the generation of the current frame. Up to our knowledge, this is the first time that models with temporal latent variables and the use of attention are proposed to handle the human motion generation task. We exhaustively evaluated our method on two widely used datasets, HumanEva and Human3.6M, and reported state-of-the-art results.
A Probabilistic Dependencies via Masked MHA
The temporal dependencies are implemented via the mask of the attention modules of the transformer decoder and encoder. The attention matrix in a Transformer layer, and is computed as follows:
Att(Q, K, V ) = Softmax M • QK T √ d k V ,(17)
where Q, K, V represent the query, key and value, and d k represents the input feature dimension of query and key. M is the attention mask and • denotes the element-wise multiplication. Obviously, an upper triangular mask without diagonal can prevent the model to see the future input. In this case, we can generate the entire sequence of x 1:T or z 1:T simultaneously. In practice, given an observed sequence with length T , we only generate x 2:T and z 2:T to bypass the estimation of initial state x 0 and z 0 .
(a) p θx (x t |x 1:T , z t , w) (b) p θx (x t |x 1:t−1 , z t , w) (c) p θx (x t |x t−1 , z t , w) Figure 4: Probabilistic dependencies on generation of x t with different mask design, the yellow block indicates the elements that will be masked in attention computation. Fig. 4 shows three cases of probabilistic dependencies when using different mask in the Transformer layer. Note that Fig. 4 (a) is a non-causal situation, thus we can not generate future motion via this dependencies. The mask in Fig. 4 (c) will make the attention computed only on one element, thus the attention mask is meaningless in this case. We choose the mask shown in Fig. 4 (b) in our proposed HiT-DVAE.
B Pseudo-code for HiT-DVAE
Here, we provide the pseudo-code for HiT-DVAE in training and generation:
Algorithm 1 HiT-DVAE in training
Inputs: Observation on human sequence x 1:T for epo in epochs do Inference: Compute posterior of w and sample w ∼ q φ w (w|x 1:T ) = N (w; µ φ w , Σ φ w ) Compute posterior z 1:T and sample z t ∼ q φ z (z t |x 1:T , w) = N (z t ; µ φ z ,t , Σ φ z ,t ) for t = 1, ..., T Generation:
Compute the distribution of x 2:T via p θx (x t |x 1:t−1 , z t , w) = N (x t ; µ θx,t , I) for t = 2, ..., T Compute the prior of z 2:T via p θz (z t |x t−1 , z 1:t−1 , w) = N (z t ; µ θz,t , Σ θz,t ) for t = 2, ..., T Compute loss and optimize via Adam end for C Action-classifier As explained in Sec. 4.1 and Sec. 4.3 of the main paper, we trained a RNN-based classifier to calculate the implicit evaluation metrics ACC and FID following [21,45]. The classifier we used is build upon 2 simple GRU layers with hidden size of 128. When training on Human3.6M dataset, we found that some action classes do not differ much from each other, which makes it difficult to train a good classifier. As our goal is to have a classifier which offers good feature, we believe the classifier with low accuracy on real data is not reliable enough, so we group the 5 similar actions, and trained the classifier on these 5 groups instead of the 15 original classes. The groups of actions are detailed in Tab 4. Note that even on the classifier trained on the 15 original classes, our method still performs better than others, as shown in Tab. 5.
Figure 1 :
1Conditional variational autoencoder (CVAE) based methods (left) aim to encode a sequence
Figure 2 :
2Overview of HiT-DVAE. The Encoder (left) inputs the observed sequence x 1:T to estimate the posterior distribution of the time-dependent z 1:T and time-independent w latent variables. Then the Decoder (right) reconstructs the data and the prior of z.
Figure 3 :
3Qualitative visualization on four different actions of the HumanEva and Human3.6M
11) performing 15 annotated actions recorded at 50 Hz. Human pose is represented in 32 skeletons, while we follow[39] to use 17 of the skeletons in our training and all testing implementations, and we use S1,5,6,7,8 as training set and the other two subjects as test set. HumanEva contains 5 actions (Box, Gesture, Jog, ThrowCatch, Walking) performed by 3 actors, recorded at 60 Hz. Each pose is represented by 15 skeletons. For both datasets, we remove the global translation and set the root joint as zero.† [18]
-
-
0
0.382 0.461
0.521
0.595
-
-
-
-
acLSTM † [35]
-
-
0
0.429 0.541
0.530
0.608
-
-
-
-
Pose-Knows † [57]
-
-
2.308 0.269 0.296
0.384
0.375
-
-
-
-
MT-VAE † [58]
-
-
0.021 0.345 0.403
0.518
0.577
-
-
-
-
HP-GAN † [6]
-
-
1.139 0.772 0.749
0.776
0.769
-
-
-
-
BoM † [8]
-
-
2.846 0.271 0.279
0.373
0.351
-
-
-
-
GMVAE † [15]
-
-
2.443 0.305 0.345
0.408
0.410
-
-
-
-
DeLiGAN † [24]
-
-
2.177 0.306 0.322
0.385
0.371
-
-
-
-
DSF † [59]
-
-
4.538 0.273 0.290
0.364
0.340
-
-
-
-
DLow [60]
52.7 3.472 4.853 0.248 0.262
0.361
0.337
0.577 0.717
0.646
0.753
gsps [39]
51.6 1.604 5.825 0.233 0.244
0.343
0.311
0.686 0.794
0.735
0.825
HiT-DVAE
72.6 0.089 4.721 0.282 0.261
0.335
0.290
0.579 0.665
0.610
0.683
4.1.1 Datasets
Following [39, 60], we train and evaluate our methods on Human3.6M [27] and HumanEva-I [50]
dataset: Human3.6M is the most commonly used dataset for motion tasks. It contains 7 actors
(S1,5,6,7,8,9,
Table 4 :
4Groups of actions of Human3.6m dataset, for training the action classifier.group number
original classes
0
Directions, Discussion, Greeting,
Photo, Posing, Purchases, WalkDog, Waiting
1
Eating, Phoning, Sitting, Smoking
2
SittingDown
3
Walking
4
WalkTogether
Table 5 :
5Implicit evaluations by different classification models on Human3.6m dataset. Our method always performs better. Acc (%) ↑ FID ↓ Acc (%) ↑ FID ↓5 groups
15 classes
Real data
85.5
-
48.1
-
DLow [60]
65.9
1.412
22.7
1.566
gsps [39]
65.0
2.030
22.2
1.915
ours
70.0
1.708
28.1
1.466
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Auto-conditioned recurrent networks for extended complex human motion synthesis. Z Li, Y Zhou, S Xiao, C He, Z Huang, H Li, arXiv:1707.05363arXiv preprintLi, Z., Zhou, Y., Xiao, S., He, C., Huang, Z., Li, H.: Auto-conditioned recurrent networks for extended complex human motion synthesis. arXiv preprint arXiv:1707.05363 (2017)
X Lin, M R Amer, arXiv:1804.10652Human motion modeling using dvgans. arXiv preprintLin, X., Amer, M.R.: Human motion modeling using dvgans. arXiv preprint arXiv:1804.10652 (2018)
Towards natural and accurate future motion prediction of humans and animals. Z Liu, S Wu, S Jin, Q Liu, S Lu, R Zimmermann, L Cheng, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionLiu, Z., Wu, S., Jin, S., Liu, Q., Lu, S., Zimmermann, R., Cheng, L.: Towards natural and accurate future motion prediction of humans and animals. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 10004-10012 (2019)
History repeats itself: Human motion prediction via motion attention. W Mao, M Liu, M Salzmann, European Conference on Computer Vision. SpringerMao, W., Liu, M., Salzmann, M.: History repeats itself: Human motion prediction via motion attention. In: European Conference on Computer Vision. pp. 474-489. Springer (2020)
Generating smooth pose sequences for diverse human motion prediction. W Mao, M Liu, M Salzmann, Proceedings of the IEEE International Conference on Computer Vision (ICCV). the IEEE International Conference on Computer Vision (ICCV)Mao, W., Liu, M., Salzmann, M.: Generating smooth pose sequences for diverse human motion prediction. In: Proceedings of the IEEE International Conference on Computer Vision (ICCV). pp. 13309-13318 (2021)
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3d human pose estimation from a single image via distance matrix regression. F Moreno-Noguer, Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR. the IEEE International Conference on Computer Vision and Pattern Recognition (CVPRMoreno-Noguer, F.: 3d human pose estimation from a single image via distance matrix regres- sion. In: Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) (2017)
Coarse-to-fine volumetric prediction for single-image 3d human pose. G Pavlakos, X Zhou, K G Derpanis, K Daniilidis, Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR)Pavlakos, G., Zhou, X., Derpanis, K.G., Daniilidis, K.: Coarse-to-fine volumetric prediction for single-image 3d human pose. In: Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR). pp. 7025-7034 (2017)
3d human pose estimation in video with temporal convolutions and semi-supervised training. D Pavllo, C Feichtenhofer, D Grangier, M Auli, Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR)Pavllo, D., Feichtenhofer, C., Grangier, D., Auli, M.: 3d human pose estimation in video with temporal convolutions and semi-supervised training. In: Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR). pp. 7753-7762 (2019)
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Learning monocular 3d human pose estimation from multi-view images. H Rhodin, J Spörri, I Katircioglu, V Constantin, F Meyer, E Müller, M Salzmann, P Fua, Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition. the IEEE International Conference on Computer Vision and Pattern RecognitionCVPRRhodin, H., Spörri, J., Katircioglu, I., Constantin, V., Meyer, F., Müller, E., Salzmann, M., Fua, P.: Learning monocular 3d human pose estimation from multi-view images. In: Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR)
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Graph attention networks. P Velickovic, G Cucurull, A Casanova, A Romero, P Liò, Y Bengio, 6th International Conference on Learning Representations. Vancouver, BC, CanadaConference Track ProceedingsVelickovic, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., Bengio, Y.: Graph attention networks. In: 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 -May 3, 2018, Conference Track Proceedings (2018)
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Mt-vae: Learning motion transformations to generate multimodal human dynamics. X Yan, A Rastogi, R Villegas, K Sunkavalli, E Shechtman, S Hadap, E Yumer, H Lee, Proceedings of the European Conference on Computer Vision (ECCV). the European Conference on Computer Vision (ECCV)Yan, X., Rastogi, A., Villegas, R., Sunkavalli, K., Shechtman, E., Hadap, S., Yumer, E., Lee, H.: Mt-vae: Learning motion transformations to generate multimodal human dynamics. In: Proceed- ings of the European Conference on Computer Vision (ECCV). pp. 265-281 (2018)
Y Yuan, K Kitani, arXiv:1907.04967Diverse trajectory forecasting with determinantal point processes. arXiv preprintYuan, Y., Kitani, K.: Diverse trajectory forecasting with determinantal point processes. arXiv preprint arXiv:1907.04967 (2019)
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Algorithm 2 HiT-DVAE in generation Inputs: Observation on human sequence x 1:O Initialization: Compute posterior of w and z 1:O for t in range(O+1, O+G) do Generateẑ t via z t ∼ p θz (z t |x t−1 , z 1:t−1 , w) Generatex t via x t ∼ p θx. x t |x 1:O ,x O+1:t−1 ,ẑ t , w) end for Output: Generated human motion sequencex O+1:O+GAlgorithm 2 HiT-DVAE in generation Inputs: Observation on human sequence x 1:O Initialization: Compute posterior of w and z 1:O for t in range(O+1, O+G) do Generateẑ t via z t ∼ p θz (z t |x t−1 , z 1:t−1 , w) Generatex t via x t ∼ p θx (x t |x 1:O ,x O+1:t−1 ,ẑ t , w) end for Output: Generated human motion sequencex O+1:O+G
|
[] |
[
"Atomic-scale effects behind structural instabilities in Si lamellae during ion beam thinning",
"Atomic-scale effects behind structural instabilities in Si lamellae during ion beam thinning"
] |
[
"E Holmström [email protected] \nDepartment of Physics\nUniversity of Helsinki\nP.O. Box 64FIN 00014HelsinkiFinland\n\nHelsinki Institute of Physics\nP.O. Box 64FIN 00014HelsinkiFinland\n",
"J Kotakoski \nDepartment of Physics\nUniversity of Helsinki\nP.O. Box 64FIN 00014HelsinkiFinland\n\nDepartment of Physics\nUniversity of Vienna\nBoltzmanngasse 5, Carl-Zeiss-Straße 561090, 73447Wien, OberkochenAustria, Germany\n",
"L Lechner \nCentral Facility of Electron Microscopy\nUniversität Ulm\nAlbert Einstein Allee 1189069UlmGermany\n",
"U Kaiser \nCentral Facility of Electron Microscopy\nUniversität Ulm\nAlbert Einstein Allee 1189069UlmGermany\n",
"K Nordlund \nDepartment of Physics\nUniversity of Helsinki\nP.O. Box 64FIN 00014HelsinkiFinland\n\nHelsinki Institute of Physics\nP.O. Box 64FIN 00014HelsinkiFinland\n"
] |
[
"Department of Physics\nUniversity of Helsinki\nP.O. Box 64FIN 00014HelsinkiFinland",
"Helsinki Institute of Physics\nP.O. Box 64FIN 00014HelsinkiFinland",
"Department of Physics\nUniversity of Helsinki\nP.O. Box 64FIN 00014HelsinkiFinland",
"Department of Physics\nUniversity of Vienna\nBoltzmanngasse 5, Carl-Zeiss-Straße 561090, 73447Wien, OberkochenAustria, Germany",
"Central Facility of Electron Microscopy\nUniversität Ulm\nAlbert Einstein Allee 1189069UlmGermany",
"Central Facility of Electron Microscopy\nUniversität Ulm\nAlbert Einstein Allee 1189069UlmGermany",
"Department of Physics\nUniversity of Helsinki\nP.O. Box 64FIN 00014HelsinkiFinland",
"Helsinki Institute of Physics\nP.O. Box 64FIN 00014HelsinkiFinland"
] |
[] |
The rise of nanotechnology has created an ever-increasing need to probe structures on the atomic scale, to which transmission electron microscopy has largely been the answer. Currently, the only way to efficiently thin arbitrary bulk samples into thin lamellae in preparation for this technique is to use a focused ion beam (FIB).Unfortunately, the established FIB thinning method is limited to producing samples of thickness above ∼20 nm. Using atomistic simulations alongside experiments, we show that this is due to effects from finite ion beam sharpness at low milling energies combined with atomic-scale effects at high energies which lead to shrinkage of the lamella. Specifically, we show that attaining thickness below 26 nm using a milling energy of 30 keV is fundamentally prevented by atomistic effects at the top edge of the lamella. Our results also explain the success of a recently proposed alternative FIB thinning method, which is free of the limitations of the conventional approach due to the absence of these physical processes.a) Corresponding author, 1 Since atomic-scale building blocks determine functionality in nanotechnology, understanding physical processes at this scale is crucial for the discipline. One of the most important means for achieving this knowledge is aberration-corrected high-resolution transmission electron microscopy (AC-HRTEM), 1,2 a method which allows determining the positions of atomic columns of a structure with accuracy higher than interatomic distance in solids.3,4HRTEM requires thin, unbent samples which ideally should contain no amorphous surface layer. Moreover, in nanotechnology, it is mandatory to achieve sample preparation with a sub-µm precision in all dimensions. These standards cannot be reached using traditional mechanical preparation methods such as mechanical milling, and thus the focused ion beam (FIB) technique, where the sample is milled into a thin lamella through sputtering, needs to be used instead.The current development in AC-HRTEM is to lower the voltage from the usual 200-300 kV to well below 100 kV, 5-10 in order to minimize displacement damage in fragile samples. 11 So far, predominantly low-dimensional materials such as graphene 5,8,10 , hexagonal boron nitride 6 , and carbon nanotubes 7 have been studied at low electron beam voltages by HRTEM. In the future, however, there will be a demand to extend these low-voltage studies to conventional materials, which similarly suffer from knock-on damage at higher voltages. This trend of lowering voltage entails an increasingly stringent requirement for sample thickness, which must not surpass the extinction length of the electron beam (∼few nm).However, the conventional FIB thinning method is limited to thicknesses above ∼20 nm.The dominant effect behind this limitation is the shrinkage of the lamella in height during milling: Thinning a lamella down to a thickness of 20 nm may induce several µm of shrinkage in the vertical direction. 12 The underlying physical reasons for this observation are currently not known, because an atomic-scale description of the thinning process has hitherto not been presented. Additionally, the conventional method is hampered by specimen warping, an uneven thickness profile, and often heavy amorphization.Although irradiation effects in Si have been studied for decades, the work has concentrated on bulk Si samples. 13-16 Moreover, earlier computational work on FIB processing of similar nanosystems has been restricted to simulation setups which severely reduce the applicability of the obtained results to the problem addressed in the present study. On the one hand, some studies have employed less detailed models based on Monte Carlo simulation of sputtering and redeposition on a three-dimensional grid. 17,18 Unfortunately, such models 2 lack real atomic resolution and hence do not account for, e.g., amorphization, which alone renders them powerless for the topic at hand. On the other hand, previous molecular dynamics (MD) simulations motivated by FIB milling have employed geometries restricted to a single surface with no deformation of the sample allowed, 19-21 thus yielding comparatively limited insight into the process. To uncover the mechanisms responsible for the deleterious effects on the lamella under conventional FIB thinning, we performed a classical MD study of the FIB thinning process in conjunction with experiments. Using the findings from our experiments and simulations, we show that the empirically observed limit of 20 nm for the conventional thinning method is a result of geometric sputtering effects due to the finite sharpness of the ion beam, dominant at low energies (∼1 keV), and of atomic-scale effects at the very top part of the lamella, significant at high energies (∼30 keV). We further explain why these mechanisms are not present in a recently proposed double-tilt method, 22 which thus allows the thinning process to proceed beyond the conventional limit.The experimental setup and the conventional FIB thinning method are described in Appendix A. The effect of ion beam tails, discussed below, is demonstrated in Appendix B.Here, we focus first on studying experimentally the deleterious shrinkage effect on a lamella during FIB thinning. In principle, creating a wedge-shaped lamella would show shrinkage versus thickness in a straightforward experiment. However, the structural stability of such a geometry is reduced fast during thinning, and warping of the sample occurs. Therefore, we instead thinned a co-planar Si lamella in a double-wedge configuration (Fig. 1): Material was removed from one side by milling at a horizontal tilt of 4 • and from the other side at 1 • but with a vertical rotation of 0.5 • . The resulting lamella thus gets gradually thinner from bottom to top and from right to left with the very thin areas restricted to the top edge to limit warping. Milling parameters for the fabrication were identical to the singleedge experiment of Appendix B. For a given material and set beam conditions the absolute thickness of the lamella can be estimated from scanning electron microscope (SEM) image contrast. 23 In the secondary electron SEM image of Fig. 1, the thickness gradient of the lamella manifests in a contrast gradient with the thin areas appearing lighter. Based on purely the milling geometry, the thin top-edge of the created double-wedge lamella should be straight, as shown in the schematic illustration of Fig. 1. Instead, we observe a strongly falling edge indicating that the removal of lamella material in the vertical direction is greatly increased when the thickness drops below a certain threshold. Further-3 5°0 .5°F IG. 1. SEM image of the double-wedge lamella. The horizontal angle is 5 • , the vertical angle is 0.5 • , making the lamella thinner from bottom to top and from right to left. Thin areas appear lighter in secondary electron imaging mode. The thin top edge is indicated by the white arrow. The dropping edge indicates that there is a thickness-dependent change in sputtering rate; otherwise a straight edge would be expected, as illustrated in the schematic in the upper right-hand corner of the figure (the green dotted line indicates the dropping edge, the white lines indicate the expected shape).more, very thin areas (< 10 nm) are notably absent in the wedge lamella. In fact, we could not find any experimental conditions that would have allowed us to create a very thin Si edge using 30 keV Ga ions. We tried shortening the line dwell time/milling depth to reduce the corner rounding -to no avail. Reducing the beam current and thus sharpening the beam did not prove a viable option either since the resulting milling times were excessive, leading to problems from stage drift. Consequentially, we can unfortunately not draw any quantifiable conclusions from the thinning experiments except that we were not able to create very thin areas, and that there is a thickness-dependent vertical shrinkage effect for the thinnest parts of the structure. 24In order to understand the unexpected shrinkage of the lamella below the threshold thickness, an atomic-level description of the irradiation process is required. Our MD simulation setup, designed for studying ion irradiation effects on the top edge of a short section of a thin Si lamella, is visualized inFig. 2. In all simulations, the studied system size was 65 nm in the vertical (z) direction and 10 nm in width (x-direction), periodic boundaries being 4 applied in the latter dimension to mimic a much wider lamella edge. The beam, assuming a Gaussian distribution with a standard deviation of 2.5 nm in both the x and y-direction, was directed at the edge of the lamella at an angle of 1 • . The thickness of the edge was controlled by the size of the system in the y-direction (S y ∈ [3, 10] nm). The simulation setup is further detailed in Appendix C, and the performed runs are described in Appendix D.FIG. 2. The simulation setup for a 10 nm thick lamella top edge, as visualized with ovito. 25 The total number of atoms in this structure is 310 000.InFig. 3, we plot sputtering yield Y as a function of edge thickness for 1 keV and 30 keV Ga + irradiation as given by two different Si-Si interaction potentials (Appendix C). The data is clearly grouped by the beam energy: The thickness of the system seems to have no systematic effect on the results. On average, the 30 keV beam produces a Y of 7.1 ± 0.2 atoms/ion, whereas the 1 keV beam gives 0.98 ± 0.05 atoms/ion, as averaged over all system sizes and potentials. To better understand whether the irradiation energy leads to other differences beyond those in Y during the milling, we carried out an extensive analysis of different physical effects occuring during the process. For all simulated cases, the initial positions of sputtered atoms were determined after each irradiation event. Secondly, the center of mass of the system in the y-direction CM y was calculated as a function of the 5 z-coordinate. Thirdly, the degree of amorphization of each system was determined likewise as a function of z using structure factor analysis. 26 30 keV EDIP 30 keV SW 1 keV EDIP 1 keV SW Edge thickness (nm)
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10.1063/1.3698411
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[
"https://arxiv.org/pdf/1201.1407v2.pdf"
] | 54,868,713 |
1201.1407
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b07070c778fb763d41cfe56f9a4440b94e87f941
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Atomic-scale effects behind structural instabilities in Si lamellae during ion beam thinning
3 Feb 2012
E Holmström [email protected]
Department of Physics
University of Helsinki
P.O. Box 64FIN 00014HelsinkiFinland
Helsinki Institute of Physics
P.O. Box 64FIN 00014HelsinkiFinland
J Kotakoski
Department of Physics
University of Helsinki
P.O. Box 64FIN 00014HelsinkiFinland
Department of Physics
University of Vienna
Boltzmanngasse 5, Carl-Zeiss-Straße 561090, 73447Wien, OberkochenAustria, Germany
L Lechner
Central Facility of Electron Microscopy
Universität Ulm
Albert Einstein Allee 1189069UlmGermany
U Kaiser
Central Facility of Electron Microscopy
Universität Ulm
Albert Einstein Allee 1189069UlmGermany
K Nordlund
Department of Physics
University of Helsinki
P.O. Box 64FIN 00014HelsinkiFinland
Helsinki Institute of Physics
P.O. Box 64FIN 00014HelsinkiFinland
Atomic-scale effects behind structural instabilities in Si lamellae during ion beam thinning
3 Feb 2012(Dated: 21 December 2013)
The rise of nanotechnology has created an ever-increasing need to probe structures on the atomic scale, to which transmission electron microscopy has largely been the answer. Currently, the only way to efficiently thin arbitrary bulk samples into thin lamellae in preparation for this technique is to use a focused ion beam (FIB).Unfortunately, the established FIB thinning method is limited to producing samples of thickness above ∼20 nm. Using atomistic simulations alongside experiments, we show that this is due to effects from finite ion beam sharpness at low milling energies combined with atomic-scale effects at high energies which lead to shrinkage of the lamella. Specifically, we show that attaining thickness below 26 nm using a milling energy of 30 keV is fundamentally prevented by atomistic effects at the top edge of the lamella. Our results also explain the success of a recently proposed alternative FIB thinning method, which is free of the limitations of the conventional approach due to the absence of these physical processes.a) Corresponding author, 1 Since atomic-scale building blocks determine functionality in nanotechnology, understanding physical processes at this scale is crucial for the discipline. One of the most important means for achieving this knowledge is aberration-corrected high-resolution transmission electron microscopy (AC-HRTEM), 1,2 a method which allows determining the positions of atomic columns of a structure with accuracy higher than interatomic distance in solids.3,4HRTEM requires thin, unbent samples which ideally should contain no amorphous surface layer. Moreover, in nanotechnology, it is mandatory to achieve sample preparation with a sub-µm precision in all dimensions. These standards cannot be reached using traditional mechanical preparation methods such as mechanical milling, and thus the focused ion beam (FIB) technique, where the sample is milled into a thin lamella through sputtering, needs to be used instead.The current development in AC-HRTEM is to lower the voltage from the usual 200-300 kV to well below 100 kV, 5-10 in order to minimize displacement damage in fragile samples. 11 So far, predominantly low-dimensional materials such as graphene 5,8,10 , hexagonal boron nitride 6 , and carbon nanotubes 7 have been studied at low electron beam voltages by HRTEM. In the future, however, there will be a demand to extend these low-voltage studies to conventional materials, which similarly suffer from knock-on damage at higher voltages. This trend of lowering voltage entails an increasingly stringent requirement for sample thickness, which must not surpass the extinction length of the electron beam (∼few nm).However, the conventional FIB thinning method is limited to thicknesses above ∼20 nm.The dominant effect behind this limitation is the shrinkage of the lamella in height during milling: Thinning a lamella down to a thickness of 20 nm may induce several µm of shrinkage in the vertical direction. 12 The underlying physical reasons for this observation are currently not known, because an atomic-scale description of the thinning process has hitherto not been presented. Additionally, the conventional method is hampered by specimen warping, an uneven thickness profile, and often heavy amorphization.Although irradiation effects in Si have been studied for decades, the work has concentrated on bulk Si samples. 13-16 Moreover, earlier computational work on FIB processing of similar nanosystems has been restricted to simulation setups which severely reduce the applicability of the obtained results to the problem addressed in the present study. On the one hand, some studies have employed less detailed models based on Monte Carlo simulation of sputtering and redeposition on a three-dimensional grid. 17,18 Unfortunately, such models 2 lack real atomic resolution and hence do not account for, e.g., amorphization, which alone renders them powerless for the topic at hand. On the other hand, previous molecular dynamics (MD) simulations motivated by FIB milling have employed geometries restricted to a single surface with no deformation of the sample allowed, 19-21 thus yielding comparatively limited insight into the process. To uncover the mechanisms responsible for the deleterious effects on the lamella under conventional FIB thinning, we performed a classical MD study of the FIB thinning process in conjunction with experiments. Using the findings from our experiments and simulations, we show that the empirically observed limit of 20 nm for the conventional thinning method is a result of geometric sputtering effects due to the finite sharpness of the ion beam, dominant at low energies (∼1 keV), and of atomic-scale effects at the very top part of the lamella, significant at high energies (∼30 keV). We further explain why these mechanisms are not present in a recently proposed double-tilt method, 22 which thus allows the thinning process to proceed beyond the conventional limit.The experimental setup and the conventional FIB thinning method are described in Appendix A. The effect of ion beam tails, discussed below, is demonstrated in Appendix B.Here, we focus first on studying experimentally the deleterious shrinkage effect on a lamella during FIB thinning. In principle, creating a wedge-shaped lamella would show shrinkage versus thickness in a straightforward experiment. However, the structural stability of such a geometry is reduced fast during thinning, and warping of the sample occurs. Therefore, we instead thinned a co-planar Si lamella in a double-wedge configuration (Fig. 1): Material was removed from one side by milling at a horizontal tilt of 4 • and from the other side at 1 • but with a vertical rotation of 0.5 • . The resulting lamella thus gets gradually thinner from bottom to top and from right to left with the very thin areas restricted to the top edge to limit warping. Milling parameters for the fabrication were identical to the singleedge experiment of Appendix B. For a given material and set beam conditions the absolute thickness of the lamella can be estimated from scanning electron microscope (SEM) image contrast. 23 In the secondary electron SEM image of Fig. 1, the thickness gradient of the lamella manifests in a contrast gradient with the thin areas appearing lighter. Based on purely the milling geometry, the thin top-edge of the created double-wedge lamella should be straight, as shown in the schematic illustration of Fig. 1. Instead, we observe a strongly falling edge indicating that the removal of lamella material in the vertical direction is greatly increased when the thickness drops below a certain threshold. Further-3 5°0 .5°F IG. 1. SEM image of the double-wedge lamella. The horizontal angle is 5 • , the vertical angle is 0.5 • , making the lamella thinner from bottom to top and from right to left. Thin areas appear lighter in secondary electron imaging mode. The thin top edge is indicated by the white arrow. The dropping edge indicates that there is a thickness-dependent change in sputtering rate; otherwise a straight edge would be expected, as illustrated in the schematic in the upper right-hand corner of the figure (the green dotted line indicates the dropping edge, the white lines indicate the expected shape).more, very thin areas (< 10 nm) are notably absent in the wedge lamella. In fact, we could not find any experimental conditions that would have allowed us to create a very thin Si edge using 30 keV Ga ions. We tried shortening the line dwell time/milling depth to reduce the corner rounding -to no avail. Reducing the beam current and thus sharpening the beam did not prove a viable option either since the resulting milling times were excessive, leading to problems from stage drift. Consequentially, we can unfortunately not draw any quantifiable conclusions from the thinning experiments except that we were not able to create very thin areas, and that there is a thickness-dependent vertical shrinkage effect for the thinnest parts of the structure. 24In order to understand the unexpected shrinkage of the lamella below the threshold thickness, an atomic-level description of the irradiation process is required. Our MD simulation setup, designed for studying ion irradiation effects on the top edge of a short section of a thin Si lamella, is visualized inFig. 2. In all simulations, the studied system size was 65 nm in the vertical (z) direction and 10 nm in width (x-direction), periodic boundaries being 4 applied in the latter dimension to mimic a much wider lamella edge. The beam, assuming a Gaussian distribution with a standard deviation of 2.5 nm in both the x and y-direction, was directed at the edge of the lamella at an angle of 1 • . The thickness of the edge was controlled by the size of the system in the y-direction (S y ∈ [3, 10] nm). The simulation setup is further detailed in Appendix C, and the performed runs are described in Appendix D.FIG. 2. The simulation setup for a 10 nm thick lamella top edge, as visualized with ovito. 25 The total number of atoms in this structure is 310 000.InFig. 3, we plot sputtering yield Y as a function of edge thickness for 1 keV and 30 keV Ga + irradiation as given by two different Si-Si interaction potentials (Appendix C). The data is clearly grouped by the beam energy: The thickness of the system seems to have no systematic effect on the results. On average, the 30 keV beam produces a Y of 7.1 ± 0.2 atoms/ion, whereas the 1 keV beam gives 0.98 ± 0.05 atoms/ion, as averaged over all system sizes and potentials. To better understand whether the irradiation energy leads to other differences beyond those in Y during the milling, we carried out an extensive analysis of different physical effects occuring during the process. For all simulated cases, the initial positions of sputtered atoms were determined after each irradiation event. Secondly, the center of mass of the system in the y-direction CM y was calculated as a function of the 5 z-coordinate. Thirdly, the degree of amorphization of each system was determined likewise as a function of z using structure factor analysis. 26 30 keV EDIP 30 keV SW 1 keV EDIP 1 keV SW Edge thickness (nm)
Ulm, Germany
The rise of nanotechnology has created an ever-increasing need to probe structures on the atomic scale, to which transmission electron microscopy has largely been the answer. Currently, the only way to efficiently thin arbitrary bulk samples into thin lamellae in preparation for this technique is to use a focused ion beam (FIB).
Unfortunately, the established FIB thinning method is limited to producing samples of thickness above ∼20 nm. Using atomistic simulations alongside experiments, we show that this is due to effects from finite ion beam sharpness at low milling energies combined with atomic-scale effects at high energies which lead to shrinkage of the lamella. Specifically, we show that attaining thickness below 26 nm using a milling energy of 30 keV is fundamentally prevented by atomistic effects at the top edge of the lamella. Our results also explain the success of a recently proposed alternative FIB thinning method, which is free of the limitations of the conventional approach due to the absence of these physical processes. Since atomic-scale building blocks determine functionality in nanotechnology, understanding physical processes at this scale is crucial for the discipline. One of the most important means for achieving this knowledge is aberration-corrected high-resolution transmission electron microscopy (AC-HRTEM), 1,2 a method which allows determining the positions of atomic columns of a structure with accuracy higher than interatomic distance in solids. 3,4 HRTEM requires thin, unbent samples which ideally should contain no amorphous surface layer. Moreover, in nanotechnology, it is mandatory to achieve sample preparation with a sub-µm precision in all dimensions. These standards cannot be reached using traditional mechanical preparation methods such as mechanical milling, and thus the focused ion beam (FIB) technique, where the sample is milled into a thin lamella through sputtering, needs to be used instead.
The current development in AC-HRTEM is to lower the voltage from the usual 200-300 kV to well below 100 kV, [5][6][7][8][9][10] in order to minimize displacement damage in fragile samples. 11 So far, predominantly low-dimensional materials such as graphene 5,8,10 , hexagonal boron nitride 6 , and carbon nanotubes 7 have been studied at low electron beam voltages by HRTEM. In the future, however, there will be a demand to extend these low-voltage studies to conventional materials, which similarly suffer from knock-on damage at higher voltages. This trend of lowering voltage entails an increasingly stringent requirement for sample thickness, which must not surpass the extinction length of the electron beam (∼few nm).
However, the conventional FIB thinning method is limited to thicknesses above ∼20 nm.
The dominant effect behind this limitation is the shrinkage of the lamella in height during milling: Thinning a lamella down to a thickness of 20 nm may induce several µm of shrinkage in the vertical direction. 12 The underlying physical reasons for this observation are currently not known, because an atomic-scale description of the thinning process has hitherto not been presented. Additionally, the conventional method is hampered by specimen warping, an uneven thickness profile, and often heavy amorphization.
Although irradiation effects in Si have been studied for decades, the work has concentrated on bulk Si samples. [13][14][15][16] Moreover, earlier computational work on FIB processing of similar nanosystems has been restricted to simulation setups which severely reduce the applicability of the obtained results to the problem addressed in the present study. On the one hand, some studies have employed less detailed models based on Monte Carlo simulation of sputtering and redeposition on a three-dimensional grid. 17,18 Unfortunately, such models 2 lack real atomic resolution and hence do not account for, e.g., amorphization, which alone renders them powerless for the topic at hand. On the other hand, previous molecular dynamics (MD) simulations motivated by FIB milling have employed geometries restricted to a single surface with no deformation of the sample allowed, 19-21 thus yielding comparatively limited insight into the process. To uncover the mechanisms responsible for the deleterious effects on the lamella under conventional FIB thinning, we performed a classical MD study of the FIB thinning process in conjunction with experiments. Using the findings from our experiments and simulations, we show that the empirically observed limit of 20 nm for the conventional thinning method is a result of geometric sputtering effects due to the finite sharpness of the ion beam, dominant at low energies (∼1 keV), and of atomic-scale effects at the very top part of the lamella, significant at high energies (∼30 keV). We further explain why these mechanisms are not present in a recently proposed double-tilt method, 22 which thus allows the thinning process to proceed beyond the conventional limit.
The experimental setup and the conventional FIB thinning method are described in Appendix A. The effect of ion beam tails, discussed below, is demonstrated in Appendix B.
Here, we focus first on studying experimentally the deleterious shrinkage effect on a lamella during FIB thinning. In principle, creating a wedge-shaped lamella would show shrinkage versus thickness in a straightforward experiment. However, the structural stability of such a geometry is reduced fast during thinning, and warping of the sample occurs. Therefore, we instead thinned a co-planar Si lamella in a double-wedge configuration ( more, very thin areas (< 10 nm) are notably absent in the wedge lamella. In fact, we could not find any experimental conditions that would have allowed us to create a very thin Si edge using 30 keV Ga ions. We tried shortening the line dwell time/milling depth to reduce the corner rounding -to no avail. Reducing the beam current and thus sharpening the beam did not prove a viable option either since the resulting milling times were excessive, leading to problems from stage drift. Consequentially, we can unfortunately not draw any quantifiable conclusions from the thinning experiments except that we were not able to create very thin areas, and that there is a thickness-dependent vertical shrinkage effect for the thinnest parts of the structure. 24 In order to understand the unexpected shrinkage of the lamella below the threshold thickness, an atomic-level description of the irradiation process is required. Our MD simulation setup, designed for studying ion irradiation effects on the top edge of a short section of a thin Si lamella, is visualized in Fig. 2. In all simulations, the studied system size was 65 nm in the vertical (z) direction and 10 nm in width (x-direction), periodic boundaries being applied in the latter dimension to mimic a much wider lamella edge. The beam, assuming a Gaussian distribution with a standard deviation of 2.5 nm in both the x and y-direction, was directed at the edge of the lamella at an angle of 1 • . The thickness of the edge was controlled by the size of the system in the y-direction (S y ∈ [3, 10] nm). The simulation setup is further detailed in Appendix C, and the performed runs are described in Appendix D.
FIG. 2. The simulation setup for a 10 nm thick lamella top edge, as visualized with ovito. 25 The total number of atoms in this structure is 310 000.
In Fig. 3, we plot sputtering yield Y as a function of edge thickness for 1 keV and 30 keV Ga + irradiation as given by two different Si-Si interaction potentials (Appendix C). The data is clearly grouped by the beam energy: The thickness of the system seems to have no systematic effect on the results. On average, the 30 keV beam produces a Y of 7.1 ± 0.2 atoms/ion, whereas the 1 keV beam gives 0.98 ± 0.05 atoms/ion, as averaged over all system sizes and potentials. To better understand whether the irradiation energy leads to other differences beyond those in Y during the milling, we carried out an extensive analysis of different physical effects occuring during the process. For all simulated cases, the initial positions of sputtered atoms were determined after each irradiation event. Secondly, the center of mass of the system in the y-direction CM y was calculated as a function of the z-coordinate. Thirdly, the degree of amorphization of each system was determined likewise as a function of z using structure factor analysis. 26 Our analysis shows that the 1 keV beam results in the smoothening of the initially abrupt upper corner of the edge, which would lead to milling of the edge upon continued irradiation. The corner is amorphized, and sputtering is mainly from the impact region.
However, the effect of the 30 keV beam, as presented in Fig. 4, is more interesting. At this beam energy, the ≥7 nm structures bend toward the edge onto which the beam is aimed. This is verified by the diagrams showing the center of mass in the y-direction as a function of the edge height, which clearly reflect what is seen in the snapshot of the final state of each system. Also, the apparent amorphization seen in the snapshots is confirmed by the quantitative amorphization analysis. Sputtering is induced over the entire front face of the system and somewhat at the back face also. For edges <7 nm, however, almost no bending is observed at the dose of 500 ions. Instead, the system simply shrinks vertically as a result of the ion bombardment. Correspondingly, sputtering for the thinnest structures is no longer predominantly off the front face but also significant off the back of the structure.
Based on the results presented above, it would seem that the 30 keV beam induces mainly bending and modest shrinking in the thicker edges (≥7 nm) and predominantly shrinking in the thinnest edges. This inference is supported by studying the evolution of the atomic structure of the 10 and 3 nm edges as a function of the 30 keV irradiation dose, as presented in Figs 5(a,b), respectively. As the dose increases, the 10 nm system bends toward the corner where the beam is incident, whereas the 3 nm system shrinks by nearly 10 nm in height as the dose is brought to 1000 ions. The explanation for these two distinct modes of behavior can be found by considering the positions of sputtered atoms in each case and taking into account the resulting surface tension. As noted above and as seen in Fig. 4, sputtering for the ≥7 nm edges occurs mainly off the front face of the structure. Therefore, the system will relax, i.e., minimize its free energy to a local minimum, by contracting the front face in the vertical dimension, hence pulling the structure into a forward-bent position.
Correspondingly, when atoms are sputtered evenly off both the front and back faces, as for the <7 nm edges, the bending behavior is taken over by the shrinkage of the system in the vertical direction, as the structure relaxes by contracting along the entire thickness.
Note that the above-described bending is not the same phenomenon as the deleterious warping observed experimentally in the sample during conventional FIB thinning. The empirical warping happens in the opposite direction relative to the beam, and throughout the 7 bulk of the lamella. Perhaps counterintuitively, the bending as described by our simulations will actually appear as shrinkage of the lamella top in the experiments. This is because the tilted structure is etched away more efficiently by the incoming ions than the originally straight-standing one. Obviously, this erosion process is very difficult to observe experimentally, and has therefore not been detected before. One example of how the erosion might proceed is presented below in Fig. 5(d). Here a large density of deposited energy on the top of the bent structure causes hundreds of atoms to sputter with a single ion hit and leads to the subsequent straightening of the lamella edge. For the extreme case of lamella edges <7 nm, sputtering from both the front and back faces simply constitutes another mechanism of shrinkage. To predict the minimum attainable thickness of the lamella top edge according to this erosion mechanism, we performed a set of SRIM runs with Ga + incident on a Si target at an angle of 1 • at energies from 1 to 30 keV. To quantify the effect of the ions in the y-direction, we calculated the sum of ion range and straggle for the different energies (Fig. 6). To first assess how these results compare with MD data, we noted that for the 10 nm thick edge, the thickness of the amorphous layer in our MD simulations in the y-direction is 3 to 5 nm for 1 keV ions, and for 8 keV ions the thickness already surpasses 10 nm. A comparison of these two points to the SRIM results in Fig. 6 shows that the SRIM results give roughly 8 half the thickness of the amorphous layer. The difference is due to the fact that in the SRIM calculations only ions incident on the front face of the structure are included, whereas in the MD runs, ions incident on the very top of the structure are simulated as well. Another factor possibly contributing to this discrepancy is a recently predicted damage flow mechanism in amorphous Si. 27 Taking into account this factor of two difference leads to the prediction that at 30 keV, lamella edges of thicknesses ≤26 nm will be subjected to bending into the beam and hence catastrophic erosion. This explains why very thin lamella edges were completely absent in our experiments using the 30 keV Ga ion beam.
Regarding the warping seen in the experiment, the likely explanation for the phenomenon is to be found from previous studies of Si wafer curvature as brought on by ion implantation: 28 As the FIB is used to mill the face of the lamella, the surface region is amorphized, which causes the region to expand and hence to bend the lamella in the observed manner. The relevant scale for the effect is much larger than that in the present MD simulations. Furthermore, the mechanism relying on surface tension producing the bending toward the beam is dominant in our study over volume changes due to amorphization. The volume change of Si upon amorphization is predicted to be positive (as in experiment) for EDIP and negative for SW, 29 but nevertheless both potentials reproduce the bending of the top part of the lamella edge in the same direction. In our simulations, this is due to the small size of the structures; the smaller a system is, the higher its surface area to volume ratio, and hence any surface effects are pronounced in smaller systems.
In conclusion, we have experimentally demonstrated the well-known fact that a FIB milling energy of 30 keV cannot be used to create very thin lamellae (<∼ 20 nm) within the conventional thinning approach. Our simulations reveal that this limitation is a fundamental one, being due to atomic-scale effects at the top edge of the lamella which lead to the erosion of the structure under the ion beam. Specifically, we predict that an edge of thickness ≤26 nm will be subject to catastrophic erosion through the lamella top edge bending into the beam. This effect can be reduced by lowering the beam energy, but the lower the energy, the more significant the effects of the beam tails become (Appendix B). Together these two mechanisms of shrinkage lead to the lower limit of 20 nm for the thickness of the lamella during conventional FIB thinning. Our results explain why the alternative doubletilt approach (Appendix E) is free of these harmful effects, and can be used for preparing extremely thin samples of good quality for current and future AC-HRTEM studies. During rough milling, a free-standing structure containing the region of interest is created by removing material around it through FIB milling. Then, for lift-out, a manipulator is attached to the structure using ion beam induced deposition (IBID). Subsequently, the structure is cut free from the bulk. Using the manipulator, the resulting lamella is transferred to a special TEM lift-out grid. Once it is firmly attached by IBID it can be thinned further using high energy ion beam milling. In order to reduce the resulting damage layer (∼30 nm for 30 keV Ga ions in Si) the Ga ion energy is reduced for one or more polishing step(s). Polishing is done analogously, until the desired final thickness is reached.
APPENDIX B: SPUTTERING GEOMETRY
In order to understand the different effects leading to the shrinking behavior of the lamella in the conventional FIB method, we studied the sputtering geometry resulting from finite beam sharpness effects on a single edge. Fig. 8 shows an SEM image of the cross-section through a FIB-prepared thick Si lamella. The sidewalls of the lamella were milled using a 30 kV Ga ion beam with a current of 10 pA at 1 • incidence angle. The beam diameter (FWHM) was ∼20 nm and the milling depth was set to 10 µm. The beam was repeatedly scanned quickly on the same line until the desired milling depth was reached. The resulting edge is perpendicular to the sample surface but with a rounded top. As seen in the figure, the rounding effect is already in the 10 nm range at a beam voltage of 30 kV, and should become much worse at lower voltages due to chromatic lens aberrations.
APPENDIX C: SIMULATION METHOD
As obtaining an atomic-level description of the FIB thinning process involves following the time evolution of hundreds of thousands of atoms for thousands of time steps for each ion impact event, MD simulation with analytical interatomic potentials, which also provides a reasonably accurate atomistic description, is currently the only feasible method for the investigation. In addition to MD, we carried out a set of calculations within the binary collision approximation scheme as implemented in the SRIM code. 31,32 These simulations were performed in order to get an estimate of how some of our MD results could be extrapolated to larger system sizes utilizing the simpler physical model of SRIM, as explained above.
Arguably, the most critical part in designing an MD simulation is selecting the interaction model. For Si, more than 30 analytical potentials have been published. 33 Out of these, the most common and well-established potentials for studying radiation damage are the Stillinger-Weber (SW), 34 , Tersoff (TS), 35 , and Environment-Dependent Interatomic Potential (EDIP). 36 Regarding irradiation effects, the displacement threshold energy T d and the sputtering yield Y can be considered to be the decisive properties in describing the relevant processes. Unfortunately, all of the three mentioned potentials have been shown to underestimate T d as compared to density-functional theory calculations. 37 SW produces the highest average T d of 29 eV, whereas EDIP gives the lowest value of 16 eV. TS falls between these two extremes with 19 eV. The DFT value is as high as 35-36 eV. Additionally, Y given by EDIP has been shown to give the closest match with experiment for Ar ions with energies up to 20 keV, 38 whereas SW and TS agree well with each other but somewhat underestimate the experimental results. Thus, as SW gives the best value for T d , whereas EDIP describes sputtering in best accordance with experiments, we started our simulations using EDIP and SW in parallel. However, it turned out that there were no significant differences in the results between these two potentials, and hence the work was completed using EDIP.
For modeling the Ga-Si interaction, a purely repulsive ZBL potential 39 was used, because the Coulombic interaction between the nuclei is the overwhelmingly dominating effect in high-energy collisions, and the finer chemistry between the two types of atoms can be neglected. Indeed, chemical effects should affect irradiation results only when the projectile energy is very low (< 100 eV). 40 We have previously used a similar approach to model Ga impacts on graphene in the keV energy range. 41 To realistically model high-energy Si-Si collisions brought on by the up to 30 keV ion impacts, a separate ZBL high-energy part was smoothly splined to the repulsive part of the Si-Si potentials at small interatomic distances. 42 Finally, to account for electronic stopping of energetic atoms, a frictional, velocity-dependent drag force was applied to all atoms with a kinetic energy higher than 5 eV.
To model the dissipation of energy deposited into the irradiated structure by an incident ion, the system was coupled to a Berendsen thermostat 43 at a temperature of 10 K with a time constant of 300 fs in a region with a thickness of 0.8 nm at the periodic boundaries and 1.5 nm from the bottom of the structure. As those parts of the system which are coupled to the thermostat constitute an effective boundary in the periodic x-direction, the system was translated over the periodic boundaries for each ion impact event so that the impact point in the x-direction was always halfway between the regions where temperature was being scaled. Also, to prevent the entire structure from moving as a result of the ion impacts, a small segment encompassing ∼3000 atoms in the lower back face of the system was fixed. Each ion impact simulation was carried out for 15 ps, which was enough to model the ballistic phase of the irradiation event. After this, the temperature of the system was slowly decreased to the initial simulation temperature of 10 K before the next impact. All simulations were performed using the parcas MD code. 44
APPENDIX D: SIMULATION RUNS
The first set of runs consisted of simulating edges of thicknesses 3-10 nm. 45 In order to simultaneously study both the role of thickness and that of beam energy, we carried out these simulations for irradiation energies of 1 keV and 30 keV up to a total beam dose of 500 ions. We also performed the same set of runs using a beam energy of 500 eV, but found no noteworthy differences to the 1 keV case. To make sure we weren't missing any important effects due to too low a dose, we at this point increased the total dose to 1000 ions. Finally, we extended the earlier simulations for the 3 and 10 nm edges up to this same dose. It is worth noting that even these 'high' doses are far from the experimental ones. However, the simulations therefore allow us to detect effects which may disappear too fast in the experiment to be directly observed.
APPENDIX E: ATTAINING THE THINNEST POSSIBLE LAMELLA
It has always been best practice to reduce the ion beam energy as the lamella becomes progressively thinner. After all, the goal of TEM sample preparation is to retain as much pristine material as possible inside the lamella. Despite this, no ultra-thin lamellae have been produced using the conventional method. The problem ultimately lies in the dominating effect of chromatic lens aberrations at low acceleration voltages which significantly increase the beam diameter. During thinning and polishing, the resulting wide beam tails result in the rounding of the edges of the lamella top. When a lamella is thinned from both sides, these tails may overlap causing shrinkage. Besides shrinkage, these tails cause the top edge to become significantly thinner than the rest of the lamella. This thin edge is eroded away through bending into the beam and sputtering from both sides, as described above by our simulations. Reducing the beam energy further does decrease these top edge effects (as seen in Fig. 6), but in turn also increases geometric shrinking of the bulk of the lamella. The resulting trade-off seems to set a fundamental lower limit of ∼20 nm to lamella thickness attainable using the conventional thinning method.
13
However it was shown recently that the conventional limit can be overcome using the double-tilt method. 22 Within this approach, the detrimental formation of a sharp edge during thinning is omitted by milling perpendicular grooves in the front and the back sides of the sample. The thin transparent area of the lamella is formed where the grooves overlap.
Geometric and top edge effects are thus effectively suppressed since there is no free edge, independent of the lamella's final thickness. In addition, amorphization of the lamella surface can be reduced to a minimum since no trade-off between beam energy, milling angle, and tolerable focus spread has to be made. Even low-kV Ar ion polishing, typically featuring beam diameters of 10 to 100 µm, can be performed at grazing incidence. Last but not least, the mechanical stability of the resulting lamella is much higher than that obtained with the conventional method since the transparent window is surrounded on all sides by a thick frame of material. These points explain why the double-tilt method can be used to obtain results superior to those of the conventional method.
a) Corresponding author, email: [email protected] b) Current address: Department of Physics, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria c) Current address: Carl Zeiss NTS GmbH, Carl-Zeiss-Straße 56, 73447 Oberkochen, Germany
Fig. 1 )
1: Material was removed from one side by milling at a horizontal tilt of 4 • and from the other side at 1 • but with a vertical rotation of 0.5 • . The resulting lamella thus gets gradually thinner from bottom to top and from right to left with the very thin areas restricted to the top edge to limit warping. Milling parameters for the fabrication were identical to the singleedge experiment of Appendix B. For a given material and set beam conditions the absolute thickness of the lamella can be estimated from scanning electron microscope (SEM) imagecontrast.23 In the secondary electron SEM image ofFig. 1, the thickness gradient of the lamella manifests in a contrast gradient with the thin areas appearing lighter.Based on purely the milling geometry, the thin top-edge of the created double-wedge lamella should be straight, as shown in the schematic illustration ofFig. 1. Instead, we observe a strongly falling edge indicating that the removal of lamella material in the vertical direction is greatly increased when the thickness drops below a certain threshold. Further-
. 1. SEM image of the double-wedge lamella. The horizontal angle is 5 • , the vertical angle is 0.5 • , making the lamella thinner from bottom to top and from right to left. Thin areas appear lighter in secondary electron imaging mode. The thin top edge is indicated by the white arrow. The dropping edge indicates that there is a thickness-dependent change in sputtering rate; otherwise a straight edge would be expected, as illustrated in the schematic in the upper right-hand corner of the figure (the green dotted line indicates the dropping edge, the white lines indicate the expected shape).
FIG. 3 .
3Sputtering yield Y as a function of edge thickness up to a dose of 500 ions. The averages and the respective errors are shown by the horizontal solid and dotted lines, respectively. Estimated uncertainties for Y , obtained from the fitting, are contained within the markers.
FIG. 4 .
4Results for the edges of all studied thicknesses after a dose of 500 ions at 30 keV. For each case, we show the snapshot of the final structure along the x-direction, the positions of sputtered ions, the center of mass CM y , and the degree of amorphization. Only the upper part of the structure is shown, as this is where all interesting effects appear.
Focusing on the described bending of the edge in more detail, it can be seen inFig. 5(c) that an 8 keV ion beam is energetic enough to induce bending in the 10 nm structure, whereas at 1 keV the edge is left standing upright. Looking at the bent structures in
Figs 4 and 5
5(a), it can be discerned that the top part of the bent structure is as if bent around a hinge at a well-defined height on the back face of the structure, as illustrated explicitly by the horizontal line inFig. 5(a). This point is roughly around the area where the amorphization starting from the front face of the structure reaches the back face, which is where the originally rigid crystalline back wall gives way to the contracting force on the front face of the system. Such a mechanism is functional only when the system is thin enough to allow for the amorphization to proceed all the way through the structure. For sufficiently thick structures this does not happen. We emphasize that this does not require a complete amorphization of the structure. Instead, it is enough if the damage reaches the back face at one location. This may happen outside the collision cascade, as was shown recently. 27
ACKNOWLEDGEMENTS
The theoretical research for this study was supported by the Academy of Finland Center of Excellence in Computational Molecular Science and the Helsinki Institute of Physics. EH is grateful to Michael Moseler and Karsten Albe for valuable comments. Generous grants of computer time from the Center for Scientific Computing in Espoo, Finland are thankfully acknowledged. The experimental research was funded by the German Research Foundation (DFG) and the State of Baden-Württemberg through the SALVE (Sub-Ångström Low-Voltage Electron Microscopy) project. APPENDIX A: CONVENTIONAL FIB THINNING METHOD Throughout the study, sample fabrication was performed in a Zeiss NVision 40 CrossBeam FIB instrument incorporating a Ga liquid metal ion source and in-situ scanning electron microscopy (SEM) imaging using a thermal field emission source. The energy of the Ga ions can be varied in the range from 1 to 30 keV. Figure 7(a) shows a schematic drawing of the microscope. TEM lamella preparation by the FIB lift-out method is carried out in four steps: 30 (1) Rough milling of a lamella, (2) lift-out, (3) thinning, and (4) polishing.
Figure 7 (
7b) shows a schematic drawing of the lamella during the thinning and polishing process of the conventional in-situ lift-out technique. The lamella faces are milled under a glancing angle of 1 to 3 • (depending on milling current) to obtain co-planar surfaces. In conventional thinning, material is removed top-down from one or both sides of the lamella.
Due to the differences observed for the 1 keV and 30 keV irradiations (as described above), we performed another set of simulations to investigate intermediate energies. These complementary runs with beam energies of 8 and 15 keV were performed on the 10 nm edge.
FIG. 5 . 18 FIG. 6 .FIG. 7 .
51867Evolution of the (a) 10 and (b) 3 nm edges as a function of the dose of a 30 keV beam. The 10 nm thick system exhibits mainly bending, whereas the 3 nm one exhibits predominantly shrinking. The dashed horizontal line in (a) depicts the hinge around which the bending occurs. The snapshot of the structure is a sideview along the x-direction. (c) The bending angle of the 10 nm edge at beam energies between 1 keV and 30 keV as a function of dose. (d) Cumulative sputtering yield i Y i for the 10 nm edge as a function of the dose between 750 to 1000 ions at 8 and 15 keV. The sharp increase close to 950 ions for the 15 keV case explains the sudden drop in the bending angle seen in panel (c) (as indicated by the arrows): An exceptionally dense region of deposited energy by one of the incoming ions craterizes the top of the structure and thus destroys the forward-bent shape (see the snapshots), leading to the experimentally observed shrinkage via the bending mechanism. Sum of range and straggle in the y-direction from SRIM calculations of Ga ions incident on Si at an angle of 1 • as a function of ion energy. On the basis of the MD simulations, this is estimated to give approximately half the maximum thickness of the amorphous layer. The inset shows the SRIM setup schematically with the coordinate axes as in the MD runs. Here θ = 1 • . (a) Schematic of the FIB/SEM microscope column arrangement. (b) Orthographic third angle projection of the lamella during conventional thinning. The direction of the Ga ions is indicated by the arrows.
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On the top edge, rounding is caused through milling by the ion beam tails. The (green) dotted line indicates the ideal edge shape an infinitely sharp beam should create. Inset (black frame): Overview of the cross-section. SEM image of a cross-section through a FIB-fabricated vertical lamella sidewall. the (white) rectangle indicates the location of the magnified viewFIG. 8. SEM image of a cross-section through a FIB-fabricated vertical lamella sidewall. On the top edge, rounding is caused through milling by the ion beam tails. The (green) dotted line indicates the ideal edge shape an infinitely sharp beam should create. Inset (black frame): Overview of the cross-section; the (white) rectangle indicates the location of the magnified view.
|
[] |
[
"Galois Conjugates of Topological Phases",
"Galois Conjugates of Topological Phases"
] |
[
"M H Freedman \nStation Q\nMicrosoft Research\nUniversity of California\n93106Santa BarbaraCAUSA\n",
"J Gukelberger \nTheoretische Physik\nETH Zurich\n8093ZurichSwitzerland\n",
"M B Hastings \nStation Q\nMicrosoft Research\nUniversity of California\n93106Santa BarbaraCAUSA\n",
"S Trebst \nStation Q\nMicrosoft Research\nUniversity of California\n93106Santa BarbaraCAUSA\n",
"M Troyer \nTheoretische Physik\nETH Zurich\n8093ZurichSwitzerland\n",
"Z Wang \nStation Q\nMicrosoft Research\nUniversity of California\n93106Santa BarbaraCAUSA\n"
] |
[
"Station Q\nMicrosoft Research\nUniversity of California\n93106Santa BarbaraCAUSA",
"Theoretische Physik\nETH Zurich\n8093ZurichSwitzerland",
"Station Q\nMicrosoft Research\nUniversity of California\n93106Santa BarbaraCAUSA",
"Station Q\nMicrosoft Research\nUniversity of California\n93106Santa BarbaraCAUSA",
"Theoretische Physik\nETH Zurich\n8093ZurichSwitzerland",
"Station Q\nMicrosoft Research\nUniversity of California\n93106Santa BarbaraCAUSA"
] |
[] |
Galois conjugation relates unitary conformal field theories (CFTs) and topological quantum field theories (TQFTs) to their non-unitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois conjugated Hamiltonians are typically non-Hermitian, we find that their ground state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such non-unitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this we rigorously prove that no local change of basis (IV.5) can transform the ground states of the Galois conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasi-local Hamiltonians. A similar statement holds for many other non-unitary TQFTs. One consequence is that the "Gaffnian" wave function cannot be the ground state of a gapped fractional quantum Hall state.
|
10.1103/physrevb.85.045414
|
[
"https://arxiv.org/pdf/1106.3267v3.pdf"
] | 54,896,614 |
1106.3267
|
de4600f3d3cfdc69f5426bf36b6d7733842fd95e
|
Galois Conjugates of Topological Phases
M H Freedman
Station Q
Microsoft Research
University of California
93106Santa BarbaraCAUSA
J Gukelberger
Theoretische Physik
ETH Zurich
8093ZurichSwitzerland
M B Hastings
Station Q
Microsoft Research
University of California
93106Santa BarbaraCAUSA
S Trebst
Station Q
Microsoft Research
University of California
93106Santa BarbaraCAUSA
M Troyer
Theoretische Physik
ETH Zurich
8093ZurichSwitzerland
Z Wang
Station Q
Microsoft Research
University of California
93106Santa BarbaraCAUSA
Galois Conjugates of Topological Phases
(Dated: July 6, 2011)PACS numbers: 0530Pr, 7343-f
Galois conjugation relates unitary conformal field theories (CFTs) and topological quantum field theories (TQFTs) to their non-unitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois conjugated Hamiltonians are typically non-Hermitian, we find that their ground state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such non-unitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this we rigorously prove that no local change of basis (IV.5) can transform the ground states of the Galois conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasi-local Hamiltonians. A similar statement holds for many other non-unitary TQFTs. One consequence is that the "Gaffnian" wave function cannot be the ground state of a gapped fractional quantum Hall state.
I. INTRODUCTION
Galois conjugation, by definition, replaces a root of a polynomial by another one with identical algebraic properties. For example, i and −i are Galois conjugate (consider z 2 + 1 = 0) as are φ = 1+ (consider z 2 − z − 1 = 0), as well as 3 √ 2, 3 √ 2e 2πi/3 , and 3 √ 2e −2πi/3 (consider z 3 − 2 = 0). In physics Galois conjugation can be used to convert nonunitary conformal field theories (CFTs) to unitary ones, and vice versa. One famous example is the non-unitary Yang-Lee CFT, which is Galois conjugate to the Fibonacci CFT (G 2 ) 1 , the even (or integer-spin) subset of su(2) 3 .
In statistical mechanics non-unitary conformal field theories have a venerable history. 1,2 However, it has remained less clear if there exist physical situations in which non-unitary models can provide a useful description of the low energy physics of a quantum mechanical system -after all, Galois conjugation typically destroys the Hermitian property of the Hamiltonian. Some non-Hermitian Hamiltonians, which surprisingly have totally real spectrum, have been found to arise in the study of P T -invariant one-particle systems 3 and in some Galois conjugate many-body systems 4 and might be seen to open the door a crack to the physical use of such models. Another situation, which has recently attracted some interest, is the question whether non-unitary models can describe 1D edge states of certain 2D bulk states (the edge holographic for the bulk). In particular, there is currently a discussion on whether or not the "Gaffnian" wave function could be the ground state for a gapped fractional quantum Hall (FQH) state albeit with a non-unitary "Yang-Lee" CFT describing its edge. [5][6][7] We conclude that this is not possible, further restricting the possible scope of non-unitary models in quantum mechanics.
We reach this conclusion quite indirectly. Our main thrust is the investigation of Galois conjugation in the simplest non-Abelian Levin-Wen model. 8 This model, which is also called "DFib", is a topological quantum field theory (TQFT) whose states are string-nets on a surface labeled by either a trivial or "Fibonacci" anyon. From this starting point, we give a rigorous argument that the "Gaffnian" ground state cannot be locally conjugated to the ground state of any topological phase, within a Hermitian model satisfying Lieb-Robinson (LR) bounds 9 (which includes but is not limited to gapped local and quasi-local Hamiltonians).
Lieb-Robinson bounds are a technical tool for local lattice models. In relativistically invariant field theories, the speed of light is a strict upper bound to the velocity of propagation. In lattice theories, the LR bounds provide a similar upper bound by a velocity called the LR velocity, but in contrast to the relativistic case there can be some exponentially small "leakage" outside the light-cone in the lattice case. The Lieb-Robinson bounds are a way of bounding the leakage outside the lightcone. The LR velocity is set by microscopic details of the Hamiltonian, such as the interaction strength and range. Combining the LR bounds with the spectral gap enables us to prove locality of various correlation and response functions. We will call a Hamiltonian a Lieb-Robinson Hamiltonian if it satisfies LR bounds.
We work primarily with a single example, but it should be clear that the concept of Galois conjugation can be widely applied to TQFTs. The essential idea is to retain the particle types and fusion rules of a unitary theory but when one comes to writing down the algebraic form of the F -matrices (also called 6j symbols), the entries are now Galois conjugated. A slight complication, which is actually an asset, is that writing an F -matrix requires a gauge choice and the most convenient choice may differ before and after Galois conjugation.
Our method is not restricted to Galois conjugated DFib G and its factors Fib G and Fib G , but can be generalized to infinitely many non-unitary TQFTs, showing that they will not arise as low energy models for a gapped 2D quantum mechan-ical system with topological order.
The 2D quantum mechanical systems which can be described by any type of TQFTs are known as topological phases. Although this concept is widely noted in the condensed matter physics literature, our introduction is not complete without providing a definition. Many authors focus on properties (e.g. existence of anyonic excitations), but we prefer to give a more fundamental definition since it is this definition which figures into our proof in Sec. IV. We say that a LR Hamiltonian describes a topological phase (or a phase is topologically ordered) if and only if its ground state manifold G satisfies the following "code" property with respect to all spacially local operators L: the composition
G inc → H L − → H inc † −−→ G(1)
is a multiplication by some scalar s(L) (possibly s(L) = 0). L local means L acts only on sites in a sufficiently small radius. This definition first appeared in Ref. 10 (see definition 3.6 there), conceptualizing the earlier formulation of Ref. 11. Recently Bravyi et al. 12 have called this axiom "TQO-1" and advocated an additional requirement that they called "TQO-2", which enforces a consistency between local and global ground states. While technically necessary, we know of no realistic case where the second axiom would be required and so have not included it in the framework of this paper. 13
II. LEVIN-WEN MODEL AND ITS GALOIS CONJUGATES
A. The Levin-Wen model
Topological quantum field theories are highly constrained mathematical constructs [14][15][16] designed to capture the low energy physics of topologically ordered systems. Chern-Simons theory 17 generates most of the known examples; the simplest of these, all chiral, being based on a Lie group and level k, G k . Starting from a set of particles and fusion rules, there is a standard construction -called the "quantum double" or "Drinfeld center" -which produces an achiral TQFT. Such quantum doubles were introduced in the physics literature by Levin and Wen 8 in the form of "string-net" Hamiltonians. If, for instance, we take the particles and fusion rules from the chiral Fib TQFT, see Eq.(2) below, and use these to label string-nets on surfaces, a "larger" TQFT DFib ∼ = Fib ⊗ Fib (with more particle types) is obtained.
The Levin-Wen model thus is a microscopic spin Hamiltonian implementing doubled topological theories. Originally, it was defined 8 on a honeycomb lattice, but its extension to any trivalent graph is straight-forward. Given a lattice graph and an anyonic theory, the model's Hilbert space is spanned by all labelings of graph edges with the theory's particle types which are consistent with a set of constraints given by the theory, the so-called fusion rules. As a simple example we first consider the Fibonacci theory Fib, where there are only two particle types, namely a trivial particle 1 and the Fibonacci anyon τ . Two particles can combine according to the fusion
1 × 1 = 1 1 × τ = τ τ × τ = 1 + τ .(2)
In the Levin-Wen model implementing the doubled Fibonacci theory DFib, this amounts to the constraint that of the three edges meeting in any single vertex never only one can carry a τ label. This Hilbert space can either be understood as that of an anyonic quantum liquid enclosing the lattice links 18 or alternatively as the the ground states of a spin model (by identifying particle types with spin directions) with a peculiar threespin interaction enforcing the vertex constraint.
Within these states, the Hamiltonian
H LW = J p plaquettes p δ φ(p),τ(3)
is a projector onto the τ -flux state of a plaquette p thus favoring the trivial flux φ(p) = 1 through each plaquette. The action of this operator on an element of the basis where the edges belonging to plaquette p carry labels α, . . . , ζ, a, . . . , f as displayed in Fig. 1 results in a superposition of states where the inner edges of the plaquette carry new labels α , . . . , ζ whereas all other edges remain unchanged. Any of the labels takes one of the values {1, τ }. The matrix elements between these basis states read explicitly (see Refs. 8 and 18 for a detailed derivation)
δ φ(p),τ = 1 − s d s D 2 F α sζ a ζ α F β sα b α β F γ sβ c β γ × F δ sγ d γ δ F sδ d δ F ζ s d ζ ,(4)
where d s denotes the quantum dimension of particle type s, i.e. d 1 = 1 and d τ = φ ≡ (1 + √ 5)/2, the golden ratio and D the total quantum dimension, D = d 2 1 + d 2 τ = √ 2 + φ for Fibonacci anyons. For different plaquette geometries this operator has an analogous form with one F -symbol for each edge of the plaquette.
The F -symbol, which can be thought of as a generalized 6j symbol, describes local basis transformations in a fusion tree as shown in Fig. 2 and is a defining property of the anyonic theory. For Fibonacci anyons, this transformation is trivial except for the case when all four outer legs of the subgraph that is to be transformed carry the τ label. Then we have the
unitary 2 × 2 matrix F τ τ τ τ = φ −1 cφ −1/2 c −1 φ −1/2 −φ −1 .(5)
where any choice of the gauge c satisfies the pentagon equations for the Fibonacci fusion rules. Choosing the gauge = 1 gives the usual unitary F -matrix for the Fibonacci theory, which we refer to as the symmetric normalization. From an algebraic point of view the natural gauge choice is = φ 5/2 which leads to
F τ τ τ τ = φ − 1 φ + 1 2φ − 3 1 − φ ,(6)
where no square roots of φ appear. We refer to this choice as the algebraic normalization and define λ = c/φ 5/2 . We remark that both normalizations give the same spectra for our models, since the corresponding Hamiltonians are conjugate by a diagonal fugacity change matrix. The Levin-Wen model can can be solved exactly since all the plaquette terms commute. 8 As a sum of projectors it counts the number of plaquettes penetrated by a nontrivial τ -flux and the spectrum hence consists of states at all nonnegative integer multiples of J p , corresponding to the number of nontrivial plaquette fluxes.
The ground states of the model correspond to all states with no plaquette fluxes, corresponding to the ground states of the topological liquid on a doubled surface around the lattice. With periodic boundary conditions in both directions this surface is a doubled torus with four degenerate ground states.
B. The doubled Yang-Lee model
Now we turn to a theory of non-unitary non-abelian anyons which are closely related to the Fibonacci ones by Galois conjugation. We start by noting that Fib is only one particular sub-theory out of a discrete set of su(2) k (for finite k) anyonic theories, specifically the integral spin half of the unitary su(2) 3 theory. The su(2) k theories are certain deformations of SU (2) characterized by the truncation level k, which defines the particle types in the theory, and additionally a deformation parameter q determining the precise values of the F -symbol. For k = 3 two sub-theories characterized by the roots of unity t = e 2πi/5 and t = e 4πi/5 are Galois conjugates of each other (see Fig. 3). Now, the former value for q produces the Fibonacci theory as described above whereas the latter leads to the non-unitary F matrix
F τ τ τ τ = d − 1 d + 1 2d − 3 1 − d ,(7)
in the algebraic normalization and
F τ τ τ τ = −φ iφ 1/2 iφ 1/2 φ .(8)
in the symmetric normalization. These are just the F -matrices of the DFib theory with
φ = −t 1/2 − t −1/2 replaced by d = −1/φ = −t 1/2 − t −1/2 .
Here we choose the fourth roots t 1/4 = ie πi/10 and t 1/4 = ie −3πi/10 , which will be needed to specify the Galois conjugation of the full theory below.
We remark on the choice of algebraic normalization for the F matrix. For the Yang-Lee theory no choice of c would make F unitary-a manifestation of the non-unitarity. While there would be no topological invariant positive definite Hermitian products on all ground state manifolds in Yang-Lee theory, there is always a topological invariant Hermitian product with possibly mixed signatures. The topological invariant inner product is Hermitian because the partition function under time reversal in a (2+1)-topological theory is Hermitian conjugated. The (1, 2)-entry of the above F -matrix is the theta symbol (the norm of a fusion basis in a fusion space) multiplied by d −2 , hence it should be a real number. The above choice of F -matrix for Fibonacci case is pleasant when we work on number theory related problems. In particular, one notices that in this case we obtain the Galois conjugate by replacing all occurrences of the golden ratio φ by d = −1/φ, which is the second solution of the quadratic equation x 2 = 1 + x.
As Galois conjugation does not change the theory's algebraic structure the doubled Yang-Lee (DYL) Levin-Wen model using the F matrix of Eq. (8), can be solved in exactly the same way as its DFib counterpart. In particular, it has exactly the same spectrum whose eigenvalues count the number of plaquettes penetrated by a non-trivial flux and the same ground state degeneracies. The DYL model also retains the topological protection of the ground state degeneracy against local perturbations.
III. HERMITIAN MODEL FROM NON-UNITARY THEORY
A. Constructing Hermitian models
While the non-Hermitian DYL model features a generalized stable topological phase and a generalized code property, discussed in more detail below, an immediately arising question is whether this phase can also be realized in a Hermitian model. There are multiple ways to obtain a Hermitian model that has the same ground states as the non-Hermitian parent model. However, as we will see in the following the question whether the topological nature of the ground state remains is a more subtle one. The simplest Hermitian model H † H is obtained by squaring the non-Hermitian parent Hamiltonian H. This model has the same right ground-state eigenvectors as the original model. Alternatively, HH † has the same left ground-state eigenvectors. The simplicity of this approach comes at the cost of a Hamiltonian which is highly non-local. To avoid non-local terms, we can take an alternative route and individually square each plaquette term of H p = δ DYL φ(p),τ , arriving at the Hamiltonian p H † p H p or p H p H † p . Since each plaquette term annihilates the ground state, squaring them in this way also annihilate the (right/left) ground state eigenvectors. Finally, we can replace the non-Hermitian plaquette operator H p with a projector onto the complement of the operator's kernel. More specifically, we diagonalize the plaquette operator and use its orthogonalized right eigenvectors 0 (r) i belonging to the eigenvalue 0 to define a projector
P p = 1 − i 0 (r) i 0 (r) i .(9)
The sum of these projectors is then used to define the Hermitian Hamiltonian
H herm = J p p P p .(10)
It turns out that all three approaches result in the same qualitative behavior -a loss of the code property and the associated stable topological order -and we will limit our discussion to the last approach.
B. Loss of the code property
We find that the non-Hermitian models are stable against local perturbations, and they satisfy a generalized code property. Keeping in mind that a non-Hermitian matrix has left and right eigenvectors, which in general are not identical, a local operator acts as a scalar multiple of an identity operator connecting the left and right ground state subspaces:
0 (l) i L 0 (r) j = λ(L)δ ij .(11)
Independent of the way we derive a Hermitian model from the parent DYL model, we find that the code property is lost for the Hermitian models: when constructing a Hermitian model, one inevitably has to decide wether to preserve left or right ground states. The code property for the Hermitian model would require expectation values of local operators of the form
0 (r) i L 0 (r) j and 0 (l) i L 0 (l) j(12)
to again be multiples of the identity. In general, this usual code property will not be satisfied, as one can see, for example, by calculating the matrix elements of a local observable such as a string tension. Perturbing any Hermitian Hamiltonian which has the (right or left) DYL ground states with an arbitrary small string tension will hence immediately lead to a splitting of the ground-state degeneracy, as we will discuss below.
C. Absence of topological order
In this section we probe whether topological order survives the construction of a Hermitian model by numerically diagonalizing the models on different lattice geometries, the honeycomb lattice of the original Levein-Wen construction 8 and the two-leg ladder geometry of Ref. 18. We diagonalized systems with up to 24 edges using a dense eigenvalue solver and employed iterative schemes for systems with up to 39 edges: the Lanczos algorithm for Hermitian models and an implicitly restarted Arnoldi method for non-Hermitian models.
Honeycomb model
Our results on the honeycomb lattice show a clear distinction between the DFib and DYL models on the one hand and the Hermitian model H herm derived from the DYL model on the other hand. While all models feature four degenerate ground states, the former two are gapped, whereas the latter one turns out to be gapless in the thermodynamic limit; see the finite-size extrapolation in Fig. 4a). Furthermore, the groundstate degeneracy is easily lifted by a local perturbation, such as a string tension -in contrast to the stability of the topological phases of the DFib and DYL models.
Ladder model
Since only small linear dimensions are accessible to exact numerical diagonalization for the honeycomb lattice, we also consider a a quasi-one-dimensional ladder geometry consisting of rectangular plaquettes as shown in Fig. 5. The DFib and DYL models on this ladder geometry were introduced and solved in Refs. 18 and 4, respectively. Both models feature topological phases with two (instead of four) degenerate ground states, but are otherwise identical to the respective honeycomb lattice models. The quasi-one dimensional geometry allows to numerically diagonalize systems up to linear system size L = 13. The finite-size gap of the Hermitian model H herm is again found to vanish in the thermodynamic limit, showing a linear dependence on the inverse system size as shown in Fig. 4b). To further demonstrate the fragility of these gapless ground states against local perturbations we add a string tension 18
H pert = J r rungs r δ l(r),τ(13)
favoring the trivial label l(r) = 1 on each rung of the ladder. We parameterize the couplings of the competing plaquette and rung terms as
J r = sin θ and J p = cos θ ,
where θ = 0 corresponds to the unperturbed Hamiltonian. The phase diagrams as a function of θ have been mapped out for both the DFib model 18 and the DYL model, 4 respectively.
Directly probing the topological order in the DYL model and its Hermitian counterpart we show the lifting of their respective ground-state degeneracies in Figs. 6 and 7 when including a string tension. We find a striking qualitative difference between these two models: For the DYL model the lifting of the ground-state degeneracy is exponentially suppressed with increasing system size -characteristic of a topological phase. For the Hermitian model, on the other hand, we find a splitting of the ground-state degeneracy proportional to J r L. The linear increase with both system size and coupling can be easily understood by the different matrix elements of the string tension term on a single rung for the two degenerate ground-states of the unperturbed model. Plotting the lowenergy spectrum in Fig. 7 clearly shows that the two-fold degeneracy of the unperturbed Hermitian model arises from a (fine-tuned) level crossing. Similar behavior is found in the honeycomb lattice model (not shown).
Considering the model in a wider range of couplings, as shown in Fig. 8, further striking differences between the non-Hermitian DYL model and its Hermitian counterpart are revealed: The DYL model exhibits two extended topological phases around θ = 0 and θ = π/2 (with two and four degenerate ground states, respectively), which are separated by a conformal critical point at precisely θ c = π/4 as discussed extensively in Refs. 4 and 18. In contrast, the Hermitian model H herm exhibits no topological phase anywhere, and the intermediate coupling θ = π/4 does not stand out. The bottom panel shows the low-energy spectrum, which clearly shows that the degeneracy at θ = 0 is due to a level-crossing.
IV. ABSENCE OF NON-UNITARY TOPOLOGICAL PHASES IN UNITARY MODELS
So far, we have considered a specific set of Hermitian models constructed to have the same ground states as a non-Hermitian parent model and found that they no longer exhibit a topological phase. This raises the question whether this observation points to a deeper principle, which we investigate in this section in rigorous mathematical terms.
A. Galois Conjugate
Let us now lay out the mathematical foundations as clearly as possible. The double DFib is isomorphic to a copy of Fib and its time reversal, DFib ∼ = Fib ⊗ Fib. Thus to Galois conjugate DFib it is sufficient to define Fib G , then
DFib G ∼ = Fib G ⊗ Fib G .
A theory such as Fibonacci can be defined using a set . Because of gauge choices, there are many different sets for the same theory. If we fix a set of data, then we can define a number field K for a theory as the number field obtained from adjoining all numbers {F ijk lmn } and {R bc a } to the rational numbers Q ( { i = ±1} are already in Q). The automorphisms of the number field K fixing Q form the Galois group of K, denoted as G K . If g is an element of G K , then by applying g to all data, we get a potentially new theory. We will call the new theory a Galois conjugate or a Galois twist. For the Fibonacci theory, the minimal number fields required for the Galois conjugation for both the algebraic normalization and unitary normalization are worked out in Ref. 20 and needed below for the discussion of the projectors for code subspace property. For the algebraic normalization, the number field is the cyclotomic number field Q(ξ 20 ), where ξ N = e 2πi/N , while for the unitary normalization, the number field is Q( √ φ, ξ 20 ). It is known in general that a theory from quantum groups such as Fibonacci can always be defined within a cyclotomic field Q(ξ N ) for some N . For the Jones representation with the algebraic normalization, this is done explicitly by Kuperberg 21 . To explain this, we digress briefly to some basic quantum topology.
The Jones representation (and polynomial) may be constructed from the Kauffman bracket:
= −t 1/4 −t −1/4 = −t 1/2 − t −1/2 , , t indeterminate.
The skein space W (n · 1, 0, t) is the vector space of formal linear combinations of arc matchings (i.e. skeins) of an even number, n, of fixed points on the top of a square (the arcs are imbedded in the interior of the square and "0" means no points marked on the bottom of the square). An n-braid b acts on W , by gluing b onto the top of the square and resolving crossings by the above rule. The trick is to define each braid generator as t 1/4 times the geometric crossing. Additionally, each skein is "even" or "odd" according to whether a 2-coloring of the skein complement (starting with the bottom of the square being colored white) has an even or odd number of black regions. All black skeins should be multiplied by a factor of t 1/2 . This results in a basis for W and a rescaling of the action so that this "W -representation" is defined over the field Q[t]. The famous quantum representations of Jones at level k will be quotients of the W -representation for t = e 2πi k+2 . Note that the rescaling cannot affect the density of the projective Jones representation, which will be important shortly.
In the case at hand, Fib, t = e 2πi/5 , and the Galois conjugate theory Fib G is obtained by replacing t by t = e 4πi/5 .
The skein space W (n·1, 0, t) carries a natural bilinear form , obtained by doubling the square (thought of as a disk) along its boundary and evaluating the union of the two skeins as a scalar using the above Kauffman relations. When |t| = 1, the form A, B is Hermitian. If further t is a root of unity, then this form has a singular subspace. X(n · 1, 0, t) is, by definition, the finite dimensional Hilbert space obtained by annihilating this kernel. The Hermitian form A, B is nonsingular on X and the braid group B n acts.
When t = e 2πi k+2 , this quotient action is the Jones representation associated to SU (2) k , whose trace leads to the Jones polynomial evaluated at t. For t = e 2πi/(k+2) , the Hermitian form A, B is positive definite. For other roots of unity, A, B may be of mixed signs (p, q), p = 0, q = 0. This happens in particular for t = e 4πi/5 when n ≥ 4 as we now check.
Well-established conventions in mathematics and physics lead to two different ways to label the particle types in SU (2) k : one by the spins of the irreps, and the other by the dimensions of irreps minus one. Unless we speak explicitly of a spin label, as in the next paragraph, the labels of particles in this section are by the dimensions minus one, which are twice of the physical spins.
Note that X(n·2, 0, e 2πi/5 ) ∼ = X(n · 1, 0, e 2πi/5 ) as Hilbert spaces via the " " automorphism of SU (2) 3 : spin 0 = spin 3/2, spin 1/2 = spin 1, spin 1 = spin 1/2, spin 3/2 = spin 0. Similarly, X(n · 2, 0, e 4πi/5 ) ∼ = X(n · 1, 0, e 4πi/5 ) as Hilbert spaces of mixed sign. The braid group actions (Galois conjugates of the Jones representation) are, of course, also identical. For t = e 4πi/5 , the loop value is
d = −e 2πi/5 − e −2πi/5 = −1/φ, φ = 1+ √ 5
2 , the golden ratio. We use trivalent graphs with the Kauffman vertex normalization:
1/2 1/2 1/2 1/2 1/2 1/2 = 1 1 1
where is the Jones-Wenzl projector P2 − = 1 d , and 1/2 and 1 are the spins of the quasi-particles, to write an orthogonal basis for X(4 · 2, 0, t) (The ".2" after n indicates the second, i.e. spin 1, nontrivial particle type).
, = = 1 d 2 ,
is a positive number. Now consider an orthogonal basis element .
We have , = = 0.
Its self pairing is negative:
, = = θ d 2 d,
a negative number where d is the loop value −1 φ and the θsymbol = −φ (simplified from the formula in Figure 19 in Ref. 22). It follows that for n ≥ 4 the Hermitian structure on X(n · 2, 0, e 4πi/5 ) ∼ = X(n · 1, 0, e 4πi/5 ) has mixed signs. The corresponding braid representations for DFib G also have mixed signs when n ≥ 4.
The doubled Fibonacci theory DFib, as with all topological phases, has the code property: the composition G inc → H L − → H P − → G is multiplication by some scalar λ L ∈ C whenever L is a (sufficiently) local operator (P = inc † is the Hermitian orthogonal projection to the Levin-Wen ground state G, see Ref. 8). In general P is Hermitian, but for Fib P is actually real symmetric.
In the Levin-Wen Hamiltonian scheme, there are two kinds of terms: the vertex type A v for each vertex v, and a plaquette type B p for each plaquette p. The code space G is the common eigenspaces of all local operators {A v } and {B p }. With the standard choices of basis, the vertex terms A v are matrices with entries 0's and 1's, while the plaquette terms B p are matrices with entries given by products of 6j-symbols. The algebraic constraints defining the code subspace transform under a Galois conjugation. Hence solving the Galois conjugates of the constraints defining G, we obtain G G and replace the matrix P with its Galois conjugate P G . Clearly 1 ) is local. Thus G G retains a "code" property but with respect to a non-Hermitian projector P G . In the symmetric normalization of Yang-Lee, P G is complex symmetric with eigenvalues 0 and 1 (since
G G inc − − → H L − → H P G − − → G G is multiplication by (λ L (G −1 ) ) G since L is local if and only if L (G −(P G ) 2 = P G , however (P G ) † =P G = P G ).
We close this paragraph by noting that our proof uses the mixed signatures in the Jones braid group representations as shown above, and as such applies only to the algebraic normalization. For the symmetric normalization of the DYL theory, the Jones representation spaces have either positive or negative definite inner products, but the mixed sign in the algebraic normalization will be sufficient to prove our theorem for any choice of normalization.
B. Lieb-Robinson bounds and local unitary evolution of a ground state under changes in the Hamiltonian
Our proof of absence below will be based on a contradiction of the above result for the Galois conjugated theory with local unitary evolution of a ground state under local changes in a Hermitian Hamiltonian. This local unitary evolution can be proven for all Hermitian Hamiltonians that satisfy Lieb-Robinson bounds.
Lieb-Robinson bounds are a mathematical way of expressing the physical fact that in local lattice Hamiltonians there is some upper bound to the velocity of excitations. These bounds can be proven for a wide-range of Hamiltonians, including what would be colloquiually referred to as "Hamiltonians with finite-range interactions" or "Hamiltonians with exponentially decaying interactions".
For a precise statement of conditions under which Lieb-Robinson bounds can be proven, we follow Ref. 23 where one sufficient condition is given as follows (see also Ref. 24). We consider lattice Hamiltonians, and use i, j, ... to label sites of the lattice, with some metric dist(i, j) on the lattice. We use X, Y, Z, ... to label sets of sites of the lattice. Let the Hamiltonian H be written as
H = Z H Z ,(14)
where the operators H Z are supported on sets Z (an operator is said to be supported on a set Z if it can be written as a tensor product of an operator on the degrees of freedom on set Z with an identity operator on the remaining degrees of freedom). Assume that the following condition holds for all sites i,
X i H X |X| exp[µ diam(X)] ≤ s < ∞,(15)
for some positive constants µ, s, where diam(X) denotes the diameter of set X, and |X| denotes the cardinality of X, and H X denotes the operator norm. Then, 23 Eq. (15)
|t| ≤ dist(X, Y )/v LR , we have [A X (t), B Y ] ≤ v LR |t| l g(l)|X| A X B Y ,(16)
where l = dist(X, Y ) and g(l) decays exponentially in l.
Given that A X (t) has small commutator with all operators B Y with sufficiently large distance dist(X, Y ), this implies 25 that the operator A X (t) can be approximated by an operator A l X (t) which is supported on the set of sites within distance l = v LR |t| of the set X up to an error in operator norm which is bounded by v LR |t| l g(l)|X| A X .
Definition IV.1. We say that a Hamiltonian is a Lieb-Robinson Hamiltonian (or that it obeys a Lieb-Robinson bound) if a bound of the form Eq. (16) holds for some v LR and some exponentially decaying g(l). A parameter dependent family of Hamiltonians H s uniformly obeys a Lieb-Robinson bound if for some v LR and g(l) the bound Eq. (16) holds for all s. Such a family is called "uniformly LR".
We also want to define what it means for a Hamiltonian to have multiple ground states and a spectral gap. Note that it is common practice in physics to refer to a system, such as a fractional quantum Hall system which has three low-lying states with an exponentially small splitting between them and then a gap to the rest of the spectrum as having a "degenerate ground state", even though the non-vanishing splitting means that the lowest eigenvalue is in fact non-degenerate. Our definition will reflect this usage, as we will not require that the states that we refer to as "ground states" be degenerate. All we will require is that the "ground states" be separated from the rest of the spectrum by a gap. Note that we did not require in the above definition that the splitting E n−1 − E 0 between the different ground states be small. For all the systems we are concerned with, this splitting will turn out to be small, but since it is not required to be small for lemma IV.3, we do not include this in our definition (in some applications of Lieb-Robinson bounds, the splitting between different ground states is important, but we don't need it here).
The next lemma expresses how the ground states evolve under changes in the Hamiltonian. We consider some parameter dependent family of Hamiltonians, H s , for 0 ≤ s ≤ 1, and imagine this family as describing some path from an initial Hamiltonian at s = 0 to some final Hamiltonian at s = 1.
Stated roughly, this lemma shows that if the Hamiltonian is gapped and local, then the change in the ground state under a local change in the Hamiltonian can be expressed by a local operator acting on the ground state.
Lemma IV.3. Let H s be a uniformly Lieb-Robinson family of Hermitian Hamiltonians, for 0 ≤ s ≤ 1, with H s differentiable with respect to s, such that ∂ s H s is supported in a disk X of radius R and such that for some J, and ∂ s H s ≤ J for all s. Let H s have uniform gap ∆E. Let P (s) denote the Hermitian projector onto the ground state subspace of H s . Then, for any l, there exists a family of unitaries U s supported on the set of sites within distance l of X such that
U s P (0)U † s −P (s) ≤ const. J ∆E exp(−l∆E/2v LR )+g(l) .(17)
Proof. The proof largely follows previous results on quasiadiabatic continuation and is given in Appendix A for completeness.
We make a few remarks. First, note the appearance of g(l) in the lemma above. For a Hamiltonian with exponentially decaying interactions, g(l) will decay exponentially in l, but for a Hamiltonian with bounded range interactions, g(l) will decay faster than exponentially in l, and the error will be dominated by the term exp(−l∆E/2v LR ). Further, in the case of exponentially decaying interactions, the length scale over which g(l) decays will be set by the decay scale of the interactions in the Hamiltonian, i.e., by the microscopic details of the interaction rather than the magnitude of the spectral gap.
Note that in the lemma above, a bound on ∂ s H s appears; that is, the bounds depend upon how rapidly the Hamiltonian changes along the path. To give a physical explanation of why this appears, consider dragging an anyon along some path. Suppose we move the anyon a distance L. Then, since we always scale the path length to unity (that is, s ranges from 0 to 1), the "velocity" at which the anyon moves along the path is proportional to L. Thus, for larger L, we are moving the anyon more rapidly along the path, and so ∂ s H s will be larger; thus, in a sense the appearance of ∂ s H s is really a way of measuring the distance we drag the anyons. Thus, it is worth restating the result in a re-scaled way: suppose that we drag an anyon a distance of order the disk radius R. Then, typically we will have ∂ s H s ≤ J for a J of order R. So, given that the error in Eq. (17) is exponentially small in l, in such a case it suffices to choose l logarithmically large in R in order to make the error of order 1. For an l of order R, the error will be exponentially small in R.
Also, note that if ∂ s H s is approximately supported in X, in that it can be approximated, up to exponentially small error, by an operator supported in X, then we can derive a similar bound to Eq. (17) which will involve the error in approximating ∂ s H s . We omit this case.
Finally, in the case that the Hamiltonian H s is a sum of commuting terms with bounded-range for all s, the Lieb-Robinson velocity is zero. In this case, it is possible to show that for sufficiently large l, the error U s P (0)U † s − P (s) is exactly zero.
C. Proof of Absence
Examined in detail, the ground state manifold G G and the projector P G which defines it depends on: 1) the number and location (Γ) within the 2-sphere S 2 of the anyons, 2) the anyon particle type-a kind of boundary condition, and 3) the
F ijk l;nm = F ijk l;nm · f (j, k, n)f (i, n, l) f (i, j, m)f (m, k, l) .
Except that it would unpleasantly cluster the notation, we should write G G n,Γ,f and P G n,Γ,f . The detailed position Γ of the anyons within the lattice model is important to us since our proof will work with the entire "braid groupoid" B n . In fact, we treat Γ as a continuous variable on a compact space of 2n (real) dimensions. This moduli space of anyon position is compact since distinct anyons are not permitted to closely approach. The elements of B n are oriented paths of n-distinct (marked and framed) points in R 2 which compose only when end points match. B n represents in a large but finite dimensional Hilbert space H of microscopic degrees of freedom on S 2 , the north pole serving as a standard ∞ for R 2 . The vertex normalization f is also important within the proof. As we have already seen, the symmetric normalization yields a TQFT with all definite Hilbert spaces (though some are positive-definite and others are negative-definite). The proof of Theorem IV.5 requires as a "kernel" a single Hilbert space on which a non-singular form of mixed signs is preserved by B n . With this kernel in hand, the proof actually covers all vertex normalizations f . Definition IV.4. We call an operator L range r if it is supported on a ball of diameter r. Also, we use the same term for sums of such operators. Similarly, an operator is called weakly range r (in either sense) if it is range r up to exponentially small corrections. We say that an operator is short range if it is supported on a ball of diameter small compared to system size.
We say an operator O is a local normalizer iff there is some constant c which is small compared to system size such that OLO −1 is range r + c whenever L is range r. We say that an operator is a weakly local normalizer iff there is some constant c which is small compared to system size such that OLO −1 is weakly range r + c whenever L is range r.
A uniform family of (weakly) local normalizers O Λ is a parameter dependent family of operators such that O Λ LO −1 Λ is (weakly) range r +c whenever L is (weakly) range r, with a uniform bound on the exponentially small corrections and on the constant c, and such that whenever
|Λ − Λ | ≤ O(1), the product O Λ O −1
Λ is a product of at most O(1) operators which are all (weakly) range r and are not necessarily the same, for some r which is O(1). An example of a local normalizer is a finite depth quantum circuit of invertible (not necessarily unitary) local operators. An example of a uniform family of local normalizers is a family of finite depth quantum circuits of invertible local operators, such that an O(1) change in the parameter changes only O(1) different operators in the circuit; for the applications we have in mind, one should imagine that the parameter Λ refers to different anyon positions and that changing Λ changes the circuit only near the anyon positions.
In the definition of weakly local normalizer, it will be important to define how we quantify the error term in the approximation by a bounded range operator. The natural way to do this would be to require that the error term be small in operator norm compared to the operator norm of OLO −1 . However, for technical reasons, for use later we will be interested in a what we call a g.s.-weakly local normalizer (g.s. stands for ground state). In this case, we consider certain operators M (i) which have the property that M (i) is bounded range and exactly maps the ground state subspace of some non-Hermitian Hamiltonian to the ground state subspace of some other non-Hermitian Hamiltonian, with M (i) † M (i) exactly preserving the ground state subspace of the first non-Hermitian Hamiltonian and having its ground state expectation value equal to its norm. Then, we require that the error term be small in operator norm compared to the norm |O Λ Proof. The theorem uses the notation of reference 21 to describe the anyons in DFib. For now, fix the algebraic vertex normalization λ = f . Below we may suppress Γ and f from the notation when they play no role.
Suppose O Γ exists, thenO Γ G G Γ is a family of code subspaces and for Γ near Γ the subspaces are connected up to ex-ponentially small discrepancy by a local unitary U Γ,Γ (these are the U s of Lemma IV. 3
. Writing DFib
G f ∼ = Fib G f ⊗ Fib G f
(one may think DFib describes a bilayer), let us recall a theorem stated in 21 for the right hand factor Fib G f , where f is the algebraic normalization (Note: while DFib G f is a theory of string-nets on the surface S 2 , with boundary conditions at anyons, Fib G f is the corresponding string-net theory 26 in the 3ball with boundary S 2 . Thus the function f gauging vertices acts compatibly in both theories.)
Now, according to Ref. 21, corollaries 1.2.4 and 1.2.6, for n ≥ 5, the Jones representation ρ on the topologically defined Hilbert space V R of ground states for Fib G is (analytically) dense in a noncompact special unitary group (preserving a Hermitian metric of mixed signs) SU (p, q) := SU ((X, n · 1, 0, e 4πi 5 )) ∼ = SU ((X, n · 2, 0, e 4πi/5 )). Recall that in subsection IV A we confirmed that the loop value for a closed string in DFib G f was − 1 φ and that with the algebraic vertex normalization the signs of the B n -invariant ground state "Hilbert" spaces are indeed mixed: p > 0 and q > 0.
We need to formulate a lemma regarding the following concept.
Definition IV.6. An ambient groupoid representation is a functor from a groupoid to the category of subspaces of a fixed Hilbert space, and linear transformations on H carrying one image subspace to another. Then this data determines a unique projective ambient groupoid representation . That is, all compositions commute with the functor up to multiplication by a nonzero scalar.
Proof. Consider A L θ | A −→ B L φ | B
−→ C, L θ and L φ : H → H being the local operators carrying A to B and B to C respectively. The composition L φ · L θ is local and L φ · L θ (A) = C. By the code property there is a scalar λ so that Π A (L φ ·L θ ) −1 · (L φ · L θ ) · inc A = λ · Id, where λ = 0 since all morphisms L in the representation are invertible. (Π A denotes the appropriate projection to A corresponding to its code property.) Thus (projectively) the homomorphic property, restricted to the subspaces {A i }, i ∈ Γ, is redundant when these subspaces have the code property.
For uniqueness consider two possible representations L(θ) and L (θ). (L ) −1 L : H → H is local and carrying A to itself. Thus Π A (L θ ) −1 L θ · inc A : A → A is multiplication by some scalar λ which, again by the invertibility of L θ and L θ , is non-zero.
By assumption the collection {O(G G Γ )} is a code with respect to the usual Hermitian projection. Using the U s of Lemma IV.3 as the generating set of morphisms, Lemma IV.7 builds a projective representation ρ : B n × H → H of DFib up to exponentially small errors which we can neglect for the moment but return to shortly. We may think of this representation as the result of (quasi-)adiabatic evolution of G G Γ inside the finite dimensional microscopic Hilbert space H. Intuitively, braiding might be realized by building and slowly moving a potential trap term added to the Hamiltonian. For plaquette excitations (e.g. τ ⊗ τ ) such a trap could have the rough form H trap = arctan(δt)B p +(π−arctan(δt)B p + σ z i for plaquettes p and p separated by edge. Such family of Hamiltonians will adiabatically braid the anyons. Formally, however, we have posited, for contradiction, the family H(Γ) and this is all we need.
Lemma IV.3 provides local unitaries intertwining G G Γ1 and G G Γ2 , and so ρ is obviously a (projective) unitary representation with respect to the standard positive definite Hermitian structure on the space H of microscopic degrees of freedom. Thus ρ preserves ordinary lengths and angles (as measured in H) and the complex structure (multiplication by i) of H as well. So this ρ manufactured from Lemma IV.7 looks geometrically quite distinct from a second representation of B n , ρ := O(ρ ⊗ ρ * )O −1 . As explained above (also see ref. 26 ), ρ and ρ * also act on the same collection of string-net spaces as ρ and the uniqueness clause of Lemma IV.7 yields a projective isomorphisms ρ proj ∼ = ρ .
We learned from Ref. 21 that ρ is dense in SU (p, q), p > 0, q > 0, and that X(n · 2, 0, e 4πi 5 ) is the fundamental representation ω 1 of SU (p, q). ω 1 preserves a form of mixed signs and thus will distort not only Euclidean length but Euclidean angles as well by an unbounded amount (For example consider the effect of boosts in O(1, 1) ⊂ U (1, 1) on Euclidean angle). This difference, that ρ and ρ preserve forms of different signatures, excludes the existence of a (g.s.-weakly) local normalizer O transforming {G G Λ } to the ground state spaces of H(Λ).
We now consider in more detail the exponentially small errors that we have neglected. First some preliminaries. We will need the following lemma which provides a useful corollary of the disk axiom:
Lemma IV.8. Let X be some set and P some projector such that for any operator A supported on X there is a scalar z such that
P AP − zP ≤ A ,(18)
for some sufficiently small . (In typical applications, we have in mind that P is the projector onto the ground state subspace of some system, X is some set of small diameter, and the above equation encodes a soft form of the disk axiom for that theory). Then, for any operator O supported on set X there is a scalar w such that
P OP − wP ≤ C (1 − P )OP P O(1 − P ) ,(19)
for some constant C of order unity (the constant C is independent of , so long as is sufficiently small).
Proof. First, assume that O is Hermitian. Consider the operator U = exp(itO), for t real. Since U = 1, P U P −zP ≤ for some z. We can expand P U P in a power series in (1 − P )OP giving
P U P = P exp(itP OP )P + O(t 2 P O(1 − P ) 2 ). (20)
Suppose P OP = w + ∆, for some traceless operator ∆ and scalar w. Pick t = 2 / ∆ . Then,
P U P = P w+2i P ∆ ∆ P +O( 2 + 2 P O(1−P ) 2 / ∆ 2 ).
(21) So, the distance from P U P to the closest scalar multiple of P is at least 2
− O( 2 + 2 P O(1 − P ) 2 / ∆ 2 ). If ∆ is sufficiently large compared to √ P O(1 − P ) , then the term O(...)
is small compared to the leading term, assuming is sufficiently small (we need sufficiently small compared to unity so that term O( 2 ) is small). However, this contradicts the assumption that P U P is within distance of some scalar multiple of P . Now, consider the general case that O need not be Hermitian. Add an additional spin-1/2 degree of freedom to the system and consider the Hermitian operatorÕ = (O − w) ⊗ σ + +(O † −w)⊗σ − , where σ + , σ − are the raising and lowering operators for that spin and w is chosen so that P (O − w)P is traceless. The assumption (18) for the original theory implies that for the system with the additional spin-1/2, for any operator A acting on set X and on the additional spin-1/2 that
P AP − P ⊗ Q ≤ 4 A ,(22)
for some 2-by-2 matrix Q acting on the additional spin (to show the above equation, expand P AP as a sum of four product operators, one operator in the product acting on X and the other on the added spin, and apply Eq. (18) to each term in the product). Construct U = exp(itÕ). Perturbatively expand the σ + component of P U P for the same t as before. This is
2i P ∆ ∆ P ⊗σ + +O( 3 + 2 P O(1−P ) (1−P )OP / ∆ 2 ).(23)
We again get a contradiction as in the Hermitian case.
The representation ρ gives a mapping from braids to matrices. We will use M to refer to such a matrix. We can also construct a matrix U by taking the U s of lemma IV.3 for the corresponding braid and projecting into the grounds state subspace. So long as the length of the braid is smaller than some quantity growing exponentially with the linear size of the system, the matrix U will be approximately unitary (the error arises from leakage out of the ground state subspace in lemma IV.3. We claim that Lemma IV.9.
U − zOM O −1 << 1 (24)
for such braids for some scalar z. Indeed, the difference in norms is exponentially small in system size.
Proof. This difference follows from the disk axiom: we decompose the braid into n segments; taking n of order L, each segment moves the anyons only a short distance. In the nonunitary theory, we can construct short range operators which move the ground state subspace before the i-segment to the ground state subspace after that segment. Call these operators M (i), so that M is equal to the projection of M (n)...M (1) into the ground state subspace. This operator M (n)...M (1) exactly preserves the ground state subspace of the Galois conjugated theory and OM O −1 exactly preserves the ground state subspace of the unitary theory. We use P (i) to denote the projector onto the ground state subspace after the i-th segment, and we use is only approximately equal to P (i)U (i − 1) (the difference is exponentially small in system size). Note that on the right-hand side of this equation the matrices act in the ground state Hilbert space, while on the left-hand side they act in the full Hilbert space. To bound the right-hand side of Eq. (26), it suffices to show that, for all i,
M (i) = O Λ(i) M (i)O −1 Λ(i−1) ,(25)P (i)U (i)P (i − 1) − z(i)P (i)M (i)P (i − 1)(27)
is exponentially small for some scalar z(i). To see this, set
z = i z(i)(28)× n−1 i=1 P (i)M (i)P (i − 1) ≤ ...
Given this norm estimate (27), then since P (i)U (i)P (i − 1) is an approximate isometry from the range of P (i − 1) to the range of P (i), the matrix P (i)z(i)M (i)P (i − 1) is also such an approximate isometry, so the product of norms P (i)M (i)P (i − 1) above is bounded. So, we must bound Eq. (27). Since U i) is an approximate isometry, it suffices to bound P (i − 1) − P (i − 1)U (i) † z(i)M (i)P (i − 1) . At first sight, this seems to follow immediately from the disk axiom: since U (i) † M (i) is short range, or at least approximately short range (note that for M (i) this follows by the definition of a family of local normalizers, but see the next paragraph for a more careful treatment of error terms), by the disk axiom it is close to a scalar when projected into the ground state subspace. Hence, choosing z(i) to be the inverse of this scalar, the desired result seems to follow. However, there is a complication: suppose
P (i − 1)U (i) † M (i)P (i − 1) is within some distance of z(i) −1 P (i − 1) for some z(i); then, we bound P (i − 1) − P (i − 1)U (i) † z(i)M (i)P (i − 1) ≤ |z(i)|.
Hence, if z(i) is large, the resulting error can be large even if is small. This is why we will need the lemma (IV.8) above. By definition of g.s.-weakly local normalizer, the operators M (i) can be approximated by operators that are short range, up to an error that is small compared to |M (i)ψ| for all ψ in the ground state subspace P (i−1) with |ψ| = 1. Since U (i) is approximately unitary and an approximate isometry between two ground state subspaces, this means that U (i) † M (i) can be approximated by an operator O(i) that is short range, up to an error that is small compared to |U (i) † M (i)ψ| for ψ in P (i − 1). (Note that if O is a local normalizer, then M (i) already is short range so we can take
O(i) = U (i) † M (i) in that case.) So, for the O(i), (1 − P (i − 1))O(i)P (i − 1)
is small compared to |O(i)ψ| for all ψ. Applying lemma (IV.8), this means that P (i − 1)O(i)P (i − 1) is close to z(i)P (i − 1), for some z(i) up to an error that is small compared to z(i).
We can find a braid such that the corresponding matrix M is diagonalizable, and with the ratio between its largest and smallest eigenvalue being at least 2 in absolute value 35 . This means that the ratio between the largest and smallest eigenvalue of zOM O −1 is at least 2 in absolute value. However, zOM O −1 is close to a unitary matrix. All eigenvalues of a unitary matrix are on the unit circle in the complex plane. Further, a small perturbation of a unitary matrix leaves all of its eigenvalues close to the unit circle, so zOM O −1 must have all of its eigenvalues close to the unit circle contradicting the assumption on the ratio of eigenvalues. Now we remove the condition on the vertex normalization or "choice of gauge". As noted as above the gauge choice is a function f : L 3 → C\0. Although this is not crucial, it is pleasant that in the Fib case f is identically 1 except for taking value f (τ, τ, τ ) = λ on the essential trivalent vertex. Recalling the A v and B p terms in the Levin-Wen model Hamiltonian, the fusion rule terms, A v , commute with and do not depend on f . The effect of f on B p may be computed (using the compatibility of gauge choice and the F -matrices used to construct B p ):
B p,λ = F λ · B p,alg. · F −1 λ ,(30)
where F λ is the "relative fugacity matrix", F λ : H → H. F λ is diagonal in the string-net basis {k} and the entry F k,k is simply λ #(k) , where #(k) denotes the number of (τ, τ, τ )vertices in the k th string-net.
As a concrete example of the above gauge dependence of B p in the Fib case
F τ τ τ τ ;sym = 1 0 0 λ −2 F τ τ τ τ ;alg 1 0 0 λ 2 .(31)
By inspecting the two F -matrices in the two normalizations, we have λ 2 = iφ 5/2 . Given a trivalent graph γ, and a labeling of its edges k γ . Let {Ψ alg.,kγ } and {Ψ sym.,kγ } be the two basis of the Hilbert space ⊗ e∈γ C #(L) of labeled graphs. Then Ψ sym.,kγ = λ #(kγ )Ψ alg.,kγ . Suppose B p,alg. Ψ alg.,kγ = k γ B p,alg.,k γ ,kγ Ψ alg.,k γ ,
noting that gauge change and recoupling are commutative, we obtain B p,sym. λ −#(kγ ) Ψ sym.,kγ = k γ B p,alg.,k γ ,kγ λ −#(k γ ) Ψ sym.,k γ .
(33) Thus B Γ,sym has the claimed conjugated form. Similarly for any vertex fugacity λ, Eq. (30) holds. Thus for a general relative fugacity λ, G Γ,λ = F λ G Γ,alg.
Observe that while F λ is not local (in the sense of having supported on a disk of bounded radius), both F λ and F −1 λ are implemented by a depth 1 invertible circuit, F λ = sites (λΠ τ τ τ + (1 − Π τ τ τ )). Thus the general local normalizer operator O may be written as O = O • F λ , where O is also local normalizer; given λ, O, and O determine each other uniquely, we have just shown that for all local normalizer O there is a local L so that L acts on OG G alg , i.e. that
inc † G G alg • O † • L • O • inc G G alg = scalar.(34)
It follows that for all O there is an L (identical to L above) so that: (1) ).
inc G G alg • F † λ • O † • L • O • F λ • inc G G alg = scalar. But inc G G alg •F † λ •O † •L•O •F λ •inc G G alg = inc G G λ •O † •L•O •inc G G λ .(35)
Although we have concentrated the discussion on the Fib TQFT, its quantum double, and their Galois conjugates, the proof requires only two ingredients: 1) finding pairs of Galois conjugate theories (with choice of vertex gauge) one of which is unitary (for the Hilbert space of a sphere or plane with fixed anyon content) and one of which is unitary with respect to a Hermitian metric of mixed signs (p, q), p > 0, q > 0, and 2) establishing denseness of the braid group representations in SU (p, q). Using just the results for SU (2)-theories obtained in Ref. 21, infinitely many other unitary theories arise which have Galois conjugates satisfying Theorem IV.5.
V. COMMENTS ON NON-HERMITIAN HAMILTONIANS
While the bulk of our paper is devoted to showing that certain wave functions cannot be the ground states of gapped Hermitian Lieb-Robinson Hamiltonians, it is worth briefly discussing what is known about non-Hermitian Hamiltonians. For non-Hermitian Hamiltonians, many of the technical tools involving Lieb-Robinson bounds are unavailable, and so many results that we know in the Hermitian case are not known here.
The first major difficulty in the non-Hermitian case is that even if the Hamiltonian is a sum of terms H Z which obey Eq. (15), if the Hamiltonian is not Hermitian, then we do not know if the Lieb-Robinson bound holds. Similarly, for a Hermitian Hamiltonian, the Lieb-Robinson bound might not hold for evolution in imaginary time. Given that the Lieb-Robinson bound fails, we are also unable to prove locality of correlation functions in a non-Hermitian Hamiltonian even if there is a gap in the spectrum.
The fact that we cannot prove locality of correlation functions is relevant to the following application of the disk axiom. Suppose we have a Hermitian Hamiltonian which obeys the disk axiom with P being the projector into the ground state subspace. Suppose operators O X and O Y are supported on small disks X and Y such that the disk axiom implies that P O X P and P O Y P are both close to scalar multiples of P . Now, let us ask whether the operator P O X O Y P is also close to a scalar multiple of P . Consider the case in which the disks X and Y are far separated such that the smallest disk containing both X and Y is too large to directly apply the disk axiom to O X O Y . Thus, the disk axiom alone does not tell us that O X O Y is close to a scalar when projected into the ground state subspace. However, if we have a gapped, Hermitian, Lieb-Robinson Hamiltonian then correlations decay exponentially in any ground state, so that P O X O Y P is close to P O X P O Y P , and then applying the disk axiom to P O X P and P O Y P implies that P O X O Y P is close to a scalar multiple of P . Unfortunately, though, in the non-Hermitian case we do not know that there is exponential decay of correlation functions, and so even if we assume the disk axiom for small disks, we do not see how to prove that O X O Y is also close to a scalar when projected into the ground state subspace of a non-Hermitian Hamiltonian obeying the disk axiom. In fact, suppose we consider two states ψ 1 , ψ 2 such that any operator supported on a small disk is equal to a scalar when projected into the space spanned by ψ 1 , ψ 2 . Let us relax any requirement that ψ 1 , ψ 2 be ground states of a Hamiltonian, whether Hermitian or not, and simply take them to be arbitrary states. Then, we can give an example in which product operators of the form O X O Y given above are not close to a scalar when projected into the ground state subspace, even though both O X , O Y are close to scalars when projected into this subspace, by considering a quantum error-correcting code on a small number of qubits, and defining the two states ψ 1 , ψ 2 on the large lattice by placing the qubits defining the code on far separated sites of the lattice, and placing all other qubits in a product state.
VI. CONCLUSIONS
To summarize, we have shown that a large class of nonunitary topological quantum field theories cannot be realized as ground states of Hermitian (quantum mechanical) Hamiltonians that satisfy a Lieb-Robinson bound. This includes but is not limited to local and quasi-local (exponentially decaying) Hamiltonians. While our proof has been formulated for quantum doubles of TQFTs, it also rules out the realization of the constituent non-doubled TQFT in a Hermitian system: if the latter were to exist, it could be used to trivially construct a Hermitian model for the corresponding quantum double.
The TQFTs covered by our proof include, in particular, the Galois conjugates of Fib and su(2) k TQFTs for k = 3 and all k ≥ 5. Among these, one case of special recent interest is the Yang-Lee TQFT (the Galois conjugate of Fib) underlying the proposed Gaffnian quantum Hall wave function. Our argument implies that this Gaffnian wave function cannot occur as ground state of a gapped fractional quantum Hall state (described by a Hermitian Hamiltonian), if one considers that the screened Coulomb interaction satisfies a Lieb-Robinson bound.
ACKNOWLEDGMENTS
We acknowledge discussions with E. Ardonne, B. Bauer, A. Ludwig, and K. Walker. We thank the Aspen Center for Physics. Our simulations used some of the ALPS libraries 28,29 and partly also the ARPACK library. 30 Data evaluation has been performed using the ALPS libraries and the VisTrails scientific workflow and provenance management system. 31 Full provenance information and workflows to recreate the figures are available by following the hyperlinks associated with each figure.
Appendix A: Proof of Lemma IV. 3 We define the "quasi-adiabatic continuation operator" D s by iD s = dtF (t) exp(iH s t) ∂ s H s exp(−iH s t), (A1) where the function F (t) is defined by (we follow 32 , while a more complicated choice of F (t) was used in 33,34 ):
i α √ 2π ∞ t du exp(−u 2 /2α 2 ),(A2)
for t > 0 and F (t) for t < 0 is defined by F (t) = −F (−t).
The quantity α in Eq. (A2) is some constant chosen below. We now show that ∂ s P (s) is close to i[D s , P (s)], and bound the difference between the two expressions in operator norm. DefineF (ω) to be the Fourier transform of F (t). One may show that |ω| ≥ ∆E → |F (ω)−1/ω)| ≤ const.×(1/∆E) exp(−α 2 ω 2 /2).
(A3) Let ψ i (s) denote eigenvectors of H s with eigenvalues E i (s), so P (s) =
We also have a bound on the time decay of F (t):
FIG. 1 .
1Edge labeling for a plaquette of the honeycomb lattice.
rules
FIG. 2 .
2The F -symbol.
3. (color online) The t-deformation parameters of Fibonacci and Yang-Lee anyons correspond to different primitive roots of unity.
FIG. 4 .FIG. 5 .
45(color online) Scaling of the finite-size gap ∆(L) (in units of Jp) with linear system size for the Hermitian projector model H herm on two different lattice geometries: the honeycomb lattice with L×W plaquettes (top panel) and 2-leg ladder systems of length L (bottom panel). Edge labeling for a plaquette of the ladder lattice.
FIG. 7 .
7-Hermitian DYL model FIG. 6. (color online) Ground-state degeneracy splitting of the non-Hermitian doubled Yang-Lee model when perturbed by a string tension (θ = 0). (color online) Ground-state degeneracy splitting of the Hermitian model H herm , the counterpart to the DYL model, when perturbed by a string tension (θ = 0) (top panel). The slope of the splitting around the unperturbed model (θ = 0) is given in the inset (top panel) for different system sizes L.
FIG. 8 .
8(color online) The low-energy spectra of the doubled Yang-Lee model (top) its Hermitian counterpart (bottom) for a wide range of coupling parameters. Data shown is for a ladder of length L = 8. of 6j-symbols {F ijk lmn }, braiding eigenvalues {R bc a } (not always necessary), and some pivotal coefficients { i = ±1}, where i, j, k, l, m, n, a, b, c are anyon types (see chapter 4 of Ref. 19)
implies the following Lieb-Robinson bound for Hermitian Hamiltonians. For any operator O, we use O(t) to denote the Heisenberg time evolution of the operator: O(t) = exp[iHt]O exp[−iHt]. Let A X , B Y be operators supported on sets X, Y , respectively. Then there is a constant v LR depending only on s, µ such that for t real with
Definition IV.2. A Hamiltonian has n ground states and a spectral gap ∆E, if E n−1 + ∆E ≤ E n where the eigenvalues of the Hamiltonian are E 0 , E 1 , ... with E 0 ≤ E 1 ≤ .... A family of Hamiltonians H s has n ground states and a uniform spectral gap ∆E if E n−1 (s) + ∆E ≤ E n (s) for all s, where the eigenvalues of H s are E 0 (s) ≤ E 1 (s) ≤ ...
(possibly non-unitary) trivalent vertex normalization f : L 3 → C\0 or gauge choice, L being the label set. For Fib, DFib, and their Galois conjugates and time reversals (represented by¯), f is always symmetric and satisfies a consistency relation with the F -symbols: suppose { F ijk l;nm } are new 6j symbols from {F ijk l;nm } by a gauge change {f (a, b, c)}, a, b, c ∈ L, then
(i+1) M (i)O −1 Λ(i) ψ| for ψ in the ground state of some other Hermitian Hamiltonian (this ground state subspace is obtained by applying O to the ground state subspace of the Hermitian Hamiltonian). Note that if O were an isometry, then the norm |O Λ(i+1) M (i)O −1 Λ(i) ψ| would equal the norm of M and so this would reduce to the more natural definition. Note also that any local normalizer is a g.s.-weakly local normalizer. Theorem IV.5. Fixing the number n ≥ 5 and particle type τ ⊗ τ of DFib anyons on S 2 and any vertex normalization f there can be no continuous uniform Γ-family of (g.s.-weakly) local normalizer operators O Γ : H → H, so that O Γ G G n,Γ,f is, for all anyon positions Γ the ground state manifold of a uniformly Lieb-Robinson and uniformly gapped family of Hermitian Hamiltonians H(Γ) defining a topological phase [see Eq. (1)].
Thus to objects a, b of the groupoid we assign spaces A ⊂ H, B ⊂ H and to a morphism a → b a unitary map H → H carrying A to B. In our case, the groupoid is B n with Γ being the objects and motions of anyons being the morphisms. The subspaces are the respective ground states for H G LW,alg.the Galois conjugated Levin-Wen Hamiltonian with algebraic vertex normalization. Lemma IV.7. Let {A i }, i ∈ Γ be the set of code subspaces of fixed Hilbert space H and B = {Γ, {→}} a groupoid. Suppose for a generating set of morphisms in B, a θ → b, there are invertible local operators L θ : H → H with LA = B (A=image a and B=image b).
where O Λ(i) is the operator O Λ corresponding to the position of the anyons after the i-th segment. so that M (i)P (i − 1) = P (i)M (i) and P (n) = P (0). In the unitary theory, let U (i) denote the unitary matrices from lemma IV.3 for the motion along the i-th segment, so that U is equal to the projection into the ground state subspace of U (n)...U (2)U (1). Note that U (n)...U (1) preserves the ground state subspace up to exponentially small exponentially small error and the matrices U(i) are bounded range. Thus, U − zOM O −1 (26) = P (0)U (n)U (n − 1)...U (1)P (0) −zP (0)M (n)M (n − 1)...M (1)P (0) ≈ P (0)U (n)P (n − 1)U (n − 1)P (n − 2)...U (1)P (0) −zP (0)M (n)P (n − 1)M (n − 1)P (n − 2)...M (1)P (0) , where the left side of this approximate equality differs from the right by inserting additional factors of P (i − 1) after every occurrence of U (i) or M (i) on the right-hand side. We have P (0)M (n)M (n − 1)...M (1)P (0) = P (0)M (n)P (n − 1)M (n − 1)P (n − 2)...M (1)P (0), but the error in Eq. (26) occurs because U (i)P (i − 1)
(
E i − E j )|Ψ j (s) Ψ j (s)|∂ s H s |Ψ i (s) Ψ i (s)| + h.c. i − E j |Ψ j (s) Ψ j (s)|∂ s H s |Ψ i (s) Ψ i (s)| + h.c.(A5)Thus, by Eqs.(A3,A4,A5)∂ s P (s) − [iD s , P (s)] ≤ const. × ( ∂ s H s /∆E) exp(−α 2 ∆E 2 /2).
So we find that for the change of variables O , L acts on O •inc G G λ . Varying O over all local normalizer operators produces a local normalizer O = O • F −1 λ . Thus for every possible local normalizer operator O there is a local L : H → H acting on O G G λ . This completes the proof of Theorem IV.5 by removing the hypothesis of algebraic vertex normalization. Theorem IV.5 immediately implies the following corollary Corollary IV.10. Let {G G Γ,f } be the ground state manifolds for the Galois conjugated Levin-Wen Hamiltonian H G Γ,f for n ≥ 5 τ ⊗ τ -anyons on the 2-sphere S 2 with positions Γ and any vertex normalization f , within a larger Hilbert space H of microscopic (lattice) degrees of freedom. There can be no continuous uniform family of local normalizer operators O Γ so that {O Γ H G Γ,f O −1 Γ } are uniformly gapped uniformly Lieb-Robinson Hamiltonians determining topological ground states {O Γ G G Γ,f }, in the sense of TQO-1 12 (i.e. satisfying the code property
We now define iD l s to be an approximation to iD s supported on the set of sites within distance l of X. To construct this approximation, we use the Lieb-Robinson bound and setwhere ∂ s H s l (t) is supported on the set of sites within distance l of X and is the approximation to exp(iH s t) ∂ s H s exp(−iH s t) given by the Lieb-Robinson bound. We now bound the difference D s − D l s . This difference is bounded byUsing the Lieb-Robinson bound and Eq. (A7) we arrive atCombining Eqs. (A6,A10), and choosing α = l/v LR ∆E we find thatFinally, we define U (s) by U (0) = I, the identity operator, and ∂ s U (s) = iD s U (s). Since iD s is Hermitian by construction, U (s) is unitary. Eq. (17) follows from Eq. (A11).
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a ratio between largest and smallest singular values which grows exponentially with l. This exponential growth actually requires some care to show since the representation is non-unitary. fact, it is possible to find a braid of length l such that the corresponding matrix M in the representation ρ ⊗ ρ * has. and it does not appear to follow directly from the fact that ρ is dense; we have been able to prove it using a form of the Solovay-Kitaev algorithm. However, since the particular exponential growth does not play any role in the proof we omit the detailsIn fact, it is possible to find a braid of length l such that the cor- responding matrix M in the representation ρ ⊗ ρ * has a ratio be- tween largest and smallest singular values which grows exponen- tially with l. This exponential growth actually requires some care to show since the representation is non-unitary, and it does not appear to follow directly from the fact that ρ is dense; we have been able to prove it using a form of the Solovay-Kitaev algo- rithm. However, since the particular exponential growth does not play any role in the proof we omit the details.
|
[] |
[
"Positive Definiteness of Paired Symmetric Tensors and Elasticity",
"Positive Definiteness of Paired Symmetric Tensors and Elasticity"
] |
[
"Zheng-Hai Huang [email protected] ",
"Liqun Qi \nDepartment of Applied Mathematics\nThe Hong Kong Polytechnic University\nHung HomKowloon, Hong Kong\n",
"\nSchool of Mathematics\nTianjin University\n300354TianjinP.R. China\n"
] |
[
"Department of Applied Mathematics\nThe Hong Kong Polytechnic University\nHung HomKowloon, Hong Kong",
"School of Mathematics\nTianjin University\n300354TianjinP.R. China"
] |
[] |
In this paper, we consider higher order paired symmetric tensors and strongly paired symmetric tensors. Elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors. A (strongly) paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite. Positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics, which plays an important role in nonlinear elasticity theory. We mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric tensors. We first show that the concerned (strongly) paired symmetric tensor is positive definite if and only if its smallest M -eigenvalue is positive. Second, we propose several necessary and sufficient conditions under which the concerned (strongly) paired symmetric tensor is positive definite. Third, we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric tensor can be written as a sum of squares of polynomials, and further, propose several necessary and/or sufficient conditions to judge whether the concerned (strongly) paired symmetric tensors are positive definite or not. Fourth, by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest M -eigenvalue of a fourth order three dimensional (strongly) paired symmetric tensor, by which we can check positive definiteness of the concerned tensor. The preliminary numerical results confirm our theoretical findings.
|
10.1016/j.cam.2018.01.025
|
[
"https://arxiv.org/pdf/1705.04315v3.pdf"
] | 4,414,569 |
1705.04315
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50174fb6b800a0d8d51d020d6e3ce4ecf3eb5f1b
|
Positive Definiteness of Paired Symmetric Tensors and Elasticity
4 Jul 2017 July 5, 2017
Zheng-Hai Huang [email protected]
Liqun Qi
Department of Applied Mathematics
The Hong Kong Polytechnic University
Hung HomKowloon, Hong Kong
School of Mathematics
Tianjin University
300354TianjinP.R. China
Positive Definiteness of Paired Symmetric Tensors and Elasticity
4 Jul 2017 July 5, 2017This author's work was partially supported by the Hong Kong Research Grant Council (Grant No. PolyU 501913, 15302114, 15300715 and 15301716). 1Paired symmetric tensorelasticity tensorpositive definiteness of tensorM -eigenvaluesemidefinite relaxation
In this paper, we consider higher order paired symmetric tensors and strongly paired symmetric tensors. Elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors. A (strongly) paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite. Positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics, which plays an important role in nonlinear elasticity theory. We mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric tensors. We first show that the concerned (strongly) paired symmetric tensor is positive definite if and only if its smallest M -eigenvalue is positive. Second, we propose several necessary and sufficient conditions under which the concerned (strongly) paired symmetric tensor is positive definite. Third, we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric tensor can be written as a sum of squares of polynomials, and further, propose several necessary and/or sufficient conditions to judge whether the concerned (strongly) paired symmetric tensors are positive definite or not. Fourth, by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest M -eigenvalue of a fourth order three dimensional (strongly) paired symmetric tensor, by which we can check positive definiteness of the concerned tensor. The preliminary numerical results confirm our theoretical findings.
Introduction
For any positive integers m and n, an m-th order n dimensional real tensor can be denoted by
a i 1 i 2 ···im x i 1 x i 2 · · · x im .(1)
We denote the degree of a polynomial p by deg(p). The polynomial p defined by (1) is said to be positive semidefinite if p(x) ≥ 0 holds for all x ∈ R n ; and p is said to be positive definite if p(x) > 0 holds for all x ∈ R n \{0}. Obviously, for nonzero tensors, m being an even integer is a necessity for positive semidefiniteness. A tensor is said to positive (semidefinite) definite if its corresponding homogeneous polynomial is positive (semidefinite) definite. Positive (semidefiniteness) definiteness of the polynomial is very important in many areas, which is related to the Hilbert seventeenth problem [24]. For an arbitrary 2m-th order n dimensional tensor denoted by A = (a i 1 j 1 i 2 j 2 ···imjm ), its indices can be divided into m adjacent blocks {i 1 j 1 }, . . . , {i m j m }. If entries of A are invariant under any permutation of indices in every block {i l j l } for l ∈ [m], i.e., a i 1 j 1 i 2 j 2 ···imjm = a j 1 i 1 i 2 j 2 ···imjm = a i 1 j 1 j 2 i 2 ···imjm = · · · = a i 1 j 1 i 2 j 2 ···jmim , then A is called a 2m-th order n dimensional paired symmetric tensor. It is well known that the most important representatives for fourth order three dimensional paired symmetric tensors are: the piezooptical tensor, the second order electrooptical effect (Kerr effect), electrostriction and second order magnetostriction [5]. Furthermore, if a paired symmetric tensor additionally satisfies block symmetry, i.e., a i 1 j 1 i 2 j 2 ···imjm = a i 2 j 2 i 3 j 3 ···i 1 j 1 = a i 3 j 3 i 4 j 4 ···i 2 j 2 = · · · = a imjmi 1 j 1 ···i m−1 j m−1 , then A is called a 2m-th order n dimensional strongly paired symmetric tensor. The most significant representative of fourth order three dimensional strongly paired symmetric tensors is the elasticity tensor, in which the pairwise permutability is based on the reversibility of mechanical deformation work [5]. In the elasticity tensor and higher order elasticity tensor [6,9,29], every entry a i 1 j 1 i 2 j 2 ···imjm is called an m-th order elastic constant, which is an important quantity in studies of elasticity theory. Positive definiteness of the elasticity tensor is called strong ellipticity, which plays an important role in elasticity theory and has been studied extensively (see, for example, [2,4,10,11,15,20,25,26,27,28,31,33]).
In this paper, we consider positive definiteness of higher order three dimensional (strongly) paired symmetric tensors. For simplicity of symbols, we only investigate some properties of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric tensors, mainly in positive definiteness of the concerned tensors. The results we obtained can be similarly extended to the case of more higher order (strongly) paired symmetric tensors. Some basic properties of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric tensors are given in the next section.
Eigenvalues of higher order tensors, introduced by Qi [18] and Lim [13], have been studied extensively in the recent years [1,7,19,21]. The concept of M -eigenvalue for fourth order paired symmetric tensor was introduced in [4,20] and further studied in [32]. In Section 3, we extend the concept of M -eigenvalue to sixth order three dimensional (strongly) paired symmetric tensors and bi-block symmetric tensors, and further discuss some related properties. In particular, we show that a sixth order three dimensional (strongly) paired symmetric tensor is positive definite if and only if its smallest M -eigenvalue is positive.
Positive definiteness of polynomials has been being an important issue in many areas, which has been discussed extensively. In [4,17,20], the authors studied positive definiteness conditions of fourth order paired symmetric tensors, which plays an important role in elasticity theory. In Section 4, following the ideas given in [4,20], we further discuss positive definiteness of fourth order three dimensional (strongly) paired symmetric tensors and propose several necessary and sufficient conditions for which the concerned tensor is positive definite. Furthermore, we extend the related results to the case of sixth order three dimensional (strongly) paired symmetric tensor.
A polynomial with real coefficients is called a sum of squares (SOS) if it can be expressed as a sum of several squares of polynomials with real coefficients [12,16]. It is obvious that a polynomial is positive semidefinite if it is an SOS polynomial. In Section 5, we investigate the SOS properties of polynomials defined by a fourth order three dimensional or a sixth order three dimensional (strongly) paired symmetric tensor. We give several necessary and/or sufficient conditions of a fourth order three dimensional (strongly) paired symmetric tensor being an SOS tensor, and propose several conditions under which a fourth order three dimensional (strongly) paired symmetric tensor is positive definite. In particular, we extend the related results to the case of sixth order three dimensional (strongly) paired symmetric tensor.
In [8], the authors introduced the tensor conic linear programming problem and proposed a sequential semidefinite programming method to solve it. As an application, they showed that their method can be applied to find the smallest Z-eigenvalue of a symmetric tensor. In Section 6, by using the special structure of the (strongly) paired symmetric tensor, we propose a sequential semidefinite programming method to compute the smallest M -eigenvalue of a fourth order three dimensional (strongly) paired symmetric tensor, which is an extension of the method proposed in [8]. By this method, we can check whether a fourth order three dimensional (strongly) paired symmetric tensor is positive definite or not.
In Section 7, we give some numerical results of our methods for judging whether a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric tensor is positive definite or not. The preliminary numerical results are consistent with our theoretical results.
Some concluding remarks are made in Section 8.
In the remaining parts of our paper, we will simply call a three dimensional strongly paired symmetric tensor an elasticity tensor as the main motivation of our paper is the strong ellipticity of elasticity and higher order elasticity tensors.
Preliminaries
In this section, we consider fourth order three dimensional and sixth order three dimensional paired symmetric (elasticity) tensor and discuss related basic properties.
Fourth order paired symmetric tensor. For any
A = (a ijkl ) ∈ T 4,3 , if a ijkl = a jikl = a ijlk = a jilk , ∀i, j, k, l ∈ {1, 2, 3},
then A is a paired symmetric tensor A ∈ T 4,3 ; and if a ijkl = a jikl = a ijlk = a jilk and a ijkl = a klij , ∀i, j, k, l ∈ {1, 2, 3}, then A is a fourth order three dimensional elasticity tensor [5].
For any tensor A = (a ijkl ) ∈ T 4,3 , the corresponding biquadratic form is defined by
A x 2 y 2 := 3 i,j,k,l=1 a ijkl x i x j y k y l , ∀x, y ∈ R 3 .(2)
Define
C := {A ∈ T 4,3 : A x 2 y 2 ≥ 0, ∀x, y ∈ R 3 }.(3)
Then, by a similar way as those in [22,23], we can obtain the following results.
Proposition 2.1. Suppose that the polynomial A x 2 y 2 and the set C are defined by (2) and (3), respectively. Then, the following statements hold.
(i) The interior of C, denoted by intC, is nonempty, and
intC = {A ∈ T 4,3 : A x 2 y 2 > 0, ∀x, y ∈ R 3 \{0}}.(4)
(ii) The set C is a pointed closed convex cone.
Proof. (i) It is obvious that C has nonempty interior. We show that (4) holds. On the one hand, if A ∈ C is not positive definite, then there exists two nonzero vectors x, y ∈ R 3 such that A x 2 y 2 = 0; and hence, for any ǫ > 0,
(A − ǫE )x 2 y 2 = −ǫ(x ⊤ x)(y ⊤ y) < 0 where E = (e ijkl ) ∈ T 4,3 is defined by e ijkl := 1 if i = j, k = l, 0 otherwise, ∀i, j, k, l ∈ {1, 2, 3}.(5)
This implies that A ∈ intC; and hence, intC ⊆ {A ∈ T 4,3 : A x 2 y 2 > 0, ∀x, y ∈ R 3 \{0}}. On the other hand, if A ∈ intC, then there exists a sequence {(B (k) , ǫ (k) )} satisfying B (k) HS = 1 and
ǫ (k) > 0 for all k ∈ {1, 2, . . .} such that A + ǫ (k) B (k) ∈ C for all k ∈ {1, 2, . . .} and lim k→∞ ǫ (k) = 0, which leads to that there exist {x (k) }, {y (k) } ⊆ R 3 satisfying x (k) = 1 and y (k) = 1 for all k ∈ {1, 2, . . .} such that A + ǫ (k) B (k) x (k) 2 y (k) 2 ≤ 0.
Let x * , y * be the limiting points of {x (k) } and {y (k) }, respectively. Then, x * = 1, y * = 1 and
A (x * ) 2 (y * ) 2 ≤ 0, which implies that A is not positive definite; and hence, {A ∈ T 4,3 : A x 2 y 2 > 0, ∀x, y ∈ R 3 \{0}} ⊆ intC. So, (4) holds.
(ii) For any A , B ∈ C and α, β ≥ 0, let C := αA + βB. Then, for any x, y ∈ R 3 , C x 2 y 2 = (αA + βB)x 2 y 2 = αA x 2 y 2 + βBx 2 y 2 ≥ 0, which implies that C is a convex cone. For any A = (a ijkl ) ∈ C, if −A ∈ C, then it follows from (2) that
3 i,j,k,l=1 a ijkl x i x j y k y l ≡ 0, ∀x, y ∈ R 3 ,
which yields that A = 0 by arbitrariness of x and y. This implies that C is a pointed cone. For any {A (k) } ⊆ C and lim k→∞ A (k) = A , it is easy to see that for any x, y ∈ R n ,
A x 2 y 2 = lim k→∞ A (k) x 2 y 2 ≥ 0,
which implies that C is a closed cone. Therefore, by combining (i) with (ii), we conclude that the results of proposition hold. In the following, we define a class of matrices which is related to tensor A ∈ T 4,3 .
Definition 2.2. For any A = (a ijkl ) ∈ T 4,3 , we define a matrix by
M = (m st ) with m st = a isitjsjt ∀s, t ∈ {1, 2, . . . , 9}(6)
where i 1 i 2 · · · i 9 and j 1 j 2 · · · j 9 are two arbitrary permutations of 123123123. We say that the matrix M defined by (6) is an unfolded matrix of tensor A with respect to indices i 1 i 2 · · · i 9 and j 1 j 2 · · · j 9 .
Obviously, there exists a unique unfold matrix of tensor A for each pair of permutations of i 1 i 2 · · · i 9 and j 1 j 2 · · · j 9 . The following result is easy to be obtained. Proposition 2.3. Suppose that A ∈ T 4,3 is a paired symmetric tensor, and M is its unfolded matrix defined by Definition 2.2. Then, the matrix M is symmetric.
For any A ∈ T 4,3 , we define
• matrix M 1 = (m st ) ∈ R 9×9 by M 1 := (m 1 st ) with m 1 st = a 3(i−1)+k,3(j−1)+l = a ijkl for any i, j, k, l ∈ {1, 2, 3};(7)
• and matrix M 2 = (m st ) ∈ R 9×9 by
M 2 := (m 2 st ) with m 2 st = a 3(k−1)+i,3(l−1)+j = a ijkl for any i, j, k, l ∈ {1, 2, 3}.(8)
Then, M 1 and M 2 are two unfolded matrices of A . Moreover, for any i, j, k, l ∈ {1, 2, 3}, we define two block sub-matrices of tensor A by
A ij := (a ijkl ) kl and B kl := (a ijkl ) ij .(9)
Then, we can easily obtain the following results.
Proposition 2.4. For any A ∈ T 4,3 , suppose that matrices M 1 , M 2 , A ij and B kl are given by (7), (8) and (9), respectively.
(i) If A ∈ T 4,3 is a paired symmetric tensor, then the following statements hold.
(a) The matrices M 1 and M 2 are symmetric, and
M 1 = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 and M 2 = B 11 B 12 B 13 B 21 B 22 B 23 B 31 B 32 B 33 ;(10)
(b) All sub-matrices A ij and B kl are symmetric.
(ii) If A ∈ T 4,3 is an elasticity tensor, then M 1 = M 2 .
2.2 Sixth order paired symmetric tensor. A = (a ijklpq ) ∈ T 6,3 is a paired symmetric tensor if its entries satisfy a ijklpq = a jiklpq = a ijlkpq = a ijklqp , ∀i, j, k, l, p, q ∈ {1, 2, 3}.
Furthermore, a paired symmetric tensor A ∈ T 6,3 is a sixth order elasticity tensor [5] if a ijklpq = a klijpq = a ijpqkl , ∀i, j, k, l, p, q ∈ {1, 2, 3}.
For any tensor A = (a ijklpq ) ∈ T 6,3 , the corresponding homogeneous polynomial is defined by
A x 2 y 2 z 2 := 3 i,j,k,l,p,q=1
a ijklpq x i x j y k y l z p z q , ∀x, y, z ∈ R 3 .
Then, similar to Proposition 2.1, we have the following results.
Proposition 2.5. Suppose that A x 2 y 2 z 2 is defined by (13). Then, the following statements hold.
(i) The interior of set D defined by D := {A ∈ T 6,3 : A x 2 y 2 z 2 ≥ 0, ∀x, y, z ∈ R 3 } is nonempty, and intD = {A ∈ T 6,3 : A x 2 y 2 z 2 > 0, ∀x, y, z ∈ R 3 \{0}}.
(ii) The set D is a pointed closed convex cone.
Similar to Definition 2.2, we define the unfolded matrix of sixth order three dimensional tensor as follows.
Definition 2.6. For any A = (a ijkl pq) ∈ T 6,3 , we define a matrix by
N = (n st ) with n st = a isitjsjtkskt ∀s, t ∈ {1, 2, . . . , 27}(14)
where i 1 i 2 · · · i 27 , j 1 j 2 · · · j 27 and k 1 k 2 · · · k 27 are three arbitrary permutations of 123123 · 123 27 . We say that the matrix N defined by (14) is an unfolded matrix of tensor A with respect to indices i 1 i 2 · · · i 27 , j 1 j 2 · · · j 27 and k 1 k 2 · · · k 27 .
Obviously, there exists a unique unfolded matrix of tensor A for each triple of permutations of i 1 i 2 · · · i 27 , j 1 j 2 · · · j 27 and k 1 k 2 · · · k 27 ; and the following result holds.
Proposition 2.7. Suppose that A ∈ T 6,3 is a paired symmetric tensor, and N is its unfolded matrix defined by Definition 2.6. Then, the matrix N is symmetric.
In the following, we give several specific examples of the unfolded matrix. That is, for any A ∈ T 6,3 , we define N 1 := (n st ) ∈ R 27×27 with n st = n 3[3(i−1)+(k−1)]+p,3[3(j−1)+(l−1)]+q = a ijklpq , ∀i, j, k, l, p, q ∈ {1, 2, 3}; N 2 := (n st ) ∈ R 27×27 with n st = n 3[3(k−1)+(i−1)]+p,3[3(l−1)+(j−1)]+q = a ijklpq , ∀i, j, k, l, p, q ∈ {1, 2, 3}; N 3 := (n st ) ∈ R 27×27 with n st = n 3[3(p−1)+(k−1)]+i,3[3(q−1)+(l−1)]+j = a ijklpq , ∀i, j, k, l, p, q ∈ {1, 2, 3};
N 4 := (n st ) ∈ R 27×27 with n st = n 3[3(i−1)+(p−1)]+k,3[3(j−1)+(q−1)]+l = a ijklpq , ∀i, j, k, l, p, q ∈ {1, 2, 3}; N 5 := (n st ) ∈ R 27×27 with n st = n 3[3(k−1)+(p−1)]+i,3[3(l−1)+(q−1)+j = a ijklpq , ∀i, j, k, l, p, q ∈ {1, 2, 3}; N 6 := (n st ) ∈ R 27×27 with n st = f 3[3(p−1)+(i−1)]+k,3[3(q−1)+(j−1)]+l = a ijklpq , ∀i, j, k, l, p, q ∈ {1, 2, 3}.(15)
Then, we have the following results.
Proposition 2.8. For any paired symmetric tensor A ∈ T 6,3 , we have the following results.
• All matrices N 1 , N 2 , . . . , N 6 defined by (15) are symmetric.
• If A is an elasticity tensor, then N 1 = N 2 = · · · = N 6 .
For any tensor A = (a ijklpq ) ∈ T 6,3 and any i, j, k, l, p, q ∈ {1, 2, 3}, we define three block sub-tensors of tensor A by A ij := (a ijklpq ) klpq , B kl := (a ijklpq ) ijpq and C pq := (a ijklpq ) ijkl .
Then, we have the following results.
Proposition 2.9. For any A ∈ T 6,3 , let sub-tensors A ij , B kl and C pq be defined by (16).
• If A is a paired symmetric tensor, then all sub-tensors A ij , B kl and C pq are paired symmetric tensors.
• If A is an elasticity tensor (i.e., its entries satisfy (11) and (12)), then A st = B st = C st for all s, t ∈ {1, 2, 3}.
M-Eigenvalue and Properties
In this section, we extend the concept of M -eigenvalues for fourth order paired symmetric tensor introduced in [4,20] to sixth order three dimensional paired symmetric (elasticity) tensor and bi-block symmetric tensor, and discuss some related properties. For any paired symmetric tensor A = (a ijklpq ) ∈ T 6,3 , the corresponding homogeneous polynomial is given in (13). For any x, y, z ∈ R 3 , we let A xy 2 z 2 , A x 2 yz 2 , A x 2 y 2 z ∈ R 3 be defined by (A xy 2 z 2 ) i := 3 j,k,l,p,q=1 a ijklpq x j y k y l z p z q , ∀i ∈ {1, 2, 3},
(A x 2 yz 2 ) k := 3 i,j,l,p,q=1
a ijklpq x i x j y l z p z q , ∀k ∈ {1, 2, 3},
(A x 2 y 2 z) p := 3 i,j,k,l,q=1
a ijklpq x i x j y k y l z q , ∀p ∈ {1, 2, 3}.
Then, it is easy to see that
x, A xy 2 z 2 = A x 2 y 2 z 2 , y, A x 2 yz 2 = A x 2 y 2 z 2 , z, A x 2 y 2 z = A x 2 y 2 z 2(17)
hold for any x, y, z ∈ R 3 .
Definition 3.1. For any paired symmetric tensor A = (a ijklpq ) ∈ T 6,3 , if there exist λ ∈ R and x, y, z ∈ R 3 such that A xy 2 z 2 = λx, A x 2 yz 2 = λy, A x 2 y 2 z = λz,
x ⊤ x = 1, y ⊤ y = 1, z ⊤ z = 1,
then λ is called an M -eigenvalue of A and x, y, z are the eigenvectors of A associated with the M -eigenvalue λ.
Theorem 3.2. For any paired symmetric (elasticity) tensor A = (a ijklpq ) ∈ T 6,3 , its Meigenvalues always exist. Moreover, if x, y, z are the eigenvectors of A associated with the Meigenvalue λ, then λ = A x 2 y 2 z 2 .
Proof. Let A x 2 y 2 z 2 be defined by (13). We consider the following optimization problem:
min A x 2 y 2 z 2 s.t. x ⊤ x = 1, y ⊤ y = 1, z ⊤ z = 1.(18)
It is easy to see that the feasible set of (18) is compact and the objective function of (18) is continuous. Thus, the optimization problem (18) has at least a minimizer, say (x * , y * , z * ), which satisfies the first order optimality condition of (18), i.e., there exist α, β, γ ∈ R such that
A x * (y * ) 2 (z * ) 2 = αx * , A (x * ) 2 y * (z * ) 2 = βy * , A (x * ) 2 (y * ) 2 z * = γz * , (x * ) ⊤ x * = 1, (y * ) ⊤ y * = 1, (z * ) ⊤ z * = 1.
This, together with (17), implies that
α = β = γ = A (x * ) 2 (y * ) 2 (z * ) 2 .
Thus, α is an M -eigenvalue of A and x * , y * , z * are the eigenvectors of A associated with the M -eigenvalue α. We complete the proof. By Theorem 3.2, we have the following result. This theorem demonstrates that positive definiteness detection of a paired symmetric (elasticity) tensor A ∈ T 6,3 can be done by computing the smallest M -eigenvalue of A . Moreover, from Definition 3.1 and Theorem 3.3, it is easy to obtain the following results. For any (λ, x, y, z) ∈ R × R 3 × R 3 × R 3 and tensor A ∈ T 6,3 , λx 2 y 2 z 2 is a rank-one sixth order paired symmetric tensor with its entries being λx i x j y k y l z p z q for all i, j, k, l, p, q ∈ {1, 2, 3}. We say that λ * (x * ) 2 (y * ) 2 (z * ) 2 is the best rank-one approximation of A if (λ * , x * , y * , z * ) ∈ R×R 3 ×R 3 ×R 3 solves the optimization problem:
min A − λx 2 y 2 z 2 2 HS s.t. λ ∈ R and x ⊤ x = 1, y ⊤ y = 1, z ⊤ z = 1, ∀x, y, z ∈ R 3 .(19)
The best rank-one approximation has wide applications in signal and image processing, wireless communication systems, independent component analysis, and so on.
Theorem 3.5. For any paired symmetric (elasticity) tensor A = (a ijklpq ) ∈ T 6,3 , if λ * is an Meigenvalue of A with the largest absolute value among all M -eigenvalues of A , and x * , y * , z * ∈ R 3 are the eigenvectors of A associated with the M -eigenvalue λ * , then λ * (x * ) 2 (y * ) 2 (z * ) 2 is the best rank-one approximation of A .
Proof. We denote the feasible set of (19) by Ω, i.e.,
Ω := {(λ, x, y, z) ∈ R × R 3 × R 3 × R 3 : x ⊤ x = 1, y ⊤ y = 1, z ⊤ z = 1}.
On the one hand, since
min{ A − λx 2 y 2 z 2 2 HS : (λ, x, y, z) ∈ Ω} = min{ A 2 HS − 2λA x 2 y 2 z 2 + λ 2 (x ⊤ x)(y ⊤ y)(z ⊤ z) : (λ, x, y, z) ∈ Ω} = min{ A 2 HS − 2λA x 2 y 2 z 2 + λ 2 : (λ, x, y, z) ∈ Ω} and when λ = A xxyyzz, min{ A − λx 2 y 2 z 2 2 HS : (λ, x, y, z) ∈ Ω} = min{ A 2 HS − (λA x 2 y 2 z 2 ) 2 : (λ, x, y, z) ∈ Ω} = A 2 HS − max{(A x 2 y 2 z 2 ) 2 : (λ, x, y, z) ∈ Ω},
it is easy to see that A − λx 2 y 2 z 2 2 HS is the smallest if and only if the absolute value of λ is the largest among all λ ∈ R satisfying λ = A x 2 y 2 z 2 and (λ, x, y, z) ∈ Ω. On the other hand, since λ * is an M -eigenvalues of A and x * , y * , z * ∈ R 3 are the eigenvectors of A associated with the M -eigenvalue λ * , from Definition 3.1 it follows that (λ * , x * , y * , z * ) ∈ Ω; and from Theorem 3
.2 it follows that λ * = A (x * ) 2 (y * ) 2 (z * ) 2 .
Therefore, by the assumption that λ * is an M -eigenvalues of A with the largest absolute value among all M -eigenvalues of A , we obtain that λ * (x * ) 2 (y * ) 2 (z * ) 2 is the best rank-one approximation of A , which completes the proof.
In the following, we introduce the bi-block symmetric tensor and study some related properties.
Definition 3.6. A = (a i 1 i 2 ···i 2m ) ∈ T 2m,3 is called a bi-block symmetric tensor if its indices {i 1 , i 2 , · · · , i 2m } are divided into two adjacent blocks {i 1 , i 2 , · · · , i t } and {i t+1 , i t+2 , · · · , i 2m } with t ∈ [1, 2m] being an even number and entries of A being invariant under any permutation of indices in every block of {i 1 , i 2 , · · · , i t } and {i t+1 , i t+2 , · · · , i 2m }, i.e.,
a i 1 i 2 ···iti t+1 i t+2 ···i 2m = a σ(i 1 i 2 ···it)σ(i t+1 i t+2 ···i 2m )(20)for all i 1 , i 2 , . . . , i 2m ∈ {1, 2, 3}, where σ(i 1 i 2 · · · i t ) denotes an arbitrary permutation of i 1 i 2 · · · i t .
For any even number t ∈ [1, 2m] and x, y ∈ R 3 , we use the following notation:
x [t] := (x t 1 , x t 2 , x t 3 ) ⊤ ; A x t y 2m−t := 3 i 1 ,··· ,it,i t+1 ,··· ,i 2m =1 a i 1 ···iti t+1 ···i 2m x i 1 · · · x t y t+1 · · · y 2m A x t−1 y 2m−t ∈ R 3 with (A x t−1 y 2m−t ) i := 3 i 2 ,··· ,it,i t+1 ,··· ,i 2m =1 a ii 2 ···iti t+1 ···i 2m x i 2 · · · x t y t+1 · · · y 2m , ∀i ∈ {1, 2, 3}, A x t y 2m−t−1 ∈ R 3 with (A x t y 2m−t−1 ) i := 3 i 1 ,··· ,it,ii t+2 ,··· ,i 2m =1 a i 1 ···itii t+2 ···i 2m x i 1 · · · x t y t+2 · · · y 2m , ∀i ∈ {1, 2, 3}.
Definition 3.7. For any bi-block symmetric tensor A = (a i 1 ···iti t+1 ···i 2m ) ∈ T 2m,3 whose entries satisfy bi-block symmetry given by (20), if there exist λ ∈ R and x, y ∈ R 3 such that
A x t−1 y 2m−t = λx, A x t y 2m−t−1 = λy, x [ t 2 ] ⊤ x [ t 2 ] = 1, y [ 2m−t 2 ] ⊤ y [ 2m−t 2 ] = 1,
then λ is called an M -eigenvalue of A and x, y are the eigenvectors of A associated with the M -eigenvalue λ.
Then, similar to Theorem 3.2, we can obtain the following results.
Theorem 3.8. For any bi-block symmetric tensor A = (a i 1 ···iti t+1 ···i 2m ) ∈ T 2m,3 whose entries satisfy bi-block symmetry given by (20), M -eigenvalues always exist. Moreover, if x, y are the eigenvectors of A associated with the M -eigenvalue λ, then λ = A x t y 2m−t .
By using Theorem 3.8, the following result holds.
Theorem 3.9. A bi-block symmetric tensor A = (a i 1 ···iti t+1 ···i 2m ) ∈ T 2m,3 whose entries satisfy bi-block symmetry given by (20) is positive definite if and only if the smallest M -eigenvalue of A is positive.
From Definition 3.7 and Theorem 3.9, we have the following results.
Theorem 3.10. For any bi-block symmetric tensor A ∈ T 2m,3 whose entries satisfy bi-block symmetry given by (20) Moreover, similar to Theorem 3.5, we can obtain the following result.
Theorem 3.11. For any bi-block symmetric tensor A = (a i 1 ···iti t+1 ···i 2m ) ∈ T 2m,3 whose entries satisfy bi-block symmetry given by (20), if λ * is an M -eigenvalues of A with the largest absolute value among all M -eigenvalues of A , and x * , y * ∈ R 3 are the eigenvectors of A associated with the M -eigenvalue λ * , then λ * (x * ) t (y * ) 2m−t is the best rank-one approximation of A .
For any bi-block symmetric tensor A = (a i 1 ···iti t+1 ···i 2m ) ∈ T 2m,3 whose entries satisfy bi-block symmetry given by (20), when m = 2 and t = 2, the tensor A reduces to a fourth order three dimensional paired symmetric tensor; and hence, the definition of M -eigenvalue and results of Theorems 3.8, 3.9 and 3.11 reduce to those given in [20]. For example, by Theorem 3.9 we have
Fourth order paired symmetric tensors.
For any x, y ∈ R 3 , we define two matrices by (21) where A ij and B kl for i, j, k, l ∈ {1, 2, 3} are defined by (9). Recall that for any tensor A = (a ijkl ) ∈ T 4,3 , the corresponding biquadratic form is given by (2), i.e.,
A(y) := y ⊤ A 11 y y ⊤ A 12 y y ⊤ A 13 y y ⊤ A 21 y y ⊤ A 22 y y ⊤ A 23 y y ⊤ A 31 y y ⊤ A 32 y y ⊤ A 33 y and B(x) := x ⊤ B 11 x x ⊤ B 12 x x ⊤ B 13 x x ⊤ B 21 x x ⊤ B 22 x x ⊤ B 23 x x ⊤ B 31 x x ⊤ B 32 x x ⊤ B 33 x ,A x 2 y 2 = 3 i,j,k,l=1 a ijkl x i x j y k y l , ∀x, y ∈ R 3 .
Thus, it follows that
A x 2 y 2 = x ⊤ A(y)x = y ⊤ B(x)y, ∀x, y ∈ R 3 .
Proposition 4.1. For any paired symmetric (elasticity) tensor A ∈ T 4,3 , suppose that the matrices A(·) and B(·) are defined by (21). Then, the following results are equivalent. Suppose that A ∈ T 4,3 is a paired symmetric tensor, the polynomial A x 2 y 2 is defined by (2), and matrices A(·) and B(·) are defined by (21). Then, the polynomial A x 2 y 2 is positive definite if and only if one of the following results holds.
(i) The polynomial A x 2 y 2 defined by (2) is positive definite. (ii) The matrix A(y) is positive definite for all y ∈ R 3 \{0}. (iii) The matrix B(x) is positive definite for all x ∈ R 3 \{0}.
(i) The matrix A 11 is positive definite, and
y ⊤ A 11 yy ⊤ A 22 y − y ⊤ A 12 yy ⊤ A 21 y > 0 and det(A(y)) > 0, ∀y ∈ R 3 \{0}.
(ii) The matrix B 11 is positive definite, and
x ⊤ B 11 xx ⊤ B 22 x − x ⊤ B 12 xx ⊤ B 21 x > 0 and det(B(x)) > 0, ∀x ∈ R 3 \{0}.
Furthermore, if A is an elasticity tensor, then the above (i) and (ii) are the same.
It is easy to see that (a) y ⊤ A 11 yy ⊤ A 22 y − y ⊤ A 12 yy ⊤ A 21 y is a special homogeneous polynomial of degree 4, and there exists a unique symmetric tensor T 1 A ∈ T 4,3 such that T 1 A y 4 = y ⊤ A 11 yy ⊤ A 22 y − y ⊤ A 12 yy ⊤ A 21 y; and (b) det(A(y)) is a special homogeneous polynomial of degree 6, and there exists a unique symmetric tensor T 2 A ∈ T 6,3 such that T 2 A y 6 = det(A(y)).
Similarly, there exist symmetric tensors T 1 B ∈ T 4,3 and T 2 B ∈ T 6,3 such that
(c) T 1 B x 4 = x ⊤ B 11 xx ⊤ B 22 x − x ⊤ B 12 xx ⊤ B 21 x and T 2 B y 6 = det(B(x)).
By combining the theory of Z-eigenvalues of symmetric tensors with Theorem 4.2, we have the following results.
Theorem 4.3. Suppose that A ∈ T 4,3 is a paired symmetric tensor, the polynomial A x 2 y 2 is defined by (2), and symmetric tensors T 1 Furthermore, if A is an elasticity tensor, then the above (i) and (ii) are the same.
A , T 2 A , T 1 B , T 2 B are defined
Sixth order paired symmetric tensors.
In this part, we consider positive definiteness of sixth order three dimensional paired symmetric tensors and elasticity tensors.
For any x, y, z ∈ R 3 , we define three matrices by
A(y, z) := A 11 y 2 z 2 A 12 y 2 z 2 A 13 y 2 z 2 A 21 y 2 z 2 A 22 y 2 z 2 A 23 y 2 z 2 A 31 y 2 z 2 A 32 y 2 z 2 A 33 y 2 z 2 , B(x, z) := B 11 x 2 z 2 B 12 x 2 z 2 B 13 x 2 z 2 B 21 x 2 z 2 B 22 x 2 z 2 B 23 x 2 z 2 B 31 x 2 z 2 B 32 x 2 z 2 B 33 x 2 z 2 , C(x, y) := C 11 x 2 y 2 C 12 x 2 y 2 C 13 x 2 y 2 C 21 x 2 y 2 C 22 x 2 y 2 C 23 x 2 y 2 C 31 x 2 y 2 C 32 x 2 y 2 C 33 x 2 y 2 .(22)
Then, it is easy to see that
A x 2 y 2 z 2 = x ⊤ A(y, z)x = y ⊤ B(x, z)y = z ⊤ C(x, y)z, ∀x, y, z ∈ R 3 .(23)
Proposition 4.4. For any paired symmetric (elasticity) tensor A ∈ T 6,3 , suppose that matrices A(·, ·), B(·, ·) and C(·, ·) are defined by (22). Then, the following results are equivalent.
(i) The polynomial A x 2 y 2 z 2 defined by (13) is positive definite.
(ii) The matrix A(y, z) is positive definite for all y, z ∈ R 3 \{0}.
(iii) The matrix B(x, z) is positive definite for all x, z ∈ R 3 \{0}.
(iv) The matrix C(x, y) is positive definite for all x, y ∈ R 3 \{0}.
Furthermore, if A is an elasticity tensor, then the above (ii), (iii) and (iv) are the same.
Furthermore, by Sylvester's criterion we have the following results.
Theorem 4.5. For any paired symmetric tensor A ∈ T 6,3 , we assume that matrices A(·, ·), B(·, ·) and C(·, ·) are defined by (22). Then, the polynomial A x 2 y 2 z 2 is positive definite if and only if one of the following results holds.
(i) A 11 y 2 z 2 > 0, (A 11 y 2 z 2 )(A 22 y 2 z 2 ) − (A 12 y 2 z 2 )(A 21 y 2 z 2 ) > 0 and det(A(y, z)) > 0 for all
y, z ∈ R 3 \{0}. (ii) B 11 x 2 z 2 > 0, (B 11 x 2 z 2 )(B 22 x 2 z 2 ) − (B 12 x 2 z 2 )(B 21 x 2 z 2 ) > 0 and det(B(x, z)) > 0 for all x, z ∈ R 3 \{0}. (iii) C 11 x 2 y 2 > 0, (C 11 x 2 y 2 )(C 22 x 2 y 2 ) − (C 12 x 2 y 2 )(C 21 x 2 y 2 ) > 0 and det(C(x, y)) > 0 for all x, y ∈ R 3 \{0}.
Furthermore, if A is an elasticity tensor, then the above (i), (ii) and (iii) are the same.
We consider the above result (i).
• It is easy to see that A 11 is a paired symmetric tensor; and hence, the positive definiteness of A 11 y 2 z 2 can be checked by the minimum M -eigenvalue of the tensor A 11 given in the above subsection.
• It is easy to see that (A 11 y 2 z 2 )(A 22 y 2 z 2 )−(A 12 y 2 z 2 )(A 21 y 2 z 2 ) is a homogeneous polynomial of degree 8 with special structure. We can define the unique paired symmetric tensor
T 1 A ∈ T 8,3 such that T 1 A y 4 z 4 = (A 11 y 2 z 2 )(A 22 y 2 z 2 ) − (A 12 y 2 z 2 )(A 21 y 2 z 2 ), where (T 1 A ) i 1 i 2 ···i 8 = (T 1 A ) σ(i 1 i 2 i 3 i 4 )σ(i 5 i 6 i 7 i 8 ) with σ(i 1 i 2 i 3 i 4 )
being an arbitrary permutation of i 1 i 2 i 3 i 4 . Thus, (A 11 y 2 z 2 )(A 22 y 2 z 2 ) − (A 12 y 2 z 2 )(A 21 y 2 z 2 ) > 0 can be checked by the minimum M -eigenvalue of the tensor T 1 A .
• It is easy to see that det(A(y, z)) is a homogeneous polynomial of degree 12 with special structure. We can define the unique paired symmetric tensor T 2 A ∈ T 12,3 such that
T 2 A y 6 z 6 = det(A(y, z)), where (T 2 A ) i 1 i 2 ···i 12 = (T 2 A ) σ(i 1 ···i 6 )σ(i 7 ···i 12 )
with σ(i 1 · · · i 6 ) being an arbitrary permutation of i 1 i 2 i 3 i 4 i 5 i 6 . Thus, det(A(y, z)) > 0 can be checked by the minimum M -eigenvalue of the tensor T 2 A .
Similarly, we can define paired symmetric tensors T 1 B , T 1 C ∈ T 8,3 and T 2 B , T 2 C ∈ T 12,3 by , z)); T 1 C x 4 y 4 = (C 11 x 2 y 2 )(C 22 x 2 y 2 ) − (C 12 x 2 y 2 )(C 21 x 2 y 2 ), T 2 C x 6 y 6 = det(C(x, y)). By combining Theorem 3.9 with Theorem 4.5, we have the following results. Furthermore, if A is an elasticity tensor, then the above (i), (ii) and (iii) are the same.
T 1 B x 4 z 4 = (B 11 x 2 z 2 )(B 22 x 2 z 2 ) − (B 12 x 2 z 2 )(B 21 x 2 z 2 ), T 2 B x 6 z 6 = det(B(x
Sum of Squares and Positive Definiteness
In this section, we investigate the SOS properties of polynomials defined by fourth order three dimensional paired symmetric (elasticity) tensors and sixth order three dimensional paired symmetric (elasticity) tensors, respectively. In particular, we give necessary and/or sufficient conditions of the concerned tensor being positive definite.
Fourth order paired symmetric tensors.
For any paired symmetric tensor A = (a ijkl ) ∈ T 4,3 , we assume that the biquadratic form is defined by (2), i.e.,
A x 2 y 2 = 3 i,j,k,l=1 a ijkl x i x j y k y l , ∀x, y ∈ R 3 .
We first investigate sufficient conditions under which the biquadratic form defined by (2) is an SOS of bilinear forms, or is positive definite.
Theorem 5.1. Let i 1 i 2 · · · i 9 and j 1 j 2 · · · j 9 be two arbitrary permutations of 123123123. For any paired symmetric (elasticity) tensor A ∈ T 4,3 , let M defined by Definition 2.2 be an unfolded matrix of A with respect to indices i 1 i 2 · · · i 9 and j 1 j 2 · · · j 9 . Then, the following results hold.
(i) If M is positive semidefinite, then the biquadratic form defined by (2) is an SOS of bilinear forms.
(ii) If M is positive definite, then the biquadratic form defined by (2) is positive definite.
Proof. (i) For any x, y ∈ R 3 , by given indices i 1 i 2 · · · i 9 and j 1 j 2 · · · j 9 , we define w ∈ R 9 by
w := (x i 1 y j 1 , x i 2 y j 2 , x i 3 y j 3 , x i 4 y j 4 , x i 5 y j 5 , x i 6 y j 6 , x i 7 y j 7 , x i 8 y j 8 , x i 9 y j 9 ) ⊤ .
Then, from Definition 2.2, it is not difficult to show that
A x 2 y 2 = 3 i,j,k,l=1 a ijkl x i x j y k y l = w ⊤ M w, ∀x, y ∈ R 3 ,(24)
and from Proposition 2.3 it follows that the matrix M is symmetric.
Since the matrix M is positive semidefinite, there are real numbers λ 1 , λ 2 , . . . , λ 9 with λ 1 ≥ λ 2 ≥ . . . ≥ λ r ≥ λ r+1 = · · · = λ 9 = 0 and an orthogonal matrix Q = (q 1 q 2 · · · q 9 ) such that
Q ⊤ M Q = diag(λ 1 , λ 2 , . . . , λ 9 ), i.e., M = r i=1 λ i q i q ⊤ i .(25)
This, together with (24), implies that
A x 2 y 2 = w ⊤ M w = r i=1 λ i w ⊤ q i q ⊤ i w = r i=1 λ i w ⊤ q i 2 ,(26)
which means that A x 2 y 2 is an SOS of bilinear forms.
(ii) Since the matrix M is positive definite, there are real numbers λ 1 , λ 2 , . . ., λ 9 with λ i > 0 for all i ∈ {1, 2, . . . , 9} and an orthogonal matrix Q = (q 1 q 2 · · · q 9 ) such that (25) holds; and hence, (26) holds. If x = 0 and y = 0, then it is easy to see that w = 0, which further implies that there exists at least an index i ∈ {1, 2, . . . , 9} such that w ⊤ q i = 0 since the matrix Q is invertible. Thus, when the matrix M is positive definite, it follows from (26) that A x 2 y 2 > 0 for any x = 0 and y = 0, i.e., A x 2 y 2 is positive definite.
Especially, we have following results.
Theorem 5.2. For any paired symmetric tensor A ∈ T 4,3 , let matrices M 1 and M 2 be defined by (7) and (8), respectively. Then, the following results hold.
• The biquadratic form defined by (2) is an SOS of bilinear forms if the matrix M 1 is positive semidefinite or the matrix M 2 is positive semidefinite. Furthermore, if A ∈ T 4,3 is an elasticity tensor, then the results mentioned above are the same.
• The biquadratic form defined by (2) is positive definite if the matrix M 1 is positive definite or the matrix M 2 is positive definite. Furthermore, if A ∈ T 4,3 is an elasticity tensor, then the results mentioned above are the same.
Proof. For any x, y ∈ R 3 , we define two vectors u, v ∈ R 9 by u := (x 1 y 1 , x 1 y 2 , x 1 y 3 , x 2 y 1 , x 2 y 2 , x 2 y 3 , x 3 y 1 , x 3 y 2 , x 3 y 3 ) ⊤ , v := (x 1 y 1 , x 2 y 1 , x 3 y 1 , x 1 y 2 , x 2 y 2 , x 3 y 2 , x 1 y 3 , x 2 y 3 , x 3 y 3 ) ⊤ .
Then, it is easy to show that
A x 2 y 2 = 3 i,j,k,l=1 a ijkl x i x j y k y l = u ⊤ M 1 u = v ⊤ M 2 v, ∀x, y ∈ R 3 .
Thus, by a similar way as Theorem 5.1 and using Proposition 2.4, we can complete the proof. Next, we define a class of tensors with the help of A ∈ T 4,3 .
Definition 5.3. For any paired symmetric tensor A ∈ T 4,3 , we define a tensor B ∈ T 4,3 whose entries satisfy
b ijkl = b jilk , b ijlk = b jikl , and b ijkl + b jilk + b ijlk + b jikl = 4a ijkl .
We say that B is a semi-paired symmetric tensor of A .
Especially, we give the following remark. J := {(i, j, k, l) ∈ {1, 2, 3} 4 : i < j, l < k or i > j, k > l}.
For any paired symmetric tensor A = (a ijkl ) ∈ T 4,3 , we define a tensor B = (b ijkl ) ∈ T 4,3 whose entries satisfy
b ijkl = 2a ijlk if (i, j, l, k) ∈ J, 0 if (i, j, l, k) ∈ I \ J, a ijlk otherwise, ∀i, j, k, l ∈ {1, 2, 3},
then B is a semi-paired symmetric tensor of A .
It is easy to see that the biquadratic forms defined by a paired symmetric tensor A ∈ T 4,3 and by its semi-paired symmetric tensor B are the same. Thus, the following results hold.
Thus, SOS property and positive (semidefiniteness) definiteness of the paired symmetric tensor
A ∈ T 4,3 can be studied by investigating SOS property and positive (semidefiniteness) definiteness of its semi-paired symmetric tensor.
By Theorems 5.1 and 5.5, we have the following results.
Theorem 5.6. For any paired symmetric (elasticity) tensor A ∈ T 4,3 , let B defined by Definition 5.3 be a semi-paired symmetric tensor of A . Suppose that i 1 i 2 · · · i 9 and j 1 j 2 · · · j 9 are two arbitrary permutations of 123123123; and S is an unfolded matrix of tensor B with respect to indices i 1 i 2 · · · i 9 and j 1 j 2 · · · j 9 . Then, the matrix S is symmetric. Furthermore, if the matrix S is positive semidefinite, then the biquadratic form defined by (2) is an SOS of bilinear forms; and if the matrix S is positive definite, then the biquadratic form defined by (2) is positive definite. In the following, we propose a necessary and sufficient condition under which a fourth order three dimensional paired symmetric (elasticity) tensor is positive semidefinite. For this purpose, we need the following lemma.
Lemma 5.8. For any paired symmetric (elasticity) tensor A ∈ T 4,3 , suppose that the biquadratic form defined by (2) can be written as
A x 2 y 2 = 3 i,j,k,l=1 a ijkl x i x j y k y l = u ⊤ M u, ∀x, y ∈ R 3 ,(29)
where the vector u is given by (27) and M = (m st ) ∈ R 9×9 is a symmetric matrix. Then, the matrix M is an unfolded matrix of some semi-paired symmetric tensor of A .
Proof. We define a tensor B = (b ijkl ) ∈ T 4,3 by
b 1111 b 1112 b 1113 b 1211 b 1212 b 1213 b 1311 b 1312 b 1313 b 1121 b 1122 b 1123 b 1221 b 1222 b 1223 b 1321 b 1322 b 1323 b 1131 b 1132 b 1133 b 1231 b 1232 b 1233 b 1331 b 1332 b 1333 b 2111 b 2112 b 2113 b 2211 b 2212 b 2213 b 2311 b 2312 b 2313 b 2121 b 2122 b 2123 b 2221 b 2222 b 2223 b 2321 b 2322 b 2323 b 2131 b 2132 b 2133 b 2231 b 2232 b 2233 b 2331 b 2332 b 2333 b 3111 b 3112 b 3113 b 3211 b 3212 b 3213 b 3311 b 3312 b 3313 b 3121 b 3122 b 3123 b 3221 b 3222 b 3223 b 3321 b 3322 b 3323 b 3131 b 3132 b 3133 b 3231 b 3232 b 3233 b 3331 b 3332 b 3333 := (m st ).(30)
By using the symmetry of matrix M , it is easy to show that b ijkl = b jilk and b ijlk = b jikl , ∀i, j, k, l ∈ {1, 2, 3}.
Moreover, by combining (29) with (30) we have
A x 2 y 2 = 3 i,j,k,l=1 a ijkl x i x j y k y l = u ⊤ M u = 3 i,j,k,l=1 b ijkl x i x j y k y l = Bx 2 y 2 ,
which, together with (31) and the paired symmetry of A , implies that
b ijkl + b jilk + b ijlk + b jikl = 4a ijkl , ∀i, j, k, l ∈ {1, 2, 3}.(32)
Thus, by Definition 5.3, (31) and (32), it follows that B is a semi-paired symmetric tensor of A . In addition, from (30) it is easy to see that 1, 1, 2, 2, 2, 3, 3, 3), (j 1 , j 2 , j 3 , j 4 , j 5 , j 6 , j 7 , j 8 , j 9 ) = (1, 2, 3, 1, 2, 3, 1, 2, 3).
m st = b isitjsjt , ∀s, t ∈ {1, 2, . . . , 9} with (i 1 , i 2 , i 3 , i 4 , i 5 , i 6 , i 7 , i 8 , i 9 ) = (1,
Thus, by Definition 2.2 we obtain that the matrix M is an unfolded matrix of tensor B with respect to indices 111222333 and 123123123.
Theorem 5.9. For any paired symmetric (elasticity) tensor A ∈ T 4,3 , the biquadratic form defined by (2) is an SOS of bilinear forms if and only if an unfolded matrix of some semi-paired symmetric tensor of A is positive semidefinite.
Proof. The sufficiency can be obtained by Theorem 5.6(i). We now show the necessity. Since A x 2 y 2 is an SOS of bilinear forms, there exists some positive integer r ≥ 1 such that
A x 2 y 2 = 3 i,j,k,l=1 a ijkl x i x j y k y l = r s=1 3 i,j=1 α s ij x i y j 2 , ∀x, y ∈ R 3
where α s ij ∈ R for any s ∈ {1, 2, . . . , r} and i, j ∈ {1, 2, 3}. For any s ∈ {1, 2, . . . , r}, we denote
q s := (α s 11 , α s 12 , α s 13 , α s 21 , α s 22 , α s 23 , α s 31 , α s 32 , α s 33 ) ⊤ .
Let the vector u ∈ R n be defined by (27). Then,
A x 2 y 2 = r s=1 q ⊤ s u 2 = u ⊤ r s=1 q s q ⊤ s u, ∀x, y ∈ R 3 .
By Lemma 5.8, we can obtain that the matrix r s=1 q s q ⊤ s is an unfolded matrix of some semipaired symmetric tensor of A , which is positive semidefinite.
In the following, we give a necessary condition of a fourth order three dimensional paired symmetric tensor being an SOS of bilinear forms, which is convenient for judging some fourth order three dimensional paired symmetric tensors not be an SOS.
Theorem 5.10. For any paired symmetric (elasticity) tensor A ∈ T 4,3 , if the biquadratic form defined by (2) is an SOS of bilinear forms, then it follows that a iikk ≥ 0, ∀i, k ∈ {1, 2, 3}.
Proof. By the assumption, we can let
A x 2 y 2 = 3 i,j,k,l=1 a ijkl x i x j y k y l = r s=1 3 i,k=1 α s ik x i y k 2 , ∀x, y ∈ R 3
where r ≥ 1 is a positive integer and α s ik ∈ R for any s ∈ {1, 2, . . . , r} and i, k ∈ {1, 2, 3}. Then, it is easy to show that
a iikk = r s=1 (α s ik ) 2 , ∀i, k ∈ {1, 2, 3},
which completes the proof.
Sixth order paired symmetric tensors.
In this part, we consider the homogeneous polynomial defined by a tensor A = (a ijklpq ) ∈ T 6,3 , which is given by (13). Similar to Theorem 5.1, we have the following results.
Theorem 5.11. Let i 1 i 2 · · · i 27 and j 1 j 2 · · · j 27 be two arbitrary permutations of 123123 · · · 123 27 .
For any paired symmetric (elasticity) tensor A ∈ T 6,3 , suppose that N defined by Definition 2.6 is an unfold matrix of A with respect to indices i 1 i 2 · · · i 27 and j 1 j 2 · · · j 27 . Then, the following results hold.
(i) If N is positive semidefinite, then the polynomial defined by (13) is an SOS of trilinear forms.
(ii) If N is positive definite, then the polynomial defined by (13) is positive definite.
Especially, we have the following results.
Theorem 5.12. For any paired symmetric tensor A = (a ijklpq ) ∈ T 6,3 , let matrices N 1 , N 2 , . . . , N 6 be defined by (15). Then, the following results hold.
(i) The polynomial defined by (13) is an SOS of trilinear polynomials if one of matrices N 1 , N 2 , . . . , N 6 is positive semidefinite. Furthermore, if A = (a ijklpq ) ∈ T 6,3 is an elasticity tensor, then the results mentioned above are the same.
(ii) The polynomial defined by (13) is positive definite if one of matrices N 1 , N 2 , . . ., N 6 is positive definite. Furthermore, if A = (a ijklpq ) ∈ T 6,3 is an elasticity tensor, then the results mentioned above are the same.
Proof. For any x, y, z ∈ R 3 , we define six vectors u i ∈ R 27 (i ∈ {1, 2, . . . , 6}) by
(u 1 ) ⊤ := (x 1 y 1 z 1 , x 1 y 1 z 2 , x 1 y 1 z 3 , x 1 y 2 z 1 , x 1 y 2 z 2 , x 1 y 2 z 3 , x 1 y 3 z 1 , x 1 y 3 z 2 , x 1 y 3 z 3 , x 2 y 1 z 1 , x 2 y 1 z 2 , x 2 y 1 z 3 , x 2 y 2 z 1 , x 2 y 2 z 2 , x 2 y 2 z 3 , x 2 y 3 z 1 , x 2 y 3 z 2 , x 2 y 3 z 3 , x 3 y 1 z 1 , x 3 y 1 z 2 , x 3 y 1 z 3 , x 3 y 2 z 1 , x 3 y 2 z 2 , x 3 y 2 z 3 , x 3 y 3 z 1 , x 3 y 3 z 2 , x 3 y 3 z 3 ); (u 2 ) ⊤ := (x 1 y 1 z 1 , x 1 y 1 z 2 , x 1 y 1 z 3 , x 2 y 1 z 1 , x 2 y 1 z 2 , x 2 y 1 z 3 , x 3 y 1 z 1 , x 3 y 1 z 2 , x 3 y 1 z 3 ,
x 1 y 2 z 1 , x 1 y 2 z 2 , x 1 y 2 z 3 , x 2 y 2 z 1 , x 2 y 2 z 2 , x 2 y 2 z 3 , x 3 y 2 z 1 , x 3 y 2 z 2 , x 3 y 2 z 3 , x 3 y 1 z 1 , x 3 y 1 z 2 , x 3 y 1 z 3 , x 3 y 2 z 1 , x 3 y 2 z 2 , x 3 y 2 z 3 , x 3 y 3 z 1 , x 3 y 3 z 2 , x 3 y 3 z 3 ); (u 3 ) ⊤ := (x 1 y 1 z 1 , x 2 y 1 z 1 , x 3 y 1 z 1 , x 1 y 2 z 1 , x 2 y 2 z 1 , x 3 y 2 z 1 , x 1 y 3 z 1 , x 2 y 3 z 1 , x 3 y 3 z 1 , x 1 y 1 z 2 , x 2 y 1 z 2 , x 3 y 1 z 2 , x 1 y 2 z 2 , x 2 y 2 z 2 , x 3 y 2 z 2 , x 1 y 3 z 2 , x 2 y 3 z 2 , x 3 y 3 z 2 , x 1 y 1 z 3 , x 2 y 1 z 3 , x 3 y 1 z 3 , x 1 y 2 z 3 , x 2 y 2 z 3 , x 3 y 2 z 3 , x 1 y 3 z 3 , x 2 y 3 z 3 , x 3 y 3 z 3 ); (u 4 ) ⊤ := (x 1 y 1 z 1 , x 1 y 2 z 1 , x 1 y 3 z 1 , x 1 y 1 z 2 , x 1 y 2 z 2 , x 1 y 3 z 2 , x 1 y 1 z 3 , x 1 y 2 z 3 , x 1 y 3 z 3 , x 2 y 1 z 1 , x 2 y 2 z 1 , x 2 y 3 z 1 , x 2 y 1 z 2 , x 2 y 2 z 2 , x 2 y 3 z 2 , x 2 y 1 z 3 , x 2 y 2 z 3 , x 2 y 3 z 3 , x 3 y 1 z 1 , x 3 y 1 z 2 , x 3 y 1 z 3 , x 3 y 2 z 1 , x 3 y 2 z 2 , x 3 y 2 z 3 , x 3 y 3 z 1 , x 3 y 3 z 2 , x 3 y 3 z 3 ); (u 5 ) ⊤ := (x 1 y 1 z 1 , x 1 y 2 z 1 , x 1 y 3 z 1 , x 2 y 1 z 1 , x 2 y 2 z 1 , x 2 y 3 z 1 , x 3 y 1 z 1 , x 3 y 2 z 1 , x 3 y 3 z 1 , x 1 y 1 z 2 , x 1 y 2 z 2 , x 1 y 3 z 2 , x 2 y 1 z 2 , x 2 y 2 z 2 , x 2 y 3 z 2 , x 3 y 1 z 2 , x 3 y 2 z 2 , x 3 y 3 z 2 , x 1 y 1 z 3 , x 2 y 1 z 3 , x 3 y 1 z 3 , x 1 y 2 z 3 , x 2 y 2 z 3 , x 3 y 2 z 3 , x 1 y 3 z 3 , x 2 y 3 z 3 , x 3 y 3 z 3 ); (u 6 ) ⊤ := (x 1 y 1 z 1 , x 2 y 1 z 1 , x 3 y 1 z 1 , x 1 y 1 z 2 , x 2 y 1 z 2 , x 3 y 1 z 2 , x 1 y 1 z 3 , x 2 y 1 z 3 , x 3 y 1 z 3 , x 1 y 2 z 1 , x 2 y 2 z 1 , x 3 y 2 z 1 , x 1 y 2 z 2 , x 2 y 2 z 2 , x 3 y 2 z 2 , x 1 y 2 z 3 , x 2 y 2 z 3 , x 3 y 2 z 3 , x 1 y 3 z 1 , x 1 y 3 z 2 , x 1 y 3 z 3 , x 2 y 3 z 1 , x 2 y 3 z 2 , x 2 y 3 z 3 , x 3 y 3 z 1 , x 3 y 3 z 2 , x 3 y 3 z 3 ).
With the help of the definitions of matrices N 1 , N 2 , . . . , N 6 by (15), it is easy to verify that A x 2 y 2 z 2 = 3 i,j,k,l,p,q=1 a ijklpq x i x j y k y l z p z q = (u 1 ) ⊤ N 1 u 1 = (u 2 ) ⊤ N 2 u 2 = · · · = (u 6 ) ⊤ N 6 u 6 .
Thus, similar to Theorem 5.2, we can obtain the desired results.
Similar to Definition 5.3, we give the concept of semi-paired symmetric tensor of A ∈ T 6,3 .
Definition 5.13. For any paired symmetric tensor A ∈ T 6,3 , we define a tensor B ∈ T 6,3 whose entries satisfy
b ijklpq = b jilkqp , b jiklpq = b ijlkqp , b ijlkpq = b jiklqp , b ijklqp = b jilkpq , and b ijklpq + b jiklpq + b ijlkpq + b ijklqp = 4a ijklpq .
We say that B is a semi-paired symmetric tensor of A .
Especially, we give the following remark.
Remark 5.14. We define several index sets by i < j, k < l and p < q; or i > j, k > l and p > q , S 2 := (i, j, k, l, p, q) ∈ T 2 : i < j, k < l and p = q; or i > j, k > l and p = q , S 3 := (i, j, k, l, p, q) ∈ T 2 : i < j, k = l and p < q; or i > j, k = l and p > q , S 4 := (i, j, k, l, p, q) ∈ T 2 : i = j, k < l and p < q; or i = j, k > l and p > q .
For any paired symmetric tensor A ∈ T 6,3 , we define a tensor B = (b ijklpq ) ∈ T 6,3 whose entries satisfy
b ijklpq = 4a ijlkpq if (i, j, l, k, p, q) ∈ S 1 , 0 if (i, j, l, k, p, q) ∈ T 1 \ S 1 , 2a ijlkpq if (i, j, l, k, p, q) ∈ S 2 S 3 S 4 , 0 if (i, j, l, k, p, q) ∈ T 2 \ {S 2 S 3 S 4 }, a ijlkpq otherwise,
∀i, j, k, l, p, q ∈ {1, 2, 3}, then B is a semi-paired symmetric tensor of A .
Similar to Theorem 5.5, we have the following results.
Theorem 5.15. For any paired symmetric (elasticity) tensor A ∈ T 6,3 , suppose that B defined by Definition 5.13 is an semi-paired symmetric tensor of A . Then, the polynomial A x 2 y 2 z 2 is an SOS of trilinear forms if and only if the polynomial Bx 2 y 2 z 2 is an SOS of trilinear forms; and tensor A is positive (semidefinite) definite if and only if tensor B is positive (semidefinite) definite.
Corollary 5.16. For any paired symmetric (elasticity) tensor A ∈ T 6,3 , suppose that B = (b ijkl ) defined by Remark 5.14 is a semi-paired symmetric tensor of A , and the unfolded matrix of B is defined by
N := (n st ) with n st = n 3[3(i−1)+(k−1)]+p,3[3(j−1)+(l−1)]+q = b ijklpq , ∀i, j, k, l, p, q ∈ {1, 2, 3};(33)
If the matrix N is positive semidefinite, then the polynomial defined by (13) is an SOS of trilinear forms; and if the matrix N is positive definite, then the polynomial defined by (13) is positive definite.
Similar to Theorem 5.9, we have the following results.
Theorem 5.17. For any paired symmetric (elasticity) tensor A ∈ T 6,3 , the polynomial defined by (13) is an SOS of trilinear forms if and only if an unfolded matrix of some semi-paired symmetric tensor of A is positive semidefinite.
The following necessary condition can be obtained in a similar way as Theorem 5.10.
Theorem 5.18. For any paired symmetric (elasticity) tensor A = (a ijklpq ) ∈ T 6,3 . If the homogeneous polynomial defined by (13) is an SOS of trilinear polynomials, then it follows that a iikkpp ≥ 0, ∀i, k, p ∈ {1, 2, 3}.
Algorithm for the Smallest M-Eigenvalue
In order to judge positive definiteness of a fourth order three dimensional paired symmetric (elasticity) tensor, from Corollary 3.12 we may compute the smallest M -eigenvalue of the concerned tensor. Some methods can be applied to do it, such as the one given in [17]. In this section, by using the special structure of the paired symmetric (elasticity) tensor, we propose a sequential sedefinite programming method for computing the smallest M -eigenvalue of a fourth order three dimensional paired symmetric (elasticity) tensor, by which we can judge whether a fourth order three dimensional paired symmetric (elasticity) tensor is positive definite or not.
Let A x 2 y 2 be defined by (2) and r be some nonnegative integer, we define a polynomial F r,s :
R 3 × R 3 → R by F r,s (x, y) := 3 i=1 x 2 i r 3 i=1 y 2 i s A x 2 y 2 ,(34)
which is a homogeneous polynomial with deg(F r ) = 2(r + s) + 4. The following result can be obtained in a similar way as those in [24,Corollary], [8,Theorem 3.2] and [14,Lemma 3.5].
Theorem 6.1. For any paired symmetric tensor A = (a ijkl ) ∈ T 4,3 , let intC and F r,s be defined by (4) and (34), respectively. If A ∈ intC, then it follows that F r,s (x, y) is an SOS for some sufficiently large integers r, s ≥ 0.
With Theorem 6.1, we define K := {A ∈ T 4,3 : F r,s (x, y) is an SOS for some r, s ≥ 0}.
Then, similar to Theorem 3.3 in [8], we can obtain the following result.
Theorem 6.2. For any paired symmetric tensor A = (a ijkl ) ∈ T 4,3 , K defined by (35) is a convex cone. In particular, it follows that intC ⊆ K ⊆ C, where C and intC are defined by (3) and (4), respectively.
For any nonnegative integer r, by the multinomial theorem we have
3 i=1 x 2 i r = r 1 +r 2 +r 3 =r r! r 1 ! × r 2 ! × r 3 ! x 2r 1 1 x 2r 2 2 x 2r 3 3 .
Thus, the polynomial F r,s (x, y) defined by (34) is a linear combination of monomials
x 2r 1 1 x 2r 2 2 x 2r 3 3 y 2s 1 1 y 2s 2 2 y 2s 3 3 x i x j y k y l
for i, j, k, l ∈ {1, 2, 3} and nonnegative integers r 1 , r 2 , r 3 , s 1 , s 2 , s 3 satisfying
3 i=1 r i = r and 3 i=1 s i = s.
We assume that the corresponding vector of monomials is denoted by v r+s+2 (x, y), which is first ordered in descending order of powers of x 1 , x 2 , x 3 and is then ordered in descending order of powers of y 1 , y 2 , y 3 given by v r+s+2 (x, y) ⊤ : = (x r+1 1 y s+1 1 , x r+1 1 y s 1 y 2 , . . . , x r+1 1 y s+1 3 , x r 1 x 2 y r+1 2 , . . . , x r+1 3 y s+1 3 ).
Then, the following result is immediate. For any given nonnegative integers r and s, we let u r+s+2 (x, y) denote the vector made up of different monomials in v r+s+2 (x, y) ⊤ v r+s+2 (x, y), which is first ordered in descending order of powers of x 1 , x 2 , x 3 and then ordered in descending order of powers of y 1 , y 2 , y 3 ; and denote the dimensions of vectors v r+s+2 (x, y) and u r+s+2 (x, y) by d vrs := dim(v r+s+2 (x, y)) and d urs := dim(u r+s+2 (x, y)), respectively.
• For r = 0 and s = 0, the vector v r+s+2 (x, y) contains monomials
x i y j , ∀i, j ∈ {1, 2, 3}, and hence, d v00 = C 1 3 C 1 3 = 9; and the vector u r+s+2 (x, y) contains monomials
x 2 i y k y l (k < l), x 2 i y 2 k , x i x j y k y l (i < j, k < l), x i x j y 2 k (i < j), ∀i, j, k, l ∈ {1, 2, 3}, and hence, d u00 = C 1 3 C 2 3 + C 1 3 C 1 3 + C 2 3 C 2 3 + C 1 3 C 2 3 = 36.
• For r = 1 and s = 0, the vector v r+s+2 (x, y) contains monomials
x α 1 1 x α 2 2 x α 3 3 y β 1 1 y β 2 2 y β 3 3 ,
where α i , β i ≥ 0 for all i ∈ {1, 2, 3}, α 1 + α 2 + α 3 = 2 and β 1 + β 2 + β 3 = 1, and hence, d v10 = C 2 3+2−1 C 1 3+1−1 = 18; and the vector u r+s+2 (x, y) contains monomials
x α 1 1 x α 2 2 x α 3 3 y β 1 1 y β 2 2 y β 3 3 ,
where α i , β i ≥ 0 for all i ∈ {1, 2, 3}, α 1 + α 2 + α 3 = 4 and β 1 + β 2 + β 3 = 2, and hence,
d u10 = C 4 3+4−1 C 2 3+2−1 = 90.
• For r = 0 and s = 1, the vector v r+s+2 (x, y) contains monomials
x α 1 1 x α 2 2 x α 3 3 y β 1 1 y β 2 2 y β 3 3 ,
where α i , β i ≥ 0 for all i ∈ {1, 2, 3}, α 1 + α 2 + α 3 = 1 and β 1 + β 2 + β 3 = 2, and hence,
d v01 = C 1 3+1−1 C 2 3+2−1 = 18
; and the vector u r+s+2 (x, y) contains monomials
x α 1 1 x α 2 2 x α 3 3 y β 1 1 y β 2 2 y β 3 3 ,
where α i , β i ≥ 0 for all i ∈ {1, 2, 3}, α 1 + α 2 + α 3 = 2 and β 1 + β 2 + β 3 = 4, and hence,
d u01 = C 2 3+2−1 C 4 3+4−1 = 90.
• For r = 1 and s = 1, the vector v r+s+2 (x, y) contains monomials
x α 1 1 x α 2 2 x α 3 3 y β 1 1 y β 2 2 y β 3 3 ,
where α i , β i ≥ 0 for all i ∈ {1, 2, 3}, α 1 + α 2 + α 3 = 2 and β 1 + β 2 + β 3 = 2, and hence, d v01 = C 2 3+2−1 C 2 3+2−1 = 36; and the vector u r+s+2 (x, y) contains monomials
x α 1 1 x α 2 2 x α 3 3 y β 1 1 y β 2 2 y β 3 3 ,
where α i , β i ≥ 0 for all i ∈ {1, 2, 3}, α 1 + α 2 + α 3 = 4 and β 1 + β 2 + β 3 = 4, and hence, d u01 = C 4 3+4−1 C 4 3+4−1 = 225. Furthermore, we introduce two operators V and W in the following:
V: For any nonnegative integers r and s, we define an operator V : R dvrs×dvrs → R durs such that [V(H)] i is the coefficient of the ith monomial in the vector u r+s+2 (x, y) of the polynomial v r+s+2 (x, y) ⊤ Hv r+s+2 (x, y) for any H ∈ R dvrs×dvrs .
W: For any nonnegative integers r and s, we define an operator W which maps a homogeneous polynomial F r,s (x, y) defined by (34) to a vector in R durs satisfying F r,s (x, y) = u r+s+2 (x, y) ⊤ W(F r,s (x, y)).
Thus, by Theorem 6.2 it follows that the set K defined by (35) can be written as
Then, the following result is immediate.
Theorem 6.4. For any paired symmetric tensor A = (a ijkl ) ∈ T 4,3 and nonnegative integers r and s, suppose that the set K r+s is defined by (38). Then, K r+s ⊆ K r+s+1 for any r, s ∈ {0, 1, 2, . . .} and lim r+s→∞ K r+s = ∪ ∞ r+s=0 K r+s = K.
For any paired symmetric tensor A = (a ijkl ) ∈ T 4,3 , it follows from Definition 3.7 and Theorem 3.8 that the smallest M -eigenvalue of A is the optimal value of the following biquadratic optimization over unit spheres [14]:
min A x 2 y 2 s.t. x ⊤ x = 1, y ⊤ y = 1,(39)
which is equivalent to
max γ s.t. A x 2 y 2 ≥ γ, ∀(x, y) ∈ {(x, y) ∈ R 3 × R 3 : x ⊤ x = 1, y ⊤ y = 1}.(40)
Let tensor E = (e ijkl ) ∈ T 4,3 be defined by (5). Then, it is easy to see that problem (40) is equivalent to
max γ s.t. A + γE ∈ K.(41)
Furthermore, by replacing the constraint A + γE ∈ K by A + γE ∈ K r+s , we obtain the r + sth order relaxation problem of (41), which can be written as max γ
s.t. V(Q) = W(F r,s (x, y)) + γW ( 3 i=1 x 2 i ) r+1 ( 3 i=1 y 2 i ) s+1 , Q ∈ S dvrs + ,(42)
where r, s ∈ {0, 1, 2, . . .}. For any r, s ∈ {0, 1, 2, . . .}, (42) is a semidefinite programming problem, which is denoted by SDP (r + s). Thus, we can solve a sequence of SDP (r + s) through increasing r + s to obtain an approximation optimal solution of (39) up to a priori precision.
Numerical Results
In this section, we present preliminary numerical results to judge whether a fourth order three dimensional or a sixth order three dimensional paired symmetric tensor is positive definite or not. All numerical experiments were run in Matlab on a PC with 2.93 GHz CPU and 2.00 GB of RAM. We divide our experiments into the following two parts.
Part 1. In this part, by using the sequential semidefinite programming method proposed in Section 6, we present preliminary numerical results for computing the smallest M -eigenvalue of the fourth order three dimensional paired symmetric tensor. We use SDPT3 [30] to solve the resulted conic linear programming problem.
The biquadratic form defined by this tensor is
f (x, y) = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 − 2(x 1 x 2 y 1 y 2 + x 2 x 3 y 2 y 3 + x 3 x 1 y 3 y 1 ) +2(x 2 1 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 ),(44)
which is a positive semidefinite biquadratic form that is not an SOS of bilinear forms [3]. Suppose λ ∈ R is an M -eigenvalue of A and x, y ∈ R 3 \ {0} are the eigenvectors of A associated with the M -eigenvalue λ, then it follows from the definition of the M -eigenvalue that
x 1 y 2 1 − (x 2 y 1 y 2 + x 3 y 3 y 1 ) + 2x 1 y 2 2 = λx 1 , x 2 y 2 2 − (x 1 y 1 y 2 + x 3 y 2 y 3 ) + 2x 2 y 2 3 = λx 2 , x 3 y 2 3 − (x 2 y 2 y 3 + x 1 y 3 y 1 ) + 2x 3 y 2 1 = λx 3 , x 2 1 y 1 − (x 1 x 2 y 2 + x 3 x 1 y 3 ) + 2x 2 3 y 1 = λy 1 , x 2 2 y 2 − (x 1 x 2 y 1 + x 2 x 3 y 3 ) + 2x 2 1 y 2 = λy 2 , x 2 3 y 3 − (x 2 x 3 y 2 + x 3 x 1 y 1 ) + 2x 2 2 y 3 = λy 3 , x 2 1 + x 2 2 + x 2 3 = 1, y 2 1 + y 2 2 + y 2 3 = 1.(45)
From (45) The numerical result is listed in Table 1. In Table 1, "ITER" means the iteration number of the SDP solver, "CPU(s)" means the total time in seconds spent for both setting up the problem and solving it, "OPT" means the approximation value computed, and "VOL" means the norm of the violation of the constraints of the approximation solution. From this table, we see that the method can find a good approximation solution even with the zero order relaxation. a 1111 = a 1122 = a 1133 = a 2211 = a 2222 = a 2233 = a 3311 = 1, a 3322 = 3, a 3333 = 3, a 3323 = a 3332 = −1, other a ijkl = 0.
(46)
The biquadratic form defined by this tensor is
f (x, y) = x 2 1 y 2 1 + x 2 1 y 2 2 + x 2 1 y 2 3 + x 2 2 y 2 1 + x 2 2 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 + 3x 2 3 y 2 2 − 2x 2 3 y 2 y 3 + 3x 2 3 y 2 3 . (47)
Suppose λ ∈ R is an M -eigenvalue of A and x, y ∈ R 3 \ {0} are the eigenvectors of A associated with the M -eigenvalue λ, then it follows from the definition of the M -eigenvalue that
x 1 y 2 1 + x 1 y 2 2 + x 1 y 2 3 = λx 1 , x 2 y 2 1 + x 2 y 2 2 + x 2 y 2 3 = λx 2 , x 3 y 2 1 + 3x 3 y 2 2 − 2x 3 y 2 y 3 + 3x 3 y 2 3 = λx 3 , x 2 1 y 1 + x 2 2 y 1 + x 2 3 y 1 = λy 1 , x 2 1 y 2 + x 2 2 y 2 + 3x 2 3 y 2 − x 2 3 y 3 = λy 2 , x 2 1 y 3 + x 2 2 y 3 − x 2 3 y 2 + 3x 2 3 y 3 = λy 3 , x 2 1 + x 2 2 + x 2 3 = 1, y 2 1 + y 2 2 + y 2 3 = 1.(48)
From the first three equalities and last two equalities in (48), we can obtain that
λ = 1 + x 2 3 y 2 2 + (x 3 y 2 − x 3 y 2 ) 2 + x 2 3 y 2 3 ≥ 1,
which implies that all M -eigenvalues of A given in (49) are greater than or equal to 1. Moreover, it is easy to see that (λ * , x * , y * ) = (1, (1, 0, 0) ⊤ , (1, 0, 0) ⊤ ) is a solution to (48). Therefore, λ * = 1 is the smallest M -eigenvalues of A given in (49). The numerical result is listed in Table 2. In Table 2, symbols "ITER", "CPU(s)", "OPT", "VOL" are the same as in Example 7.1. From this table, we see that the method can find a good approximation solution even with the zero order relaxation. Part 2. In this part, by using some results obtained in Section 5, we compute the smallest eigenvalue of an unfolded matrix of some semi-paired symmetric tensor to judge whether the concerned paired symmetric tensor is positive definite or not.
The biquadratic form defined by this tensor is f (x, y) = 2x 2 1 y 2 1 + 2x 2 2 y 2 2 + 2x 2 3 y 2 3 − 2x 1 x 3 y 1 y 3 − 2x 2 x 3 y 2 y 3 .
It is not difficult to obtain that f (x, y) = x 2 1 y 2 1 + x 2 2 y 2 2 + (x 1 y 1 − x 3 y 3 ) 2 + (x 2 y 2 − x 3 y 3 ) 2 and f (x * , y * ) = 0 where x * = (1, 0, 0) ⊤ and y * = (0, 1, 0). These demonstrate that the biquadratic form in (50) is an SOS of bilinear forms but not positive definite.
Let A be given by (49) It is easy to obtain that eigenvalues of M are 0, 0, 0, 0, 0, 0, 0.5858, 2, 3.4142, respectively, which implies that the matrix M is positive semidefinite but not positive definite. Thus, by Corollary 5.7 we obtain that the quadratic form (50) is an SOS of bilinear forms but not positive definite.
The homogeneous polynomial defined by this tensor is f (x, y, z) = x 2 1 y 2 1 z 2 1 + x 2 1 y 2 2 z 2 3 + x 2 2 y 2 1 z 2 1 + x 2 2 y 2 2 z 2 2 + x 2 3 y 2 1 z 2 2 + x 2 3 y 2 3 z 2 3 −2(x 2 1 y 1 y 2 z 1 z 3 + x 2 2 y 1 y 2 z 1 z 2 + x 2 3 y 1 y 3 z 2 z 3 ),
It is not difficult to obtain that f (x, y, z) = (x 1 y 1 z 1 − x 1 y 2 z 3 ) 2 + (x 2 y 1 z 1 − x 2 y 2 z 2 ) 2 + (x 3 y 1 z 2 − x 3 y 3 z 3 ) 2 and f (x * , y * , z * ) = 0 where x * = (0, 0, 1) ⊤ , y * = (0, 1, 0) and y * = (1, 0, 0). These demonstrate that the homogeneous polynomial in (52) is an SOS of bilinear forms but not positive definite. Let A be given by (51) and B defined by Remark 5.14 be a semi-paired symmetric tensor, then It is easy to obtain that eigenvalues of N are 0, 0, · · · , 0 24 , 2, 2, 2, respectively, which implies that the matrix N is positive semidefinite but not positive definite. Thus, by Corollary 5.16 we obtain that the homogeneous polynomial in (52) is an SOS of trilinear forms but not positive definite.
Concluding Remarks
In this paper, we investigated positive definiteness of the fourth order three dimensional and sixth order three dimensional paired symmetric (elasticity) tensor. With the help of the M -eigenvalue we gave several necessary and/or sufficient conditions under which a fourth order three dimensional or sixth order three dimensional paired symmetric (elasticity) tensor is positive definite. By introducing semi-paired symmetric tensor and the unfolded matrix of that tensor, we showed that positive definiteness of a fourth order three dimensional and sixth order three dimensional paired symmetric (elasticity) tensor can be obtained by investigating positive definiteness of some unfolded matrix of a semi-paired symmetric tensor of that tensor. We also proposed a necessary and sufficient condition under which a fourth order three dimensional or sixth order three dimensional paired symmetric (elasticity) tensor is positive semidefinite. These results can be extended to the case of higher order higher dimensional (strongly) paired symmetric tensors. We reported preliminary numerical results which confirm our theoretical findings.
We derived several necessary and sufficient conditions for positive definiteness of a sixth order three dimensional paired symmetric tensor in Section 4, where the most main condition needs to judge positive definiteness of a fourth order paired symmetric tensor, an eighth order biblock symmetric tensor and a twelfth order bi-block symmetric tensor. It is well known that it is difficult to solve the polynomial optimization problem with the involved polynomials being defined by higher order higher dimensional tensors. Thus, one of further issues is how to design effective methods to judge positive definiteness of higher order (strongly) paired symmetric tensors or higher order elasticity tensors. Moreover, in this paper, we mainly gave some analysis for positive definiteness of the fourth order three dimensional and sixth order three dimensional paired symmetric (elasticity) tensor. It is worth investigating effective algorithms to check positive definiteness of paired symmetric (elasticity) tensors arised from practical mechanical problems.
A
:= (a i 1 i 2 ···im ), where a i 1 i 2 ···im ∈ R for all i j ∈ [n] with j ∈ [m], here [l] := {1, 2, . . . , l} for any positive integer l. We use T m,n to denote the set of all m-th order n dimensional real tensors. For any A = (a i 1 i 2 ···im ), B = (b i 1 i 2 ···im ) ∈ T m,n , we will use the inner product defined by A , B = n i 1 ,··· ,im=1 a i 1 i 2 ···im b i 1 i 2 ···im and the Hilbert-Schmidt norm defined by A HS = A , A ∈ T m,n is said to be symmetric if its entries are invariant under any permutation of indices {i 1 , i 2 , . . . , i m }. For any x = (x 1 , . . . , x n ) ∈ R n , a tensor A = (a i 1 i 2 ···im ) ∈ T m,n
Theorem 3. 3 .
3A paired symmetric (elasticity) tensor A = (a ijklpq ) ∈ T 6,3 is positive definite if and only if the smallest M -eigenvalue of A is positive.
Theorem 3 . 4 .
34For any paired symmetric (elasticity) tensor A = (a ijklpq ) ∈ T 6,3 , it follows that λ is an M -eigenvalue of A if and only if −λ is an M -eigenvalue of −A ; and furthermore, A is positive definite if and only if the largest M -eigenvalue of −A is negative.From the point of view of numerical calculation, Theorem 3.4 is useful since positive definiteness detection of a paired symmetric (elasticity) tensor A ∈ T 6,3 can be done by computing the largest M -eigenvalue of −A .
, it follows that λ is an M -eigenvalue of A if and only if −λ is an Meigenvalue of −A ; and furthermore, A is positive definite if and only if the largest M -eigenvalue of −A is negative.
Corollary 3.12. A paired symmetric (elasticity) tensor A = (a ijkl ) ∈ T 4,3 is positive definite if and only if the smallest M -eigenvalue of A is positive.
It is well known that a symmetric matrix M is positive definite if and only if all leading principal minors of M are positive, which is the Sylvester's criterion. Thus, we have the following results, where det(M ) denotes the determinant of the matrix M .
by the above (a)-(c). Then, the polynomial A x 2 y 2 is positive definite if and only if one of the following results holds. (i) The symmetric matrix A 11 is positive definite, and the smallest Z-eigenvalues of the symmetric tensors T 1 A and T 2 A are positive.(ii) The symmetric matrix B 11 is positive definite, and the smallest Z-eigenvalues of the symmetric tensors T 1 B and T 2 B are positive.
Theorem 4. 6 .
6For any paired symmetric tensor A ∈ T 6,3 , the polynomial A x 2 y 2 z 2 defined by(13) is positive definite if and only if one of the following results holds.(i) The smallest M -eigenvalues of tensors A 11 , T 1 A and T 2 A are positive.(ii) The smallest M -eigenvalues of tensors B 11 , T 1 B and T 2 B are positive.(iii) The smallest M -eigenvalue of tensors C 11 , T 1 C and T 2 C are positive.
Remark 5. 4 .
4We define two sets of indices by I := {(i, j, k, l) ∈ {1, 2, 3} 4 : i = j, k = l};
Theorem 5 . 5 .
55For any paired symmetric (elasticity) tensor A ∈ T 4,3 , let B defined by Definition 5.3 be a semi-paired symmetric tensor of A . Then, the quadratic form A x 2 y 2 is an SOS of bilinear forms if and only if the quadratic form Bx 2 y 2 is an SOS of bilinear forms; and tensor A is positive (semidefinite) definite if and only if tensor B is positive (semidefinite) definite.
Corollary 5. 7 .
7For any paired symmetric (elasticity) tensor A ∈ T 4,3 , suppose that B = (b ijkl ) defined by Remark 5.4 is a semi-paired symmetric tensor of A , and the unfolded matrix of B is defined by M := (m st ) with m st = m 3(i−1)+k,3(j−1)+l = b ijkl for any i, j, k, l ∈ {1, 2, 3}; (28) If the matrix M is positive semidefinite, then the biquadratic form defined by (2) is an SOS of bilinear forms; and if the matrix M is positive definite, then the biquadratic form defined by (2) is positive definite.
ii
j, k, l, p, q) ∈ {1, 2, 3} 6 : i = j, k = l and p = q , = j, k = l and p = q; or i = j, k = l and p = q; or i = j, k = l and p = q = j, k = l and p = q; or i = j, k = l and p = q; or i = j, k = l and p = q , j, k, l, p, q) ∈ {1, 2, 3} 6 : i = j, k = l and p = q and S 1 := (i, j, k, l, p, q) ∈ T 1 :
Theorem 6. 3 .
3For any paired symmetric tensor A = (a ijkl ) ∈ T 4,3 and nonnegative integers r and s, suppose that the homogeneous polynomial F r,s and the vector of monomials v r+s+2 (x, y) are defined by (34) and (36), respectively. Then, the polynomial F r,s (x, y) is an SOS if and only if F r,s (x, y) = v r+s+2 (x, y) ⊤ Hv r+s+2 (x, y), where H is a positive semidefinite matrix.
K
:= A ∈ T 4,3 : W(F r,s (x, y)) = V(Q), r, s ∈ {0, 1, 2, . . .}, Q ∈ S dvrs + .(37)For any r, s ∈ {0, 1, 2, . . .}, we define K r+s := A ∈ T 4,3 : W(F r,s (x, y)) = V(Q), Q ∈ S dvrs + .
a
1111 = 1, a 2222 = 1, a 3333 = 1, a 1122 = 2, a 2233 = 2, a 3311 = 2, a 1212 = a 1221 = a 2112 = a 2121 = − 1 2 , a 2323 = a 2332 = a 3223 = a 3232 = − 1 2 , a 3131 = a 3113 = a 1331 = a 1313 = − 1 2 .
it is not difficult to show that λ = 0 is an M -eigenvalue of A , and ((x 1 , x 2 , x 3 ), (y 1 , y 2 , y 3 )) = (eigenvectors of A associated with the M -eigenvalue λ = 0. Furthermore, λ = 0 is the smallest M -eigenvalue of A since the biquadratic form f (x, y) in (44) is positive semidefinite.
Example 7. 3 .
3Let A = (a ijkl ) ∈ T 4,3 be given by a 1111 = 2, a 2222 = 2, a 3333 = 2, a 1313 = a 3113 = a 1331 = a 3131 = − 1 2 , a 2323 = a 3223 = a 2332 = a 3232 = − 1 2 , other a ijkl = 0.
and B defined by Remark 5.4 be a semi-paired symmetric tensor, thenb 1111 = 2, b 2222 = 2, b 3333 = 2, b 1313 = d 3131 = −1, b 2323 = b 3232 = −1, other b ijkl = 0. Suppose M ∈ R 9×9 defined by(28)is the unfolded matrix of B, then m 11 = 2, m 55 = 2, m 99 = 2, m 19 = m 91 = −1, m 59 = m 95 = −1, other m ij = 0.
Example 7 . 4 .
74Let A = (a ijklpq ) ∈ T 6,3 be given by a 111111 = a 112233 = a 221111 = a 222222 = a 331122 = a 333333 = 1, a 111213 = a 111231 = a 112113 = a 112131 = − 1 2 , a 221212 = a 221221 = a 222112 = a 222121 = − 1 2 , a 331323 = a 331332 = a 333123 = a 333132 = − 1 2 , other a ijklpq = 0.
b 111111 = b 112233 = b 221111 = b 222222 = b 331122 = b 333333 = 1, b 111213 = b 112131 = −1, b 221212 = b 222121 = −1, b 331323 = b 333132 = −1, other b ijklpq = 0. Suppose N ∈ R 27×27 defined by(33)is the unfolded matrix of B, then n 11 = n 66 = n 10,10 = n 14,14 = n 20,20 = n 27,27 = 1, n 16 = n 61 = n 10,14 = n 14,10 = n 20,27 = n 27,20 = −1, other n ij = 0.
Table 1 :
1The numerical results of the problem in Example 7.1r
s
ITER
CPU(s)
OPT
VOL
0
0
9
1.313
-1.5243441995e-09
2.7246124908e-12
1
0
11
1.719
-2.6041279810e-10
5.7978715466e-12
0
1
11
1.813
-2.6041187999e-10
5.7989670655e-12
1
1
13
4.594
-2.4538473770e-10
1.5420658730e-11
2
1
14
8.797
-2.3617078649e-10
1.3527256926e-11
1
2
14
8.859
-2.3616477957e-10
1.3665548633e-11
2
2
16
29.344
-2.8204474346e-10
1.0935180661e-11
Table 2 :
2The numerical results of the problem in Example 7.2r
s
ITER
CPU(s)
OPT
VOL
0
0
10
1.219
9.9999999970e-01
1.4472115766e-14
1
0
12
1.734
9.9999999963e-01
1.8617918560e-11
0
1
11
1.844
9.9999999504e-01
2.6733847183e-10
1
1
13
4.906
9.9999999954e-01
1.7099952719e-11
2
1
14
9.000
9.9999999949e-01
1.4822319679e-11
1
2
14
8.938
9.9999999812e-01
3.8117223117e-11
2
2
16
29.734
9.9999999749e-01
6.7937294113e-12
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|
[] |
[
"Quasiclassical Asymptotics and Coherent States for Bounded Discrete Spectra",
"Quasiclassical Asymptotics and Coherent States for Bounded Discrete Spectra"
] |
[
"K Górska \nInstitute of Physics\nNicolaus Copernicus University\nul. Grudziadzka 5/787-100ToruńPLPoland\n\nLaboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance\n",
"K A Penson [email protected] \nLaboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance\n",
"A Horzela \nH. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland\n",
"G H E Duchamp \nInstitut Galilée\nUMR 7030\nUniversité Paris XIII\nLIPN\nCNRS\n99 Av. J.-B. ClementF 93430VilletaneuseFrance\n",
"P Blasiak [email protected] \nH. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland\n",
"A I Solomon [email protected] \nLaboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance\n\nPhysics and Astronomy Department Milton\nThe Open University\nMK7 6AAKeynesUnited Kingdom\n"
] |
[
"Institute of Physics\nNicolaus Copernicus University\nul. Grudziadzka 5/787-100ToruńPLPoland",
"Laboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance",
"Laboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance",
"H. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland",
"Institut Galilée\nUMR 7030\nUniversité Paris XIII\nLIPN\nCNRS\n99 Av. J.-B. ClementF 93430VilletaneuseFrance",
"H. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland",
"Laboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance",
"Physics and Astronomy Department Milton\nThe Open University\nMK7 6AAKeynesUnited Kingdom"
] |
[] |
We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V σ (x) = −|V 0 | |x| −σ , 0 < σ ≤ 2. For these potentials the quasiclassical approximation for n → ∞ predicts quantized energy levels e σ (n) of a bounded spectrum varying as e σ (n) ∼ −n −2σ/(2−σ) . We construct collective quantum states using the set of wavefunctions of the discrete spectrum assuming this asymptotic behaviour. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For σ in the range 0 < σ ≤ 2/3 we present exact implementations of such states for the parametrization σ = 2(k − l)/(3k − l), with k and l positive integers satisfying k > l.
|
10.1063/1.3503775
|
[
"https://arxiv.org/pdf/1007.2617v2.pdf"
] | 119,642,553 |
1007.2617
|
37adaa27cbf5d804bd74714b5a035013bdc684ce
|
Quasiclassical Asymptotics and Coherent States for Bounded Discrete Spectra
5 Jan 2011
K Górska
Institute of Physics
Nicolaus Copernicus University
ul. Grudziadzka 5/787-100ToruńPLPoland
Laboratoire de Physique Théorique de la Matière Condensée
UMR 7600
Université Pierre et Marie Curie
CNRS
Tour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance
K A Penson [email protected]
Laboratoire de Physique Théorique de la Matière Condensée
UMR 7600
Université Pierre et Marie Curie
CNRS
Tour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance
A Horzela
H. Niewodniczański Institute of Nuclear Physics
Polish Academy of Sciences
ul. Eliasza-Radzikowskiego 15231342KrakówPLPoland
G H E Duchamp
Institut Galilée
UMR 7030
Université Paris XIII
LIPN
CNRS
99 Av. J.-B. ClementF 93430VilletaneuseFrance
P Blasiak [email protected]
H. Niewodniczański Institute of Nuclear Physics
Polish Academy of Sciences
ul. Eliasza-Radzikowskiego 15231342KrakówPLPoland
A I Solomon [email protected]
Laboratoire de Physique Théorique de la Matière Condensée
UMR 7600
Université Pierre et Marie Curie
CNRS
Tour 13 -5ièmeét., 4 pl. JussieuB.C. 12175252, Cedex 05ParisFrance
Physics and Astronomy Department Milton
The Open University
MK7 6AAKeynesUnited Kingdom
Quasiclassical Asymptotics and Coherent States for Bounded Discrete Spectra
5 Jan 2011Quasiclassical Asymptotics and Coherent States for Bounded Discrete Spectra 2numbers: 4250Ar0365Sq0365Ge
We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V σ (x) = −|V 0 | |x| −σ , 0 < σ ≤ 2. For these potentials the quasiclassical approximation for n → ∞ predicts quantized energy levels e σ (n) of a bounded spectrum varying as e σ (n) ∼ −n −2σ/(2−σ) . We construct collective quantum states using the set of wavefunctions of the discrete spectrum assuming this asymptotic behaviour. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For σ in the range 0 < σ ≤ 2/3 we present exact implementations of such states for the parametrization σ = 2(k − l)/(3k − l), with k and l positive integers satisfying k > l.
Introduction
The construction of collective quantum states characterizing the whole spectrum of a quantum system is a challenging problem which depends sensitively on the nature of the potential involved. The standard coherent states (CS) are closely related to the harmonic oscillator potential and are defined, for complex z, by
|z = exp(−|z| 2 /2) ∞ n=0 z n √ n! |n ,(1)
where {|n } ∞ n=0 is the Fock space of eigenfunctions of operator H = p 2 /(2m) + mω 2 x 2 /2 satisfying H|n = ω(n + 1/2)|n , n|n ′ = δ n,n ′ . There have been many attempts to generalize the construction of Eq. (1) for potentials other than ∼ x 2 [1,2]. These efforts were hampered by the fact that the exact eigenstates and spectra of general potentials V (x) are known in only a few cases, and for a very special form of V (x). In fact for purely power-law type potentials of the form V σ (x) = ±|V 0 | |x| −σ the only exact solutions we know are for |σ| = 1, 2. The attractive case σ > 2 is considered in a certain sense as unphysical [3]. This paucity forces one either to resort to approximations or to construct trial generalizations of (1) in which some known ingredients are built in, whereas other features are not accounted for. In this context, the quasiclassical approach may play a prominent role [4,5,6,7].
In this work we will be concerned with attractive power-law potentials and the discrete part of their spectrum, which is known to be bounded [3] and which we assume here to be nondegenerate. Here V σ (x) = −|V 0 | |x| −σ , 0 < σ ≤ 2 and the quasiclassical estimate for the spectrum E σ (n) is obtained from the Bohr-Sommerfeld quantization rule 1 b a 2m [E σ (n) − V σ (x)] dx = π(n + 1/2), n = 0, 1, . . .
which can be evaluated as (with = 1)
E σ (n) = − π 2 n + 1/2 √ 2m D(σ) |V 0 | −2σ/(2−σ) → −n −2σ/(2−σ) , n → ∞,(3)
see [6] for derivation and [4,5] for various refinements. In Eq. (2) x = a and x = b are turning points of the potential, defined by p(a) = p(b) = 0, and
D(σ) = b a √ x σ − 1dx. We now use dimensionless units in which H σ (p, x) ⇒ h σ (p, x) = p 2 − x −σ .
Then, we fix the value e σ (0) = 0 and thus arrive at the quasiclassical form of the spectrum
e σ (n) ≈ 1 − c n −2σ/(2−σ) + . . . , 0 < σ ≤ 2, n → ∞,(4)
where the constant c > 0. We shall now follow the approach developed in [8,9,10] and incorporate the form of Eq. (4) to construct the generalization of states defined in Eq. (1) which are specially adapted to attractive potentials proportional to −|x| −σ .
(For a recent treatment of CS for continuous spectra, see [8,11].)
To do so we use the method elaborated for discrete spectra and propose the specific form of a collective quantum state spanned by a set of eigenfunctions |n, σ of h σ (p, x) satisfying (with n, σ|n ′ , σ = δ n,n ′ ) asymptotically, as n → ∞ h σ (p, x)|n, σ = e σ (n)|n, σ .
Note that the equality sign in Eq. (5) is in general not valid for small values of n.
Although |n, σ are not known in general, we still construct a trial wave function [8,9,12,13] in the form generalizing Eq. (1):
|J, γ, σ = N −1/2 σ (J) ∞ n=0 J n/2 exp(−i γ e σ (n)) ρ σ (n) |n, σ ,(6)
where J > 0, and γ are real and ρ σ (n) are so chosen as to assure the convergence of the normalization N σ (J) > 0,
N σ (J) = ∞ n=0 J n ρ σ (n) < ∞, 0 ≤ J ≤ R ≤ ∞.(7)
The choice of ρ σ (n) in Eq. (6) is dictated by an "action identity" of the form [8,9] J, γ, σ|h σ (p, x)|J, γ, σ = J
which directly implies
ρ σ (n) = n j=1 e σ (j), ρ σ (0) = 1,(9)
and consequently the basic relation follows:
e σ (n) = ρ σ (n) ρ σ (n − 1) , n = 1, 2, . . . ,(10)
e σ (0) = 0, which, within our approach should be understood in the asymptotic sense. The states |J, γ, σ should satisfy the resolution of identity with a weight function W σ (J) > 0:
dJ dγ |J, γ, σ W σ (J) J, γ, σ| = ∞ n=0
|n, σ n, σ| = I
which reduces (vide Eq. (102) of Ref. [9]) to an infinite set of integral equations for an unknown positive function W σ (x):
R 0 x n W σ (x) N σ (x) dx = R 0 x n W σ (x) dx = ρ σ (n), n = 0, 1, . . .(12)
where ρ σ (0) = 1. If R < ∞, Eqs. (12) is the Hausdorff moment problem [14]. It is known that if for a given set of ρ(n)'s the positive solution of Eqs. (12) exists then it is always unique [14]. The situation is very different for Hamiltonians with unbounded discrete spectra which lead to the Stieltjes moment problem with R = ∞ in Eq. (12). In this case the solutions can be either unique or non-unique, see [15]. Observe that apart from their orthogonality no specific knowledge of the |n, σ 's is required to derive Eq. (12). The state |J, γ, σ defined by Eq. (6), with ρ σ (n) satisfying Eqs. (10) and (12), is a generalized CS state in a sense of Klauder [9], asymptotically relevant for V σ (x) ∼ − |x| −σ .
Generating solutions of Hausdorff moment problems
Our strategy from now on is: a) identify the form of ρ σ (n)'s to assure that Eq. (12) can be solved for W σ (x) > 0; b) calculate the associated energy spectrum from Eq. (10); and c) identify the exponent σ obtained from Eq. (4) and thus link the potential V σ (x) to ρ σ (n) and |J, γ, σ . Evidently the correspondence in c) above is not unique: one may give different ρ
(s) = M[f (x); s] = ∞ 0 x s−1 f (x)dx, s complex [16]: M W σ (x); s = ρ σ (s − 1), s ≥ 1,(13)
or equivalently
W σ (x) = M −1 [ρ σ (s − 1); x] ,(14)
where M −1 is the inverse Mellin transform. We observe that the moments used in this work will always be of such a nature that the integration range R in Eqs. (12) will be equal to 1. Then the moment sequences will be decreasing functions of n. (We stress that this is not a general rule: there exist Hausdorff moment problems necessitating R > 1 for which the moment sequences are increasing [17,18].) In our search for positive solutions of Eqs. (12) we were greatly helped by a relation between the Laplace transform and a special case of Mellin transform [20]. To elucidate this link suppose that a function F (x) is considered for which its Laplace transform is known:
L[F (x); p] ≡ F (p) = ∞ 0 e −px F (x) dx, p > 0.(15)
We now perform a change of variable x = ln(1/y) in Eq. (15), which gives
F (p) = ∞ 0 y p−1 F (ln(1/y)) H(1 − y) dy,(16)
where H(z) is the Heaviside function. Through a formal renaming p ↔ s we treat Eq. (16) as a Mellin transform
F (s) = M[F (ln(1/x)) H(1 − x); s].(17)
The relations of Eqs. (15)-(17) allow one to search for possible solutions of the Hausdorff moment problem (12) for R = 1 via the method of the inverse Laplace transform [19], with a succession of following steps: i) choose a strictly decreasing sequence of moments ρ(n), n = 0, 1, . . .; ii) rename them as F (p + 1), s.Eq. (16); iii) search for the inverse Laplace transform F (x) corresponding to F (p) and check whether
W (x) = F (ln(1/x)) H(1 − x)
is a positive function on [0, 1]; then, if so, W (x) is the solution of the Hausdorff moment problem Eqs. (12). One is helped here by the fact if
that F (x) is positive on [0, ∞) then F (ln(1/x)) is positive on [0, 1].
Illustrative example
We illustrate this approach with an example directly related to our construction. We choose the moments as
ρ(n) = e exp(− √ n + 1), ρ(0) = 1; the relabelling ρ(n) ↔ F (p+1) gives F (p) = e exp(− √ p) which,e exp(− √ p) = ∞ 0 e −px e 2 √ π x 3/2 exp(−1/4x) dx.(18)
In the next step we verify that e
2 √ π [ln(1/x)] −3/2 exp − 1 4 ln(1/x) is a positive function on [0, 1] and consequently 1 0 x n e 2 √ π [ln(1/x)] 3/2 exp − 1 4 ln(1/x) dx = e · e − √ n+1 ,(19)
n = 0, 1, . . ., is a complete solution of the Hausdorff moment problem Eqs. (12). With the above moments the spectrum Eq. (10) is
e(n) = e √ n− √ n+1 ,(20)
and its n → ∞ asymptotics is
e(n) → 1 − 1 2 n 1/2 + 1 8 n + . . . ,(21)
which with Eq. (4) determines σ = 2/5 and c = 1/2. The generalized coherent state describing such a spectrum is given by Eq. (6) with the normalization
N (J) = 1 e ∞ n=0 e (n+1) 1/2 J n , 0 ≤ J < 1.(22)
The CS |J, γ, 2/5 is then asymptotically relevant for motion in the potential V 2/5 (x) ∼ −|x| −2/5 . The formula Eq. (19) can also be cross-checked by referring to the tables of inverse Mellin transforms (see formula 3.7 on p. 174 for α = 1 of Ref. [20]). In the spirit of Eqs. (12) we call the weight function in Eq.
(19) W 2/5 (x), 0 ≤ x ≤ 1, which can also be derived from eL(1/2, ln(1/x)), where L(γ, x) = γ 2π x −3/2 e −γ/(2x)
is the so-called one-sided Lévy stable distribution [21].
We now present a more general case which can be treated with the above method. For that purpose we again stress that we are looking for special sequences of moments ρ(n) satisfying the Hausdorff moment equations which at the same time possess very specific asymptotic properties implied by Eqs. (4) and (10). These are very restrictive conditions indeed. The search for such solutions may proceed by exclusion and at the beginning it was not certain at all if a general solution existed. It is all the more satisfying that a parametrization can be given that produces a vast ensemble of solutions, at least for some range of values of σ.
2.2.
Full solutions for σ rational in the range 0 < σ ≤ 2/3 and for σ = 1
Let us define a sequence of moments, parametrized by a, ν, k and l, and given by:
ρ (a,ν) (k, l; n) = e a (n + 1) −ν e −a (n+1) l/k , n = 0, 1, . . . ,(23)
with conditions: k and l positive integers; k > l; a > 0 and ν ≥ 0. The last condition assures the positivity of the weight function, see Appendix A. We use now the formula 2.2.1.19 listed without proof on p. 53 of [19]:
∞ 0 e −px √ k l 1/2−ν (2π) (k−l)/2 x ν−1 G k,0 l,k a k l l k k x l ∆(l,ν) ∆(k,0) dx = p −ν e −a p l/k ,(24)
for p > 0, where G m,n p,q (z| · · ·) is Meijer's G function [22]. The detailed demonstration of Eq. (24) will be given elsewhere.
We transform the Eq. (24) with x = ln(1/y) and arrive at the expression of the type of Eq. (16), namely
∞ 0 y p−1 e a √ k (2π) (k−l)/2 l 1/2−ν [ln(1/y)] ν−1 (25) × G k,0 l,k a k l l k k [ln(1/y)] l ∆(l,ν) ∆(k,0) H(1 − y) dy = e a p −ν e −a p l/k , p > 0.
In Eqs. (24) and (25) we use a compact notation for special lists of k elements [19]: ∆(k, a) = a k , a+1 k , . . . , a+k−1 k . The Meijer G function is defined as an inverse Mellin transform [22,23]:
G m,n p,q z α 1 ...αp β 1 ...βq = M −1 m j=1 Γ(β j + s) n j=1 Γ(1 − α j − s) q j=m+1 Γ(1 − β j − s) p j=n+1 Γ(α j +z = 0, 0 ≤ m ≤ q, 0 ≤ n ≤ p;(28)
α j ∈ C, j = 1, . . . , p; β j ∈ C, j = 1, . . . , q.
For a full description of integration contours in Eq. (26), general properties and special cases of the G functions see [22,23]. In Eq. (27) we present a transparent notation, which we will use henceforth, inspired by computer algebra [24]. With this notation Eq. (25) now becomes
Observe that in Eq. (30) only the second and third lists of parameters are non empty in the G functions (vide the notation of Eq. (27)), as may be inferred from the conditions of Eq. (28). Eqs. (30) and (31) have a very rich ensemble of solutions which will be studied for various values of the parameters a, ν, k and l. The role played by a and ν is fundamentally different from that played by k and l. The asymptotic behaviour as n → ∞ of ρ (a,ν) (k, l; n) does not depend on a and ν and the exponent of the power-law n dependence of spectra is a function of l/k only: e (a,ν) (k, l; n) = 1 − l k a n (k−l)/k + . . . ,
which, by Eqs. (3) and (4) immediately implies that σ may be "fine-tuned" with the parametrization σ = 2(k − l) 3k − l , k > l; k, l = 1, 2, . . . .
Eq. (33) confines σ to a possible range of 0 < σ < 2/3. In Eq. (32) the constant c of Eq. (4) is equal to c = al/k. The above analysis indicates that the weights W (a,ν) (k, l; y) for different a and ν give the same asymptotics. We shall give examples of such an asymptotic "degeneracy" with explicit forms of W (a,ν) (k, l; y) = W (a,ν ′ ) (k, l; y) for a few ν = ν ′ . For simplicity, from now on a fixed value a = 1 will be used for all examples derived from Eq. (24). We shall observe the onset of complexity with increasing values of k and l: starting with k = 4 and l = 1 we leave the realm of standard special functions as then the corresponding Meijer's G functions can be only converted to finite sums of generalized hypergeometric functions of type p F q . They are however available through computer algebra systems [24] and their properties are, to a large extend, readily accessible. Since a = 1 we shall denote W (1,ν) (k, l; y) ≡ W (ν) (k, l; y).
Special cases:
1. k = 2, l = 1 and ν = 0:
W (0) (
where K ν (z) is the modified Bessel function of the second kind. Eq. (36) can also be checked with formula 3.13, p. 175 of [20]. With Eq. (32) the corresponding spectrum varies as
e (1,0) (3, 1; n) → 1 − 1 3 n 2/3 + . . . , n → ∞,(37)
yielding σ = 1/2 and c = 1/3. 3. k = 3, l = 1 and ν = 1/2: It is instructive to compare weight functions leading to the same spectrum asymptotics. To this end we present the weight functions from Eqs. (37) and (38) in Fig. 1 . 4. k = 3, l = 2 and ν = 0: which leads to the asymptotics:
W (1/2) (3, 1; y) = e √ 3/(2π) [ln(1/y)] 1/2 G [[ ], [1/2]],W (0) (3, 2; y) = e 3/π ln(1/y) G [[ ],e (1,0) (3, 2; n) → 1 − 2 3 n 1/3 + . . . ,(40)
with σ = 2/7 and c = 2/3. which has a representation in terms of a sum of three hypergeometric functions of type 2 F 2 , which will not be quoted here. The asymptotics is that of Eq. (40). The weight functions from Eqs. (39) and (41) which share the same spectrum asymptotics are compared in Fig. 2. 6. k = 4, l = 1, ν = 0: The corresponding weight function W (0) (4, 1; y) has an exact representation in terms of a sum of three hypergeometric functions of type 0 F 2 which we will not given here.
This pattern extends to higher values of k and l for which the weight functions become increasingly complicated. They can however be fully handled analytically and graphically with a relative ease: we always deal with finite sums of hypergeometric functions.
We shall go over to further examples and employ a wealth of formulae of Ref. [19] and [20], different from that of Eq. (24). 7. We now choose a function differing from the exponential which nevertheless yields asymptotic behaviour which is close to Eq. (21): this is provided by the formula 3.16.6.7, p. 358 of [19] or, alternatively by Eq. 7.69, p. 230 (for a = 1) of [20], namely:
e (K) (n) ∼ 1 − 1 2n 1/2 − 1 8n + . . . ,(43)
leading to c = 1/2 and σ = 2/5. The weight functions from Eqs. (19) or (35), and Eq. (44) display the same spectrum asymptotics and are illustrated in Fig. 3. 8. We use now Eq. 2.2.2.1, p. 53 of [19] for the choice ν = 4/3 and a = 1, which leads to the normalized Hausdorff moment problem:
giving the asymptotics with σ = 2/3 and c = 4/3:
e(n) ∼ 1 − 4 3n + . . . .(45)
In Eq. (45) I ν (z) is the modified Bessel function of the first kind. Note that the value σ = 2/3 cannot be obtained from Eq. (33). 9. In this final example we address the problem of the one-dimensional Coulomb potential ∼ −|x| −1 for which the exact spectrum is [3] e c (n) = 1 − 1 (n + 1) 2 , n = 0, 1, . . . , ∞.
The CS for this case have been constructed in [8] and the corresponding moments are given by ρ c (n) = n+2 2(n+1) , n = 0, 1, . . ., and they lead to the exact form for the corresponding weight function
W c (y) = 1 2 1 + δ(y − 1 − ) H(1 − y).(47)
The Coulomb problem, even in its simplified version treated here, presents a particularity in that its resolution of unity is distributional in character as it involves, in Eq. (47), the Dirac delta function δ(z). Although the solution Eq. (47) is unique for the exact moments ρ c (n) defined above, we are able to construct another weight function V c (y) which will asymptotically reproduce e c (n) of Eq. (46). To this end we use the formula 2.2.2.8, p. 54 of [19] for a = 1, which gives
with asymptoticsẽ c (n) → 1 − n −2 + 3n −3 + . . . . Note that V c (y) displays a non-trivial dependence on y ∈ [0, 1] but still retains a Dirac peak at y = 1. We are not aware of existence of any weight function without Dirac's delta leading to Eq. (46). The different weight functions relevant for the Coulomb interaction, both of them involving Dirac's delta function at y = 1, are schematically displayed on Fig. 4.
Discussion and Conclusion
In this work we used a semiclassical approximation for the spectra of one-dimensional inverse power-law potentials to construct approximate coherent states, relevant for these potentials. Our goal in using this construction was to achieve the resolution of unity for a given potential characterized by an exponent σ in V σ (x) ∼ −|x| −σ . In this sense the ρ σ (n)'s of Eq. (9) should be perceived as a sort of trial parameters which should asymptoticaly reproduce e σ (n) via Eq. (10). This leads, of course, to a multiplicity of possible choices of ρ σ (n) leading to the same e σ (n). The price of this approximation is the fact, that the temporal stability characterizing exact Gazeau-Klauder CS [8,11] denoted by |J, γ, σ GK (i.e. the equality e −ihσt |J, γ, σ GK = |J, γ + t, σ GK ) will be only approximately satisfied by the states of Eq. (6). On the contrary, the basic Gazeau-Klauder axiom of the resolution of unity is fully maintained in our construction.
For certain values of σ, more precisely for rational σ such that 0 < σ ≤ 2/3 and for σ = 1, we are able to produce many resolutions of unity that are asymptotically relevant for one and the same σ. This is due in the first place to two key formulae, Eq. (24) and (33) which involve the use of the inverse Laplace transform. For other values of σ in the range 2/3 < σ < 2 (except σ = 1) our approach does not produce any required solutions for the resolution of unity.
We should comment here on the nature of approximation involved in the formulation of Eq. (6). Since for general σ and n neither exact spectra nor exact eigenfunctions |n, σ are known we strike a compromise in Eq. (6) by retaining the exact orthonormal eigenfunctions and replacing the energy spectrum by its quasi-classical form of Eq. (4).
In Eq. (6) we substitute the quasiclassical approximation for e σ (n) instead of the exact spectrum. This means that we are using the asymptotic approximation in the low energy region, where its use is not a priori justified. However, for certain values of σ the quasiclassical approximation is very successful even down to the low energy: in fact, for the linear repulsive potential V (x) ∼ x (x ≥ 0; V (x) = ∞, x < 0) it predicts the correct energies and wave functions right down to the ground state [5,6]. For attractive potentials, for which in general one does not have exact solutions, the agreement is less spectacular [6] but in general the quasiclassical approximation works well in large parts of the spectra and not only for n → ∞, in which region it tends to the exact solution. Therefore we believe that the use of Eq. (4) in Eq. (6)
σ
(n), r = 1, 2, . . . which yield the same asymptotics via Eq. (4). We consider Eq. (12) as a Mellin transform f *
=
G([[α 1 , . . . , α n ], [α n+1 , . . . , α p ]], [[β 1 , . . . , β m ], [β m+1 , . . . , β q ]], z), (27) where in Eq. (26) empty products are taken to be equal to one. In Eqs. (26) and (27) the parameters are subject of conditions:
−1 e a √ k l 1/2−ν (2π) (k−l)/2 [ln(1/y)] ν−1 (29) × G [[ ], [∆(l, ν)]], [[∆(k, 0)], [ ]], a k l l k k [ln(1/y)] l dy= e a p −ν e −ap l/k W (a,ν) (k, l; y)dy, n = 0, 1, . . . , ∞.
the same n-dependence as in Eq. (37).
Figure 1 .
1Plot of the weight functions W (y); the line I corresponds to W (0) (3, 1; y), see Eq. (36) and the line II represents W (1/2) (3, 1; y), see Eq. (38). 5. k = 3, l = 2, ν = 1/4: This case can be expressed by the G function
Figure 2 .
2Plot of the weight functions W (y); the line I corresponds to W (0) (3, 2; y), see Eq. (39) and the line II represents W (1/4) (3, 2; y), see Eq. (41).
Bessel functions K ν (z) in Eq. (42) for ν = p/2, p = 1, 2, . . . are not elementary functions. The weight function in the l.h.s. of Eq. (42) is positive on [0, 1] and normalized as ρ (K) ν (0) = 1. The asymptotic behaviour for ν = 4/3 is close to that of Eq. (21):
e −1 [ln(1/y)] 1/6 I 1/3 2 ln(1/y) dy = e −1 e 1/(n+1) (n + 1) 4/3 ,
Figure 3 .
3Plot of the weight functions W (y): the line I corresponds to W (0) (2, 1; y), see Eqs. (34) and (19) and the line II represents the weight function in Eq. (42).
δ(y − 1 − ) dy = e −1 e 1/(n+1) , n = 0, 1, . . . ,
Figure 4 .
4Plot of the weight functions W (y): the line I corresponds to W c (y), see Eqs. (47) and the line II represents the weight function V c (y), see Eq. (48). The delta peak at y = 1 are represented by a vertical line. Observe that this is a schematic plot: different coefficients (1/2 and e −1 , s. Eq. (47) and (48) respectively) in front of delta peaks at y = 1 cannot be accounted for.
be viewed as a multiple, k-fold Mellin convolution of positive functions, which by (A.3) is itself positive. The proof of positivity of W (1,ν) (y) is completed by remarking that if F (y) is positive on [0, ∞) then for α and β arbitrary ln the ν < 0 case destroys the positivity of W (1,ν) (y) and is not relevant for our purposes.
which can be neatly expressed in terms of modified Bessel functions K ν (z):[1/2]], [[1/3, 2/3], [ ]],
4
27[ln(1/y)] 2 ,
W (0) (3, 2; y) =
e 2
√
3
27π
[ln(1/y)] −3 exp
2
27[ln(1/y)] 2
× K 1/3
2
27[ln(1/y)] 2 + K 2/3
2
27[ln(1/y)] 2
(39)
AcknowledgementsThe authors acknowledge support from Agence Nationale de la RechercheAppendix A.We give here a streamlined proof of positivity W (1,ν) (k, l; y), for 0 ≤ y ≤ 1 from Eq. (30) under the conditions k, l positive integers, k > l, and ν ≥ 0.Consider first Meijer's G function of Eq. (24), with ∆(k, a) = a k , a+1 k , . . . , a+k−1The convolution property for M[f (x); s] = f * (s) and M[g(x); s] = g * (s)for f (x) > 0 and g(x) > 0 clearly conserves the positivity, see[15]and references therein. M −1 Γ(s + j k ) Γ(s + ν+j l )
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O I Marichev, Handbook of Integral Transforms of Higher Transcendental Functions. Theory and Algorithmic Tables. ChichesterEllis Horwood LtdO. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions. Theory and Algorithmic Tables (Ellis Horwood Ltd, Chichester, 1983).
. A P Prudnikov, Yu A Brychkov, O I Marichev, Integrals and Series. 3Gordon and BreachA. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 3: More Special Functions (Gordon and Breach, New York, 1998).
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[] |
[
"Chiral symmetry and mesons in hot and dense matter: recent developments HOT AND DENSE RESONANCES IN UNITARIZED CHIRAL PERTURBATION THEORY",
"Chiral symmetry and mesons in hot and dense matter: recent developments HOT AND DENSE RESONANCES IN UNITARIZED CHIRAL PERTURBATION THEORY"
] |
[
"A Gómez Nicola ",
"D Fernández-Fraile \nDepartment of Physics\nNuclear Theory Group\nBrookhaven National Laboratory\nUptonNY-11973USA\n",
"\nDepartamento de Física Teórica II. Univ. Complutense\n28040MadridSpain\n"
] |
[
"Department of Physics\nNuclear Theory Group\nBrookhaven National Laboratory\nUptonNY-11973USA",
"Departamento de Física Teórica II. Univ. Complutense\n28040MadridSpain"
] |
[] |
We review recent results on properties of the meson gas relevant for Heavy Ion Collision and Nuclear Matter experiments, within the framework of chiral lagrangians. In particular, we describe the temperature and density evolution of the σ and ρ poles and its connection with chiral symmetry restoration, as well as the chemical nonequilibrated phase and transport coefficients.It is well established nowadays that QCD undergoes a chiral restoration transition, as confirmed by different lattice simulations[1,2]. The transition is presumably a crossover for 2+1 massive light flavors, which means that the critical point could be different for different observables. In the lattice, the main order parameters are the quark condensate and the light scalar susceptibility. Nevertheless, there have been several efforts in the literature to try to find signals of chiral symmetry restorations in physical observables which could be eventually measured in Heavy Ion or Nuclear Matter experiments. The most promising signals are related to the behaviour of resonances in hot and dense matter, such as the ρ − a 1 degeneracy[3,4]or the role of the ρ resonance in the dilepton spectrum[3,5]. In the latter case, a significant dropping of the ρ mass would be interpreted as a signal of chiral restoration according to the Brown-Rho scaling[6]or QCD sum rules [7] scenarios. However, this does not seem to be the case experimentally in Heavy Ion Collisions, where the alternative explanation of broadening dominance [3] is equally valid to explain the data and it is actually favored in recent experiments, such as NA60 with dimuons [8]. At RHIC energies (PHENIX), even the broadening picture is insufficient to explain the observed excess in the region of low invariant mass[9,10]. The situation is less clear in cold nuclear matter experiments. The E325-KEK collaboration[11]has reported a measurable shift in the masses of vector mesons compatible with theoretical predictions based on Brown-Rho scaling and QCD sum rules. On the other hand, the JLab-CLAS experiment[12]has obtained results compatible with vanishing mass shift, as predicted by most in-medium hadronic manybody analysis, where broadening is the dominant effect[13,14].The σ / f 0 (600) meson is also a suitable candidate to study signals of chiral restoration
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10.1063/1.3542028
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"https://arxiv.org/pdf/1011.3920v1.pdf"
] | 40,487,225 |
1011.3920
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665e9999bea0caa7c6f3d69c624b3fe4836f19e7
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Chiral symmetry and mesons in hot and dense matter: recent developments HOT AND DENSE RESONANCES IN UNITARIZED CHIRAL PERTURBATION THEORY
17 Nov 2010
A Gómez Nicola
D Fernández-Fraile
Department of Physics
Nuclear Theory Group
Brookhaven National Laboratory
UptonNY-11973USA
Departamento de Física Teórica II. Univ. Complutense
28040MadridSpain
Chiral symmetry and mesons in hot and dense matter: recent developments HOT AND DENSE RESONANCES IN UNITARIZED CHIRAL PERTURBATION THEORY
17 Nov 2010Chiral SymmetryMesonsChiral LagrangiansTransport Coefficients PACS: 1110Wx1239Fe2575-q2165Jk
We review recent results on properties of the meson gas relevant for Heavy Ion Collision and Nuclear Matter experiments, within the framework of chiral lagrangians. In particular, we describe the temperature and density evolution of the σ and ρ poles and its connection with chiral symmetry restoration, as well as the chemical nonequilibrated phase and transport coefficients.It is well established nowadays that QCD undergoes a chiral restoration transition, as confirmed by different lattice simulations[1,2]. The transition is presumably a crossover for 2+1 massive light flavors, which means that the critical point could be different for different observables. In the lattice, the main order parameters are the quark condensate and the light scalar susceptibility. Nevertheless, there have been several efforts in the literature to try to find signals of chiral symmetry restorations in physical observables which could be eventually measured in Heavy Ion or Nuclear Matter experiments. The most promising signals are related to the behaviour of resonances in hot and dense matter, such as the ρ − a 1 degeneracy[3,4]or the role of the ρ resonance in the dilepton spectrum[3,5]. In the latter case, a significant dropping of the ρ mass would be interpreted as a signal of chiral restoration according to the Brown-Rho scaling[6]or QCD sum rules [7] scenarios. However, this does not seem to be the case experimentally in Heavy Ion Collisions, where the alternative explanation of broadening dominance [3] is equally valid to explain the data and it is actually favored in recent experiments, such as NA60 with dimuons [8]. At RHIC energies (PHENIX), even the broadening picture is insufficient to explain the observed excess in the region of low invariant mass[9,10]. The situation is less clear in cold nuclear matter experiments. The E325-KEK collaboration[11]has reported a measurable shift in the masses of vector mesons compatible with theoretical predictions based on Brown-Rho scaling and QCD sum rules. On the other hand, the JLab-CLAS experiment[12]has obtained results compatible with vanishing mass shift, as predicted by most in-medium hadronic manybody analysis, where broadening is the dominant effect[13,14].The σ / f 0 (600) meson is also a suitable candidate to study signals of chiral restoration
through its in-medium behaviour, since it has the quantum numbers of the vacuum. A simple O(4) model description would suggest that the dropping of σ ∼ qq driven by chiral restoration should imply a significant reduction of the σ mass, at least in the chiral limit. Such a reduction could produce experimentally an enhancement of ππ scattering and cross section near the point where M σ → 2m π and the phase space shrinks. This is the so called threshold enhancement effect [15], whose signals are observed in cold nuclear matter reactions πA → ππA ′ [16,17] and γA → ππA ′ [18].
Our approach is to consider the temperature and density modifications of the lightest meson resonances in the unitarized chiral lagrangian framework [19,20,21,22]. We rely mostly on Chiral Perturbation Theory (ChPT), which provides a model-independent low-energy expansion for a given ππ scattering partial wave with total isospin I and angular momentum J as t IJ (s; T ) = t IJ 2 (s) + t IJ 4 (s; T ) + O(p 6 ) with s the center of mass energy squared and T the temperature. According to the usual ChPT scheme, t 2 is the tree level contribution from the lowest order lagrangian L 2 , while t 4 includes both the tree level from L 4 and the one-loop diagrams from L 2 . The latter contain the dependence with the temperature [19], as well as the imaginary part demanded by unitarity, which at finite temperature reads Imt 4 = σ T |t 2 | 2 , where:
σ T (s) = σ 0 (s)[1 + 2n( √ s/2)](1)
is the two-pion thermal phase space, with n(x) = [exp(x/T ) − 1] −1 the Bose-Einstein distribution function and σ 0 (s) = 1 − 4m 2 π /s. The ChPT series cannot reproduce a resonant behaviour, which can be recovered by using a unitarization method, which amounts to find a unitary amplitude t satisfying exactly Imt = σ T |t| 2 and matching the ChPT series when expanded at low energies or temperatures. We follow the Inverse Amplitude Method (IAM) which reads:
t IAM = t 2 (s) 2 t 2 (s) − t 4 (s; T ) + A(s; T )(2)
where A(s; T ) is a known function constructed from t 2 and t 4 ensuring that the zeros of the unitarized amplitude coincide with those of the perturbative one and which can be alternatively obtained using dispersion relations [23]. It is irrelevant for poles far from the real axis but should be included in medium analysis where, as commented above, chiral symmetry may induce a vanishing imaginary part and in fact the A function in that case prevents the appearance of spurious poles [21]. Once we have a unitarized ππ amplitude, the σ and ρ are generated as poles in the second Riemann sheet dynamically, i.e., without introducing them explicitly in the lagrangian nor assuming anything about their nature. The results with the finite-T IAM are showed in Fig.1. The so-called low-energy constants of L 4 are fixed to yield the mass and width of the ρ at their physical values at T = 0. The main conclusions of our analysis are: i) The main effect for the ρ is thermal broadening, in agreement with dilepton data, the mass being reduced only slightly for temperatures below chiral restoration, ii) such broadening does not only come from phase space increasing σ T /σ 0 but also from the increasing of the effective ρππ vertex, extracted from the residue of the amplitude at the pole position, iii) in the σ channel, mass reduction dominates, presumably driven by chiral restoration, and at some point overcomes the thermal width phase space increasing so that the imaginary part (width) reaches a maximum value and then starts decreasing, iv) Near chiral restoration, the width of the σ remains sizable (its spectral function is not peaked around the mass as in a Breit-Wigner resonance) so that no threshold enhancement is expected in a Heavy Ion environment, v) The mass dropping pattern in both the σ and ρ cases does not scale with the condensate as expected from Brown-Rho scaling or simple O(4) model arguments for the σ , not even near the chiral limit. The next question is how to incorporate the effect of the nuclear density ρ in our approach. At T = 0, a simple procedure [21] which encodes most of the relevant corrections is to scale the pion decay constant f π according to [24]:
f 2 π (ρ) f 2 π (0) ≃ qq (ρ) qq (0) ≃ 1 − σ πN m 2 π f 2 π (0) ρ ≃ 1 − 0.35 ρ ρ 0 (3)
where ρ is the nuclear density, σ πN ≃ 45 MeV is the pion-nucleon sigma term and ρ 0 ≃ 0.17 fm −3 is the normal or saturation nuclear matter density. In this way we are actually examining the scaling properties of the resonances when we scale f 2 π (ρ), or equivalently the quark condensate. Actually, in this simple approach there is no broadening source, so that only the masses change. The results of the mass scaling in our IAM approach are showed in Fig.2. In the I = J = 0 channel, the scaling corresponds to the σ of the O(4) model. When the pole approaches the real axis, threshold enhancement is clearly observed and when it reaches it a pole-doubling occurs: one of the poles remains near
f Π 2 Ρ f Π 2 0 fΠ Ρ fΠ 0 M Ρ M 0 I J 0 0.5 0.6 0.7 0.8 2 mΠ M 0 mΠ M 0 fΠ Ρ fΠ 0 f Π 2 Ρ f Π 2 0 fΠ Ρ fΠ 0 M Ρ M 0 I J 1 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.4 0.6 0.8 fΠ Ρ fΠ 0 FIGURE 2.
Scaling of the mass in the IAM, compared with that of f π (ρ). In the I = J = 0 channel, the masses displayed correspond to the lowest masses of the second-sheet poles.
threshold and eventually jumps into the first Riemann sheet becoming a ππ bound state, while the other one (plotted in Fig.2) tends to degenerate with the pion as its chiral partner. Although this phenomenon takes place at rather high ρ ∼ 2ρ 0 for this approach to be fully trusted, it has been obtained also in [25] and reveals the molecular-like character of the f 0 state [21]. The chiral degeneracy between the dynamically generated σ and the pion for f π → 0 had been also noticed in [26]. In the ρ channel, the f π scaling induces a linear M ρ scaling in agreement with the experimental value of [11] and the theoretical predictions of [6] and [7], which however must be taken with a pinch of salt because all the broadening effects coming from many-body interactions are neglected. The picture that emerges from our analysis is then that when broadening effects are ignored, chiral restoration in the σ channel takes place more or less along the expected O(4) pattern. However, the T = 0 study shows that thermal broadening distorts the picture of aqq-like narrow state and, despite experiencing a large mass reduction, the state is still wide near the transition. This is confirmed by the detailed many-body calculation performed in [21] for the σ channel, including both nuclear density and temperature effects. The unitarization scheme used is the Bethe-Salpeter one, more suitable for nuclear many-body analysis, and takes into account the pion self-energy in the nuclear medium plus all the other relevant diagrams compatible with chiral symmetry at the same order in density. The results in Figure 3 for the imaginary part of the amplitude show some distinctive features of this full approach: the abrupt change with respect to the vacuum or thermal case, originated by physical in-medium excitations such as particle-hole or ∆-hole, which produce also strength below threshold, the sizable threshold enhancement, even though the pole is still rather away from the axis like in the thermal case, and finally the amplification of the threshold strength by the combined effect of temperature and density, which in fact accelerates the migration of the pole towards threshold by sudden mass decrease.
PIONS OUT OF CHEMICAL EQUILIBRIUM
In the hadron gas formed in a Relativistic Heavy Ion Collision, it is phenomenologically well justified to consider a phase where thermal equilibrium still prevails but inelastic collisions involving particle number change are negligible, so that chemical equilibrium is lost. For instance, according to the estimates in [27], at T = 150 MeV the relaxation time of elastic ππ collision is τ el ∼ 2fm/c, whereas that of the process ππ ↔ ππππ is τ in ∼ 200 fm/c, the hadronic phase lifetime being about 10 fm/c. In other words, there is a temperature window between the chemical freeze-out temperature T chem ∼ 180 MeV, below which approximately only elastic collisions remain, and the thermal freeze-out one T ther ∼ 100-120 MeV of hadron decoupling, where the chemical potential associated to pion number conservation µ π is not zero [28,29,30]. Actually, in this phase µ π is a function of T so that µ π (T → T chem ) = 0. On the other hand, different phenomenological fits of particle yields and ratios at SPS and RHIC energies predict µ π (T ther ) ∼ 70-100 MeV [29,30].
In a recent work [31], we have considered a ChPT description of this phase in the pion gas, within a Quantum Field Theory treatment of chemical nonequilibrium for boson fields. A crucial point is that the total pion number is only approximately conserved, unlike for instance the electric charge or the third isospin component. This makes the usual path-integral description in terms of field states inconvenient for this case. We have used instead an holomorphic representation, which allows to develop the formalism in terms of creation and annihilation operators, in terms of which one can easily express the free particle number. Another technical complication is that there are no KMS-like periodicity conditions, which prevents the imaginary-time thermal formalism to be used. As in other nonequilibrium formulations, the appropriate choice is a real-time contour in complex time. This implies the doubling of the pion degrees of freedom, so that the propagators have a two-dimensional matrix structure. One of the two types of fields is unphysical and only appears in internal lines. After a detailed analysis, we obtain the generating functional, which can be read from the usual one at µ π = 0 with the following replacements for the free propagators and for the free partition function:
D 11 (p 0 , E p ) = i p 2 0 − E 2 p + iε + 2πδ (p 2 0 − E 2 p )n(|p 0 | − µ π ) D 22 (p 0 , E p ) = −i p 2 0 − E 2 p + iε + 2πδ (p 2 0 − E 2 p )n(|p 0 | − µ π )D 12 (p 0 , E p ) = 2πδ (p 2 0 − E 2 p ) [θ (−p 0 ) + n(|p 0 | − µ π )] = D 21 (−p 0 , E p ) log Z 0 β = −V d 3 p (2π) 3 β E p 2 + log 1 − e −β (E p −µ)(4)
with V the system volume and E 2 p = | p| 2 + m 2 π . With the above ingredients, we can calculate the different thermodynamical variables of the pion gas in this regime, by evaluating the corresponding closed diagrams, whose topology is the same as the µ π = 0 case [32]. We have extended our analysis up to O(T 8 ). At that order, particle-changing processes can be seen to become already important, but the perturbative scheme remains consistent since an additional physical condition has to be imposed in order to regulate the plasma dynamical evolution from thermal to chemical equilibrium, namely the µ π (T ) function. A simple way to model this is to demand that the ratio of entropy density to pion particle number density s/n remains constant [28]. This isentropic condition is nothing but the combination of entropy conservation without dissipation and approximate particle number conservation in this phase. The crucial point is that this ratio is a decreasing function of both T and µ π , so that as the systems cools down, the chemical potential grows in order to keep it constant. The precise value of s/n can be fixed either by the chemical freeze-out temperature at which µ π (T chem ) = 0 or by the value µ π (T ther ) at thermal freeze-out. We show in Figure 4 our results for different chiral orders, comparing also with the virial expansion approach followed in [33]. We see that one of the consequences of including the pion interactions is to lower the value of T chem by about 25 MeV with respect to the ideal gas.
Our diagrammatic approach allows also to calculate the pion self-energy corrections in T and µ π . The detailed analysis and results are given in [31]. To leading order in the pion density (dilute gas regime) the self-energy obeys a Luscher-like relation:
m 2 π (T, µ π ) − m 2 π (0, 0) = − d 3 p (2π) 3 n(E p − µ π ) 2E p Re T f ππ (s = (E p + m π ) 2 − | p| 2 ) + O(n 2 ) Γ p (T, µ π ) = 1 2E p d 3 k (2π) 3 n(E k − µ π ) 2E k Im T f ππ s = (E p + E k ) 2 − | p + k| 2 + O(n 2 ) (5)
where T f ππ (s) is the forward ππ scattering amplitude. The O(p 4 ) and unitarized (IAM) amplitudes produce a decreasing pion mass with both T and µ π . Actually, at a given temperature, the mass can reach the chemical potential, as showed in Figure 4. This implies the interesting possibility that pion Bose-Einstein condensation could be reached dynamically, i.e., driven by interactions. The corresponding µ BE (T ) where this condition is met is above but not far from the isentropic curves for realistic chemical freeze-out conditions. On the other hand, from the pion width Γ p , we can estimate the elastic mean collision time τ el and the thermal freeze-out temperature where τ el (T ther ) ∼ 10 fm/c. Taking into account the µ π dependence of τ el through the isentropic µ π (T ) yields a reduction ∆T ther ∼ -20 MeV with respect to neglecting chemical nonequilibrium.
TRANSPORT COEFFICIENTS
Transport coefficients measure the linear response of the system to a deviation from equilibrium induced by an external source. Their knowledge is essential to describe many phenomenological aspects regarding dissipation effects in Heavy Ion Collisions. Recently, we have developed an extensive programme for the calculation of transport coefficients within diagrammatic ChPT, with direct application to the mesonic phase of the plasma expansion [34,35,36]. The diagrammatic treatment is technically nontrivial because transport coefficients involve the zero external frequency and momentum limit of retarded correlators. This implies the appearance of the so called pinching poles, proportional to 1/Γ with Γ the collisional width of the particles in the medium. This generates nonperturbative contributions, since Γ is proportional to the collision amplitude (see e.g. eq.(5)) and therefore perturbatively small. Thus, the usual chiral power counting of ChPT has to modified to account properly for these contributions [34].
The viscosity coefficients are particularly interesting for phenomenology. The shear viscosity to entropy density ratio η/s can be determined through elliptic flow analysis, the present RHIC data pointing to a small value η/s < 0.5 [37], an almost perfect fluid. Our ChPT result is greatly influenced by a correct unitarized description of the scattering amplitudes entering the pion width. In fact, without unitarization, at O(p 4 ), η/s is a monotonically decreasing function of T , violating the so called KSS bound η/s > 1/4π [38], as showed in Figure 5. The unitarized result satisfies that bound, gives phenomenologically compatible values at the relevant temperatures and is compatible with the existence of a minimum for this ratio near the transition [39]. The case of the bulk viscosity ζ is also important because it is directly sensitive to variations of conformal symmetry. Thus, it should have a similar behaviour [40] to the trace anomaly:
θ T ≡ T µ µ T = T 5 d dT P T 4(6)
where T µ ν is the energy-momentum tensor and P the thermodynamic pressure. The trace anomaly has a maximum at the QCD transition, originated mostly from the anomalous glue contribution to the beta function and observed in the lattice [2]. Therefore, significantly large values for ζ could be observed near the critical point. Our results in Figure 5 show a clear correlation between these quantities, both developing a twopeak structure. The first maximum comes from the explicit conformal breaking due to the quark mass and thus it disappears in the chiral limit. Actually, in that temperature range δ θ T ∼ 2m q δ qq T [35]. The second maximum is the critical one and is almost unchanged in the chiral limit, not being related to the quark condensate or to the quark mass, but to deconfinement effects encoded in the gluon condensate anomalous conformal breaking. We see again that incorporating properly unitarization is crucial to describe correctly the high temperature region. It is also essential to consider the speed of sound squared c 2 s to O(T 8 ) in order to capture its minimum at the transition arising from anomalous conformal breaking. Our analysis does not rely on assumptions on the behaviour of the θ θ spectral function, as it is the case of the original proposal in [40], which has been recently corrected [41]. It is therefore unclear from general considerations how the correlation between the trace anomaly and the bulk viscosity should actually hold. In fact, it does not hold in simple models mimicking QCD [42]. We believe then that our pion gas description can be useful to understand this and other issues regarding the behaviour of transport coefficients below the critical temperature.
FIGURE 1 .
1Results for the ρ (upper panels) and σ (lower panels) thermal mass and width extracted from the IAM poles at finite temperature.
FIGURE 3 .
3Imaginary part of the ππ amplitude in the I = J = 0 channel at finite temperature and nuclear density in the many-body BS approach.
FIGURE 4 .
4Left: the function µ π (T ) with the isentropic condition for different approximations. Right: the pion mass dependence with T and µ π .
FIGURE 5 .
5Viscosity coefficients for the pion gas in ChPT. Up: shear viscosity. Down: Bulk viscosity and trace anomaly.
ACKNOWLEDGMENTSWork partially supported by the Spanish research contracts FPA2008-00592, FIS2008-01323, UCM-BSCH GR58/08 910309.
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. D Fernandez-Fraile, arXiv:1009.2741hep-phD. Fernandez-Fraile, arXiv:1009.2741 [hep-ph].
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[] |
[
"Supermassive black hole spin-flip during the inspiral",
"Supermassive black hole spin-flip during the inspiral"
] |
[
"Lászlóá Gergely *[email protected] \nDepartment of Theoretical Physics\nUniversity of Szeged\nHungary\n\nDepartment of Experimental Physics\nUniversity of Szeged\nHungary\n",
"Peter L Biermann †[email protected] \nMPI for Radioastronomy\nBonnGermany\n\nDepartment of Physics & Astronomy\nUniversity of Bonn\nGermany\n\nDepartment of Physics & Astronomy\nUniversity of Alabama\nTuscaloosaALUSA\n\nDepartment of Physics\nUniversity of Alabama at Huntsville\nALUSA\n\nPhysics Department\nUniversity of Karlsruhe\nKarlsruheGermany\n",
"† ",
"Laurenţiu I Caramete ‡[email protected] \nInstitute for Space Sciences\nBucharestRomania\n"
] |
[
"Department of Theoretical Physics\nUniversity of Szeged\nHungary",
"Department of Experimental Physics\nUniversity of Szeged\nHungary",
"MPI for Radioastronomy\nBonnGermany",
"Department of Physics & Astronomy\nUniversity of Bonn\nGermany",
"Department of Physics & Astronomy\nUniversity of Alabama\nTuscaloosaALUSA",
"Department of Physics\nUniversity of Alabama at Huntsville\nALUSA",
"Physics Department\nUniversity of Karlsruhe\nKarlsruheGermany",
"Institute for Space Sciences\nBucharestRomania"
] |
[] |
During post-Newtonian evolution of a compact binary, a mass ratio ν different from 1 provides a second small parameter, which can lead to unexpected results. We present a statistics of supermassive black hole candidates, which enables us first to derive their mass distribution, then to establish a logarithmically even probability in ν of the mass ratios at their encounter. In the mass ratio range ν ∈ (1/30, 1/3) of supermassive black hole mergers representing 40% of all possible cases, the combined effect of spin-orbit precession and gravitational radiation leads to a spin-flip of the dominant spin during the inspiral phase of the merger. This provides a mechanism for explaining a large set of observations on X-shaped radio galaxies. In another 40% with mass ratios ν ∈ (1/30, 1/1000) a spin-flip never occurs, while in the remaining 20% of mergers with mass ratios ν ∈ (1/3, 1) it may occur during the plunge. We analyze the magnitude of the spin-flip angle occurring during the inspiral as function of the mass ratio and original relative orientation of the spin and orbital angular momentum. We also derive a formula for the final spin at the end of the inspiral in this mass ratio range.
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10.1088/0264-9381/27/19/194009
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[
"https://arxiv.org/pdf/1005.2287v2.pdf"
] | 118,597,235 |
1005.2287
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bc4c3e92095496bdaee8f3fe9c6f4b8b38117679
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Supermassive black hole spin-flip during the inspiral
26 Jul 2010
Lászlóá Gergely *[email protected]
Department of Theoretical Physics
University of Szeged
Hungary
Department of Experimental Physics
University of Szeged
Hungary
Peter L Biermann †[email protected]
MPI for Radioastronomy
BonnGermany
Department of Physics & Astronomy
University of Bonn
Germany
Department of Physics & Astronomy
University of Alabama
TuscaloosaALUSA
Department of Physics
University of Alabama at Huntsville
ALUSA
Physics Department
University of Karlsruhe
KarlsruheGermany
†
Laurenţiu I Caramete ‡[email protected]
Institute for Space Sciences
BucharestRomania
Supermassive black hole spin-flip during the inspiral
26 Jul 2010
During post-Newtonian evolution of a compact binary, a mass ratio ν different from 1 provides a second small parameter, which can lead to unexpected results. We present a statistics of supermassive black hole candidates, which enables us first to derive their mass distribution, then to establish a logarithmically even probability in ν of the mass ratios at their encounter. In the mass ratio range ν ∈ (1/30, 1/3) of supermassive black hole mergers representing 40% of all possible cases, the combined effect of spin-orbit precession and gravitational radiation leads to a spin-flip of the dominant spin during the inspiral phase of the merger. This provides a mechanism for explaining a large set of observations on X-shaped radio galaxies. In another 40% with mass ratios ν ∈ (1/30, 1/1000) a spin-flip never occurs, while in the remaining 20% of mergers with mass ratios ν ∈ (1/3, 1) it may occur during the plunge. We analyze the magnitude of the spin-flip angle occurring during the inspiral as function of the mass ratio and original relative orientation of the spin and orbital angular momentum. We also derive a formula for the final spin at the end of the inspiral in this mass ratio range.
* [email protected]; † [email protected]; ‡ [email protected] Abstract. During post-Newtonian evolution of a compact binary, a mass ratio ν different from 1 provides a second small parameter, which can lead to unexpected results. We present a statistics of supermassive black hole candidates, which enables us first to derive their mass distribution, then to establish a logarithmically even probability in ν of the mass ratios at their encounter. In the mass ratio range ν ∈ (1/30, 1/3) of supermassive black hole mergers representing 40% of all possible cases, the combined effect of spin-orbit precession and gravitational radiation leads to a spin-flip of the dominant spin during the inspiral phase of the merger. This provides a mechanism for explaining a large set of observations on X-shaped radio galaxies. In another 40% with mass ratios ν ∈ (1/30, 1/1000) a spin-flip never occurs, while in the remaining 20% of mergers with mass ratios ν ∈ (1/3, 1) it may occur during the plunge. We analyze the magnitude of the spin-flip angle occurring during the inspiral as function of the mass ratio and original relative orientation of the spin and orbital angular momentum. We also derive a formula for the final spin at the end of the inspiral in this mass ratio range.
Introduction
During galaxy mergers, following a regime of slow approach due to dynamical friction, eventually the central supermassive black holes (SMBHs) approach each other to a separation of the order of 10 3 Schwarzschild radii, when gravitational radiation takes over as the leading order dissipative effect. The Laser Interferometer Space Antenna (LISA, see [1]) is expected to detect merging binary SMBHs with masses m 1 + m 2 ≤ 10 7 solar masses (M ⊙ ) up to redshift z ≈ 30. A post-Newtonian approach is well suited to describe their forthcoming inspiral, a regime we define in terms of the post-Newtonian (PN) parameter ε = Gm/c 2 r ≈ v 2 /c 2 ∈ (ε in = 10 −3 , ε f in = 10 −1 ), where r and v characterize the orbital separation (from the center of mass) and speed of the reduced mass particle, G is the gravitational constant and c the speed of light. Various corrections to the conservative dynamics add up to 2 PN, while the gravitational radiation results in dissipation of energy, angular momentum and orbital angular momentum at 2.5 PN.
The leading order conservative correction to the Newtonian dynamics in a compact binary, which results in a change of the orbital plane (defined by the directionL N of the Newtonian orbital angular momentum L N = µr×v of the reduced mass particle µ) is the spin-orbit (SO) interaction [2], [3]. The precessional time-scale (the time during which the normal to the orbitL N undergoes a full rotation) is longer than the orbital period, however shorter than the characteristic time-scale of gravitational radiation (defined as L/L, where L is the magnitude of the total orbital angular momentum). Combined with the leading order gravitational radiation backreaction averaged over one quasicircular orbit, the SO correction provides a fair approximation to orbital dynamics, explored in Refs. [3], [4]. X-shaped radio galaxies (XRGs) exhibit two pairs of radio lobes and jets [5], [6]. A recent review [7] summarizes the four different models explaining XRGs: galaxy harbouring twin AGNs, back-flow diversion models, rapid jet reorientation models, finally a new jet-shell interaction model. A large subset of the observations (excepting cases, when the jets are aligned with the optical axes of the host ellipticals [8]) are well-explained by the jet reorientation model, which in turn implies a spin-flip [5], [9] of the dominant black hole.
The details of how this would occur were worked out in Ref. [4]. A key element was the determination of the typical mass ratio at SMBH mergers by a series of estimates, which resulted in mass ratios ν = m 2 /m 1 = 1/30 to ν = 1/3. Because the spin scales with the mass squared, the second spin was neglected and only the dominant spin S 1 (with magnitude S 1 ) kept. We summarize the consequences of this model as follows.
a) For the typical mass ratio the dominance of L over S 1 is reversed as the separation in the binary decreases throughout the inspiral. In the last stages of the inspiral the spin dominates over the orbital angular momentum S 1 ≫ L.
b) The angle α between the orbital angular momentum and total angular momentum J (with magnitude J), also the angle β between the dominant spin and total angular momentum evolve as:
α = −L J sin α > 0 ,(1)β =L J sin α < 0 .(2)
c) The approximate expression relating α to the post-Newtonian parameter ε, mass ratio ν and initial angle α + β span by the dominant spin with the orbital angular momentum (this angle being a constant during the inspiral) is:
tan α ≈ sin (α + β) ε −1/2 ν + cos (α + β) .(3)
(In Eq. (41) of Ref. [4] the left hand side was given as sin 2α/ (1 + cos 2α).) In a criticism to the work presented in Ref. [4], Gopakumar recently argued that "it is unlikely that the spin-flip phenomenon will occur during the binary black hole inspiral phase" [10]. This misconception comes from mixing up the instantaneous change in the direction of the total angular momentum, dĴ/dt = L /J L − L ·Ĵ Ĵ = 0 with its averaged expression dĴ/dt = 0 over the precessional time-scale. The angles α and β are not constants during the post-Newtonian evolution, as claimed in Ref. [10], they rather change as given in Eqs. (1)- (2). ‡
In the present paper we revisit some of the arguments of the spin-flip mechanism and also provide more details on it as compared to Ref [4]. In Section 2 we revisit the typical mass ratio argument, following a recent statistics of supermassive black hole candidates, resulting a newly established mass distribution. We comment on how these findings would affect the typical mass ratio range at SMBH encounters. In Section 3 we analyze how the spin-flip angle depends on the mass ratio and relative orientation of the spin and orbital angular momentum. We also derive a formula for the final spin during the inspiral. Finally we present our Concluding Remarks.
2. The sky in black holes: new statistics, consequences for the mass ratio at SMBH encounters and chances of the spin-flip during the inspiral First we summarize the arguments of Ref. [4] on the mass ratios at SMBH encounters. The mass distribution Φ BH (M BH ) of the galactic central SMBHs in the mass range 3 × 10 6 ÷ 3 × 10 9 M ⊙ is well described by a power-law with an exponential cutoff, but for our purposes can be adequately approximated by a broken power-law [11]- [13] (confirmed by an observational survey [14]). The break is at about 10 8 M ⊙ . In agreement with these arguments and observations we assume Φ BH (M BH ) ∝ M −k BH , with k ∈ (1, 2) below, and Φ BH (M BH ) ∝ M −h BH , with h ≥ 3 above the break. Then the probability for a specific mass ratio arose as an integral over the black hole mass distribution, folded with ‡ Only when the total and orbital angular momenta are aligned, become the angles α and β individually constant, as they identically vanish. Therefore in the aligned configuration no spin-flip could ever occur by the combined mechanism of SO precession and gravitational radiation.
the rate F to merge, and by adopting the lower values of the exponents. For the merger rate we assumed that it scales with the capture cross section S (the dependence on the relative velocity of the two galaxies was neglected, as the universe is not old enough for mass segregation). For the capture cross-section we assumed S ∝ ν −1/2 , motivated by the following arguments:
• for galaxies an increase with a factor of 10 in radius (10 2 in cross-section) accounts for an increase with a factor of 10 4 in mass (from the comparison of our Galaxy with dwarf spheroidals [15]- [16],
• there is a well established correlation between the SMBH mass and the mass of the host bulge [17],
• the mass of the central SMBH scales with both the spheroidal galaxy mass component and the total, dark matter dominated mass of a galaxy [18].
As a result of these considerations we have found that most likely the mass ratio is in the range ν ∈ (1/30 , 1/3). A typical value to consider would be ν = 1/10, thus one of the SMBHs being 10 times as massive as the other.
New work on the statistical analysis of 5,895 NED candidate sources [19] has been carried out in the mass range from 10 5 M ⊙ to above 10 9 M ⊙ . Below about 10 6 M ⊙ all candidates are probably compact star clusters, however the rest are likely SMBHs. This work shows that the SMBH mass function is a broken power law with M −2 BH at low masses, and M −3 BH at high masses, with a break near 1.25 × 10 8 M ⊙ ; this general behaviour has been long known, and has now been rederived with a very large sample.
The key difference with respect to previous work was the careful attention paid in order to have equal probability for detecting a SMBH in a galaxy, regardless to the Hubble type. The mass distribution of the SMBHs is represented on Fig 1. This particular distribution can be interpreted in the context of the merger model [20] with a merger rate scaling as (mass) +2 , very much stronger than what we favored in Ref [4]. The extreme mass dependence describes well a M −3 BH black hole mass distribution consistent with the high end of the mass distribution; on the other hand a mass dependence of the merger rate close to (mass) +4/3 , suggested by gravitational focusing arguments [20] describes well the lower mass distribution nearer to M −2 BH . It remains to be seen, whether all details of the mass function can be understood using either of these mass ratio dependences. Of course these simple merger rate calculations assume an environment without cosmological expansion. However, for the densest part of the cosmos the local expansion is very weak [21], and that is where most of the mergers occur.
However, for the determination of the typical mass ratio the essential result is only slightly changed. A merger rate running with mass +2 analytically gives a M −3 BH mass function (see [20]), as observed; we use this rate to estimate here the typical mass ratios for high BH masses. Redoing the integrals of Section 2 of Ref. [4] with k = 2, h = 3 (denoted there α, β) and ξ = 2 (as in [20], so much more extreme than what was assumed in [4]), then all four integrals are still dominated by the lower bound; only the second of the integrals has q ≡ ν −1 in its lower bound, and so the four integrals have the q-dependencies of q 0 , q +1 , q −1 and again q −1 . We can ignore the second integral, since it all refers to lower masses merging with lower masses. The most important integrals are those combining a SMBH above the break with a SMBH either below or above the break. Then the distribution in q is found as q −1 , a logarithmically even distribution in (dq)/q over a range of q from 1 to 1000, so a logarithmic average of 30. Weighting the two parts of the distribution, the larger mass ratios are favored, which would skew the logarithmic average of the mass ratio to q > 30, thus ν < 0.03.
The logarithmically even distribution means that the mass ratio ranges ν from 1 to 1/3, from 1/3 to 1/10, from 1/10 to 1/30, from 1/30 to 1/100, from 1/100 to 1/300 and finally from 1/300 to 1/1000 are roughly equal likely. A glance at Table 1 of Ref. [4] shows, that concerning the behaviour of the ratio of the dominant spin and orbital angular momentum magnitudes, we have three regimes:
(1) ν ∈ (1/3, 1) when S 1 < L throughout the inspiral, (2) ν ∈ (1/30, 1/3) when the initial S 1 < L is reversed to S 1 > L during the inspiral and
(3) ν ∈ (1/1000, 1/30) when S 1 > L holds throughout the inspiral. For the mass ratio ranges (1) and (3) no spin-flip can occur during the inspiral, while for (2) it should. For (1) there is chance for a spin-flip to occur during the plunge, as some numerical simulations have already found this for equal masses [22]. For (3) by contrast there is no possibility for a spin-flip by the combined mechanism of SO precession and gravitational radiation. These mass ratio ranges then occur with (1) 20%, (2) 40% and (3) again 40% probability. This means that the spin-flip still typically occurs during the inspiral.
Spin-flip angle distribution
In this section we will present an analysis of the spin-flip angle occurring during the inspiral phase in the mass ratio range ν ∈ (1/30, 1/3) as a function of the mass ratio.
The spin-flip model be understood as follows. Initially the galactic SMBH has conserved spin, along which the primary jet can form. When the two galaxies collide, the SO induced spin precession starts, while gravitational radiation is diminishing the orbital angular momentum. The direction of the total angular momentum stays unchanged. The constancy ofĴ over the precessional time-scale is due to the fact, that the change in the total angular momentumJ =LL is about the orbital angular momentum, which (disregarding gravitational radiation) undergoes a precessional motion about J. This shows that the averaged change in J is along J (simple precession, [3]). This conclusion, however, depends strongly on whether the precessional angular frequency Ω p is larger thanα andβ. Indeed, if these are comparable, the component perpendicular to J in the changeJ =LL will not average out during one precessional cycle, as due to the increase of α it can significantly differ at the beginning and at the end of the same precessional cycle. Such a situation would occur, when the spin and the orbital angular momentum are of comparable magnitude (S 1 ≈ L, a regime through which a binary with typical Figure 1. Aitoff projection in galactic coordinates of 5,895 NED SMBH candidate sources. The complete sample is complete in a sensitivity sense, in order to derive densities one needs a volume correction. In the electronic version the colour code is Orange, Green, Blue, Red, Black corresponding to masses above 10 5 M ⊙ , 10 6 M ⊙ , 10 7 M ⊙ , 10 8 M ⊙ , 10 9 M ⊙ , respectively. With the exception of the less numerous first range (Orange), representing compact star clusters, the rest are SMBHs. mass ratio will pass through during the inspiral) and also roughly antialigned, a low probability regime known as transitional precession. During simple precession Eqs. (1)- (2) governing the evolution of the angles α and β also hold in an average sense over the precessional time-scale. In what follows, we assume simple precession.
The magnitude of the spin is unaffected by gravitational radiation, therefore by the simple rule of addition of vectors the spin has to align close to theĴ direction. The second jet then can start to form. In the intermediate phase when the spin precesses, instead of jet formation the precessing magnetic field creates a wind, sweeping away the base of the old jet, which in many cases can be observed.
Spin and orbital angular momentum orientations, final spin formula
The key equation to start with is Eq. (3). In order to see the validity of this equation, also to generalize it to the cases of non-extreme rotation, we need to evaluate
S 1 ≈ m 1 RV 1 ≈ m 1 Gm 1 c 2 c V 1 c ≈ G c m 2 1 χ 1 , L ≈ L N ≈ µrv = G c v c c 2 r Gm µm = G c ε −1/2 m 1 m 2 = G c m 2 1 ε −1/2 ν .(4)
Here V is some characteristic rotational velocity, R the radius of the SMBH (of the order of its Schwarzschild radius) and χ 1 ∈ (0, 1) is the dimensionless (χ 1 = 1 for extreme rotation). Therefore
S 1 L ≈ χ 1 ε 1/2 ν −1 .(5)
Next we express J/S 1 first from J = L cos α + S 1 cos β by rewriting β = (α + β) − α, and secondly from the equality of the projections perpendicular toL of the total and spin angular momenta J sin α = S 1 sin (α + β), so that we can equal them. By also employing Eq. (5) and basic trigonometry we obtain
tan α ≈ sin (α + β) χ −1 1 ε −1/2 ν + cos (α + β) .(6)
This is the desired generalization of Eq. (3). It is worth to note that when applied to the final configuration ε f in , Eq. (6) also stands as a formula for the final spin at the end of the inspiral, giving the polar angle of the final spin α f in with respect to the axisĴ in terms of the mass ratio, spin magnitude and angle span with the orbital angular momentum. Related formulae based on numerical runs were advanced in Refs. [23]. These results are not immediate to compare with ours, as Eq. (6) could at most be applied at the end of the inspiral; although in the mass ratio range where it is valid, one would intuitively expect that as not much orbital angular momentum is left at the end of the inspiral in comparison with the dominant spin, the direction of the latter will not be significantly changed during the plunge.
3.1.1. Particular cases. There are three particular configurations worth to mention:
i) The spin is aligned with the orbital angular momentum: α + β = 0, thus from Eq. (6) α = 0 and there is no room for any spin-flip. This would be the situation for perfectly wet mergers, which align the spin with the orbital angular momentum.
ii) The spin is anti-aligned with the orbital angular momentum, α + β = π. Therefore depending on which of the S 1 and L are larger, the angle α is either 0 or π.
iii) For the parameter ranges when the denominator vanishes α + β = arccos −χ −1 1 ε −1/2 ν , from Eq. (6) we obtain α = π/2, therefore β is also determined.
3.1.2. Discussion as function of mass ratios. Keeping in mind that due to Eqs. (1)-(2) the angle α+β is a constant during the inspiral (a parameter), and the dimensionless spin χ 1 behaves similarly, also regarding the mass ratio as a third parameter characterizing the particular merger, the angle α in general remains a function of ε, thus it evolves together with the orbital separation r and velocity v.
For the mass ratio ν = 1/10 we have (S 1 /L) in ≈ χ 1 ε 1/2 in ν −1 = 0.316χ 1 and (S 1 /L) f in ≈ χ 1 ε 1/2 f in ν −1 = 3.162χ 1 . As tan α ≤ S 1 /L and tan β ≤ L/S 1 (the equalities arising when the spin and orbital angular momentum are perpendicular) we have tan α in ≤ 0.316χ 1 and tan β f in ≤ 0.316χ −1 1 . For extreme rotation (χ 1 = 1) we obtain α in , β f in ≤ 0.316 = 18.105 • .
For ν = 1/3 we obtain tan α in ≤ (S 1 /L) in ≈ χ 1 ε 1/2 in ν −1 = 0.095χ 1 and for extreme rotation α in ≤ 0.095 = 5.44 • . In fact at the beginning of the inspiral this latter condition (meaning that the orbital angular momentum is roughly the total angular momentum) holds in the whole mass ratio range ν ∈ (1/3, 1). Under these conditions Eq. (6) can be approximated as
α in ≈ χ 1 ε 1/2 in ν −1 sin β in = 0.032χ 1 ν −1 sin β in .(7)
For ν = 1/30 we obtain tan
β f in ≤ (L/S 1 ) f in ≈ χ −1 1 ε −1/2 f in ν = 0.105χ −1 1
and for extreme rotation β f in ≤ 0.105 = 6.04 • . In fact at the beginning of the inspiral this latter condition (meaning that the dominant spin is roughly the total angular momentum) holds in the whole range ν ∈ (1/1000, 1/30). Under these conditions Eq. (6) can be expanded (to first order in β f in , with
χ −1 1 ε −1/2 f in ν of the order of β) as β f in ≈ χ −1 1 ε −1/2 f in ν sin α f in = 3.162χ −1 1 ν sin α f in .(8)
For slowly rotating SMBHs with χ 1 ≈ 0.1 the above formula would hold only in the range ν ∈ (1/1000, 1/300) .
The spin-flip angle during the inspiral
A minimal value for the spin-flip angle σ arises by forming the difference between the angles β, characterizing the orientation of the spin with respect to the inertial direction J. Thus
σ min = β in − β f in = α f in − α in .(9)
In the second equality we have used that α in + β in = α f in + β f in . However we have to take into account, that the above is only true in a 2-dimensional picture. In reality the 3-dimensional SO precession will complicate the situation, and the above angle emerges only if the number of precessions during the inspiral is an integer multiple of 2π. If instead is of the type (2k + 1) π the spin-flip angle will be maximal, to be calculated as
σ max = β in + β f in − lπ = 2 (α in + β in ) − lπ − (α in + α f in ) ,(10)
where l = 0 if β in + β f in ≤ π and l = 1 if π < β in + β f in < 2π. The difference between σ max and σ min is due to the fact, that the realignment of the spin alongĴ is not perfect. The closer S f in 1 is toĴ, the less their difference ought to be due a more perfect alignement, therefore σ max − σ min = β f in − β in should go to 0 with decreasing ν.
For generic mass ratios ν ∈ (1/30, 1/3), Eqs. (6), (9) and (10) give the range of allowed spin-flip angle for each relative orientation α + β of the spin with respect to the plane of motion and each χ 1 . The generic numerical solution for σ min in the case χ 1 = 1 is represented on Fig 2 as function of the relative orientation of the spin and orbital angular momentum α + β and mass ratio ν. For a given mass ratio the spin-flip angle has a maximum shifted from π/2 towards the anti-aligned configurations. The figure confirms the prediction, that significant spin-flip will occur during the inspiral in the mass ratio range ν ∈ (1/30, 1/3). For mass ratios smaller than 1/100 the spin does not flip at all, as the infalling SMBH acts as a test particle. Figure 2. The spin-flip angle σ min as function of the relative orientation of the spin and orbital angular momentum α + β (a constant during inspiral), and mass ratio ν. For a given mass ratio the spin-flip angle has a maximum shifted from π/2 towards the anti-aligned configurations. The mass ratios ν = 1; 1/3; 1/30 and 1/1000 are located on the log ν −1 axis at 0; 1.09; 3.40 and 6.91, respectively, confirming the prediction, that a significant spin-flip will occur in the mass ratio range ν ∈ (1/30, 1/3). For mass ratios smaller than 1/100 the spin does not flip at all, as the infalling SMBH acts as a test particle.
Concluding Remarks
In light of the new data on a large sample of SMBH candidates we have established that the mass ratios obey an even logarithmic distribution in ν. In the mass ratio range ν ∈ (1/30, 1/3) of SMBH mergers representing 40% of all possible cases, we have investigated the SO precession driven conservative and gravitational radiation driven dissipative contributions to the orbital evolution during the inspiral, averaged over the precession time-scale. In this mass range the ratio of the dominant spin magnitude and orbital angular momentum magnitude S/L changes from less than 1 to larger than 1 during the inspiral. As the direction of the total angular momentum is unchanged on all time-scales larger than the precession time-scale, while the magnitude of the the orbital angular momentum decreases due to gravitational radiation and the magnitude of the spin stays constant, the spin direction has to change. The spin-flip of the dominant spin therefore occurs during the inspiral. If jet activity is involved, X-shaped radio galaxies arise by this mechanism and a large set of observations on X-shaped radio galaxies could be explained.
In another 40% of the mergers with mass ratios ν ∈ (1/1000, 1/30) the spin-flip never occurs by this mechanism, while in the remaining 20% of mergers with mass ratios ν ∈ (1/3, 1) it may occur during the plunge.
SMBH mergers of equal mass to ν = 1/3 are only half as likely as the mass ratios 1/30 to 1/3, therefore the occurrence of the spin-flip can be considered typical during the inspiral. We analyzed the magnitude of the spin-flip angle occurring during the inspiral as function of the mass ratio and original relative orientation of the spin and orbital angular momentum and supported by numerical analysis the theoretical prediction ( Fig 2). We also derived a formula for the final spin at the end of the inspiral in this mass ratio range.
During the inspiral the following relations among the relevant time-scales hold: tilt / spin-flip time-scale ≥ inspiral time-scale ≫ precession time-scale ≫ orbital time-scale (for all mass ratios in the typical range). Interestingly enough, the spin-flip time-scale for a typical mass ratio of 1/10 is only about three years, while the precession time-scale is less then a day [4]. Thus rapidly rotating relativistic jets coming close to our line of sight could produce significant variability at all wavelengths years before the coalescence. Therefore electromagnetic counterparts / precursors to the strongest gravitational wave emission are also likely to occur.
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|
[] |
[
"Near-Infrared Properties of Metal-Poor Globular Clusters in the Galactic Bulge Direction ⋆",
"Near-Infrared Properties of Metal-Poor Globular Clusters in the Galactic Bulge Direction ⋆"
] |
[
"S.-H Chun [email protected] \nDepartment of Astronomy\nYonsei University\n120-749SeoulKorea\n",
"J.-W Kim \nInstitute for Computational Cosmology\nDepartment of Physics\nDurham University\nSouth RoadDH1 3LEDurhamUK\n",
"I.-G Shin \nDepartment of Astronomy\nYonsei University\n120-749SeoulKorea\n",
"C Chung \nDepartment of Astronomy\nYonsei University\n120-749SeoulKorea\n",
"D.-W Lim \nDepartment of Astronomy\nYonsei University\n120-749SeoulKorea\n",
"J.-H Park \nKorea Astronomy and Space Science Institute\n305-348DaejeonKorea\n",
"H.-I Kim \nKorea Astronomy and Space Science Institute\n305-348DaejeonKorea\n",
"W Han \nKorea Astronomy and Space Science Institute\n305-348DaejeonKorea\n",
"Y.-J Sohn [email protected] \nDepartment of Astronomy\nYonsei University\n120-749SeoulKorea\n\nKorea Astronomy and Space Science Institute\n305-348DaejeonKorea\n"
] |
[
"Department of Astronomy\nYonsei University\n120-749SeoulKorea",
"Institute for Computational Cosmology\nDepartment of Physics\nDurham University\nSouth RoadDH1 3LEDurhamUK",
"Department of Astronomy\nYonsei University\n120-749SeoulKorea",
"Department of Astronomy\nYonsei University\n120-749SeoulKorea",
"Department of Astronomy\nYonsei University\n120-749SeoulKorea",
"Korea Astronomy and Space Science Institute\n305-348DaejeonKorea",
"Korea Astronomy and Space Science Institute\n305-348DaejeonKorea",
"Korea Astronomy and Space Science Institute\n305-348DaejeonKorea",
"Department of Astronomy\nYonsei University\n120-749SeoulKorea",
"Korea Astronomy and Space Science Institute\n305-348DaejeonKorea"
] |
[] |
Aims. J, H, and K ′ images obtained from the near-infrared imager CFHTIR on the Canada-France-Hawaii Telescope are used to derive the morphological parameters of the red giant branch (RGB) in the near-infrared color-magnitude diagrams for 12 metal-poor globular clusters in the Galactic bulge direction. Using the compiled data set of the RGB parameters for the observed 12 clusters, in addition to the previously studied 5 clusters, we discuss the properties of the RGB morphology for the clusters and compare them with the calibration relations for the metal-rich bulge clusters and the metal-poor halo clusters. Methods. The photometric RGB shape indices such as colors at fixed magnitudes of M K = M H = (−5.5, −5, −4, and −3), magnitudes at fixed colors of (J − K) o = (J − H) o = 0.7, and the RGB slope are measured from the fiducial normal points defined in the nearinfrared color-magnitude diagrams for each cluster. The magnitudes of RGB bump and tip are also estimated from the differential and cumulative luminosity functions of the selected RGB stars. The derived RGB parameters have been used to examine the overall behaviors of the RGB morphology as a function of cluster metallicity. Results. The correlations between the near-infrared photometric RGB shape indices and the cluster metallicity for the programme clusters compare favorably with the previous observational calibration relations for metal-rich clusters in the Galactic bulge and the metal-poor halo clusters. The observed near-infrared magnitudes of the RGB bump and tip for the investigated clusters are also in accordance with the previous calibration relations for the Galactic bulge clusters.
|
10.1051/0004-6361/200913070
|
[
"https://arxiv.org/pdf/1004.3156v1.pdf"
] | 55,118,330 |
1004.3156
|
9818a9577114e3c86ec5c15ff5fcc88ae73b7aa1
|
Near-Infrared Properties of Metal-Poor Globular Clusters in the Galactic Bulge Direction ⋆
19 Apr 2010 April 20, 2010
S.-H Chun [email protected]
Department of Astronomy
Yonsei University
120-749SeoulKorea
J.-W Kim
Institute for Computational Cosmology
Department of Physics
Durham University
South RoadDH1 3LEDurhamUK
I.-G Shin
Department of Astronomy
Yonsei University
120-749SeoulKorea
C Chung
Department of Astronomy
Yonsei University
120-749SeoulKorea
D.-W Lim
Department of Astronomy
Yonsei University
120-749SeoulKorea
J.-H Park
Korea Astronomy and Space Science Institute
305-348DaejeonKorea
H.-I Kim
Korea Astronomy and Space Science Institute
305-348DaejeonKorea
W Han
Korea Astronomy and Space Science Institute
305-348DaejeonKorea
Y.-J Sohn [email protected]
Department of Astronomy
Yonsei University
120-749SeoulKorea
Korea Astronomy and Space Science Institute
305-348DaejeonKorea
Near-Infrared Properties of Metal-Poor Globular Clusters in the Galactic Bulge Direction ⋆
19 Apr 2010 April 20, 2010Received dd August 2009 / Accepted dd Monthber 2010Astronomy & Astrophysics manuscript no. bulge˙manuscriptGalaxy: structure -globular clusters: general -stars: evolution -infrared: stars -techniques: photometric
Aims. J, H, and K ′ images obtained from the near-infrared imager CFHTIR on the Canada-France-Hawaii Telescope are used to derive the morphological parameters of the red giant branch (RGB) in the near-infrared color-magnitude diagrams for 12 metal-poor globular clusters in the Galactic bulge direction. Using the compiled data set of the RGB parameters for the observed 12 clusters, in addition to the previously studied 5 clusters, we discuss the properties of the RGB morphology for the clusters and compare them with the calibration relations for the metal-rich bulge clusters and the metal-poor halo clusters. Methods. The photometric RGB shape indices such as colors at fixed magnitudes of M K = M H = (−5.5, −5, −4, and −3), magnitudes at fixed colors of (J − K) o = (J − H) o = 0.7, and the RGB slope are measured from the fiducial normal points defined in the nearinfrared color-magnitude diagrams for each cluster. The magnitudes of RGB bump and tip are also estimated from the differential and cumulative luminosity functions of the selected RGB stars. The derived RGB parameters have been used to examine the overall behaviors of the RGB morphology as a function of cluster metallicity. Results. The correlations between the near-infrared photometric RGB shape indices and the cluster metallicity for the programme clusters compare favorably with the previous observational calibration relations for metal-rich clusters in the Galactic bulge and the metal-poor halo clusters. The observed near-infrared magnitudes of the RGB bump and tip for the investigated clusters are also in accordance with the previous calibration relations for the Galactic bulge clusters.
Introduction
The current view of the Galaxy formation is mainly focused on the hierarchical merging paradigm in the cold dark matter cosmology. Globular clusters, as tracers of the early formation and the current structure of the Galaxy, play a key role in studies of the paradigm, because they are present from the central bulge to the outer halo with various metallicities. Particularly, the Galactic bulge harbors a globular cluster population with a broad metallicity distribution that extends from about twice solar to less than one-tenth solar abundance (Ortolani 1999), while most field stars in the bulge have near-solar metallicity (McWilliam & Rich 1994;Zoccali et al. 2003). The metalrich globular clusters in the the Galactic bulge share the kinematics, spatial distribution, and composition of the bulge field stars (e.g., Minniti & Zoccali 2008;and references therin). This indicates that metal-rich globular clusters are associated with the Galactic bulge recognized as the dominant proto-Galactic building block (e.g., Côté et al. 2000). On the other hand, the origin of the metal-poor globular clusters in the Galactic bulge direction is still a subject of debate since accurate measurements of kinematics and high resolution chemical abundances are lacking.
In the hierarchical model of the Galaxy formation, however, old metal-poor field stars in the bulge form via merging and accretion events in the early Universe (Nakasato & Nomoto 2003). In this sense, the metal-poor clusters currently located in the central region of the Galaxy might be the oldest objects which did not form originally in the Galactic bulge. Thus, the metal-poor clusters in the bulge region can play a key role to understand the early epoch of the formation of the Galactic bulge.
In the Milky Way, about 150 globular clusters are listed in the database of Harris (1996), which was revised in 2003. Recently, new faint clusters and cluster candidates have also been found (e.g., Carraro 2005;Kobulnicky et al. 2005;Willman et al. 2005;Froebrich et al. 2007). Out of 43 globular clusters located within 3 kpc of the Galactic center (Harris 1996), 22 are metal-poor ([Fe/H] < −1.0) and 21 are metal-rich ([Fe/H] > −1.0). Recently, Valenti et al. (2007) presented near-infrared color-magnitude diagrams (CMDs) and physical parameters for a sample of 24 globular clusters toward the Galactic bulge and located within |b| ≦ 10 • and |l| ≦ 20 • . They discussed the near-infrared properties of the red giant branch (RGB) for 12 observed clusters, in addition to those previously studied by their group (e.g., Ferraro et al. 2000;Valenti et al. 2004abc 2005Origlia et al. 2005). We note, however, that their sample of the clusters have high priority to the metal-rich population, i.e., 17 out of the 24 are relatively metal-rich with [Fe/H] > −1.0, taking into account of a bulge origin for the metal-rich globular clusters.
In our research, we have focused on obtaining a moderately deep homogeneous photometric data set in the near-infrared regime for the metal-poor clusters in the bulge direction. Nearinfrared photometry offers advantages for a study of the cool population of the RGB stars in the Galactic globular clusters, because of its high sensitivity to low temperature. In addition, high extinctions toward the bulge can be reduced by observing the near-infrared wavelengths, as the extinction in the K band is only ∼ 10 percent of that in the V band (Rieke & Lebofsky 1985). Using these bases, Kim et al. (2006) presented the morphological properties of the RGB in the near-infrared CMDs for five metal-poor clusters of the Galactic bulge (NGC 6541,NGC 6642,NGC 6681,NGC 6717,and NGC 6723) and also for three halo clusters.
In this paper, we report new results of the near-infrared photometry for 12 metal-poor clusters and present a homogeneous photometric data set of the RGB morphology for 17 globular clusters, covering ∼ 75% of the total 22 metal-poor globular clusters in the Galactic bulge direction. The results of the RGB morphology for the programme clusters are compared with the previously published calibrations of Valenti et al. (2004ab 2007. The observations, procedures for data reduction, and phtometric measurements are described in Sect. 2. In Sect. 3, we describe the near-infrared CMDs and the fiducial normal points of target clusters. In Sect. 4, the morphological properties of CMDs such as RGB shape feature, RGB bump, and RGB tip are presented. Finally, the results are discussed and summarized in Sect. 5.
OBSERVATIONS, DATA REDUCTION, AND PHOTOMETRIC MEASUREMENTS
Observations were obtained during the nights of June 1, 2002, andApril 20-21, 2003. Using the CFHTIR imager mounted on the f/8 Cassegrain focus of the Canada-France-Hawaii telescope (CFHT), the fields centered on each cluster were observed with J, H and K ′ filters. The CFHTIR contains a 1024 × 1024 Hg:Cd:Te array. Its angular scale is 0.211 ′′ /pixel, so that each image covers a total field-of-view of 3.6 ′ × 3.6 ′ . The observations were split into short and long exposures for each filter in order to optimize the photometry of bright and faint stars. The images by short and long exposures are combinations of four 1-second or 2-second exposures, and of eight 30-second exposures, respectively. A four-points square dither pattern was used to identify and reject bad pixels and cosmic rays in the observed images. In both run the UKIRT standard stars and M13 were also observed for a photometric standardization. The summary of observations for the target clusters is presented in Table 1.
Calibration frames of darks, flats, and blank sky backgrounds were also obtained during the runs. Dark frames were recorded at the beginning and the end of each run. Dome flats were made by subtracting exposures of the dome white spot taken with the lamps off from those taken with the lamps on. Thermal emission patterns were constructed by combining flat-fielded images of blank sky regions.
The process of data reduction consists of subtracting a dark frame, dividing by the normalized flat image for each filter, and subtracting the thermal signature and the sky background level estimated by the mode of pixel intensity distribution. Then, the processed images were combined for each exposure after aligning the dither offsets. The seeing conditions of the reduced images range between 0 ′′ .6 ∼ 0 ′′ .9. The brightness of stars in the clusters was measured with the point-spread function fitting routine DAOPHOT II/ALLSTAR (Stetson 1987;Stetson & Harris 1988). The brightness of stars around the RGB tip was measured only in short-exposure images because of saturation in long-exposure images, while faint stars were detected in only long-exposure images. For stars detected in both short and long exposures, measurements with smaller photometric error were assigned to the brightness. To avoid false detection, only stars detected in all filters with detection errors of less than 0.2 mag were considered for the photometric analysis. The photometric calibration equations obtained from UKIRT standard stars were then applied to the magnitudes of the stars on the target clusters. Standardizations were also double-checked in direct starto-star comparison with the photometry of bright stars in M13 of Kim et al. (2006). As shown in Kim et al. (2006), there are only small photometric offsets, △ K = 0.03 ± 0.01 and △ (J − K) = 0.04 ± 0.01, between the photometric data with the UKIRT system and those with the 2MASS system of Valenti et al. (2004c). Note that the offsets will become negligible after the transformation of the fiducial normal points for the observed near-infrared CMD into the 2MASS photometric system (see Sec. 3). Here, we also note that the measured photometric data in the southwest quarter part of the images for the runs of 2002, and those in the south-east quarter part of the images for the runs of 2003, were not used for the subsequent photometric analyses, because of possible readout anomalies of the CFHTIR imager during the runs. in this study. As can be seen, all of the observations are deep enough to reach the base of the RGB at ∆K ′ ∼ ∆H ≈ 8 mag fainter than the RGB tip. As we expected in the near-infrared CMDs for metal-poor globular clusters, the horizontal branch sequences are aslant compared to the RGB sequences. Scattering in the near-infrared CMDs of the target clusters might be due to both photometric errors and contamination by foreground field stars toward the Galactic bulge. Apparently in the case of the highly reddened tiny cluster Terzan 4, for which Bonatto & Bica (2008) derived the tidal radius of 7.6 ′ ± 1.3 ′ and the concentration parameter of c = 0.9 ± 0.2 from the 2MASS images, the CMDs contain significant noise owing to a strong field star contamination in the observed field.
COLOR-MAGNITUDE DIAGRAMS AND FIDUCIAL NORMAL POINTS
To examine the relationship between the RGB morphological parameters in CMDs of the absolute plane and cluster's metallicity, the values of metallicity, reddening, and distance modulus were estimated for each cluster using the method adopted in Kim et al. (2006). Metallicities for target clusters are used in the Carreta & Gratton (1997) scale, [Fe/H] CG97 , to directly compare the photometric properties of the measured RGB morphology with the results presented in Valenti et al. (2004aValenti et al. ( 2007. Metallicities [Fe/H] CG97 of two clusters NGC 6266 and NGC 6333 were adopted from Ferraro et al. (1999). For the other clusters, we obtained [Fe/H] CG97 by transforming the data given in Zinn (1985) into the scale of Carreta & Gratton (1997) as per Valenti et al. (2004a). Note that we assigned [Fe/H] CG97 = −1.62 ± 0.08 to Terzan 4, for which Stephens & Frogel (2004) measured the metallicity of 7 stars in the cluster. We also estimated global metallicities [M/H] of the target clusters by using the equation for the α elements enhanced theoretical evolutionary sequence (Salaris et al. 1993 Table 2.
Distance moduli of two clusters NGC 6266 and NGC 6333 were adopted from Ferraro et al. (1999), in which a new methodology is presented to derive distance moduli of globular clusters by matching the observed visual magnitude of the zero-age horizontal branch (V ZAHB ) and the theoretical synthetic horizontal branch (HB) models. For the other ten clusters, a similar procedure to that of Ferraro et al. (1999) was applied to determine the distance moduli from the synthetic and observed ZAHB levels. Synthetic HBs for each cluster with different metallicities were generated by the method used in Lee et al. (1994) with the HB evolutionary tracks of Yi et al. (1997). Details of the generated synthetic HBs with various metallicities are described in Kim et al. (2006). The synthetic HBs in absolute plane were transformed into the observed HBs in the CMDs of the target clusters from Rich et al. (1998) for NGC 6558, Ortolani et al. (1997) for Terzan 4, and Piotto et al. (2002) for the other eight clusters. The extinction correction was calculated by using the latest compilation of E(B − V) in Harris (1996) and by applying the reddening ratios of Schlegel et al. (1998). The dis-tance modulus for each cluster was then estimated by measuring the ZAHB levels in the synthetic and observed CMDs of HB stars, taking into account the extinction values for each cluster. Note that we determined the reddening E(B − V) = 2.06 for the highly reddened cluster Terzan 4 from the synthetic and observed CMDs of HB stars. This seems to be slightly smaller than E(B − V) = 2.35 in Harris (1996) and E(B − V) = 2.31 in Ortolani et al. (1997), but comparable to the reddening value E(B − V) = 2.05 of Valenti et al. (2010). The determined distance moduli µ o for the target clusters are listed in Table 2 with reddening E(B − V) and extinction values A J , A H , and A K in the near-infrared wavelengths.
Prior to the derivation of the morphological parameters of the RGB sequence, we obtained the RGB fiducial normal points for the near-infrared CMDs of the sample clusters, following the same strategy as in Kim et al. (2006). As shown in Figure 1, the CMD of Terzan 4 shows a significant field star contamination. In order to minimize field star contamination of the tiny cluster Terzan 4, we determined the fiducial normal points of the RGB with stars only within 16 ′′ of the cluster center. Note that Valenti et al. (2010) derived the RGB ridge line of Terzan 4 using stars within 40 ′′ of the cluster center to derive the morphological parameters. For the other clusters, the resolved stars within 30 ′′ from the cluster center were used to construct the fiducial normal points of the RGB. We first determined the mean magnitude and color in the 0.25 mag bin of the CMDs, excluding asymptotic giant branch stars, slanted HB stars, and highly scattered foreground stars. Subsequently, we rejected stars with colors larger than ±2σ of the mean, and the mean values of the magnitude and color were calculated again in the assigned magnitude bin. The procedure with a 2σ rejection criterion was repeated until the mean values of the magnitude and color are stable at constant values. This iterative process statistically removed the asymptotic giant branch stars, HB stars, and field stars from the RGB stars in the obtained near-infrared CMDs for the central region of the target clusters. Then, the cluster reddening and distance were used to convert the determined fiducial normal points into the absolute plane. Finally, the color and magnitude of the fiducial normal points in the UKIRT system were transformed into the 2MASS system by using equations (37)-(39) from Carpenter (2001) to compare the results directly with those of Valenti et al. (2004aValenti et al. ( 2007.
Target (J − K) −5.5 o (J − K) −5 o (J − K) −4 o (J − K) −3 o (J − H) −5.5 o (J − H) −5 o (J − H) −4 o (J − H) −
MORPHOLOGY OF THE NEAR-INFRARED CMDS
In this section, we present and discuss the morphological properties of the near-infrared CMDs for the programme clusters. The near-infrared RGB morphology for each cluster are characterized by parameters of the RGB location in colors at fixed magnitudes and in magnitudes at fixed colors, the slopes of the RGB, and the absolute magnitudes of the RGB bumps and tips. The RGB parameters for 12 clusters in this paper which together with the 5 clusters in Kim et al. (2006) have been used to examine the overall behaviors of the RGB morphology in the nearinfrared CMDs as a function of cluster metallicity for the metalpoor globular clusters in the Galactic bulge direction. The results were compared with the previous observational calibrations of Valenti et al. (2004aValenti et al. ( 2007 and the theoretical predictions of the Yonsei-Yale isochrones (Kim et al. 2002;Yi et al. 2003).
The RGB Shape
To characterize the overall behaviors of the RGB morphology in the near-infrared and optical CMDs of globular clusters, Ferraro et al. (2000) defined a new set of photometric indices for the RGB location, i.e., colors at fixed magnitudes and magnitudes at fixed colors. In a similar fashion, Kim et al. (2006) measured the photometric color and magnitude indices of the RGB morphology for five metal-poor globular clusters in the bulge direction, and compared the results with calibrations of the RGB morphology for 28 bulge clusters from Valenti et al. (2004aValenti et al. ( 2005. The representative morphological parameters of the RGB include (1) In the present study, we also measured the same parameters for the observed clusters. To derive the RGB location parameters in color and in magnitude, we applied a second-or thirdorder polynomial fit to adjacent 10 fiducial normal points of CMDs in Figure 2 at the given magnitude and color. The RGB slope has usually been determined by fitting an equation of the form J − K = aK + b to the upper part of the RGB in the (J − K, K) CMD, where the RGB morphology is less curved than in the other lower faint ranges. In the same manner as Kim et al. (2006), the fiducial normal points in a magnitude range between 0 and 5 mag fainter than the brightest point were used to determine the RGB slope. Table 3 Table 4. In Table 3 and Table 4, we also list previously studied RGB shape parameters by our group (i.e., Kim et al. 2006) for 5 metal-poor clusters in the bulge direction.
In Figure 3 and slope increases progressively toward the RGB tip. Theoretical predictions of the RGB location parameters were extracted from the Yonsei-Yale isochrones (Kim et al. 2002;Yi et al. 2003) in order to compare with the observed relations of the RGB colors and cluster metallicity. The dotted and dashed lines are the theoretically estimated (J − K) o and (J − H) o values of the RGB location as a function of metallicity at t = 12 Gyr and 10 Gyr, respectively. While the overall trends of the theoretical models show a good correlation with the observed data, it appears that there are systematic shifts in the RGB colors in our results from the relations inferred from the Yonsei-Yale isochrones. Indeed, the theoretical model colors of the RGB at [Fe/H]= −1.5 seem to be ∼ 0.06 − 0.11 mag redder in (J − K) and ∼ 0.04 − 0.09 mag redder in (J − H) than the empirical results of Valenti et al. (2004a). In addition, the shifts of the model colors become larger toward the RGB tip. The shifts can be understood as a combination of the uncertainties involved in the color calibration of logT into (J − K) o colors (Kim 2009, private communications) and in the magnitude transformation of K into 2MASS system, the errors in the abundance determinations, and the photometric errors in the observed colors. Figure 5 shows the dependence of the absolute magnitudes of M K and M H at fixed colors of (J − K) o = (J − H) o = 0.7 on metallicity of the clusters investigated in this study. The values measured in our sample fit well with the calibration relations (solid lines) of Valenti et al. (2004a). We note, however, the observed clusters in this paper show a larger scattered distribution of the M K and M H magnitudes, compared with the distribution of metal-rich bulge globular clusters and halo clusters of Valenti et al. (2004a). This is possibly due to the uncertainty in the derived absolute magnitudes associated with errors in the distance and reddening, and errors in the polynomial fitting measurements on the fiducial normal points. In fact, Valenti et al. (2004a) noted that errors in color of a few hundredths of a magnitude produce uncertainties of about 0.2 − 0.3 in K magnitude, depending on the RGB region intercepted. On the other hand, Valenti et al. (2004a). Here, we attribute the discrepancy to the uncertainties in the color calibration of the theoretical isochrone models.
The RGB slope is a useful parameter as it provides a photometric estimate of cluster metallicity. Indeed, the RGB slope becomes progressively flatter with increasing metallicity, mainly because the enhanced molecular blanketing could result in redder colors at a constant temperature in the coolest and brightest stars (Ortolani et al. 1991;Kuchinski et al. 1995). Moreover, the RGB slope in a CMD is independent of reddening and the distance of a cluster. Figure 6 shows the measured RGB slopes as a function of metallicity for the 17 programme clusters with the empirical calibration relations (solid lines) from Valenti et al. (2004a) and the theoretical predictions (dotted lines) from Ivanov & Borissova (2002). It is apparent in Figure 6 that the trends for the dependency of the RGB slopes on the metallicity is consistent with previous observational calibrations and theoretical predictions, i.e., the steeper the RGB slope, the lower the metallicity of the cluster. We also find a good consistency in the theoretical predictions of the distribution of the observed RGB slopes for the metal-poor bulge globular clusters. However, the estimated values of the RGB slopes for the observed metal-poor clusters tend to be flatter at a given metallicity than the corresponding values in the previous empirical calibrations of Valenti et al. (2004a) for the metal-rich bulge clusters and the halo clusters. This disagreement between our results and the relations found by Valenti et al. (2004a) is presum-ably due to the difference in the methods used to determine the RGB slope. Indeed, Valenti et al. (2004a) fit the fiducial ridge line of the RGB in a magnitude range between 0.5 and 5 mag fainter than the brightest stars of each cluster, while we used the fiducial normal points in a magnitude range between 0 and 5 mag fainter than the brightest point to keep consistency with the results in Kim et al. (2006). We also note that the discrepancy might stem from the difficulty in estimating the RGB slope for the metal-poor globular clusters in (J − K, K) plane, especially where the RGB is steeper than in any other plane, as mentioned in Valenti et al. (2004a). In particular, the near-infrared CMDs in the magnitude range that fits the RGB slope for a metal-poor globular cluster can be contaminated by HB stars despite statistical decontaminations from the RGB stars. This is because the HB is not horizontal at all but slanted close to the RGB in the near-infrared CMDs.
The RGB Bump and Tip
The RGB bump on the CMDs has a crucial astrophysical significance for the post-main-sequence evolution of low-mass stars in a globular cluster. The position of the RGB bump for a globular cluster depends on the chemical composition, the age, and other parameters controlling the internal evolution of a star. Theoretical models of stellar evolution (e.g., Thomas 1967;Iben 1968) predict that, at some level in the hydrogen burning shell stage in the RGB after the first dredge-up in a star, the innermost penetration of the convective envelope inside star generates a discontinuity in the hydrogen distribution profile. When the advancing hydrogen burning shell passes through the generated discontinuity, a star is expected to experience an evolutionary hesitation revealed as a temporary drop in luminosity and a change in the evolutionary rate along the RGB. This yields the RGB bump on the CMDs of stars in a globular cluster.
The detection of the RGB bump has been the subject of many studies from an empirical point of view (e.g., Fusi Pecci et al. 1990;Ferraro et al. 1999;Cho & Lee 2002;Valenti et al. 2004bValenti et al. 2007Kim et al. 2006), suggesting that the combined use of the differential and integrated luminosity functions (LFs) of the RGB stars is the best way to properly detect the RGB bump. We note, however, it is more difficult to detect the RGB bumps in the metal-poor globular clusters than in the metal-rich ones, because of the small number of stars along the bright part of the RGB sequence. Indeed, as mentioned in Valenti et al. (2004b), the RGB bumps for the metal-poor clusters occur in the brightest portion of the RGB, which is poorly populated sequence because of the high evolutionary rate of stars at the very end of the RGB. Using the near-infrared LFs of the RGB stars, Valenti et al. (2007) recently determined the RGB bumps for Galactic bulge globular clusters with metallicities in the range of −1.73
[Fe/H] −0.17, and presented new calibrations of the relation between the cluster metallicity and the brightness of the RGB bump in the K and bolometric magnitudes, which differ from those in Valenti et al. (2004b) only in the metal-rich ends.
To construct the LF of the RGB stars for the observed globular clusters in this study, we used the RGB stars selected to define the fiducial normal points of the (J − K, K) CMDs for each cluster. As mentioned in Sect. 3, the selected RGB stars used to estimate the fiducial normal points include only RGB samples within 2σ deviation of the mean color for a given magnitude bin, from which we properly avoided contaminations from other populations of stars, such as asymptotic giant branch star, HB stars, and foreground field stars. Considering the sample size of the RGB stars, we adjusted the size of the magnitude bins of the LFs for each cluster, which enabled us to detect the RGB bump with an appropriate measurement error. Figure 7 shows the differential LF and the logarithmic cumulative LF of the RGB stars for the observed 11 globular clusters. We defined the RGB bump at a significant peak in the differential LF with a break in slope of the logarithmic cumulative LF for the RGB stars in a cluster. In the case of Terzan 4, the RGB bump could not be measured, as the RGB sample is not sufficiently large to reach a safe detection of the bump. Magnitudes of RGB bumps for NGC 6287 and NGC 6626 are not clearly detected in the differential LFs. Instead, the clusters show breaks in the slopes of the cumulative LFs at the magnitudes of which we assign the RGB bumps for the clusters. Applying the distance modulus and the reddening value for each cluster in Table 3, the determined K magnitudes of the RGB bumps were transformed into the absolute M K magnitudes. Then, the bolometric corrections for population II giant stars provided by Montegriffo et al. (1998) were used to convert the absolute magnitudes M K of the RGB bump into the bolometric magnitude M bol .
In columns (2)-(4) of Table 5, we list the observed K, the absolute M K , and the bolometric M bol magnitudes of the RGB bumps for the observed clusters in addition to those for 5 bulge clusters in Kim et al. (2006). The magnitude values of RGB bumps for NGC 6287 and NGC 6626, which were determined from their cumulative LFs of the RGB stars, are in parenthesis. Errors in K and M K are measurement errors, and those in M bol are a combination of measurement errors and the global uncertainty of the distance moduli, which is assumed to be 0.2 mag (e.g., Cho & Lee 2002). Figure 8 plots the determined M K and M bol of the RGB bumps versus cluster metallicity [Fe/H] CG97 and global metallicity [M/H], indicating that the RGB bump moves to fainter locations with increasing cluster metallicity. As shown in Figure 8, the determinations of the RGB bumps for the metal-poor clusters are consistent with the new calibrations for the Galactic bulge clusters (solid curves) of Valenti et al. (2007). The dotted and dashed lines indicate the theoretical predictions of the RGB bump magnitudes as a function of metallicity from the Yonsei-Yale isochrones at t = 12 Gyr and 10 Gyr (Kim et al. 2002;Yi et al. 2003), showing a good agreement with the observations. The RGB tip (TRGB) is the evolution along the RGB ends with helium ignition in the stellar core. Because the luminosity of the TRGB depends on the helium core mass which is fairly constant over a large part of the low mass star range (Salaris et al. 2002), the TRGB has a roughly constant brightness unrelated to the age of the population. Thus, the luminosity of the TRGB is widely used as a standard candle to estimate the distance to galaxies of any morphological type (e.g., Lee et al. 1993;Madore & Freedman 1995;Walker 2003). Recently, this method has also been carried out in nearinfrared observations to estimate the distances of nearby galax- ies and Galactic globular clusters (e.g., Montegriffo et al. 1995;Cioni et al. 2000;Cioni & Habing 2005;Bellazzini et al. 2004). In this paper, the TRGB K magnitude of the observed globular clusters were determined from the brightness measurements of the brightest RGB stars and the bright end of the observed LF of the RGB stars. We note however, we were able to determine the K magnitudes of the TRGBs only for 6 clusters (NGC 6273, NGC 6287, NGC 6293, NGC 6333, NGC 6626, and Terzan 4), because the brightest RGB is too poorly populated to define the TRGB in the limited area of the other observed clusters. Following the case of the RGB bumps, we estimated the absolute M K and the bolometric M bol magnitudes of the TRGB for the observed clusters.
The measured K, M K , and M bol magnitudes of the TRGB are listed in the columns (5)-(7) of Table 5. Similar to the M bol of the RGB bumps, errors in M bol for the TRGB are a combination of measurement errors and the global uncertainty of 0.2 mag of the distance moduli. Figure 9 shows the relationship between the M K and M bol of the TRGB and the cluster metallicity of the observed 6 clusters in addition to those of the 5 clusters in Kim et al. (2006), indicating a good correlation with the previous calibrations (solid lines) of Valenti et al. (2004a). As noted in Kim et al. (2006), the values of the TRGB for a compact post-core-collapse cluster NGC 6717 show a significant deviation from the calibration relations, because the number of bright RGB stars are still too small to accurately measure the TRGB on the observed CMDs. In Figure 9 we overlay the theoretical predictions of the TRGB magnitudes as a function of metallicity estimated from the Yonsei-Yale isochrones (Kim et al. 2002;Yi et al. 2003), which also seems to be consistent with the observations.
Summary and Conclusions
Detailed analyses of the RGB morphology for 12 metal-poor ([Fe/H] ≤ −1.0) globular clusters in the Galactic bulge direction have been performed using the high-quality near-infrared JHK ′ photometry. From the study of the RGB shapes in the near-infrared CMDs for each cluster, we measured photometric parameters, such as, the colors at different magnitude levels, the magnitudes at different colors, and the RGB slopes. The magnitudes of the RGB bump and tip, as major RGB evolutionary features, have also been determined from the LFs of the selected RGB stars in each cluster. The determined indices of the RGB morphology for the 12 observed clusters have been combined with the results for 5 bulge clusters in Kim et al. (2006), thus the entire dataset comprises ∼ 75% of the total 22 metal-poor ([Fe/H] ≤ −1.0) globular clusters within 3 kpc from the Galactic center. The behavior of the RGB morphology for the programme clusters has been compared with the previous empirical calibration relations as a function of cluster metallicity for the Galactic bulge globular clusters by Valenti et al. (2004aValenti et al. ( 2007 and theoretical predictions of the Yonsei-Yale isochrones (Kim et al. 2002;Yi et al. 2003). The results are summarized as follows: Our results indicate that the correlations between the derived RGB indices and the cluster metallicity for the metal-poor globular clusters in the Galactic bulge direction are consistent with previous observational calibration relations for a sample of the metal-rich bulge clusters and the halo clusters (Valenti et al. 2004a). The trends of the theoretical models reliably represent the observed RGB color and magnitude indices, although there appears to be systematic shifts in color and magnitude, as a result of the uncertainties in the theoretical calculations and observational measurements. 2. The RGB slopes have been estimated from the determined fiducial normal points at the magnitude range between 0 and 5 magnitude fainter than the brightest point of the RGB. The distribution of the RGB slopes for the observed clusters show an expected evolutionary feature, i.e., the lower metallicity of the cluster, the steeper the RGB slope, while the RGB slopes for the programme clusters tend to be slightly flat-ter than those in the previous calibrations of Valenti et al. (2004a). 3. The absolute M K and bolometric M bol magnitudes of the RGB bump and tip for the observed clusters have been determined from the differential and cumulative LFs of the selected RGB stars. The correlations between the cluster metallicity and the derived magnitudes of the RGB bump and tip for the metal-poor clusters in the Galactic bulge direction are consistent with the recent calibration relations for the Galactic bulge clusters (Valenti et al. 2007).
Of a total of 17 metal-poor clusters presented in this paper, only two clusters NGC 6266 and NGC 6723 have the cluster's orbital data (Dinescu et al. 1999(Dinescu et al. 2003, indicating that NGC 6723 is a halo member passing the Galactic bulge at this moment and NGC 6266 is associated with the motion of the Galactic thick disk. Together with the derived RGB morphological properties, further information about detailed orbital data will provide more robust constraints on the role of the metal-poor globular clusters in the formation of the Galactic bulge.
Fig. 1 .
1The upper and lower panels are (J − K ′ , K ′ ) and (J − H, H) CMDs of the observed 12 clusters.
Figure 1
1shows (J − K ′ , K ′ ) and (J − H, H) CMDs of the resolved stars in the observed area for the clusters investigated
Fig. 2 .
2Fiducial normal points of target clusters in (J − K) o − M K (upper) and (J − H) o − M H (lower) planes. For clarity, the colors are given zero-point offsets; from left to right, these are const. = 0.8, 1.6, 2.4, 3.2, 4.0, 4.8, 5.6, 6.4, 7.2, 8.0, 8.8 and 9.6 magnitudes.Open circles indicate fiducial normal points determined from the 2MASS data.
), i.e., [M/H]=[Fe/H] CG97 + log(0.638 f α + 0.362) with f α = 10 0.30 , where f α is the enhancement factor of the α elements. The determined metallicity values of [Fe/H] CG97 and [M/H] for each cluster are listed in
Figure 2
2shows the fiducial normal points in (J − K) o − M K and (J − H) o − M H planes for the target clusters. In the case of bright stars saturated around the RGB tip, we estimated the fiducial normal points from the 2MASS catalog data of the area observed in this study, and those are represented in Figure 2 by open circles.
0.826±0.02 0.778±0.02 0.686±0.02 0.628±0.02 0.680±0.03 0.644±0.03 0.584±0.03 0.537±0.03 NGC 6717 0.968±0.03 0.887±0.03 0.777±0.03 0.714±0.03 0.820±0.02 0.796±0.02 0.727±0.02 0.650±0.02 NGC 6723 0.940±0.02 0.899±0.02 0.787±0.03 0.696±0.03 0.751±0.03 0.717±0.03 0.665±0.04 0.580±0.03
(J − K) o and (J − H) o colors at four fixed absolute magnitude levels of M K = M H = (−5.5, −5, −4, and −3), (2) the absolute magnitudes of M K and M H at fixed colors of (J − K) o = (J − H) o = 0.7, and (3) the slope in the (J − K, K) color-magnitude plane.
lists the determined (J − K) o and (J − H) o colors at different magnitude levels. Furthermore, the absolute magnitudes M K at (J − K) o = 0.7 and M H at (J − H) o = 0.7, and the RGB slope are presented in
Figure 4 ,
4we present the colors at fixed magnitudes of M K = M H = (−5.5, −5, −4, and −3) as functions of cluster metallicity [ Fe/H] CG97 and global metallicity [M/H] for 17 programme clusters. Filled and open circles represent samples of clusters in this paper and Kim et al. (2006), respectively. Solid lines are the calibration relations of Valenti et al. (2004a) for globular clusters spanning a metallicity range of −2.12 [Fe/H] −0.49 in the Galactic bulge and halo. As can be seen
Fig. 3 .
3RGB (J−K) o color indices at fixed magnitudes of M K as a function of the metallicity [Fe/H] CG97 and the global metallicity [M/H]. Filled circles and open circles represent 12 metal-poor bulge clusters observed here and 5 bulge clusters inKim et al. (2006), respectively. Solid lines are the calibration relations ofValenti et al. (2004a). Dotted and dashed lines are the theoretical predictions of the Yonsei-Yale isochrones(Kim et al. 2002;Yi et al. 2003) at t = 12 Gyr and 10 Gyr, respectively.
Fig. 4 .
4RGB (J−H) o color indices at fixed magnitudes of M H as a function of the metallicity [Fe/H] CG97 and the global metallicity [M/H]. Symbols are the same as Figure 3. in Figure 3 and Figure 4, the trends of the RGB color indices of (J − K) o and (J − H) o as a function of metallicity agree well with the calibrations put forward by Valenti et al. (2004a). The RGB color indices of (J − K) o and (J − H) o linearly scale with the cluster metallicity as the RGB color indices are bluer for the metal-poor clusters than metal-rich clusters. In addition the fit
Fig. 5 .
5RGB magnitude indices M K and M H at fixed color (J − K) o = (J − H) o = 0.7 as a function of the metallicity [Fe/H] CG97 and the global metallicity [M/H]. Symbols are the same as Figure 3.
Fig. 6 .
6The RGB slope as a function of metallicity. The dotted lines are the relations found byIvanov & Borissova (2002). The other symbols are the same asFigure 3. theoretical predictions of the absolute magnitudes of M K and M H at fixed colors of (J −K) o = (J −H) o = 0.7 from Yonsei-Yale isochrones (dotted and dashed lines) seem to be much fainter than the observed calibrations. The magnitude shifts of the theoretical models at [Fe/H]= −1.5 are ∼ 1.3 − 1.4 mag fainter in M K and M H than the empirical results of
Fig. 7 .
7The logarithmic cumulative (upper) and differential (lower) LFs for RGB stars in the observed clusters. The arrows indicate the RGB bump position. The dashed lines in the cumulative LF are the linear fit to the regions above and below the RGB bump.
Fig. 8 .
8The behavior of the M K and M bol magnitudes of the RGB bumps for the observed clusters as a function of metallicity [Fe/H] CG97 and global metallicity [M/H]. Symbols are the same as Figure 3.
Fig. 9 .
9The behavior of the M K and M bol magnitudes of the RGB tip for the observed clusters as a function of metallicity [Fe/H] CG97 and global metallicity [M/H]. Symbols are the same as Figure 3..
1 .
1Photometric indices for the RGB color at fixed magnitudes, M K = M H = (−5.5, −5, −4, and −3), and the RGB magnitudes at fixed colors, (J − K) o = (J − H) o = 0.7 have been measured from the fiducial normal points of the near-infrared (J − K, K) and (J − H, H) CMDs.
Table 1 .
1Observational log of the target clustersTarget
Filter Exp. time (sec) FWHM(
′′ ) Year
NGC 6333
J
4×1, 8×30
0.60, 0.67 2002
H
4×1, 8×30
0.61, 0.58
K
Table 2 .
2Metallicity, distance modulus, reddening, and extinction values of the observed 12 globular clusters in the Galactic bulge.Target
[Fe/H] CG97 [M/H]
µ o
E(B − V)
A J
A H
A K
NGC 6235
-1.17
-0.97
15.05
0.36
0.325 0.207 0.132
NGC 6266
-1.08
-0.87
14.26
0.47
0.424 0.271 0.172
NGC 6273
-1.45
-1.24
14.58
0.41
0.370 0.236 0.150
NGC 6287
-1.90
-1.68
15.61
0.60
0.541 0.346 0.220
NGC 6293
-1.73
-1.53
14.79
0.41
0.370 0.236 0.150
NGC 6325
-1.21
-0.99
14.10
0.89
0.803 0.513 0.327
NGC 6333
-1.56
-1.36
14.67
0.38
0.343 0.219 0.139
NGC 6355
-1.26
-1.07
14.60
0.75
0.677 0.432 0.275
NGC 6401
-0.97
-0.74
14.61
0.72
0.649 0.415 0.264
NGC 6558
-1.21
-0.99
14.30
0.44
0.397 0.253 0.161
NGC 6626
-1.21
-0.99
13.60
0.40
0.361 0.230 0.147
Terzan 4
-1.62
-1.41
15.10
2.06
1.858 1.187 0.756
Table 3 .
3The RGB colors of the observed bulge clusters at different magnitudes.
Table 4 .
4The RGB magnitudes at different colors, and the RGB slopes for the observed bulge clusters.Target
M (J−K)o=0.7
K
M (J−H)o =0.7
H
RGB slope
This paper
NGC 6235 -3.55±0.64 -5.56±1.11 -0.068±0.006
NGC 6266 -3.49±0.57 -4.90±0.45 -0.098±0.008
NGC 6273 -4.60±0.36
...
-0.084±0.009
NGC 6287 -4.45±1.11 -6.17±0.76 -0.069±0.008
NGC 6293 -4.03±0.42 -5.63±0.16 -0.066±0.004
NGC 6325 -3.10±0.51 -4.30±0.35 -0.106±0.010
NGC 6333 -4.55±0.46 -6.06±0.26 -0.064±0.003
NGC 6355 -3.48±0.64 -5.14±0.59 -0.078±0.008
NGC 6401 -2.09±0.97 -3.72±0.47 -0.087±0.010
NGC 6558 -3.22±0.58 -3.73±0.63 -0.083±0.004
NGC 6626 -3.93±0.53 -5.01±0.47 -0.095±0.005
Terzan 4
-4.68±0.37 -6.28±0.79 -0.075±0.005
Kim et al. (2006)
NGC 6541 -4.14±0.23 -5.56±0.34 -0.072±0.003
NGC 6642 -3.71±0.28 -4.51±0.37 -0.104±0.006
NGC 6681 -4.10±0.25 -5.94±0.24 -0.075±0.003
NGC 6717 -2.73±0.33 -3.66±0.25 -0.077±0.004
NGC 6723 -3.05±0.37 -4.87±0.50 -0.082±0.003
Table 5 .
5Magnitudes of the RGB bump and tip for the observed bulge clusters.Target
K Bump
M Bump
K
M Bump
bol
K T ip
M T ip
K
M T ip
bol
This paper
NGC 6235
13.39±0.06
-1.79±0.06
0.06±0.21
...
...
...
NGC 6266
12.67±0.05
-1.76±0.05
0.08±0.21
...
...
...
NGC 6273
12.82±0.05
-1.91±0.05
-0.28±0.21
8.78±0.05
-5.96±0.05 -3.43±0.21
NGC 6287 (13.56±0.07) (-2.27±0.07) (-0.33±0.21)
9.88±0.07
-5.96±0.07 -3.32±0.21
NGC 6293
12.80±0.08
-2.14±0.08
-0.30±0.22
9.28±0.08
-5.66±0.08 -3.25±0.22
NGC 6325
12.55±0.10
-1.88±0.10
0.1±0.22
...
...
...
NGC 6333
12.83±0.10
-1.98±0.10
-0.09±0.22
8.82±0.10
-5.99±0.10 -3.53±0.22
NGC 6355
13.14±0.09
-1.74±0.09
0.09±0.22
...
...
...
NGC 6401
13.29±0.06
-1.58±0.06
0.55±0.21
...
...
...
NGC 6558
12.83±0.09
-1.63±0.09
0.21±0.22
...
...
...
NGC 6626 (11.96±0.10) (-1.79±0.10)
(0.01±0.20)
7.45±0.10
-6.29±0.10 -3.69±0.22
Terzan 4
...
...
...
10.06±0.03 -5.80±0.03 -3.38±0.20
Kim et al. (2006)
NGC 6541
11.74±0.09
-2.36±0.09
-0.48±0.22
8.59±0.09
-5.67±0.09 -3.22±0.22
NGC 6642
13.23±0.10
-1.41±0.10
0.30±0.22
8.29±0.10
-6.35±0.10 -3.53±0.22
NGC 6681
13.13±0.04
-1.90±0.04
-0.02±0.20
8.69±0.04
-6.34±0.04 -3.78±0.20
NGC 6717
12.98±0.05
-1.53±0.05
0.57±0.21
9.02±0.05
-5.49±0.05 -2.84±0.21
NGC 6723
13.19±0.05
-1.54±0.05
0.50±0.21
8.73±0.05
-6.00±0.05 -3.37±0.21
Acknowledgements. This work has been supported by the Korea Research Foundation Grant funded by the Korea Government (KRF 2007-313-C00321), and also partly supported by Korea Astronomy and Space Science Institute (KASI 2009220000 and Yonsei-KASI Joint Research Program for Frontiers of Astronomy and Space Science), for which we are grateful.
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[] |
[
"Long-range correlations of density in a Bose-Einstein condensate expanding in a random potential",
"Long-range correlations of density in a Bose-Einstein condensate expanding in a random potential"
] |
[
"N Cherroret \nLaboratoire de Physique et Modélisation des Milieux Condensés\nUniversité Joseph Fourier\nCNRS\n25 rue des Martyrs, BP 16638042GrenobleFrance\n",
"S E Skipetrov \nLaboratoire de Physique et Modélisation des Milieux Condensés\nUniversité Joseph Fourier\nCNRS\n25 rue des Martyrs, BP 16638042GrenobleFrance\n"
] |
[
"Laboratoire de Physique et Modélisation des Milieux Condensés\nUniversité Joseph Fourier\nCNRS\n25 rue des Martyrs, BP 16638042GrenobleFrance",
"Laboratoire de Physique et Modélisation des Milieux Condensés\nUniversité Joseph Fourier\nCNRS\n25 rue des Martyrs, BP 16638042GrenobleFrance"
] |
[] |
We study correlations of atomic density in a weakly interacting Bose-Einstein condensate, expanding diffusively in a random potential. We show that these correlations are long-range and that they are strongly enhanced at long times. Density at distant points exhibits negative correlations.
|
10.1103/physrevlett.101.190406
|
[
"https://arxiv.org/pdf/0807.2954v2.pdf"
] | 12,236,888 |
0807.2954
|
8dceaccde1ce75c3232f556cf6eec91c5ddd5d7e
|
Long-range correlations of density in a Bose-Einstein condensate expanding in a random potential
4 Oct 2008 (Dated: October 4, 2008)
N Cherroret
Laboratoire de Physique et Modélisation des Milieux Condensés
Université Joseph Fourier
CNRS
25 rue des Martyrs, BP 16638042GrenobleFrance
S E Skipetrov
Laboratoire de Physique et Modélisation des Milieux Condensés
Université Joseph Fourier
CNRS
25 rue des Martyrs, BP 16638042GrenobleFrance
Long-range correlations of density in a Bose-Einstein condensate expanding in a random potential
4 Oct 2008 (Dated: October 4, 2008)arXiv:0807.2954v2 [cond-mat.dis-nn]
We study correlations of atomic density in a weakly interacting Bose-Einstein condensate, expanding diffusively in a random potential. We show that these correlations are long-range and that they are strongly enhanced at long times. Density at distant points exhibits negative correlations.
The behavior of Bose-Einstein Condensates (BECs) in disordered potentials has attracted growing interest of physicists during the last few years. In particular, expansion of BECs in random potentials has been investigated in detail [1,2,3,4,5,6]. The main interest of using BEC systems to study disorder-related phenomena is that their physical parameters (such as the number of atoms, the parameters of the random potential, or the strength of inter-atom interactions) can be controlled quite precisely. During the last two years, considerable efforts were undertaken to study the atomic density n(r, t) averaged over disorder in 1D [2], 2D [3], and 3D [3,4] geometries, with special interest in the phenomenon of Anderson localization [2,4,5,6]. At the same time, little is known about statistical fluctuations of n(r, t) due the randomness of the potential. Recently, Henseler and Shapiro [7] have shown that a BEC expanding in a random potential is characterized by a complicated, highly irregular density pattern reminiscent to what we know as "speckle" in optics [8]. According to Ref. [7], multiple scattering from the potential completely randomizes n(r, t) and reduces the correlation length of atomic speckle pattern to a value of the order of the healing length of the initial trapped condensate, which is the minimal possible correlation length for a coherent matter wave. The purpose of this Letter is to show that the macroscopic coherence of the condensate gives rise to genuine interference effects that were ignored in Ref. [7]. These interference effects are similar in nature to those leading to Anderson localization of the condensate at large disorder [4]. They result in stronger fluctuations and long-range correlations of n(r, t), akin to the long-range correlations of conductance fluctuations in disordered metals [9] and the long-range correlations of intensity in optical speckle patterns [10,11]. For a BEC expanding inside an optical waveguide, the longrange correlations grow in absolute value with time and become dominant in the long-time limit. They can take both positive (for relatively close points) and negative (for distant points) values.
We consider a dilute BEC of N ≫ 1 atoms of mass m inside an infinitely long optical waveguide of diam-eter d and cross-section A = πd 2 /4, parallel to the z axis and described by a 2D potential V ⊥ (x, y), see Fig. 1. The waveguide geometry is rather popular in BEC experiments and was, in particular, used in the recent work on Anderson localization [5]. In the longitudinal direction, the condensate was initially confined by a tight 1D parabolic trap potential V z (z) = mω 2 z z 2 /2 that has been switched off to let the condensate expand along the z axis. A time T 1/ω z later the role of interactions between atoms in the condensate becomes negligible [2,5,13] and a weak 3D random potential V (r) is switched on; we refer to this moment as t = 0. V (r) is assumed to have a white-noise Gaussian statistics: V (r) = 0 and V (r)V (r ′ ) = uδ(r − r ′ ), where the horizontal bar denotes averaging over realizations of the random potential. The associated mean free path is ℓ = 4 π/um 2 [12] and the "weakness" of the random potential is quantified by a condition k µ ℓ ≫ 1, where k µ = √ 2mµ/ and µ is the chemical potential of the initial trapped condensate. For t > 0, the macroscopic wave function of the condensate ψ(r, t) obeys the linear Schrödinger equation [13]:
i ∂ψ ∂t = − 2 2m ∆ + V ⊥ (x, y) + V (r) ψ.(1)
In this Letter we assume that the diameter d of the waveguide to which the expansion of the BEC is constrained obeys 2π/k µ ≪ d ℓ. This corresponds to what is known as a "quasi-1D" geometry in the multiplescattering literature [12]: at distances larger than ℓ, the average atomic densityn(r, t) = |ψ(r, t)| 2 is described by a 1D diffusion process, unlike the recent work [2,5] where 1D (and not 3D) disorder was considered and diffusive propagation didn't appear. In the quasi-1D geometry, the localization length at energies of the order of µ is typically ξ µ ∼ ℓ(k µ d) 2 , exceeding the mean free path ℓ by a factor (k µ d) 2 ≫ 1. Hence, the condensate expands by diffusion until |z| ∼ ξ µ ≫ ℓ before it starts to be affected by Anderson localization effects. This regime was not accessible in the recent experiment [5] and has not been studied theoretically yet. For weak disorder k µ ℓ ≫ 1 and at large distances z ≫ ℓ and long times t ≫ ℓ/v µ (with v µ = k µ /m being the typical velocity of an atom with kinetic energy µ), the average atomic density is independent of x, y and can be written as [3,4,7]
n(z, t) = ∞ −∞ dk 2π |φ(k)| 2 P ǫ k (z, t),(2)
where |φ(k) [2,13,14], with H(x) the Heaviside step function, describes the momentum distribution of atoms in the condensate at t = 0, ǫ k = 2 k 2 /2m, and P ǫ (z, t) is the probability to find a particle of energy ǫ, initially located at the origin, in the vicinity of r = (x, y, z) after a time t [12]. The Fourier transform of the latter
| 2 ∝ (1 − k 2 /2k 2 µ )H(1 − |k| / √ 2k µ )is P ǫ (z, Ω) = G ǫ+ Ω/2 (r, 0)G * ǫ− Ω/2 (r, 0)/2πν ǫ , where G ǫ (r, r ′ )
is the Fourier transform of the Green's function of Eq. (1) and ν ǫ is the local density of states at the energy ǫ. In the hydrodynamic limit Ω ≪ ǫ, P ǫ is a solution of a 1D diffusion equation:
P ǫ (z, Ω) = exp(− |z| −iΩ/D ǫ )/2A √ −iΩD ǫ [12], where D ǫ = v ǫ ℓ/3 is the diffusion coefficient for atoms at energy ǫ. This yieldsn (z, t) = N A D µ t f z D µ t ,(3)
where f (x) can be expressed through special functions and f (x) ≃ 0.6 for x ≫ 1.
As the condensate expands, its typical size grows with time according to z 2 ≃ D µ t; profiles of atomic density are plotted as functions of z in the inset of Fig. 2. In the main plot of Fig. 2 we shown as a function of time. The density reaches a maximum at the "arrival time" t arrival ≃ 2z 2 /D µ and decays as 1/ √ t at long times t > t arrival . This long-time limit is the most interesting regime to which we restrict our analysis from here on. It is worthwhile to note that Eq. (3) breaks down at very long times, when Anderson localization comes into play. Indeed, Eq. (3) predictsn → 0 for t → ∞, whereas localization will "freeze"n at values of the order of N/Aξ µ (for z < ξ µ ), starting from some localization time t loc . The latter can be estimated by comparing N/Aξ µ with the long-time limit of Eq. (3) or, equivalently, by equating z 2 and ξ 2 µ . One obtains t loc ∼ (ℓ/v µ )(k µ d) 4 , which shows that a considerable interval of validity exists for Eq. (3) between the typical mean-free time ℓ/v µ and t loc ≫ ℓ/v µ , when k µ d ≫ 1.
Let us now study correlations of atomic density in the expanding condensate, which is the primary subject of this Letter. We define the correlation function of density fluctuations δn(r, t) = n(r, t) −n(r, t) as C(r, t; r ′ , t ′ ) = δn(r, t)δn(r ′ , t ′ ) n(r, t) × n(r ′ , t ′ ) .
Using the relation ψ(r, t) = d 3 r 1 G(r, r 1 , t)φ(r 1 ), we can write the numerator of Eq. (4) as
δn(r, t)δn(r ′ , t ′ ) = 1 (2π ) 4 × 4 j=1 d 3 r j dǫ j e − i (ǫ1−ǫ2)t− i (ǫ3−ǫ4)t ′ × K (r, t; r ′ , t ′ ; {r j } , {ǫ j }) φ(r 1 )φ * (r 2 )φ(r 3 )φ * (r 4 ),(5)
where the 6-point kernel K is given by the connected part of a product of 4 Green's functions, averaged over disorder:
K = G ǫ1 (r, r 1 )G * ǫ2 (r, r 2 )G ǫ3 (r ′ , r 3 )G * ǫ4 (r ′ , r 4 ) − G ǫ1 (r, r 1 )G * ǫ2 (r, r 2 ) × G ǫ3 (r ′ , r 3 )G * ǫ4 (r ′ , r 4 ). (6)
The largest contribution to K is obtained by decoupling GG * GG * in the first line of Eq. (6) as if G were a circular complex Gaussian random field: K 1 = G ǫ1 (r, r 1 )G * ǫ4 (r ′ , r 4 ) × G ǫ3 (r ′ , r 3 )G * ǫ2 (r, r 2 ). When K = K 1 is inserted into Eqs. (5) and (4), the short-range correlation function C 1 studied in Ref. [7] is obtained. C 1 is of order 1 for r = r ′ , but rapidly decays to zero already for |r − r ′ | ∼ 1/k µ . The long-range part of the correlation function -C 2 -can be obtained by using a next-order contribution to K: K 2 given by the diagram depicted in Fig. 3. This diagram represents an interference process between four matter waves that propagate in pairs to some point, where they interchange partners before continuing to the measurement points r and r ′ . A proper treatment of such a crossing of wave paths is guaranteed by the use of the so-called Hikami box diagram [15,16] the H box in Fig. 3 -that ensures conservation of particle number in the final result. A simplified version of the diagram of Fig. 3, corresponding to equal energies ǫ j = ǫ 0 and identical initial points r j = r 0 , yields the long-range correlation function of intensity fluctuations for a scalar wave emitted by a point source in a disordered medium [11]. Evaluating the diagram of Fig. 3 and inserting the result into Eq. (5) gives
δn(r, t)δn(r ′ , t ′ ) = 2πℓ 2 A 3m 2 (2π) 4 × ∞ −∞ dz 1 2 j=1 dk j dΩ j |φ(k j )| 2 P ǫ k j (z 1 , Ω j ) ×∂ z1 P ǫ− (z − z 1 , Ω − )∂ z1 P ǫ+ (z ′ − z 1 , Ω + )e −i(Ω+t+Ω−t ′ ) ,(7)
where ǫ ± = [ǫ k1 + ǫ k2 ± (Ω 1 − Ω 2 )/2]/2 and Ω ± = ±(ǫ k1 − ǫ k2 ) + (Ω 1 + Ω 2 )/2. This equation can be evaluated numerically and allows for analytical analysis in some special cases, as we now show. The simplest quantity that Eq. (7) allows us to study is the correction to the (normalized) variance of the atomic density fluctuations: where the unity on the right-hand side originates from the C 1 term and
δn 2 (r, t) n 2 (r, t) = 1 + C 2 (r, t; r, t),(8)C 2 (r, t; r, t) = 2 k 2 µ A k µ ℓ c(ζ, τ )(9)
with a combinatorial factor 2 added. Here we introduced dimensionless variables ζ = z k µ ℓ/ℓ and τ = tµ/ that feature natural spatial and temporal scales of the problem. The dependence of C 2 on position ζ and time τ appears to be given by a universal function c(ζ, τ ) that does not depend on any parameters of the problem. We plot this function in Fig. 4 for three fixed values of τ . It is quadratic in ζ for small ζ ≪ τ 1/4 : c(ζ, τ ) ≃ Bτ 3/4 [1 − C(ζ/τ 1/4 ) 2 ] and decays as Dτ /ζ for ζ ≫ τ 1/4 . Here B ≃ 6.3, C ≃ 0.3 and D ≃ 1.9 are constants that had to be calculated numerically. c(ζ, τ ) grows with time as τ 3/4 for ζ ≪ τ 1/4 as we also show in the inset of Fig. 4. This amplification of c(ζ, τ ) with time can make C 2 significant for large τ , despite the small prefactor in front of c in Eq. (9). C 2 (r, t; r, t) becomes of order 1 for t ∼ (ℓ/v µ )(k µ d) 8/3 (k µ ℓ) −1/3 , which is still smaller than the localization time t loc that limits the validity of our analysis. The long-range correlation of the fluctuations of atomic density is obtained from Eq. (7) with ∆z = |z − z ′ | ≫ ℓ. Note that C 1 correlation is negligible for such large spatial separations. In this work we restrict ourselves to equal-time correlations: t = t ′ . As an example, we consider correlations of density at two points symmetric with respect to the origin: z = −z ′ = ∆z/2, but results are qualitatively similar for any sufficiently distant z and z ′ of opposite sign. We obtain We show c 2 in Fig. 5 as a function of ∆ζ for three different times. For small distances ∆ζ ≪ τ 1/4 , the correlation decays linearly with ∆ζ:
C 2 (r, t; r ′ , t) = 1 k 2 µ A k µ ℓ c 2 (∆ζ, τ ).(10)c 2 (∆ζ, τ ) ≃ Bτ 3/4 (1 − E∆ζ/τ 1/4 ), where E ≃ 2.
At ∆ζ ∼ τ 1/4 , it becomes negative and reaches a minimum. For longer distances ∆ζ ≫ τ 1/4 , c 2 remains negative whereas its magnitude decays only algebraically:
c 2 ≃ −F τ /∆ζ with F ≃ 1.2.
In addition to having long range is space, C 2 correlation grows in magnitude with time (see the inset of Fig. 5), similarly to the variance of δn. This should facilitate its experimental observation. Negative correlations of atomic density in an expanding BEC could be anticipated from the conservation of atom number N which implies d 3 rδn(r, t)δn(r ′ , t) = 0 and thus requires that the integrand must change sign. (Note that our Eq. (7) obeys this condition exactly.) The important result of our work is to show that negative correlations occur at large distances between points r and r ′ and that they become increasingly important as the condensate expands. Negative correlations of similar origin were predicted to exist in reflection of waves from a thick disordered slab [17]. The analysis of long-range correlations introduces a new characteristic length scale ζ * ∼ τ 1/4 or z * ∼ (D µ t · ℓ/k µ ) 1/4 which is much smaller than the root mean square size of the condensate z 2 1/2 . It determines the typical separation between two points, situated symmetrically with respect to the initial location of the condensate, at which density correlations change sign.
In conclusion, the macroscopic coherence of the condensate prevents the breakdown of the atomic speckle pattern n(r, t) into small independent spots, imposing correlations between distant points. As a consequence, atomic speckles appear to have a much more complex and nontrivial spatial structure than just a random arrangement of small regions of high and low density put forward in Ref. [7]. Although in the present paper we consider an uncorrelated, white-noise potential, our results can be generalized to correlated potentials in a standard way [12,18]. We estimate that the long-range correlations that we study in this Letter should be directly observable under conditions of the experiment of Ref. [5] for an isotropic random potential with the correlation length equal to the longitudinal correlation length of Ref. [5] and a slightly weaker transverse confinement. It would be extremely interesting to extend our calculation to the Anderson localization regime (z > ξ µ ), where, by analogy with microwave experiments [19], one expects anomalously large density fluctuations.
FIG. 1 :
1(color online). Cartoon of a BEC expanding in a 3D random potential and confined transversally to a waveguide of typical diameter d. We assume 2π/kµ ≪ d ℓ.
FIG. 2 :
2(color online). Ensemble-averaged atomic densityn of a BEC expanding in a 3D random potential inside a quasi-1D optical waveguide. The main plot showsn as a function of time for three different distances z. The dashed line is a 1/ √ t asymptote. The inset showsn as a function of z.
FIG. 3 :
3Diagram for the kernel K2 generating the long-range correlation function of density fluctuations C2 through Eq.(5). The solid and dashed lines represent the Green's functions G and it complex conjugates G * , respectively. The two parallel G-lines connected by dotted "ladders" symbolize averages of products of two Green's functions, GG * . H = (ℓ 5 m 3 /3π 3 ǫ1)∇r 5 · ∇r 7 is the Hikami box[15,16]. Integration should be performed over r5 = r6 = r7 = r8.
FIG. 4 :
4(color online). Position dependence of the correction c to the variance of atomic density in an expanding BEC for three different times τ = tµ/ . The dimensionless distance is ζ = z p kµℓ/ℓ. The dashed lines show analytic results Bτ 3/4 [1 − C(ζ/τ 1/4 ) 2 ] (small ζ) and Dτ /ζ (large ζ) for τ = 3 × 10 4 . The numerical constants B, C and D given in the text. The inset shows the time dependence of c for a given (small) ζ = 0.4.
online). Long-range correlation function c2 versus dimensionless distance ∆ζ = 2ζ (solid lines) for three different times τ = tµ/ . The dashed lines show asymptotes Bτ 3/4 (1 − E∆ζ/τ 1/4 ) and −F τ /∆ζ for τ = 3 × 10 4 . The numerical constants B, E and F are given in the text. The inset shows the time dependence of the correlation for ∆ζ = 0.8.
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"N H Bingham ",
"Tasmin L Symons "
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We begin with isotropic Gaussian random fields, and show how the Bochner-Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs). We then turn from this continuous setting to approximating discrete settings, Gaussian Markov random fields (GMRFs), and the computational advantages they bring in handling large data sets, by exploiting the sparseness properties of the relevant precision (concentration) matrices.
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10.1090/tpms/1163
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"https://arxiv.org/pdf/2111.11960v1.pdf"
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2111.11960
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5feb1b47d9e57f975d25da1c74dbc5f781bcc455
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23 Nov 2021
N H Bingham
Tasmin L Symons 23 Nov 2021GAUSSIAN RANDOM FIELDS: WITH AND WITHOUT COVARIANCES To Mikhailo Iosifovich Yadrenko, on his 90th birthdayGaussian random fieldcovariance functionBessel potentialstochastic partial differential equationMatérn processGaussian Markov random fieldprecision matrixsparsenessnumerical linear algebra
We begin with isotropic Gaussian random fields, and show how the Bochner-Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs). We then turn from this continuous setting to approximating discrete settings, Gaussian Markov random fields (GMRFs), and the computational advantages they bring in handling large data sets, by exploiting the sparseness properties of the relevant precision (concentration) matrices.
Introduction
In this survey, we study Gaussian random fields -Gaussian processes parametrized by some space Ω, which will usually be Euclidean space R d of dimension d or the sphere S d of dimension d as a Riemannian manifold, embedded in R d+1 ; we normalise to radius 1 for convenience. Gaussian processes are specified by their mean function (which we shall take to be zero for simplicity), and covariance function. So specifying Gaussian processes is the same as specifying covariance functions, equivalently, positive-definite functions. All this is in a continuous setting, where we can use our most powerful weapon, calculus.
Data, however, and the computation needed to handle it, are discrete. The tension between these two inspired the title of the 2012 paper [SimLR1]: 'In order to make spatial statistics computationally feasible, we need to forget about the covariance function' (of course, the authors do not dismiss covariances for theory, only for computational statistics with large data sets). Regarding Euclidean space and spheres, compare their later comment [SimLP2]:
'the great tragedy of spatial statistics is that the Earth turned out not to be flat'.
We begin in §1 with a study of covariance functions in symmetric spaces (this class, aptly named, includes both Euclidean space and spheres). Our main tool is the powerful and under-utilised Bochner-Godement theorem. We continue in §2 with Bessel potentials (again, powerful and under-utilised in this area), stochastic partial differential equations (SPDEs), and Matérn processes, ubiquitous in the field.
Quoting the above authors again: 'In contrast to traditional statistical modelling, practical problems in spatial statistics are, by and large, computational in nature'. We turn in §3 to Gaussian Markov random fields (GMRFs), which arise [LinRL1] via graphs obtained by triangulation of the space Ω. Dependence occurs only between neighbouring vertices of the graph. This means that the precision (or concentration) matrix (the inverse of the covariance matrix) is sparse: most of its elements are zero. The computational burden is then carried by sparse numerical linear algebra.
For further background on random fields, see [Mal], [MarP], [Yad].
1. Covariances; the Bochner-Godement theorem
The Bochner and Bochner-Schoenberg theorems
In Euclidean space, the positive definite functions (normalised to be 1 at the origin) are the characteristic functions, by Bochner's theorem of 1933. These are also the covariance functions [Kal1,50]: for a process ξ parametrised by T , ρ ij = cov(ξ(τ i ), ξ(τ j )), ij a i a j ρ ij = a i a j cov(ξ(τ i ), ξ(τ j )) = var( i a i ξ(τ i )) ≥ 0, for all a i ∈ R, τ i ∈ T . This result was extended from processes on Euclidean space to isotropic processes on spheres by Bochner and Schoenberg in 1940-42 (see [BinS4] for references). Here one needs the ultraspherical (or Gegenbauer) polynomials P λ n (u) of index λ (u ∈ [−1, 1], λ = 1 2 (d − 1)); these are classical orthogonal polynomials on the interval [−1, 1]; see e.g. Szegő [Sze,§4.7]. The Bochner-Schoenberg theorem gives the general isotropic covariance on S d to within scale by a convex combination (or mixture) of Gegengauer polynomials: c ∞ 0 a n P λ n (u), a n ≥ 0,
∞ 0 a n = 1, u = d(x, y), x, y ∈ S d , λ = 1 2 (d − 1)
(BS) (here and below, isotropic means that the covariance between the values at two points depends only on their geodesic distance u = d(x, y) as here).
Symmetric spaces
One looks for a general framework to include both these classical results, and this is given by symmetric spaces. This field, which stems from Elie Cartan in the 1920s, belongs to differential and Riemannian geometry and Lie theory rather than to probability. Below we summarise briefly what we need; for background and detail, we refer to the standard works by Helgason [Hel1,2,3,4] and Wolf [Wol1,Ch. 11]. Readers without a geometric background may prefer to think of what are for us the prime examples: spheres, lines and half-lines, and their products. For the relevant harmonic analysis (extending the Fourier transform to the spherical transform), see §1.3, §1.4 and §1.8 below.
A symmetric space is a Riemannian manifold M with curvature tensor invariant under parallel translation. Equivalently, a geodesic symmetry exists: this fixes some base point o (the origin in R d , the North Pole in S d ), and reverses the direction of the geodesics through o. This gives an involutive isomorphism, mapping a point into its 'mirror image' on the geodesic through it and o. Then M is a homogeneous space G/K: the coset space of G, a closed subgroup of the isometry group of M (containing the transvections, [Wol2,11.1A]) and K, the isotropy subgroup of G fixing o.
The spherical transform and spherical dual
To proceed, we need a version of the Fourier transform on symmetric spaces. This is provided by the spherical transform, involving spherical measures and spherical functions. A spherical measure m is a K-bi-invariant multiplicative linear functional on C c (K\G/K) (continuous functions of compact support on the double-coset space). The spherical functions (originally, zonal spherical functions) are the continuous functions ω :
G → C such that the measure m ω (f ) := G f (g)ω(g −1 )dµ G (g) is spherical. The map f →f (ω) := m ω (f ) = G f (g)ω(g −1 )dµ G (g)
is the spherical transform. The positive definite spherical functions φ on (G, K) are in bijection with the irreducible unitary representations π of G with a K-fixed unit vector u by φ(g) = u, π(g)u .
These form the spherical dual, Λ. We confine ourselves to the rank-one case [Hel1,V.6]. These fall into three classes, the compact and non-compact (dual to each other) and the Euclidean (self-dual) [Hel1,V.2]. These are the two-point homogeneous spaces, the spaces of constant curvature κ: spheres (curvature κ > 0), Euclidean (κ = 0) and hyperbolic spaces (κ < 0) respectively [Hel1, IX.5, X.3], [Wol2].
In the compact rank-one case, the π and φ above, and Λ, are in bijection with the Cartan-Weyl dominant weights [Wol1,§6.3], and with a subset of R, which we may again write as Λ, by the Cartan-Helgason theorem [Hel3,V.1.1,[534][535][536][537][538], [Wol1,11.4B].
The Bochner-Godement theorem
That Bochner's theorem may be extended beyond R d and S d to symmetric spaces goes back to Godement in 1957, giving the Bochner-Godement theorem. In its modern formulation, this very useful and powerful result is as follows.
Theorem BG (Bochner-Godement theorem). The general positive definite function ψ on a symmetric space M is given (to within scale c) by a mixture of positive-definite spherical functions φ(λ) over the spherical dual Λ by a probability measure µ:
ψ(m) = c Λ φ λ (m) dµ(λ) (m ∈ M) : ψ = c Λ φ λ dµ(λ).(BG)
For background and details, see e.g. Wolf [Wol1,Th. 9.3.4], van Dijk [Dij,Prop. 8.3.3] (there as the 'Bochner-Godement-Schwartz theorem'). An early use in a probability context (Gaussian processes on compact symmetric spaces) is Askey and Bingham [AskB] in 1976.
Bochner's theorem is the special case G = R d , K = {0} (here the spherical functions are the characters, the complex exponentials e ix. = [t → e ixt ]). The Bochner-Schoenberg theorem is the case
S d = SO(d + 1)/SO(d) ↔ (SO(d + 1), SO(d)).
Here the spherical functions are the Gegenbauer polynomials above.
Products of symmetric spaces; spatio-temporal and geotemporal processes
By the Schur product theorem (1911; [Sch], [HorJ,§7.5]), products of positive definite functions are positive definite: the class P(M) of positive definite functions is closed under pointwise products (which correspond to tensor products of representations). For product spaces, if f i ∈ P(M i ), then f 1 f 2 ∈ P(M 1 × M 2 ). The spherical dual of the product is the product of the spherical duals:
M = M 1 × M 2 , (G, K) = (G 1 , K 1 ) × (G 2 , K 2 ), Λ = Λ 1 × Λ 2 .
The products of interest here are those in which the first factor is spatial (Euclidean or spherical) and the second is time. We refer to the first case as spatio-temporal and the second as geotemporal. Both are of great importance, in both theory and applications. We recall Whittle's advice, that one should think of a spatial process as the equilibnrium process of a spatio-temporal one [Whi3,Preface].
Gelfand pairs
For G a locally compact group (Lie group will suffice here) with K a compact subgroup, (G, K) is called a Gelfand pair if the convolution algebra of K-bi-invariant continuous measures on G with compact support is commutative. For symmetric spaces M = (G, K), (G, K) is a Gelfand pair ([Hel1, X, Th. 2.9, 4.1], [Hel3,IV,Th. 3,1]). We use the language of symmetric spaces rather than Gelfand pairs, but Gelfand pairs (which can be discrete or continuous) are very interesting; see van Dijk [Dij] for a monograph treatment. Their relevance here is shown by there being a form of the Bochner-Godement theorem for them ( [Dij,Prop. 8.3.3] as noted above).
Sphere cross line; the Berg-Porcu theorem
From 1.5 above, one can produce geotemporal covariances by taking an isotropic covariance on the sphere (from the Bochner-Schoenberg theorem), a stationary covariance on the line (by Bochner's theorem) and taking their product. This, though a valid covariance on sphere cross line, is separable (one might as well deal with the two separately). The interesting and useful covariances here are inseparable. The question was raised by Mijatović and the authors [BinMS, §4.4] in 2016 of finding the general (isotropic, stationary) covariance on sphere cross line, and answered by Berg and Porcu [BerP] in 2017.
We state the Berg-Porcu theorem below, and illustrate the power and usefulness of the Bochner-Godement theorem by using it to give a very short and simple proof (indeed, two proofs).
Theorem BP (Berg-Porcu Theorem). The class of isotropic stationary covariances on sphere cross line coincides (to within scale) with the class of mixtures of products of Gegenbauer polynomials P λ n (u) and characteristic functions φ n (t) on the line: with u, λ as before, they are given by
c ∞ 0 a n P λ n (u)φ n (t), c > 0, a n ≥ 0, ∞ 0 a n = 1. (BP )
Proof. In the notation above, with the first factor the sphere S d the spherical dual Λ 1 is the set of Gegenbauer polynomials P λ n ; with the second factor the line, Λ 2 is the set of characters, which can also be identified with the line:
t ↔ e it. = (x → e ixt ).
We can now proceed using the language of either measure theory or probability theory. 1. Measure theory. We use disintegration of measures in (BG) (Fubini's theorem extended beyond product measures: see e.g. [Kal1,Th. 6.4], Bogachev [Bog,§10.6]). Here, Λ 1 = N 0 := N ∪ {0}, Λ 2 = R. Integrate the probability measure µ on Λ over the second (time) variable for fixed n, then sum over n. This disintegrates µ into a sequence of probability measures µ n on the line, and a probability measure (a n ) on N 0 . Now integrating e itx over µ n (dx) gives its characteristic function φ n (t), the second factor in (BP ). The remaining integration is a sum over n of the product of this and the first factor P λ n (u) with weight a n , giving the result. 2. Probability theory. Take λ = (λ 1 , λ 2 ) in (BG) as a random variable with law µ. Condition on its first coordinate, and use the Conditional Mean Formula [Wil2,390] (a special case of the tower property [Wil1,9.7i] or chain rule [Kal1,105]).
Products of spheres.
There is a similar result due to Guella, Menegatto and Peron in 2016 [GueMP]. Its proof is immediate from the Bochner-Godement theorem. See [BinMS] for details, [BinS3,4] for background.
Transforms: commutative and non-commutative settings
The prototypical transform for us is the Fourier transform. There are two major extensions to commutative settings: Pontryagin duality on locally compact groups (1934), and the Gelfand transform on commutative Banach algebras in the 1940s [Rud,Ch. 11]. The spherical transform above derives from Gelfand in 1950 [Gel] (see its review by Godement [God]). The commutativity in a Gelfand pair captures what is needed to extend to a genuinely non-commutative setting.
Bessel potentials, SPDEs and Matérn processes
Riesz potentials
The two most lasting contributions of Marcel Riesz (1886Riesz ( -1969, younger brother of Frederick Riez) are Riesz means (typical means) in summability theory, and his work of 1937-38 on fractional potentials [Rie]. As potential theory is intimately linked to the Laplacian, it is natural that one can treat Riesz potentials in terms of fractional Laplacians (an important type of pseudodifferential operators, [Tay]). See the now classic treatment in Stein [Ste, V]. Now the Laplacian ∆ has non-positive eigenvalues (as D 2 e iω. = −ω 2 e iω. ), so it is preferable to deal with −∆, which has non-negative eigenvalues, before taking fractional powers. The operators
I α := (−∆) −α/2 are called Riesz potentials, and in d dimensions R d are well-behaved for α ∈ (0, d). Then for suitable functions f , (−∆) −α/2 (f ) = Γ( d 2 − α 2 ) π d/2 2 α Γ(α/2) R d |x − y| −d+α f (y)dy (0 < α < d) = R d k(x, y)f (y)dy, say, where k(., .) is called the kernel of the operator I −α/2 [Ste, V.1].
Bessel potentials
The restriction α < d is often inconvenient [Ste V.3]. We can avoid it by working instead with (I − ∆), which has positive eigenvalues. For suitable f , the Fourier transforms of −∆f and (I − ∆)f are 4π 2 |x| 2f (x) and (1 + 4π 2 |x| 2 )f (x). The corresponding fractional powers
J α := (I − ∆) −α/2
are called Bessel potentials (the calligraphic J α is to avoid confusion with J α , the Bessel function of the first kind). The kernel of J α is the function G α with Fourier transform
(1 + 4π 2 |x| 2 ) −α/2 .
The name Bessel potential derives from the fact that the kernel G α involves a Bessel function of the third kind with imaginary argument, more briefly a Macdonald function [Wat,3.7(6)], K ν say. With all three kinds of Bessel function, J ν , I ν , K ν , the standard notation for the order is ν [Wat].
We now have [Den], [AroS] G α (x) := K (d−α)/2 (|x|) |x| (α−d)/2 2 (d+α−2)/2 π d/2 Γ(α/2) .
The functions in the two last displays are essentially Fourier transforms of each other; see below.
The dimension d is fixed; the parameter α governs smoothness, as we shall see. It is convenient, following Simpson. Lindgren and Rue [SimLR1], to use
η := (α − d)/2(eta)
instead as smoothness parameter. We can then write the kernel G α as
G α (x) = |x| η K η (|x|)/c η , c η := 2 (d+α−2)/2 π d/2 Γ(α/2).
Note that η may be negative (not a problem, as K −η = K η ; see below).
Macdonald functions.
As Watson remarks, the importance of the Macdonald function is as a Bessel function with exponential decay at infinity [Wat,3.7 and 7.23]. The integral representation
K ν (x) = 1 2 ∞ 0 u ν exp {x(u + 1/u)} du/u
is often useful [Erd,II,7.12(23)], [Jør]. The behaviour of K ν near the origin is important, as we shall see. We note here that K 0 has a logarithmic singularity at the origin [Wat,(3.71) (14)]: [Wat,3.7(2),(6)]:
K 0 (x) ∼ log(1/x) (x → 0+). For ν = 0K ν (z) := 1 2 π I −ν (z) − I ν (z) sin πν , I ν (z) = ∞ 0 ( 1 2 z) ν+2m m!Γ(ν + m + 1)
.
(using the obvious 'L'Hospital convention' for ν = 0), so
K ν = K −ν .
For small z K ν (z) ∼ c ν z ν for constant c ν = 0.
Macdonald-Student pairs
Stein gives the following useful integral formula for G α [Ste,V.3(26)]:
G α (x) = 1 (4π) α/2 Γ(α/2) ∞ 0 e −π|x| 2 /u e −u/4π u (−d+α)/2 du/u (G α )
(as he points out, this is just a rephrasing of the definition of the Gamma function). Thus G α is positive, even, integrable, and decreasing on (0, ∞). So it is (to within a scale factor) a probability density, let us call it the Macdonald density. So the Macdonald and Student-t densities form a Fourier pair ('self-reciprocal pair' in Feller's terminology [Fel,503]): each is the other's characteristic function to within scale (as one is always integrable [Fel,XV.3 Th. 3]). For this Macdonald-Student Fourier pair (Macdonald-Student pair for brevity), see e.g. Guttorp and Gneiting [GutG]. They give a historical study of of the family below, studied in 1960 by the Swedish statistician Bertil Matérn (1921Matérn ( -2010 [Mat], and now named after him.
The Matérn family of correlation functions; Matérn processes
Guttorp and Gneiting [GutG,(1)] give the correlation function in the Matérn family between values of a spatial random function at locations s ∈ R d apart as
ρ ν (s) := 2 1−ν Γ(ν) (κ|s|) ν K ν (κ|s|) ∝ R d exp{is T x} (κ 2 + |x| 2 ) (2ν+d)/2 dx(Mat)
(κ > 0 is the scale parameter, ν > 0 is their notation for the smoothness parameter; the variance is scaled so that ρ ν (0) = 1).
Bochner's theorem of 1933 shows that characteristic functions of probability distributions are exactly the continuous positive definite functions taking the value 1 at the origin. Khintchine's theorem of 1934 shows that for stationary processes, the correlation function is the Fourier transform of a probability distribution, the spectral measure. More is true: by the Cramér representation of 1942, the process itself is the Fourier transform of a stochastic process with orthogonal increments (for details and references, see e.g. Doob [Doo, X, XI], Cramér and Leadbetter [CraL,§7.5]). Thus the Matérn family of correlations are those of stationary processes, the Matérn processes, with spectral densities of Student type. Note that here the correlation function does not depend on the dimension d, while the spectral density does.
Matérn processes have long been widely used, and have become 'the workhorses of spatial statistics'. Their theoretical study is thus justified, not only by its intrinsic interest, but also by the decades of practical experience of practitioners. This is well summarised by M. L. Stein, whose advice in his book on spatial data and kriging is 'use the Matérn model' [Stei,14]. He 'showed that the behaviour of the covariance function near the origin has fundamental implications on predictive distributions, particularly predictive uncertainty. The key feature of the Matérn is the inclusion of a smoothness parameter that directly controls correlation at small distances', and 'The smoothness parameter is aptly named as it implies levels of mean-square differentiability of the random process, with large ν yielding very smooth processes that are many times differentiable, and small ν yielding rough processes'. There is also a link with Hausdorff dimension [GenK,§3].
Bessel potentials and the Matérn family
The kernel G α of J α involves a Macdonald function K η . Crucial for us is the stochastic partial differential equation (SPDE)
(κ 2 − ∆) α/2 X(s) = σ W (s) (s ∈ R d ), (SP DE)
where W is white noise. We note here that white noise involves generalised functions (Schwartz distributions) rather than ordinary ones. For background on white noise analysis, we refer to the standard work by Hida et al. [HidKPS], and for background on generalized random fields to Gelfand and Vilenkin [GelV]. Following Simpson, Lindgren and Rue [SimLR1], we write the Matérn covariance as
c η (s i , s j ) = C η ( s i −s j ) := σ 2 Γ(η + d/2)(4π) d/2 κ 2η 2 η−1 (κ s i −s j ) η K η (κ s i −s j 2 )
(this differs slightly from the notation of [GutG]: κ is a scale parameter as there; σ appears as this is a covariance, not a correlation; we use η for the smoothness parameter here rather than the traditional ν to show the link with the dimension d and the Bessel parameter α in (eta) above). It is clear from the effect of the Bessel potential being to introduce a multiplier into the Fourier transform that a convolution is involved, and clear from the stochastic context that this convolution will involve a stochastic process. So it is to be expected that the deterministic factor is the kernel c η above and the stochastic factor is white noise.
Convolution (kernel) representations; Whittle's theorem
The link between Matérn processes and SPDEs goes back to Whittle [Whi1] [Whi2] in 1954 and1963. This was the starting-point of the 2011 study [LinRL], and its 2012 sequel [SimLR1]. So we shall call the result below Whittle's theorem; we follow [SimLR1] in the proof. The result is as in Simpson's contribution to the discussion of [LinRL,[65][66].
Theorem W (Whittle's theorem). The stationary solutions to (SP DE) above are the convolutions
X(s) = R d c η (s, t) dW (t) (s ∈ R d )
with c η the Matérn covariance above and W white noise. That is,
R d ds ψ(s)X(s) = R d ds ψ(s) R d c η (s, t) dW (t)
for all test functions ψ.
Proof. Following Walsh [Wal], we define a solution of (SP DE) to be any random field X(s) satisfying
X(s)(κ 2 − ∆) α/2 φ(s)ds = σ φ(s)dW (s) for every test function φ. Choose φ to solve (κ 2 − ∆) α/2 φ(s)ds = ψ(s) : then φ(s) = (κ 2 − ∆) −α/2 ψ(s)
is smooth [Ste]. Substituting in the above gives
X(s)ψ(s)ds = σ (κ 2 − ∆) −α/2 ψ(s)dW (s).
But as the kernel of the Bessel potential is the Matérn covariance c η ,
(κ 2 − ∆) −α/2 ψ(s) = c η (s, t)ψ(t)dt.
Substituting and using Fubini's theorem gives the result.
As above, Whittle's result [Whi1], [Whi2] was the starting-point for [LinRL]. The authors proceeded, as in [GutG], to use the above results on Fourier transforms to identify the finite-dimensional distributions of the two sides of (SP DE) by showing that they integrate the same way over each finite set of test functions. Note. 1. One important advantage of the approach above is that the parameter α > 0 is not subject to the restriction to half-integer values in [LinRL], [SimLR1]. Such situations are typical of this area, where the relevant analysis and/or special-function theory involves a continuous parameter, which has geometric significance as a dimension for integer (or half-integer) values. See e.g. [Bin1,2,3], [BinS1,2]. 2. The representation of the process X in Whittle's Theorem as a convolution of the function (kernel) C η and the process W is a prototype of a wider class, the convolution processes (process convolutions, kernel convolutions). These were introduced by Higdon [Hig1,2] in 1998, in a climate-modelling setting, and have been widely used since; see e.g. Rodrigues and Diggle [RodD]. 3. Kernel convolutions have the great advantage that one can generalise by replacing the Matérn kernel c η by any L 2 -kernel k:
X(s) = R d k(s, t) dW (t)
still gives a Gaussian random field. This gives one great flexibility when modelling. One can also replace white noise by any independently scattered random measure (measures of disjoint sets are independent: [Kal2]), and hence one can construct non-Gaussian random fields [AbeP]. 4. For more on smoothness of paths (regularity) and SPDEs, see [LanS], [BinS1], [BroKLO].
Matern processes on spheres, manifolds and graphs
Much of the theory above, motivated by the needs of spatial statistics, has involved Fourier analysis in Euclidean space, which is self-dual under the Fourier transform. While this is the right framework for a locality or region, it is not on the larger scale ('maps and globes'). We live on Planet Earth, which is (approximately) a sphere, which we may scale to be the unit sphere, S := S 2 as in §1.
As on any compact Riemannian manifold M, there exists a complete orthonormal basis {φ n } of L 2 (S) of eigenfunctions of −∆ = −∆ S , with eigenvalues λ j with 0 = λ 0 < λ 1 ≤ · · · ≤ λ j · · · ↑ +∞ One can expand a suitable function on M in an eigenexpansion, the Sturm-Liouville expansion (see e.g. Chavel [Cha1,VI.1], where this is done for the heat kernel):
−∆f = ∞ 0 λ n f, φ n φ n .
For suitable Φ, one can extend this to
Φ(−∆)f = ∞ 0 Φ(λ n ) f, φ n φ n ,
an example of the functional calculus (or symbolic calculus, of F. Riesz and N. Dunford); see e.g. [Rud,[290][291][292].
To exploit the geometry, one expands in a basis of interest; for the sphere here, one uses the spherical harmonics ( [AndAR Ch.9], [SteW,IV.2]) as in [BinS1 §3].
Using F for the Fourier transform, the last display in the proof above may be summarised as [LinRL,B.3.2] {F ((κ 2 − ∆) α/2 )φ}(k) = (κ 2 + k 2 ) α/2 (F φ)(k) (k ∈ R d ).
The analogous statement for the sphere is [LinRL,B.3.2] {F ((κ 2 − ∆) α/2 )φ}(k) = (κ 2 + λ 2 k ) α/2 (F φ)(k) (k = 0, 1, 2, · · ·).
For such fractional calculus on spheres, see [BinS1, §4 Remark 2] and the references there (to Askey and Wainger, Bavinck etc.) For Matérn Gaussian processes on general Riemannian manifolds, see Borovitskiy et al. [BorTMD]. For graphs, one proceeds similarly using the graph Laplacian (see e.g. [Chu]). For Matérn processes on graphs, see Borovitskiy et al. [BorATMDD,§3].
Extreme values
The area of spatial (or spatio-temporal, or geotemporal) extremes is vast and topical. For reasons of space, we can only refer here to [AsaDE], [CooNN], [HaaRo], [ShaW] and the first author's survey [BinO] with Ostaszewski.
Gaussian Markov random fields; sparse numerical linear algebra
Precision matrices and sparseness
If the space Ω is discretised by triangulation, the vertices and edges of the triangles form a graph, G = (V, E), V = {v 1 , · · · , v n } say. We may thus use the powerful graphical models of statistics; see e.g. Lauritzen [Lau].
The values x i of our Gaussian process X at the vertices v i are multivariate Gaussian, with covariance matrix Σ = (σ ij ), say, and precision (concentration, inverse covariance) matrix K := Σ −1 = (k ij ) ('K for Konzentration'). Two Gaussian variables x i and x j are independent if and only if their covariance σ ij = 0, i.e. they are uncorrelated (for the simple proof, see e.g. [Kal1,Lemma 13.1]).
That x i , x j are conditionally independent given all the others if and only if k ij = 0 is much more recent. Now taking x 1 as the random 2-vector and x 2 as the (p − 2)-vector we condition on, the Gaussian Regression Formula (GRF; [Lau,Prop. C5], [BinK,5.2]) shows us that the conditional density f 1|2 of x 1 |x 2 has (conditional) covariance matrix K −1 11 . So one has conditional independence if and only if K −1 11 is diagonal, that is (the matrices being 2×2), K 11 is diagonal, that is, k 12 = 0.
This important result derives from Dempster [Dem2,p.159], [Dem1], so as it needs a name let us call it Dempster's theorem. It owes its great importance in modern statistics to the role it plays in graphical models [Lau,Prop. 5.2]. Sparseness properties of concentration matrices are highly revealing about structure, as in [HasTW], as well as being numerically very convenient (we note in passing the great importance of numerical linear algebra, here and elsewhere; see e.g. [GolV]). Conditional independence statements of this sort are important in Markov properties on undirected graphs (Hammersley-Clifford theorem; [Lau,3.2.1]), multivariate normal models [Lau,5.1.3], covariance selection models ( [Lau,5.2], following [Dem2]), etc. They come into their own with Gaussian Markov random fields; see [RueH].
Building on [RueH], [LinRL] proceed by triangulating the space Ω. For this one can use Delaunay triangulation, giving the dual of the Voronoi diagram (see e.g. [PrepS,Ch. 5,6]). Here one maximises the minimum angle in a triangle, avoiding 'long thin triangles' which are numerically (and geometrically) awkward.
Although the flexibility given by the smoothness parameter ν in the Matérn model to reflect smoothness of the process being modelled is valuable, 'ν usually fixed, since it is poorly identified in typical applications. A more natural interpretation of the scale parameter κ is as a range parameter' [LinRL,§2.1]. From the nature of the data, one may be able to select a range ρ beyond which one can neglect dependence between two sites (data points x i , x j at vertices v i , v j ). An empirical rule used here is ρ = √ 8ν/κ. We have stressed the importance of the smoothness parameter ν (or η in 2.2, 2.5), yet as we have just seen it is difficult to estimate from data. This situation is not surprising, and is reminiscent of that in density estimation [Sil]. A discrete distribution can be approximated arbitrarily closely by an arbitrarily smooth one (in any reasonable metric), and vice versa. So the choice of what smoothness to assume is to be made by the statistician, with the specifics of the particular situation in mind, and the resulting flexibility is an important advantage.
Discretization: From Gaussian process to Gaussian Markov random field
Having chosen the range parameter ρ, one can now assume that the data points distant more than ρ apart are conditionally independent given the rest, that is, by Dempster's theorem, that the corresponding entries in the precision matrix K are zero. As this will be the case for most entries, the n×n matrix K is sparse. This in contrast to the covariance matrix Σ = K −1 , which will be dense (most of its entries non-zero). So matrix operations on K will have a computational cost of O(n 3/2 ), while those on Σ have cost O(n 3 ).
While the 'arms race' between increasing computing power and increasing size of data sets is ongoing, such a dramatic difference in computational efficiency is and will remain decisive. This explains the deliberately dramatic titles of the papers [SimLR1,2] -which in turn suggested our own.
In the above, only those edges e ij for which the corresponding κ ij = 0 need to be retained (thus the graph will itself be 'sparse', reflecting the interpoint distances). As above, the conditional independence corresponds to the Markov property. So the continuous Gaussian process has been approximated by a Gaussian (as all distributions are still multivariate normal) Markov random field (GMRF); for background on GMRFs, see Rue and Held [RueH].
Although Σ and its inverse K are both symmetric and positive definite, have a Cholesky decomposition etc., they are diametrically opposite computationally as only one is sparse. Numerical linear algebra for sparse matrices is a subject in its own right; see e.g. Davis et al. [DavRS] and its many references. We are at liberty to permute the order of the vertices v i , and this can lead to worthwhile improvements in computational efficiency, etc. Statistical learning with sparsity is also a field in its own right, in which the lasso (least absolute shrinkage and selection operator) plays a central role. For a monograph treatment, see [HasTW].
Discretization of Riemannian manifolds is widely practised; see e.g. Chavel [Cha2,§4.4]. There are connections with rough isometries.
There is much more to say here on the numerics side, particularly on the finite-element method (FEM) and integrated nested Laplace approximation (INLA) [RueMC], but for reasons of space we must refer to the sources cited above, particularly the RSS discussion papers [LinRL], [RueMC]. There is also much to be said about related areas in which sparsity is the key, such as wavelets [Dau] and compressed sensing [FouR]. We mention some of the many application areas. The first is digitization and reproduction of images: see e.g. [FouR Fig. 1] for a photograph of one of the authors' children, and a near-perfect reconstruction of it using only the 1% of the wavelet coefficients largest in magnitude. The second is automatic fingerprint (and more recently, iris) identification systems; see e.g. [MeyC,, [RajWMC,. Related applications include automatic number plate recognition for vehicles, bank card recognition for cashless transactions, etc.
Dedication and acknowledgement
It is a pleasure to dedicate this paper to Mikhailo Iosifovich Yadrenko on his 90th birthday, in acknowledgement of his long and productive scientific career and his many contributions to random fields.
We thank the editors for their kind invitation to contribute to this special volume, and the referees for their helpful comments.
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"The Herschel view of the dense core population in the Ophiuchus molecular cloud",
"The Herschel view of the dense core population in the Ophiuchus molecular cloud"
] |
[
"B Ladjelate [email protected] \nInstituto de Radioastronomía Milimétrica\nIRAM Avenida Divina Pastora 7, Local 2018012GranadaSpain\n\nLaboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance\n",
"Ph André \nLaboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance\n",
"V Könyves \nLaboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance\n\nJeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPreston, LancashireUK\n",
"D Ward-Thompson \nJeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPreston, LancashireUK\n",
"A Men'shchikov \nLaboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance\n",
"A Bracco \nLaboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université\nUniversité de Paris\nParisFrance\n",
"P Palmeirim \nInstituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nCAUP\nRua das EstrelasPT4150-762PortoPortugal\n",
"A Roy \nLaboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance\n",
"Y Shimajiri \nDepartment of Physics and Astronomy\nGraduate School of Science and Engineering\nKagoshima University\n1-21-35 Korimoto890-0065KagoshimaKagoshimaJapan\n\nNational Astronomical Observatory of Japan\nOsawa 2-21-1, Mitaka, Tokyo 181-8588Japan\n",
"J M Kirk \nJeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPreston, LancashireUK\n",
"D Arzoumanian \nInstituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nCAUP\nRua das EstrelasPT4150-762PortoPortugal\n",
"M Benedettini \nINAF -Istituto di Astrofisica e Planetologia Spaziali\nVia Fosso del Cavaliere 100, I00133RomaItaly\n",
"J Di Francesco \nHerzberg Astronomy & Astrophysics Research Centre\n10 I. Physik. Institut\nNational Research Council of Canada\n5071 West Saanich Road Victoria, BC V9E 2E7Canada\n\nUniversity of Cologne\n50937CologneGermany\n",
"E Fiorellino \nDipartimento di Fisica\nUniversità di Roma 'Tor Vergata' Via della Ricerca Scientifica 100133RomaItaly\n\nINAF-Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly\n\nEuropean Southern Observatory\nKarl-Schwarzschild-Strasse 285748Garching bei MünchenGermany\n",
"N Schneider \nLaboratoire d'Astrophysique de Bordeaux\nUMR\nCNRS/INSU\nUniversité de Bordeaux\n5804France\n",
"S Pezzuto \nNational Astronomical Observatory of Japan\nOsawa 2-21-1, Mitaka, Tokyo 181-8588Japan\n",
"Herschel Gould Belt ",
"Survey Team "
] |
[
"Instituto de Radioastronomía Milimétrica\nIRAM Avenida Divina Pastora 7, Local 2018012GranadaSpain",
"Laboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance",
"Laboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance",
"Laboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance",
"Jeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPreston, LancashireUK",
"Jeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPreston, LancashireUK",
"Laboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance",
"Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université\nUniversité de Paris\nParisFrance",
"Instituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nCAUP\nRua das EstrelasPT4150-762PortoPortugal",
"Laboratoire d'Astrophysique (AIM)\nCEA/DRF\nCNRS\nUniversité Paris-Saclay\nUniversité Paris Diderot\nSorbonne Paris Cité\n91191Gif-sur-YvetteFrance",
"Department of Physics and Astronomy\nGraduate School of Science and Engineering\nKagoshima University\n1-21-35 Korimoto890-0065KagoshimaKagoshimaJapan",
"National Astronomical Observatory of Japan\nOsawa 2-21-1, Mitaka, Tokyo 181-8588Japan",
"Jeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPreston, LancashireUK",
"Instituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nCAUP\nRua das EstrelasPT4150-762PortoPortugal",
"INAF -Istituto di Astrofisica e Planetologia Spaziali\nVia Fosso del Cavaliere 100, I00133RomaItaly",
"Herzberg Astronomy & Astrophysics Research Centre\n10 I. Physik. Institut\nNational Research Council of Canada\n5071 West Saanich Road Victoria, BC V9E 2E7Canada",
"University of Cologne\n50937CologneGermany",
"Dipartimento di Fisica\nUniversità di Roma 'Tor Vergata' Via della Ricerca Scientifica 100133RomaItaly",
"INAF-Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly",
"European Southern Observatory\nKarl-Schwarzschild-Strasse 285748Garching bei MünchenGermany",
"Laboratoire d'Astrophysique de Bordeaux\nUMR\nCNRS/INSU\nUniversité de Bordeaux\n5804France",
"National Astronomical Observatory of Japan\nOsawa 2-21-1, Mitaka, Tokyo 181-8588Japan"
] |
[] |
Context. Herschel observations of nearby clouds in the Gould Belt support a paradigm for low-mass star formation, starting with the generation of molecular filaments, followed by filament fragmentation, and the concentration of mass into self-gravitating prestellar cores. In the case of the Ophiuchus molecular complex, a rich star formation activity has been documented for many years inside the clumps of L1688, the main and densest cloud of the complex, and in the more quiescent twin cloud L1689 thanks to extensive surveys at infrared and other wavelengths. Aims. With the unique far-infrared and submillimeter continuum imaging capabilities of the Herschel Space observatory, the closeby (d = 139 pc) Ophiuchus cloud was extensively mapped at five wavelengths from 70 µm to 500 µm with the aim of providing a complete census of dense cores in this region, including unbound starless cores, bound prestellar cores, and protostellar cores. Methods. Taking full advantage of the high dynamic range and multi-wavelength nature of the Herschel data, we used the multi-scale decomposition algorithms getsources and getfilaments to identify an essentially complete sample of dense cores and filaments in the cloud and study their properties. Results. The densest clouds of the Ophiuchus complex, L1688 and L1689, which thus far are only indirectly described as filamentary regions owing to the spatial distribution of their young stellar objects (YSOs), are now confirmed to be dominated by filamentary structures. The tight correlation observed between prestellar cores and filamentary structures in L1688 and L1689 supports the view that solar-type star formation occurs primarily in dense filaments. While the sub clouds of the complex show some disparities, L1689 being apparently less efficient than L1688 at forming stars when considering their total mass budgets, both sub clouds share almost the same prestellar core formation efficiency in dense molecular gas. We also find evidence in the Herschel data for a remarkable concentric geometrical configuration in L1688 which is dominated by up to three arc-like compression fronts and presumably created by shockwave events emanating from the Sco OB2 association, including the neighboring massive (O9V) star σ Sco. Conclusions. Our Herschel study of the well-documented Ophiuchus region has allowed us to further analyze the influence of several early-type (OB) stars surrounding the complex, thus providing positive feedback and enhancing star formation activity in the dense central part of the region, L1688.
|
10.1051/0004-6361/201936442
|
[
"https://arxiv.org/pdf/2001.11036v1.pdf"
] | 210,966,291 |
2001.11036
|
d2aa7c05613a28a73e1a76efcd89cc32a7bf9099
|
The Herschel view of the dense core population in the Ophiuchus molecular cloud
January 31, 2020 January 31, 2020
B Ladjelate [email protected]
Instituto de Radioastronomía Milimétrica
IRAM Avenida Divina Pastora 7, Local 2018012GranadaSpain
Laboratoire d'Astrophysique (AIM)
CEA/DRF
CNRS
Université Paris-Saclay
Université Paris Diderot
Sorbonne Paris Cité
91191Gif-sur-YvetteFrance
Ph André
Laboratoire d'Astrophysique (AIM)
CEA/DRF
CNRS
Université Paris-Saclay
Université Paris Diderot
Sorbonne Paris Cité
91191Gif-sur-YvetteFrance
V Könyves
Laboratoire d'Astrophysique (AIM)
CEA/DRF
CNRS
Université Paris-Saclay
Université Paris Diderot
Sorbonne Paris Cité
91191Gif-sur-YvetteFrance
Jeremiah Horrocks Institute
University of Central Lancashire
PR1 2HEPreston, LancashireUK
D Ward-Thompson
Jeremiah Horrocks Institute
University of Central Lancashire
PR1 2HEPreston, LancashireUK
A Men'shchikov
Laboratoire d'Astrophysique (AIM)
CEA/DRF
CNRS
Université Paris-Saclay
Université Paris Diderot
Sorbonne Paris Cité
91191Gif-sur-YvetteFrance
A Bracco
Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université
Université de Paris
ParisFrance
P Palmeirim
Instituto de Astrofísica e Ciências do Espaço
Universidade do Porto
CAUP
Rua das EstrelasPT4150-762PortoPortugal
A Roy
Laboratoire d'Astrophysique (AIM)
CEA/DRF
CNRS
Université Paris-Saclay
Université Paris Diderot
Sorbonne Paris Cité
91191Gif-sur-YvetteFrance
Y Shimajiri
Department of Physics and Astronomy
Graduate School of Science and Engineering
Kagoshima University
1-21-35 Korimoto890-0065KagoshimaKagoshimaJapan
National Astronomical Observatory of Japan
Osawa 2-21-1, Mitaka, Tokyo 181-8588Japan
J M Kirk
Jeremiah Horrocks Institute
University of Central Lancashire
PR1 2HEPreston, LancashireUK
D Arzoumanian
Instituto de Astrofísica e Ciências do Espaço
Universidade do Porto
CAUP
Rua das EstrelasPT4150-762PortoPortugal
M Benedettini
INAF -Istituto di Astrofisica e Planetologia Spaziali
Via Fosso del Cavaliere 100, I00133RomaItaly
J Di Francesco
Herzberg Astronomy & Astrophysics Research Centre
10 I. Physik. Institut
National Research Council of Canada
5071 West Saanich Road Victoria, BC V9E 2E7Canada
University of Cologne
50937CologneGermany
E Fiorellino
Dipartimento di Fisica
Università di Roma 'Tor Vergata' Via della Ricerca Scientifica 100133RomaItaly
INAF-Osservatorio Astronomico di Roma
via di Frascati 3300078Monte Porzio CatoneItaly
European Southern Observatory
Karl-Schwarzschild-Strasse 285748Garching bei MünchenGermany
N Schneider
Laboratoire d'Astrophysique de Bordeaux
UMR
CNRS/INSU
Université de Bordeaux
5804France
S Pezzuto
National Astronomical Observatory of Japan
Osawa 2-21-1, Mitaka, Tokyo 181-8588Japan
Herschel Gould Belt
Survey Team
The Herschel view of the dense core population in the Ophiuchus molecular cloud
January 31, 2020 January 31, 2020Astronomy & Astrophysics manuscript no. oph_herschel_1stgen c ESO 2020stars: formation -ISM: clouds -ISM: structure -ISM: individual objects (Ophiuchus complex) -submillimeter
Context. Herschel observations of nearby clouds in the Gould Belt support a paradigm for low-mass star formation, starting with the generation of molecular filaments, followed by filament fragmentation, and the concentration of mass into self-gravitating prestellar cores. In the case of the Ophiuchus molecular complex, a rich star formation activity has been documented for many years inside the clumps of L1688, the main and densest cloud of the complex, and in the more quiescent twin cloud L1689 thanks to extensive surveys at infrared and other wavelengths. Aims. With the unique far-infrared and submillimeter continuum imaging capabilities of the Herschel Space observatory, the closeby (d = 139 pc) Ophiuchus cloud was extensively mapped at five wavelengths from 70 µm to 500 µm with the aim of providing a complete census of dense cores in this region, including unbound starless cores, bound prestellar cores, and protostellar cores. Methods. Taking full advantage of the high dynamic range and multi-wavelength nature of the Herschel data, we used the multi-scale decomposition algorithms getsources and getfilaments to identify an essentially complete sample of dense cores and filaments in the cloud and study their properties. Results. The densest clouds of the Ophiuchus complex, L1688 and L1689, which thus far are only indirectly described as filamentary regions owing to the spatial distribution of their young stellar objects (YSOs), are now confirmed to be dominated by filamentary structures. The tight correlation observed between prestellar cores and filamentary structures in L1688 and L1689 supports the view that solar-type star formation occurs primarily in dense filaments. While the sub clouds of the complex show some disparities, L1689 being apparently less efficient than L1688 at forming stars when considering their total mass budgets, both sub clouds share almost the same prestellar core formation efficiency in dense molecular gas. We also find evidence in the Herschel data for a remarkable concentric geometrical configuration in L1688 which is dominated by up to three arc-like compression fronts and presumably created by shockwave events emanating from the Sco OB2 association, including the neighboring massive (O9V) star σ Sco. Conclusions. Our Herschel study of the well-documented Ophiuchus region has allowed us to further analyze the influence of several early-type (OB) stars surrounding the complex, thus providing positive feedback and enhancing star formation activity in the dense central part of the region, L1688.
Introduction
Our general observational understanding of star formation in the Milky Way has greatly advanced in the past decade with the exploitation of extensive surveys at infrared and submillimeter wavelengths with the Spitzer and Herschel space observatories (e.g., Evans et al. 2009, André et al. 2010, Molinari et al. 2010, Dunham et al. 2015. In particular, Herschel has made way for Article number, page 1 of 24 arXiv:2001.11036v1 [astro-ph.GA] 29 Jan 2020 A&A proofs: manuscript no. oph_herschel_1stgen a new horizon in the observational characterization of the density structure of molecular clouds thanks to its unique imaging capabilities in the far-infrared and submillimeter regime, which probe the structure of cold cloud material down to low column densities and unveil some aspects of the star formation process at relatively high resolution inside the most deeply embedded areas of molecular clouds.
Prestellar cores represent the initial stages of low-mass star formation (see di Francesco et al. 2007, Ward-Thompson et al. 2007) and studying their connection with the properties and structure of the parent molecular clouds is crucial in understanding the processes leading to populations and clusters of young stars. The Herschel Gould Belt Survey (HGBS - André et al. 2010) has provided a significant step forward in our understanding of the link between cloud structure and the formation of prestellar cores. This survey of nearby molecular clouds, at distances between ∼ 130 pc and ∼ 500 pc from the Sun, led to important first-look results, which were then confirmed by more extensive studies in Aquila, Taurus, and Corona Australis for example (Könyves et al. 2015, Marsh et al. 2016, Bresnahan et al. 2018, Benedettini et al. 2018. Overall, HGBS studies confirm the ubiquity of ∼ 0.1 pc-wide filaments in nearby molecular clouds (Arzoumanian et al. 2011 and highlight the key role of filaments in the core and star formation process .
Here, we report the results of the HGBS census of dense cores and molecular filaments in the Ophiuchus cloud complex (L1688/L1689). The outline of the paper is as follows. Section 2 introduces the region and summarizes the results of previous observations on the Ophiuchus molecular cloud. In Sect. 3, we describe the Herschel observations as well as the data reduction performed to provide scientific images. Section 4 is dedicated to a complete description of the Herschel GBS map products, including dust temperature and column-density maps. This section also provides details on the procedure adopted to obtain a census of filaments and to extract and characterize the dense cores of the region. In Sect. 5, the properties of the Ophiuchus cloud complex and its objects are discussed, including the close relationship between prestellar cores and filaments. A star formation scenario relying on the external influence of various early-type (OB) stars in the surrounding area is also explored. Section 6 concludes the paper.
The Ophiuchus molecular cloud
The Ophiuchus molecular cloud complex is a nearby region of low-mass star formation in the Gould Belt, located at a distance of ∼139±6 pc from the Sun (Mamajek 2008). Using VLBA radio observations, Ortiz-León et al. (2017) derived a distance of 137.3 ± 1.2 pc for L1688 and 147.3 ± 3.4 pc for L1689. Ortiz-León et al. (2018) recently confirmed these distances with Gaia DR2 data, finding 138.4 ± 2.6 pc for L1688 and 144.2 ± 1.3 pc for L1689. These recent VLBA and Gaia results are in excellent agreement with our adopted distance of 139 pc. The stellar content of the Ophiuchus star-forming complex has been deeply studied for a long time at almost every wavelength of the spectrum, revealing a rich environment of young stellar objects (YSOs), protostellar sources, and prestellar cores at various evolutionary stages (e.g., Wilking et al. 1989;Leous et al. 1991;Casanova et al. 1995;Motte et al. 1998;Pattle et al. 2015, andWilking et al. 2008 for a review). The complex includes two concentrated clouds, L1688 and L1689, extended by large-scale streamers stretching toward the northeast direction (Loren et al. 1990, see also Fig. 1). These streamers, visible with as dark lanes in optical photographs and detected in CO observations, have been part of the cobweb description of the Ophiuchus molecular cloud complex (Loren 1989a,b). Despite being larger scale (e.g., more than 10 pc in length) than most filaments seen with Herschel in nearby clouds and being located outside the densest areas, L1688 and L1689, they provided the first direct evidence of the presence of filamentary structures in the complex.
While L1688 harbors dense star formation activity, L1689 is more quiescent ("the dog that didn't bark", Nutter et al. 2006). Despite its significant mass, L1689 does not have as much star formation activity as L1688 or other Gould Belt regions such as Aquila (e.g., André et al. 2010, Könyves et al. 2015. This peculiar distribution of star formation efficiency for independent, yet neighboring regions, sharing similar physical initial conditions, provides an interesting testbed to investigate the influence of potential triggers of star formation and the origin of stellar masses (cf. Motte et al. 1998).
The molecular gas of the Ophiuchus star-forming complex is also well documented and has been scrutinized by groundbased and space-borne instruments for many years. In the present study, we use the naming conventions of Lynds (1962) and Loren et al. (1990) to refer to the different parts and clumps within the complex (cf. Fig. 1). In particular, the clumpy structure of L1688 in the form of several dense gas clumps (Oph A to Oph F) was initially described by Loren et al. (1990) based on DCO + observations. Oph A is a very high extinction clump with several prominent prestellar and protostellar cores (André et al. 1993, Motte et al. 1998, while Oph B is a more quiescent region regarding dense core properties. Oph C, E, F have lower column densities, but many infrared YSOs have been detected in this area, while Oph D is an isolated quiescent clump with only a few objects (e.g., Wilking et al. 1989, Bontemps et al. 2001.
The whole Ophiuchus complex is under the heavy influence of, and feedback from, the Sco OB2 association (e.g., Loren & Wootten 1986;Motte et al. 1998) enhancing the star formation activity in the L1688 cloud (Nutter et al. 2006). While the young B3 star S1 may regulate the star formation activity locally in Oph A by driving a compact HII region and an associated photodissociation region, the B2V star HD 147889 influences the cloud to the west of Oph A (Grasdalen et al. 1973, André et al. 1988, Abergel et al. 1996. Furthermore, Nutter et al. (2006) emphasized the influence on the cloud of the nearby O9V star σ Scorpii (or σ Sco for short) of the Sco OB2 association.
Herschel observations and data reduction
The Herschel Space Observatory (Pilbratt et al. 2010) was the facility used in this study, performing broadband parallel-mode observations thanks to its two imaging cameras, PACS (Poglitsch et al. 2010) and SPIRE (Griffin et al. 2010). The Herschel Gould Belt survey (HGBS) 1 observations of L1688 and L1689 were taken on 25 sept. 2010, with the Herschel Science Archive Observations IDs (OBSIDs): 1342205093, 1342205094. The SPIRE 250 µm data taken in parallel mode are saturated in the close vicinity of IRAS16293-2422 and a correction patch was observed in SPIRE-only mode, as OBSID 1342239773. The SPIRE images used in the present study result from the combination of the parallel-mode data with the SPIRE-only patch around IRAS 16293-242 (Fig. 1). The scanning speed of the parallelmode observations was 60 .s −1 , with the same overall observing strategy as for every other HGBS region (André et al. 2010, 16 . The red rectangle shows the area encompassing the Ophiuchus main cloud covered by the Herschel data discussed in the paper. The effective resolution ranges from 18.2 in this area to 5 in the outer parts. (Bottom) Composite three-color image of the field outlined by the red rectangle in the top panel, combining Herschel/PACS 160 µm data as a blue layer, Herschel/SPIRE 250 µm data as a green layer, and Herschel/SPIRE 350 µm data as a red layer. The main clumps and sub-regions of the field are marked. The green dashed polygons highlight the regions defined as L1688 and L1689+L1709 in the rest of the paper. Fig. 3. Herschel dust temperature map of the Ophiuchus molecular cloud at a HPBW resolution of 36.3 The dust temperature was derived by SED fitting from 160 µm to 500 µm. The warmest regions are located around S1, east of Oph A, and HD 147889, while the warm shell to the northwest is driven by the star ρ Ophiuchi. These three objects are B-type stars, S1 and HD 147889 being both young members of the L1688 infrared star cluster. Könyves et al. 2015). For further reference and more details, the full processing pattern is described in Könyves et al. (2015).
PACS data reduction
The parallel-mode PACS data at 70 µm and 160 µm were reduced with HIPE (Ott 2011), version 10, provided by the Herschel Science Center. The path from the Level-0 data to the level-1 stage followed the standard steps of the Herschel parallel-mode pipeline, and flux calibration for PACS was applied using the last version of the correction factors (PACS CAL 45 0). A deglitching method was used in HIPE to remove cosmic-ray hits on affected pixels. High-level processing of the PACS data, including map making, was performed with Scanamorphos 20 (Roussel 2013) following Könyves et al. (2015) for scanning artifacts. While the calibration error is estimated to be ∼5% in our maps for point sources by default, we adopt a calibration error value of 10% for the 70 µm band, and 20% for the 160 µm band. The final products are maps reprojected on the same 3 -pixel grid at all wavelengths. The half-power beam width (HPBW) resolution of the maps is given in Table 1 at each wavelength.
SPIRE data reduction
The reduction of the SPIRE maps was performed with HIPE version 10, where the scans in both orthogonal directions were combined. From level-0 to level-1, the standard calibration tree SPIRE CAL 10 1 provided by HIPE was used. We used the destriping module in HIPE for map correction by baseline subtraction. The initial maps had pixel sizes of 6 , 10 , and 14 for the SPIRE 250 µm, 350 µm, and 500 µm bands, respectively. The maps were finally reprojected to a 3 -pixel grid for consistency with the PACS maps. The calibration error is estimated to be around 10% for all SPIRE bands.
Saturation correction around IRAS 16293-2422
With a flux of more than 50000 MJy/sr at 250 µm, IRAS 16293-2422, a Class 0 object in L1689 (e.g., Walker et al. 1986;Mundy et al. 1986), saturated the bolometers of SPIRE in the parallelmode observations at 250 µm. A small field around this source was thus re-observed with Herschel using a different observing configuration for the SPIRE camera (OBSID: 1342239773). An image combination between the whole parallel-mode map of Ophiuchus and this particular patch was performed by detecting the saturation, which are pixels fixed at a −99 value, and replacing the saturated data by the data from the correction patch.
Results and analysis
Dust temperature and column density maps
The multi-wavelength Herschel data were used to derive a column density map and a dust temperature map for the Ophiuchus cloud. All Herschel maps were first smoothed to a common 36.3 beam corresponding to the resolution of the SPIRE 500 µm data. Zero-level offsets, derived from Planck and IRAS data (cf. Bernard et al. 2010), were added to the Herschel maps (see Table 1 for the offset values). To obtain a column density map at a resolution of 36.3 , we then fit a modified blackbody function to the 160 µm, 250 µm, 350 µm, and 500 µm data points on a pixel by pixel basis:
I ν (x, y) = B ν [T d (x, y)] × κ ν × Σ(x, y) ,(1)
where I ν (x, y) is the surface brightness at frequency ν for pixel (x, y), T d (x, y) the dust temperature, B ν [T d (x, y)] is the corresponding Planck function, κ ν the dust opacity per unit mass of dust+gas, and Σ(x, y) the local gas+dust surface density of the cloud. We also used a multi-scale decomposition scheme to add small-scale information from the SPIRE 350 µm/250 µm and PACS 160 µm data to the 36.3 -resolution column-density map (see Appendix A of Palmeirim et al. 2013), obtaining a "highresolution" column-density map with an effective HPBW resolution of ∼18.2 as a final result (see Fig. 2). Each point of the local spectral energy distribution (SED) was weighted by the absolute calibration error (20% for PACS 160 µm and 10% for the SPIRE bands), and the same dust-opacity law as in other HGBS papers was assumed, namely κ λ = 0.1 × (λ/300µm) −β cm 2 /g as a function of wavelength in the submillimeter regime, with a dust emissivity index β fixed to 2 (cf. Hildebrand 1983). The same multi-scale combination scheme of Palmeirim et al. (2013) was also adapted to merge the Herschel column-density data with the Planck optical depth map converted to column density using the HGBS dust opacity assumptions, as presented in Fig. 1 (Bracco et al. in prep).
When smoothed to a common 5 resolution, both the standard and the high-resolution column density map derived from Herschel data are consistent with a column density map derived from Planck data assuming the above-mentioned dust opacity law. More precisely, the column density ratio Herschel/Planck has a median value of 1.11, a lower quartile of 1.07, and an upper quartile of 1.16, for the material above A V = 3.
A similar comparison between our Herschel column-density map(s) and the 2MASS/NICER near-IR extinction map of Lombardi (2009), both regridded and convolved to the same resolution, yields a median ratio (Herschel/Extinction) of 0.65 in the range 2 ≤ A V ≤ 10, assuming the standard conversion N H 2 (cm −2 ) = 0.94 × 10 21 A V (mag) derived by Bohlin et al. (1978) for the diffuse interstellar medium. Since in dense molecular clouds (typically for A V > ∼ 6), the conversion between A V and column density changes and becomes N H 2 (cm −2 ) = 0.69 × 10 21 A V (mag) (cf. Draine 2003 andEvans et al. 2009), the calibration of our column-density maps actually agrees with that of the 2MASS/NICER extinction map within ∼ 10%-30%. We conclude that the absolute scaling of the Herschel columndensity maps used in this paper is likely accurate to better than a factor of 1.5 (see also Roy et al. 2014).
Distribution of mass in the Ophiuchus cloud
The detailed distribution of mass in the map of Fig. 2 is closely related to the star formation process and other physical processes happening in the Ophiuchus molecular cloud, including the large-scale effect of the Sco OB2 association, and the shorterscale effects of the shells formed by the stars ρ Ophiuchi, HD 147889, and S1 (see Fig. 3 and Motte et al. 1998). Our columndensity maps allow us to directly trace the distribution of mass in the cloud, assuming it is the only significant cloud along the line of sight. This is a safe assumption given the position of the Ophiuchus molecular cloud above the Galactic plane (b = +16.5 • ). We therefore calculated the mass of the Ophiuchus cloud enclosed within a given contour as
M cl = δA pixel µ H 2 m H × pixels N H 2 ,(2)
where δA pixel = (s pixel × d) 2 is the physical area covered by one pixel, a square of s pixel = 3 on a side at d = 139 pc, µ H 2 = 2.8 is the mean molecular weight per hydrogen molecule, m H is the hydrogen atom mass, and the sum is taken over the pixels inside the contour. In the following, we will define as dense molecular gas, the gas and dust material that lies above a column-density of 7 × 10 21 cm −2 (i.e., A V > 7), the typical column-density level marking the transition between a regime of very low core and star formation efficiency and a regime of high core and star formation efficiency in nearby clouds (André et al. 2010;Lada et al. 2010;Könyves et al. 2015). We obtained dense gas masses of ∼415 M for L1688 (over the area shown in Fig 4) and ∼165 M for the complex formed by L1689 and L1709 (over the field shown in Fig 5), both above 7 × 10 21 cm −2 . If we consider all of the gas, regardless of column density, the total masses become ∼980 M for L1688 and ∼890 M in L1689+L1709. A first observation is that the dense gas masses differ in the two sub clouds by more than a factore of two, but the total gas masses are on par with each other. The material reservoir is therefore the same in both clouds, but they have different compression status.
Given the column density map, we can derive the probability density function (PDF) of column density in the cloud and display this function separately for each sub-cloud (Fig. 6). While Loren et al. (1990). The effective HPBW resolution is 18.2 . The colorscale has been adjusted to transition from red to blue in the range of A V from 6 to 10. the low column density part of the PDF is usually well fit by a lognormal distribution, either due to turbulence (Kainulainen et al. 2009, Schneider et al. 2013), or to incompleteness effects (Alves et al. 2017), we find that the situation in Ophiuchus is more complex due to the presence of various star-forming cloud environments, each contributing separately to the PDF. Table 2 provides the results of piecewise power-law fits to the PDF of each subcloud in Ophiuchus. The break points of each part of the PDF were defined by fitting two power-laws in the high column density tail above a fiducial level of A V = 7, which again defines the dense gas within which most prestellar cores (André et al. 2010;Könyves et al. 2015) and young stars (e.g., Lada et al. 2010;Heiderman et al. 2010) are observed in nearby clouds.
A simple relationship exists between the high mass slope of the PDF derived in a region, and the associated density profile (see, e.g., Federrath & Klessen 2013). For a spherical cloud with a density profile ρ ∝ r −α , that is, a column density profile Σ ∝ r −α+1 , the logarithmic slope s of the column density PDF, defined by dN dlogN H 2 ∝ N s H 2 can be expressed as:
α sph = − 2 s + 1,(3)
while for a cylindrical cloud the logarithmic slope is:
α cyl = − 1 s + 1.(4)
Given the morphology of the cloud, with multiple clumps gathered in a single, large molecular cloud, we know that the overall PDF is the sum of contributions from individual clumps. It remains difficult to characterize the column-density PDF of each clump separately given the relatively low number of independent beams per clump. The typical values reported in Table 2 for the power-law fits to the PDF are compatible with α sph = 2 and α cyl = 1.5, which is physically reasonable for a self-gravitating cloud and consistent with results found in other regions (e.g., Schneider et al. 2012Schneider et al. , 2013Benedettini et al. 2015).
Filamentary texture of the Ophiuchus molecular cloud
The Ophiuchus cloud reveals a wealth of extended filamentary structure at parsec scale when observed with the high sensitivity and dynamic range of Herschel (Fig. 4, Fig. 5). Using getfilaments (Men'shchikov 2013), we extracted and identified the main filamentary structures in the mapped area. Based on a multi-scale filtering approach, getfilaments can separate filaments from the background and measure their properties. Filtered images accumulating transverse angular scales up to an upper limit of interest can be constructed. In Fig. 7, we adopted an upper angular scale of 150 , corresponding to a linear scale of ∼ 0.1 pc, which is the typical inner width of Herschel filaments in nearby clouds (Arzoumanian et al. 2011. We independently extracted filaments in the Ophiuchus region using the DisPerSE algorithm (Sousbie 2011) with a persistence threshold of 0.2 × 10 21 cm −2 , a robustness threshold of 2 × 10 21 cm −2 , and an assembling angle of 65 • . DisPerSE allows the filaments to be reconnected and filtered by "persistence", while getfilaments enhances the contrast of all filamentary structures present in the cloud via a multi-scale decomposition. The observation of widespread filamentary structures in L1688 and L1689 is quite remarkable since these two centrally-condensed clouds are not known to be filamentary in the literature, in contrast to the elongated streamers, L1709 and L1712 (cf. Loren 1989a,b). 2.0 1.5
α sph α cyl (cm −2 ) dN/dlogN H 2 ∝ M(> N H 2 ) ∝ L1688 (single PL fit † † ) 7×10 21 ≤ N(H 2 ) N H 2 −2.
Notes. ( †) Power-law fits to both the differential and the cumulative PDF were performed for each column-density range given in Col. 2. ( † †) T he results of single power-law (PL) fits are given for the whole range of column densities above N H 2 ∼ 7×10 21 cm −2 . Broken PL fits are also provided for the L1688 and L1689 PDFs. The break points were left free and identified using the Multivariate Adaptive Regression Splines technique (Friedman 1991) via its python implementation, py-earth.
Core extraction with getsources
Multiwavelength detections and measurements with getsources
To identify compact emission sources corresponding to candidate dense cores within the structured background emission traced by Herschel at all wavelengths, we used the source extraction algorithm getsources 2 (Men'shchikov et al. 2012). This multi-wavelength, multi-scale algorithm has been adopted for all HGBS "first-generation" catalog studies (e.g., Könyves et al. 2015, Marsh et al. 2016, Bresnahan et al. 2018). The algorithm extracts sources from the data in two main steps, first a detection step and second a measurement step. These detection and measurement steps are fully described in Men'shchikov et al. (2012) and Könyves et al. (2015), and the processing has been standardized for easier use in every HGBS study.
For source extraction, we used the Herschel maps in the five bands at 70, 160, 250, 350, and 500 µm, as well as the high resolution column-density map (18.2 resolution) and a temperaturecorrected 160 µm map. The latter map was obtained by converting the original 160 µm map to column density assuming the dust temperature given by the color-temperature ratio between 160 µm and 250 µm. We performed two sets of source extractions, one based on detections between 160 µm and 500 µm to identify dense cores, and the other based on detections at 70 µm to identify YSOs including protostellar objects. Although the extraction process is fully automated and the getsources code itself performs a pre-selection of sources, several additional criteria remain crucial for selecting reliable core detections (see next subsection).
Automatic selection and derivation of physical properties
Following the getsources extractions, we applied additional selection criteria to ensure the reliability and physical robustness of the catalog of candidate dense cores returned by getsources (cf. Könyves et al. 2015):
-The global detection significance over all bands [see Eq (18) of Men'shchikov et al. 2012] must be greater than 10; -The column density detection significance must be greater than 5 in the high-resolution column density map; -The global goodness parameter must be greater than 1; -The measurement signal-to-noise ratio must be greater than 1 in the high-resolution column-density map; -The monochromatic detection significance must be greater than 5 in at least two bands between 160 µm and 500 µm, and the measurement signal-to-noise ratio must be greater than 1 in at least one of these bands. We refer the reader to Men'shchikov et al. (2012) for the exact definition of "significance" and "goodness" for getsources. Table 2 for the results of power-law fits to the high-column density tails of the PDF.) The PDF of L1688 shows an overdensity around N H 2 ∼ 2 × 10 22 cm −2 , probably as a consequence of the Sco OB2 influence. (Right) Normalized cumulative mass fraction as function of column density for the Ophiuchus molecular cloud using the same columndensity map as in the top figure. The dashed line represents the fiducial dividing line between dense star-forming gas and lower-density gas. In the various parts of the cloud, the vast majority of the gas is found at low column density (A V ≤ 2).
resulting, final catalog were then derived by SED fitting (see Sect. 4.5 below). After a first pass to fit a simple modified blackbody model to the photometric data points provided by the getsources extractions, we selected a total of 513 candidate dense cores, including 459 starless cores and 54 protostellar cores or more evolved objects (e.g., Class II YSOs). Protostellar cores and YSOs were identified based on the detection of point-like emission at 70 µm (cf. Dunham et al. 2008). In the rest of this paper, we are not discussing evolved YSOs since they are not closely associated with dense cores.
Dense core properties
Exploiting the wavelength coverage of Herschel and multiwavelength capability of getsources, we constructed the SED of each source from the integrated flux densities measured at 160-500 µm. By fitting these SEDs with a modified blackbody function, in a manner similar to the method employed for deriving the column-density map (cf. Sect. 4.1 and Eq. 1), we obtained estimates of the average line-of-sight temperature and mass of each candidate core. For consistency with other HGBS studies, we derived a deconvolved radius for each source, calculated as the square root of the quadratic difference between the geometric mean of the major/minor angular sizes and the HPBW size of the high-resolution column density map (18.2 ). For each core, we also computed the geometric mean of the major and minor FWHM sizes, the peak column density, the beam-averaged column density, the central beam volume density, and the average volume density (cf. Fig. 8 for their distributions), assuming a Gaussian density distribution within each object (cf. Könyves et al. 2015). All of these quantities are provided in online Table A.2 (see Appendix A for a sample portion). From the derived properties, we selected self-gravitating prestellar cores among the group of starless cores based on a criterion related to the critical Bonnor-Ebert (BE) mass. The latter can be expressed as:
M BE,crit ≈ 2.4 × R BE c 2 s G ,(5)
where R BE is the BE radius, c s ≈ 0.19 km/s the isothermal sound speed (assuming a temperature of ∼ 10 K for the dense gas in the ambient cloud), and G the gravitational constant (Bonnor 1956). We estimated the outer radius, R obs , of each core from the FWHM source diameter measured in the high-resolution column-density map, deconvolved from the effective HPBW resolution of the map (18.2 ). We then compared the mass derived from Herschel observations, M obs , to the critical BE mass estimated as M BE,crit = 2.4 × R obs c 2 s /G, and selected self-gravitating cores based on the evaluated BE mass ratio:
α BE ≡ M BE,crit M obs ≤ 2.(6)
This provided a first set of prestellar cores, hereafter called robust prestellar cores, with a total of 93 objects after visual post-selection checks (see filled blue triangles in the top panel of Fig. 9). We also used an alternative, less stringent criterion to select candidate prestellar cores among starless cores, corresponding to the empirical lower envelope of self-gravitating objects in the mass-size diagram of Fig. 9 according to Monte-Carlo simulations similar to those described in Sect. 4.6 below and Appendix B (see Könyves et al. 2015):
α BE ≤ 5 × (HPBW N H 2 /FWHM N H 2 ) 0.4 ,(7)
where HPBW N H 2 = 18.2 is the HPBW resolution of the column density map and FWHM N H 2 is the FWHM size of the source measured in the column density map. This second criterion allows us to consider as candidate prestellar cores unresolved objects whose physical radius is more uncertain than that of resolved cores and which do not satisfy Eq. (6). Using this alternative criterion, 51 additional candidate prestellar cores were selected (see open green triangles in the top panel of Fig. 9) To summarize, we identified a total of 144 candidate prestellar cores satisfying Eq. (7), including 93 robust prestellar cores matching the strict Bonnor-Ebert criterion of Eq. (6). The observed and derived properties of all dense cores are given in two accompanying online catalogs (cf. Tables A.1 and A.2 in Appendix A, respectively). The bottom panel of Fig. 9 shows that there is an interesting correlation between the masses of the cores and the local pressure in the ambient cloud, P cl . Assuming the ambient cloud is in approximate virial equilibrium, the latter can be estimated as
P cl ≈ 0.88 G Σ 2 cl (McKee & Tan 2003), where Σ cl ≡ µ H 2 m H N cl H 2
is the local background cloud column density of each core as estimated from Herschel data with getsources. As can be seen in Fig. 9, there is a clear trend for most self-gravitating prestellar cores to lie in high-pressure, high-column-density parts of the cloud and for low-mass unbound starless cores to lie in lowpressure areas. We will come back to this trend in Sect. 5 below in connection with the spatial distribution of the various subsets of dense cores.
Completeness of the prestellar core survey
Monte-Carlo simulations were performed to estimate the completeness of our census of prestellar cores in the Ophiuchus region (see Appendix B). Clean background emission maps of the L1688 region were first constructed at all Herschel wavelengths (including a column density plane), by subtracting the emission of all compact sources identified with getsources from the observed maps. A population of 285 model Bonnor-Ebert-like cores were then inserted throughout the clean-background images to generate a full set of synthetic Herschel and column density images for L1688. The model cores were given a flat mass distribution such as ∆N/∆logM ∝ M 0 from 0.05 M to 2.0 M . An M ∝ R mass versus size relation appropriate for isothermal Bonnor-Ebert spheres was adopted. The dust continuum emission from the synthetic Bonnor-Ebert cores in all Herschel bands was simulated using an extensive grid of spherical dust radiative transfer models constructed by us with the MODUST code (e.g., Bouwman et al. 2000Bouwman et al. , 2001.
Based on the results of these simulations, our Herschel census of candidate prestellar cores in Ophiuchus is estimated to be > 80% complete above a true core mass of ∼0.1 M (cf. Fig. B.1), corresponding to an observed core mass of ∼0.2 M on average given that observed masses appear to be typically underestimated by a factor of ∼ 2 near the completeness limitdue to the internal temperature gradient within starless cores (see top and bottom panels of The simple model of the core extraction process described in Appendix B of Könyves et al. (2015) was also used to independently assess the completeness level and its dependence on the local column density of the ambient cloud. This model shows that the completeness of prestellar core extractions does decrease as background cloud column density and cirrus noise increase but suggests that the global completeness curve of the prestellar core survey in Ophiuchus is consistent with that inferred from the Monte-Carlo simulations (Fig. B.1).
A&A proofs: manuscript no. oph_herschel_1stgen
Discussion
A bimodal distribution of filament orientations
The shape of the Ophiuchus molecular cloud seems to be organized as a hub of filaments converging toward the center of mass of the main cloud L1688 as marked by a red star in Fig. 7. One of the interesting features of this region is the U-shape of the densest parts of the cloud (cf. Fig. 12). This shape is inverted toward the likely influence of the O9V star σ Sco (see Nutter et al. 2006) and is related to an interesting feature in the distribution of position angles for the Ophiuchus filaments. Figure 10 shows the distribution of median position angles for the sample of filaments identified with getfilaments. It can be seen that this distribution is bimodal, with a group of filaments roughly parallel to the large-scale streamers (cf. Sect. 2 and Fig. 1), and a group of filaments roughly perpendicular to the streamers, the latter group being associated with the L1688 main cloud. We compared the properties of starless cores embedded in parallel filamentswith those of cores embedded in perpendicular filaments, but did not find any clear evidence of differences between the two groups of cores.
The two modes of filament orientations observed here are roughly centered around position angles P.A. ∼ 50 deg and P.A. ∼ 130 deg (measured east of north), respectively, and may possibly be correlated with changes in the orientation of the magnetic field within the Ophiuchus cloud complex. On large scales, Planck polarization data (Planck Collaboration et al. 2016) show that the magnetic field orientation in the complex is mostly parallel to the streamers (P.A. ∼ 45 deg) and to the direction of the large-scale flows from the Sco OB2 association. Close to L1688, however, the Planck data suggest that the magnetic field becomes almost perpendicular to the streamers, possibly as a result of large-scale compression by the Sco OB2 flows. While the angular resolution of Planck polarization data is insufficient to constrain the orientation of the field on small scales within the densest regions, optical and near-IR polarization results do suggest that the magnetic field has a bimodal distribution of position angles within L1688, centered at P.A. ∼ 50 deg and P.A. ∼ 150 deg (Vrba et al. 1976;Sato et al. 1988;Goodman & Heiles 1994). This is strongly reminiscent of the two modes of filament orientations in Fig. 10, pointing to a common origin.
Filaments and cores
The ubiquity of filaments in star-forming regions and the intimate link between prestellar cores and dense filaments is a crucial result of the Herschel HGBS survey (André et al. 2010). As prestellar cores are believed to be the direct precursors of protostars, this result suggests that dense molecular filaments may play a crucial role in the origin of protostars and stars themselves.
The high sensitivity and dynamic range of the Herschel images allow us to investigate this link between cores and filaments in the Ophiuchus complex, where there was no strong evidence of filamentary structures up to now, except for the large-scale streamers.
As observed with Herschel in other Gould Belt regions (André et al. 2010, and indirectly hinted at in previous studies of Ophiuchus (Motte et al. 1998), the prestellar cores of L1688 and L1689 tend to lie in elongated cloud structures or filaments. . Two modes of filament orientations are clearly visible. The "parallel" filaments (with P.A. from 20 to 80 degrees east of north) are following the general direction of the large-scale streamers (cf. Fig. 1) and may be associated with material swept-up by the winds of the Sco OB2 association (whose general direction is about 45 degrees east of north), while the "perpendicular" filaments (with P.A. from 100 to 160 degrees) may have formed as a result of large-scale compression from the Sco OB2 winds.
To quantify this trend, we carried out an analysis of the separations between dense cores and filaments, considering both unbound starless cores and prestellar cores. Figure 11 shows that prestellar cores indeed lie within the ∼ 0.1 pc inner portions of the densest filaments extracted with DisPerSE. The population of unbound starless cores lies inside dense filaments, but also outside of them, displaying a higher median separation to the nearest filaments. This may have distinct explanations. First, unbound starless cores may in fact be split into two populations, with a subset of cores that will become prestellar cores by accumulating more mass on one hand, and transient or "failed" cores (cf. Ballesteros-Paredes et al. 2007) that will eventually disperse on the other hand. Second, our filament sample may be partly in- complete, especially in the low-density parts of the cloud where a significant fraction of unbound starless cores are found, and this may lead to a significant bias in the distribution of coreto-filament separations for unbound starless cores. Nevertheless, the result that prestellar cores tend to lie closer to the crests of their nearest filaments than unbound starless cores provides an interesting clue suggesting that the star-forming activity of the cloud has not stopped, and that the lower-density material may possibly continue to form star-forming structures in the future, evolving toward another generation of prestellar cores and protostars by cloud contraction (cf. Huff & Stahler 2007). Figures 12 and 13 show the spatial distribution of the candidate dense cores identified as explained in Sect. 4.4, in relation with the distribution of cloud material as traced by our Herschel column density maps for the various subregions of the Ophiuchus complex. It can be seen that prestellar cores (yellow symbols) tend to be mostly clustered inside the densest regions, while unbound starless cores (blue symbols) are more widely spread out.
Core clustering and distribution of core separations
Following the detection of DCO + gas clumps by Loren et al. (1990), the L1688 cloud has been described as a collection of cold dense molecular clumps (called "cores" at the time), labeled with letters A to F. Based on our results, the most active regions for prestellar core growth appear to be Oph A and Oph B, while Oph C, E, F harbor fewer prestellar cores but contain a higher number of Class I protostars (Wilking et al. 1989, André & Montmerle 1994. No significant difference in dense core properties is observed between L1688 and L1689. The sub clouds of L1688 (A, B, C, D, E, F, G) contain 40, 22, 8, 5, 5, 11, 7 prestellar cores, respectively, and 79, 23, 11, 7, 3, 17, 11 unbound starless cores. Two prestellar cores and 60 unbound starless cores could not be assigned to any specific sub-region. Indeed, this historical breakdown of L1688 in sub-regions has been based on studies focusing on the densest gas in the cloud, and is therefore not well suited for classifying objects lying in lower-density areas. L1689 contains 35 prestellar cores and 79 unbound starless cores. L1709 contains 9 prestellar cores and 24 unbound starless cores. Independently of background cloud structures and filaments, prestellar and protostellar cores tend to form in clusters, and therefore retain pristine properties of the parent molecular clouds in their distribution of separations. We can try to test this idea based on a simple statistical analysis of the distribution of core spacings. In Fig. 14, we compare the distributions of nearestneighbor core separations for the populations of prestellar cores and starless cores, in both L1688 and L1689. Even if the statistics are low, the different figures and modes of clustering show very little difference from one another, indicating that there may be only one underlying statistics in this sample, and probably a main physical explanation for the observed distribution of cores separations. Linked to molecular cloud structure, we observe a peak separation of ∼ 0.05 pc in both panels of Fig. 14 Table A.1 are represented by yellow ellipses overlaid on the Herschel high-resolution column density map derived from Herschel data (grayscale background image). Different colors are used for the separate dense clumps of L1688 (L1689 in red, L1709 in green). (Right) Compared spatial distribution of prestellar cores (yellow squares) and unbound cores (blue circles) overlaid on the high resolution column-density map.
median nearest neighbor separation of ∼ 0.04 pc for prestellar cores. This is slightly smaller than, but comparable to, the characteristic inner width of Herschel filaments found by Arzoumanian et al. 2011, 2019 3 . More specifically, Arzoumanian et al.
recently measured a median inner width of 0.06±0.02 pc (FWHM) for 57 filaments in the same Herschel field of Ophiuchus. It is also consistent with the typical Jeans length c 2 s /(GΣ cl ) ∼ 0.04 pc in cold dense molecular gas at a column density N cl H 2 ≡ Σ cl /(µ H 2 m H ) ∼ 10 22 cm −2 . This result is physically not trivial, because Inutsuka & Miyama (1997) found, through both analytical calculations and numerical simulations of cylindrical cloud fragmentation, that the most unstable wave-length initiated in a gas cylinder would enhance the production of core structures with a characteristic separation of ∼ 4 times the filament diameter, or about 0.4 pc for filaments of ∼ 0.1 pc width. While there is no clear explanation yet for this discrepancy in a seemingly filament-dominated mode of core formation, external factors such as large-scale compression or other physical effects like magnetic fields (Nakamura et al. 1993) or turbulence (Clarke et al. 2017) may possibly account for this result.
Using the core positions derived from Herschel data in each sub-cluster, we can also compare the areas occupied by the population of prestellar cores and the population of unbound starless cores. To this purpose, we derived the convex hull areas corresponding to prestellar cores and unbound starless cores for each sub-cloud, and this led to a characteristic radius in each case given in Table 3. For most sub clouds, the characteristic area A&A proofs: manuscript no. oph_herschel_1stgen occupied by prestellar cores is significantly smaller than the area occupied by unbound starless cores. This result is consistent with the overall correlation found in Sect. 4.5 between core mass and ambient cloud pressure (see bottom panel of Fig. 9) and shows us that prestellar cores are intimately related to the densest parts of the molecular cloud, while unbound starless cores are also found on the outskirts of the densest regions. This raises the issue of the physical nature of unbound starless cores. On one hand, many of these objects lie in portions of the cloud where the quantity of dense gas material may not be large enough for the cores to reach the critical Bonnor-Ebert mass even if they continue accumulating mass (cf. Fig. 12 and Fig. 13). Such unbound starless cores may be "failed" cores in the sense of Ballesteros-Paredes et al. (2007). On the other hand, the whole L1688 cloud may be globally contracting toward its center of mass (see Huff & Stahler 2007 and Sect. 5.6 below). If this is indeed the case, both the converging network of filaments and the starless cores of L1688 may become significantly denser on a typical timescale ranging from ∼ 3 × t cloud ff ∼ 1.5 Myr to the sound crossing time R cloud /c s ∼ 3-4 Myr, and a significant fraction of the unbound starless cores identified here with Herschel may evolve into selfgravitating prestellar cores (and later on protostars) on a similar timescale.
Prestellar core formation efficiency in the Ophiuchus molecular cloud
Section 4 described how we derived accurate mass estimates for both the cloud and the dense cores. Both are important for characterizing the future star formation potential of the cloud. Indeed, the ratio of the total mass in the form of cores to the total cloud mass is directly related to the efficiency of core formation process. Furthermore, the fraction of cloud mass that is not involved in core formation at the present time can either form other cores in the future or remain as a quiescent background, not participating in the star formation process. As mentioned in Sect. 4.2 (see also Table 4), the fractions of dense molecular gas in L1688 and L1689 are different, L1688 being denser, confirming the idea that L1688 may be a shockcompressed region (e.g Loren & Wootten 1986), where the accumulation of gas is enhanced by diverse external factors, while L1689 has significantly less dense gas. But the total gas masses in the two clouds are similar, and L1689 should be able to form, by large-scale contraction and fragmentation, enough clumps and cores to match the star formation activity of L1688, if it is characterized by the same core-formation efficiency. We find a dense molecular gas fraction, above N H 2 ∼ 7 × 10 21 cm −2 (or A V > ∼ 7), as high as 42% in L1688 alone, compared to a dense gas fraction of only 18.5% in L1689 and L1709. In the whole field (Fig 2) we find a dense gas fraction of 15%, similar to the values obtained in the Herschel studies of Aquila (24%, Könyves et al. 2015) and Orion B (13%, Könyves et al. 2019), but significantly higher than the 1% value found in the Musca and Pipe nebula clouds (cf. Table 1 of Arzoumanian et al. 2019). However, we observe very similar prestellar core formation efficiencies in the dense gas of L1688 and L1689+L1709, with 12% and 14% of dense gas mass in the form of prestellar cores, respectively.
More precisely, following Könyves et al. (2015), we can define the differential prestellar core formation efficiency as a function of "background" cloud column density as follows: viding the mass ∆M cores (A V ) of the candidate prestellar cores identified with Herschel in a given bin of background column densities (expressed in A V units) by the total cloud mass ∆M cloud (A V ) in the same bin. Figure 15 shows a comparison of CFE obs vs. A V for L1688, L1689, and Ophiuchus as a whole. It can be seen that all three CFE obs (A V ) plots have similar shapes and exhibit similar values for A V > ∼ 7. Moreover, the differential CFE obs (A V ) plots level off above A V of 7-8 with typical values of ∼ 20%. In contrast, if we consider the total mass of molecular gas in each cloud, including low-density material, then L1688 appears significantly more efficient at forming prestellar cores, since the fraction of prestellar core mass to total gas mass is 5.1% there compared to only 2.7% in L1689+L1709 (i.e., almost a factor of ∼ 2 lower than in L1688). This is most likely only due to the fact that L1688 contains more dense molecular gas (A V > 7) than L1689+L1709, however. These results are summarized in Table 4.
A plot of the number of prestellar cores as a function of background column density (Fig. 16) shows a main peak around A V ∼10, as well as a secondary peak at higher extinction A V ∼ 30 in L1688 and A V ∼ 40 in L1689. The secondary peak is related to the dense clumps Oph A and Oph B in L1688, and to the immediate vicinity of IRAS16293 in L1689.
Interestingly, while the physical meaning of this secondary peak remains difficult to assess, it appears to be related to features seen in the column density PDFs of the clouds (Fig. 6) and in the differential core formation efficiency plot against back- for the candidate prestellar cores L1688 prestellar cores L1689 prestellar cores Fig. 16. Distribution of background column densities for the L1688 (orange) and L1689 (green) prestellar cores. A clear peak in column density is visible, suggestive of a column-density threshold for prestellar core formation. However, this peak lies at a significantly higher column density in L1688 (orange) and in L1689 (green) than in other Gould Belt regions such as Aquila (cf. Könyves et al. 2015). The vertical dashed line marks the same fiducial threshold at A bg V ∼ 7 as in Fig. 15.
ground column density (Fig. 15). The main peak is related to a break in the column density PDF and a flattening in the differential CFE obs (A V ) plot. The secondary peak may result from a relative depletion of intermediate column density gas in L1688 because the corresponding material may be trapped between the shells around S1 and HD147889 and compressed to higher column densities. Figure 15 and the main peaks in Fig. 16 are consistent with the presence of a "threshold" for the formation of prestellar cores at a fiducial A V value of ∼ 7-8, as seen in other HGBS regions (e.g., Könyves et al. 2015;Marsh et al. 2016). This column density threshold should not be viewed as a strict step function but rather as a sharp transition between a regime of very low prestellar core formation efficiency (< 1%) at A V << 7 and a regime of roughly constant (or slowly varying), moderately high core formation efficiency (≈ 20%) at A V >> 7 (cf. Fig. 15). The details of the transition differ slightly in Ophiuchus compared to Aquila (Könyves et al. 2015) and Taurus (Marsh et al. 2016), due to the presence of secondary peaks at higher column-densities in L1688 and L1689 which have not been seen in other regions. This slight difference may be due to the fact that the Ophiuchus cloud is the nearest region of the Gould Belt influenced by strong external compression, where prestellar cores can be detected (and resolved) in higher column-density areas than in previously studied HGBS regions. The overall transition seen in Fig. 15 for L1688 and L1689 is nevertheless very similar to that observed in Aquila and Taurus. A physical interpretation of this transition in prestellar core formation efficiency around A V ≈ 7 (or a cloud surface density Σ cl ≈ 150 M /pc 2 ) was proposed by André et al. (2010) in terms of the filamentary structure of the parent clouds. Indeed, given the typical inner width ∼ 0.1 pc of molecular filaments (Arzoumanian et al. 2011, the transition at Σ cl ≈ 150 M /pc 2 closely corresponds to the critical mass per unit length M line,crit = 2 c 2 s /G ∼ 16 M /pc (see also André et al. 2014), around which nearly isothermal gas filaments at T ∼ 10 K are known to fragment efficiently (Inutsuka & Miyama 1997).
Starless and prestellar core mass functions
The mass distribution of our sample of 144 candidate prestellar cores provides an estimate of the prestellar core mass function (CMF) in the Ophiuchus cloud (Fig. 17).
Core Mass Function of candidate dense cores
Starless cores Prestellar cores Robust prestellar cores Fig. 17. Mass distribution of robust prestellar cores (blue), candidate prestellar cores (green), and unbound starless cores (red). The cyan area marks the difference between robust and candidate prestellar cores. To left of the vertical dashed line, the sample of prestellar cores is less than 80% complete according to Monte-Carlo simulations (see Sect. 4.6 and Appendix B). For comparison, the black dashed curve shows the system IMF (Chabrier 2005).
The prestellar CMF in Ophiuchus shows an apparent peak at about 0.3 M , which is very similar to the CMFs derived from HGBS data in Taurus/L1495 (Marsh et al. 2016) and is also consistent within uncertainties with previous results in Aquila (Könyves et al. 2015) where the prestellar CMF seemed to peak around 0.4-0.6 M . It should be stressed that the prestellar core masses derived here are likely underestimated by a factor of ∼ 2 on average, due to the extraction process and the effect of temperature gradients within the cores.This has been evaluated based on the Monte-Carlo simulations described in Appendix B where synthetic prestellar cores were injected injecting in the Herschel maps.
The prestellar CMF also has a direct relationship with the IMF, emphasizing the importance of core formation studies in molecular clouds (cf. Motte et al. 1998). The close resemblance of the prestellar CMF to the stellar system IMF (Chabrier 2005) is clearly confirmed in Ophiuchus (cf. Fig. 17), like in Aquila (Könyves et al. 2015) and Taurus (Marsh et al. 2016). However, while the total number of prestellar cores identified here (144) is significantly higher than in previous studies of Ophiuchus (e.g Motte et al. 1998;Johnstone et al. 2000;Stanke et al. 2006;Pattle et al. 2015), the core statistics in this cloud are still not high enough to allow robust, final conclusions to be drawn about the prestellar CMF.
Long-distance external effects and positive feedback
There appears to be an imprint of at least three compression fronts in the Herschel column density map of L1688 (see Fig. 18), which can be confirmed by performing a circular average of the cloud emission about a central point located close to Oph D. These compression fronts are detected in the column density profile of the cloud, with three consecutive local peaks in the cloud profile as a function of radius from a center near Oph D (see Fig. 19). The two most prominent peaks correspond first to an arc-like area crossing Oph B2 (named Arc 3 in Fig. 18), and second to an arc-like area (named Arc 1) following the well-known dense C 18 O ridge of L1688 (Wilking & Lada 1983). A third, shallower peak seen in the circularly-averaged column density profile of Fig. 19 corresponds to the arc-like area named Arc 2 in Fig. 18, which includes Oph B1. A hint of the presence of this arc was already visible in the form of a weak filamentary structure in the ground-based 1.2 mm continuum map of Stanke et al. (2006, see their Fig. 1).
The three consecutive arc-like structures Arcs 1-3 may originate from compression by large-scale shockwaves propagating through the cloud from the Sco OB2 association (Loren & Wootten 1986;Motte et al. 1998). There are at least two possibilities for the origin of these multiple arc-like features. First, the presence of multiple compression fronts is reminiscent of the theoretical scenario proposed by Inutsuka et al. (2015) for the formation of molecular clouds based on multiple shockwave compressions of interstellar atomic gas. Second, the multiple compression fronts may trace the interaction of a single shockwave with multiple preexisting structures within the L1688 cloud. In the latter case, however, we would expect to observe an age gradient from northeast to southwest with younger YSOs associated with Arc 3 (compressed last) and more evolved YSOs associated with Arc 1 (compressed first). In contrast, the highest concentration of Class I (Wilking et al. 1989;Bontemps et al. 2001) and even Class 0 (André et al. 1993) objects is found in the C 18 O ridge (Arc 1), while a number of relatively evolved Class II/III objects and X-ray emitting weak-line T Tauri stars are observed close to Arc 3 (Casanova et al. 1995). For this reason, we favor the former interpretation of multiple compression events and shockwaves. The interaction of each shockwave with an overdensity cloud structure created in a previous compression event can generate a bow-shape or U-shape feature similar to the arcs seen in Fig. 18 (cf. Fig. 1 shape of the Oph D dense clumps is also consistent with this scenario. The Ophiuchus complex, and the L1688 cloud in particular, are known to be influenced and possibly shock-compressed by an expanding shell of HI gas powered by stellar winds and supernova explosions in the Sco OB2 association, including the effect of the neighboring massive (O9V) star σ Sco (Loren & Wootten 1986;de Geus 1992;Nutter et al. 2006). Based on Fig. 18, we suggest that the main filaments of L1688 may result from three consecutive shock compressions. The region studied here is in fact enclosed between Antarès (α Sco) and σ Sco (cf. Fig. 1). The feedback influences of these stars may be significant as suggested by the positions of these OB stars relative to the three arcs identified here (Fig. 1, Fig. 18).
Given the estimated age of < ∼ 2 Myr for the L1688 embedded cluster (Bontemps et al. 2001) and the different number ratios of prestellar to unbound starless cores observed in the three arcs (see Table 5), it seems reasonable to assume a typical age difference of only < ∼ 1 Myr between these arc-like compression fronts, similar to the typical timescale adopted by McKee & Ostriker (1977) and Inutsuka et al. (2015) for the dynamics of the Galactic interstellar medium resulting from supernova explosions and expanding HII regions. The large-scale HI shell propagating into the Ophiuchus cloud complex expands at a velocity of about 10-15 km s −1 according to de Geus (1992), but the velocity of the transmitted shock within the L1688 dense cloud may be only < ∼ 1 km s −1 based on the range of line-of-sight velocities observed in optically thin tracers of molecular gas (Loren 1989b;Loren & Wootten 1986). This kind of low-velocity shock would not destroy the cloud (Loren & Wootten 1986;Herbst & Warner 1981;Foster & Boss 1996) and may ideally permit the formation of gravitationally-unstable condensations such as the candidate pre-brown dwarf Oph B-11 (Greaves et al. 2003;André et al. 2012) in regions of the cloud with relatively low initial density. Notes. ( †) Comparison between the number of Class I protostars detected with Spitzer in Dunham et al. (2013) and the number of unbound starless cores in the present paper.
While large-scale flows may be shaping the whole complex, looking for local hints of their effect on the small-scale structure of L1688 is interesting. The spatial distribution of the dense cores identified in this paper shows global relationships, such as overall spreads and mean separations, but also contains information about the structure of the molecular cloud at a given time. Also relevant are the substructures identified in the column density map of the complex. Historically divided into clumps (e.g., Loren et al. 1990), the Ophiuchus complex had never been observed with high enough sensitivity and dynamic range up to now to unveil the underlying filamentary texture of the cloud. While many clumps (A to G) were apparent in this region, the Herschel observations now allows us to distinguish three nearly concentric arcs in the main cloud (Fig. 18). This leads us to pro- pose another classification of YSOs, prestellar cores, and unbound starless cores in L1688, not divided into dense clumps as in Loren et al. (1990)but divided into three concentric arcs Fig. 18. Simple statistics about the ratios of bound to unbound cores in the three arcs are summarized in Table 5. We notice that the second or central arc (Arc 2) seen in the azimuthal average of L1688 (Fig. 19) is significantly more populated with unbound starless cores, in the middle of the cloud. In contrast, Arc 1 and Arc 3 are seemingly more evolved than Arc 2 with more YSOs and prestellar cores inside of them. With signs of global contraction in this cloud from self-absorbed CO profiles skewed to the blue (Encrenaz 1974, Lada & Wilking 1980, it is tempting to speculate that some of the unbound starless cores observed within Arc 2 will be further compressed in the future and accumulate more mass, eventually leading to the formation of new prestellar cores and a new generation of stars in L1688.
Conclusions
Parallel-mode (SPIRE and PACS) maps obtained as part of the Herschel Gould Belt survey were used to carry out an extensive study of the population of dense cores in the Ophiuchus molecular cloud (L1688, L1689, L1709). We summarize our main results as follows:
1. Using the Herschel multi-wavelength data, we derived a high-resolution (18.2 ) column density map of the Ophiuchus complex (Figs. 1 & 2). This map clearly reveals for the first time the presence of filaments inside the main cloud, L1688, which was not previously known to be a filamentary region. L1688 appears to be structured in the form of a converging network of filaments pointing toward its center of mass near Oph B2 (see Fig. 7-top). The filaments have a bimodal distribution of orientations (cf. Fig. 10), either roughly parallel or roughly perpendicular to the direction of the largescale streamers of the complex. 2. We identified a total of 144 candidate prestellar cores, including 93 robust prestellar cores, in L1688 and L1689/L1709, using the getsources source extraction algorithm after a careful selection and inspection of each source. In addition, 315 unbound starless cores and 54 protostellar cores were also detected. The main properties (e.g., radius, mass, temperature, density) of all of these cores were characterized (cf. online Table A.2) thanks to a good sampling of the SEDs in the far-infrared and submillimeter range. 3. The mass distribution of the prestellar cores closely resembles the IMF of stellar systems (Fig. 17) confirming earlier results. 4. The prestellar cores lie preferentially within dense filaments and are mostly found very close (< 0.1 pc) to the crests of their parent filaments (Fig. 11). They are also significantly more concentrated in the densest parts of the complex, occupy a significantly smaller area of the total field (Figs. 12 & 13), and tend to lie in regions of higher ambient cloud pressure ( Fig. 9) than the unbound starless cores. 5. The median nearest neighbor separation between prestellar cores is ∼ 0.04 pc, which is comparable to the typical Jeans length in cold dense molecular gas at N H 2 ∼ 10 22 cm −2 . The typical projected core separation is also comparable to the median inner filament width ∼ 0.06 pc measured by Arzoumanian et al. (2019) in Ophiuchus but significantly shorter than the characteristic core spacing of ∼ 4 times the filament width (∼ 0.24 pc here) expected from standard cylinder fragmentation models. This difference may result from the effects of magnetic fields and turbulence, or the fact that the Ophiuchus filaments are not isolated systems. 6. We confirmed the presence of a "threshold" or sharp transition in column density around a fiducial A V value of ∼ 7 for the formation of prestellar cores (Figs. 15 & 16). Moreover, by comparing our results in L1688 and L1689, we found clear evidence that only a fraction of the total mass budget of the complex is directly participating in the star formation process. Indeed, L1689 is less efficient at forming stars than L1688, but shares almost the same total mass of molecular gas. When only considering the dense molecular gas above A V ∼ 7, L1689 and L1688 appear to be equally efficient at forming dense cores. 7. The prestellar core formation threshold is strongly correlated to the presence of dense filaments in the Ophiuchus complex. Most of the dense gas material in the cloud and most of the prestellar cores are distributed in dense filaments within several clumps. Furthermore, the majority of prestellar cores lie within the ∼ 0.1 pc inner portions of their parent filaments (Fig. 11). The close connection between prestellar cores and filaments is especially remarkable here in Ophiuchus, as there was no direct detection of filamentary structures up to now in the main region, L1688. 8. Three shock compression fronts can be detected in the column density map of the L1688 cloud, locally enhancing the quantity of dense molecular gas and supercritical filaments, hence the formation of prestellar cores.
A&A proofs: manuscript no. oph_herschel_1stgen
(3)
(6) ±
(8)
(10) ±
Notes.
Catalog entries are as follows: (1) Core running number;
(2)
Core name = HGBS_J prefix directly followed by a tag created from the J2000 sexagesimal coordinates; (3) and (4) Core name = HGBS_J prefix directly followed by a tag created from the J2000 sexagesimal coordinates; (3) and (4):
Right ascension and declination of core center; (5) and (6): Geometrical average between the major and minor FWHM sizes of the core (in pc), as measured in the high-resolution column density map after deconvolution from the 18.2 HPBW resolution of the map and before deconvolution, respectively. (NB: Both values provide estimates of the object's outer radius when the core can be approximately described by a Gaussian distribution, as is the case for a critical Bonnor-Ebert spheroid); (7)
Estimated core mass (M ) assuming the dust opacity law advocated by Roy et al. (2014);
(9) SED dust temperature (K); (8) & (10) Statistical errors on the mass and temperature, respectively, including calibration uncertainties, but excluding dust opacity uncertainties; (11)
Peak H 2 column density, at the resolution of the 500 µm data, derived from a graybody SED fit to the core peak flux densities measured in a common 36.3 beam at all wavelengths; (12)
Average column density, calculated as N ave H 2 = Mcore πR 2 core 1 µm H , where M core is the estimated core mass (col. 7), R core the estimated core radius prior to deconvolution (col. 6), and µ = 2.8; (13) Average column density calculated in the same way as for col. 12 but using the deconvolved core radius (col. 5) instead of the core radius measured prior to deconvolution; (14)
Beam-averaged peak volume density at the resolution of the 500 µm data, derived from the peak column density (col. 11) assuming a Gaussian spherical distribution: Appendix B: Completeness of the prestellar core sample
To estimate the completeness of the present Herschel census of prestellar cores in the Ophiuchus complex, we used both Monte-Carlo simulations (cf. Sect. 4.6) and the same model of the core identification process as described in Appendix B.2 of Könyves et al. (2015). The same extraction steps with getsources and postextraction selection criteria as described in Sect. 4.4 for the real Herschel data were applied to the simulated images. The resulting catalog of identified sources was then compared with the "truth table" of input synthetic cores injected in the simulated images to assess the completeness level of the Herschel survey. Figure B.1) plots the fraction of successfully identified synthetic cores as a function true core mass. As can be seen in this figure, our census of prestellar cores appears to be > 80% complete above a true core mass of ∼0.1 M . In addition, the properties found for the cores identified in the simulated data were also compared to the true properties of the injected synthetic cores to estimate the accuracy of the main derived parameters (e.g., core mass, radius, temperature) given in Table A.1 for the real cores. Figure B.2 (top panel) suggests that the derived core masses typically underestimate the true core masses by ∼ 50% on average for observed masses between ∼ 0.1 M and ∼ 1 M , around the peak of the observed prestellar CMF (Fig. 17). This is most likely due to a slight (∼ 20%) underestimate of the core sizes (cf. middle panel of Fig. B.2) and an overestimate of the massaveraged dust temperatures by ∼ 1.5 K on average (cf. bottom panel of Fig. B.2).
Fig. 4 .
4Zoom on the Herschel column-density map ofFig. 2a, showing the various dense clumps of L1688 named as in
Fig. 5 .
5Zoom on the Herschel column-density map ofFig. 2, showing several sub-regions around L1689. The effective HPBW resolution is 18.2 . The colorscale has been adjusted to transition from red to blue in the range of A V from 6 to 10.
Fig. 6 .
6(Left) Probability density function (PDF) of column density in the Ophiuchus molecular cloud and its subclouds L1688, L1689 and L1709, as derived from the high-resolution column density map at a resolution of 18 2. (See
Fig. 7 .
7Filtered versions of the Herschel column density maps of L1688 (top) and L1689 (bottom), where the contrast of filamentary structures has been enhanced by accumulating small-scale emission up to a transverse scale of 150 (∼ 0.1 pc) using the getfilaments algorithm (seeSect. 4.3). The grayscale represents effective column density within the skeleton masks generated by getfilaments. The skeleton of the filaments independently extracted with the DisPerSE algorithm is displayed in yellow. In the top panel, the red star marks the center of mass of L1688.
Fig. 8 .
8Distributions of beam-averaged column densities (top) and beam-averaged volume densities (bottom), at the resolution of the SPIRE 500 µm data, for the populations of 459 starless cores (dashed) and 144 candidate prestellar cores (solid) identified in the Ophiuchus complex.
Fig. B.2 in Appendix B).
Fig. 9 .
9(Left) Mass versus size diagram for the entire population of 459 starless cores identified with Herschel in the Ophiuchus molecular cloud. The core FWHM sizes were measured with getsources in the high-resolution column density map (Fig. 2) and deconvolved from an 18 2 (HPBW) Gaussian beam. The core masses were derived via graybody fitting to the getsources-estimated fluxes (see text). The 93 robust prestellar cores (for which α BE ≤ 2) are shown as filled blue symbols, the other (candidate) prestellar cores as open green symbols, and the other (unbound) starless cores as open red symbols. For comparison, models of critical isothermal Bonnor-Ebert spheres at T = 7 K and T = 20 K are plotted as dashed black lines. The mass-size correlation observed for diffuse CO clumps (cf. Elmegreen & Falgarone 1996) is displayed as a shaded yellow band. (Right) Same mass versus size diagram for all 459 starless cores, where the color coding of the data points is related to the ambient cloud pressure, P cl ≈ 0.88 G Σ 2 cl , estimated from the local background column density Σ cl assuming a cloud in virial equilibrium (McKee & Tan 2003).
Fig. 10 .
10(Top) Skeleton map of filament orientations (colorscale) overlaid on a filtered version of the Herschel column density map of the Ophiuchus cloud (grayscale) obtained with the getfilaments algorithm (see Fig. 7). The filamentary structures identified with getfilaments are displayed in red or blue depending on their position angles measured east of north (see colorscale). (Bottom) Distribution of median position angles (P.A.) for the sample of filaments identified with getfilaments (cf. Sect. 4.3)
Fig. 11 .
11Distribution of core-to-filament separations in the Ophiuchus cloud, for both prestellar cores and (unbound) starless cores. For each core, the projected separation to the nearest filament in the sample of filamentary structures extracted with DisPerSE (cf. Sect. 4.3) was considered. For comparison, the red dashed curve shows an example of a normalized, crest-averaged radial column density profile for a filament traced with DisPerSE in Ophiuchus.
)
25 m 00.00 s 26 m 00.00 s 27 m 00.00 s 28 m 00.00 s 29 m 00.00 s 30 m 00.00 s RA (J2000) 25 m 00.00 s 26 m 00.00 s 27 m 00.00 s 28 m 00.00 s 29 m 00.00 s 30 m 00.00 s RA (J2000) Fig. 12. (Top) Spatial distribution of prestellar cores in L1688. The centroid positions, sizes, and position angles estimated with getsources for the prestellar and unbound cores in
)
Fig. 13. (Left) Spatial distribution of prestellar cores in L1689 and L1709. The centroid positions, sizes, and position angles estimated with getsources for the prestellar and unbound cores in
Fig. 14 .
14(Top) Distributions of nearest-neighbor separations for the populations of starless and prestellar cores. (Bottom) Comparison of the distributions of nearest-neighbor prestellar core separations in L1688 and L1689.
CFE obs (A V ) = ∆M cores (A V )/∆M cloud (A V ). This is obtained by di-B. Ladjelate et al.: Herschel Gould Belt survey results in Ophiuchus
Fig. 15 .
15Observed differential core formation efficiency (CFE) as a function of background column density expressed in A V units (blue histograms with error bars) in the Ophiuchus molecular cloud, in blue for the whole cloud, in orange for L1688, and in green for L1689. The CFE if obtained by dividing the total mass of prestellar cores in a given column density bin by the cloud mass in this column density bin. The vertical dashed line marks a fiducial threshold at A bg V ∼ 7.
Fig. 18 .
18of Inoue & Fukui 2013). The cometary 16 h 25 m 00.00 s 26 m 00.00 s 27 m 00.00 s 28 m 00.00 s 29 m 00.00 s 30 m 00.00 s RA (J2000) Sketch of three putative compression fronts in dashed blue (Arc 1), dashed yellow (Arc 2), and dashed green (Arc 3), corresponding to density enhancements, overlaid on the Herschel high-resolution column density map of L1688 (grayscale). All three compression arc-like structures point roughly toward the direction of the O9V star σ Sco (cf.Fig. 1), and are possibly skewed by the influence of the bubble around the star ρ Ophiuchi. The yellow cross marks the rough center of the three arcs and the red star indicates the center of mass of L1688.
Fig. 19 .
19(Left) Filtered column density map of L1688 where concentric annuli associated with the shaded ranges are overplotted, each annulus marking the presence of one arc-like over-compression as seen in the right panel. The yellow cross marks the center of the concentric annuli and the red star indicates the center of mass of L1688. (Right) Radial column density profile of the L1688 cloud, circularly averaged about a center close to Oph D. The shaded ranges highlight the three compression fronts seen in the column density image and associated with the three arcs in the left panel. The solid curve shows the profile obtained from an azimuthal average of the unfiltered column density map shown in Fig 18 over the whole 190 • of the shaded sectors in the left panel. The dashed curve shows the profile obtained from an azimuthal average of the filtered column density map over the same rangeof the shaded sectors in the left panel. The dotted line shows the profile averaged over the angles excluding the part around Oph B, outside of the two black solid lines in the left panel, with evidence of an overdensity highlighting the second arc (Arc 2).
declination of core center; (5), (15), (25), (35), and (45): Detection significance from monochromatic single scales, in the 70, 160, 250, 350, and 500 µm maps, respectively. (NB: the detection significance has the special value of 0.0 when the core is not visible in clean single scales); (6)±(7), (16)±(17) (26)±(27) (36)±(37) (46)±(47): Peak flux density and its error in Jy/beam as estimated by getsources; (8), (18), (28), (38), (48): Contrast over the local background, defined as the ratio of the background-subtracted peak intensity to the local background intensity (), (19), (29), (39): Peak flux density measured after smoothing to a 36.3 beam; (10)±(11), (20)±(21), (30)±(31), (40)±(41), (49)±(50): Integrated flux density and its error in Jy as estimated by getsources; (12)-(13), (22)-(23), (32)-(33), (42)-(43), (51)-(52): Major & minor FWHM diameters of the core (in arcsec), respectively, as estimated by getsources. (NB: the special value of −1 means that no size measurement was possible); (14), (24), (34), (44), (53): Position angle of the core major axis, measured east of north, in degrees; estimated by getsources in the high-resolution column density image; (56) Column density contrast over the local background, as estimated by getsources in the high-resolution column density image; (57) Peak column density measured in a 36.3 beam; (58) Local background H 2 column density as estimated by getsources in the high-resolution column density image; (59)-(60)-(61): Major & minor FWHM diameters of the core, and position angle of the major axis, respectively, as measured in the high-resolution column density image; (62) Number of Herschel bands in which the core is significant (Sig λ > 5) and has a positive flux density, excluding the column density plane; (63) Core type: starless, prestellar, or protostellar; (64) Closest counterpart found in SIMBAD, if any, up to 6 from the Herschel peak position;
Fig. B. 1 .
1Completeness curves for our Herschel census of candidate prestellar cores based on Monte-Carlo simulations using a population of synthetic prestellar cores detected by getsources and classified as candidate prestellar cores (green solid curve) or classified as (bound or unbound) starless cores (red solid curve). For comparison, the dashed blue curve shows the completeness curve predicted by the simple model of the core extraction process described in Appendix B.2 ofKönyves et al. (2015), scaled to the distance of the Ophiuchus cloud (d = 139 pc).
Fig. B. 2 .
2Comparison between the derived and the true values of the core masses (top panel), sizes (middle panel), and dust temperatures (bottom panel) in the Monte-Carlo simulations described here and in Sect. 4.6.
Fig. 1. (Top) Multi-resolution column density map of the Ophiuchus complex derived from Planck/Herschel data (see text)h 15 m
20 m
25 m
30 m
35 m
40 m
RA (J2000)
−26 •
−25 •
−24 •
−23 •
Dec (J2000)
σ Sco
α Sco
τ Sco
ρ Ophiuchi
HD 147889
Ophiuchus Main Cloud
S c o O B 2 i n fl u e n c e
S t re a m e rs d ir e c t io n
5 pc
0.2
0.4
0.6
0.8
1.0
Column density (N(H2).cm −2
)
×10 23
2 pc
O p h A
O p h C
O p h E & F
O p h B
O p h D
L1709
I R A S 1 6 2 9 3
L 1 6 8 9 A
L1689B
L1688
L1689
L1689+L1709
L1688
Herschel column-density map of the Ophiuchus molecular cloud at an effective HPBW resolution of 18.2 . The column-density map is given in units of H 2 .cm −2 , calculated on a pixel-by-pixel basis usin a multi-scale graybody fitting procedure (see Appendix A of Palmeirim et al. 2013). (Right) Same as the left panel with the candidate prestellar cores identified in Sect. 4.4 overlaid as green triangles and the unbound starless cores as orange triangles. The bar at the lower right indicates a scale of 2 pc.Article number, page 3 of 24
A&A proofs: manuscript no. oph_herschel_1stgen
16 h 20 m
24 m
28 m
32 m
36 m
RA (J2000)
26
25
24
23
22
Dec (J2000)
2 pc
16 h 20 m
24 m
28 m
32 m
36 m
RA (J2000)
2 pc
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Column density (N(H2).cm 2
)
⇥10 22
Fig. 2. (Left) 16 h 20 m
24 m
28 m
32 m
36 m
RA (J2000)
−26 •
−25 •
−24 •
−23 •
−22 •
Dec (J2000)
S1
HD 147889
Rho Ophiuchi
S c o
O
B 2
i n fl u e n c e
σ Sco
2 pc
12
14
16
18
20
22
24
Temperature (K)
Table 1 .
1Observed Herschel bands and zero-level offsetsInstrument Wavelength HPBW Map Offset
(microns)
MJy/sr
PACS
70
8.4
4.7
PACS
160
13.5
86.8
SPIRE
250
18.2
119.9
SPIRE
350
24.9
59.8
SPIRE
500
36.3
24.2
Table 2 .
2Results of power-law fits to the column density PDFs in the Ophiuchus subclouds.Region
Range
Power law fit † Cumulative power law fit †
Table A.1 are represented by yellow ellipses overlaid on the Herschel high-resolution column density map derived from Herschel data (grayscale background image). Different colors are used for the separate dense clumps of L1688 (Oph A in blue, Oph B in orange, Oph C in green, Oph D in red, Oph E in purple, Oph F in brown, and Oph G in pink). (Bottom) Compared spatial distribution of prestellar cores (yellow squares) and unbound cores (blue circles) overlaid on the high resolution column-density map.
Table 3 .
3Characteristic extents of various starless core populations † √ N with N the number of objects used to derive the convex hull of each subset of cores.Cloud Characteristic radius of
Characteristic radius of
bound-core population unbound-core population
(pc)
(pc)
L1688
0.43 ± 0.02 † †
0.50 ± 0.02
Oph A
0.20 ± 0.015
0.25 ± 0.015
Oph B
0.10 ± 0.015
0.12 ± 0.015
Oph C
0.05 ± 0.01
0.08 ± 0.01
Oph D
0.03 ± 0.005
0.10 ± 0.02
Oph E
0.02 ± 0.005
0.01 ± 0.002
Oph F
0.12 ± 0.02
0.13 ± 0.015
Oph G
0.09 ± 0.015
0.12 ± 0.015
L1689
0.38 ± 0.035
0.58 ± 0.035
L1709
0.20 ± 0.035
0.28 ± 0.03
Notes. ( †) Calculated from the equivalent diameter of the convex hull of
each subset of cores. ( † †) Uncertainties are calculated as proportional to
1
2
Table 4 .
4Main properties of the Ophiuchus subclouds Cloud Total Cloud Cloud mass Number of M cores /M gas M cores /M dense R gas n †gas
R dense n dense
†
Table 5 .
5Ratios of various types of cores in the three arcsArc
Number ratio of
Mass ratio
Number ratio of
bound to
bound to
protostars † to
unbound cores unbound cores unbound cores
Arc 1
1.2
21.8
0.5
Arc 2
0.6
7.4
0.1
Arc 3
0.7
12.2
0.13
Table A . 1 .
A1Catalog of dense cores identified in the HGBS maps of the Ophiuchus complex (template, full catalog only provided online).rNO
Core name
RA
2000
Dec
2000
Sig
070
S
peak
070
S
peak
070 /S
bg
S conv,500
070
S tot
070
FWHM a
070 FWHM b
070 PA
070
HGBS_J*
(h m s)
( •
)
(Jy/beam)
(Jy/beam
500 )
(Jy)
( )
( )
( •
)
Table A . 2 .
A2Derived properties of the dense cores identified in the HGBS maps of the Ophiuchus region (template, full table only provided online).Notes.Table entries are as follows:(1)rNO
Core name
RA
2000
Dec
2000
R
core
M
core
T
dust
N
peak
H
2
N ave
H
2
n
peak
H
2
n ave
H
2
α
BE
Core type
Comments
HGBS_J*
(h m s)
( •
)
(pc)
(M )
(K)
(10 21
cm −2
)
(10 21
cm −2
)
(10 4
cm −3
)
(10 4
cm −3
)
(1)
(2)
(3)
(4)
(5)
(6)
(7) ±
(8)
(9) ±
(10)
(11)
(12) (13)
(14)
(15) (16)
(17)
(18)
(19)
· · ·
89 162626.6-242431 16:26:26.68 -24:24:31.0 8.1e-03 1.4e-02 2.234 0.318 13.6 0.6
190.64
127.12 402.69
225.87
192.86 1087.41 0.07 protostellar
VLA1623
· · ·
91 162627.6-242359 16:26:27.65 -24:23:59.3 1.1e-02 1.7e-02 7.868 1.062 12.7 0.6
378.77
322.26 753.76
423.33
414.80 1483.83 0.03
prestellar
SM1
· · ·
458 163223.1-242836 16:32:23.16 -24:28:36.5 6.1e-03 1.2e-02 4.659 0.910 16.6 1.6
302.91
368.05 1472.20
375.98
657.90 5263.18 0.03 protostellar IRAS 16293
· · ·
Core running number; (2)
H , using the estimated core radius prior to deconvolution; (16) Average volume density, calculated in the same way as for col. 15 but using the deconvolved core radius (col. 5) instead of the core radius measured prior to deconvolution;(17)Bonnor-Ebert mass ratio: α BE = M BE,crit /M obs (see text for details); (18) Core type: starless, prestellar, or protostellar; (19) Comments may be no SED fit or tentative bound (see text for details).Article number, page 23 of 24 A&A proofs: manuscript no. oph_herschel_1stgenn peak
H
2
= 4 ln 2
π
N
peak
H
2
FWHM
500
; (15)
Average volume density, calculated as
n ave
H
2 = Mcore
4/3πR 3
core
1
µm
http://gouldbelt-herschel.cea.fr Article number, page 2 of 24 B. Ladjelate et al.: Herschel Gould Belt survey results in Ophiuchus
The HGBS first-generation catalog of cores presented in this paper (see Appendix A) was produced with the "November 2013" major release of getsources (v1.140127), which is publicly available from http://gouldbelt-herschel.cea.fr/getsources.
As a final, post-selection check, the 740 candidate cores automatically selected on the basis of these criteria were all inspected visually. The main properties of each dense core in the Article number, page 7 of 24 A&A proofs: manuscript no. oph_herschel_1stgen
This is also similar to the widths of the N 2 H + fiber-like structures observed byHacar et al. (2018) in the Orion integral filament region, but slightly smaller than the typical width of the fibers identified byHacar et al. (2013) in Taurus.
Acknowledgements. We thank Shu-ichiro Inutsuka for insightful discussions on cloud compression and the referee, Thomas Stanke, for useful comments which improved the clarity of the paper. SPIRE has been developed by a consortium of institutes led by Cardiff Univ. (UK) and including: Univ. LethbridgeAppendix A: A catalog of dense cores identified with Herschel in the Ophiuchus cloudBased on our Herschel SPIRE/PACS parallel-mode imaging survey of the Ophiuchus molecular cloud, we identified a total of 513 dense cores, including 459 starless cores (144 of them being self-gravitating and prestellar in nature), and 54 protostellar cores. The master catalog listing the observed properties of all of these Herschel cores is available in onlineTable A.1 (see below for an illustrative portion of this table and http://gouldbeltherschel.cea.fr/archives for the full table). The derived properties (physical radius, mass, SED dust temperature, peak column density at the resolution of the 500 µm data, average column density, peak volume density, and average density) of the same cores are provided in onlineTable A.2 (see also below for an illustrative portion and http://gouldbelt-herschel.cea.fr/archives for the full table).
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|
[] |
[
"PROPERTIES OF KOSZUL HOMOLOGY MODULES",
"PROPERTIES OF KOSZUL HOMOLOGY MODULES"
] |
[
"Uwe Nagel ",
"Tony J Puthenpurakal "
] |
[] |
[] |
We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, S 2 -property and their Bass numbers
| null |
[
"https://arxiv.org/pdf/0809.3853v1.pdf"
] | 17,940,305 |
0809.3853
|
468bdb4db4fe0a3d13553660c66e83cd3de8cf71
|
PROPERTIES OF KOSZUL HOMOLOGY MODULES
23 Sep 2008
Uwe Nagel
Tony J Puthenpurakal
PROPERTIES OF KOSZUL HOMOLOGY MODULES
23 Sep 2008
We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, S 2 -property and their Bass numbers
In particular H l−g−1 (y, M ) satisfies S 2 .
If (R, m) is Cohen-Macaulay and I is a Cohen-Macaulay ideal, Vasconcelos notes that H n−g−1 (y, R) is S 2 ; see [7, 1.3.2]. We prove: Theorem 1.3. Let (R, m) be a Gorenstein and I be a Cohen-Macaulay ideal. Then H n−g−1 (y) ∼ = Ext g R (H 1 (y), R). Next we consider the projective dimension of Koszul homology modules. Note that if I is perfect then H l−g (y, R) ∼ = Ext g R (R/I, R) has finite projective dimension. We now briefly describe the organization of this paper. In Section 2 we introduce some notation. In Section 3 we prove Theorems 1.2, 1.3 and 1.4. In Section 4 we prove Theorem 1.1. We conclude with an estimate of certain Bass numbers in Section 5.
Notation
In this paper all rings are commutative Noetherian. Let R be a ring, I an ideal in R and let M be an R-module (not-necessarily finitely generated).
2.1.
Let y = {y 1 , . . . , y l } be a set of generators of I and let K • (y, R) be the Koszul complex with respect to y. Set
K • = K • (y, R) : 0 → K l → · · · K 1 → K 0 → 0.
Let K • = Hom R (K • , R) be the Koszul co-chain complex with respect to y. Let K • (y, M ) and K • (y, M ) be respectively the Koszul complex and co-chain complex with respect to y with coefficients in M .
2.2.
If D • is a chain complex of R-modules then we set H i (D • ) to denote its i th homology module. Likewise if D • is a co-chain complex of R-modules then we set H i (D • ) to denote its i th cohomology module. Let H i (y, M ) and H i (y, M ) denote respectively the i th Koszul homology and cohomology module with respect to y with coefficients in M .
3. Proof of theorems 1.2, 1.3 and 1.4
In this section we prove Theorems 1.2, 1.3 and 1.4
3.1. Let I • be a "deleted" injective resolution of M ;
I • : 0 → I 0 → I 1 → · · · → I n → I n+1 → · · ·
We consider the Hom co-chain complex C = Hom(K • , I • ); see [10, 2.7.4]. Set
C = {C pq } p,q≥0 where C pq = Hom(K p , I q ). Set T • = Tot L (C) where T n = p+q=n C pq .
Proof. Notice
I E pq 1 = H q (Hom R (K p , I • ) = Ext q R (K p , M ) = Hom R (K p , M ), for q = 0; 0 for q > 0.
The last equality is true since K p is a free R-module. It follows that
I E pq 2 = H p (y, M ), for q = 0; 0 for q > 0.
Hence the spectral sequence collapses at E 2 and the claim follows.
Proposition 3.3. [with hypothesis as in 3.1] II E pq 2 = Ext p R (H q (y), M ). Proof. Notice II E pq 1 = H q (Hom R (K • , I p )) = Hom R (H q (y), I p )) .
The last equality is true since I p is an injective R-module. Thus II E pq 2 = H p (Hom R (H q (y), I • )) = Ext p R (H q (y), M )) ; as claimed.
3.4. II E 2 -page in three Special Cases: We will consider the following three special cases (i) (R, m) is a d-dimensional Gorenstein local ring, A = R/I is CM and M = R.
(a) By local duality we get Ext p R (H 0 (I), M ) = 0 for p > g. So II E p,0 2 = 0 for p > g. (b) Since grade H q (y) = g for all q and M = R is maximal Cohen-Macaulay.
So
II E pq 2 = 0 for p < g. (ii) (R, m) is a d-dimensional
Cohen-Macaulay local ring with a canonical module, the ideal I is perfect (in particular A = R/I is CM) and M is a Maximal Cohen-Macaulay R-module.
(a) Since H 0 (y) = R/I has projective dimension g we get that Ext p R (H 0 (I), M ) = 0 for p > g. So II E p,0 2 = 0 for p > g.
(b) Since grade H q (y) = g for all q and M is maximal Cohen-Macaulay we get
II E pq 2 = 0 for p < g. (iii) (R, m)
is Noetherian local of dimension d and M is a finitely generated Rmodule. The ideal I is perfect and has l = µ(I) = grade(I) + 2. Finally, y is a minimal set of generators of I.
(a) We have that H 2 (I) ∼ = Ext g R (R/I, R) has projective dimension g. So Ext p R (H 2 (I), M ) = 0 for p > g. II E p,l−g 2 = 0 for p > g.
(b) Similarly as R/I has projective dimension g we get Ext p R (H 0 (I), M ) = 0 for p > g. So II E pq 2 = 0 for p < g. We now establish Theorem 1.2.
Proof of Theorem 1.2. Assume R, I, A, M are as in 3.4(ii). Recall II E r has differential of degree (r, −r + 1). Using the vanishing results in 3.4(ii) we get that
II E g,1 ∞ = II E g,1 2 = Ext g R (H 1 (y), M ) and II E g+1,0 ∞ = II E g+1,0 2 = 0. The only non-zero term in II E ∞ with total degree g + 1 is II E g,1 ∞ . It follows that H g+1 (T • ) ∼ = II E g,1 ∞ = Ext g R (H 1 (y), M ). Proposition 3.2 provides our claim. In particular H l−g−1 (y, M ) is S 2 .
We now prove our second main result.
Proof of Theorem1.3. Assume R, I, A are as in 3.4(i). We use the spectral sequence with M = R. Recall II E r has differential of degree (r, −r + 1). Using the vanishing results in 3.4(i) we get that
II E g,1 ∞ = II E g,1 2 = Ext g R (H 1 (y), R) and II E g+1,0 ∞ = II E g+1,0 2 = 0.
The only non-zero term in II E ∞ with total degree g + 1 is
II E g,1 ∞ . It follows that H g+1 (T • ) ∼ = II E g,1 ∞ = Ext g R (H 1 (y), R)
. Proposition 3.2 provides our claim.
We now establish Theorem 1.4
Proof of Theorem 1.4. Recall II E r has differential of degree (r, −r + 1). Also recall that g ≤ d.
Using the vanishing results in 3.4iii we get
II E d+3,1 ∞ = II E d+3,1 2 = Ext d+3 R (H 1 (y), M ). By Proposition 3.2; H d+4 (T • ) = 0. It follows that Ext d+3 R (H 1 (y), M ) = 0, where the module M is an arbitrary R-module. So we get projdim R H 1 (I) < ∞.
An easy consequence to 1.4 is the following result.
Corollary 3.5. Let (R, m) be a Gorenstein local ring and let I be a perfect ideal in R with ν(I) = grade(I) + 2. Set A = R/I. Then H 1 (I) is a perfect R-module and a self-dual A-module
Proof. By a result due to Avramov and Herzog [1,Supplement], the ideal I is strongly CM. Using 1.4 we get that H 1 (I) is a perfect R-module.
Let ω be the canonical module of A. By 1.
3 we have H 1 (I) ∼ = Ext g R (H 1 (I), R). Notice Ext g R (H 1 (I), R) ∼ = Hom A (H 1 (I), ω). Thus H 1 (I) is a self-dual A-module.
Proof of Theorem 1.1
Throughout this section (R, m) is Cohen-Macaulay local ring of dimension d and I is an ideal in R. Let u = u 1 , . . . , u l be a system of minimal generators for I. Let K • = K • (u, R) be the Koszul complex with respect to u. Let g = grade(I) and let x = x 1 , . . . , x d−g ∈ m be such that (1) x is a R-regular sequence.
(2) x is a system of parameters for R/I.
Let C • be theČech complex on x. We write C • [−(d − g)] homologically and call it D • . So D • : 0 → D d−g → · · · → D 1 → D 0 → 0 and H i (D ⊗ M ) = H d−g−i x (M ) for a R-module M .
Consider the double complex X = K • ⊗ D • and set W • = Tot(X). We look at the two standard spectral sequences associated to X. Proposition 4.1. The spectral sequence { I E r pq } collapses; hence, for each i ≥ 0,
we have H i (W • ) = H i (u, H d−g x (R)). Proof. I E 0 pq = K p ⊗ D q . So I E 1 pq = H q (K p ⊗ D • ) = H d−g−q x (K p ) = H d−g−q x (R) ⊗ K p = 0 for q > 0; H d−g x (R) ⊗ K p for q = 0. Therefore I E 2 pq = 0 for q > 0; H p (u, H d−g x (R)) for q = 0.
The result follows. Proof.
II E 0 pq = K q ⊗ D p . So II E 1 pq = H q (K • ⊗ D p ) = H q (K • ) ⊗ D p ; since D p is a flat R-module = H q (I) ⊗ D p Therefore II E 2 pq = H p (H q (I) ⊗ D • ) = H d−g−p x (H q (I)) = H d−g−p m (H q (I))
.
Surprisingly we have the following vanishing result. To prove this result the following Lemma is needed.
Lemma 4.4. Let (R, m) be a Cohen-Macaulay local ring. Let x 1 , . . . , x r , y 1 , . . . , y s be an R-regular sequence. Then (1) y = y 1 , . . . , y s is a weak H r x (R)-regular sequence.
(2) H r x (R)/yH r x (R) = H r x (R/yR) Proof.
It is sufficient to prove it for s = 1. Set y = y 1 and R = R/yR. Consider the exact sequence 0 −→ R y − → R −→ R −→ 0 Notice x is a R ⊕ R-regular sequence. Therefore taking local cohomology with respect to x we obtain
0 −→ H r x (R) y − → H r x (R) −→ H r x (R) −→ 0. So y is H r x (R)-regular and H r x (R)/yH r x (R) = H r x (R)
. Proof of Proposition 4.3. Choose y = y 1 , . . . , y g in I such that x, y is a system of parameters for R and hence a R-regular sequence. The result follows from Lemma 4.4 and [2, 1.6.16].
The following example shows that the result in Theorem 1.1 cannot be improved in general. By [3], we get that height I = 3, so dim A = 5. Let y be the set of minimal generators of I. So l = 6, g = 3. Using MACAULAY [4], one verifies that depth H 2 (y) = 2. Here depth H 3 (y) = 5.
We now give the proof of Theorem 1.1.
Proof. We have to show the following (1) If depth H l−g (I) ≥ 3 then depth H l−g−1 (I) ≥ 1.
(2) If dim H l−g (I) ≥ 4 then depth H l−g−1 (I) ≥ 2. We use the two standard spectral sequences induced on the above double complex
X = K • ⊗ D • .
Recall II E r has differential of degree (−r, r − 1). We also notice that
II E 2 pq = H d−g−p m (H q (I)) = 0 for p > d − g.
If depth H l−g (I) ≥ i + 1 then II E 2 d−g−j,l−g = 0 for j = 0, . . . , i. (1) If depth H l−g (I) ≥ 3 then using the above vanishing results we get that
II E ∞ d−g,l−g−1 = II E 2 d−g,l−g−1 = H 0 m (H l−g−1 (I)) Since dim H l−g (I) = d − g ≥ depth H l−g (I) = 3, the total degree of II E ∞ d−g,l−g−1 is r = d − g + l − g − 1 ≥ l − g + 2. As II E ∞ d−g,l−g−1 is a subquo- tient of H r (W • ) = 0 (by Proposition 4.3), we get that II E ∞ d−g,l−g−1 = 0. Thus H 0 m (H l−g−1 (I)) = 0. Therefore depth H l−g−1 (I) ≥ 1.
(2) If depth H l−g (I) ≥ 4 then similarly as above we get
II E ∞ d−g−1,l−g−1 = II E 2 d−g−1,l−g−1 = H 1 m (H l−g−1 (I)) The total degree of II E d−g−1,l−g−1 ∞ is r = d − g − 1 + l − g − 1 ≥ l − g + 2.
By an argument similar to (1) it follows that H 1 m (H l−g−1 (I)) = 0. By (1) we also have that H 0 m (H l−g−1 (I)) = 0. Therefore depth H l−g−1 (I) ≥ 2.
Bass numbers
In this section (R, m) is a Gorenstein local ring. Let ν(E) denote the minimal number of generators of an R-module E and let µ i (m, E) = ℓ Ext i R (k, E) denote the i-th Bass number of E (with respect to m). Proof. Let F • be a "deleted" minimal free resolution of k = R/m, let I • be a "deleted" minimal injective resolution of R and let K • be the Koszul complex on a set of minimal generators of I. Consider the double co-chain complexes
X = Hom R (Tot(F • ⊗ R K • ), I • ) and Y = Hom R (F • , Tot(Hom R (K • , I • ))).
Since all complexes involved are first quadrant complexes we have X ∼ = Y; cf. [10, 2.7.3]. Both the cases considered involve computing the cohomology of Z • = Tot(X). We use the second standard spectral sequence associated to X to compute cohomology of Z • . Notice Tot(X) ∼ = Tot(Y). We use the first standard spectral spectral sequence on Y to derive our results.
Set D • = Tot(F • ⊗ R K • ) and T • = Tot(Hom R (K • , I • )). By Proposition 3.2 we have that H i (T • ) = H i (I); the i-th Koszul cohomology of I. Now we compute the homology of D • . We use the second standard spectral sequence on F
• ⊗ R K • . So E 0 pq = F q ⊗ K p . So we get E 1 pq = H q (F • ⊗ K p ) = Tor R q (k, K p ). Since K p is free we have E 1 pq = 0 for q = 0 k ⊗ K p for q = 0
It follows that
E 2 pq = 0 for q = 0
k ( l p ) for q = 0 Now we compute the cohomology of Z • = Tot(X),
X = Hom R (Tot(F • ⊗ R K • ), I • )
and using the second standard spectral sequence for X. So E pq 0 = Hom R (D q , I p ). Therefore
E pq 1 = H q (Hom(D • , I p ) = Hom R (H q (D • ), I p ) ; since I p is injective, = Hom R k ( l q ) , I p = Hom R (k, I p ) ( l q ) Therefore E pq 2 = H p (Hom(k, I • ) ( l q ) = Ext p R (k, R) ( l q ) = 0 for p = d k ( l q ) for p = d
Thus this spectral sequence collapses. It follows that
(5.1.1) H i (Z • ) = k ( l d−i ) for d ≤ i ≤ d + l 0 otherwise
We now use the fact that X ∼ = Y. So Z • ∼ = Tot(Y). We compute the cohomology of Z • by using the first standard spectral sequence on Y. So E pq 0 = Hom R (F p , T q ). Therefore we get
E pq 1 = H q (Hom R (F p , T • )) = Hom R (F p , H q (T • )) ; since F p is free = Hom R (F p , H q (I))
Therefore E pq 2 = H p (Hom R (F • , H q (I))) = Ext p R (k, H q (I)) Now we distinguish the two cases considered in the statement of the theorem Case (I): Assume the ideal I is strongly Cohen-Macaulay. Since H q (I) = 0 for q < g we have E pq 2 = 0 for q < g. Also since if H q (I) = 0 it is a Cohen-Macaulay R-module of dimension d − g we get E pq 2 = 0 for p < d − g. We look at elements of total degree d+1 There are only two terms of total degree d + 1. They will make up the filtration for H d+1 (Z • ) = k l . So H g (I)). (ii) We look at E pq 3 when p = d − g and q = g + 1. Notice we have an exact sequence
(5.1.2) l = ℓ E d−g+1,g ∞ + ℓ E d−g,g+1 ∞ . Notice (i) E d−g+1,g ∞ = E d−g+1,g 2 = Ext d−g+1 R (k,(5.1.3) 0 −→ E d−g,g+1 3 −→ E d−g,g+1 2 −→ E d−g+2,g 2
Recall that E r has differential of degree (r, −r + 1). It follows that
E d−g,g+1 3 = E d−g,g+1 ∞
The result follows. Case (II): Assume projdim R H i (I) is finite for all i. Since R is Gorenstein we have injdim R H i (I) = d for all i. It follows that E pq 2 = Ext p R (k, H q (I)) = 0 for all p > d. (a.) The only term with total degree d + l is E d,l ∞ . Notice E d,l ∞ = E d,l 2 = Ext d R (k, H l (I)) Now the result follows from 5.1.1.
(b) We look at the two terms of total degree d+ l − 1. The proof is almost similar to that of Case (I). Except that here we have an exact sequence Remark 5.2. We wonder if it is possible to relax the assumption in (II) and still have the conclusion that µ d (m, R/I) = 1.
Theorem 1 . 4 .
14Let (R, m) be a local Noetherian and let I be a perfect ideal with ν(I) = grade(I) + 2, where ν(I) denotes the number of minimal generators of I. Then projdim R H 1 (I) is finite. We conclude with an estimate of certain Bass numbers of H 1 (I). Let ν(E) denote the minimal number of generators of an R-module E and let µ i (m, E) = ℓ Ext i R (k, E) denote the i-th Bass number of E (with respect to m). If (R, m) is regular local of dimension d then one can verify µ d (m, R/I) = 1. Theorem 1.5. Let (R, m) be a regular local ring and let I be an ideal in R. Then l − µ d−1 (m, R/I) ≤ µ d (m, H 1 (I)) ≤ l − µ d−1 (m, R/I) + µ d−2 (m, R/I).
Proposition 3. 2 .
2[with hypothesis as in 3.1] The spectral sequence { I E pq r } collapses; hence, for each i ≥ 0, we have H i (T • ) = H i (y, M ).
Proposition 4.2. II E 2 pq = H d−g−p m(H q (I)).
Proposition 4. 3 .
3Adopt the above assumptions. Then H i (I, H d−g x (R)) = 0 for i > l − g.
Example 4. 5 .
5Let R = Q[x, y, z, w, a, b, c, d] and let I be the ideal generated by the maximal minors of ψ where
Theorem 5 . 1 .
51Let (R, m) be a Gorenstein local ring of dimension d. Set l = ν(I), g = grade(I) and assume that l ≥ g + 2. We have the following (I) Assume I be a strongly Cohen-Macaulay ideal in R. Set c = d − g. Then l − µ c+1 (m, H g (I)) ≤ µ c (m, H g+1 (I)) ≤ l − µ c+1 (m, H g (I)) + µ c+2 (m, H g (I)). (II) Assume projdim R H i (I) is finite for all i. (Notice I need not be strongly Cohen-Macaulay). Then (a) µ d (m, R/I) = 1. (b) l − µ d−1 (m, R/I) ≤ µ d (m, H 1 (I)) ≤ l − µ d−1 (m, R/I) + µ d−2 (m, R/I).
Ext d− 2 R
2(k, H l (I)) −→ Ext d R (k, H l−1 (I)) −→ E d,
We now give a proof of Theorem 1.5.Proof of Theorem 1.5 . Since R is regular local we have projdim R H i (I) is finite for all i. The result follows from Theorem 5.1
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[] |
[
"NORI FUNDAMENTAL GERBE OF ESSENTIALLY FINITE COVERS AND GALOIS CLOSURE OF TOWERS OF TORSORS",
"NORI FUNDAMENTAL GERBE OF ESSENTIALLY FINITE COVERS AND GALOIS CLOSURE OF TOWERS OF TORSORS"
] |
[
"Marco Antei ",
"Indranil Biswas ",
"Michel Emsalem ",
"Fabio Tonini ",
"Lei Zhang "
] |
[] |
[] |
We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack X over a field k. This is done by describing the Nori fundamental gerbe of an essentially finite cover of X. A similar result is also obtained for the S-fundamental gerbe.* pY, yq ÝÑ pX , xq π N pY, yqwhere in the right side we consider inclusions as arrows. Moreover ‚ an essentially finite cover pY, yq ÝÑ pX , xq with Y inflexible is a torsor under a finite group G if and only if π N pY, yq is a normal subgroup of π N pX , xq and in this case there is an exact sequence 1 ÝÑ π N pY, yq ÝÑ π N pX , xq ÝÑ G ÝÑ 1‚ an essentially finite cover pY, yq ÝÑ pX , xq with Y inflexible is étale if and only if the finite scheme π N pX , xq{π N pY, yq is étale over k.
|
10.1007/s00029-019-0449-z
|
[
"https://arxiv.org/pdf/1706.00739v2.pdf"
] | 119,310,704 |
1706.00739
|
df09eb1a8632e4fa9ee614afc7cebbe7e6f83b6b
|
NORI FUNDAMENTAL GERBE OF ESSENTIALLY FINITE COVERS AND GALOIS CLOSURE OF TOWERS OF TORSORS
2 Jun 2017
Marco Antei
Indranil Biswas
Michel Emsalem
Fabio Tonini
Lei Zhang
NORI FUNDAMENTAL GERBE OF ESSENTIALLY FINITE COVERS AND GALOIS CLOSURE OF TOWERS OF TORSORS
2 Jun 2017
We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack X over a field k. This is done by describing the Nori fundamental gerbe of an essentially finite cover of X. A similar result is also obtained for the S-fundamental gerbe.* pY, yq ÝÑ pX , xq π N pY, yqwhere in the right side we consider inclusions as arrows. Moreover ‚ an essentially finite cover pY, yq ÝÑ pX , xq with Y inflexible is a torsor under a finite group G if and only if π N pY, yq is a normal subgroup of π N pX , xq and in this case there is an exact sequence 1 ÝÑ π N pY, yq ÝÑ π N pX , xq ÝÑ G ÝÑ 1‚ an essentially finite cover pY, yq ÝÑ pX , xq with Y inflexible is étale if and only if the finite scheme π N pX , xq{π N pY, yq is étale over k.
Introduction
Let G and H be finite group schemes over a field k. A pG, Hq-tower of torsors over a k-algebraic stack X consists in maps Z h Ý ÝÑ Y g Ý ÝÑ X where h and g are an H-torsor and a G-torsor respectively. The initial motivation for this work is the following question: given a tower of torsors as before does there exist a "Galois closure" under a finite group scheme which would dominate the tower? A Galois closure is a torsor P ÝÑ X under a finite group scheme over k together with a map λ : P ÝÑ Z over X satisfying some equivariance conditions (see Definition 3.8). Previous attempts to give an affirmative answer to this question failed: both [Ga] and (unpublished) [ABE] contain mistakes in the proof of their main theorems.
The étale case is not difficult and one can prove that there is even a canonical Galois closure.
Theorem I. If X is an algebraic stack over k and G or H is étale then all pG, Hqtowers of torsors Z ÝÑ Y ÝÑ X admit a canonical Galois closure P λ Ý ÝÑ Z by a torsor under the group scheme Q representing the sheaf of automorphisms of the trivial tower GˆH ÝÑ G ÝÑ Spec k. Moreover λ is a torsor under a finite subgroup scheme of Q.
Thus the problem of finding a Galois closure is a positive characteristic problem and it is related with the notion of essentially finite covers. A cover f : Y ÝÑ X of an algebraic stack (i.e. an affine map f such that f˚O Y is a vector bundle) is essentially finite if f˚O Y is essentially finite (in the category of vector bundles of X (see Definition 1.5)).
In order to study Galois closures we introduce the stack BpG, Hq of pG, Hq-tower of torsors, which is locally of finite type over k (see Proposition 3.3). We say that a tower Z ÝÑ Y ÝÑ X is pointed if Z has a given rational point; it is Nori reduced if the torsors are Nori reduced (see Definition 1.12). A Galois closure P λ Ý ÝÑ Z is pointed if P comes with a rational point and λ preserves the given rational points. The following is a general result proved here.
Theorem II. Let X be a pseudo-proper (see Definition 1.6) and inflexible (see Definition 1.1) algebraic stack of finite type over k. If char k ą 0 assume that dim k H 1 pX , Eq ă 8 for all vector bundles E on X . If Z h Ý ÝÑ Y g Ý ÝÑ X is a pG, Hq-tower of torsors then Z ÝÑ X is an essentially finite cover. If z P Zpkq then the tower admits a pointed Nori reduced Galois closure pP, pq λ Ý ÝÑ pZ, zq such that:
(1) λ is faithfully flat if and only if the torsors in the tower are Nori reduced, or equivalently Z and Y are inflexible, and in this case it is a Nori reduced torsor under a finite group scheme.
(2) P ÝÑ X is a torsor under a finite subgroup of the affine and of finite type k-group scheme Aut BpG,Hq pξq, where ξ P BpG, Hqpkq is the tower fiber of the given tower over x " ghpzq : Spec k ÝÑ X .
Examples 5.3 and 5.4 shows that there are examples of towers without a Galois closure if we remove the pseudo-properness assumption or the condition on the first cohomology groups. This last condition is needed to ensure that every G a -torsor over a cover of X is induced by a torsor under a finite subgroup of G a . Notice that the cohomological assumption is met for proper algebraic stack of finite type over k ( [Fal]). Moreover a geometrically connected and geometrically reduced algebraic stack over k is inflexible (see Remark 1.2).
The idea of the proof of Theorem II is simple. Given an essentially finite cover one can define its "Galois closure" and, applying this procedure on gh : Z ÝÑ X , one get the desired closure. So one must show that Z ÝÑ X is essentially finite and, since all torsors are essentially finite, the key point is to prove that the pushforward of an essentially finite vector bundle under an essentially finite cover is again essentially finite.
This motivates the study of essentially finite covers in general and, because of their strict relation with the notion of essentially finite vector bundles, their Nori fundamental groups or gerbes, which are denoted by π N p´,˚q and Π Ń respectively. In particular we use the machinery introduced in [No1] and later generalized in [BV] (see also Definition 1.1 and Theorem 1.9).
Theorem III. Let X be a pseudo-proper and inflexible algebraic stack of finite type over k and let f : Y ÝÑ X be an essentially finite cover. If char k ą 0 assume that either f is étale or dim k H 1 pX , Eq ă 8 for all vector bundles E on X .
Then there exists a unique (up to equivalence) finite map Π ÝÑ Π N X {k whose base change along X ÝÑ Π N X {k is Y f Ý ÝÑ X and Y is inflexible over k if and only if Π is a gerbe over k.
In this case Y ÝÑ Π is the Nori fundamental gerbe of Y, that is the diagram
Y Π N Y{k X Π N X {k f
is 2-Cartesian, and we have EFinpVectpYqq " tV P VectpYq | f˚V P EFinpVectpX qqu.
If f is étale then Y is inflexible over k if and only if H 0 pY, O Y q " k and in this case the following diagrams are 2-Cartesian
Y Π N Y{k Π N,ét Y{k X Π N X {k Π N,ét X {k f
If f is a torsor under a finite group scheme G over k the following are equivalent: ‚ Y is inflexible over k; ‚ H 0 pY, O Y q " k; ‚ f is Nori reduced over k. Under these conditions the following diagrams are 2-Cartesian
Y Π N Y{k Spec k X Π N X {k B G β π α f
For pointed covers Theorem III also yields the following Galois correspondence.
Corollary I. Let X be a pseudo-proper and inflexible algebraic stack of finite type over k with a rational point x P X pkq. If char k ą 0 assume that dim k H 1 pX , Eq ă 8 for all vector bundles E on X . Then there is an equivalence of categories " Pointed essentially finite covers pY, yq ÝÑ pX , xq with Y inflexible * " Subgroups H ă π N pX , xq with π N pX , xq{H finite
The above exact sequence has already been proved in [EHS,Thoerem 2.9] under the assumption that G is étale. The crucial point in the above results is again to show that, if f : Y ÝÑ X is an essentially finite cover, then f˚preserves essentially finite vector bundles. This is false without some properness and cohomological assumptions on X (see Lemmas 2.8, 2.10 and, for a counter-example, Example 5.5). A key tool in the proof is a characterization of essentially finite vector bundles given in [TZ2] which generalizes previous results of [BdS] and [AM].
We also study a similar problem for the S-fundamental gerbe (see [BPS], [La1], [La2] and Definition 4.1). The following analog of Theorem III is proved for S-fundamental gerbes.
Theorem IV. Let X be a pseudo-proper algebraic stack of finite type over k and with an S-fundamental gerbe. Then X is inflexible and the profinite quotient of the S-fundamental gerbe of X over k is the Nori fundamental gerbe of X over k. Let also f : Y ÝÑ X be an essentially finite cover with Y inflexible and, if char k ą 0, assume that dim k H 1 pX , Eq ă 8 for all vector bundles E on X . Then
NspYq " tV P VectpYq | f˚V P NspX qu and Y has an S-fundamental gerbe fitting in the 2-Cartesian diagram
Y Π S Y Π N Y X Π S X Π N X
In particular if y P Ypkq and x " f pyq P X pkq then there is a Cartesian diagram of affine group schemes π S pY, yq π N pY, yq π S pX , xq π N pX , xq
When f : Y ÝÑ X is pointed torsor under a finite group scheme G, then the following sequence is exact 1 ÝÑ π S pY, yq ÝÑ π S pX , xq ÝÑ G ÝÑ 1.
The paper is divided as follows. In the first section we recall part of the machinery about Nori fundamental gerbes and prove some preliminary results. In the second section we study essentially finite covers and their Nori fundamental gerbes, proving in particular Theorem III and Corollary I. In the third section we study towers of torsors and their Galois closures, proving Theorems I and II, while in the fourth section we study the S-fundamental gerbe and prove Theorem IV. Finally in the last section we collect some counter-examples.
Acknowledgement
We would like to thank Hélène Esnault and Angelo Vistoli for helpful conversations and suggestions received.
Notation and Preliminaries
1.1. Notation. By a fibered category over a scheme S we will always mean a category fibered in groupoids over the category Aff{S of affine schemes over S.
A cover of a fibered category X is a finite, flat and finitely presented morphism or, equivalently, an affine map f : Y ÝÑ X with the property that f˚O Y is locally free of finite rank. If X is defined over a field k a pointed cover over k pY, yq ÝÑ pX , xq is a cover f : Y ÝÑ X with x P X pkq and y P Y x pkq, where Y x denotes the fiber of f over x, which is a finite k-scheme; equivalently y P Ypkq with a given isomorphism f pyq » x.
Given a morphism of schemes U ÝÑ V and a functor F : Aff{U ÝÑ pSetsq, the Weil restriction of F along U ÝÑ V is the functor
W U pF q : Aff{V ÝÑ pSetsq, Z Þ ÝÑ HompZˆV U, F q .
Given any functor G : Aff{V ÝÑ pSetsq, we set W U pGq " W U pGˆV Uq.
Injectivity and surjectivity of morphisms of group schemes always mean the corresponding properties for fpqc sheaves. For affine group schemes over a field an injective morphism is a closed immersion and a surjective morphism is faithfully flat ( [Wat,Theorem 15.5] for surjectivity).
1.2. Preliminaries. We will now recall some results used in later sections.
Fix a base field k.
For properties of affine gerbes over a field (often improperly called just gerbes) and Tannakian categories used here the reader is referred to [TZ1,Appendix B].
Definition 1.1 ( [BV,Definition 5.1,Definition 5.3]). For a fibered category X over k, the Nori fundamental gerbe (respectively, Nori étale fundamental gerbe) of X {k is a profinite (respectively, proétale) gerbe Π over k together with a map X ÝÑ Π such that for all finite (respectively, finite and étale) stacks Γ over k the pullback functor Hom k pΠ, Γq ÝÑ Hom k pX , Γq is an equivalence of categories. Furthermore, if this gerbe exists, it is unique up to a unique isomorphism and in that case it will be denoted by Π N X {k (respectively, Π N,ét X {k q; sometimes {k will be dropped if it is clear from the context.
We call X inflexible if it is non-empty and all maps from it to a finite stack over k factors through a finite gerbe over k.
Remark 1.2. By [BV,p. 13,Theorem 5.7] X admits a Nori fundamental gerbe if and only it is inflexible; in this case, the Nori étale fundamental gerbe of X is the maximal proétale quotient of the Nori fundamental gerbe of X . If X is reduced, quasi-compact and quasi-separated, then X is inflexible if and only if k is algebraically closed in H 0 pO X q [TZ1,Theorem 4.4]. In particular if X is geometrically connected and geometrically reduced, then it is inflexible. Definition 1.3. If X is an inflexible fibered category over k with a rational point x P X pkq and Nori gerbe ψ : X ÝÑ Π N X {k , the Nori fundamental group scheme π N pX {k, xq of pX , xq over k is the sheaf of automorphisms of ψpxq P Π N X {k pkq. Again {k will often be dropped if it is clear from the context. Remark 1.4. The Nori fundamental group scheme π N pX , xq is a profinite group scheme and B π N pX , xq » Π N X (the trivial torsor is sent to ψpxq). The universal property of Π N X translates into the following: for all finite group schemes G over k the map
Hom k-groups pπ N pX , xq, Gq tpointed G-torsors pP, pq ÝÑ pX , xqu{ » π N pX , xq ÝÑ G X ÝÑ B π N pX , xq ÝÑ B G is bijective.
Definition 1.5 ( [BV,p. 21,Definition 7.7]). Let C be an additive and monoidal category. An object E P C is called finite if there exist polynomials f ‰ g P NrXs and an isomorphism f pEq » gpEq; the object E is called essentially finite if it is a kernel of a homomorphism of finite objects of C.
Let EFinpCq denote the full subcategory of C consisting of essentially finite objects.
Definition 1.6 ( [BV,p. 20,Definition 7.1]). A category X fibered in groupoid over a field k is pseudo-proper if it satisfies the following two conditions:
(1) there exists a quasi-compact scheme U together with a morphism U ÝÑ X which is representable, faithfully flat, quasi-compact, and quasi-separated, and (2) for all vector bundles E on X the k-vector space H 0 pX , Eq is finite dimensional.
Example 1.7 ( [BV,p. 20,Example 7.2]). Examples of pseudo-proper fiber categories are proper algebraic stacks, finite stacks and affine gerbes.
Remark 1.8. Let X be a pseudo-proper algebraic stack of finite type over k. If X is inflexible then H 0 pO X q " k (see [BV,Lemma 7.4]), while the converse holds if X is reduced (see Remark 1.2). Theorem 1.9 ( [BV,p. 22,Theorem 7.9, Corollary 7.10]). Let X be an inflexible pseudoproper fibered category over a field k. Then the pullback along X ÝÑ Π N X {k induces an equivalence of categories VectpΠ N X {k q ÝÑ EFinpVectpX qq. Let C be a Tannakian category. Then EFinpCq is the Tannakian subcategory of C of objects whose monodromy gerbe is finite.
Definition 1.10. A cover f : Y ÝÑ X is essentially finite if f˚O Y is an essentially finite vector bundle.
Definition 1.11. Let X be an inflexible and pseudo-proper fiber category over a field k. Given an object V of EFinpVectpX qq, the gerbe corresponding to the full Tannakian subcategory of EFinpVectpX qq generated by V will be called the monodromy gerbe of V . When f : Y ÝÑ X is an essentially finite cover, the monodromy gerbe of the cover is by definition the monodromy gerbe of f˚O Y . Definition 1.12 ( [BV,Definition 5.10]). A map X ÝÑ Γ from a fibered category over k to a finite gerbe over k is called Nori reduced over k if any faithful morphism Γ 1 α Ý ÝÑ Γ that fits in a factorization X ÝÑ Γ 1 α Ý ÝÑ Γ, where Γ 1 is a gerbe, is an isomorphism. A torsor P ÝÑ X under a finite group scheme G over k is called Nori reduced over k if the map X ÝÑ B G is Nori reduced over k.
Remark 1.13. If X is inflexible, then any map from X to a finite gerbe factors uniquely through a Nori reduced map (see [BV,Lemma 5.12]). Moreover Π N X can be seen as the projective limit of the Nori reduced maps X ÝÑ Γ (see [BV,Theorem 5.7] and its proof).
If X is an inflexible and pseudo-proper fibered category, and φ : X ÝÑ Γ is a map to a finite gerbe, then φ is Nori reduced if and only if the induced map Π N X ÝÑ Γ is a quotient; in this case VectpΓq ÝÑ EFinpVectpX qq is a sub Tannakian category. This is a direct consequence of Theorem 1.9 and the universal property of Π N X . Moreover φ˚O X » O Γ (see [BV,Lemma 7.11]).
One of the key ingredient in the paper is the following result.
Theorem 1.14 ([TZ2, Corollary I]). Let X be a pseudo-proper and inflexible algebraic stack of finite type over a field k of positive characteristic, and let f : Y ÝÑ X be a surjective cover. If V P VectpX q, and f˚V is free, then V is essentially finite in VectpX q.
Remark 1.15. If X is a pseudo-proper and inflexible algebraic stack of finite type over a field k (the characteristic is allowed to be 0), f : Y ÝÑ X is a surjective étale cover and V P VectpX q is trivialized by f , then it follows that V is essentially finite with étale monodromy gerbe in EFinpVectpX qq. Indeed, by standard theory of étale covers one can assume that f is an étale Galois cover. This case is exactly [TZ3,Lemma 1.4].
Lemma 1.16. Let T 2 a Ý ÝÑ T and T 1 b Ý ÝÑ T be two maps of affine group schemes over k, and let
R B T 1 B T 2 B T
be the corresponding 2-Cartesian diagram. Then the following two hold.
(1) The functor Ψ : BpT 2ˆT T 1 q ÝÑ R mapping a T 2ˆT T 1 -torsor to the associated T 2 and T 1 torsors is fully faithful and it is an equivalence if and only if the map T 1ˆT 2 ÝÑ T , pt 1 , t 2 q Þ Ñ bpt 1 qapt 2 q is an fpqc epimorphism (e.g. if a or b is surjective). In this case a quasi-inverse is obtained mapping an object of R given by torsors P 2 , P 1 , P under T 2 , T 1 , T respectively and equivariant maps P 1 ÝÑ P and P 2 ÝÑ P to the fiber product P 2ˆP P 1 . (2) If T 2 " Spec k, so that B T 2 " Spec k, and T 1 ÝÑ T is injective, then R " T {T 1 , where T {T 1 ÝÑ B T 1 is induced by the T 1 -torsor T ÝÑ T {T 1 . In particular, if T 1 is a finite subgroup of T , then B T 1 ÝÑ B T is an affine map.
Proof. The functor Ψ maps the trivial torsor to pT 2 , T 1 , idq P Rpkq. A direct computation shows that the sheaf of automorphisms of this object is exactly T 2ˆT T 1 (via Ψ). This means that Ψ is an equivalence onto the full-substack R 1 of R of objects locally isomorphic to pT 2 , T 1 , idq. Thus we have to understand when R 1 " R. All objects of R are locally isomorphic to an object of the form pT 2 , T 1 , cq P RpUq where U is an affine scheme and c P T pUq is thought of as multiplication on the left T ÝÑ T . An isomorphism pT 2 , T 1 , 1q ÝÑ pT 2 , T 1 , cq is given by t 2 P T 2 pUq and t 1 P T 1 pUq such that capt 2 q " bpt 1 q. Thus pT 2 , T 1 , cq is locally isomorphic to pT 2 , T 1 , 1q if and only if c is in the (fpqc) image of T 1ˆT 2 ÝÑ T . The last claim of p1q follows because if P is a T 2ˆT T 1 -torsor inducing torsors P 2 , P 1 , P under T 2 , T 1 , T respectively then the commutative diagram P P 1 P 2 P is automatically Cartesian: locally, after choosing a section of P , the above diagram is the one yielding T 2ˆT T 1 . Notice that the T 2ˆT T 1 -space given in the last part of p1q is not a torsor in general because it may not have sections locally. For p2q, R is the sheaf of T 1 -torsors P together with an equivariant map P ÝÑ T , which is represented by T {T 1 .
Remark 1.17. If f : Y ÝÑ X is a cover of algebraic stacks then f˚preserves vector bundles. Moreover since f˚is exact we have
H i pY, Eq " H i pX , f˚Eq for all i ě 0, E P QCohpYq .
In particular, if X is pseudo-proper over k, then Y is pseudo-proper over k. Moreover we will often use also the following property: if for all vector bundles E on X one has dim k H 1 pEq ă 8 then the same holds for vector bundles on Y.
Lemma 1.18 ([TZ2, Lemma 2.5]). Let X be an algebraic stack over a field k, of positive characteristic, such that dim k H 1 pX, Eq ă 8 for all vector bundles E on X . Let
G 0 ÝÑ G 1 ÝѨ¨¨ÝÑ G N´1 ÝÑ G N " 0
be a sequence of surjective maps of quasi-coherent sheaves on X such that KerpG l´1 ÝÑ G l q is free of finite rank for all 1 ď l ď N. Then there exists a surjective cover f : X 1 ÝÑ X such that f˚G l is free of finite rank for all l.
Remark 1.19 ([TZ1, Example 1.5, Corollary 1.7]). Let Γ be a finite stack or an affine gerbe over k. For all fiber categories Z over k, the pullback of vector bundles establishes an equivalence of categories between Hom k pZ, Γq and the groupoid of functors VectpΓq ÝÑ VectpZq which are k-linear, monoidal and preserves short exact sequences in the category of quasi-coherent sheaves.
Remark 1.20. Let us comment on the relationship between the essentially finite vector bundles on a fibered category X over k and the vector bundles pullback from a finite stack. When X is inflexible and pseudo-proper, these two notions agree as a consequence of [BV]. One of the key observation in [BV] is that if Γ is a finite stack over k and V P VectpΓq, then V is essentially finite. More precisely, there is a finite vector bundle E on Γ and an exact sequence in CohpΓq
0 ÝÑ V ÝÑ E 'a ÝÑ E 'b ÝÑ E 1 ÝÑ 0
for some a, b P N and E 1 P VectpΓq. In particular, if φ : X ÝÑ Γ is any map from a fibered category, then φ˚V is an essentially finite vector bundle on X . The proof of it is the same as that of [BV,Lemma 7.15] together with the following clarification. First we can assume that Γ is connected. Let ρ : T ÝÑ Γ be a surjective cover from a finite connected k-scheme T . The direct image E " ρ˚O T is finite by [BV,Lemma 7.15]. Since the cokernel of V ÝÑ ρ˚ρ˚V » E ' rk V is a vector bundle one can easily construct the above sequence.
Now let X be a fibered category and V P VectpX q. If V is essentially finite, one might argue that there is a homomorphism between two finite vector bundles q : E 1 ÝÑ E 2 whose kernel is V . This is actually misleading. Since the definition of essentially finite in the category VectpX q is intrinsic to this category, V has to be a kernel of q inside the category VectpX q. This does not imply that V coincides with the kernel K of q in QCohpX q. This equality holds if X is pseudo-proper and inflexible. If X has the resolution property, that is all quasi-coherent sheaves are quotient of sum of vector bundles (e.g. when X is a quasiprojective scheme or a smooth separated scheme), and if the kernel V exists in the category VectpX q, then V " K.
In order to avoid the above mentioned issue, if X is pseudo-proper but not inflexible it seems to us that the "correct" essentially finite vector bundles to use are vector bundles coming from a finite stack or at least that are kernel in QCohpX q of a map of finite vector bundles. Although this is not an intrinsic notion it would be a good working definition. This should also explain why Lemma 2.10 and Lemma 2.8 should be understood as results assuring that pushforward preserves essentially finite vector bundles. In any case in the present paper we consider essentially finite vector bundles only on pseudo-proper and inflexible fibered categories, but we maintain the notion of essentially finite in Definition 1.5.
There is a partial converse to the fact that vector bundles coming from finite stacks are essentially finite. If X is a fibered category over k with dim k H 0 pO X q ă 8 and V P VectpX q is a finite vector bundle, then there exist a map φ : X ÝÑ Φ to a finite stack and W P VectpΦq such that V » φ˚W . This is essentially proved in [BV,p. 19] (just after the proof of Lemma 7.11). We recall here the construction for the convenience of the reader. We can assume that V has rank r and take f ‰ g P Nrxs such that f pV q » gpV q. The group GL r acts on the scheme I " Isopf pk r q, gpk r qq " GL N with N " f prq " gprq. The isomorphism f pV q » gpV q gives a factorization of the vector bundle V : X ÝÑ B GL r through rI{ GL r s and we have Cartesian diagrams
Ω I X rΩ{ GL r s rI{ GL r s B GL r Spec H 0 pO X q I{GL r
Here we are using that I ÝÑ I{ GL r is a geometric quotient and I{ GL r is affine because GL r is geometrically reductive. Thus we must show that Φ " rΩ{ GL r s is a finite stack.
As the geometric fibers of I ÝÑ I{ GL r consist (topologically) of one orbit and H 0 pO X q is a finite k-algebra one sees that Φpkq has finitely many isomorphism classes. The action of GL r on I has finite stabilizers by [BV,Lemma 7.12] and hence it follows that the diagonal of Φ is quasi-finite. By [BV,Proposition 4.2] it follows that Φ is a finite stack.
Lemma 1.21. Let X be a quasi-compact and quasi-separated algebraic stack and u : X ÝÑ Γ a map to an affine gerbe. Then u˚: VectpΓq ÝÑ VectpX q is fully faithful if and only if
u # : O Γ ÝÑ u˚O X is an isomorphism. Proof. For V, V 1 P VectpΓq the composition Hom Γ pV, V 1 q uÝ ÝÑ Hom X pu˚V, u˚V 1 q » Hom Γ pV, u˚u˚V 1 q » Hom Γ pV, V 1 b u˚O X q is the map induced by u # : O Γ ÝÑ u˚O X .
In particular, if this map is an isomorphism then u˚: VectpΓq ÝÑ VectpX q is fully faithful. Conversely, setting V 1 " O Γ above, we have that the homomorphism
Hom Γ pV, O Γ q ÝÑ Hom Γ pV, u˚O X q
induced by u # is an isomorphism for all vector bundles V over Γ. Since u˚O X is a quasicoherent sheaf, it is a quotient of a sum of vector bundles [De,p. 132,Corollary 3.9]. Therefore, above isomorphism implies that u # is surjective. Since u : X ÝÑ Γ is faithfully flat we also have that u # injective.
Remark 1.22. If X is a pseudo-proper and inflexible algebraic stack over k and α : X ÝÑ Π N X is the structure map, then using Theorem 1.9 and Lemma 1.21 we conclude that α˚O X » O Π N X . The same holds for the Nori étale fundamental gerbe.
Essentially finite covers and their Nori gerbes
Let k be a base field. In this section we study the notion of an essentially finite cover, which generalizes the notion of torsor under a finite group scheme. Moreover we are going to prove Theorem III and Corollary I.
Recall that an essentially finite cover f : Y ÝÑ X of fibered categories is a cover such that f˚O Y is essentially finite as object of VectpX q (see Definition 1.10).
First observe that a torsor under a finite group scheme is an essentially finite cover. Indeed, let f : Y ÝÑ X be a torsor under a finite group scheme G over k corresponding to u : X ÝÑ B G. Then u˚pkrGsq » f˚O Y , where krGs is the regular representation. Applying [BV,Lemma 7.15] to Spec k ÝÑ B G we see that krGs is finite in VectpB Gq " Rep G and thus f˚O Y is a finite vector bundle.
Proposition 2.1. Let X be a pseudo-proper and inflexible fibered category over k. Then there are equivalences of categories "
Stacks finite over Π N X * " Essentially finite covers of X * " Finite k-schemes with an action of π N pX , xq * Φ Ψ
where Φ is the pullback along X ÝÑ Π N X and Ψ is the pullback along Spec k
x Ý ÝÑ X with x P X pkq. Furthermore, Ψ extends the correspondence between pointed Nori reduced torsor of X and quotient group schemes of π N pX , xq.
Proof. The functor Φ is the equivalence mapping ring objects of VectpΠ N X q to ring objects of EFinpVectpXqq. If x P X pkq, then Π N X " B π N pX , xq, and the ring objects of VectpΠ N X q " Rep π N pX , xq are precisely the finite k-algebras with an action of π N pX , xq. This easily implies that Ψ˝Φ is an equivalence. The last claim follows by construction.
Lemma 2.2. Let X be a pseudo-proper and inflexible algebraic stack of finite type over k and f : Y ÝÑ X an essentially finite cover; let u : X ÝÑ Γ be the monodromy gerbe of f˚O Y P EFinpVectpX qq. Then there exists a unique extension of f
Y ∆ X Γ v f u and also v˚O Y » O ∆ . If Y is inflexible then ∆ is a finite gerbe.
Proof. The multiplication map of f˚O Y and its unit map lie in VectpΓq Ď VectpX q and therefore determine a cover ∆ ÝÑ Γ
(1) extending f as claimed in the statement. Uniqueness of the extension follows from the fact that VectpΓq ÝÑ VectpX q is fully faithful. As u is Nori reduced we have that u˚O
X » O Γ . Since ∆ ÝÑ Γ is flat we also conclude that v˚O Y » O ∆ .
Assume now that Y is inflexible. By definition, Y ÝÑ ∆ factors through a finite gerbe ∆ 1 , which can be chosen as closed substack
∆ 1 Ď ∆. But v˚O Y » O ∆ which implies that ∆ " ∆ 1 as required.
Here are some technical lemmas which will be used in proving Theorem III.
Lemma 2.3. Consider a 2-Cartesian diagram Y Ψ X Φ v f π u
where X is a quasi-compact and quasi-separated algebraic stack, π is affine and faithfully flat, Φ is a stack such that u˚O X » O Φ and u is faithfully flat. Then v˚O Y » O Ψ , and the two functors u˚: VectpΦq ÝÑ VectpX q and v˚: VectpΨq ÝÑ VectpYq are fully faithful.
A vector bundle V P VectpX q lies in the essential image of u˚: VectpΦq ÝÑ VectpX q if and only if f˚V comes from a vector bundle on Ψ.
If π is a surjective cover, a vector bundle V P VectpYq lies in the essential image of v˚: VectpΨq ÝÑ VectpYq if and only if f˚V comes from a vector bundle on Φ.
Proof. By [BV,Lemma 7.17] and flat base change it follows that ‚ v˚O Y » O Ψ , ‚ u˚: VectpΦq ÝÑ VectpX q and v˚: VectpΨq ÝÑ VectpYq are fully faithful.
Denote by D and C the essential images of these u˚and v˚respectively.
Let V P VectpX q. We must show that V P D if and only if f˚V P C. The "only if" part is clear. Conversely, suppose that f˚V " v˚W with W P VectpΨq, and consider the canonical homomorphism u˚u˚V ÝÑ V ; pulling back by f one gets v˚v˚pf˚V q " v˚v˚pv˚W q ÝÑ v˚W " f˚V .
This homomorphism is an isomorphism because v˚O Y » O Ψ . As f is faithfully flat, one concludes that V » u˚u˚V , and as u is faithfully flat, it follows that u˚V is a vector bundle. Thus we have V P D.
Assume now that π is a surjective cover; consequently f is also a surjective cover. In particular, π˚and f˚send vector bundles to vector bundles. Given V P VectpYq we must show that V P C if and only if f˚V P D. The "only if" part is easy: if W P VectpΨq then f˚pv˚W q » u˚π˚W because π is affine.
For the converse, assume that f˚V P D, meaning f˚V comes from a vector bundle on Φ. Since u˚O X » O Φ it follows that u˚pf˚V q is a vector bundle and the canonical homomorphism u˚u˚pf˚V q ÝÑ pf˚V q is an isomorphism. This homomorphism can also be obtained by applying f˚to the canonical homomorphism v˚v˚V ÝÑ V . Since f is affine this means that the previous homomorphism is an isomorphism. To conclude that V P C it suffices to show that v˚V is a vector bundle. But v is faithfully flat and v˚pv˚V q is a vector bundle. Now by descent it follows that v˚V is also a vector bundle.
Remark 2.4. Consider a G-torsor f : Y Ñ X for an affine group scheme G, where X is a quasi-compact and quasi-separated algebraic stack, and the corresponding 2-Cartesian diagram
Y Spec k X B G v f u We see that H 0 pO Y q " k if and only if u˚O X » O B G .
In this case, applying Lemma 2.3, we conclude that u˚: VectpB Gq ÝÑ VectpX q is fully faithful with essential image the category of vector bundles V such that f˚V is trivial. No2,p. 264,Lemma 1]). Let X be a quasi-compact and quasi-separated algebraic algebraic stack and Z Y
Lemma 2.5 ([
X h g f a 2-commutative diagram, where f and g are torsors for affine group schemes G and H respectively. Suppose that H 0 pO Z q " k. Then there exists a homomorphism ϕ :
G ÝÑ H inducing h.
Moreover, h is faithfully flat if and only if ϕ : G ÝÑ H is faithfully flat, in which case h : Z ÝÑ Y is a torsor for the kernel of ϕ. If H{ Impϕq is affine (e.g. if H or G are finite) then this is also equivalent to the statement that H 0 pO Y q " k.
Proof. Consider the morphisms u : X ÝÑ B G and v : X ÝÑ B H corresponding to the torsors f and g respectively. Since H 0 pO Z q " k we have that u˚O X » O B G , and hence the pullback functor u˚: VectpB Gq ÝÑ VectpX q is fully faithful. The objects of the essential image of v˚: VectpB Hq ÝÑ VectpX q are trivialized by g and thus by f . From Remark 2.4 we obtain a factorization v˚: VectpB Hq ÝÑ VectpB Gq Ď VectpX q which, by Tannakian duality, is induced by a factorization v :
X u Ý ÝÑ B G γ Ý ÝÑ B H. Con- sider the 2-Cartesian diagrams Z Spec k Y U Spec k X B G B H w u a g g 1 h γ
We claim that there exists a dashed arrow w as above making the upper diagram 2-Cartesian. This would imply that the functor γ : B G ÝÑ B H is induced by a group homomorphism ϕ : G ÝÑ H. Set f 1 : Spec k ÝÑ B G. Consider the map of O X -algebras λ : g˚O Y ÝÑ f˚O Z . Applying u˚to λ and using a˚O Y » O U , H 0 pO Z q " k, we get that
g 1˚O U -g 1˚a˚O Y -u˚g˚O Y ÝÑ u˚f˚O Z -f 1˚O Spec pkq .
Applying Spec B G p´q on both sides we get the arrow w.
To prove that h is the pullback of w we just have to show that the adjunction maps u˚u˚g˚O Y ÝÑ g˚O Y and u˚u˚f˚O Z ÝÑ f˚O Z are isomorphisms. But the adjunction map for g˚O Y coincides with the following composition:
u˚u˚g˚O Y -u˚g 1˚a˚O Y -u˚g 1˚O U -g˚O Ywhich
is an isomorphism, and the same is true for f˚O Z .
The map h is faithfully flat if and only if w is faithfully flat. By Lemma 1.16 this is the case if and only if ϕ : G ÝÑ H is surjective, so that U " BpKerpϕqq. The condition that H 0 pO Y q " k is equivalent to the condition that v˚O X » O B H and, by Lemma 1.21, to the fully faithfulness of the functor γ˚: VectpB Hq ÝÑ VectpB Gq. This last condition is equivalent to the surjectivity of ϕ : G ÝÑ H when H{ Impϕq is affine (see [TZ1,Remark B.7]).
Pointed essentially finite covers admit a natural "Galois closure" by a torsor, as explained in the following proposition.
Proposition 2.6. Let X be a pseudo-proper and inflexible fibered category over k and f : Y ÝÑ X an essentially finite cover with a rational point y P Ypkq. Denote by Γ the monodromy gerbe of f˚O Y in EFinpVectpX qq and by ∆ ÝÑ Γ the cover in (1) that extends f . Then there are Cartesian diagrams
Spec k P f Spec k Y ∆ X Γ p λ y x f u
The map π : P f ÝÑ X is a pointed Nori reduced torsor for the finite group scheme G f " Aut Γ pupxqq, and Γ is the monodromy gerbe of π˚O P f . The map λ : P f ÝÑ Y is faithfully flat if and only if ∆ is a gerbe in which case it is a Nori reduced torsor for a subgroup scheme of G f .
In the general case λ : P f ÝÑ Y factors as η˝λ 1 , where λ 1 : P f ÝÑ Y 1 is faithfully flat and a Nori reduced torsor for a finite subgroup scheme of G f , and η : Y 1 ÝÑ Y is a closed immersion.
Finally the torsor π : P f ÝÑ X has the following universal property: for any pointed Nori-reduced torsor g : pT , tq ÝÑ pX , xq for a finite group scheme G, and any pointed faithfully flat X -morphism h : pT , tq Ñ pY, yq, there is a unique factorization h " λ˝j, where j : pT , tq ÝÑ pP f , pq is equivariant with respect to a surjective homomorphism G ÝÑ G f .
Proof. The existence of the diagram is clear. Since u : X ÝÑ Γ " B G f is Nori reduced so is the G f -torsor P f ÝÑ X . The claim about the monodromy gerbe of π˚O P f follows from the following fact: if G is a finite group scheme over k, the regular representation krGs generates Rep G because every finite G-representation is a sub object of some krGs n .
The morphism λ is faithfully flat if and only if Spec k ÝÑ ∆ is faithfully flat. This is the case if and only if ∆ is a gerbe. In such a situation we have
∆ " B H, where H is a subgroup of G f and P f λ Ý ÝÑ Y is an H-torsor. The map v : Y ÝÑ ∆ " B H is Nori reduced because v˚O Y » O ∆ .
The factorization λ " η˝λ 1 arises from the factorization Spec k ÝÑ ∆ 1 ÝÑ ∆, where ∆ 1 is a subgerbe of the finite stack ∆.
For the last statement, h induces an inclusion O Y Ă h˚O T , and as f is affine, an inclusion f˚O Y Ă g˚O T . If we denote by xg˚O T y (respectively, xπ˚O P f y) the full Tannakian subcategory of EFinpVectpX qq generated by the object g˚O T (respectively, π˚O P f ), one gets the following 2-Cartesian diagram:
VectpB G f q / / u˚ VectpB Gq v xπ˚O P f y x˚' ' P P P P P P P P P P P / / xg˚O T y xẘ w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ VectpSpec kq
where v : X ÝÑ B G corresponds to the torsor g : T ÝÑ X , the horizontal arrows are inclusions and the vertical arrows are equivalences. As the torsors are pointed above x, the functors x˚˝u˚and x˚˝v˚are equivalent to the forgetful functors. Therefore, the commutativity of this last diagram proves the existence of a surjective morphism ϕ : G ÝÑ G f such that u˚» v˚˝ϕ˚: VectpB G f q ÝÑ VectpX q. Thus implies that u " ϕ 1˝v , where ϕ 1 : B G ÝÑ B G f the morphism of gerbes induced by ϕ.
Lemma 2.7. Let X be an algebraic stack over a field k, A a finite and local k-algebra with residue field k and F P VectpXˆSpec Aq. Let X i Ý ÝÑ XˆSpec A f Ý ÝÑ X be the maps corresponding to k ÝÑ A ÝÑ k. Then there is a sequence of surjective maps of vector bundles f˚F " G N ÝÑ G N´1 ÝѨ¨¨ÝÑ G 1 ÝÑ G 0 " 0 such that KerpG l ÝÑ G l´1 q » i˚F for all 1 ď l ď N.
Proof. Consider a decomposition series of A-modules
A " A N ÝÑ A N´1 ÝѨ¨¨ÝÑ A 1 ÝÑ A 0 " 0 ;
the above maps are surjective with KerpA l ÝÑ A l´1 q » k as A-modules for all 1 ď l ď N. Let p : XˆSpec A ÝÑ Spec A be the projection; consider the functor
Ψ " f˚pF b p˚p´qq : ModA ÝÑ QCohpX q .
Since p is flat, F is a vector bundle, and as f is affine the functor Ψ is exact. Moreover ΨpAq " f˚F and
Ψpkq " f˚pF b p˚kq » f˚pF b i˚O X q » f˚i˚i˚F » i˚F .
Applying Ψ to the above sequence of A-modules we find the desired sequence.
Lemma 2.8. Let X be a pseudo-proper and inflexible algebraic stack of finite type over a field k of positive characteristic, such that dim k H 1 pX , Eq ă 8 for all vector bundles E on X . Let f : Y ÝÑ X be an essentially finite cover. Then for all maps φ : Y ÝÑ Φ to a finite stack over k, and for all W P VectpΦq, the vector bundle f˚φ˚W is essentially finite in VectpX q.
Proof. To avoid problems with different ranks, we first observe that the rank of W can be assumed to be constant, for instance, by considering the connected components of Φ. By Lemma 2.2 we have a Cartesian diagram
Y ∆ X Γ v u f
where Γ is a finite gerbe. We will prove that there exists a surjective cover s : X 1 ÝÑ X such that s˚f˚φ˚W is free on the connected components of X 1 ; this will be done by only assuming there is a Cartesian diagram as above with Γ a finite gerbe and f finite, but without requiring that X is inflexible, as this will be more useful. In the case when X is inflexible the above claim follows from Theorem 1.14.
Having weakened the hypothesis, if s 1 : X 1 ÝÑ X is a surjective cover we can always replace X by X 1 . Let L{k be a finite field extension with a map Spec L ÝÑ Y. The base change of Spec L ÝÑ Y ÝÑ ∆ ÝÑ Γ along X ÝÑ Γ is a surjective cover Q ÝÑ X . Replacing X by Q we can assume that X ÝÑ Γ factors as X ÝÑ Spec L ÝÑ Γ. This means that Y f Ý ÝÑ X is the projection XˆL A ÝÑ X , where A{L is a finite L-algebra. Since φ factors as Y ÝÑ Φˆk L ÝÑ Φ we can assume that L " k. Extending again k we can further assume that A is a product of local k-algebras with residue field k. Splitting Y according to the decomposition of A we can assume that A is also local. Let i : X ÝÑ Y " XˆA be the inclusion corresponding to A ÝÑ k.
Finally consider a surjective cover T ÝÑ Φ from a finite scheme and the 2-
Cartesian diagram Z T X Y Φ i φ
Replacing X by Z we can assume that φ˝i : X ÝÑ Φ factors through a finite scheme, which in particular implies that i˚φ˚W is free. Applying Lemma 2.7 to F " φ˚W we obtain a sequence of surjective homomorphisms of quasi-coherent sheaves on X f˚pφ˚W q " G N ÝÑ G N´1 ÝѨ¨¨ÝÑ G 1 ÝÑ G 0 " 0 with free kernels. The final cover trivializing f˚pφ˚W q exists due to Lemma 1.18.
Remark 2.9. Assume char k " 0. It is unclear whether Lemma 2.8 continues to hold and under which hypothesis on X it may hold. Clearly the finiteness of H 1 does not help since Lemma 1.18 only holds in positive characteristic. Let Y ÝÑ X be an essentially finite cover over a pseudo-proper and inflexible algebraic stack of finite type over k. Following the proof of Lemma 2.8, taking into account Remark 1.15 we have the possibility of extending the base field [BV, Proposition 6.1]. Consequently, considering L " k and noticing that the cover constructed Q ÝÑ X is a Nori reduced étale torsor (so that Q is inflexible by Theorem III we are going to prove) it follows that we can reduce the problem to the case where f is the projection Y " XˆA ÝÑ X and A is a finite and local k-algebra with residue field k. If i : X ÝÑ Y corresponds to A ÝÑ k, and F " φ˚W , then i˚F is essentially finite in X . Thus replacing again X by the total space of an étale Nori reduced torsor we can further assume that i˚F free. By Lemma 2.7 we obtain a sequence of surjective homomorphisms
f˚F " G N ÝÑ G N´1 ÝѨ¨¨ÝÑ G 1 ÝÑ G 0 " 0
with KerpG l ÝÑ G l´1 q free for all 1 ď l ď N. Missing Lemma 1.18 we can't go further. Moreover there are sequences like this with G N not essentially finite, for instance when X " B G a with V being the 2-dimensional representation given by G a " U 2 . The representation V is a nontrivial extension of k by itself and is not essentially finite because the profinite quotient of G a and therefore the Nori gerbe of B G a are trivial. Set ξ : V ÝÑ k ÝÑ V . Since ξ 2 " 0, ξ determines a sheaf F on B G aˆp krxs{px 2 qq whose pushforward to B G a is V . It is also possible to show that F is an invertible sheaf. On the other hand it seems unlikely that F is essentially finite. At least with some further computation one can exclude the possibility of being finite.
Lemma 2.10. Let X be a pseudo-proper and inflexible algebraic stack of finite type over k, and let f : Y ÝÑ X be an étale surjective cover. Then for all maps φ : Y ÝÑ Φ to a finite (respectively, finite and étale) stack over k and for all W P VectpΦq, the vector bundle f˚pφ˚W q is essentially finite (respectively, essentially finite with étale monodromy gerbe) in VectpX q. In particular, f is essentially finite.
Proof. There exists a Cartesian diagram
\ i Z Y Z X a r f b
where r : Z ÝÑ X is an étale surjective cover. Since a˚φ˚W also comes from the finite stack Φ, taking a finite atlas of Φ we can find a surjective cover λ : U Ñ Ů i Z which trivializes it. Denote by U i the inverse image of the i-th piece of Ů i Z under λ. Then r˚f˚φ˚W " b˚a˚φ˚W is trivialized by the surjective cover U 1ˆZˆ¨¨¨ˆZ U n ÝÑ Z. Thus, by Remark 1.15, f˚φ˚W is essentially finite and it has an étale monodromy gerbe if Φ is étale (so that λ can also be chosen to be étale).
Proof of Theorem III. By Lemma 2.2 there are 2-Cartesian diagrams
Y Π ∆ X Π N X Γ β α f π
where Γ is the monodromy gerbe of f˚O Y P EFinpVectpX qq. Notice that, since Π N X ÝÑ Γ is a quotient, the stack ∆ is a gerbe if and only if Π is a gerbe.
It follows from Lemma 2.2 that Π is a gerbe if Y is inflexible.
For the converse assume that Π is a gerbe. By Remark 1.22 and Lemma 2.3, the pullback functor β˚: VectpΠq ÝÑ VectpYq is fully faithful with essential image the full subcategory C of VectpYq of vector bundles V such that f˚V P EFinpVectpX qq. Since Π ÝÑ Π N X is faithful the gerbe Π is profinite, so we have C Ď EFinpVectpYqq. We will show that this is an equality and that Y is inflexible. This would immediately imply that Y ÝÑ Π is the Nori fundamental gerbe. Notice that if char k " 0 and Π is a gerbe then ∆ is a gerbe and therefore ∆ ÝÑ Γ and f : Y ÝÑ X are étale covers. Let φ : Y ÝÑ Φ be a map to a finite stack. Using Lemma 2.10 and Lemma 2.8 it follows that φ˚W P C for all W P VectpΦq. Thus the pullback by φ : Y ÝÑ Φ has a factorization VectpΦq ÝÑ VectpΠq » C Ď VectpYq .
By Remark 1.19 one gets a factorization Y ÝÑ Π ÝÑ Φ as required. This shows that Y ÝÑ Π is a Nori fundamental gerbe and, in particular, Y is inflexible. Since Y is pseudoproper, all essentially finite vector bundles on Y are pullbacks from some finite gerbe. Thus the above factorization also implies the equality C " EFinpVectpYqq.
Notice that if Y is inflexible then one always has H 0 pO Y q " k (see Remark 1.8).
The étale case. Assume that f is étale and that H 0 pO Y q " k. By Lemma 2.3 we have that H 0 pO ∆ q " k. In particular, ∆ is geometrically connected. Since ∆ ÝÑ Γ is étale, it follows that ∆ is also geometrically reduced. Using [BV,Proposition 4.3] we conclude that ∆ and therefore Π are gerbes. Thus Y is inflexible. By Lemma 2.10 we see that Γ is étale.
In particular there are 2-Cartesian diagrams Y Π N Y Π 1 ∆ X Π N X Π N,ét X Γ f
We have to show that Π 1 " Π N,ét Y . By Lemma 2.3 the pullback VectpΠ 1 q ÝÑ VectpYq is fully faithful and its essential image C consists of vector bundles V P VectpYq such that f˚V comes from Π N,ét X . One must show that the essential image D of the fully faithful map VectpΠ N,ét Y q ÝÑ VectpYq coincides with C. The vector bundles in D are the essentially finite vector bundles with étale monodromy gerbes. Since the map Π 1 ÝÑ Π N,ét X is faithful, it follows that Π 1 is proétale, so that C Ď D. The opposite inclusion instead follows from Lemma 2.10.
The torsor case. Assume that f is a torsor for a finite group scheme G. Since the regular representation krGs generates Rep pGq as k-Tannakian category, there are 2-Cartesian diagrams
Y β / / f Π π / / ∆ / / Spec pkq X α / / Π N X / / / / Γ / / B G
and the map Γ ÝÑ B G is faithful and, by [TZ1,Remark B.7], affine. In particular ∆ is a finite scheme, and by Lemma 2.3 we can conclude that ∆ " Spec pH 0 pO Y qq. Now the theorem follows because Y is inflexible if and only if ∆ is a gerbe.
Proof of Corollary I. The Nori gerbe of X is Π N X " B π N pX , xq, and X ÝÑ B π N pX , xq maps x to the trivial torsor. If pY, yq f Ý ÝÑ pX , xq is an essentially finite cover, then the extension Π ÝÑ Π N X defined in Theorem III is described by the 2-Cartesian diagrams
Spec k π N pX , xq r X Spec k Y x Y rY x {π N pX , xqs Spec k X B π N pX , xq 1 y x f that is Π " rY x {π N pX , xqs.
Again by Theorem III we have that Y is inflexible if and only if Π is a gerbe. On the other hand, the following three conditions are equivalent: ‚ Π is a gerbe, ‚ Spec k ÝÑ Π is faithfully flat, and ‚ the orbit map π N pX , xq ÝÑ Y x of y is faithfully flat. When these equivalent conditions hold, by Theorem III we have that Π " B π N pY, yq and Y x » π N pX , xq{π N pY, yq. The equivalence of categories in the statement follows easily from Proposition 2.1.
Let f : pY, yq ÝÑ pX , xq be an essentially finite cover with Y inflexible. If f is a torsor for a group G, using Lemma 1.16 the last diagram in Theorem III tells that π N pY, yq is normal in π N pX , xq with quotient G. Conversely, if π N pY, yq is normal in π N pX , xq with quotient G, then using again Lemma 1.16, B π N pY, yq ÝÑ B π N pX , xq and its base change f : Y ÝÑ X is torsor for G.
For the last claim, considering the above Cartesian diagrams we see that the following three are equivalent:
‚ f is étale, ‚ Π " B π N pY, yq ÝÑ π N pX , xq is étale, ‚ Y x » π N pX , xq{π N pY, yq ÝÑ Spec k is étale. This completes the proof.
Theorem 2.11. Let X be a pseudo-proper and inflexible fibered category over k and f : Y ÝÑ X be a cover. The following are equivalent:
(1) Y is inflexible, f is essentially finite and f˚O Y has étale monodromy gerbe in EFinpVectpX qq.
(2) f is étale and H 0 pO Y q " k.
Proof. The implication p2q ùñ p1q follows from Lemma 2.10 and Theorem III. For the converse, let ∆ ÝÑ Γ be as in Lemma 2.2. Since Y is inflexible it follows that ∆ is a gerbe.
Moreover by hypothesis Γ is étale. Thus ∆ is étale too because ∆ ÝÑ Γ is faithful. By base change it follows that f is étale.
Tower of torsors and their Galois closure
Let k be a base field and G and H be finite group schemes over k. In this section we introduce the notion of tower of torsors and Galois closure of a tower of torsors. At the end of the section Theorems I and II will be proved.
Definition 3.1. A pG, Hq-tower of torsors over a fibered category X over k is a sequence of map of fibered categories Z h Ý ÝÑ Y g Ý ÝÑ X where g is a G-torsor and h is an H-torsor. When G, H are clear from the context we will just talk about a tower of torsors. The pG, Hq-tower is called pointed over k if Z ÝÑ X (and therefore Y ÝÑ X ) is a pointed cover over k.
We define the stack BpG, Hq as the stack over Aff{k whose section over an affine scheme U is the groupoid of pG, Hq-tower of torsors over U. The isomorphisms of towers are defined in the obvious way.
Let us start with a preliminary remark:
Remark 3.2. A tower of torsors over a fibered category X is the same as a map X ÝÑ BpG, Hq. Moreover BpG, Hq has a universal tower given by
W Spec k Hom k pG, B Hq B H Spec k BpG, Hq B G u
where u is the restriction along 1 : Spec k ÝÑ G. The pullback of the above tower along a map X ÝÑ BpG, Hq yields exactly the tower encoded in the map X ÝÑ BpG, Hq.
Proposition 3.3. The stack BpG, Hq is algebraic, locally of finite type over k and has affine diagonal.
Proof. The stack BpG, Hq is algebraic and locally of finite type over the field k because Hom k pG, B Hq is so by [HR,Theorem 3] and the fact that there is a finite and flat map Hom k pG, B Hq ÝÑ BpG, Hq by Remark 3.2. Now let P i ÝÑ P 1 i ÝÑ T , i " 1, 2, be two towers ξ i P BpG, HqpT q for some affine scheme T , and set I " Iso BpG,Hq pξ 1 , ξ 2 q. We need to show that I is affine. We denote by p´q T the base change to T . Base changing along P 1 1ˆT P 1 2 ÝÑ T allows us to assume that P 1 i " G T . In particular, there is a map a : I ÝÑ G T " Aut G T pG T q and we want to show that it is affine. If W is the fiber of a along a map T g Ý ÝÑ G T , it is enough to show that W is affine. Let P 2 ÝÑ G T be the H-torsor base change of P 2 ÝÑ G T along the multiplication G T ÝÑ G T by g and set J " Iso H G T pP 1 , P 2 q ÝÑ G T . It is easy to see that W " W G T pJq, where W G T pJq is the Weil restriction of J along G T ÝÑ T . The scheme J is affine and finitely presented over G T . Since G T ÝÑ T is a cover using a presentation of J it is easy to write W as a closed subscheme of an affine space over T .
Lemma 3.4. Let X be a pseudo-proper and inflexible algebraic stack of finite type over k, and let Z h Ý ÝÑ Y g Ý ÝÑ X be a pG, Hq-tower of torsors. If char k ą 0 assume that either g is étale or dim k H 1 pX , Eq ă 8 for every vector bundle E on X . Then g˝h is an essentially finite cover.
Proof. Consider the morphism v : Y ÝÑ B H corresponding to the H-torsor h : Z ÝÑ Y. One knows that h˚O Z " v˚pkrHsq, where krHs denotes the regular representation. Thus we have pg˝hq˚O Z " g˚v˚pkrHsq. When char k ą 0, one concludes the proof by Lemma 2.8 and when char k " 0 by Lemma 2.10.
Lemma 3.5. Let P ÝÑ G ÝÑ Spec k be a rational point ξ of BpG, Hq with a point p P P pkq mapping to 1 P Gpkq, and let Q ξ " Aut BpG,Hq pξq. Then Q ξ is an affine group scheme of finite type over k, and there are exact sequences
0 ÝÑ W G pAut H G pP qq ÝÑ Q ξ α Ý ÝÑ G, 0 ÝÑ W 1 ÝÑ W G pAut H G pP qq β Ý ÝÑ H
where α forgets the automorphism of P , W G denotes the Weil restriction and β is the evaluation at 1 : Spec k ÝÑ G.
There is a fully faithful map B Q ξ ÝÑ BpG, Hq sending the trivial torsor to ξ and whose image consist of the towers locally isomorphic to ξ.
Proof. The scheme Q ξ is affine of finite type by Proposition 3.3. The first sequence is clear. The map β is well defined because the point p P P pkq gives an H-equivariant isomorphism between H and the base change of P ÝÑ G along the identity: the base change of Aut H G pP q ÝÑ G along the identity if Aut H k pHq " H. The last claim is standard.
More can be said in the case of the trivial tower.
Lemma 3.6. Let Q be the sheaf of automorphisms of the trivial tower GˆH ÝÑ G ÝÑ Spec k in BpG, Hq. Then Q is an affine group scheme of finite type,
Q " W G pHq ⋉ G
where W G pHq is the Weil restriction of the group scheme H over G along G ÝÑ Spec k with G acting on W G pHq via automorphisms of the base. Evaluation at 1 P G yields a map W G pHq ÝÑ H and, if W 1 is the kernel, then
W G pHq " W 1 ⋉ H .
The fully faithful map B Q ÝÑ BpG, Hq of Lemma 3.5 corresponds to the pointed tower of Nori reduced torsors B W 1 ÝÑ B W G pHq ÝÑ B Q .
If G or H is étale over k then Q is a finite group scheme.
Proof. Consider the maps α and β defined in Lemma 3.5. We have P " GˆH, so that Aut H G pP q " HˆG ÝÑ G . Moreover it is easy to see that both α and β are surjective. The map
G ÝÑ Q , g Þ ÝÑ pt gˆi d H , t g q ,
where t g is the multiplication by g, produces the first decomposition. The map HpT q ÝÑ HpGˆT q " W G pHqpT q produces the second decomposition.
The claim about the tower of B Q is an easy consequence of Lemma 1.16. Now assume that G or H are étale. We can also assume that k is algebraically closed, so that G or H are constant. If G is constant then W G pHq " H #G is a finite scheme and therefore Q is finite. Thus assume H constant. The map G red ÝÑ G, where p´q red denotes the reduction, is a nilpotent closed immersion and, since H is étale, it follows that the map W G pHq ÝÑ W G red pHq is an isomorphism. But G red " G ét and again we conclude that W G red pHq is finite.
Proposition 3.7. If G or H is étale then the map B Q ÝÑ BpH, Gq is an equivalence, where Q is the sheaf of automorphisms of the trivial tower GˆH ÝÑ G ÝÑ Spec k in BpG, Hq. In particular, BpG, Hq is a finite neutral gerbe over k.
Proof. In view of Lemma 3.6 it suffices to show that any tower is fpqc locally trivial.
Let P ÝÑ P 1 ÝÑ U be a tower over an affine scheme. Using base changing along P ÝÑ U we may assume this map has a section, so that, in particular, P 1 " UˆG. We can also assume that k is algebraically closed, so that G or H is constant. If G is constant, P ÝÑ UˆG is given by #G many H-torsors over U and, trivializing those torsors, one gets a trivialization of P ÝÑ UˆG. Now consider H to be étale. Since P ÝÑ UˆG is étale and UˆG red ÝÑ UˆG is a nilpotent closed immersion we conclude that P ÝÑ UˆG has a section if and only if its restriction to UˆG red is trivial. Thus we reduce to the known case where G " G red " G ét is étale.
We now move to the problem of finding a Galois closure for a given tower of torsors.
Definition 3.8. Let X be a fibered category, and let Z ÝÑ Y ÝÑ X be a pG, Hq-tower of torsors. A Galois closure for the pG, Hq-tower consists of the following data:
‚ a finite group scheme G with homomorphisms of group schemes α : G ÝÑ G and Kerpαq ÝÑ H, ‚ a G-torsor P ÝÑ X together with a factorization P ÝÑ Z such that P ÝÑ Y is G-equivariant and P ÝÑ Z is Kerpαq-equivariant. We say that the Galois closure is Nori reduced if the G-torsor P ÝÑ Z is Nori reduced.
Theorem I is now easy to deduce.
Proof of Theorem I. By Lemma 3.6 and Proposition 3.7 we have BpG, Hq " B Q, with universal tower B W 1 ÝÑ B W G pHq ÝÑ B Q. The Q-torsor Spec k ÝÑ B Q with splitting Spec k ÝÑ B W 1 gives a Galois closure of the universal tower. Now the proof is completed using universality.
The following lemma shows that a pointed tower Z ÝÑ Y ÝÑ X has a Galois closure.
Lemma 3.9. Let X be a pseudo-proper and inflexible algebraic stack of finite type over k and Z ÝÑ Y ÝÑ X a pointed tower of torsors. If char k ą 0, assume that either g is étale of dim k H 1 pX , Eq ă 8 for every vector bundle E on X . Then the following hold:
(1) f : Z ÝÑ X is essentially finite, (2) the tower ω : X ÝÑ BpG, Hq factors through the monodromy gerbe X ÝÑ Γ " B G f of f˚O Z in EFinpVectpX qq and (3) the G f -torsor P f ÝÑ X and the factorization P f ÝÑ Z introduced in Proposition 2.6 define a Nori reduced Galois closure for Z ÝÑ Y ÝÑ X . Moreover, the group scheme G f is a finite subgroup of the affine and finite type k-group scheme Aut BpG,Hq pωpxqq, where x is the given rational point of X .
Proof. The cover f : Z ÝÑ X is essentially finite by Lemma 3.4. We want to extends the given tower along X ÝÑ B G f . Using the notation of Proposition 2.6, according to Lemma 2.5 there exists a morphism of group schemes G f ÝÑ G inducing the morphism P f ÝÑ Y. From its construction it follows that the cover Z ÝÑ X extends to a cover ∆ ÝÑ B G f . We are in the following situation
P f Z Y X Spec k ∆ U B G f a b
Dashed arrows exist because all the covers of X involved correspond to locally free sheaves of algebras in the essential image of the fully faithful functor VectpB G f q ÝÑ VectpX q. We must equip b : ∆ ÝÑ U with a compatible structure of H-torsor. Notice that, by Lemma 2.3, a vector bundle on Y whose pullback to Z is free (and thus comes from a vector bundle on ∆) comes from a vector bundle on U. Moreover Lemma 2.3 also tells us that VectpUq ÝÑ VectpYq is fully faithful. This shows that we get a factorization
VectpB Hq ÝÑ VectpUq ÝÑ VectpYq and, by Tannakian duality, a factorization Y ÝÑ U ÝÑ B H. This determines an H-torsor ∆ 1 ÝÑ U extending Z ÝÑ Y. Since VectpUq ÝÑ VectpYq is fully faithful one concludes that ∆ 1 » ∆ over U.
Let K be the kernel of G f ÝÑ G. The map Spec k ÝÑ U factors through a closed immersion B K ÝÑ U and the composition B K ÝÑ U ÝÑ B H preserves trivial torsors, meaning it is induced by a homomorphism K ÝÑ H. It is easy to show that all the data constructed define a Galois closure of the original tower.
For the last claim, set ξ " ωpxq and Q " Aut BpG,Hq pξq. By hypothesis the tower ξ is pointed and therefore, by Lemma 3.5, the group scheme Q is affine and of finite type and there is a fully faithful map B Q ÝÑ BpG, Hq whose essential image is the full substack of BpG, Hq of towers which are fpqc locally isomorphic to ξ. Since ω factors as X ÝÑ B G f ÝÑ BpG, Hq, all objects in the image of B G f ÝÑ BpG, Hq are locally isomorphic to ξ. Thus the previous morphism factors through B G f ÝÑ B Q. This map preserves trivial torsors and it is therefore induced by a map G f q Ý ÝÑ Q. Let G be the image of q. The factorization
X ÝÑ B G f ÝÑ B G ÝÑ B Q Ď BpG, Hq
tells us that the tower over X extends to a tower over B G and therefore f˚O Z comes from a vector bundle on B G. But B G f is exactly the monodromy gerbe of f˚O Z in EFinpVectpX qq, which implies that G f " G Ď Q.
Proof of Theorem II. The closure we consider is the one in Lemma 3.9. Statement (1) follows by applying Theorem III and lemma 2.5 to both X and Y as bases. Statement (2) follows from the last statement of Lemma 3.9.
The S-fundamental gerbe of essentially finite covers
The aim of this section is to prove Theorem IV. We start by introducing the S-fundamental gerbe, which generalizes the notion of S-fundamental group (see [BPS], [La1], [La2]).
Definition 4.1. A vector bundle V on a fibered category X is called Nori semistable if for all smooth projective curves C over an algebraically closed field and all maps i : C ÝÑ X the pullback i˚V is semistable of degree 0.
We denote by NspX q the full subcategory of VectpX q of Nori semistable vector bundles. The S-fundamental gerbe over k of a fibered category X over k is an affine gerbe Π over k together with a map u : X ÝÑ Π whose pullback u˚: VectpΠq ÝÑ VectpX q is fully faithful with essential image NspX q. The S-fundamental gerbe is unique when it exists; it is denoted by Π S X {k when it exists. If X has an S-fundamental gerbe Ψ : X ÝÑ Π S X {k and x P X pkq, then the S-fundamental group π S pX {k, xq of pX , xq over k is the sheaf of automorphisms of ψpxq P Π S X {k . We will usually drop the {k when k is clear from the context. Remark 4.2. A fiber category X over k has an S-fundamental gerbe if and only if H 0 pO X q " k and NspX q is an abelian subcategory of QCohpX q. The "only if" is clear. For the converse observe that NspX q is a rigid monoidal category. If it is also an abelian subcategory of QCohpX q then NspX q is k-Tannakian and the map NspX q ÝÑ VectpX q sends exact sequences to exact sequences in QCohpX q. By Tannakian duality NspX q » VectpΠq, where Π is an affine gerbe, and by Remark 1.19, the inclusion NspX q Ď VectpX q is realized as the pullback of a map X ÝÑ Π, that is Π " Π S X . Remark 4.3. If X is a fibered category with an S-fundamental gerbe, then its profinite quotient is a Nori fundamental gerbe. In particular X is inflexible. Indeed since finite vector bundles on X are Nori semistable, we have that EFinpVectpX qq " EFinpNspX qq is also a k-Tannakian category by Theorem 1.9 and that it is an abelian subcategory of QCohpX q. From Remark 1.19 and Remark 1.20 it follows that the affine gerbe associated to EFinpNspX qq, which is the profinite quotient of Π S X , is a Nori fundamental gerbe for X . Remark 4.4. Let φ : X 1 ÝÑ X be a map of fibered categories and F P VectpX q. If F is Nori semistable then φ˚F is Nori semistable too. The converse holds if φ is representable (by a scheme), proper and surjective. Indeed one can assume that X is a proper, smooth, integral curve over an algebraically closed field k and must prove that F is semistable of degree 0. Considering a closed point in the generic fiber of φ and taking the normalization of its closure one can moreover assume that X 1 is also a proper, smooth, integral curve over k. In particular φ is a cover. In this case the result follows because the pullback of a subbundle destabilizing F actually destabilizes φ˚F .
Example 4.5. If X is a smooth, geometrically connected and geometrically projective scheme over k then X has an S-fundamental gerbe over k (see [BPS], [La1], [La2]).
Proposition 4.6. An affine gerbe Γ over k has an S-fundamental gerbe over k.
Proof. In view of Remark 4.2 we need to show that NspΓq is an abelian subcategory of QCohpΓq. So let 0 ÝÑ K ÝÑ F 1 ÝÑ F 2 ÝÑ Q ÝÑ 0 be an exact sequence in QCohpΓq with F 1 , F 2 P NspΓq. We must show that K, Q P NspΓq. Let i : C ÝÑ Γ be a map from a smooth projective curve over some algebraically closed field. Since Γ is a gerbe, both K and Q are vector bundles. Thus i˚K and i˚Q are respectively kernel and cokernel, in QCohpCq, of a homomorphism between Nori semistable vector bundles on C. Since NspCq is an abelian subcategory of QCohpCq, it follows that i˚K and i˚Q are in NspCq.
Lemma 4.7. Let X be a pseudo-proper and inflexible category over k, and let f : Y ÝÑ X be an essentially finite cover. Then
NspYq " tV P VectpYq | f˚V P NspX qu Proof. Given F P VectpYq we have to prove that F P NspYq ðñ f˚F P NspX q .
Nori semistability is tested on curves. Thus we can assume that X " C is a smooth, integral, projective curve over an algebraically closed field k: for "ùñ" we know that F P NspYq and we must prove that f˚F P NspCq; for "ðù" we have a section C i Ý ÝÑ Y, we know that f˚F P NspCq and we must prove that i˚F P NspCq. Here we are using the following: since X is pseudo-proper and inflexible, the pullback of f˚O X along the curve C ÝÑ X is still essentially finite; see Remark 1.20.
Let C ÝÑ Γ be the monodromy gerbe of f˚O Y in EFinpVectpCqq and ∆ ÝÑ Γ the extension given in Lemma 2.2. We have Γ " B G for some finite group scheme G so that the map C ÝÑ B G is given by a G-torsor Q ÝÑ C. Let D be the normalization of an irreducible component of Q surjecting onto C. It follows that g : D ÝÑ C is a surjective cover. Since a vector bundle on C is Nori semistable if and only if its pullback via g is so (see Remark 4.4), we can replace C by D, that is assume that C ÝÑ B G factors through Spec k. Since the cover Y ÝÑ C extends to B G we know that f : Y " CˆA ÝÑ C is the projection, where A{k is a finite k-algebra. Splitting Y according to a decomposition of A we can moreover assume A local. In the case "ðù" the inclusion i : C ÝÑ Y " CˆA is induced by A ÝÑ k. This map is also defined in the case "ùñ" and we denote it with the same symbol i.
We may replacing C by another test curve. Hence it is enough to prove that f˚F is semistable of degree 0 if and only if i˚F is so. From Lemma 2.7 we obtain a sequence of surjective homomorphisms of vector bundles f˚F " G N ÝÑ G N´1 ÝѨ¨¨ÝÑ G 1 ÝÑ G 0 " 0 such that KerpG l ÝÑ G l´1 q » i˚F . By induction we have detpf˚F q » pdet i˚F q N so that f˚F has degree 0 if and only if i˚F has degree 0. Again by induction we also see that all G l have the same slope as that of i˚F . In particular if f˚F is semistable so is i˚F Ď f˚F . The converse is deduced from the following fact: if 0 ÝÑ E 1 ÝÑ E ÝÑ E 2 ÝÑ 0 is an exact sequence of vector bundles on C with equal slope then E is semistable if E 1 and E 2 are semistable.
Proof of Theorem IV. The first claim follows from Remark 4.3. In particular by Theorem III we have 2-Cartesian diagrams
Y Π Π N Y X Π S X Π N X v u
Since Π S X ÝÑ Π N X is a quotient it follows that Π is a gerbe. As NspX q is a full subcategory of VectpX q, by Lemma 1.21, Lemma 2.3 and Lemma 4.7 we have that VectpΠq ÝÑ VectpYq is fully faithful with essential image NspYq. Thus we have Π " Π S Y . The last claim follows from the Cartesian diagram and Lemma 1.16.
Counterexamples
In this section we collect various examples. We start by showing that, under the assumption of Theorem III, the condition H 0 pO Y q " k in general does not imply that Y is inflexible, even when f˚O Y has étale monodromy gerbe (e.g. if char k " 0).
Example 5.1. Let k be an algebraically closed field. We show an example of an elliptic curve X over k with an essentially finite cover f : Y ÝÑ X of degree 2 such that H 0 pO Y q " k but Y is not inflexible. Clearly here Y is not reduced. The monodromy gerbe of f˚O Y is B µ 2 . Let X be an elliptic curve together with a non trivial line bundle L such that L b2 » O X . This is the data of a Nori reduced map u : X ÝÑ B µ 2 . Let A " krǫs " krxs{px 2 q equipped with the µ 2 " Spec pkrys{py 2´1 qq action
A ÝÑ A b pkrys{py 2´1 qq, ǫ Þ ÝÑ ǫ b y and set ∆ " rSpec A{µ 2 s. Define Y ÝÑ X with the 2-Cartesian diagram Y ∆ X B µ 2 v u f
Since u is Nori reduced we have u˚O X » O B µ 2 and v˚O Y » O ∆ by flat base change. In particular H 0 pO Y q " H 0 pO ∆ q " A µ 2 " k where the last equality follows from a direct computation. On the other hand, Y is not inflexible because v˚O Y » O ∆ ; this implies that v : Y ÝÑ ∆ does not factor through a gerbe.
We now show examples of towers without a Galois closure. The lemma below will be our method to exclude that a given tower has a Galois closure.
A torsor under a finite group scheme G over k is called minimal if it does not come from a torsor under a proper subgroup of G. For instance Nori reduced torsors are minimal.
Lemma 5.2. Let X be a fibered category, and let Z ÝÑ Y ÝÑ X be a pG, Hq-tower of torsors with Y ÝÑ X minimal. If the tower has a Galois closure with group G then the map X ÝÑ BpG, Hq factors through a map B G ÝÑ BpG, Hq.
Proof. Let P ÝÑ X and P ÝÑ Z be the Galois closure. We must shows that there is a tower over B G whose pullback along X ÝÑ B G is the original tower. Since Y ÝÑ X is minimal the map G ÝÑ G is surjective. By Lemma 1.16 the G-torsor over B G induced by This example shows that the condition on the cohomology groups H 1 in Theorem II is necessary.
G ÝÑ G is B H ÝÑ B G, where H is the kernel of G ÝÑ G. The pullback of the G-torsor B H ÝÑ B G along X ÝÑ B G is, by construction, Y ÝÑ X .
Example 5.3. Assume that G and H are not étale. We give an example of a pseudoproper, inflexible and smooth algebraic stacks X with a pointed pG, Hq-tower of Nori reduced torsors Z ÝÑ Y ÝÑ X without a Galois closure. In particular Z ÝÑ X cannot be an essentially finite cover by Lemma 3.9.
Using notations of Lemma 3.6 set X " B Q with the tower B W 1 ÝÑ B W G pHq ÝÑ B Q. If this tower has a Galois closure then by Lemma 5.2 the map B Q ÝÑ BpG, Hq factors through a finite gerbe. Since the map B Q ÝÑ BpG, Hq is fully faithful, this means that Q has to be a finite group scheme. Thus we must show that Q is not a finite group scheme. In particular we can assume k to be algebraically closed, so that G is a disjoint union of copies of the connected component G 0 . In particular W G pHq " pW G 0 pHqq #G ét , so that we can assume G local but not trivial. Moreover there is an injective map W G pH 0 q ÝÑ W G pHq, where H 0 is the connected component of H. Thus we can also assume that H is local but not trivial. If krǫs " krxs{px 2 q there is a map krǫs Ď krGs. Thus one get an injective map of group schemes W U pHq ÝÑ W G pHq where U " Spec krǫs. Similarly one can find a closed embedding U ÝÑ H, which yields a monomorphism of schemes W U pUq ÝÑ W U pHq. Moreover there is a monomorphism A 1 pBq ÝÑ W U pUqpBq " Hom B-algebras pBrǫs, Brǫsq, b Þ ÝÑ pǫ Þ Ñ bǫq
In conclusion we find a monomorphism φ : A 1 ÝÑ W G pHq. If W G pHq is finite, the image of φ must be connected, reduced, finite and with a rational point, that is Spec k, so that φ is not a monomorphism. Therefore, W G pHq is not finite.
The next example shows the importance of the pseudo-properness assumption on X in Theorem II.
Example 5.4. We give an example of a smooth, integral and affine scheme X over k with a pointed pG, Hq-tower of Nori reduced torsors without a Galois closure.
Assume that k is an algebraically closed field of characteristic 2 and let H " µ 2 and G be either µ 2 or α 2 . Recall that if B is a k-algebra and b P B˚then Brxs{px 2´b q has an action by µ 2 and an action by α 2 and it is a torsor over B for both actions. Since k is algebraically closed and α 2 and µ 2 are simple we have that Brxs{px 2´b q is Nori reduced if b is not a square in B and it is trivial otherwise.
Let K be the separable closure of the field of fractions kptq and consider K 1 " Krxs{px 2´t q and K 2 " Krx, ys{px 2´t , y 2´x q
The rings K 1 Ď K 2 are fields. Thus Spec K 2 ÝÑ Spec K 1 ÝÑ Spec K is a pG, Hq-tower of non trivial torsors which defines a map ξ : Spec K ÝÑ BpG, Hq. Consider a smooth map X ÝÑ BpG, Hq from a connected affine scheme and whose image contains the point ξ. We claim that the corresponding tower is pointed and Nori reduced. It is pointed because k is algebraically closed. It is Nori reduced because, since K is separably closed, the map ξ factors through a map Spec K ÝÑ X and the torsors in the tower ξ are not trivial. Let α : X ÝÑ BpG, Hq and β : Y ÝÑ BpG, Hq be smooth maps from connected affine schemes and assume that their images contain ξ and the trivial tower respectively. If the tower α does not have a Galois closure we have our counter-example. Otherwise, by 5.2, the map is (topologically) constant and ξ is an open point in BpG, Hq. If BpG, Hq is irreducible then ξ would be its generic point and, since the image of β is open, it would contain ξ. In this case if the tower β has a Galois closure it would follow that ξ is the trivial tower, that is x P K 1 become a square extending the field K, which is not true.
Thus it is enough to show that BpG, Hq is a smooth and connected algebraic stack. Consider the tower W ÝÑ Hom k pG, B Hq ÝÑ BpG, Hq described in 3.2. It is enough to show that W is a smooth connected algebraic stack. The objects of WpBq are Htorsors over GˆB with a trivialization over Spec B 1 Ý ÝÑ GˆB. We think µ 2 -torsors as line bundles with an isomorphism between its square and the trivial bundle. Set also Brǫs " Brxs{px 2 q " BrGs. Thus WpBq is the stack of triples pL, φ, ψq where L is a Brǫsline bundle, φ : L 2 ÝÑ Brǫs is an isomorphism and ψ : L{ǫL ÝÑ B is an isomorphism such that ψ b2 " φ modulo ǫ. If L " Brǫs then φ " a`bǫ P Brǫs˚and ψ " c P B˚with a " c 2 and, up to an isomorphism in WpBq, one can always assume c " 1. Thus the map A 1 ÝÑ W, mapping b P A 1 pBq to pBrǫs, 1`bǫ, 1q, is an epimorphism in the Zariski topology. In particular A 1 ÝÑ W is an fppf covering if A 1ˆW A 1 ÝÑ A 1 is. A direct computation shows that an isomorphism pBrǫs, 1`bǫ, 1q ÝÑ pBrǫs, 1`cǫ, 1q exists if and only if b " c and in this case it is the multiplication for 1`λǫ for λ P B. This means that A 1ˆW A 1 ÝÑ A 1 coincides with the projection pr 1 : A 1ˆk A 1 ÝÑ A 1 which is an fppf covering.
We conclude the section with an example showing that Corollary I (and thus Theorem III) as well as Lemma 2.8 fails without the finiteness assumption on the first cohomology group of vector bundles.
Example 5.5. We give an example of a Nori reduced torsor f : Y ÝÑ X between pseudoproper, inflexible and smooth algebraic stacks over k with y P Ypkq such that the following hold:
‚ f˚does not map essentially finite vector bundles to essentially finite vector bundles, ‚ π N pX , f pxqq is finite, ‚ f : Y ÝÑ X is the universal torsor, but ‚ π N pY, yq is not trivial. Let k be a field of characteristic 2, H " µ 2 , and let G be either µ 2 or α 2 . Consider the pG, Hq-tower of pointed Nori reduced torsors
Z " B W 1 h Ý ÝÑ Y " B W G pµ 2 q f Ý ÝÑ X " B Q
introduced in Example 5.3. Since Z ÝÑ X is not essentially finite it follows that h˚O Z is essentially finite while f˚ph˚O Z q is not. Consider y and x " f pyq as the trivial torsors. Since the Nori fundamental group of an affine gerbe B S pointed at the trivial torsor is the profinite quotient p S of S, we must show that p Q " G and { W G pµ 2 q " µ 2 . Given a k-algebra B set Brǫs " Brxs{px 2 q. We have W G pµ 2 qpBq " µ 2 pBrǫsq " ta`bǫ | a 2 " 1u from which it is easy to conclude that W G pµ 2 q " G aˆµ2 . Since any homomorphism from G a to a profinite group are trivial we conclude that { W G pµ 2 q " µ 2 . Similarly, denoting by K be the kernel of Q ÝÑ p Q, we have G a Ď K Ď W G pµ 2 q. Since µ 2 is simple we just have to check that G a is not normal in Q. Note that G " Spec krǫs; for g P GpBq, let ψ g : Brǫs ÝÑ Brǫs be the multiplication by g. Then,
g ‹ x ‹ g´1 " ψ g pxq for x P µ 2 pBrǫsq " W G pµ 2 qpBq where ‹ denotes the multiplication in Q. If G " µ 2 , so that g P B˚with g 2 " 1, an easy computation shows that ψ g pǫq " gǫ`pg´1q. Thus ψ g p1`ǫq " g`gǫ is in G a if and only if g " 1, which is not always the case.
If G " α 2 , so that g P B with g 2 " 0, then ψ g pǫq " ǫ`g. Thus ψ g p1`ǫq " 1`g`ǫ is in G a if and only if g " 0, which is not always the case.
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06108 NICE Cedex 2, France E-mail address: Marco. J A Laboratoire, Umr Dieudonné, Cnrs-Uns, [email protected] School of Mathematics. 7351Homi Bhabha RoadUniversité de Nice Sophia-Antipolis, Parc Valrose ; Tata Institute of Fundamental ResearchIndia E-mail address: [email protected] J.A.Dieudonné, UMR CNRS-UNS No 7351, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 NICE Cedex 2, France E-mail address: [email protected] School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India E-mail address: [email protected]
Université des Sciences et des Technologies de Lille 1, 59 655 Villeneuve d'Ascq, France E-mail address: emsalem@math. Laboratoire Paul Painlevé, U F R De Mathématiques, Laboratoire Paul Painlevé, U.F.R. de Mathématiques, Université des Sciences et des Technologies de Lille 1, 59 655 Villeneuve d'Ascq, France E-mail address: [email protected]
. F B Freie Universität Berlin, Mathematik, Informatik, Arnimallee. 314195Freie Universität Berlin, FB Mathematik und Informatik, Arnimallee 3, Zimmer 112A, 14195
. Deutschland Berlin, [email protected], Deutschland E-mail address: [email protected]
. F B Freie Universität Berlin, Mathematik, Informatik, Arnimallee. 314195Freie Universität Berlin, FB Mathematik und Informatik, Arnimallee 3, Zimmer 112A, 14195
. Deutschland E Berlin, demail address: l.zhang@fu-berlinBerlin, Deutschland E-mail address: [email protected]
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[] |
[
"Implications for New Physics in b → sµµ transitions after recent measurements by Belle and LHCb",
"Implications for New Physics in b → sµµ transitions after recent measurements by Belle and LHCb"
] |
[
"Kamila Kowalska *[email protected]†[email protected]‡[email protected] \nNational Centre for Nuclear Research\nPasteura 702-093WarsawPoland\n",
"Dinesh Kumar \nNational Centre for Nuclear Research\nPasteura 702-093WarsawPoland\n",
"Enrico Maria Sessolo \nNational Centre for Nuclear Research\nPasteura 702-093WarsawPoland\n"
] |
[
"National Centre for Nuclear Research\nPasteura 702-093WarsawPoland",
"National Centre for Nuclear Research\nPasteura 702-093WarsawPoland",
"National Centre for Nuclear Research\nPasteura 702-093WarsawPoland"
] |
[] |
We present a Bayesian analysis of the implications for new physics in semileptonic b → s transitions after including new measurements of R K at LHCb and new determinations of R K * and R K * + at Belle. We perform global fits with 2, 4, and 8 input Wilson coefficients, plus one CKM nuisance parameter to take into account uncertainties that are not factorizable. We infer the 68% and 95.4% credibility regions of the marginalized posterior probability density for all scenarios and perform comparisons of models in pairs by calculating the Bayes factor given a common data set. We then proceed to analyzing a few well-known BSM models that can provide a high energy framework for the EFT analysis. These include the exchange of a heavy Z boson in models with heavy vector-like fermions and a scalar field, and a model with scalar leptoquarks. We provide predictions for the BSM couplings and expected mass values.
|
10.1140/epjc/s10052-019-7330-2
|
[
"https://arxiv.org/pdf/1903.10932v1.pdf"
] | 85,518,335 |
1903.10932
|
7ef8067f0182536791a362f04eccfc24530f3c96
|
Implications for New Physics in b → sµµ transitions after recent measurements by Belle and LHCb
26 Mar 2019
Kamila Kowalska *[email protected]†[email protected]‡[email protected]
National Centre for Nuclear Research
Pasteura 702-093WarsawPoland
Dinesh Kumar
National Centre for Nuclear Research
Pasteura 702-093WarsawPoland
Enrico Maria Sessolo
National Centre for Nuclear Research
Pasteura 702-093WarsawPoland
Implications for New Physics in b → sµµ transitions after recent measurements by Belle and LHCb
26 Mar 20191
We present a Bayesian analysis of the implications for new physics in semileptonic b → s transitions after including new measurements of R K at LHCb and new determinations of R K * and R K * + at Belle. We perform global fits with 2, 4, and 8 input Wilson coefficients, plus one CKM nuisance parameter to take into account uncertainties that are not factorizable. We infer the 68% and 95.4% credibility regions of the marginalized posterior probability density for all scenarios and perform comparisons of models in pairs by calculating the Bayes factor given a common data set. We then proceed to analyzing a few well-known BSM models that can provide a high energy framework for the EFT analysis. These include the exchange of a heavy Z boson in models with heavy vector-like fermions and a scalar field, and a model with scalar leptoquarks. We provide predictions for the BSM couplings and expected mass values.
Introduction
The LHCb Collaboration has recently presented a new measurement of the observable R K , the ratio of the branching fraction of B-meson decay into a kaon and muons, over the decay to electrons, from the combined analyses of the Run 1 and partially of Run 2 data set [1], which reads R K = 0.846 +0.060+0.016 −0.054−0.014 .
This new measurement of R K is compatible with the Standard Model (SM) prediction at 2.5 σ significance. At the same time, the Belle Collaboration has presented new results for the observable R K * in B 0 -meson decays, as well as the first measurement of its counterpart R K * + in B + decays [2]. These new results, listed in Table 1, are consistent with the SM at 1 σ, mainly due to large experimental uncertainties. The rare decays of B mesons are known to provide fertile testing ground for physics beyond the Standard Model (BSM), as in the SM they are highly suppressed by the smallness of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and by helicity. While for many observables an anomalous determination does not necessarily imply the presence of new physics, as the QCD uncertainties can be sizable, ratios like R K and R K * provide fairly 61 −0.36 ± 0.10 1.40 +1.99 −0.68 ± 0.11 Table 1: R K * and R K * + results from Belle. clean probes, with parametric uncertainties that cancel out to high precision. Additionally, a deviation from the SM in these observables would imply a violation of lepton-flavor universality (LFUV), a purely BSM phenomenon. The rare decays of B-meson involving b → sll transitions have attracted a lot of attention for the search of New Physics (NP) beyond the SM over the last few years. The update of the LHCb results had been long awaited. The Run 1 determinations of R K and R K * [3,4] both featured a 2-3 σ deficit with respect to the SM and were part of a broader set of anomalous measurements in rare semileptonic B decays obtained at LHC and Belle [5,6,7,8,9,10], which involved b → s transitions and muons in the final state, and which in global statistical analyses [11,12,13,14,15,16] had been shown to favor strongly the presence of a few new physics operators over the SM, with a significance that according to some studies reached the ∼ 6 σ level.
q 2 in GeV 2 R K * R K *
In light of the recent measurements of R K and R K * by LHCb and Belle, in this paper we perform a global Bayesian analysis of the BSM effects appearing in the combination of the new data with previous results from the LHC and Belle. In this sense the paper follows the spirit of Run 1 global fits [11,12,13,14,15,16,17,18], and of earlier Bayesian analyses of radiative B-mesons decays [19,20]. The effect of incorporating new data has been also analyzed in Refs. [21,22,23,24,25].
We first consider model-independent fits to the global set of flavor observables, within the framework of the electroweak effective field theory (EFT), assuming as input the Wilson coefficients of four-fermion vector operators that were shown to be able to accommodate the observed data in Run 1 better than the SM. We perform global fits with 2, 4, and 8 independent input parameters, plus one nuisance parameter, the V cb element of the CKM matrix, which takes into account uncertainties that are not factorizable. For each scan we infer the 68.0% and 95.4% credibility regions of the marginalized posterior probability density function (pdf). We then compare models in pairs and provide the Bayes factor for a given set of data. Additionally, we make contact with frequentist analyses by providing for each scan the best-fit point, its pull from the SM, and an estimate of the goodness of fit.
We then proceed to analyzing a few well-known models that can provide a high energy framework for the EFT analysis. Depending on the case and on which set of observables is included in the global fit, BSM interpretations have involved the exchange of a heavy Z boson (see [26,27,28,29,30,31] for early studies) or of a leptoquark (early studies include [32,33,34,35,36,37,38,39]), both with non-universal and flavor-violating couplings to leptons and quarks. Moving one step further, such couplings can be generated assuming heavy vector-like (VL) fermions that can mix with the SM fermions [40,41]. We perform a few global scans for some of the Z models with VL fermions, and for some of the leptoquark models that have been shown to be consistent with the flavor anomalies.
The paper is organized as follows. In Sec. 2, we describe the methodology which is used in our analysis. In Sec. 3, we perform a model-independent global fit to a set of b → s observables. Predictions for several extensions of the SM that can accommodate the observed anomalies are presented in Sec. 4. Finally, we summarize our findings in Sec. 5.
Fit methodology
For each model described by a set of input parameters we map out the regions of the parameter space that are in best agreement with all relevant experimental constraints. To this end we use Bayesian statistics, whose main features we briefly summarize here.
In the Bayesian approach, for a theory described by some parameters m, experimental observables ξ(m) can be compared with data d and a pdf p(m|d), of the model parameters m, can be calculated through Bayes' Theorem. This reads
p(m|d) = p(d|ξ(m))π(m) p(d) ,(2)
where the likelihood p(d|ξ(m)) ≡ L(m) gives the probability density for obtaining d from a measurement of ξ given a specific value of m, and the prior π(m) parametrizes assumptions about the theory prior to performing the measurement. The evidence, p(d) ≡ Z, is a function of the data that depends globally on the model's parameter space. As long as one considers only one model the evidence is a normalization constant, but it serves as a comparative measure for different models or scenarios.
Bayes' theorem provides an efficient and natural procedure for drawing inferences on a subset of r variables in the parameter space, ψ i=1,..,r ⊂ m. One just needs to marginalize, or integrate, the posterior pdf over the remaining parameters,
p(ψ i=1,..,r |d) = p(m|d)d n−r m ,(3)
where n denotes the dimension of the full parameter space. In this work we will be interested in drawing the 68% (1 σ) and 95.4% (2 σ) marginalized 2-dimensional credible regions of the posterior pdf for each model under consideration. We will also compare in pairs different models fitting to the same data set, to determine which one is favored by the data distribution. We do this by computing the Bayes factor, defined as the ratio of evidences for two arbitrary models, M 1 and M 2 :
p(d) M 1 /p(d) M 2 .
We estimate the significance of Bayes factors according to Jeffrey's scale [42,43]. The central object in our statistical analysis is the likelihood function, constructed using the following prescription. Given the set m of input parameters, which can be, depending on the case, Wilson coefficients, particle masses, coupling constants, or other, the likelihood function is
L(m) = exp − 1 2 O th (m) − O exp T (C exp + C th ) −1 O th (m) − O exp ,(4)
where O th gives a vector of theoretical predictions of the observables of interest and O exp is the vector of the experimental measurements of those observables. We have taken into account the available experimental correlation which is encoded in the matrix C exp . The experimental correlation is available in angular observables for B → K * µµ [6] and B s → φµµ [7]. The theoretical correlation is given by the matrix C th , which is computed using flavio [44]. The theoretical uncertainties, including possible correlations, are estimated by the standard deviation of values of observables, calculated by taking N random values of all input parameters (form factors, bag parameters, decays constants, masses of the particles) distributed according to their probability distribution [44]. In this procedure, the precision depends on the number of random points and we take N = 2000 random points which gives a precision of about 2%. The V cb element of the CKM matrix is treated as a real nuisance parameter. We scan it together with the models' input parameters, following a Gaussian distribution around its central Particle Data Group (PDG) value [45], and adopting PDG uncertainties.
The statistical analysis performed in this study takes into account a large set of experimental measurements involving b → s transitions. The full list of all observables included in the likelihood function can be found in Appendix A. In the following we summarize them briefly:
• R K and R K * (Table 6) • B 0 → K * 0 µ + µ − : CP-averaged angular observables S i=3, 4,5,7,8,9 , fraction of longitudinal polarization of the K * 0 meson F L , and forward-backward asymmetry of the dimuon system A F B (alternatively, CP-averaged optimized P i=1,4,5 observables can be used), binned differential branching ratio dBR/dq 2 (Tables 7-10) (Tables 11-13) • B 0 s → φµ + µ − : time-and CP-averaged angular observables S i=3,4,7 , time-averaged fraction of longitudinal polarization F L , and differential branching ratio dBR/dq 2 (Tables [14][15] • Λ 0 b → Λµ + µ − : binned forward-backward asymmetries and binned differential branching ratios (Tables 16-17) • B + → K + µ + µ − : binned forward-backward asymmetry A F B (Table 18) • B 0 → K * 0 e + e − : CP-averaged angular observables P 4,5 , binned longitudinal polarization fraction F L and binned differential branching ratio dBR/dq 2 (Tables 19-20) • B + → K + e + e − : binned differential branching ratio (Table 21) • binned branching ratios BR(B 0 s → X s µ + µ − ) and BR(B 0 s → X s e + e − ) (Table 22) (Table 23).
• B + → K + µ + µ − , B + → K * + µ + µ − , B 0 → K 0 µ + µ − : binned differential branching ratios dBR/dq 2• time-integrated branching ratio BR(B 0 s → µ + µ − )
Effective field theory analysis
In the model-independent approach we adopt the weak EFT framework. The effective Hamiltonian for the b → sll transition can be written as:
H ef f = − 4G F √ 2 V tb V * ts i,l (C l i O l i + C l i O l i ) + H.c. ,(5)
where G F is the Fermi constant and V tb , V ts are elements of the CKM matrix. In Eq. (5) the short-distance physics is encoded in the Wilson coefficients C ( )l i after integrating out the heavy degrees of freedom, whereas the long-distance physics is described by the four-fermion dimension-six interaction operators O ( )l i , invariant under the SU(3) c ×U(1) em gauge group. In this study we will assume the presence of new physics in the following semi-leptonic operators:
O l 9 = (s L γ µ b L )(lγ µ l), O l 9 = (s R γ µ b R )(lγ µ l),(6)O l 10 = (s L γ µ b L )(lγ µ γ 5 l), O l 10 = (s R γ µ b R )(lγ µ γ 5 l),(7)
where the lepton l can be an electron or a muon. We restrict ourselves to the analysis of CP-conserving new physics effects, so that the Wilson coefficient are assumed to be real. We do not consider here new physics in scalar and pseudoscalar operators, O ( ) S and O ( ) P , as they are severely constrained by the B s → µ + µ − measurement [46,47]. Similarly, the electromagnetic dipole operator O ( ) 7 is tightly constrained by radiative decays [48]. The remaining dimension-six operators, chromomagnetic dipole operators, and four-quark operators, at the leading order can only contribute to the semi-leptonic decays through the mixing into semi-leptonic operators. All other BSM contributions enter at a higher order, so that it is safe to consider them as negligible for the purposes of this analysis.
The Wilson coefficients defined in Eq. (5) contain both the SM and new physics (NP) contributions, which can be written as, e.g.,
New physics contributions to the primed operators can in principle be significant and as such can be considered a smoking gun of BSM phenomena. We perform in this work 6 separate EFT scan fits to the data, each with a different combination of Wilson coefficients as input parameters. We summarize their input ranges and prior distributions in the first 6 lines of Table 2 (here and in what follows we drop the superscript "NP" from the Wilson coefficients' names, but we always take as parameters the new physics contribution). In each determination we also simultaneously scan over the CKM matrix element V cb , which we treat as a nuisance parameter for the Bayesian analysis. We have checked with several preliminary scans that the latter is the only CKM matrix element that can substantially interfere with NP effects due to its large uncertainty, as was pointed out, e.g., in Ref. [11].
This section is dedicated to the discussion of the EFT fits. In Sec. 4, we will instead investigate the implications of the new data for a few popular BSM models, involving a new gauge boson Z or a leptoquark, which are able to provide the favored parameter space regions for the Wilson coefficients. The input parameters, ranges and priors of two of those models, which will be described in the next section, occupy the remaining two lines of Table 2.
Discussion of results
All the observables are calculated with flavio [44], according to the procedure outlined in Sec. 2. In order to efficiently scan the multidimensional parameter space we used MultiNest Parameter Range Prior v.2.7 [49] and pyMultiNest [50] for sampling. The 68% (1 σ) and 95.4% (2 σ) credible regions of the marginalized posterior pdf are computed and plotted with the public tool Superplot [51]. In Fig. 1(a) we show the 1 σ and 2 σ credible regions of the posterior pdf for the model in the first row of Table 2, parametrized by C µ 9 , C µ 10 and the nuisance parameter. The posterior is compared to the one obtained with the previous data, pre LHCb Run 2. The overall value of R K , higher than in the previous determination, has the effect of bringing the 2 σ region closer to the axes origin. The modification of the posterior pdf is not large, but visible. One encounters a less substantial modification of the pdf in the case of the scan parametrized by C µ 9 , C µ 9 (second row of Table 2), for which the credible regions are shown in Fig. 1
C µ 9 , C µ 10 (−3, 3) Flat C µ 9 , C µ 9 (−3, 3) Flat C µ 9 , C µ 10 , C µ 9 , C µ 10 (−3, 3) Flat C µ 9 , C µ 10 , C e 9 , C e 10 (−3, 3) Flat C µ 9 , C µ 9 , C e 9 , C e 9 (−3, 3) Flat C µ 9 , C µ 9 , C µ 10 , C µ 10 (−3, 3) Flat C e 9 , C e 9 , C e 10 , C e 10 m Z /g X 500-5000 GeV Log M Q /λ Q , M D /λ D 0.1-500 TeV Log m Z /g X 500-5000 GeV Log M Q /λ Q , M E /λ E 0.1-500 TeV Log Nuisance parameter Central value, error (×10 −2 ) CKM matrix element V cb (4.22, 0.08) [45] Gaussian(b).
The overall effect appears to be a very slight detachment of the 2 σ region from the C µ 9 = 0 axis. A fit to the new data with 4 input NP parameters, C µ 9 , C µ 10 , C µ 9 , C µ 10 (third row of Table 2) shows that the introduction of C µ 10 as a free parameter leads to an interesting interference with C µ 9 . The latter can be made thus comfortably consistent with zero, at the price of introducing a substantial negative value of C µ 10 . This is presented in Fig. 2. In Fig. 2(a) we show a comparison between the marginalized pdf in the (C µ 9 , C µ 10 ) plane for the scan with 2 input NP parameters (first row of Table 2), and the one with 4 NP parameters (third row of Table 2). Larger negative values of C µ 9 are favored by the data with 4 parameters. In Fig. 2(b) we show an equivalent comparison in the (C µ 9 , C µ 9 ) plane, between the marginalized posterior pdf of the 2 parameter scan (second row of Table 2), versus the 4 parameter scan (third row of Table 2). An ample region of the parameter space with C µ 9 ≤ 0 appears, due to the introduction of the parameter C µ 10 . The correlation with C µ 9 is explicitly shown in Fig. 2(c).
−1.8 −1.2 −0.6 0.0 0.6 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10 −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10 Best-fit point 2σ region 1σ region (a) −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −0.6 0.0 0.6 1.2 1.8 C µ 9 −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −0.6 0.0 0.6 1.2 1.8 C µ 9
Best-fit point 2σ region 1σ region (b) Figure 1: (a) In green, the 1 σ (dark) and 2 σ (light) credible regions of the posterior pdf for the scan in the input parameter C µ 9 , C µ 10 (first row of Table 2), marginalized over the nuisance parameter. The red star marks the position of the best-fit point. The gray solid (dashed) line shows the 1 σ (2 σ) credible region of the pdf corresponding to the data pre-LHCb Run 2. The associated best-fit point is also shown in gray. (b) Same as (a) for the scan parametrized by C µ 9 , C µ 9 (second row of Table 2).
The 2-dimensional regions of the posterior pdf undergo less dramatic modifications if one scans a different set of 4 input parameters: C µ 9 , C µ 10 , C e 9 , C e 10 , see the fourth row of Table 2, or C µ 9 , C µ 9 , C e 9 , C e 9 , see the fifth row of Table 2. We perform the comparison between the relative marginalized 2-dimensional regions of these different models in Fig. 3(a) and Fig. 3(b). The details of the plots are explained in the caption. Note, that the Wilson coefficients of the electron sector, whose pdf's are presented in Fig. 3(c) and Fig. 3(d) remain consistent at 2 σ with zero, implying that the global data set can be easily explained by the presence of NP in the muon sector only.
We finally present in shades of blue the marginalized pdf of the 8-parameter scan introduced in the sixth row of Table 2 in the most relevant planes, (C µ 9 , C µ 10 ) in Fig. 4(a), and (C µ 9 , C µ 9 ) in Fig. 4(b). The posterior regions are compared to the 1 σ and 2 σ regions of the 2-parameters scans in the same plane, presented in shades of green. As one can see, the figures do not differ significantly from Fig. 2(a) and Fig. 2(b), as can be expected given the limited impact the Wilson coefficient of the electron sector bring to the fit.
We summarize the main characteristics of the six fits analyzed in this section in Table 3. The main Bayesian quantity is the negative logarithm of the evidence Z, featured in the second column. We use Jeffrey scale [42,43] to quickly assess the Bayes factor, defined in Sec. 2, which will point to which model is favored by the data. In general, models that are characterized by a smaller number of input parameters tend to fare significantly better than those with a larger input set, as the latter are penalized by volume effects. Unless, of course, these volume effects are counterbalanced by a significantly higher likelihood function.
−1.8 −1.2 −0.6 0.0 0.6 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10
Best-fit (4 pars.) 2σ region 1σ region
−1.8 −1.2 −0.6 0.0 0.6 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10 Best-fit (2 pars.) 2σ region 1σ region (a) −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −0.6 0.0 0.6 1.2 1.8 C µ 9
Best-fit (4 pars.) 2σ region 1σ region
−1.8 −1.2 −0.6 0.0 0.6 C µ 9 −0.6 0.0 0.6 1.2 1.8 C µ 9
Best-fit (2 pars.) 2σ region 1σ region
(b) −1.2 −0.6 0.0 0.6 1.2 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10
Best-fit (4 pars.) 2σ region 1σ region (c) Figure 2: (a) In green, the 1 σ (dark) and 2 σ (light) credible regions of the posterior pdf for the scan in the input parameter C µ 9 , C µ 10 (first row of Table 2), compared with the marginalized 2-dimensional regions in the same parameters for the scan with C µ 9 , C µ 10 , C µ 9 , C µ 10 all floating (third row of Table 2), which are shown in brown (1 σ) and orange (2 σ). The red stars mark the position of the best-fit points. (b) A similar comparison of the posterior pdf for the scan in C µ 9 , C µ 9 (shades of green) and the one with C µ 9 , C µ 10 , C µ 9 , C µ 10 all floating (orange/brown). (c) The marginalized 2-dimensional credible regions in C µ 9 , C µ 10 for the scan with C µ 9 , C µ 10 , C µ 9 , C µ 10 all floating (third row of Table 2).
In the specific cases considered here we see that, for example,
p(d) C µ 9 ,C µ 9 p(d) C µ 9 ,C µ 10 ≡ Z C µ 9 ,C µ 9 Z C µ 9 ,C µ 10 = 6.7 (Positive) (9) −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10
Best-fit (4 pars.) 2σ region 1σ region
−1.8 −1.2 −0.6 0.0 0.6 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10 Best-fit (2 pars.) 2σ region 1σ region (a) −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −0.6 0.0 0.6 1.2 1.8 C µ 9
Best-fit (4 pars.) 2σ region 1σ region
−1.8 −1.2 −0.6 0.0 0.6 C µ 9 −0.6 0.0 0.6 1.2 1.8 C µ 9
Best-fit (2 pars.) 2σ region 1σ region Best-fit (4 pars.) 2σ region 1σ region (d) Figure 3: (a) In green, the 1 σ (dark) and 2 σ (light) credible regions of the posterior pdf for the scan in the input parameter C µ 9 , C µ 10 (first row of Table 2), compared with the marginalized 2-dimensional regions in the same parameters for the scan with C µ 9 , C µ 10 , C e 9 , C e 10 all floating (fourth row of Table 2), which are shown in brown (1 σ) and orange (2 σ). The red stars mark the position of the best-fit points. (b) A similar comparison of the posterior pdf for the scan in C µ 9 , C µ 9 (second row of Table 2, in shades of green) and the one with C µ 9 , C µ 9 , C e 9 , C e 9 all floating (fifth row of Table 2, in orange/brown). (c) The marginalized 2dimensional credible regions in C e 9 , C e 10 for the scan with C µ 9 , C µ 10 , C e 9 , C e 10 all floating. (d) The marginalized 2-dimensional credible regions in C e 9 , C e 9 for the scan with C µ 9 , C µ 9 , C e 9 , C e 9 all floating.
(b)Z C µ 9 ,C µ 10 ,C µ 9 ,C µ 10 Z C µ 9 ,C µ 10 = 4.5 (Positive) (10) −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10 Best-fit (8 pars.) 2σ region 1σ region −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −1.2 −0.6 0.0 0.6 1.2 C µ 10 Best-fit (2 pars.) 2σ region 1σ region (a) −1.8 −1.2 −0.6 0.0 0.6 C µ 9 −0.6 0.0 0.6 1.2 1.8 C µ 9
Best-fit (8 pars.) 2σ region 1σ region
−1.8 −1.2 −0.6 0.0 0.6 C µ 9 −0.6 0.0 0.6 1.2 1.8 C µ 9
Best-fit (2 pars.) 2σ region 1σ region (b) Figure 4: (a) In green, the 1 σ (dark) and 2 σ (light) credible regions of the posterior pdf for the scan in the input parameter C µ 9 , C µ 10 (first row of Table 2), compared with the marginalized 2-dimensional regions in the same parameters for the scan with all 8 NP parameters floating (sixth row of Table 2), which are shown in shades of blue. The stars mark the position of the best-fit points. (b) A similar comparison of the posterior pdf for the scan in C µ 9 , C µ 9 (second row of Table 2, shades of green) and the one with 8 parameters floating (sixth row of Table 2, shades of blue). Z C µ 9 ,C µ 9 Z C µ 9 ,C µ 10 ,C µ 9 ,C µ 10 = 1.5 (Barely worth mentioning)
Z C µ 9 ,C µ 10 ,C µ 9 ,C µ 10 Z C µ 9 ,C µ 10 ,C e 9 ,C e 10 = 221 (Very strong)
Z C µ 9 ,C µ 10 ,C µ 9 ,C µ 10 Z C µ 9 ,C µ 9 ,C e 9 ,C e 9 = 109 , (Strong) (13) which brings us to the conclusion that the models favored by the data are the ones in the second and third rows of Table 2.
In the third and fourth column of Table 3 we make contact with frequentist approaches by presenting the pull of the best-fit point from the SM, the minimum chi-squared χ 2 TOT , the minimum chi-squared per degree of freedom, and the relative chi-squared of the muon observables, χ 2 µ , electron observables, χ 2 e , and LFUV observables, χ 2 R K and χ 2 R * K , of the six EFT scans analyzed here. We calculate the minimum chi-squared per degree of freedom very roughly, neglecting all correlations, as an indicative measure of the relative goodness of fit:
χ 2 TOT d.o.f. = χ 2 TOT num.constraints + 1 -(num.input + 1) ,(14)
where the ±1 is placed as a reminder of the nuisance parameter. The full list of constraints is collected in Appendix A. Finally, again to favor comparison with frequentist analyses, we show in Table 4 the numerical value of the Wilson coefficients and of the most important observables at the best-fit points.
Simple models for b → sll anomalies
In the previous section we determined the preferred 1σ and 2σ ranges for the NP Wilson coefficients relevant to explaining b → s anomalies. In the following we will discuss several simple BSM scenarios that are known to naturally lead to the desired EFT operator structure due to an exchange of a BSM boson with flavor-violating couplings to the SM quarks and leptons. Since we confirmed in our model-independent fit (see Table 3) that the presence of the electron Wilson coefficients does not improve the goodness of the fit, we will only consider models in which the BSM boson couples exclusively to muons.
Heavy Z
As a first example, we discuss the SM extended by an additional vector boson, commonly denoted as Z . The most generic Lagrangian, parametrizing non-universal couplings of Z to the b-s current and the muons reads
L ⊃ Z α ∆ sb Ls L γ α b L + ∆ sb Rs R γ α b R + H.c. + Z α (∆ µµ Lμ L γ α µ L + ∆ µµ Rμ R γ α µ R ) .(15)
The relevant Wilson coefficients are then given by
C µ 9,NP = −2 ∆ sb L ∆ µµ 9 V tb V * ts Λ v m Z 2 , C µ 9,NP = −2 ∆ sb R ∆ µµ 9 V tb V * ts Λ v m Z 2 , C µ 10,NP = −2 ∆ sb L ∆ µµ 10 V tb V * ts Λ v m Z 2 , C µ 10,NP = −2 ∆ sb R ∆ µµ 10 V tb V * ts Λ v m Z 2 ,(16)
where ∆ µµ 9 ≡ (∆ µµ R + ∆ µµ L )/2, ∆ µµ 10 ≡ (∆ µµ R − ∆ µµ L )/2, m Z is the mass of the Z boson, and
Λ v = π √ 2G F α em 1/2 ≈ 4.94 TeV,(17)
is the typical effective scale of the new physics. If the heavy Z is the gauge boson of a new U(1) X gauge group, its couplings to the gauge eigenstates must be family-universal, and an additional structure is required to generate ∆ sb L and ∆ sb R . Thus, in this work we also consider the impact of the new LHCb and Belle data on the masses and couplings of a few simplified but UV complete models.
In Eqs. (18)- (20) and in the following text we label SM fields by lower-case letters and BSM matter with capital ones. Besides Z , we also add to the SM a scalar singlet field S to spontaneously break the U(1) X symmetry and one family of VL quark pairs Q, Q and D, D to create the flavorchanging couplings ∆ bs L,R [55]:
S : (1, 1, 0, −1) ,(21)
Q : (3, 2, 1/6, −1) Q : (3, 2, −1/6, 1) ,
With the above choice of the U (1) X charges, the Lagrangian features new Yukawa couplings λ Q,i and λ D,i that mix the SM and BSM fields, as well as VL mass terms M Q,D :
L ⊃ (−λ Q,i SQ q i − λ D,i SD d R,i + H.c.) − M Q Q Q − M D D D ,(24)
where i = 1, 2, 3 label the SM generations. After spontaneously breaking U(1) X by a new scalar vev v S , and rotating the Lagrangian (24) to the mass basis, one obtains the flavor-generating couplings of the form
∆ sb L ≈ −g X λ Q,2 λ Q,3 v 2 S 2M 2 Q + λ 2 Q,2 + λ 2 Q,3 v 2 S ,(25)∆ sb R ≈ g X λ D,2 λ D,3 v 2 S 2M 2 D + λ 2 D,2 + λ 2 D,3 v 2 S ,(26)
and ∆ µµ 9 = g X ∆ µµ 10 = 0 ,
where g X is the gauge coupling of the U(1) X group. By recalling that v S ≡ m Z /g X one finally obtains
C µ 9,NP = 2Λ 2 v V tb V * ts λ Q,2 λ Q,3 2M 2 Q + λ 2 Q,2 + λ 2 Q,3 v 2 S , C µ 9,NP = − 2Λ 2 v V tb V * ts λ D,2 λ D,3 2M 2 D + λ 2 D,2 + λ 2 D,3 v 2 S ,(28)
while C µ 10,NP = C µ 10,NP = 0. Without loss of generality one can assume that the couplings of the second and third generations are unified, denoted as λ Q,D . Therefore, the model can be parametrized in terms of only 3 free parameters: m Z /g X , M Q /λ Q , and M D /λ D . Their scanning ranges imposed in the global fit are shown in Table 2. We scan m Z /g X for values larger than 500 GeV to evade the strong bound from neutrino trident production [56].
Recall, finally, that any scenario with a non-universal ∆ bs L,R coupling is subject to the strong constraints from B s mixing [11,57]: R BB ≤ 0.014. In our VL model the latter can be expressed in terms of the Wilson coeffiecints as [26]:
R BB = g 2 2 S 0 16π 2 −1 v 2 h v 2 S 4Λ 4 v C µ 9,NP 2 + C µ 9,NP 2 + 0.094 C µ 9,NP C µ 9,NP ,(29)
where S 0 ≈ 2.3 is a loop factor. For this reason, we impose an upper bound on the prior range of m Z /g X ≤ 5 TeV.
Model 2.
Another realization of the L µ − L τ model we consider is an extension of the SM characterized by one pair of VL quark doublets Q, Q , to generate the flavor-violating coupling of the Z in the quark sector, ∆ bs L , and one pair of VL U(1) X neutral leptons E, E [58,59], which have to be SU(2) singlets for reasons that will be clear below. Thus,
S : (1, 1, 0, −1) ,(30)
Q : (3, 2, 1/6, −1) Q : (3, 2, −1/6, 1) ,
E : (1, 1, 1, 0) E : (1, 1, −1, 0) .(31)
The Yukawa and VL mass part of the Lagrangian relative to the new leptons reads
L ⊃ −λ E,2 S * E µ R − λ E,3 SE τ R − Y E φ † l 1 E + H.c. − M E E E .(33)
Assuming that in Eq. (33) only second-generation Yukawa couplings are nonzero one gets
C µ 9,NP = Λ 2 v V tb V * ts λ Q,2 λ Q,3 2M 2 Q + λ 2 Q,2 + λ 2 Q,3 v 2 S 1 + 2M 2 E 2M 2 E + λ 2 E,2 v 2 S (34) C µ 10,NP = Λ 2 v V tb V * ts λ Q,2 λ Q,3 2M 2 Q + λ 2 Q,2 + λ 2 Q,3 v 2 S −1 + 2M 2 E 2M 2 E + λ 2 E,2 v 2 S .(35)
Note that if we had chosen a VL lepton SU(2) doublet instead of a singlet we would have obtained C µ 9,NP and C µ 10,NP of the same sign, which is disfavored by the data. Again we parametrize this model in terms of 3 free parameters: m Z /g X , M Q /λ Q , and M E /λ E,2 , where λ Q represent equal couplings to the second and third generation quarks. Their scanning ranges imposed in the global fit are shown in Table 2. We also apply the bounds from B s mixing.
We present in Fig. 5(a) the marginalized 2-dimensional posterior pdf in the (m Z /g X , M Q /λ Q ) plane for Model 2. The VL mass range is determined by the 2 σ range in C µ 9,NP and lies around a 20-30 TeV scale for a coupling λ Q of order unity. Within 2 σ probability, the m Z /g X mass is limited to values below 5 TeV, as a result of the B s mixing constraint, which directly depends on v 2 S . Note that the 2-dimensional posterior pdf in the (m Z /g X , M Q /λ Q ) plane for Model 1 looks very similar to Fig. 5(a), as a direct consequence of the bounds on C µ 9,NP and R BB . In both Model 1 and Model 2, the second VL mass is unbounded from above at the 2 σ level, as depicted in Fig. 5(b), where we show scatter plots of the χ 2 distributions of Model 1 (red points) and Model 2 (blue points) versus the VL mass rescaled by the Yukawa coupling. This is a consequence of the fact that C µ 9,NP in Model 1 and, especially C µ 10,NP in Model 2, are consistent with zero at the 2 σ level. On the other hand, the values of M D /λ D and M E /λ E,2 emerging at the best-fit point are very different for the two cases.
We summarize the main characteristics of the scans of Model 1 and Model 2 in Table 5. In analogy to what we observed in the EFT analysis, the Bayes factor favors Model 1 over Model 2 by 4.9:1, which reads "positive" evidence on Jeffrey's scale.
A model with U(1) X charged VL leptons
We finally consider an alternative to the L µ − L τ model, obtained if one charges the VL leptons under the U(1) X symmetry, and leaves the SM leptons uncharged, see, e.g., [60]. Model 3. We add to the SM the following particle content
S : (1, 1, 0, −1) ,(36)Z + VL −ln Z Pull χ 2 TOT χ 2 TOT d.o.f m Z /g X M Q /λ Q M VL /λ VL
The lepton Yukawa and VL mass part of the Lagrangian reads
L ⊃ (λ L,i S * L l i + H.c.) − M L L L .(39)
After rotating to the quark and lepton mass bases one finds
C µµ 9 = −C µµ 10 = 2Λ 2 v V tb V * ts λ Q,2 λ Q,3 2M 2 Q + λ 2 Q,2 + λ 2 Q,3 v 2 S λ 2 L,2 v 2 S 2M 2 L + λ 2 L,2 v 2 S .(40)
This model can be parametrized in terms of 3 parameters: m Z /g X , M Q /λ Q , and the hierarchical , defined such that M L /λ L,2 = M Q /λ Q .
The 2 σ region of the profile likelihood reads for this 1-dimensional model
C µ 9 = −C µ 10 ∈ (−0.68, −0.27) .(41)
We apply this bound to Model 3, together with the bound from B s mixing. The favored 2 σ regions are shown in Fig. 5(c), with different color code for different . The severe bound on R BB limits this model to steep hierarchies between VL quark and lepton masses.
Leptoquarks
A second, well known class of models that can easily generate NP contributions to the Wilson coefficients of the EFT are leptoquarks. Much work has been done in the past few years on the phenomenology of leptoquarks in relation to the flavor anomalies, for some early references, see, e.g., [32,33,34,35,36,37,38,39]. Here, we limit ourselves to the analysis of but one of these models, the scalar SU(2) triplet S 3 , which can generate a C µ 9 = −C µ 10 contribution at the tree level, like Model 3 of the previous subsection. We introduce the scalar leptoquark
S 3 : (3, 3, 1/3) .(42)
The Lagrangian acquires a Yukawa term
L ⊃ Y ij Q T i (iσ 2 )S 3 L j + H.c. ,(43)
where
S 3 = S 1/3 √ 2S 4/3 √ 2S −2/3 −S 1/3 ,(44)
and where the electric charge is indicated in the subscript. After rotating to the mass basis one writes
L ⊃ −Ŷ ij S 1/3 d L i ν L j + √ 2S 4/3 d L i e L j − Y ij S 1/3 u L i e L j − √ 2S −2/3 u L i ν L j +H.c. ,(45)
where indices ij span the SM generations, and Y T ij =Ŷ T ik (V † CKM ) kj . Note that couplings of the type QS † 3 Q, which are very dangerous for proton decay, are allowed in the SM, so that in UV complete models one should make sure they are forbidden by an additional symmetry.
By matching to the EFT one finds
C µµ 9 = −C µµ 10 = πv 2 V tb V * ts α emŶ bµŶ * sµ m 2 S 3 .(46)
where in this and what follows we have assumed that the mass of the triplet states is the same, m S 3 . The constraint from the 1-dimensional EFT at 2 σ is given in Eq. (41). This leads to
0.4 × 10 −3 m S 3 TeV 2 ≤Ŷ bµŶ * sµ ≤ 1.0 × 10 −3 m S 3 TeV 2 .(47)
If one starts withŶ bµŶ * sµ = 0, the CKM matrix generates additional Yukawa couplings,
Y uµ = (V * CKM ) 12Ŷ22 + (V * CKM ) 13Ŷ32 (48) Y cµ = (V * CKM ) 22Ŷ22 + (V * CKM ) 23Ŷ32 (49) Y tµ = (V * CKM ) 32Ŷ22 + (V * CKM ) 33Ŷ32 .(50)
so, that possible complementary constraints come from B → K ( * ) νν, b → cµ −ν decays, t → c µ + µ − , and t → c νν. The most dangerous constraint is possibly given by B → K ( * ) νν decay. The bound can be expressed as [61,33]
Br (SM) · 1 + 4πv 2 3αV tb V * ts m 2 S 3 C SM L (Ŷ sµŶ * bµ ) + 1 3|C SM L | 2 2πv 2 αV tb V * ts m 2 S 3 2 |Ŷ sµ | 2 |Ŷ bµ | 2 ≤ Br (90% CL) ,(51)
where Br (SM) = (4.0 ± 0.5) × 10 −6 , Br (90% CL) = 1.6 × 10 −5 , and C SM L = −6.38 ± 0.06. We get the limit
(Ŷ bµŶ * sµ ) 2.2 × 10 −2 m S 3 TeV 2 ,(52)
which does not constrain the parameter space emerging in Eq. (47).
Summary and conclusions
In this paper we have presented a global Bayesian analysis of the new physics effects on effective operators of semileptonic b → s transitions after the very recent measurement of R K at LHCb and new results for the observable R K * in B 0 -meson decays, as well as the first measurement of its counterpart R K * + in B + decays at Belle. We have performed global fits with 2, 4, and 8 Wilson coefficients as inputs, plus one CKM nuisance parameter to take into account uncertainties that are not factorizable with the NP effects. From the fits, we then inferred the 68% and 95.4% credibility regions of the marginalized posterior probability density for all models.
The new measurement of R K is closer in central value to the SM prediction than the Run 1 determination, but the much improved precision of the new data keeps it at 2.5σ from the SM. As a result the high-probability region of the posterior pdf in the NP Wilson coefficients C µ 9 and C µ 10 shifts slightly towards the zero value with respect to the scans with the Run 1 determination of R K , but the overall pull remains quite large, at the level of 4 − 5 σ, quite independently of the number of scanned input coefficients.
We have confirmed previous observations that the impact of the Wilson coefficients of the electron sector on the data is negligible with respect to the muon sector. Moreover, a pair-like comparison of the Bayes factors of different models has allowed us to determine that the two scans characterized by the inputs C µ 9 , C µ 9 , and C µ 9 , C µ 10 , C µ 9 , C µ 10 are strongly or even very strongly favored by the data, with respect to all other combinations, even if frequentist measures of the goodness of fit like the minimum chi-squared per degree of freedom show a less pronounced preference.
Finally, we have also analyzed a few well-known BSM models that can provide a high energy framework for the EFT analysis. These include the exchange of a heavy Z gauge boson in models with heavy vector-like fermions and a scalar field whose vev breaks spontaneously the new symmetry, and a model with scalar leptoquarks. But despite the introduction of new constraints that are specific to the model-dependent analysis, when it comes to determining which hypotheses are strongly favored by the data, the Bayes factors mirror the results of the EFT fits, i.e., models that can generate the C µ 9 and C µ 9 Wilson coefficients after integrating out heavy degrees of freedom are preferred with respect to other combinations.
A List of observables used in the global analysis
In this appendix we provide a tabularized list of all the observables included in our global analysis as components of the likelihood function. For each of them we show the experimental measurement and the SM prediction derived with flavio, which includes the theoretical error obtained by calculating the spread of values for a given observable, when a set of input parameters (form factors, bag parameters, decays constants, masses of the particles) were randomly generated for 2000 times. In the last column we also present a deviation of the measurement from the SM prediction that quantifies the significance of a potential anomaly.
LFUV observables
Observable SM prediction Experimental value Deviation Table 9: Angular observables of B 0 → K * 0 µ + µ − included in the global fit. [15,19] 0.00 ± 0.00 0.13 +0.11 −0.11 ± 0.01 1.2σ [15,20] 0.71 ± 0.08 1.20 +0.09 −0.09 ± 0.25 1.7σ [15,20] 0.14 ± 0.01 0.25 ± 0.04 ± 0.01 2.7σ A F B [1,6] 0.00 ± 0.00 −0.14 +0.07 −0.06 ± 0.03 1.8σ
LHCb (B + → K + l + l − ) [62] R [B 0 → K * 0 µ + µ − differential
F H [2,4.3] 0.02 ± 0.00 0.85 +0. 34 −0.31 ± 0.14 2.2σ 2.5 ± 0.4 3.1 +0.9 −0.9 ± 0.2 0.6σ
, again, l = e, µ. In the SM, the Wilson coefficients at the scale µ = 4.2 GeV are lepton-flavor-universal and read:
of the L µ − L τ model Model 1. A U(1) X model that has proven to be quite popular is the traditional X = L µ − L τ model[52,53,54,28], in which the SM leptons carry an additional charge and are characterized by the following SU(3)×SU(2) L ×U(1) Y ×U(1) X quantum numbers:l 1 : (1, 2, −1/2, 0) e R : (1, 1, 1, 0) (18) l 2 : (1, 2, −1/2, 1) µ R : (1, 1, 1, −1) (19) l 3 : (1, 2, −1/2, −1) τ R : (1, 1, 1, 1) .
Figure 5 :
5(a) The marginalized 2-dimensional posterior pdf in the (m Z /g X , M Q /λ Q ) plane in Model 2 (the pdf is very similar in Model 1). (b) A scatter plot of the χ 2 distribution as a function of VL mass rescaled by the Yukawa coupling for Model 1 (red, M D /λ D on the x-axis) and Model 2 (blue, M E,2 /λ E,2 on the x-axis). (c) The 2 σ regions of the profile likelihood in the (m Z /g X , M Q /λ Q ) plane of Model 3 for different values of (M L /λ L,2 = M Q /λ Q ). The gray area is excluded by the upper bound on R BB from B s mixing.
ACKNOWLEDGMENTS KK and DK are supported in part by the National Science Centre (Poland) under the research Grant No. 2017/26/E/ST2/00470. EMS is supported in part by the National Science Centre (Poland) under the research Grant No. 2017/26/D/ST2/00490. The use of the CIS computer cluster at the National Centre for Nuclear Research in Warsaw is gratefully acknowledged.
Table 2 :
2Input parameters, their ranges, and prior distributions for the 8 scans we run in this study.
Input parameters −ln Z Pull χ 2TOT
χ 2
TOT
d.o.f
χ 2
µ
χ 2
e
χ 2
RK
χ 2
R K *
SM
−
−
176.4
1.31
151.1 6.2
7.0
12.0
−
−
177.7
1.26
151.1 6.6
6.2
13.6
C µ
9 = −C µ
10
70.5
5.6 σ 141.6
1.06
132.1 6.2
0.6
2.6
73.4
5.1 σ 147.7
1.06
133.2 6.6
0.8
7.1
C µ
9 , C µ
10
71.1
5.5 σ 139.2
1.05
129.7 6.2
0.3
3.2
74.1
5.0 σ 145.6
1.06
130.2 6.6
1.1
7.7
C µ
9 , C µ
9
71.3
5.5 σ 139.4
1.06
126.9 6.2
2.0
4.3
72.2
5.4 σ 141.2
1.02
126.6 6.6
0.6
7.3
C µ
9 , C µ
10 , C µ
9 , C µ
10
71.2
5.6 σ 132.4
1.02
123.7 6.2
0.2
2.3
72.6
5.3 σ 136.7
1.00
123.1 6.6
0.1
6.8
C µ
9 , C µ
10 , C e
9 , C e
10
75.4
5.1 σ 138.6
1.07
129.7 6.3
0.4
2.2
78.0
4.8 σ 142.6
1.05
129.7 6.9
0.2
5.8
C µ
9 , C e
9 , C µ
9 , C e
9
75.1
5.3 σ 135.5
1.04
125.9 6.8
0.3
2.5
77.3
5.1 σ 139.8
1.03
125.9 6.6
0.4
6.7
Table 3 :
3Evidence, pull to the SM, and chi-squared statistics for the best-fit points of the considered scenarios. Gray highlighted rows correspond to the new data, while the white ones show the previous determinations.
Table 4 :
4Wilson coefficients at the best-fit points, as well as the values there of R K and R K * . Gray highlighted rows correspond to the new data, while the white ones show the previous determinations.
Table 5 :
5Evidence, pull to the SM, chi-squared statistics, and input parameters at the best-fit points of Model 1 and Model 2.
Table 6 :
6LFUV observables included in the global fit.B 0 → K * 0 µ + µ − angular observables
Observable
SM prediction
Experimental value
Pull
LHCb [6]
F L
[1.1,2.5]
0.761 ± 0.044
0.666 +0.083
−0.077 ± 0.022
1.0σ
F L
[2.5,4]
0.796 ± 0.036
0.876 +0.109
−0.097 ± 0.017
0.7σ
F L
[4,6]
0.711 ± 0.049
0.611 +0.052
−0.053 ± 0.017
1.4σ
F L
[15,19]
0.340 ± 0.022
0.344 +0.028
−0.030 ± 0.008
0.1σ
A F B
[1.1,2.5]
−0.137 ± 0.030 −0.191 +0.068
−0.080 ± 0.012 0.6σ
A F B
[2.5,4]
−0.017 ± 0.032 −0.118 +0.082
−0.090 ± 0.007 1.1σ
A F B
[4,6]
0.123 ± 0.042
0.025 +0.051
−0.052 ± 0.004
1.5σ
A F B
[15,19]
0.368 ± 0.021
0.355 +0.027
−0.027 ± 0.009
0.4σ
S 3
[1.1,2.5]
0.002 ± 0.005
−0.077 +0.087
−0.105 ± 0.005 0.8σ
S 3
[2.5,4]
−0.011 ± 0.004
0.035 +0.098
−0.089 ± 0.007
0.5σ
S 3
[4,6]
−0.025 ± 0.009
0.035 +0.069
−0.068 ± 0.007
0.9σ
S 3
[15,19]
−0.205 ± 0.016 −0.163 +0.033
−0.033 ± 0.009 1.1σ
S 4
[1.1,2.5]
−0.026 ± 0.017 −0.077 +0.111
−0.113 ± 0.005 0.4σ
S 4
[2.5,4]
−0.152 ± 0.022 −0.234 +0.127
−0.144 ± 0.006 0.6σ
S 4
[4,6]
−0.224 ± 0.020 −0.219 +0.086
−0.084 ± 0.008 0.1σ
S 4
[15,19]
−0.300 ± 0.006 −0.284 +0.038
−0.041 ± 0.007 0.4σ
S 5
[1.1,2.5]
0.053 ± 0.035
0.137 +0.099
−0.094 ± 0.009
0.8σ
S 5
[2.5,4]
−0.194 ± 0.039 −0.022 +0.110
−0.103 ± 0.008 1.5σ
S 5
[4,6]
−0.337 ± 0.035 −0.146 +0.077
−0.078 ± 0.011 2.2σ
S 5
[15,19]
−0.281 ± 0.017 −0.325 +0.036
−0.037 ± 0.009 1.1σ
S 7
[1.1,2.5]
−0.027 ± 0.030 −0.219 +0.094
−0.104 ± 0.004 1.8σ
S 7
[2.5,4]
−0.020 ± 0.041
0.068 +0.120
−0.112 ± 0.005
0.7σ
S 7
[4,6]
−0.013 ± 0.051 −0.016 +0.081
−0.080 ± 0.004 0.0σ
S 7
[15,19]
−0.001 ± 0.001
0.048 +0.043
−0.043 ± 0.006
1.1σ
S 8
[1.1,2.5]
−0.007 ± 0.013 −0.098 +0.108
−0.123 ± 0.005 0.7σ
S 8
[2.5,4]
−0.006 ± 0.014
0.030 +0.129
−0.131 ± 0.006
0.3σ
S 8
[4,6]
−0.005 ± 0.015
0.167 +0.094
−0.091 ± 0.004
1.8σ
S 8
[15,19]
0.000 ± 0.000
0.028 +0.044
−0.045 ± 0.003
0.6σ
S 9
[1.1,2.5]
−0.001 ± 0.005 −0.119 +0.087
−0.104 ± 0.005 1.1σ
S 9
[2.5,4]
−0.001 ± 0.002 −0.092 +0.105
−0.125 ± 0.007 0.7σ
S 9
[4,6]
−0.001 ± 0.005 −0.032 +0.071
−0.071 ± 0.004 0.4σ
S 9
[15,19]
0.000 ± 0.000
−0.053 +0.039
−0.039 ± 0.002 1.4σ
Belle [8]
P 4
[0.1,4]
−0.03 ± 0.03
−0.38 +0.50
−0.48 ± 0.12
0.7σ
P 4
[14.18,19]
−0.63 ± 0.01
−0.10 +0.39
−0.39 ± 0.07
1.3σ
P 5
[0.1,4]
0.15 ± 0.06
0.42 +0.39
−0.39 ± 0.14
0.6σ
P 5
[14.18,19]
−0.63 ± 0.03
−0.13 +0.39
−0.35 ± 0.06
1.3σ
Table 7 :
7Angular observables of B 0 → K * 0 µ + µ − included in the global fit. B 0 → K * 0 µ + µ − angular observablesObservable SM prediction Experimental value PullATLAS [9]
F L
[0.04,2]
0.39 ± 0.06
0.44 ± 0.08 ± 0.07
0.4σ
F L
[2,4]
0.80 ± 0.04
0.64 ± 0.11 ± 0.05
1.3σ
F L
[4,6]
0.71 ± 0.05
0.42 ± 0.13 ± 0.12
1.6σ
S 3
[0.04,2]
0.01 ± 0.01
−0.02 ± 0.09 ± 0.02 0.3σ
S 3
[2,4]
−0.01 ± 0.00
−0.15 ± 0.10 ± 0.07 1.2σ
S 3
[4,6]
−0.02 ± 0.01
0.00 ± 0.12 ± 0.07
0.2σ
S 4
[0.04,2]
0.06 ± 0.01
0.15 ± 0.20 ± 0.10
0.4σ
S 4
[2,4]
−0.13 ± 0.02
−0.37 ± 0.15 ± 0.10 1.3σ
S 4
[4,6]
−0.22 ± 0.02
0.32 ± 0.16 ± 0.09
3.0σ
S 5
[0.04,2]
0.20 ± 0.01
0.33 ± 0.13 ± 0.08
0.9σ
S 5
[2,4]
−0.16 ± 0.04
−0.16 ± 0.15 ± 0.06 0.0σ
S 5
[4,6]
−0.34 ± 0.04
0.13 ± 0.18 ± 0.09
2.3σ
S 7
[0.04,2]
−0.02 ± 0.02
−0.09 ± 0.10 ± 0.02 0.7σ
S 7
[2,4]
−0.02 ± 0.04
0.15 ± 0.14 ± 0.09
1.0σ
S 7
[4,6]
−0.01 ± 0.05
0.03 ± 0.13 ± 0.07
0.3σ
S 8
[0.04,2]
0.00 ± 0.01
−0.14 ± 0.24 ± 0.09 0.5σ
S 8
[2,4]
−0.01 ± 0.01
0.52 ± 0.20 ± 0.19
1.9σ
S 8
[4,6]
0.00 ± 0.02
−0.12 ± 0.21 ± 0.05 0.5σ
CMS 2015 [63]
F L
[1,2]
0.73 ± 0.05
0.64 +0.10
−0.09 ± 0.07
0.7σ
F L
[2,4.3]
0.79 ± 0.04
0.80 +0.08
−0.08 ± 0.06
0.1σ
F L
[4.3,6]
0.70 ± 0.05
0.62 +0.10
−0.09 ± 0.07
0.6σ
A F B
[1,2]
−0.16 ± 0.03
−0.27 +0.17
−0.40 ± 0.07
0.3σ
A F B
[2,4.3]
−0.02 ± 0.03
−0.12 +0.15
−0.17 ± 0.05
0.5σ
A F B
[4.3,6]
0.13 ± 0.04
0.01 +0.15
−0.15 ± 0.03
0.8σ
CMS 2017 [64]
P 1
[1,2]
0.05 ± 0.05
0.12 +0.46
−0.47 ± 0.10
0.2σ
P 1
[2,4.3]
−0.11 ± 0.04
−0.69 +0.58
−0.27 ± 0.23
1.0σ
P 1
[4.3,6]
−0.18 ± 0.05
0.53 +0.24
−0.33 ± 0.19
1.9σ
P 5
[1,2]
0.29 ± 0.07
0.10 +0.32
−0.31 ± 0.07
0.5σ
P 5
[2,4.3]
−0.45 ± 0.10
−0.57 +0.34
−0.31 ± 0.18
0.3σ
P 5
[4.3,6]
−0.77 ± 0.08
−0.96 +0.22
−0.21 ± 0.25
0.7σ
Table 8 :
8Angular observables of B 0 → K * 0 µ + µ − included in the global fit.B 0 → K * 0 µ + µ − angular observables Observable SM prediction Experimental value PullCDF [65]
F L
[0,2]
0.39 ± 0.06
0.26 +0.14
−0.13 ± 0.04
0.8σ
F L
[2,4.3]
0.79 ± 0.04
0.72 +0.15
−0.17 ± 0.09
0.4σ
A F B
[0,2]
−0.10 ± 0.01
0.07 +0.29
−0.28 ± 0.11
0.6σ
A F B
[2,4.3]
−0.03 ± 0.03
−0.11 +0.34
−0.45 ± 0.16
0.2σ
Table 10 :
10Binned differential branching ratio of B 0 → K * 0 µ + µ − included in the global fit.B 0 → K 0 µ + µ − differential branching ratioObservable
SM prediction Experimental value Pull
CDF [65]
10 8 × dBR
dq 2
[0,2]
3.28 ± 0.57
2.45 ± 1.59 ± 0.21
0.5σ
10 8 × dBR
dq 2
[2,4.3]
3.25 ± 0.56
2.55 ± 1.70 ± 0.35
0.4σ
CMS [5]
10 8 × dBR
dq 2
[0.1,2]
3.28 ± 0.57
1.22 +0.59
−0.52 ± 0.06
2.5σ
10 8 × dBR
dq 2
[2,4]
3.25 ± 0.55
1.87 +0.55
−0.49 ± 0.09
1.7σ
10 8 × dBR
dq 2
4,6]
3.21 ± 0.54
1.73 +0.53
−0.48 ± 0.09
1.9σ
10 8 × dBR
dq 2
15,22]
1.39 ± 0.16
0.95 +0.16
−0.15 ± 0.05
1.9σ
Table 11 :
11Binned differential branching ratio of B 0 → K 0 µ + µ − included in the global fit.B + → K + µ + µ − differential branching ratioObservable
SM prediction Experimental value Pull
CDF [65]
10 8 × dBR
dq 2
[0,2]
3.52 ± 0.61
1.80 ± 0.53 ± 0.12
2.1σ
10 8 × dBR
dq 2
[2,4.3]
3.50 ± 0.59
3.16 ± 0.54 ± 0.18
0.4σ
CMS [5]
10 8 × dBR
dq 2
[1.1,2]
3.53 ± 0.61
2.33 ± 0.15 ± 0.12
1.9σ
10 8 × dBR
dq 2
[2,3]
3.51 ± 0.61
2.82 ± 0.16 ± 0.14
1.1σ
10 8 × dBR
dq 2
[3,4]
3.49 ± 0.59
2.54 ± 0.15 ± 0.13
1.5σ
10 8 × dBR
dq 2
[4,5]
3.47 ± 0.59
2.21 ± 0.14 ± 0.11
2.0σ
10 8 × dBR
dq 2
[5,6]
3.45 ± 0.57
2.31 ± 0.14 ± 0.12
1.9σ
10 8 × dBR
dq 2
[15,22]
1.51 ± 0.17
1.21 ± 0.04 ± 0.06
1.6σ
Table 12 :
12Binned differential branching ratio of B + → K + µ + µ − included in the global fit.B + → K * + µ + µ − differential branching ratio ObservableSM prediction Experimental value PullCDF [65]
10 8 × dBR
dq 2
[0,2]
8.63 ± 1.23
7.50 ± 4.68 ± 0.88
0.2σ
10 8 × dBR
dq 2
[2,4.3]
4.90 ± 0.74
4.94 ± 3.58 ± 0.63
0.0σ
CMS [5]
10 8 × dBR
dq 2
[0.1,2]
7.93 ± 1.09
5.92 +1.44
−1.30 ± 0.40
1.1σ
10 8 × dBR
dq 2
[2,4]
4.87 ± 0.73
5.59 +1.59
−1.44 ± 0.38
0.4σ
10 8 × dBR
dq 2
4,6]
5.43 ± 0.82
2.49 +1.10
−0.96 ± 0.17
2.1σ
10 8 × dBR
dq 2
15,19]
6.42 ± 0.67
3.95 +0.80
−0.73 ± 0.28
2.3σ
Table 13 :
13Binned differential branching ratio of B + → K * + µ + µ − included in the global fit.B 0
s → φµ + µ − differential branching ratio
Observable
SM prediction
Experimental value
Pull
LHCb [7]
10 8 × dBR
dq 2
[1,6]
5.39 ± 0.65
2.58 +0.33
−0.31 ± 0.08 ± 0.19 3.7σ
10 8 × dBR
dq 2
[15,19]
5.57 ± 0.47
4.04 +0.39
−0.38 ± 0.13 ± 0.30 2.2σ
Table 14 :
14Binned differential branching ratio of B 0 s → φµ + µ − included in the global fit.B 0 s → φµ + µ − angular observables Observable SM prediction Experimental value PullLHCb [7]
F L
[0.1,2]
0.50 ± 0.04
0.20 +0.08
−0.09 ± 0.02
3.0σ
F L
[2,5]
0.81 ± 0.02
0.68 +0.16
−0.13 ± 0.03
0.8σ
F L
[15,19]
0.34 ± 0.01
0.29 +0.07
−0.06 ± 0.02
0.6σ
S 3
[0.1,2]
0.02 ± 0.01
−0.05 +0.13
−0.13 ± 0.01
0.5σ
S 3
[2,5]
−0.01 ± 0.00
−0.06 +0.19
−0.23 ± 0.01
0.2σ
S 3
[15,19]
−0.21 ± 0.01
−0.09 +0.11
−0.12 ± 0.01
1.0σ
S 4
[0.1,2]
0.06 ± 0.01
0.27 +0.28
−0.18 ± 0.01
0.8σ
S 4
[2,5]
−0.15 ± 0.02
−0.47 +0.30
−0.44 ± 0.01
0.7σ
S 4
[15,19]
−0.30 ± 0.00
−0.14 +0.11
−0.11 ± 0.01
1.5σ
S 7
[0.1,2]
−0.02 ± 0.02
0.04 +0.12
−0.12 ± 0.00
0.5σ
S 7
[2,5]
−0.02 ± 0.04
−0.03 +0.18
−0.23 ± 0.01
0.0σ
S 7
Table 15 :
15Angular observables of B 0 s → φµ + µ − included in the global fit. Λ 0 b → Λµ + µ − differentialbranching ratio Observable SM prediction Experimental value PullLHCb [67]
10 7 × dBR
dq 2
[1,6]
0.10 ± 0.06
0.09 +0.06
−0.05 ± 0.02
0.2σ
10 7 × dBR
dq 2
Table 16 :
16Binned differential branching ratio of Λ 0 b → Λµ + µ − included in the global fit. Λ 0 b → Λµ + µ − angular asymmetries Observable SM prediction Experimental value PullLHCb [10]
A l
F B
[15,20]
−0.36 ± 0.02
−0.39 ± 0.04 ± 0.01 0.8σ
A h
F B
[15,20]
−0.27 ± 0.01
−0.30 ± 0.05 ± 0.02 0.5σ
A lh
F B
Table 17 :
17Angular observables of Λ 0 b → Λµ + µ − included in the global fit. B + → K + µ + µ − angular observablesObservable SM prediction Experimental value PullCMS [68]
Table 18 :
18Angular observables of B + → K + µ + µ − included in the global fit. B 0 → K * 0 e + e − angular observablesObservable
SM prediction Experimental value Pull
Belle [8]
P 4
[1,4]
−0.01 ± 0.03
0.34 +0.41
−0.45 ± 0.11
0.8σ
P 4
[14.18,19]
−0.63 ± 0.01
−0.15 +0.41
−0.40 ± 0.04
1.2σ
P 5
[1,4]
0.17 ± 0.06
0.51 +0.39
−0.46 ± 0.09
0.8σ
P 5
[14.18,19]
−0.62 ± 0.03
−0.91 +0.36
−0.30 ± 0.03
0.9σ
LHCb [69]
F L
[0.002,1.12]
0.18 ± 0.04
0.16 ± 0.06 ± 0.03
0.3σ
Table 19 :
19Angular observables of B 0 → K * 0 e + e − included in the global fit.B 0 → K * 0 e + e − differential branching ratio Observable SM prediction Experimental value PullLHCb [70]
10 7 × dBR
dq 2
[0.003,1]
Table 20 :
20Binned differential branching ratio of B 0 → K * 0 e + e − included in the global fit.B + → K + e + e − differential branching ratioObservable
SM prediction Experimental value Pull
LHCb [3]
10 7 × dBR
dq 2
[1,6]
0.349 ± 0.059
0.312 +0.040
−0.031
0.5σ
Table 21 :
21Binned differential branching ratio of B + → K + e + e − included in the global fit.B → X s l + l − branchingratio Observable SM prediction Experimental value Pull BaBar [71]10 6 × BR (B → X s µ + µ − ) [1,6] × BR (B → X s µ + µ − ) [14.× BR (B → X s e + e − ) [14.1.68 ± 0.17
0.66 +0.87
−0.80 ± 0.07
1.1σ
10 6 2,25]
0.34 ± 0.04
0.60 +0.31
−0.29 ± 0.00
0.8σ
10 6 × BR (B → X s e + e − ) [1,6]
1.74 ± 0.18
1.93 +0.51
−0.48 ± 0.18
0.3σ
10 6 2,25]
0.29 ± 0.04
0.56 +0.19
−0.18 ± 0.00
1.5σ
Table 22 :
22Binned B → X s l + l − branching ratio included in the global fit.B 0
s → µ + µ − branching ratio
Observable
SM prediction Experimental value Pull
LHCb+CMS [72]
10 9 × BR(B 0
s → µ + µ − )
3.67 ± 0.15
2.80 +0.70
−0.60
1.2σ
Table 23 :
23B 0 s → µ + µ − branching ratio included in the global fit.
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[] |
[
"STRANGE DUALITY REVISITED",
"STRANGE DUALITY REVISITED"
] |
[
"Pauly Christian "
] |
[] |
[] |
We give a proof of the strange duality or rank-level duality of the WZW models of conformal blocks by extending the genus-0 result, obtained by Nakanishi-Tsuchiya in 1992, to higher genus curves via the sewing procedure. The new ingredient of the proof is an explicit use of the branching rules of the conformal embedding of affine Lie algebras sl(r) × sl(l) ⊂ sl(rl). We recover the strange duality of spaces of generalized theta functions obtained by Belkale, Marian-Oprea, as well as by Oudompheng in the parabolic case.
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10.4310/mrl.2014.v21.n6.a8
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[
"https://arxiv.org/pdf/1204.1186v1.pdf"
] | 96,460,730 |
1204.1186
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95159d5b2329ab1724c7be0a835e4d3182270092
|
STRANGE DUALITY REVISITED
5 Apr 2012
Pauly Christian
STRANGE DUALITY REVISITED
5 Apr 2012
We give a proof of the strange duality or rank-level duality of the WZW models of conformal blocks by extending the genus-0 result, obtained by Nakanishi-Tsuchiya in 1992, to higher genus curves via the sewing procedure. The new ingredient of the proof is an explicit use of the branching rules of the conformal embedding of affine Lie algebras sl(r) × sl(l) ⊂ sl(rl). We recover the strange duality of spaces of generalized theta functions obtained by Belkale, Marian-Oprea, as well as by Oudompheng in the parabolic case.
Introduction
One of the most significant recent results in the theory of vector bundles over curves is the proof of the strange duality or rank-level duality given by Belkale and Marian-Oprea. For a survey of the results we refer e.g. to [MO2], [Pa2], [Po]. In this note we give another proof of that duality in the framework of conformal blocks. In fact, the duality statement for conformal blocks over the projective line was proved by Nakanishi-Tsuchiya in 1992 [NT]. We generalize their statement to a smooth projective curve of any genus.
In order to state the Main Theorem we need to introduce some notation. Let r, l ≥ 2 be integers. We denote by P l (r) the finite set of dominant weights at level l of the Lie algebra sl(r) and for λ ∈ P l (r) we denote by H λ,l (r) the irreducible integrable sl(r)-module of weight λ and level l, where sl(r) is the affine Lie algebra associated to sl(r). The new ingredient of the proof is the use of the branching rules for the conformal embedding sl(r) × sl(l) ⊂ sl(rl) of the irreducible integrable sl(rl)-module H λ,1 (rl), when λ is a (possibly zero) fundamental dominant weight sl(rl). The branching rule [H] gives the decomposition
H λ,1 (rl) = Y ∈Y af f r,l (λ) H µ,l (r) ⊗ Ht µ,r (l),
where Y varies over a finite set Y af f r,l (λ) ⊂ Y af f r,l of Young diagrams of type (r, l) and of size λ (for the definitions see section 3.1). The dominant weights µ and t µ of sl(r) and sl(l) are naturally associated to Y and to its transpose t Y . The above decomposition is an infinite-dimensional analogue of the classical Skew Cauchy Formula (see e.g. [Pr]) giving the branching rule of the representation Λ λ C r ⊗ C l for the embedding of finite Lie algebras sl(r)×sl(l) ⊂ sl(rl). Given a smooth projective curve C with n marked points and a collection λ = (λ 1 , . . . , λ n ) ∈ P l (r) n we denote by V † λ,l (C, r) the corresponding conformal block. With this notation we now can state the 2000 Mathematics Subject Classification. Primary 14D20, 14H60, 17B67, 81T40. This research was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme.
Main Theorem. Let C be a smooth projective complex curve with n marked points and let λ = (λ 1 , . . . , λ n ) ∈ P 1 (rl) n = {0, 1, . . . , rl − 1} n be a labelling of the marked points with fundamental weights of sl(rl) satisfying the condition n i=1 λ i ≡ 0 mod rl. For any collection of Young diagrams
Y = (Y 1 , . . . , Y n ) ∈ n i=1 Y af f r,l (λ i )
we denote by µ = π( Y ) ∈ P l (r) n and t µ = π( t Y ) ∈ P r (l) n the collections of associated dominant weights of sl(r) and sl(l) respectively. Then the natural linear map between spaces of conformal blocks over the pointed curve C obtained via the conformal embedding sl(r) × sl(l) ⊂ sl(rl)
α : V † λ,1 (C, rl) −→ V † µ,l (C, r) ⊗ V † t µ,r (C, l) induces an injective linear map SD Y : V µ,l (C, r) −→ V λ,1 (C, rl) ⊗ V † t µ,r (C, l).
We recall that there is a canonical isomorphism (up to homothety) between the space of conformal blocks associated to a 1-pointed curve labelled with the trivial weight λ 1 = 0 (1) V † 0,l (C, r) ∼ = H 0 (SU C (r), L l ) and the so-called space of generalized theta functions of rank r and level l, i.e., the space of global sections of the l-th power of the determinant line bundle L over the coarse moduli space SU C (r) of semi-stable rank-r vector bundles with fixed trivial determinant over the curve C. In the special case of a 1-pointed curve labelled with the trivial weight and Y 1 = 0, µ 1 = 0, t µ 1 = 0 the above theorem combined with the isomorphism (1) states that the linear map
SD 0 : H 0 (SU C (r), L l ) † −→ H 0 (SU C (rl), L) † ⊗ H 0 (SU C (l), L r )
is injective. We denote by U * C (r) the coarse moduli space of semi-stable rank r and degree r(g − 1) vector bundles over C and by Θ the divisor
{E ∈ U * C (r) | dim H 0 (C, E) = 0} ⊂ U * C (r). The tensor product map U * C (1) × SU C (l) → U * C (l) induces an inclusion H 0 (U * C (l), O(rΘ)) ⊂ H 0 (U * C (1), O(rlΘ))⊗H 0 (SU C (l), L r ) ∼ = H 0 (SU C (rl), L) † ⊗H 0 (SU C (l), L r ).
The last isomorphism is proved in [BNR]. It is shown in [Be2] that the image of the linear map SD 0 is contained in H 0 (U * C (l), O(rΘ)). Hence, assuming the well-known fact that both vector spaces have the same dimension, we obtain a new proof of the following theorem.
Theorem 1.1 ( [Be1], [MO1]). For any smooth curve C, the linear map
SD 0 : H 0 (SU C (r), L l ) † −→ H 0 (U * C (l), O(rΘ)) is an isomorphim.
Note the map SD 0 is defined here in terms of conformal blocks, but it coincides (see [Be2] Proposition 5.2) under the isomorphism (1) with the one defined at the level of moduli spaces of vector bundles.
The parabolic version of Theorem 1.1 proved by Oudompheng [O] Theorem 4.10 can be similarly deduced from our Main Theorem by using the parabolic version of the isomorphism (1) proved in [Pa1]. We leave the details to the reader.
The paper is organised as follows. In sections 2,3 and 4 we collect for the reader's convenience some known results on conformal blocks, on the branching rules and on the WZW-connection. The proof of the Main Theorem is given in section 5.
I would like to thank Laurent Manivel for helpful comments.
Conformal blocks and factorization
2.1. Definition and properties of conformal blocks. Given an integer l ≥ 1 called the level, we introduce the finite set of dominant weights of sl(r)
P l (r) = {λ = r−1 i=1 a i ̟ i | r−1 i=1 a i ≤ l; a i ≥ 0},
where the ̟ i denote the r − 1 fundamental weights of sl(r). We also consider the involution of the set P l (r)
(2) λ = r−1 i=1 a i ̟ i → λ † = r−1 i=1 a r−i ̟ i .
We denote by V λ the irreducible sl(r)-module with dominant weight λ. Then λ † is the dominant weight of the dual V † λ . For the level l = 1 we will often identify P 1 (r) with the set of integers {0, 1, . . . , r−1} mapping the trivial weight to 0 and the i-th fundamental weight ̟ i to i. Under this identification the above involution (2) becomes 0 † = 0 and i † = r − i for i > 0.
Given an integer n ≥ 1, a collection λ = (λ 1 , . . . , λ n ) ∈ P l (r) n of dominant weights of sl(r) and a family (3) F = (π : C → B; s 1 , . . . , s n ; ξ 1 , . . . , ξ n ) of n-pointed stable curves of arithmetic genus g parameterized by a base variety B with sections s i : B → C and formal coordinates ξ i at the divisor s i (B) ⊂ C, one constructs (see [TUY] section 4.1) a locally free sheaf V † λ,l (F , r) over the base variety B, called the sheaf of conformal blocks or the sheaf of vacua for the Lie algebra sl(r) and the markings λ at level l. We recall that
V † λ,l (F , r) is a subsheaf of O B ⊗ H † λ,l , where H †
λ,l denotes the dual of the tensor product H λ,l = H λ 1 ,l ⊗ · · · ⊗ H λn,l of the integrable highest weight representations H λ i ,l of level l and weight λ i of the affine Lie algebra sl(r). The formation of the sheaf of conformal blocks commutes with base change. In particular, we have for any point
b ∈ B V † λ,l (F , r) ⊗ O B O b ∼ = V † λ,l (C, r), where C denotes the data (C b = π −1 (b); s 1 (b), . . . , s n (b); ξ 1|C b , . . . , ξ n|C b )
consisting of a stable curve C b with n marked points s 1 (b), . . . , s n (b) and formal coordinates ξ i|C b at the points s i (b). We recall that the sheaf of conformal blocks V † λ,l (F , r) does not depend (up to a canonical isomorphism) on the formal coordinates ξ i (see e.g. [U] Theorem 4.1.7). We therefore omit the formal coordinates in the notation.
We have the following factorization theorem.
Proposition 2.1 ( [TUY]). Let C be a nodal curve with a node n and let π : C → C be the partial desingularization at n. Then we have the direct sum decomposition
V † λ,l (C, r) = µ∈P l (r) V † λ∪µ∪µ † ,l ( C, r),
where we put the weights µ and µ † at the two points a, b ∈ C lying over the node n ∈ C.
Lemma 2.2. Let C be a stable curve of genus g with n marked points labelled with the dominant weights λ = (λ 1 , . . . , λ n ) ∈ P 1 (rl) n . Then
dim V † λ,1 (C, rl) = (rl) g if n i=1 λ i ≡ 0 mod rl 0 otherwise
Proof. This is a straightforward consequence of the factorization of conformal blocks when degenerating the genus-g curve to a rational curve with g nodes and taking the desingularization -we iterate Proposition 2.1 g times. The fomula states that
dim V † λ,1 (C, rl) = µ∈P 1 (rl) g dim V † λ∪ µ∪ µ † ,1 (P 1 , rl)
On the other hand n i=1 λ i + g j=1 µ j + µ † j ≡ n i=1 λ i mod rl, since µ j + µ † j ≡ 0 mod rl. Thus, it is sufficient to prove the dimension formula for g = 0. Then use once more the factorization formula to reduce to the case of P 1 with three marked points, i.e. g = 0 and n = 3. That calculation is standard, see e.g. [G] Formula 2.14 or [NT] Section 4.
Branching rules
In this section we review the results on the branching rules of [H] used in the proof of the Main Theorem.
3.1. Young diagrams. Given two positive integers r and l, we will denote by Young diagram of type (r, l) a decreasing sequence of r positive integers Y = (y 1 ≥ y 2 ≥ . . . ≥ y r−1 ≥ y r ≥ 0) such that y 1 − y r ≤ l
We denote the (infinite) set of Young diagrams of type (r, l) by Y r,l and consider the map to the set of dominant weights of sl(r)
π : Y r,l −→ P l (r), Y → π(Y ) = r−1 i=1 (y i − y i+1 )̟ i .
We introduce the finite subsets:
Y af f r,l = {Y ∈ Y r,l | y r ≤ l − 1} and Y f in r,l = {Y ∈ Y af f r,l | y 1 ≤ l}.
Note that all fibers of the map π : Y af f r,l −→ P l (r) have cardinality l. We will now define several maps between these finite sets of Young diagrams. We can think of a Young diagram of type (r, l) as a collection of r rows, where we put into the i-th row y i boxes. We distinguish two cases:
Case 1: Y ∈ Y f in r,l . We have y 1 ≤ l and y r ≤ l − 1. We put t Y ∈ Y f in l,r the Young diagram of size (l, r) defined by taking the transpose of Y , i.e. by putting the y i boxes into the i-th column, and Y † ∈ Y f in r,l the complement (after a 180 degree rotation) of Y in the full rectangle consisting of r rows having l boxes each.
Case 2: Y ∈ Y af f r,l \ Y f in r,l . We have l + 1 ≤ y 1 ≤ 2l − 1 and y r ≤ l − 1. We can write the Young diagram Y as a union of two diagrams Y 1 ∪ Y 2 with Y i ∈ Y f in r,l , where the first one Y 1 has in its i-th row min(l, y i ) boxes and the second one Y 2 has in its i-th row max(y i − l, 0) boxes. We then define the transpose t Y to be the union of the transposes t Y 1 ∪ t Y 2 , where the t Y i are defined as in Case 1. Note that the Young diagram fits into two full rectangles consisting of r rows having 2l boxes each. We then define Y † ∈ Y af f r,l to be the complement (after a 180 degree rotation) of Y in that double rectangle.
We thus have constructed two involutive bijective maps
Y af f r,l −→ Y af f r,l Y → Y † and Y af f r,l −→ Y af f l,r Y → t Y
These two maps preserve the subsets Y f in r,l ⊂ Y af f r,l and Y f in l,r ⊂ Y af f l,r . Note that t (Y † ) = ( t Y ) † and that π(Y † ) = π(Y ) † , where π(Y ) † is defined as in (2). We also note the equalities |Y af f r,l | = |Y af f l,r | = l|P l (r)| = r|P r (l)|.
Given a Young diagram Y ∈ Y r,l we define its size |Y | = r i=1 y i mod rl. Moreover we identify Z/rlZ = {0, . . . , rl −1}, so that |Y | ∈ {0, . . . , rl −1}. It is clear from the definition that |Y † | + |Y | = rl and that | t Y | = |Y |. We will denote by Y f in r,l (λ) and Y af f r,l (λ) the corresponding subsets of Young diagrams of size λ.
Example 3.1. We consider the following example for r = 3 and l = 4: Y = (6, 4, 3) ∈ Y af f 3,4 \ Y f in 3,4 . Then we have the following table.
Y = (6, 4, 3) t Y = (4, 4, 3, 2) Y † = (5, 4, 2) t Y † = (4, 3, 2, 2) π(Y ) = 2̟ 1 + ̟ 2 π(Y † ) = ̟ 1 + 2̟ 2 π( t Y ) = ̟ 2 + ̟ 3 π( t Y † ) = ̟ 1 + ̟ 2 Moreover |Y | = | t Y | = 1 and |Y † | = | t Y † | = 11.
3.2. Finite-dimensional case. We now recall the classical Skew Cauchy Formula (see e.g. [Pr] Theorem 8.4.1. Chapter 9), which gives the branching rule of the fundamental sl(rl)modules under the embedding sl(r) × sl(l) ⊂ sl(rl). Let λ be in {0, . . . , rl − 1}. Under the identification {0, . . . , rl − 1} = P 1 (rl) the sl(rl)-module Λ λ C rl = Λ λ C r ⊗ C l corresponds to the fundamental weight λ ∈ P 1 (rl) and decomposes as sum of irreducible sl(r) × sl(l)-modules
(4) V λ = Λ λ C r ⊗ C l = Y ∈Y f in r,l (λ) V µ ⊗ Vt µ ,
where V µ and Vt µ denote the sl(r) and sl(l)-modules with dominant weights µ = π(Y ) ∈ P l (r) and t µ = π( t Y ) ∈ P r (l).
3.3.
Infinite-dimensional case. The analogue of the above Skew Cauchy Formula for the embedding of affine Lie algebras sl(r) × sl(l) ⊂ sl(rl) is worked out in [H] Theorem 4.2. With the above notation we have the decomposition as sum of irreducible sl(r) × sl(l)-modules
H λ,1 = Y ∈Y af f r,l (λ) H µ,l ⊗ Ht µ,r .
The Virasoro operator L 0 associated to sl(rl) induces a decomposition (see [TUY] or [U]) into eigenspaces of the sl(rl)-module
H λ,1 = ∞ d=0 H λ,1 (d) with H λ,1 (0) = V λ .
For every Young diagram Y ∈ Y af f r,l (λ) we have an inclusion (5) H µ,l ⊗ Ht µ,r ֒→ H λ,1 .
Note that the Virasoro operators L 0 associated to the two Lie algebras sl(r) × sl(l) ⊂ sl(rl) coincide since these Lie algebras form a conformal pair (see [KW] Proposition 3.2 (c)). Hence restricting the previous inclusion to the 0-eigenspace we obtain an inclusion
H µ,l ⊗ Ht µ,r (0) = V µ ⊗ Vt µ ֒→ H λ,1 (n Y )
for some positive integer n Y . It follows from the Skew Cauchy Formula (4) that
Y ∈ Y f in r,l (λ) ⇐⇒ n Y = 0.
4. The projective WZW-connection 4.1. Definition of the projective WZW-connection. We now outline the definition of the projective WZW-connection on the sheaf V † λ,l (F , r) over the smooth locus B s ⊂ B parameterizing smooth curves and refer to [TUY] or [U] for a detailed account. Let D ⊂ B be the discriminant locus and let S = n i=1 s i (B) be the union of the images of the n sections. We recall the exact sequence
(6) 0 −→ π * Θ C/B ( * S) −→ π * Θ ′ C ( * S) π θ −→ Θ B (−log D) −→ 0,
where Θ C/B ( * S) denotes the sheaf of vertical rational vector fields on C with poles only along the divisor S, and Θ ′ C ( * S) π the sheaf of rational vector fields on C with poles only along the divisor S and with constant horizontal components along the fibers of π. There is an O B -linear map
p : π * Θ ′ C ( * S) π −→ n i=1 O B ((ξ i )) d dξ i ,
which associates to a vector field ℓ in Θ ′ C ( * S) π the n Laurent expansions ℓ i d dξ i around the divisor s i (B). Abusing notation we also write ℓ for its image under p
ℓ = (ℓ 1 d dξ 1 , · · · , ℓ n d dξ n ) ∈ n i=1 O B ((ξ i )) d dξ i .
We then define for any vector field ℓ in Θ ′
C ( * S) π the endomorphism D( ℓ) of O B ⊗ H † λ,l by D( ℓ)(f ⊗ u) = θ( ℓ).f ⊗ u + n i=1 f ⊗ (T [ℓ i ].u)
for f a local section of O B and u ∈ H † λ,l . Here T [ℓ i ] denotes the action of the energy-momentum tensor on the i-th component H
† λ i ,l , i.e., expanding ℓ i = ∞ j=−n 0 α j ξ j+1 i with α j local sections of O B , the operator T [ℓ i ] equals ∞ j=−n 0 α j L j .
Here the operators L j are the Virasoro operator acting linearly on H † λ i ,l via the Sugawara representation. It is shown in [TUY] that D( ℓ) preserves V † λ,l (F , r) and that D( ℓ) only depends on the image θ( ℓ) up to homothety. One therefore obtains a projective connection ∇ on the sheaf V † λ,l (F , r) over B s given by (7) ∇ θ( ℓ) = θ( ℓ) + T [ ℓ].
4.2.
Conformal embedding and projective flatness.
Proposition 4.1. Let F be a family of smooth projective n-pointed curves as in (3) and let λ ∈ P 1 (rl) n be a labelling of the marked points with fundamental weights of sl(rl). Then for every collection of Young diagrams Y = (Y 1 , . . . , Y n ) ∈ n i=1 Y af f r,l (λ i ) the tensor product of the n inclusions (5) H µ,l ⊗ Ht µ,r ֒→ H λ,1 , induces a natural homomorphism between sheaves of conformal blocks
(8) (V † λ,1 (F , rl), ∇) −→ (V † µ,l (F , r), ∇) ⊗ (V † t µ,r (F , l), ∇)
, which is projectively flat for the WZW connections.
Proof. We consider the embedding of semi-simple Lie algebras p = sl(r) × sl(l) ⊂ g = sl(rl). Since this embedding is conformal, we have by [KW] Proposition 3.2(c) that for any integer n the two Virasoro operators L n associated to p and g coincide. The proposition now follows since the two linear parts of the connections (7) given by the energy-momentum tensor also coincide.
Remark 4.2. Note that the WZW-connection is only defined for a family of smooth curves.
Remark 4.3. The previous proposition actually holds for any pair p ⊂ g of conformal embeddings of semi-simple Lie algebras. For the list of conformal embeddings, see e.g. [BB].
We have the following Corollary 4.4. With the above notation the homomorphism obtained from (8)
SD Y : (V µ,l (F , r), ∇) −→ (V λ,1 (F , rl), ∇) ⊗ (V † t µ,r (F , l), ∇)
is projectively flat and has constant rank.
Proof. Both assertions are valid for any vector bundles equipped with projective connections. We refer e.g. to [Be2] Lemma A.1 and Lemma A.2 for a proof.
Proof of the Main Theorem
We prove the Main Theorem by induction on the genus g of the curve. For g = 0 the Theorem coincides with [NT] Theorem 4.4, since in that case dim V λ,1 (C, rl) = 1 by Lemma 2.2. Note that for g = 0 the map SD Y is an isomorphism.
We now assume that the Theorem holds for any smooth marked curve of genus g − 1. First of all we notice that the map SD Y as defined in the Main Theorem for smooth curves can be defined as well for a nodal curve. Next, we observe that it is enough to show injectivity of the map SD Y for a curve C of genus g with one node. In fact, by upper semi-continuity of the rank of a homomorphism between vector bundles, we then obtain injectivity of SD Y for a general smooth curve. Then, since the rank of SD Y is constant for smooth families, as shown in Corollary 4.4, we obtain injectivity for any smooth curve.
We consider the desingularization π : C → C of the nodal curve C of genus g. Note that C is smooth of genus g − 1. The linear map α decomposes under the factorization given by Proposition 2.1 as follows
V † λ,1 (C, rl) α − −− → V † µ,l (C, r) ⊗ V † t µ,r (C, l) ∼ = ∼ = λ 0 ∈P 1 (rl) V † λ∪λ 0 ∪λ † 0 ,1 ( C, rl) − −− → (µ 1 ,µ 2 )∈P l (r)×Pr(l) V † µ∪µ 1 ∪µ † 1 ,l ( C, r) ⊗ V † t µ∪µ 2 ∪µ † 2
,r ( C, l) For any triple (λ 0 , µ 1 , µ 2 ) ∈ P 1 (rl) × P l (r) × P r (l) we define the map α λ 0 ,µ 1 ,µ 2 : V † λ∪λ 0 ∪λ † 0 ,1 ( C, rl) −→ V † µ∪µ 1 ∪µ † 1 ,l ( C, r) ⊗ V † t µ∪µ 2 ∪µ † 2 ,r ( C, l) as the (λ 0 , µ 1 , µ 2 ) component of α in the above decomposition.
Definition 5.1. We say that a triple (λ 0 , µ 1 , µ 2 ) ∈ P 1 (rl) × P l (r) × P r (l) is admissible if there exists a Young diagram Y ∈ Y af f r,l (λ 0 ) such that π(Y ) = µ 1 ∈ P l (r) and π( t Y ) = µ 2 ∈ P r (l). It is clear from the definition that if (λ 0 , µ 1 , µ 2 ) is admissible then the corresponding Young diagram Y ∈ Y af f r,l (λ 0 ) is uniquely determined either by the pair (λ 0 , µ 1 ) or by the pair (λ 0 , µ 2 ).
We now recall [BP] Proposition 4.4 in our context, i.e., for the conformal embedding p = sl(r) × sl(l) ⊂ g = sl(rl).
Proposition 5.2. Given a triple (λ 0 , µ 1 , µ 2 ) ∈ P 1 (rl) × P l (r) × P r (l) the linear map α λ 0 ,µ 1 ,µ 2 isWe denote by,r ( C, l) the linear map induced by α λ 0 ,µ 1 ,µ 2 . If we are in the second case of Proposition 5.2, i.e., if (λ 0 , µ 1 , µ 2 ) is admissible and Y ∈ Y f in r,l (λ 0 ), then Proposition 5.2 and the induction hypothesis applied to the curve C implies that the map SD(λ 0 , µ 1 , µ 2 ) is injective. In all other cases this map is zero. Now we observe that for any pair (λ 0 , µ 2 ) ∈ P 1 (rl) × P r (l) there exists at most one µ 1 ∈ P l (r) such that the triple (λ 0 , µ 1 , µ 2 ) is admissible. We introduce the set S(µ 1 ) = {(λ 0 , µ 2 ) | (λ 0 , µ 1 , µ 2 ) admissible}. In order to show injectivity of SD Y it is enough to show for each µ 1 ∈ P l (r) injectivity of the mapBut this follows from the fact that the map SD(λ 0 , µ 1 , µ 2 ) is injective for an admissible triple with Y ∈ Y f in r,l (λ 0 ). Since π : Y f in r,l → P l (r) is surjective, we can find for each µ 1 ∈ P l (r) such a pair (λ 0 , µ 2 ) ∈ S(µ 1 ). This completes the proof.
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. J A Laboratoire De Mathématiques, Dieudonné, 6108Université de Nice -Sophia AntipolisLaboratoire de Mathématiques J. A. Dieudonné, Université de Nice -Sophia Antipolis, 06108
Nice Cedex 02, France E-mail address: [email protected]. Nice Cedex 02, France E-mail address: [email protected]
|
[] |
[
"Sample Recycling Method -A New Approach to Efficient Nested Monte Carlo Simulations",
"Sample Recycling Method -A New Approach to Efficient Nested Monte Carlo Simulations"
] |
[
"Runhuan Feng [email protected] \nDepartment of Mathematics\nSchool of Finance, Nanjing University of Finance and Economics\nUniversity of Illinois at Urbana-Champaign. †\n\n",
"Peng Li [email protected] \nDepartment of Mathematics\nSchool of Finance, Nanjing University of Finance and Economics\nUniversity of Illinois at Urbana-Champaign. †\n\n"
] |
[
"Department of Mathematics\nSchool of Finance, Nanjing University of Finance and Economics\nUniversity of Illinois at Urbana-Champaign. †\n",
"Department of Mathematics\nSchool of Finance, Nanjing University of Finance and Economics\nUniversity of Illinois at Urbana-Champaign. †\n"
] |
[] |
Nested stochastic modeling has been on the rise in many fields of the financial industry. Such modeling arises whenever certain components of a stochastic model are stochastically determined by other models. There are at least two main areas of applications, including (1) portfolio risk management in the banking sector and (2) principle-based reserving and capital requirements in the insurance sector. As financial instrument values often change with economic fundamentals, the risk management of a portfolio (outer loop) often requires the assessment of financial positions subject to changes in risk factors in the immediate future. The valuation of financial position (inner loop) is based on projections of cashflows and risk factors into the distant future. The nesting of such stochastic modeling can be computationally challenging.Most of existing techniques to speed up nested simulations are based on curve fitting. The main idea is to establish a functional relationship between inner loop estimator and risk factors by running a limited set of economic scenarios, and, instead of running inner loop simulations, inner loop estimations are made by feeding other scenarios into the fitted curve. This paper presents a non-conventional approach based on the concept of sample recycling. Its essence is to run inner loop estimation for a small set of outer loop scenarios and to find inner loop estimates under other outer loop scenarios by recycling those known inner loop paths. This new approach can be much more efficient when traditional techniques are difficult to implement in practice.
|
10.1016/j.insmatheco.2022.04.012
|
[
"https://arxiv.org/pdf/2106.06028v1.pdf"
] | 235,417,275 |
2106.06028
|
7f3c7c8cae4743bf3541f21f3e640a6798a46a72
|
Sample Recycling Method -A New Approach to Efficient Nested Monte Carlo Simulations
10 Jun 2021
Runhuan Feng [email protected]
Department of Mathematics
School of Finance, Nanjing University of Finance and Economics
University of Illinois at Urbana-Champaign. †
Peng Li [email protected]
Department of Mathematics
School of Finance, Nanjing University of Finance and Economics
University of Illinois at Urbana-Champaign. †
Sample Recycling Method -A New Approach to Efficient Nested Monte Carlo Simulations
10 Jun 2021arXiv:2106.06028v1 [q-fin.CP]Nested simulationrisk estimationchange of measuredensity-ratio estima- tionsample recycling method *
Nested stochastic modeling has been on the rise in many fields of the financial industry. Such modeling arises whenever certain components of a stochastic model are stochastically determined by other models. There are at least two main areas of applications, including (1) portfolio risk management in the banking sector and (2) principle-based reserving and capital requirements in the insurance sector. As financial instrument values often change with economic fundamentals, the risk management of a portfolio (outer loop) often requires the assessment of financial positions subject to changes in risk factors in the immediate future. The valuation of financial position (inner loop) is based on projections of cashflows and risk factors into the distant future. The nesting of such stochastic modeling can be computationally challenging.Most of existing techniques to speed up nested simulations are based on curve fitting. The main idea is to establish a functional relationship between inner loop estimator and risk factors by running a limited set of economic scenarios, and, instead of running inner loop simulations, inner loop estimations are made by feeding other scenarios into the fitted curve. This paper presents a non-conventional approach based on the concept of sample recycling. Its essence is to run inner loop estimation for a small set of outer loop scenarios and to find inner loop estimates under other outer loop scenarios by recycling those known inner loop paths. This new approach can be much more efficient when traditional techniques are difficult to implement in practice.
Introduction
Many problems in portfolio risk measurement and financial reporting require nested stochastic modeling. Standard nested Monte Carlo methods can be costly and time consuming to reach a reasonable degree of accuracy. There has been growing demand in the financial industry for methods to speed up the nested simulation procedure.
In portfolio risk management, nested simulations are applied in a wide variety of risk assessments. Current use of Monte Carlo simulations are typically divided into two stages: outer loops and inner loops. In outer loops, Monte Carlo simulations are performed on all relevant risk factors over a specific risk horizon; the objective is often to calculate some risk measure of a portfolio consisting of multiple financial instruments. In inner loops, those financial instruments are evaluated conditional on risk factors generated from outer scenarios. As mentioned earlier, standard nested Monte Carlo simulations impose heavy computational burden. To tackle this problem, Gordy and Juneja (2010) analyzed the optimal allocation of computational resources between the inner and the outer stage. By minimizing the mean square error of the resultant estimator, they estimated multiple portfolio risk measures such as probability of large losses, Value-at-Risk(VaR), and expected shortfall. Moreover, Lan et al. (2010) constructed confidence intervals based on statistical theory of empirical likelihood and ranking-and-selection method. Broadie et al. (2011) developed a sequential allocation method in the inner stage based on marginal changes of the risk estimator in each scenario. Following their earlier work, Broadie et al. (2015) introduced the least square Monte Carlo in the inner stage to estimate the portfolio risk, and Hong et al. (2017) expanded on the Nadaraya-Watson kernel smoothing method in the inner stage. Recently, Giles and Haji-Ali (2019) used the multilevel Monte Carlo method in the nested simulation of risk estimation.
Nested simulations are also commonly used in the insurance literature when financial reporting procedures, such as reserving and capital requirement calculation, are performed under various stochastically determined economic scenarios. Reynolds and Man (2008b) pointed out that the need of nested stochastic is driven by a number of changes in the regulatory and accounting world and explain the move from stochastic to nested stochastic by a few examples under various accounting standards. A review of various circumstances under which nested simulation arises in financial reporting can be found in Feng et al. (2016). Standard nested Monte Carlo simulations were studied under different accounting requirements, such as the Solvency Capital Requirement(SCR) in Solvency II (Bauer et al., 2012;Morgan and Slutzky, 2006), reserve and capital with a principle-based approach (Reynolds and Man, 2008a), and the dynamic hedging under Actuarial Guideline (AG) 43 (Feng et al., 2016). In the context of AG-43, Li and Feng (2021) replaced the inner stage simulation with PDE numerical approximation in the dynamic hedging. Additionally, universal kriging method and machine learning method improved the efficiency in the stochastic pricing of a large variable annuity portfolio (Gan, 2013;Gan andLin, 2015, 2017). Most recently, a neural network approach has been used in the SCR of a large portfolio of variable annuity (Hejazi and Jackson, 2017). A surrogate modeling approach is developed by (Lin and Yang, 2020b) where the functional relationship between input and output of VA valuation models can be approximated by various statistical models. The work is further extended for dynamic hedging of variable annuity portfolio in (Lin and Yang, 2020a).
All the existing methods to speed up nested simulations can be summarized in three categories: (1) optimal allocation of resources between outer and inner loops (Broadie et al., 2011;Giles and Haji-Ali, 2019;Gordy and Juneja, 2010;Lan et al., 2010), (2) reduction of inner loops through approximation techniques (Broadie et al., 2015;Feng et al., 2016), and (3) volume reduction of nested simulation (Gan, 2013;Gan and Lin, 2015;Hejazi and Jackson, 2017). The second category is more common used in financial reporting due to the ease of implementation. Note that there exist many other fitting methods in the inner stage of the nested simulations, such as the exponential fitting technique (Beylkin and Monzón, 2005) adapted for actuarial applications in (Feng and Jing, 2017), the multivariate interpolation techniques (Hardy, 2003), and polynomial approximation.
The new technique proposed in this paper is based on an entirely different strategy. The basic idea is to reduce the number of inner loops by recycling a small set of them for different inner loop estimators. Hence we call this new method sample recycling method (SRM). In contrast with existing methods in the second category, this method completely avoids approximating functional relationship between inner loop estimator and risk factors. Once inner loop paths are generated for an inner loop estimator at some reference point (in state space), we reuse them to compute the estimators at other target points. Estimation with recycled samples requires the distorted weight (density-ratio) based on change of measures. In most well-known Markov models, we can calculate analytical expressions of distorted weights. In general case, one can estimate these distorted weights by non-parametric methods. Through a variety of examples, we will demonstrate the efficiency and applications of both parametric and non-parametric SRMs.
It was recently brought to our attention that a similar concept to sample recycling was developed in an independent work by Feng and Staum (2017), which is called the Green simulation method. Their work promotes reusing the output from previous simulation experiments to answer new questions based on simulations. The work of Feng and Staum (2017) and this paper differ in the problem set-up and implementation details. Their work focuses on general stochastic models, whereas this paper frames sample recycling methods in the context of nested stochastic modeling. The Green simulation method uses sample from all previous experiments and do not necessarily use particular sample sets. A mixture likelihood ratio estimator based on samples of all previous experiments is used to estimate quantities with a new input. Hence, in their setting, it is less of an issue to choose appropriate reference samples. In the context of nested simulation, we assume a pre-processed set of sample points. The aim of this paper is to reduce the number of inner loop simulations in a nested stochastic model. Hence the strategy of the sample recycling method is to identify a set of reference outer loop scenarios from which inner loop samples are obtained and to recycle them for the purpose of estimating quantities for other (target) outer loop scenarios. We propose a block method to ensure that sufficient and relevant sample paths are collected to improve the accuracy and efficiency of inner loop estimations. In this method, one reference point is chosen for each block, which effectively control the difference of distributions under reference scenarios and target scenarios. The mixture likelihood ratio method is further studied and extended in the context of tail event estimation in Dang (2021).
The rest of the paper is organized as follows. Section 2 provides a brief introduction to the standard nested Monte Carlo simulation. Section 3 describes the proposed sample recycling method, estimator, accuracy, and computational efforts. To further illustrate this method, it gives some examples to explain the calculations of inner loops and the estimation of risk measure. Section 4 continues to expand on the sample recycling framework by discussing a data-driven (non-parametric) likelihood estimation method. In both methods, numerical examples are given to compare with the standard nested Monte Carlo simulation and nested simulation via regression. Details of mathematical derivations and experiments are presented in the Appendix.
Standard nested Monte Carlo method
In a typical setting of nested simulations, we are interested in the risk measure of a portfolio's loss or gain at some future time τ . This value depends on the evolution of various financial risk factors over the period [0, τ ]. Common risk factors may include but are not limited to short-term 0 0 Figure 1: Comparison of standard nested MC and sample recycling methods yield rates, long-term yield rates, equity values, equity volatilities, exchanges rates, etc. Let Ω be a set of all possible sample paths for risk factors, P be the physical measure under which data are observable in financial markets, Q be the risk-neutral measure for market consistent valuation. Typically, all valuations on portfolio risk management are done under risk-neutral measures. In insurance applications, however, risk measure may be considered under physical measure in financial reporting. As far as the methodology itself is concerned, it does not matter under which measure the application is performed.
Outer loop estimation
For example, we may consider the risk measure for the valuation of a portfolio
ρ = E[f (L)], (2.1)
where L is the future loss of the portfolio over the period [0, τ ], and f is a real-valued function such that the expectation exists. Examples of such risk measure may include the probability of a large loss where f (x) = 1(x ≥ c), the expected excess loss where f (x) = (x − c) + , and the present value of loss where f (x) = e −rτ x.
In other applications, one may be interested in risk measures such as the Value-at-Risk
VaR α [f (L)] = inf{x ∈ R|P(f (L) ≤ x) > α},
or the conditional tail expectation
CTE α [f (L)] = E f (L)|f (L) > VaR α [f (L)] ,
neither of which conforms to the form in (2.1), which we will focus on for analysis. Nevertheless, it is worthwhile to point out that this sample recycling technique is not restricted to the exact form of (2.1) and can be extended to other risk measures.
Inner loop estimation
In practice, the computational challenge arises as neither the risk measure in (2.1) nor the loss random variable L is explicitly expressed by an algebraic formula. Instead, the quantity is estimated in two steps, i.e., a "nested" setting. The outer layer of the simulation approximates the distribution of the loss L by its empirical distribution as a result of Monte Carlo sampling. In particular, the risk measure can be estimated by the standard statistic given independent and identically distributed samples based on the physical measure,
1 n n i=1 f (L i ), (2.2)
where (L 1 , L 2 , · · · , L n ) is an i.i.d sample of random variable L.
Note that the portfolio loss L i in i-th scenario is difficult to compute, as it is usually dependent on paths and cashflows over the period [τ, T ] where T is the specified maturity time. The purpose of inner level simulation is exactly to avoid this difficulty. In practice, the portfolio loss L is often viewed as a conditional expectation on the information of the risk horizon [0, τ ]. Let F τ be a field that contains all the information available to investors at time τ . This conditional expectation can be written as
L = E Q [g(Z)|F τ ], (2.3)
where Z is a random element of R d describing the performance of portfolio on [τ, T ], and g(·) is a known function from R d to R. 1 To obtain the sample (L 1 , L 2 , . . . , L n ), we typically obtain from each outer loop simulation values of the underlying risk factors and generate inner loop sample paths over the period [τ, T ] under the risk-neutral measure. Let us denote by (Z i,1 , Z i,2 , ..., Z i,m ) an independent and identically distributed sample of cash flows corresponding to the risk factors for L i . One can think of {Z i,j , j = 1, · · · , m} for each fixed i = 1, · · · , n as a set of inner loop paths that emanate from the same initial position determined by the i-th outer loop scenario. See the sets of black lines in Figure 1
Standard nested MC estimator
Returning to the outer loop, the risk measure ρ can be estimated by
ρ SN = 1 n n i=1 f ( L i ). (2.5)
As mentioned earlier, there are two issues with the standard nested simulation: computation and accuracy. Previous studies present many methods to accelerate nested simulations, which can be summarized in three categories. (1) Optimal allocation of computation between outer and inner levels. Such methods are dedicated to decision-making on the number of outer and inner loops given a fixed budget. It is shown that risk estimators with optimal allocation of computational resources presenrs a faster convergence order compared to the uniform allocation schemes (Broadie et al., 2011;Giles and Haji-Ali, 2019;Gordy and Juneja, 2010;Lan et al., 2010). (2) Reduction of inner levels through curve fitting techniques. The main principle is to find replace the mapping between inner loop estimators and outer loop risk factors. Since the inner-level calculation brings most computational challenge, these methods focus on the approximation of inner loop estimates the proxy functional relationship. A relatively small set of sample is used to estimate the proxy function, which is then used to produce values of inner loop estimator under a wide range of outer loop scenarios. (Broadie et al., 2015;Feng et al., 2016). (3) Reduction of the the volume of nested simulation. The central idea of this category is to strike a balance between computational efficiency and model granularity. (Gan, 2013;Gan and Lin, 2015;Hejazi and Jackson, 2017).
Sample Recycling Method
Here we introduce a new technique that belongs to the second category: reduction of inner levels. However, the proposed method aims to reduce the number of inner loop simulations based on an entirely different philosophy from curve fitting techniques, such as least square Monte Carlo or pre-processed inner loops. This approach avoids redundant computations in the inner loops by re-sampling a few sets of inner loop paths.
Inner loop estimation
Bear in mind that the outer loop procedure is kept the same as (2.2) and the proposed method differs from the standard nested simulation and other methods in the inner loop estimation. To consider the new estimator, we typically generate inner loop paths to estimate the loss under a particular scenario. The initial position of risk factors under the particular outer loop scenario is referred to as the reference point. See the initial position from which the middle set of black lines is generated in Figure 1(b) as an example of the reference point. Without loss of generality, we consider the reference point to be generated under the 1-st outer loop scenario. Recall that the inner loop estimation is carried out for the loss random variable
L 1 = E Q 1 [g(Z)],
where Q 1 is the measure under which inner loop sample paths are generated from some initial position determined by the 1-st outer loop scenario. Then we can determine the inner loop estimator under the 1-st scenario by
L 1 = 1 m m j=1 g(Z 1,j ),(3.1)
where (Z 1,1 , Z 1,2 , . . . , Z 1,m ) are i.i.d samples generated for the random element Z conditioned on the 1-st outer loop scenario (under measure Q 1 ). Note that this estimator is the same as the one for standard nested MC method (2.4). For simplicity, the 1-st outer loop scenario is referred to as a reference point and other scenarios as target points. We intend to reuse the inner paths(Z 1,1 , Z 1,2 , . . . , Z 1,m ) and evaluations g(Z 1,1 ), g(Z 1,2 ), ..., g(Z 1,m ) for the reference point to estimate loss L i for other target points i > 1. Denote by Q i the probability measure under which the underlying process starts from the i-st outer loop scenario at time τ . The loss for any target point i can be written as
L i = E Q i [g(Z)] = E Q 1 [p i|1 (Z)g(Z)],(3.2)
where p i|1 (·) is the Radon-Nikodym derivative of measure Q i with respect to Q 1 . If the random element Z has conditional probability density p i (·) under Q i , then the Radon-Nikodym derivative can be given by
p i|1 (·) = dQ i dQ 1 = p i (·) p 1 (·) . (3.3)
The sample version of the portfolio loss (3.2) can be written as
L i := 1 m m j=1 p i|1 (Z 1,j )g(Z 1,j ). (3.4)
Under the original measure Q 1 each inner loop sample path Z 1,j carries equal weight 1/m in (3.1). In contrast, the evaluation of each inner loop sample path under the measure Q i is given a "distorted" weight in (3.4). In general, we can interpret the weights in the following way. If the recycled path deviates far from the target point, the Radon-Nikodym derivative p i|1 gives a small weight, as it is unlikely to observe such a path eminating from the target point. If the recycled path is close to the target point, the derivartive p i|1 offers a large weight to reflect its high likelihood.
Sample recycling estimator
Then the estimation of risk measure ρ by the sample recycling method is given by
ρ SR = 1 n n i=1 f ( L i ). (3.5)
A quick comparison of (2.4) and (3.4) shows their differences. Observe that in (2.4) each estimator under scenario i uses a new sample (Z i,1 , Z i,2 , · · · , Z i,m ), whereas in (3.4) estimators for all i = 1, · · · , n only use the same sample (Z 1,1 , Z 1,2 , · · · , Z 1,m ). Because all random variables (Z i,1 , Z i,2 , · · · , Z i,m ) are drawn independently under the measure Q 1 , all evaluations in (2.4) are done with equal weight 1/m. In contrast, these random variables no longer appear with equal probability under another measure Q i for i > 0. For this reason, we shall refer to the probability adjustment p i|1 as "distorted" probability. Now the question is shifted to evaluating the distorted weight p i|1 (·). In the discussion above, we assumed for simplicity that Z is R d -valued, but the ideas extend to Z taking values in more general sets. Also, we have assumed that Z conditioned on ω i has a conditional probability density p i (·) under Q i , so that the weight p i|1 (·) is the ratio of two density functions of multidimensional random variable. The following subsection gives a simplified method to determine the weights under Markov models.
Distorted weights
For portfolio management, it is natural to think of Z as the price of underlying assets. To illustrate the calculation on the distorted weights p i|1 (·), we only consider one risk factor and use a Markov process {F t } t≥0 to represent the price of underlying asset.
We consider the discrete path of {F t } t≥0 on the interval [0, T ]. For simplicity, let F h := F t h , h = 0, 1, 2, ..., K with t 0 = 0 and t K = T , and the risk horizon τ = t k ∈ [0, T ]. In this special example, we denote the asset prices under the i-th outer loop scenario by (F
(i) 1 , F (i) 2 , · · · , F (i) k ) for i = 1, · · · , n.
Under the i-th scenario, we can further generate inner loop sample paths
Z i,j = (F (i,j) k+1 , F (i,j) k+2 , · · · , F (i,j)
K ) for j = 1, · · · , m. Suppose that we use the 1-st scenario as the reference point. We shall recycle sample paths from the reference point, i.e. (F
(1,j) k+1 , F (1,j) k+2 , · · · , F (1,j) K ).
We now consider the sample recycling estimator. Observe that Q i is the measure under which F
(i) k is realized, i.e. Q i (F k = F (i) k ) = 1.
In view of (3.2), we can obtain that
L i = E Q i [g(F k+1 , F k+2 , ..., F K )] = E Q 1 p i|1 (F k+1 , F k+2 , ..., F K )g(F k+1 , F k+2 , ..., F K ) , (3.6) where p i|1 (y k+1 , y k+2 , ..., y K ) = p i (y k+1 , y k+2 , ..., y K ) p 1 (y k+1 , y k+2 , ..., y K ) , (3.7)
and p i (y k+1 , y k+2 , ..., y K ) is the conditional probability density of (F k+1 , F k+2 , ..., F K ) under Q i . The approximation of this weight has high computational cost because it is a ratio of multidimensional density functions. Note that the process of inner simulation is based on the Markov property, indicating that the inner path simulation (F
(i,j) k+1 , F (i,j) k+2 , · · · , F (i,j) K ) for j = 1, · · · , m is conditioned on F (i)
k . This Markov property can also be used in the simulation of the samples of F t , t > t k . In other words, we can simulate the path of (F k+1 , F k+2 , . . . , F K ) through a recursion, for some function G,
F h+1 = G(F h , X h+1 ), h ≥ k, (3.8)
which is driven by i.i.d. risk factors X k+1 , X k+2 , ..., X K . Then the "distorted" weight can be reduced to
p i|1 (y k+1 , y k+2 , ..., y K ) = p i (y k+1 )f (y k+2 , ..., y K |y k+1 , F k = x i ) p 1 (y k+1 )f (y k+2 , ..., y K |y k+1 , F k = x 1 )
where f (·|·) is the conditional density function of (F k+2 , ..., F K ) given (F k+1 , F k ). Thanks to the Markov property, it has f (y k+2 , ..., y K |y k+1 , F k = x i ) = f (y k+2 , ..., y K |y k+1 ), which has no dependence on F k . Hence, the "distorted" weight can be simplified to
p i|1 (y k+1 , y k+2 , ..., y K ) = p i (y k+1 ) p 1 (y k+1 ) . (3.9)
Hence, according to (3.4), the inner loop estimator for the target point can be written as
1 m m j=1 p i (F (1,j) k+1 ) p 1 (F (1,j) k+1 ) g(F (1,j) k+1 , F (1,j) k+2 , ..., F (1,j) K ).
(3.10)
We give the following three examples to further illustrate the simplified weights.
Example 3.1. In this example, we assume that the price of underlying asset {F t } t≥0 follows a geometric Brownian motion. The portfolio only has one underlying asset, and the asset price at the risk horizon τ (the outer scenario) is driven by, under the real-world measure P
dF t = µF t dt + σF t dB t , F 0 > 0,
where {B t } t≥0 is a standard Brownian motion. The loss of portfolio is evaluated under risk-neutral measure Q, under which the asset price is determined by,
dF t = rF t dt + σF t dW t , F 0 > 0, (3.11)
where {W t } t≥0 is a standard Brownian motion under risk-neutral measure Q. There are n scenarios for the asset price before risk horizon τ . We define the corresponding prices at the risk horizon τ as x 1 , x 2 , ..., x n where x i := F τ (ω i ). In each scenario, we can simulate the path of (F k+1 , F k+2 , . . . , F K ) through the following recursion
F (i) h+1 = F (i) h exp((r − σ 2 /2)∆t + σ √ ∆tX h+1 ), h = k, k + 1, . . . , K,
where X 1 , ..., X K are independently draw from standard normal distribution with density function φ. This gives the distribution
ln F (i) h+1 F (i) h ∼ N ((r − σ 2 2 )∆t, σ 2 ∆t).
We use x 1 as the reference point, then the weights (3.9) can be written as
p i|1 (y) = φ ln(y/x i )−(r−σ 2 /2)∆t σ √ ∆t φ ln(y/x 1 )−(r−σ 2 /2)∆t σ √ ∆t
, and the weights can be simplified as p i|1 (y) = Ay B , where coefficients are given by
A = exp ln x i /x 1 σ 2 ∆t − 1 2 ln(x 1 x i ) − (r − 1 2 σ 2 )∆t , B = ln x i /x 1 σ 2 ∆t .
If we insert parameters x i = x 1 , i.e., using oneself as a reference, p i|1 (y) = 1 for any y, as expected.
Example 3.2. Suppose we have a portfolio exposed to interest rate risk, and let the rate follow a Vasicek model (Vasicek, 1977) under a risk neutral measure,
dF t = κ(θ − r t )dt + σdW t ,
where constants κ, θ, σ denote the speed of reversion, the long-term mean level, and the instantaneous volatility respectively. Here {W t } t>0 is a pure Brownian motion under the risk neutral measure. Given the risk horizon τ and the outer scenarios x 1 , x 2 , ..., x n where x i := F τ (ω i ), we can simulate the path of F t on the interval [τ, T ) in each scenario with the following recursion (Glasserman, 2003)
F (i) h+1 = e −κ∆t F (i) h + θ(1 − e −κ∆t ) + σ 1 2κ (1 − e −2κ∆t )X h+1 ,
where X 1 , ...X K are independent draws from a standard normal distribution. Similarly, we can get the weight as follows
p i|1 (y) = φ y − (e −κ∆t x i + θ(1 − e −κ∆t )) σ 1 2κ (1 − e −2κ∆t ) φ y − (e −κ∆t x 1 + θ(1 − e −κ∆t )) σ 1 2κ (1 − e −2κ∆t ) ,
which can be simplified to p i|1 (y) = A exp(By), where coefficients are given by
A = exp − κe −κ∆t (x i − x 1 ) e −κ∆t (x i + x 1 ) + 2θ(1 − e −κ∆t ) σ 2 (1 − e −2κ∆t ) , B = 2κ(x i − x 1 )e −κ∆t σ 2 (1 − e −2κ∆t ) .
Example 3.3. Assume that the equity return process is modeled by a two-state regime switching log-normal model (Hardy, 2001) with parameters Θ = {µ 1 , σ 1 , µ 2 , σ 2 , p 12 , p 21 }. In such a model, the equity process switches between two regimes with low and high volatilities. Let s k := s t k denote the regime at time t k and F k := F t k be the equity return at time t k . The two regimes are represented by 1 and 2, i.e. s k ∈ {1, 2}. There are two risk factors in this model, which are modeled by the bivariate process {(F k , s k ), k = 1, 2, · · · }. The equity return is log-normally distributed, i.e.
ln F k+1 F k s k+1 ∼ N (µ s k+1 ∆t, σ 2 s k+1 ∆t).
The transition probability from regime m to l is given by p ml = P(s h+1 = l|s h = m), m, l = 1, 2.
Given the risk horizon τ and the outer scenarios
x 1 , x 2 , ..., x n where x i := F (i) k .
We need to simulate the path of (F
(i) h , s (i) h ) for h > k, which is determined by F (i) h+1 = F (i) h exp(µ s h ∆t + σ s h √ ∆tX h+1 ), h > k,
where X 1 , · · · , X K are independent draws from a standard normal distribution and s h is the regime applying in the interval [t h , t h+1 ). The regime s h+1 is simulated by a uniform random variable, and is determined by s h and the transition probability. Then the weight is based on the regime applying in the interval [t k , t k+1 ) in each scenario. Let q represent the density function of (F k+1 , s k+1 ) conditioned on (F k , s k ), then we have
q(y, s k+1 |x, s k ) = P(s k+1 |s k )f (y|s k+1 , x), where f (y|s k+1 , x) = φ log y x − µ s k+1 ∆t σ s k+1 √ ∆t ,
and φ is the standard normal probability density function. In such a model, the distorted weight is given by p (i,m)|(1,l) where l, m are the states of reference point and target point, respectively. Define q i,m (y, s) := q(y, s|F k = x i , s k = m) and f i (y|s) = f (y|s, F k = x i ). Therefore, for m, l = 1, 2 the weights can be written as
p (i,m)|(1,l) (y, s) = q i,m (y, s) q 1,l (y, s) ,
which can be simplified to
p (i,m)|(1,l) = p ms p ls f i (y|s) f 1 (y|s) .
Analysis of Estimators
Bias and variance
In this subsection, we analyze the bias and variance of estimator L i under Q i and the convergence of estimator ρ SN . The error analysis of statistic L i is similar to the importance sampling method (Hesterberg, 1995;Øivind Skare et al., 2003), and the convergence of ρ SN is an extension of the work on the standard nested Monte Carlo (Rainforth et al., 2018).
Proposition 3.1. The asymptotic bias and variance of L i are given by
Bias( L i ) = 0, Var Q 1 ( L i ) = O 1 m , as m → ∞. (3.12)
Proof. It follows from (3.2) and (3.4) that the estimator L i is unbiased. Since (Z 1,1 , Z 1,2 , ..., Z 1,m ) is a sample of i.i.d. random variables generated from the random element Z conditioned on the 1-st outer loop scenario. The variance of L i under Q can be written as
Var Q 1 L i = 1 m Var Q 1 [p i|1 (Z 1 )g(Z 1 )] = 1 m E Q i [p i|1 (Z 1 )g 2 (Z 1 )] − L 2 i .
Here we provide some comparison of the variances ofρ SN andρ SR . In particular, we focus on the special case that f (x) = x in (2.1) and ρ = E(L). For brevity, we denote for i = 1, 2,
A l := E p i|1 (Z 1,1 )g(Z 1,1 ) l , B l := E[(g(Z 1,1 )) l ], C := E p i|1 (Z 1,1 ) (g(Z 1,1 )) 2 , D := E p i|1 (Z 1,1 )p j|1 (Z 1,1 ) (g(Z 1,1 )) 2 ,
Proposition 3.2. The variances can be written as
Var( ρ SN ) = O 1 mn , (3.13) Var( ρ SR ) = O 1 m . (3.14) Proof. Since L i , i = 1, 2, ..., n are i.i.d. estimators and Z i,1 , Z i,2 , ..., Z i,m are i.i.d. random variables, it follows from (2.4) and (2.5) that Var( ρ SN ) = 1 n Var L 1 = 1 mn Var(g(Z 1,1 )) = 1 mn (B 2 − B 2 1 ).
In view of (3.4), and (3.5), we can write
ρ SR = 1 n n i=1 L i = 1 n n i=1 1 m m j=1 p i|1 (Z 1,j )g(Z 1,j ) = 1 m m j=1 1 n n i=1 p i|1 (Z 1,j )g(Z 1,j ) .
Therefore,
Var( ρ SR ) = 1 m Var 1 n n i=1 p i|1 (Z 1,1 )g(Z 1,1 ) = 1 mn 2 Var g(Z 1,1 ) 1 + n i=2 p i|1 (Z 1,1 ) (3.15) = 1 mn 2 E g(Z 1,1 ) 1 + n i=2 p i|1 (Z 1,1 ) 2 − E g(Z 1,1 ) 1 + n i=2 p i|1 (Z 1,1 ) 2 = 1 mn 2 B 2 + (n − 1)A 2 + 2(n − 1)C + (n 2 − 3n + 2)D − (B 1 + (n − 1)A 1 ) 2 .
The asymptotics follow immediately from the results above.
To illustrate convergence rates, we consider an example where X is uniformly distributed on [−1, 1] and Z has a standard normal distribution in (2.3). In such a case, we can calculate the exact loss L = E[g(Z)|X] = E[ 2/π exp(−2(Z − X))|X]. Details of the calculation are left in Appendix A. The left panel in Figure 2 shows the changes in variances of ρ SN and ρ SR with the increasing number of outer loops n and the fixed number of inner loops m = 1, 000. When n = 1, both estimators are precisely the same as there is only one set of inner loop paths and hence they have the same variance, i.e. 1/mVar(g(Z 1,1 )). When n = 2, the jump in the variance of SRM estimator is due to the presence of error from using the inner loop sample of a reference point for the target point. As n increases, while the variance of ρ SN decreases, it does not diminish as quickly as that of ρ SR . This is because all target points on outer loop scenarios use exactly the same set of inner loop paths from the reference point. All portfolio loss estimatorsL i 's are driven by the same source of randomness (Z 1,1 , · · · , Z 1,m ). Therefore, they tend to overestimate or underestimate all in the same direction and the sample errors in L i do not offset each other. In contrast, each estimate of L i is based on an independent sample of (Z i,1 , · · · , Z i,m ) and hence the sample errors in ρ SN average out. The right panel of Figure 2 shows the convergence of variances of ρ SN and ρ SR with an increasing number of inner loop paths m and a fixed number of outer loop scenarios n = 1, 000. In such an experiment, the increased inner loop sample size significantly improves the accuracy of estimation involving the reference point and hence in turn reduces the error in the estimation of other target points. The value of the difference Var( ρ SN ) − Var( ρ SR ) converges to the constant 1.0150 × 10 −4 which is given by (D − A 2 1 )/m. This numerical example confirms the observation earlier that the standard nested Monte Carlo estimator tends to converge faster than the sample recycling method. The real purpose of the sample recycling method is to give up some accuracy in exchange for high efficiency for any fixed computational budget. The comparison of computational effort is discussed in the next subsection.
Computational efforts
To compare the computational effort, we should first look at algorithms of both standard nested Monte Carlo and sample recycling methods.
Algorithm 1 Estimate risk measure ρ using ρ SN Generate n outer scenarios for i = 1 to n do Conditioned on scenario ω i , generate m i.i.d. inner pathes Z i,1 , Z i,2 , . . . , Z i,m Calculate the sequence g(Z i,1 ), g(Z i,2 ), ..., g(Z i,m )
L i ← (1/m) m j=1 g(Z i,j ) end for ρ SN ← (1/n) n i=1 f ( L i )
Algorithm 2 Estimate risk measure ρ using ρ SR Generate n outer scenarios Conditioned on the 1-st scenario, generate m i.i.d. inner pathes Z 1,1 , Z 1,2 , ..., Z 1,m Calculate the sequence g(Z 1,1 ), g(Z 1,2 ), ..., g(Z 1,m )
for i = 1 to n do Calculate the sequence p i|1 (Z 1,1 ), p i|1 (Z 1,2 ), ..., p i|1 (Z 1,m ) L i ← 1 m m j=1 p i|1 (Z 1,j )g(Z 1,j ) end for ρ SR ← (1/n) n i=1 f ( L i )
According to these algorithms, the estimation of ρ SN requires generating a total of nm inner paths and evaluating the g(Z i,j ) for a total of nm times, while the estimation of ρ SR uses only m inner paths, the computation of g(Z 1,j ) for m times and that of p i|1 (Z 1,j ) for (n − 1)m times. In other words, we can measure the computational efforts with the following units:
• γ := simulation time of each inner path Z i,j + calculation time of each g(Z i,j ),
• δ := calculation time of each p i|1 (Z 1,j ).
We use CE to denote the computational effort of each method. Then the computational efforts required by the two methods are given by
CE SN = nmγ (3.16) CE SR = mγ + (n − 1)mδ. (3.17)
The main computational difference depends on the sizes of γ and δ. It is clear that when γ = δ the two methods require exactly the same amount of computational resources. Note, however, that γ includes the computation of each inner loop and cash flow projection. If the financial instrument is path-dependent, then such a calculation can be very time-consuming. While the value of δ is determined by a likelihood, the Radon-Nikodym derivative is not path-dependent in Markov models as shown in (3.9). The real advantage of sample recylcing method is only shown when γ far exceeds δ, which is often the case with long-term products and very sophisticated evaluation of cash flows.
Extension to multiple reference points
It follows from Theorem 3.2 that for a fixed number of outer loop scenarios the sample recycling estimator achieves the same rate of convergence, O(1/m) as the standard Monte Carlo estimate. Nonetheless, the main advantage of this method is to enhance efficiency by reducing computational efforts. In order to improve the accuracy of this method, one can introduce multiple reference points for variance reduction. For example, consider b reference points x 1 , x 2 , . . . , x b and we want to estimate the portfolio loss for the target point x i (i > b). We can take a weighted average of estimates based on individual reference points given in (3.4),
L i := b k=1 w ik 1 m m j=1 p i|k (Z k,j )g(Z k,j ) ,
where the weights shall satisfy w ik ≥ 0 and b k=1 w ik = 1 for each target point i = b + 1, · · · , n. A simple approach is to use equal weights, i.e. w ik = 1/b for k = 1, · · · , b, which corresponds to the simple average of L i estimated using each reference point. Since samples generated for reference points are mutually independent, an advantage of this approach is the reduction of variance of L i due to the increase of sample size to mb,
Var b k=1 1 b 1 m m j=1 p i|k (Z k,j )g(Z k,j ) = 1 mb Var p i|k (Z k,j )g(Z k,j ) .
Another approach is to apply a proximity rule. We can break the entire range of scenarios into a number of blocks and select one reference point in each block. Then we generate a set of inner risk paths for each reference point. Inner loop estimation for other target points in each block uses only the reference point in that block, i.e., w ik = 1 if i is in the block with k and w ik = 0 otherwise. This consideration is inspired from potential higher variance due to reference points being far from target, which is reflected in the numerator terms in Var( L i ) in Theorem 3.1,
Var( L i ) = 1 m E[p i|k (Z k )g 2 (Z k )] − (L i ) 2 v.s. Var[ L i ] = 1 m E[g 2 (Z i )] − (L i ) 2 .
If reference point k is properly chosen for each i, then we can achieve a reduction in variance.
In the following examples, we consider a single risk factor for simplicity and use the absolute difference as a metric to assign target scenarios into blocks. In higher dimensional or more complicated cases, one can define more suitable distance metrics on the sample space of risk factor F for the assignment of reference points. There are two common methods for block partitioning. (1) Equidistant partition: keep the same distance between boundary points in each block; (2) Quantile partition: use order statisics or empirical quantiles to allocate F k into blocks, each of which contains the same number of points. For each block, we shall choose one reference point, for example, the midpoint or a boundary point.
Numerical examples
As a trade-off between sampling variance and computational effort, we observe that the sample recycling method tends to reduce computational effort at the expense of increased variance. We offer a number of examples where the inner simulation and the evaluation of g(Z i,j ) can be computationally much more challenging than that of p i|1 ().
We assume that all of the underlying asset prices {F t } t≥0 follow geometric Brownian motion processes and that asset prices at the risk horizon τ (outer scenarios) are evaluated under a realworld measure P. While in theory we can use a single reference point to estimate portfolio losses for all other target points, our experiments show that more reference points can significantly improve accuracy. There are many methods to determine the reference points. For example, the reference points can be chosen equidistantly. In each trial, we sort the samples (x k = F (k) τ , k = 1, 2, . . . , n) and calculate the difference between the maximum and the minimum. Let s be the block number, then samples can be divided equidistantly into s intervals, and each interval has same range. In each block, the intermediate point or endpoints can be chosen as the reference points. In the following numerical example, we implement the estimation by dividing blocks.
Example 3.4. Consider an asset with initial price F 0 = 100, real-world drift µ = 8%, and instantaneous volatility σ 0 = σ 1 = 20%. Let the risk-free continuously compounding interest rate be r = 3%. Construct a portfolio of three partial-time barrier options that can only be knocked in or out on the interval [τ, T ] where risk horizon is τ = 1/52 year and maturity time T = 1/12 year. This model has been studied with least squares Monte Carlo method in Broadie et al. (2015). The portfolio consists of the following positions:
1. Long one down-and-out put option with strike K 1 = 101 and barrier H 1 = 91.
2. Long one down-and-out put option with strike K 2 = 110 and barrier H 2 = 100.
3. Short one down-and-out put option with strike K 3 = 114.5 and barrier H 3 = 104.5.
We aim to estimate the risk measure α = E[(L τ − c) + ], where the threshold is the 95-th percentile of the portfolio loss L τ , i.e. c = VaR 0.95 (L τ ) = 0.3608. Let F T define the minimum asset price on [τ, T ], and F T define the final asset price, then the portfolio loss at time τ is given by
L τ = e −r(T −τ ) E (K 3 − F T ) + I(F T > H 3 ) − (K 2 − F T ) + I(F T > H 2 ) −(K 1 − F T ) + I(F T > H 1 )|F τ − (P 3 − P 1 − P 2 ),
where I(·) is the indicator function, and P 1 , P 2 , P 3 define the purchase prices of three options at time τ , that is
P i = e −r(T −τ ) E E (K i − F T ) + I(F T > H i )|F Q τ , i = 1, 2, 3.
where F Q τ means the outer scenario generated by risk-free interest rate r.
We compare efficiency and accuracy of three methods: (1) standard nested Monte Carlo simulation, (2) least squares Monte Carlo introduced in Broadie et al. (2015), and (3) sample recycling method proposed in this paper. Note that the payoff of a barrier option depends only on the minimum underlying asset price and the final asset price on time [τ, T ]. In the first numerical calculation, we shall simulate these two quantities instead of sampling the entire sample path (c.f. Becker (2010)). Also note that the closed form expression for the portfolio losses L τ given a risk factor scenario can be found in Haug (2007). Therefore, the risk measure α can be precisely computed by the simulation in the outer stage. Details of each method can be found below.
• Standard nested Monte Carlo simulation
It is known from Broadie et al. (2015) that, given a fixed computing budget k = mn, the asymptotically optimal choice to minimize the MSE of the estimator is given by n * = βk 2/3 outer stage scenarios and m * = k 1/3 /β inner stage paths, where β is determined by minimizing the asymptotic MSE and is difficult to derive. In this example, we use the optimized parameter value β * = 0.076 suggested by Broadie et al. (2015). In this numerical example, we set k = 10 6 for the budget allocation, which results in m * = 1, 316 inner paths and n * = 760 outer scenarios. Each scenario or path is based on the simulation of (F T , F T ), and the simulation method can be found in Haug (2007).
• Risk estimation via regression (least squares Monte Carlo)
The reference points are chosen equidistantly. We break the range of asset values from the 760 scenarios into 10 intervals of equal length. We select the right boundary points as sample outer scenarios and generate corresponding inner paths. The portfolio loss is evaluated under each outer scenario and the corresponding set of inner paths. Then we apply the method introduced in Broadie et al. (2015) with the basis function set Φ (2) , including
1, F τ , (F τ − H 1 ) + , (F τ − H 2 ) + , (F τ − H 3 ) + ,
and their corresponding squared functions. We then use the approximate functional relationship by regression to determine portfolio loss under each of the 760 outer scenarios.
• Sample recycling method: Since we can simulate the minimum asset price F T and the final asset price F T on [τ, T ] directly, then the performance of portfolio on [τ, T ] can be determined entirely by the pair Z = (Z 1 , Z 2 ) = (F T , F T ). The joint density function of (F T , F T ) is already known (see for example Becker (2010)) as follows,
p i (z 1 , z 2 ) = 2 √ 2π exp − z 2 − r − 0.5σ 2 σ 1 √ T − τ 2 /2 ( z 2 − 2 z 1 ) exp (−2 z 1 ( z 1 − z 2 )) , where z 2 = ln(z 2 /x i ) σ 1 (T − τ ) , z 1 = ln(z 1 /x i ) σ 1 (T − τ ) .
Then the likelihood can be calculated by (3.3) as follows
p i|1 (z 1 , z 2 ) = exp ln x 1 x i ln √ x 1 x i z 2 /z 2 1 + (r − 0.5σ 2 1 )(T − τ ) σ 2 1 (T − τ ) ln x i z 2 /z 2 1 ln x 1 z 2 /z 2 1 , where (x k = F (k)
τ , k = 1, 2, . . . , n) is an i.i.d sample of F τ and x 1 is the referred scenario. We use the same method to determine the 10 referred outer scenarios as in the least squares Monte Carlo method. The evaluation of portfolio loss is carried out in the way outlined in Section 3.3. Table 1 displays the MSE and run time for the above-mentioned methods over 1000 independent trials. It shows that both sample recycling method and regression consume significantly less time than Monte Carlo approach with MSE of the same order. While the sample recycling method in this example requires more time than the least squares Monte Carlo but achieves higher accuracy. It should be pointed out that the determination of optimal parameter β * for the budget allocation method requires searching over a set of different potential values. These values are dependent on the specific form of risk measure under consideration and only known for a limited number of risk measures. It is often difficult to determine such values for general risk measures.
Estimator
MSE
Time ( It should also be pointed out that the sample recycling method requires fewer reference points to approximate losses. Table 2 shows the MSE and time consumption for two, five and eight reference points. There are a total of 1, 000 independent trials for each case. In comparison, the regression method needs at least 10 sample points because there are 9 basis functions in this example. mγ mδ Time(Sec) 2.2160 × 10 −3 6.8562 × 10 −5 To tie it to the earlier discussion on computational effort, we show in Table 3 computational efforts required for nested simulations. It is clear that in this case that, the simulation time of inner paths and the computation time for inner loop evaluation, mγ, is greater than the computation time of the likelihood, mδ. In this example, we use 5 reference points with the sample recycling method. By definition (3.16) and (3.17), we calculate computational efforts of 1, 000 independent trials given by CE SN = 1000 × 1000 × mγ = 2216.00346 CE SR = 1000 × (5 × mγ + 995 × mδ) = 11.4800 + 68.2193 = 79.6993.
(3.18) Figure 3 shows a comparison of estimations for expected excess loss by regression and the sample recycling method. The light blue line represents the true value of expected excess loss α as a function of asset price F τ . The symbol + shows estimates by the sample recycling method and the dashed line provides estimates by the regression. Both methods produce quite accurate estimates. The regression tends to overestimate for large asset values and underestimate in modest small asset values (between 96 and 100). In this graph, the regression approach is based on 20 equidistant sample points, whereas the sample recycling method uses 5 reference points, which are shown by the symbol •. For the sample recycling methods, we break the range of asset prices into five blocks of equal lengths and use the right-end point as the reference point for each block. One would notice that expected excess losses are either all overestimated or underestimated in each block.
Example 3.5. Consider a portfolio of financial derivatives written on five underlying assets. Assume that the initial assets prices are all F 0 = 100, and that the assets share common real-world drifts of µ = 8% and annual volatility of σ = 20%. The risk-free continuously compounding interest rate is r = 3.5%. The asset price processes are assumed to be mutually independent. Suppose that the portfolio consists of 10 short positions of at-the-money (average price) Asian call options on five underlying assets. All options have the same maturity date T = 1/12 years and the portfolio is evaluated at τ = 1/52 years from now. We want to estimate the expected excess loss ρ = E[(L τ − c) + ] with threshold c is the the 99-th percentile of the portfolio loss, i.e. c = VaR 0.99 = 114.8151. Let F j,t , j = 1, 2, . . . , 5 represent the five underlying assets prices andF j represent the arithmetic price on [τ, T ], then the portfolio loss can be given by
L τ = E 10e −r(T −τ ) 5 j=1 (F j − 100) + |F τ − C
where C is the purchase price of the portfolio, the price of Asian option can be approximated by built-in function of Matlab.
In this example, we use the built-in function asianbylevy of Matlab to approximate the closed form pricing solution of continuous arithmetic Asian options (Lévy, 1992), which give rise to the true value of the loss of the portfolio. Detailed specification of each method is described below.
• Nested Monte Carlo simulation: In this numerical example, we set k = 10 6 for the budget allocation and set n = 1, 000 outer scenarios and m = 1, 000 inner paths to estimate the expected excess loss ρ. The portfolio loss is estimated by simulating the entire sample path of F t as Example 3.1.
• Risk estimation via regression: We choose basis functions up to fifth order polynomials. Specifically, let F j,t , j = 1, 2, . . . , 5 represent the five underlying assets prices, the basis functions contain all the following functions:
1, F j,τ , (F j,τ ) 2 , (F j,τ ) 3 , (F j,τ ) 4 , (F j,τ ) 5 , j = 1, 2, . . . , 5.
We use 50 simulated sample points (each with the inner path number m = 1000) to perform the regression and to get the proxy function. The loss L τ on the sample points is simulated by Monte Carlo, and the rest 950 loss value is approximated by the proxy function.
• Sample recycling method: To calculate the loss of the portfolio, we simulate the entire sample path of the underlying assets. Recall the discussion in Section 3.1, p i|1 (·) can be determined by the density function of Z j = F j,τ +∆t , j = 1, 2, · · · , 5 because of the Markov property. For each underlying asset, the weight used in the evaluation of Asian options is same as Example 3.1
p i|1 (z j ) = exp ln x i x 1 −0.5 ln(x 1 x i ) + (r − 0.5σ 2 )(∆t) + ln(z j ) σ 2 (∆t) , j = 1, 2, . . . , 5,
where ∆t is the time step used to simulate the entire sample path of underlying assets. Finally, we take ∆t = 1/624 in this numerical example. We divide 10 blocks for each underlying asset and choose the intermediate point as the reference point. Then the total number of reference points is 10 × 5 = 50. Table 4 shows that both sample recycling method and regression are more efficient than nested simulation method. The efficiency of sample recycling method is due to the computational effort mδ < mγ (see Table 5 ). From (3.16) and (3.17), we can calculate the follows results CE SN = 5 × 1000 × 1000 × mγ = 5682.7 CE SR = 5 × 1000(10 × mγ + 990 × mδ) = 55.6827 + 290.8549 = 346.5376
In Table 4, the MSE of sample recycling method has the same magnitude with nested simulation method, but the regression method often leads to wrong results because the sample points are insufficient for a basis set containing 26 basis functions. The accuracy of the regression method can be improved by increasing the number of sample points (see Table 7), but each sample points needs computational efforts mγ to simulate the value in once trial, then Table 7 shows that increasing the number of sample also significantly increases the computational efforts. On the contrary, Table 6 shows the MSE and run time when increasing the reference points for each Asian option from 15 to 30 in the sample recycling method, and the results showed that the sample recycling method performs stably using different number of reference points, and the number of reference has less influence to the computational effort . Example 3.6. Another common application of nested Monte Carlo simulation is on the calculation of risk measure for variable annuity guaranteed benefits. Consider one of the most common investment guarantees on variable annuity products, known as the guaranteed minimum withdrawal benefit (GMWB). Suppose that the instantaneous change in fund value is the net effect of proportional return from equity-linking less percentage rider charges and fixed withdrawal where S t is the equity-index driven by (3.11), m f > 0 be the rate per time unit of total fees charged by the insurer, and w be the guaranteed rate of withdrawal per time unit. Let G be the initial deposit, the GMWB rider provides safeguards to the continuous withdrawal until the initial deposit is completely refunded, i.e. the GMWB matures at time T = G/w. In this example, we take F 0 = G, meaning that the policyholder is guaranteed to receive a full refund of his or her premium payments. It is only when the account value is depleted prior to the maturity T that the maximum withdrawal rate w is paid at the cost of the insurer. Therefore, the present value of the cost to an insurer of GMWB rider is given by T 0 e −rs wI(F s ≤ 0)ds, where I(·) is an indicator function. On the other hand, the insurer receives the distribution of fees from the third party fund manager, which are often a fixed percentage of the policyholder's account until the account value hits zero. Thus the accumulated present value of the fee income is given by T 0 e −rs m f F s I(F s > 0))ds. Therefore, the liability of insurer at time τ is given by
dF t = F t S t dS t − m f F t dt − wdt = ((r − m f )F t − w)dt + σF t dW t ,L τ = E T τ e −r(s−τ ) (wI(F s ≤ 0) − m f F s I(F s > 0))ds F τ .
(3.19)
We calculate the risk measure ρ = VaR 0.7 [L τ ] by Monte Carlo and sample recycling method. For the numerical calculation, we take F 0 = G = 1, µ = 0.08, r = 0.05, σ = 0.2, w = 0.1, m f = 0.01. Suppose that the withdrawal benefit expires in T = 10 years and the risk measure is evaluated in τ = 5 years.
• Nested Monte Carlo simulation: We use n = 1000 outer scenarios and m = 1000 inner paths to estimate the risk measure. We estimate the liability by simulating the entire sample path of F t on [τ, T ) with ∆t = 0.05, and risk factor F t is simulated by the following recursion
F h+1 = F h exp((r − m f − 0.5σ 2 )∆t + σ √ ∆tX h+1 ) − w∆t, h = k, k + 1, ..., K. (3.20)
where X 1 , ..., X K are independent draws from a standard normal distribution.
• Risk estimation via regression: We choose basis functions up to fifth order polynomials: 1, F τ , (F τ ) 2 , (F τ ) 3 , (F τ ) 4 , (F τ ) 5 . We use 50 sample points (each with the inner path number m = 1000) to perform the regression. The sample points are chosen the right endpoints equidistantly in each trial.
• Sample recycling method: The liability is determined by the entire sample path of Z = F t , t ≥ τ . From the recursion equation (3.20), we have
(F h+1 + ω∆t)/F h ∼ N ((r − m f − σ 2 2 )∆t, σ 2 ∆t).
then we can calculate the weight as follows
p i|1 (z) = exp ln(x i /x 1 ) σ 2 ∆t − 1 2 ln(x 1 x i ) − (r − m f − 1 2 σ 2 )∆t + ln(z + w∆t) .
The first reference point x ref τ , k = 1, 2 · · · , n, and the rest of reference points are determined by x ref
k = inf{x ∈ (x 1 , x 2 , · · · , x n )|x ref k−1 /x < 1.1}.
In this example, the number of reference points is 50 or so.
Estimator
VaR 0.7 Stand. Dev. Time (secs) Standard nested estimator 1.0050 × 10 −2 5.6411 × 10 −3 3890.5233 Sample recycling method 1.0152 × 10 −2 6.0046 × 10 −3 299.9947 Regression 4.5300 × 10 −2 9.8774 × 10 −3 220.3021 Table 9: The values of mγ and mδ.
In Table 8, we calculate the risk measure VaR 0.7 by three methods. In this GMWB example, we cannot find the analytical solution, then we use the standard deviation (SD) of 1000 trials to present the stability. Both nested Monte Carlo method and sample recycling method show high accuracy. However, the sample recycling method can be less time consuming than the nested Monte Carlo method. Table 9 gives the values of mγ and mδ, the average time of 1000 independent trials. It is shown that the simulation and calculation process of g(Z i,j ) of standard nested Monte Carlo method is equal to 64 times of that of p i|1 (·). It is important to note that we ran 1000 independent trials and used 50 sample points in the sample recycling method. It follows from (3.16) It is clear that the main time consumption is from the simulation and calculation of g(Z i,j ).
Non-parametric method
In practice, equity scenarios are typically generated from a sophisticated economic scenario generator. The underlying stochastic models are sometimes unknown to end users of equity scenarios. Therefore, it is possible that the likelihood (distorted weight) in (3.4) is not known by analytical formula. In this section, we develop a non-parametric sample recycling method, which does not require prior knowledge about the underlying stochastic model. It is particularly useful when the likelihood cannot be derived explicitly or when underlying asset paths are generated by the empirical data rather than a specific model. .
Likelihood ratio estimation
In this section, we introduce a naive estimation method for the likelihood ratio function p i|1 (·). Despite its simplicity, this method demonstrates high accuracy for loss estimation by numerical examples.
Algorithm 3 Estimate likelihood ratio p i|1 (·) usingp i|1 (·)
Generate n outer scenarios Conditioned on i-th ( i = 1, · · · , n) outer scenario, generate m i.
1 , n (1) 2 , · · · , n (1) l ) for i = 1 to n do
Find the counts of target sample points in each set (n
(i) 1 , n (i) 2 , · · · , n (i) l ) for j = 1 to m do if y k+1 = F (i,[j]) k+1 in a-th set then p i|1 (y k+1 ) = n (i) a n (1) a end if end for p i|1 (·) ←p i|1 (·) end for
To illustrate this method, we only consider one risk factor {F t } t≥0 as Section 3.1 and set an independent and identically distributed inner loop sample {F (i,j) k+1 } m j=1 for i = 1, · · · , n, generated from outer scenario F shall be used as a reference point, while others are considered as target points. The method can be broken down into the following steps.
1. Sort the data set for the sample point in an increasing order:
{F (i,[j]) k+1 } m j=1 where [j]
indicates the j-th order statistic. In other words, F
(i,[1]) k+1 ≤ F (i,[2]) k+1 ≤ · · · ≤ F (i,[m]) k+1 .
2. Seperate the set of integers {1, · · · , m} into l sets with break points m 0 = 0, m 1 , m 2 , · · · , m l = m. Denote the a-th interval of risk factor by
I a := F (1,[m a−1 ]) k+1 , F (1,[ma]) k+1
.
3. Count the number of observations of samples from target scenario and reference scenario in each interval respectively. Denote the counts by (n
l ).
We can construct the following likelihood ratio estimator p i|1 (y k+1 ) = n (i) a n (1) a for y k+1 ∈ I a , (4.1) resulting in an empirical likelihood ratio function. If we choose intervals based on quantiles, we could construct the intervals such that there are equal numbers of points in each interval, i.e. n
(1) a = m/l for each a, further simplifying the estimator to
p i|1 (y k+1 ) = l · n (i) a m for y k+1 ∈ I a .
Proposition 4.1.
Given i.i.d sample {F (i,j)
k+1 } m j=1 for i = 1, · · · , n, the estimator (4.1) converges pointwise to its true value for each valid input y k+1 ∈ I a :
p i|1 (y k+1 ) = n (i) a n (1) a → p i|1 (y k+1 ) as m → +∞, l → +∞.
Proof. By Glivenko-Cantelli Theorem, an empirical distribution function uniformly converges to the true cumulative density function as the number of i.i.d observations approaches infinity. Let Q i denote the cumulative distribution function for F k+1 under measure Q i and Q i be the empirical distribution function. Then for each scenario i,
|| Q i − Q i || ∞ → 0.
Notice that (4.1) can be written as
p i|1 (F k+1 ) = n (i) a n (1) a = n (i) a /m n (1) a /m = Q i (F (1,[ma]) k+1 ) − Q i (F (1,[m a−1 ]) k+1 ) Q 1 (F (1,[ma]) k+1 ) − Q 1 (F (1,[m a−1 ]) k+1 ) ,
As m, l → ∞, we obtain
Q i (F (1,[ma]) k+1 ) − Q i (F (1,[m a−1 ] k+1 ) F (1,[ma] k+1 − F (1,[m a−1 ] k+1 → p i (F k+1 ).
Therefore, their ratio approaches the ratio of limit at each given F k+1 ,
p i|1 (F k+1 ) → p i (F k+1 ) p 1 (F k+1 ) = p i|1 (F k+1 ).
To illustrate the estimation, we choose two outer scenarios F τ = 99 (reference) and F τ = 99.2 (target) with τ = 1/52 in the geometric Brownian motion, and we generate 1000 sample points for each scenario. The histogram and the empirical likelihood ratio function demonstrate the result of Algorithm 3 in Figure 4. Five intervals are constructed based on the 20-th, 40-th, 60-th, 80-th quantiles of reference sample points, which means l = 5. In this example, the left figure shows that (n (i) 1 , · · · , n (i) 5 ) = (122,174,184,243,272) and n (1) k = 200 for k = 1, · · · , 5. We overlay the histograms and theoretical density functions for both reference point (blue) and target point (red) to show their differences in the left plot while the empirical likelihood ratio function (solid-line) Figure 4: The calculation of weight by nonparametric method and the true theoretical likelihood ratio (dashed-line) are shown on the right. Keep in mind that the true likelihood ratio function is typically not known in advance. The graph shows a reasonable estimate from empirical data.
Note that estimating probability density is a common question in machine learning. While this paper only discusses a naive method, we believe that many other methods can be used to estimate the likelihood, such as least square importance fitting (Kanamori et al., 2009), kernel mean matching (Huang et al., 2007), Kullback-Leibler importance estimation procedure and so on. Sugiyama et al. (2012) offers detailed accounts of machine learning methods.
Non-parametric sample recycling method
In the non-parametric setting, we estimate the theoretical likelihood ratio p i|1 by an estimated p i|1 () in the estimator (3.4). Therefore, We obtain the empirical sample recycling estimator of L i ,
L i = 1 m m j=1 p i|1 (Z 1,j )g(Z 1,j ),(4.2)
and the non-parametric sample recycling estimator of the risk measure ρ is given bỹ
ρ NSR = 1 n n i=1 f (L i ).
Note that there is an additional source of randomness in this estimator -likelihood ratio estimation. As the estimate requires no information about the underlying stochastic model, we do not expect this estimator to outperform the sample recycling method in the previous section. Nevertheless, the estimator (4.2) offers an appealing non-parametric framework when equipped with a reasonably fast and accurate algorithm to estimate likelihood ratios.
To test the accuracy and efficiency of this non-parametric method, we re-run inner loop estimations in Examples 3.4-3.6 and compare results from the non-parametric sample recycling (NSR) Figure 5 that the SR method leads to fairly accuracy results even with only 5 reference points and that the NSR method produce results with larger estimation errors. However, as we increase the number of reference points to ten, we observe from the right panel of Figure 5 that results from the NSR method are much closer to those from the SR method and hence improve significantly. We can apply the same technique to the other two examples. Figure 6 shows the comparison of results by both methods for the Asian option portfolio and the GMWB liability. In the estimation of portfolio loss in the basket of Asian options, we use 10 reference points to estimate losses on 100 equidistant equity values over the range [95.1, 105] and midpoints as the reference points for all intervals. In the estimation of the GMWB liability, we use 30 reference points to estimate the GMWB liability for 100 equidistant equity values over the range [0.03, 6]. However, for this example we take a different approach to choose reference points. The reference points are chosen by right points in intervals of length determined by a geometric series. We first set the first reference point x 1 = 6, and the rest of reference points are given by x k = inf{x ∈ (x 1 , x 2 , · · · , x n )|x k−1 /x < 1.1}. The right panel of Figure 6 shows estimated GMWB liabilities based on 100 equidistant points of asset prices. The graph clearly shows that both methods produce very similar results. Figure 6: Non-parametric sample recycling estimations of portfolio loss in Examples 3.5 and 3.6
Number of reference points MSE Time (secs) 5 × 5 6.5915 × 10 −3 283.6537 Table 11: MSE and time ofρ NSR for Example 3.5.
paring Table 10 with Table 2, we observe results by both the non-parametric estimatorρ NSR and the original estimatorρ SR . We use 30 reference points in the inner estimation of non-parametric method to guarantee the accuracy. Table 10 indicates that it takes more time than sample recycling method because of the increased number of reference points. Nonetheless, the NSR method still outperforms the standard nested Monte Carlo. Table 11 is the analogue of Table 4 for the non-parametric method. We use 5 reference points to estimate each Asian option and hence the total number of reference points is 5 × 5. Table 12 corresponds to Table 8 with the non-parametric method. Both of these examples show that the non-parametric method has higher efficiency and accuracy than standard Monte Carlo. As shown in previous numerical examples, the non-parametric sample recycling method is easy to implement. While it does not achieve the same level of accuracy as the original sample recycling method given a fixed set of reference points, one may have to resort to the non-parametric approach as the underlying model is unknown. The examples provide evidence to show that the non-parametric approach is a viable alternative whose accuracy improves with the size of reference points.
VaR
Stand Dev
Time (secs) 0.83757 × 10 −2 6.1827 × 10 −3 368.8039 Table 12: Standard deviation and run-time ofρ NSR for Example 3.6.
Conclusion
Most of existing techniques to reduce run-time for nested simulation are based on the replacement of inner loop simulations with curve fitting. The essence of these techniques is to develop a functional relationship between risk factors (equity values, interest rates, etc) and target features (insurance liability, Greek values) of inner loop calculations. Such a functional relationship can be approximated by multivariate interpolation or smoothing techniques such as least squares Monte Carlo. Nonetheless, these techniques often require a large size of economic scenarios to develop accurate enough functional relationships, which could also be costly to begin with. This paper proposes a new approach based on an entirely different strategy, which is to avoid approximate functional relationship and instead to save time by reducing repeated re-sampling of economic scenarios. The technique is to generate sample of risk factors under a small set of probability measures and recycle them by twisting likelihood ratios under other probability measures. The advantage of this approach is to reduce the number of sample generation for risk factors and subsequent inner loop evaluations. The disadvantage of such an approach is that the reduction of computational burden is achieved at the expense of increased sampling errors. This method is particularly suitable for long term products that require heavy computation for inner loop evaluation. While we have shown analytical solutions to distorted weights for various parametric models, we also consider the application of non-parametric sample recycling method to settings where the underlying model is either unknown or too complicated. The non-parametric is shown to be able to reproduce results, free of any information about the underlying model. It is less accurate than the sample recycling method but can be improved with an increased number of reference points. We only present a naive version of non-parametric likelihood ratio estimation as a proofof-concept. However, there is a rich body of literature on machine learning techniques that can be used to estimate density ratios. Future work is needed to improve the naive method with more sophisticated machine learning for better accuracy and efficiency.
A Calculations
Example A.1. In (2.1) and (2.3), we assume that X is given by a uniform random variable on [−1, 1] and Z is a standard normal random variable. Consider expected value of loss where the loss is determined by L = E[g(Z)|X] = E[ 2/π exp(−2(Z − X))|X]. It follows from Proposition 3.2 that Var( ρ SN ) = 1 nm 1 √ π Φ( 8/9) − 0.5 − (Φ(2/ √ 5) − 0.5) 2
and Var( ρ SR ) = 1 mn 2 [B 2 + (n − 1)A 2 + 2(n − 1)C + (n 2 − 3n + 2)D − (B 1 + (n − 1)A 1 ) 2 ], where A l = 1 −1 1 −1 2 π l 1 4 √ 4l + 1 exp − 1 8l + 2 (3l 2 + l)x 2 i + (3l − 13l 2 )x 2 1 + 10l 2 x 1 x i dx 1 dx i B l = 2 π l π 2l Φ( 4l/4l + 1) − 0.5 , l = 1, 2
C = 1 −1 1 −1 1 6π exp − 4 9 x 2 i + x 2 1 − x 1 x i dzdx 1 dx i . D = 1 −1 1 −1 1 −1 1 12π exp − 4 9 (x 2 i + x 2 j ) − x 1 (x i + x j ) + 3 2 x 2 1 + x i x j 9 dx 1 dx i dx j .
Consider an independent and identically distributed sample of X denoted by (X 1 , X 2 , ...X n ). For each given X i , we have (Z − X i ) follows a normal distribution with mean −X i and variance 1. We can therefore determine the coefficients.
p i|1 (Z) = φ(Z + X i ) φ(Z + X 1 ) .
The follows gives the calculations of A l , B l , C, D.
B l = 1 −1 1 2 +∞ −∞ 2 π l 1 √ 2π exp −2l(x − z) 2 − z 2 2 dzdx = 1 −1 1 2 +∞ −∞ 2 π l 1 √ 2π exp − 1 2 √ 4l + 1z − 4l √ 4l + 1 x 2 − 2l 4l + 1 x 2 dzdx = 1 −1 2 π l 1 2 √ 4l + 1 exp − 2l 4l + 1 x 2 dx = 2 π l π 2l √ 4l/4l+1 − √ 4l/4l+1 1 √ 2π exp − x 2 2 dx
(a) as examples. Then, we can approximate the loss L i under the i-th outer loop scenario by
Figure 2 :
2Variances of ρ SN and ρ SR .
Figure 3 :
3Illustration of approximation of loss in barrier options.
1 is the maximum value of the i.i.d samples x k =: F (k)
2
, . . . , n
Figure 5 :
5Non-parametric sample recycling estimation of portfolio loss in Example 3.4 method with those by the sample recycling (SR) method in (3.4). For Examples 3.4, we consider losses of the barrier option portfolio for 100 equidistant points in the range of equity price [91, 110.8]. For both SR and NSR methods, we choose the same set of reference points to estimate the corresponding L τ of other points in each example. We always use l = 5 intervals for counting observations to estimate the corresponding likelihood ratios in all examples. Figure 5 compares estimations of L i by both methods for the barrier option portfolio. The right endpoint is chosen as the reference point in each interval. It is clear from the left panel of
Table 2 :
2MSE and run time for sample recycling method with various reference points
Table 3 :
3The values of mγ and mδ.Underlying Asset Price F
τ (ω)
90
95
100
105
110
115
Portfolio Loss L(ω)
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
True Loss
Sample Recycling Method
Regression
Reference Points of SRM
Table 4 :
4MSE and run time for Asian option portfolio with 1, 000 independent trials.mγ
mδ
Time (secs) 1.113654 × 10 −3 6.1233 × 10 −5
Table 5 :
5Comparison of computational efforts
Table 6 :
6MSE and run time for sample recycling method with different reference pointsNumber of reference points MSE
Time (secs)
75
5.3095
403.114
100
1.5625
580.9215
125
5.7268 × 10 −2
826.877
150
3.8298 × 10 −2
1022.6205
Table 7 :
7MSE and run time for regression with different sample points
Table 8 :
8The standard deviation and time for the GMWB example with 1000 independent trials.mγ
mδ
Time (secs) 3.8349 × 10 −3 5.9691 × 10 −5
CE SN = 1000 × 1000 × mγ = 3834.95 CE SR = 1000(50 × mγ + 950 × mδ) = 191.745 + 56.70645 = 248.45.and (3.17)
that computational efforts are given by
i.d. sample points {F(i,j)
k+1 } m
j=1 ,
Sort the sample points in increasing order {F
(i,[j])
k+1 } m
j=1
Separate the reference sample points {F
(1,[j])
k+1 } m
j=1 into l sets and find the counts of each set
(n
(1)
Table 10: MSE and time ofρ NSR for Example 3.4.To further illustrate the implement of the NSR method, we extend these numerical examples further to show the computation of risk measures ρ by the non-parametric estimatorρ NSR . Com-Number of reference points
MSE
Time(Sec)
30
2.9717 × 10 −5
92.0662
95
100
105
F
-20
-10
0
10
20
30
40
Asian option
SR
NSR
Reference points
0
2
4
6
F
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
GMWB
SR
NSR
Reference points
Note that Z is defined for simplicity; in general, it can also contain path-dependent situations, for example, the average underlying price in Asian options.
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|
[] |
[
"UNIFIED STATISTICAL INFERENCE FOR A NOVEL NONLINEAR DYNAMIC FUNCTIONAL/LONGITUDINAL DATA MODEL",
"UNIFIED STATISTICAL INFERENCE FOR A NOVEL NONLINEAR DYNAMIC FUNCTIONAL/LONGITUDINAL DATA MODEL"
] |
[
"Lixia Hu \nShanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †\n\n",
"Tao Huang \nShanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †\n\n",
"Jinhong You \nShanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †\n\n"
] |
[
"Shanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †\n",
"Shanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †\n",
"Shanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †\n"
] |
[] |
In light of recent work studying massive functional/longitudinal data, such as the resulting data from the COVID-19 pandemic, we propose a novel functional/longitudinal data model which is a combination of the popular varying coefficient (VC) model and additive model. We call it Semi-VCAM in which the response could be a functional/longitudinal variable, and the explanatory variables could be a mixture of functional/longitudinal and scalar variables. Notably some of the scalar variables could be categorical variables as well. The Semi-VCAM simultaneously allows for both substantial flexibility and the maintaining of one-dimensional rates of convergence. A local linear smoothing with the aid of an initial B spline series approximation is developed to estimate the unknown functional effects in the model. To avoid the subjective choice between the sparse and dense cases of the data, we establish the asymptotic theories of the resultant Pilot Estimation Based Local Linear Estimators (PE-BLLE) on a unified framework of sparse, dense and ultra-dense cases of the data. Moreover, we construct unified consistent tests to justify whether a parsimony submodel is sufficient or not. These test methods also avoid the subjective choice between the sparse, dense and ultra dense cases of the data. Extensive Monte Carlo simulation studies investigating the finite sample performance of the proposed methodologies confirm our asymptotic results. We further illustrate our methodologies via analyzing the COVID-19 data from China and the CD4 data.
| null |
[
"https://arxiv.org/pdf/2007.01784v1.pdf"
] | 220,347,078 |
2007.01784
|
cd1b78b8ae3e737c4e3c9566d1780a9ee799b822
|
UNIFIED STATISTICAL INFERENCE FOR A NOVEL NONLINEAR DYNAMIC FUNCTIONAL/LONGITUDINAL DATA MODEL
Lixia Hu
Shanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †
Tao Huang
Shanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †
Jinhong You
Shanghai Lixin University of Accounting and Finance * and Shanghai University of Finance and Economics †
UNIFIED STATISTICAL INFERENCE FOR A NOVEL NONLINEAR DYNAMIC FUNCTIONAL/LONGITUDINAL DATA MODEL
arXiv:2007.01784v1 [stat.ME] 3 Jul 2020 Submitted to the Annals of Statistics
In light of recent work studying massive functional/longitudinal data, such as the resulting data from the COVID-19 pandemic, we propose a novel functional/longitudinal data model which is a combination of the popular varying coefficient (VC) model and additive model. We call it Semi-VCAM in which the response could be a functional/longitudinal variable, and the explanatory variables could be a mixture of functional/longitudinal and scalar variables. Notably some of the scalar variables could be categorical variables as well. The Semi-VCAM simultaneously allows for both substantial flexibility and the maintaining of one-dimensional rates of convergence. A local linear smoothing with the aid of an initial B spline series approximation is developed to estimate the unknown functional effects in the model. To avoid the subjective choice between the sparse and dense cases of the data, we establish the asymptotic theories of the resultant Pilot Estimation Based Local Linear Estimators (PE-BLLE) on a unified framework of sparse, dense and ultra-dense cases of the data. Moreover, we construct unified consistent tests to justify whether a parsimony submodel is sufficient or not. These test methods also avoid the subjective choice between the sparse, dense and ultra dense cases of the data. Extensive Monte Carlo simulation studies investigating the finite sample performance of the proposed methodologies confirm our asymptotic results. We further illustrate our methodologies via analyzing the COVID-19 data from China and the CD4 data.
In light of recent work studying massive functional/longitudinal data, such as the resulting data from the COVID-19 pandemic, we propose a novel functional/longitudinal data model which is a combination of the popular varying coefficient (VC) model and additive model. We call it Semi-VCAM in which the response could be a functional/longitudinal variable, and the explanatory variables could be a mixture of functional/longitudinal and scalar variables. Notably some of the scalar variables could be categorical variables as well. The Semi-VCAM simultaneously allows for both substantial flexibility and the maintaining of one-dimensional rates of convergence. A local linear smoothing with the aid of an initial B spline series approximation is developed to estimate the unknown functional effects in the model. To avoid the subjective choice between the sparse and dense cases of the data, we establish the asymptotic theories of the resultant Pilot Estimation Based Local Linear Estimators (PE-BLLE) on a unified framework of sparse, dense and ultra-dense cases of the data. Moreover, we construct unified consistent tests to justify whether a parsimony submodel is sufficient or not. These test methods also avoid the subjective choice between the sparse, dense and ultra dense cases of the data. Extensive Monte Carlo simulation studies investigating the finite sample performance of the proposed methodologies confirm our asymptotic results. We further illustrate our methodologies via analyzing the COVID-19 data from China and the CD4 data.
cases of different countries are recorded and made available. When a variable is measured or observed at different times, the variable is usually treated as a function of time. As a result, the variable is called a functional variable, the data for the variable are called functional data and the related statistical analysis is called functional data analysis (FDA) ( [38]). The functional data and corresponding FDA have been successfully applied to explore the interactions and co-movements among a group of temporally evolving subjects. Several monographs by [31], [32] and [9] provide comprehensive discussions on the methods and applications. More recent work about FDA could refer to [38].
According to [38], usually, the functional data could be divided into two cases: sparse and dense. Sparse functional data usually occurs in longitudinal studies where subjects are measured at different time points and the number of measurements for each subject is often bounded away from infinity. Inversely, in the dense functional data the number of measurements of each subject tends towards to infinity. In theory, the difference between sparse (longitudinal) and dense function data is clear. However, due to the limitations of humans, the observations in real data sets could not be infinite and are definitely finite. Therefore, the edge of sparse (longitudinal) and dense function data in practice is vague in some scenarios, especially when the number of measurement of each subject is moderate or different subjects have different numbers of measurements.
In many functional/longitudinal studies, repeated measurements within each subject are possibly correlated with each other, but different subjects can be supposed to be independent. One approach to take intra-subject variation into account is the mixed-effects model [42], which decomposes regression function into a fixed population mean and a subject-specific random trajectory with zero mean. For sparse and dense functional data, [19] considered a mixed-effects nonparametric regression model absence of covariates, and showed that the asymptotic distributions of kernel estimators are essentially different in these two situations. Therefore, a subjective choice between sparse and dense cases may lead to erroneous conclusion. To evade this problem, they proposed a self-normalized method, which can deal with sparse and dense functional data in a unified framework. Furthermore, [4] generalized the results of [19] to a mixed-effects VCM presence of covariates with sparse or dense functional data. Lately, [47] provided a comprehensive perspective that deals with a general weighing scheme on a unified paltform for all types of sampling plan, including sparse, dense and ultra dense case. Motivated by a monotone relationship between gray matter volume and age in the older population, [5] considered sparse and dense cases on a unified framework under monotone constraint of the mean function. The research work of [47,5] has focused on the statistical inference about mean function of the underlying process. To the best of our knowledge, there exists no further development about unified inference parallel to [47] for nonparametric regression model presence of covariates, a common case in practice.
In the analysis of longitudinal data, a varying-coefficient models (VCM) enjoying flexibility, parsimony and interpretability, is a widely-used nonparametric regression method. One can refer to [8,10,12,15,16,17,29,35]. An additive model (AM) is another popular nonparametric regression method, which has been studied by [1,3,25,27,28,30,43,44,34]. Recently, [13,14,46,48] have investigated a novel nonparametric regression method, named the varying-coefficient additive model (VCAM), which can be viewed as a generalization of the VCM and AM. Let T ij be the observation time when the jth measurement of the ith subject is made, Y ij and X i (T ij ) := X ij be the response and p-covariates for the ith subject at time T ij , respectively. Then {(Y ij , X ij , T ij ) ; i = 1, ..., n; j = 1, ..., m i } constitutes a longitudinal/functional sample from n randomly selected subjects with m i repeated measurements of the ith subject. The VCAM for longitudinal/functional data is proposed by [14] as below
(1.1) Y ij = α 0 (T ij ) + p k=1 α k (T ij ) β k (X ijk ) + ν i (T ij ) + ε ij ,
with the abuse of notations. Here ν i (T ij ) is the subject-specific random trajectory at observation time T ij , and {ε ij } are i.i.d. random measurement errors. The multiplicative factors α k (k = 1, ..., p) and β k (k = 1, ..., p) are called to be varying-coefficient component functions and additive component functions, respectively, and α 0 is a trend term. Obviously, the VCAM (1.1) reduces to an AM provided that each α k (k = 0, ..., p) is time-invariant, whilst it becomes a VCM if each β k (k = 1, ..., p) has a simple linear form. Therefore, it can be said that the VCAM is a kind of hybird of an AM and a VCM, enjoying more flexibility, which can greatly decrease the bias of model misspecification. On the other hand, it is hard to address how to choose between an AM and a VCM in practice. The general type of a VCAM provides a data-driven method to decide which model may be more suitable for the real-life data at hand. However, the product forms of α k and β k in (1.1) exclude the discrete covariates from this model. It will vastly limit the scope of applications because categorical variables are often important influence factors in the practical fields. To accommodate both discrete and continuous covariates in regression model, in this paper we consider a mixed-effects semi varying-coefficient additive model (Semi-VCAM) to analyze longitudinal data. Let Z i (T ij ) := Z ij = (1, Z ij,1 , ..., Z ij,q ) τ be a {q + 1}-vector of discrete covariates observed at time T ij , and α 0 (t) = (α 00 (t) , α 01 (t) , ..., α 0q (t)) τ is the vector of varying-coefficient functions for Z that is i.i.d. with Z i , and α 00 denotes the trend function. Then, we generalize the VCAM (1.1) to a Semi-VCAM as below,
(1.2) Y ij = Z τ ij α 0 (T ij ) + p k=1 α k (T ij ) β k (X ijk ) + ν i (T ij ) + σ (T ij ) ε ij ,
where the subject-specific random trajectory
ν i (t) satisfies E [ν i (t)] = 0 and covariance function γ (t, t ′ ) = E [ν i (t) ν i (t ′ )]
, {ε ij } are random errors such that E (ε ij ) = 0 and E(ε 2 ij ) = 1, and σ (t) is a smooth standard deviation function of process ε (t). Note that (1.2) allows a mixture of functional/longitudinal predictors and scalar covariates, and it reduces to a partial linear additive model (PLAM), if each varying-coefficient function is time-invariant. Compared with the model (1.1) studied in [14], (1.2) allows categorical covariates and heteroscedasticity as time elapsed. Meanwhile, in this paper we also take into account intra-subject correlation, which was merged into random errors in [14]. Therefore, Semi-VCAM is a more refined nonparametric model than VCAM (1.1) in the analysis of longitudinal data.
As a global smoothing technique, spline method is widely used to fit a smooth nonparametric function because of its merit of cost saving. But it usually has no asymptotic distribution due to absence of decomposition of bias part and variance part, unless the asymptotic bias is smaller of high order than the asymptotic variance. All of the existing research literatures about VCAM are based upon a spline method, [46,48] provide no asymptotic distributions of estimators, whilst [13,14] obtain the asymptotic distributions under the condition that the asymptotic bias can be ignored. Alternatively, kernel method is a local smoothing tool, based upon which we can construct the involved asymptotic distribution presence of asymptotic bias, and make statistical inference on certain interested function. Specially, local linear smoothing is popular due to its nice properties, such as design adaption, good boundary performance, and statistical efficiency in an asymptotic minimax sense, see [7] for more details.
In this paper, we build a pilot estimation based local linear estimator (PEBLLE) for varying-coefficient component functions and additive component functions, respectively. The proposed estimation method has wide applicability, including sparse data and dense data, and the data presence of functional/longitudinal covariates and scalar variables. We have shown the consistency of PEBLLE, and as a main contributor of this paper, we con-struct the asymptotic distributions on a unified framework for sparse, dense and ultra dense data. For the convenience of concise presentation, we only consider the same weight to each subject (SUBJ), and our theoretical results can be viewed as a generalization of [47] to nonparametric regression model presence of covariates with SUBJ scheme. Another intriguing question is how to judge a general Semi-VCAM or a submodel is sufficient. To this end, we develop two hypothesis testing to decide whether each varying-coefficient component functions is time-invariant (i.e., a PLAM or especially, an AM if absence of Z covariates), or whether each additive component function has linear form (i.e., a VCM). It has been shown that the proposed testing procedure is consistent on a unified framework of sparse, dense and ultra dense case of data.
In the empirical studies, we consider the new coronavirus disease (COVID-19) breaking out in December 2019, and apply our method to analyze the growth rate of cumulative confirmed (GRCC) cases in China except Hubei Province, Tibet, Macao, Taiwan and Hong Kong. We collect the data from https://github.com/CSSEGISandData/COVID-19, and take sample period from January 22th, 2020 and April 8th, 2020. To model GRCC, four function covariates and one scalar covariate (population size) are chosen. The testing procedures show that a Semi-VCAM is necessary for this dataset. Another example is CD4 data from the Multicenter AIDS Cohort Study (a data set in the R package "timereg"), which has been studied by [15,45]. In this model, smoke status (1 for smoker and 0 for nonsmoker) is included. Employing Semi-VCAM, the testing procedure shows a VCM is sufficient, which verifies the rationality of the research results in [15].
The rest of this paper is organized as follows. Section 2 proposes a pilot estimation based local linear smoothing method and Section 3 presents a series of the asymptotic theories. In Section 4, we propose a testing procedure to justify whether a VCM or a PLAM is sufficient or not, and show its asymptotic properties. Section 5 speaks about the implementation of the proposed method. Extensive simulation studies investing the finite-sample performance and real data applications illustrating our methodologies are considered in Section 6. Brief remarks are concluded in Section 7. The requirements for validity of the asymptotic theories are presented in the Appendix, and the main proofs are relegated to the Supplementary Material.
Estimation Method.
We assume that observation time {T ij } are i.i.d. copies of T , which has a density function f T with a bounded support, say [a, b]. The vector of covariates X i = (X i1 , ..., X im i ) τ for the i-th subject is randomly drawn from a p-dimension stochastic process X(T ), of which the k-th element X k (T ) has a marginal density function f X k with support S k . To identify the trend term and product terms in model (1.2), we impose the conditions E [α k (T )] = 1 and E [β k (X k )] = 0 (k = 1, ..., p), a similar practice with [48,14].
In this section, we develop pilot estimation based local linear estimators (PEBLLEs) for α k and β k . Suppose that β k 's are known, then Semi-VCAM (1.2) become a VCM, and the LLE of α k 's are easily obtained. Let
a 0t = {a 00 (t) , ..., a 0q (t)}, a t = {a 1 (t) , ..., a p (t)} τ , b 0t = {b 00 (t) , ..., b 0q (t)}, b t = {b 1 (t) , ..., b p (t)}, where t is any interior point on the interval [a, b].
We solve the optimization problem as below
Q 1 (â t ,b t ) = min t∈(a,b) Q (a t , b t ) = n i=1 1 m i m i j=1 Y ij − q l=0 Z ijl {a 0l (t) + b 0l (t) (T ij − t)} − p k=1 {a k (t) + b k (t) (T ij − t)} β k (X ijk ) 2 k h C (T ij − t) , (2.1)
where k h (·) = k (·/h) /h for certain kernel function k. Then the LLE of varying-coefficient component functions are given byα 0l (t) =â 0l for l = 0, ..., q andα k (t) =â k for k = 1, ..., p.
On the other hand, if α k 's are known, then Semi-VCAM reduces to an AM. Suppose that we have got estimation of additive component functions except β k , denoted asβ l for l = k, and consider the following minimum problem
Q 2 (â x ,b x ) = min S k Q 2 (a x , b x ) = n i=1 1 m i m i j=1 Ŷ ij,−k − α k (T ij ) {a x + b x (X ijk − x)} 2 k h A (X ijk − x) , where x is any interior point of support S k of X k , andŶ ij,−k = Y ij − Z τ ij α 0 (T ij ) − l =k α l (T ij )β l (X ijl ). Then, the LLE of β k is given byβ k (x) = a x .
However, both α k and β k are unknown, implying the above-mentioned estimation methods are infeasible. To this end, we propose pilot estimations of additive component functions. Similar to [14], we view multiplicative term α k (t) β k (x) as a general bivariate function, say g k (t, x), and estimate it using tensor B-spline method. Specifically, for any given t and x, the tensor product is defined as
T (t, x) = B k,A (x) ⊗ b C (t),
where ⊗ means the Kronecker product of matrices or vectors, and b C (t) and B k,A (x) denote the B-spline basis approximating α k (t) and β k (x), respectively.
Then, we approximate α 0l (t) ≈ γ τ 0l b C (t) for l = 0, ..., q, and g k (t,
x k ) ≈ γ τ k T k (t, x k ) for k = 1, .
.., p. Solving the following optimization problem
(2.2) min γ n i=1 1 m i m i j=1 Y ij − γ τ 0 Z ij ⊗ b C (T ij ) − p k=1 γ τ k T k (T ij , X ijk ) 2 , we got he estimator of g k asĝ k (t, x k ) =γ τ k T k (t, x k ), whereγ k is given by (2.2). Furthermore, the identification condition E[α k (T ij )] = 1 implies β k (x) = b a g k (t, x)f (t)dt.
Hence, a pilot estimator of additive component function β k can be given by
(2.3)β k,P (x) = 1 N n i=1 m i j=1γ τ k T k (t ij , x) , k = 1, ..., p,
where N = n i=1 m i is total observation, and subscript 'P' means pilot estimator. Now, we can define the PEBLLEs of varying-coefficient component functions and additive component functions.
• Substituting the pilot estimatorsβ k,P (k = 1, ..., p) into the objective function Q 1 , we obtain the PEBLLE of α 0l for l = 0, ..., q and α k for k = 1, ..., p, and still denote them asα 0l andα k , respectively. • In the objective function Q 2 , we take the PEBLLEs of varying-coefficient component functions as their pilot estimations, andβ l,P (l = k) as the pilot estimators of additive component functions, and yield the PE-BLLE of β k , still write asβ k .
Remark 1. Compared to the spline-based estimators of [14], the PE-BLLE can provide asymptotic distribution with the specific expression of asymptotic bias, and make inference on the confidence interval of component functions. Meanwhile, our estimation methodologies adapt to both sparse and dense longitudinal/functional data, and have wide application in the real word.
3. Asymptotic Results. In this section, we will present the asymptotic distribution and convergence rate of PEBLLE on a unified platform for different sampling plans.
Asymptotic Properties of Varying-coefficient Component Functions
. LetN H = 1 n n i=1
1 m i −1 be the harmonic mean of {m 1 , ..., m n }, and denote the interior knots number of B-spline basis b C (t) and B k,A (x) (k = 1, ..., p) as K C and K A , respectively. Then, based upon the result of Proposition 1 presented in Supplement A, Theorem 3.1 shows the uniform convergence rates of PEBLLEs of varying-coefficient component functions.
Theorem 3.1. Under Assumption (A1) -(A6) and (A9), if K C K A = o nN h 4 C and K −r+1/2 A + K −r C = o h 2 C , then we obtain that sup t∈(a,b) |α 0l (t) − α 0l (t)| = O p h 2 C + K A K −r C + K −r A + log n n 1 + 1 N H h C , sup t∈(a,b) |α k (t) − α k (t)| = O p h 2 C + K A K −r C + K −r A + log n n 1 + 1 N H h C ,
where l = 0, ..., q and k = 1, ..., p.
Remark 2. From Theorem 3.1, we notice that the variance term obtains a nonparametric rate of convergence log n/(nN H h C ) provided that Remark 3. Similar to [47], we split data into sparse, dense or ultra dense according to the ratioN H /n 1 2r tends to 0, a nonzero constant or ∞ as n → ∞. In fact, we give a more general method of partitioning data in the sense that the same split with [47] is used if r = 2.
N H /n 1 2r → 0 and K C ≍ nN H 1 2r+1 , where "a ≍ b"
Let α (t) = {α 00 (t) , α 01 (t) , ..., α 0q (t) , α 1 (t) ..., α p (t)} τ , andα (t) the PEBLLE of α (t). Furthermore, we introduced the following symbols:
F ij = Z τ ij , β τ ij τ with β ij = {β 1 (X ij1 ) , ..., β p (X ijp )} τ , Ξ (t) = E F ij F τ ij |T ij = t := [v 1 (t) v 2 (t)] and G (t, t) = lim t ′ →t G (t, t ′ ) with G (t, t ′ ) = E F ij F τ ij ′ |T ij = t, T ij ′ = t ′ . In addition, we define κ = K 2 (v) dv, κ 2 = v 2 K (v) dv, κ 4 = v 4 K (v) dv, κ 22 = v 2 K 2 (v) dv, and g ′′ denotes the second deriva- tive of function g.
Theorem 3.2 presents a unified asymptotic normality ofα (t), which can be applied to sparse, dense and ultra dense cases of the data.
Theorem 3.2. Under the assumption of (A1) -(A9), if max 1 n 3 h 2 C n i=1 1 m 2 i , 1 n 3 hC n i=1 1 m 2 i (m i − 1) , 1 n 3 n i=1 1 − 1 mi 1 − 2 mi 1 nNHhC + 1 n 1 − 1 NH 3/2
holds. Then, for any an interior t in (a, b), we obtain the asymptotic distribution ofα(t) as below:
(3.1) Γ −1/2 C (t) α (t) − α (t) − 1 2 h 2 C κ 2 Ξ −1 (t) ρ 1 (t) D − → N (0, I p+q+1 ) , where ρ 1 (t) = q l=0 α ′′ 0l (t) v 1l (t) + p k=0 α ′′ k (t) v 2k (t) with v 1l (t) being the lth column of v 1 (t) and v 2k (t) being the kth column of v 2 (t), and Γ C (t) = κ nN H h C f T (t) Σ 1,S (t) + 1 n 1 − 1 N H Σ 1,D (t) with Σ 1,S = Ξ −1 (t) γ (t, t) + σ 2 (t) and Σ 1,D = Ξ −1 (t) γ (t, t) G (t, t) Ξ −1 (t).
According to the method of partitioning data defined in Remark 3 and (3.1), Corollary 1 lists the asymptotic distributions for sparse, dense and ultra dense cases of the data as follows.
Corollary 1. Suppose that the conditions of Theorem 3.2 hold and t is a fixed interior point on the interval (a, b).
(i) Sparsity Case (N H /n 1 2r → 0). If h C ≍ nN H −1/(2r+1) , then (3.2) nN H h C f T (t) α (t) − α (t) − 1 2 h 2 C κ 2 Ξ −1 (t) ρ 1 (t) D − → N (0, κΣ 1,S ) . (ii) Dense Case (N H /n 1 2r → C 1 < ∞). If h C = O n −1/(2r) , then (3.3) √ n α (t) − α (t) − 1 2 h 2 C κ 2 Ξ −1 (t) ρ 1 (t) D − → N 0, κ f T (t)C 1 Σ 1,S + Σ 1,D . (iii) Ultra Dense Case (N H /n 1 2r → ∞). If h C = o n −1/(2r) , then (3.4) √ n α (t) − α (t) − 1 2 h 2 C κ 2 Ξ −1 (t) ρ 1 (t) D − → N (0, Σ 1,D ) . LetΞ −1 (t),ρ 1 (t),γ (t, t),σ 2 (t),f T (t),Ĝ (t, t),v 1l (t) andv 2l (t) are kernel smoothing of Ξ −1 (t), ρ 1 (t) , γ (t, t), σ 2 (t), f T (t), G (t, t), v 1l (t)
and v 2l (t). Then, the naive consistent estimators of asymptotic bias ρ 1 (t) and asymptotic variance Γ C (t) are given byρ 1
(t) = q l=0α ′′ 0l (t)v 1l (t) + p k=1α ′′ k (t)v 2k (t) and Γ C (t) = κ nN HĥCfT (t)Σ 1,S (t) + 1 n 1 − 1 N H Σ 1,D (t) , whereΣ 1,S =Ξ −1 (t) γ (t, t) +σ 2 (t) andΣ 1,D =Ξ −1 (t)γ (t, t)Ĝ (t, t)Ξ −1 (t). Based upon (3.1), we can construct a (1 − α) % confidence interval of varying-coefficient component functions as beloŵ α 0l (t) − 1 2 h 2 C κ 2 Ξ −1 (t)ρ 1 (t) l+1 ± z 1−α/2 Γ 1/2 C (t) l+1,l+1 , α k (t) − 1 2 h 2 C κ 2 Ξ −1 (t)ρ 1 (t) q+1+k ± z 1−α/2 Γ 1/2 C (t) q+k+1,q+k+1 , (3.5)
where z 1−α/2 is the 1−α/2 standard normal quantile, the subscript k denotes the k-th element of involved vector, and the subscript (k, k) means the k-th diagonal element of a given matrix. Note that (3.5) is a unified confidence interval suitable for sparse, dense and ultra dense cases of the data.
Asymptotic Properties of Additive Component Functions.
In this subsection, we focus on the asymptotic results of PEBLLE of additive component functions. Theorem 3.3 gives the uniform rates of convergence of β k .
Theorem 3.3. Suppose that (A1) -(A6) and (A9) hold. If K C K A = o nN h 4 C and K −r A + K −r C = o h 2 C , and x is any interior in S k , then sup x∈S k |β k (x) − β k (x) | is bounded by O p h 2 A + h 2 C + K A K −r C + K −r A + log n n 1 + 1 N H h A . Denote µ k = E α 2 k (T ij ) , ψ k,1 = E α 2 k (T ij ) γ (T ij , T ij ) + σ 2 (T ij ) and ψ k,2 = E α k (T ij ) α k T ij ′ γ T ij , T ij ′ .
Theorem 3.4 presents the asymptotic normality ofβ k on a unified formwork for different types of data.
Theorem 3.4. Under the condition (A1) -(A9), if h C = o (h A ) and max 1 n 3 h 2 A n i=1 1 m 2 i , 1 n 3 hA n i=1 1 m 2 i (m i − 1) , 1 n 3 n i=1 1 − 1 mi 1 − 2 mi 1 nNHhA + 1 n 1 − 1 NH 3/2
hold. Then, for any an interior x in S k , we have
(3.6) Γ −1/2 A (x) β k (x) − β k (x) − 1 2 β ′′ k (x) h 2 A κ 2 /µ k D − → N (0, 1) , where Γ A,k (x) = κψ k,1 nN H h A f X k (x) + 1 n 1 − 1 N H ψ k,2 µ 2 k .
As a corollary, we get different asymptotic results for sparse, dense and ultra dense data.
Corollary 2. Suppose that the conditions of Theorem 3.4 hold and x is a fixed interior point in S k .
(i) Sparsity Case. If h A ≍ nN H − 1 2r+1 , then it follows that (3.7) nN H h A f X k (x) β k (x) − β k (x) − 1 2 β ′′ k (x) h 2 A κ 2 /µ k D − → N (0, κψ k,1 ) . (ii) Dense Case. If h A = O n − 1 2r , then (3.8) √ n β k (x) − β k (x) − 1 2 β ′′ k (x) h 2 A κ 2 /µ k D − → N 0, κψ k,1 f X k (x) C 1 + ψ k,2 µ 2 k . (iii) Ultra Dense Case. If h A = o n − 1 2r , then (3.9) √ n β k (x) − β k (x) − 1 2 β ′′ k (x) h 2 A κ 2 /µ k D − → N 0, ψ k,2 /µ 2 k .
Letμ k ,f X k (x) andψ k,j , j = 1, 2 be consistent estimators of µ k , f X k (x) and ψ k,j . Then, the asymptotic variance Γ A (x) can be consistently estimated byΓ
A (x) = κψ k,1 nN HĥAf X k (x) + 1 n 1 − 1 N H ψ k,2 µ 2 k ,
which gives a (1 − α)% pointwise confidence interval of β k in a unified forms for sparse, dense and ultra dense data. That is,
(3.10)β k (x) − 1 2 β ′′ k (x)h 2 A κ 2 /μ k ± z 1−α/2Γ −1/2 A (x).
Testing of Model Specification.
For the sake of parsimony, it is essential to test time-varying property of varying-coefficient component functions and to test linearity of additive component functions.
Time-varying Testing of Varying-coefficient Component Functions.
In this subsection, we propose a consistent testing to judge whether the varying-coefficient component functions are really time-varying or not. It is a problem of model selection between a general Semi-VCAM and a submodel PLAM or an AM in the practical applications.
We denote δ ij = ν i (T ij ) + ε ij in Semi-VCAM (1.2), and consider a mixed-
effect nonparametric model Y ij = m (T ij , Z ij , X ij ) + δ ij , where m (t, z, x) = E [Y ij |T ij = t, Z ij = z, X ij = x].
The time-varying testing postulates m as
m (t, z, x) = z τ a 0 + p k=1 a k β k (x k ) := g 0 (z, x; a, β (x))
under null hypothesis H 0,C , where a = (a τ 0 , a 1 , ..., a k ) τ is a unknown constant vector, and β (x) = (β 1 (x 1 ), ..., β p (x p )) τ . Whilst under alternative hypothesis H 1,C , m is the regression function of Semi-VCAM (1.2), denoted as g (t, z, x; α (t) , β (x)). Then, the interested hypothesis is given as below
H 0,C : m (t, z, x) = g 0 (t, z, x; α, β (x)) a.s. ↔ H 1,C : m (t, z, x) = g (t, z, x; α (t) , β (x)) a.s. (4.1)
Under H 0,C , we replace β k (x) with PEBLLEβ k (x), and obtain the parametric estimator of vector a as follows
(4.2)ã = n i=1Ŝ iŜ τ i −1 n i=1Ŝ i Y i , whereŜ i = (Ŝ i1 , ...,Ŝ im i ) τ withŜ ij = Z τ ij ,β 1 (X ij1 ), ...,β p (X ijp ) τ , and Y i = (Y i1 , ..., Y im i ) τ .
For the i-th subject and the j-th subject, we introduce the weight ma-
trix W ij = w (l,v) ij m i ×m j , where w (l,v) ij = k h C (T il , T jv ) K h A (X il , X jv ), with k h (t, z) = k t−z h and K h (x 1 , x 2 ) = Π p i=1 k h (x 1i , x 2i ) for x k = (x k1 , ..., x kp ), k = 1, 2. Letê ij = Y ij − g 0 (T ij , Z ij , X ij ;ã,β (X ij ))
, and we propose a testing statistic based upon the quadratic form of residuals as follows
(4.3)Ĵ n = 1 n 2N 2 H |H| n i=1 n j =iê τ i W ijêj ,
whereê i = (ê i1 , ...,ê im i ) τ and |H| = h C h p A . Furthermore, we assumes additional conditions as follows.
(T1) lim n→∞ 1 n n i=1 E [S i S τ i ] = Ω,
and Ω F is bounded away from zero and infinity, where S i is the analogue ofŜ i afterβ k being replaced by β k , and · F is the Frobenius norm of the involved matrix. under H 0,C , wherê
σ 2 1 = 1 n 2N H |H| n i=1 n j =i m i l=1 m j v=1ê 2 ilê 2 jv w (l,v) ij 2
is a consistent estimator of the asymptotic variance of Let h 0 (t, z, x; α(t)) = z τ α 0 (t) + p k=1 α k (t)x k . It is expected to test H 0,A : m (t, z, x) = h 0 (t, z, x; α (t)) a.s. ↔ H 1,A : m (t, z, x) = g (t, z, x; α (t) , β (x)) a.s. (4.4) Denoteα (t) as the LLE of α (t) under null hypothesis H 0,A . Then, the testing statistics is given by
n 2N 2 H √ N 2 −nN 2 |H|Ĵ n , i.e., σ 2 1 = κ p+1 E {γ (T, T ) + σ 2 (T )} 2 f T (T ) Π p k=1 E [f k (X k )] .(4.5)Î n = 1 n 2N 2 H |H| n i=1 n j =iς τ i W ijς j , whereς i = (ς i1 , ...,ς im i ) τ withς ij = Y ij − h 0 (T ij , Z ij , X ij ;α (T ij )
). The asymptotic distributions ofÎ n under H 0,A and H 1,A are presented in the following two theorems.
Implementation.
In this section, we address the practical issues that arise in the newly-proposed methodologies.
• B-spline method of pilot estimation As a common practice in spline smoothing, we predetermine the order of the B-spline functions and then select optimal interior knots number through BIC criterion BIC (K C , K A ) = log (RSS) + N log n/n,
where RSS = 1 n n i=1 1 mi mi j=1 [Y ij − γ τ 0 Z ij ⊗ b C (T ij ) − p k=1 γ τ k T k (T ij , X ijk )] 2
and N = (q + 1) J C + pJ C J A , with J C and J A being the dimension of Bspline basis space b C (t) and B k,A (x k ). Then the optimal interior knots number is given by (K C ,K A ) = arg min BIC (K C , K A ).
• LLE based on pilot estimation
In local linear smoothing, we use Epanechnikov kernel function k (u) = 0.75 1 − u 2 I |u|≤1 , and select the optimal bandwidths using "leave-one-out" cross-validation procedure suggested by [33]. Define the subject-based crossvalidation (CV) criterion as below
CV (h C , h A ) = n i=1 1 m i m i j=1 Y ij − Z τ ijα0,−i (T ij ) − p k=1α k,−i (T ij )β k,−i (X ijk ) 2 ,
where the subscript "−i" represents the estimator using the data with all repeated measurements except the ith subject. The optimal bandwidth is the unique minimizer of CV(h C , h A ).
In simulation studies, we also can use the average squared error (ASE) as follows
ASE (h C , h A ) = n i=1 1 m i m i j=1 Z τ ij (α 0 (T ij ) −α 0 (T ij )) + p k=1 α k (T ij ) β k (X ijk ) − p k=1α k (T ij )β k (X ijk ) 2 . (5.1)
Similar to Remark 2.3 of [40], it is not difficult to show that the CV bandwidths approximately minimize ASE.
6. Numerical Studies.
Simulation Studies.
In this subsection, we consider simulation examples to investigate the finite-sample performance of the proposed estimation method in Section 2 and the testing procedure in Section 4.
U i (1 + T ij ) + ϑ ij , where U i ∼ U (−0.4, 0.4) and ϑ ij ∼ N 0, 0.2 2 .
The response Y ij is generated by a mixed-effects Semi-VCAM as below
Y ij = α 00 (T ij ) + α 01 (T ij ) Z ij + α 1 (T ij ) β 1 (X ij ) + ν i (T ij ) + ε ij ,
for i = 1, ..., n and j = 1, ..., m, where the measurement error ε ij are i.i.d from N (0, 1), and the subject-specific random trajectory ν i (T ij ) = η i1 + √ 2η i2 sin (2πT ij ) + √ 2η i3 cos (2πT ij ) with η ij ∼ N (0, w j ) for j = 1, 2, 3 and (w 1 , w 2 , w 3 ) = (0.6, 0.2, 0.2). The univariate smooth component functions are given by α 00 (t) = 6t, α 01 (t) = 2.5 cos (2πt), α 1 (t) = t(1−t) We select 20 equally-spaced points on the range of T ij and X ij , and define the mean prediction integrated squared error (MPISE) based on Q replications,
MPISE (f ) = 1 Q Q q=1 f q (u) − f (u) 2 du,
wheref q is the PEBLLE of the estimated function f in the q-th replication. Under different combinations of n and m, based upon Q = 300 Monte Carlo replications, Table 1 gives the MPISEs of PEBLLE of component functions, and the standard deviation is shown in parentheses. We also list the optimal bandwidths according to (5.1). The result exhibits a good finite-sample performance whatever the data is sparse or dense. It is also found that MPISEs decrease markedly as the total observations increase. We also investigate the performance of asymptotic distribution given in (1) the AECPs of unified method (bold tags in tables) are superior to the other three methods, whatever the data is sparse, dense or ultra dense; (2) the AECPs of sparse method decrease as m grows, and they are inferior to the unified method even for sparse data; (3) the AECPs of dense and ultra dense method increase as m grows, and they are comparable to that of the unified method.
Example 2. Now we investigate the performance of hypothesis testing constructed in Section 4. To this end, we consider the following two DGP:
• DGP I: In this case, we test the time-varying property of varyingcoefficient component functions, that is to decide whether a PLAM is sufficient. We take the same settings with Example 1 for T ij , X ij , Z ij , ε ij , ν i and β 1 (x). The time-varying testing of conditional mean function m (t, z, x) is as follows where g 0 (z, x) = 6 + 2.5z + β 1 (x) and
(6.1) H 0 : m (t, z, x) = g 0 (z, x) a.s. ↔ H 1 : m (t, z, x) = g 1 (t, z, x) a.s.,g 1 (t, z, x) = g 0 (z, x) + θ (t + z cos (2πt) + t (1 − t) β 1 (x)) ,
with θ = 0.2, 0.4, 0.6, 0.8, 1.0. • DGP II: Here we consider the linearity testing of additive component functions to judge whether a VCM is sufficient. Let T ij , Z ij , α 00 (t) and α 01 (t) be given in Example 1, α 1 (t) = sin (πt)/ 1 0 sin (πt)dt, and N (0, 1). The interested hypothesis is given by
X ij = U i (1 + T ij ) + ϑ ij , where U i ∼ U (−0.5, 0.5) and ϑ ij ∼(6.2) H 0 : m (t, z, x) = h 0 (t, z, x) a.s. ↔ H 1 : m (t, z, x) = h 1 (t, z, x) a.s.,
where h 0 (t, z, x) = α 00 (t) + α 01 (t) z + α 1 (t) x and h 1 (t, z, x) = h 0 (t, z, x) + 1.5θα 1 (t) sin (πx) with θ = 0.2, 0.4, 0.6, 0.8, 1.0.
We consider different combinations of n = 30, 50, 100 and m = 5, 10, 30, 60, 100, and generate Q = 300 Monte Carlo replications and B = 300 bootstrap samples for each simulated data set. Under 5% and 10% significance levels, based upon bootstrap critical value, Table 4 and 5 present power of hypothesis (6.1) and (6.2) for different deviation parameters θ ranging from 0 to 1 with the span of 0.2, respectively. The results show that the proposed testing procedure all performs well for sparse, dense and dense data. In fact, the power for θ = 0 is size of hypothesis, which is close to the theoretical significance level 0.05 or 0.1. As expected, the power increases to one as θ ascends whatever significance levels and sampling plans. Moreover, Figures 2 and 3 plot the rejection rates of testing (6.1) and (6.2) at the 5% and 10% significance levels for some combinations of n and m, respectively.
Real Data Analysis.
Example 3. Now we apply our method to the new coronavirus disease (COVID-19) mentioned in Section 1. We collected the daily cumulative confirmed cases (Z i,t ) and the daily cumulative cured cases from https://github.com/CSSEGISandData/COVID-19, the daily movement population from Wuhan to other provinces (https://qianxi.baidu.com/), the maximum daily temperature (http://www.weather.com.cn) and the population data (https://zh.wikipedia.org/wiki/).
The response variable GRCC, denoted by Y i,t , is measured by log (Z i,t ) − log (Z i,t−1 ), which is presented in Figure 4 (a) for 29 provinces in China from January 23th to April 8th. We notice that the big values of GRCC (above 0.5) mainly concentrate on the period from January 23th to February 3th. It is a strong evidence that the intervention policy of China's government plays a positive role in controlling the spread of Coronavirus disease.
To explore the influence factor of GRCC, we used five covariates: X 1,it being the movement population from Wuhan (MPFW), which is measured by the proportion of the population moving from Wuhan to the ith province out of moving out population at day t − 14; X 2,it the daily cumulative cured cases (CUCC) at day t − 1; X 3,it the daily cumulative confirmed cases (CFCC) at day t − 1; X 4,it the maximum daily temperature at day t, and X 5,i the population of ith province.
We normalize the covariate X 1,it , and make the logarithm transformation for X k,it , k = 2, 3 and X 5,i . Based on 500 bootstrap sampling, we do the time-varying testing (4.1) and linearity testing (4.4), obtaining the p values 0.028 and 0.038, respectively. Therefore, we reject the AM and VCM at significant level 0.05, and adopt the general model as below:
(6.3) Y it = α 0 (t/T ) + 4 k=1 α k (t/T ) β k (X k,it ) + α 5 (t/T ) β 5 (X 5,i ) ,
where i = 1, ..., 29, t = 1, ..., T with T = 77. Figure 4 gives the PEBLLE of component functions, and 95% pointwise confidence bands according to (3.5) and (3.10). From Figure 4, we conclude that the trend term α 0 , varying-coefficient function α 1 for MPFW, α 3 for CCFC and α 4 for MDT have similar properties, i.e., they drop rapidly until about February 29th, and then maintain on the level close to zero; α 2 for CUCC decreases until about February 22th, and increases until about March 9th, and thereafter levels near zero; α 5 for POP decreases slowly until about February 9th, then increases until about February 29th, and decreases thereafter.
For the medium values of the normalized MPFW, the effect increases as MPFW grows, and some fluctuations appears for the large value (above 2), since large MPFW usually takes place in the early stage and the period of work resumption. The influence of CUCC increases as it grows, and the rate of increases become slower above 2; while the effect of CCFC ascends as it increases, and levels out above 3. The effect of MDT drops under 0℃, and increases until 10℃, and almost no influence between 10℃and 20℃, then ascends rapidly above 20℃. The trend of effect of POP grows as the population size ascends, especially when log-POP is larger than 7.5.
Example 4. We revisit a CD4 data from the Multicenter AIDS Cohort Study, which contains 1817 observations from 283 homosexual men infected with HIV between 1984 and 1991. [4,15] have analyzed this data set using a VCM. Now, we apply our method to this dataset. The response variable Y ij is the i-th subject's CD4 percentage at time T ij . Following the covariates of [15], we let X 1i be the i-th subject's smoke status, a dichotomous variable, X 2i the i-th subject's centred age, and X 3i the i-th subject's centred preinfection CD4 percentage. The relationship between response and covariates are modeled by a Semi-VCAM as below
(6.4) Y ij = α 0 (T ij ) + α 1 (T ij ) X 1i + α 2 (T ij ) β 1 (X 2i ) + α 3 (T ij ) β 2 (X 3i ) ,
where the covariates are all time-invariant. Based on 500 bootstrap sampling, we do the time-varying testing (4.1) and the linearity testing (4.4), obtaining the p values 0.028 and 0.457, respectively. That means, at significant level 0.05, VCM is a reasonable choice, which verifies that the model used in [15] is appropriate.
7. Concluding Remarks. In this paper, we have considered a Semi-VCAM for the functional/longitudinal data with different sampling plan. The Semi-VCAM is an extension of the existing VCAM. We have developed a pilot estimation based local linear estimation for the Semi-VCAM and have presented asymptotic distribution on a unified platform for sparse, dense and ultra dense cases of the data. The virtue of unified asymptotic results is to help us avoid deciding the types of data in advance, which is a subjective choice and may lead to wrong conclusions. From the viewpoint of model parsimony, we also have developed consistent testing procedures to justify whether a VCM or PLAM, especially an AM is sufficient for the real-life data. These test methods also avoid the subjective choice between the sparse, dense and ultra dense cases of the data.
Our model and inference methods may be extended in various directions. We close the paper by outlining some of them. In many application areas, data may be collected on a count or binary response. For example, daily death toll, suspected and confirmed cases of COVID-19. As a result, it is useful to extend our proposed model and inference to the generalized Semi-VCAM to accommodate the discrete functional/longitudinal responses. Data in the form of samples of densities or distributions are increasingly encountered in practice and same as [11] there is a need for flexible regression models that accommodate random densities as responses. We believe our proposed model could also be used to model the data in which the responses are random densities. In addition, due to the fact that the proposed test method in our paper is based on the local smoothing, it may suffer the curse of dimensionality, struggle to maintain the significance level and lose its power to an extent as the dimension of explanatory variables increases. Same as [20] and [22], we may use projection technique, or bridging between local smoothing and global smoothing methods to avoid this. Due to the complication of our model, extending the methods in [20] and [22] to our scenario is not simple.
|β k,I (x) − β k (x)| = O p K A (K −r A + K −r C ) + K C K 2 A nN H + K A n .
means that a and b have the same order. On the other hand, a parametric rate of convergence is implied ifN H /n 1 2r → C (0 < C < ∞) and K C
( T2 )N 2 H
T22|H| → 0, nN H |H| → ∞ and nN H |H|h the asymptotic distribution of the proposed test statisticĴ n under H 0,C and H 1,C , respectively.
Theorem 4. 2 .
2Under the conditions of Theorem 4.1, If H 1,C holds, then P r nN H |H|Ĵ n /σ 1 ≥ M n → 1 as n → ∞, where M n is any non-stochastic positive sequence such that M n = o nN H |H| . 4.2. Linearity Testing of Additive Component Functions. In this subsection, we check whether each additive component function in Semi-VCAM (1.2) reduces to a linear form, which yields a more parsimonious VCM.
. 4 .
4Suppose that the conditions of Theorem 4.1 holds. Then under H 1,A , we have P r(nN H |H|Î n /σ 1 ≥ E n ) → 1 as n → ∞, where E n is any non-stochastic positive sequence such that E n = o(nN H |H|).
Example 1 .
1Here we consider a mixed-effects Semi-VCAM. Let T ij are uniformly distributed on [0, 1], Z ij are i.i.d. Bernoulli random variable with the probability of success p = 0.5, and X ij =
Figure 1
1visualizes the PEBLLE for (n, m) = (100, 10). The solid curve plots true component function, the dashed line figures the PEBLLE, and the dash-dotted lines give 95% confidence bands based on the asymptotic distribution. The figure shows that our estimator is close to the true function even under the medium total observations N = 1000.
Fig 1 .
1Theorems 3.2 and 3.4. After doing 300 Monte Carlo replications, we compare the average empirical coverage percentages (AECPs) based on four methods, that is, the unified method (U) given in (3.1) and (3.6), sparse method (S) in (3.2) and (3.7), dense method (D) in (3.3) and (3.8), and ultra dense method (UD) in (3.4) and (3.9). We take n = 50, 100 and m = 5, 10, 30, 80, 200. Table 2 and 3 list the AECPs and the average empirical length (AEL) of confidence interval under the significance level 90% and 95%, respectively. From the resultant tables, we make a conclusion that: Estimation of component functions in Example 1. The solid curve represents the true function, and the dashed line plots the PEBLLE, and the dash-dotted lines gives the 95% pointwise confidence bands based on (3.5) and (3.10).
Fig 2 .Fig 3 .
23′ ) Density for (n, m) = (50, 5) (b ′ ) Density for (n, m) = (50, 30) (c ′ ) Density for (n, m) = (30, 60) Power of time-varying testing (6.1) in Example 2. For three combinations of n and m, (a) -(c) figure power at the level α = 0.05 and 0.1; whist (a ′ ) -(c ′ ) give the simulated density of standardized test statistics (thick black) and five bootstrap approximations (thin). for (n, m) = (100, 10) (b) Power for (n, m) = (50, 30) (c) Power for (n, m) = (30, 100) ′ ) Density for (n, m) = (100, 10) (b ′ ) Density for (n, m) = (50, 30) (c ′ ) Density for (n, m) = (30, 100) Power of linearity testing (6.2) in Example 2. For three combinations of n and m, (a) -(c) figure power at the level α = 0.05 and 0.1; whist (a ′ ) -(c ′ ) give the simulated density of standardized test statistics (thick black) and five bootstrap approximations (thin).
′ ) β 3 (f ′ ) β 4 (g ′ )β 5
Fig 4 .
4Analysis Results for COVID-19 Data
(http://www.e-publications.org/ims/support/dowload/imsart-ims.zip). Proposition 1. Under Assumption (A1) -(A6) and (A9), it follows that sup x∈[a k ,b k ]
Table 1
1The MPISEs(standard deviation in parentheses) of component functions in Example 1.n
mĥCĥAα00α01α1β
50
5
0.1763 0.4132
0.0880
0.1794
0.0048
0.1529
(0.0605)
(0.1129)
(0.0028)
(0.1122)
10
0.1600 0.3950
0.0603
0.0986
0.0028
0.0957
(0.0400)
(0.0585)
(0.0016)
(0.0646)
30
0.1447 0.3500
0.0278
0.0439
0.0019
0.0821
(0.0194)
(0.0245)
(0.0012)
(0.0574)
50
0.1237 0.3395
0.0276
0.0278
0.0016
0.0662
(0.0193)
(0.0137)
(0.0010)
(0.0516)
100 0.1132 0.2921
0.0264
0.0182
0.0014
0.0641
(0.0263)
(0.0092)
(0.0010)
(0.0441)
100
10
0.1553 0.3553
0.0311
0.0655
0.0023
0.0657
(0.0200)
(0.0369)
(0.0013)
(0.0395)
30
0.1500 0.2831
0.0193
0.0318
0.0016
0.0597
(0.0153)
(0.0129)
(0.0009)
(0.0405)
60
0.1111 0.2278
0.0174
0.0170
0.0009
0.0553
(0.0147)
(0.0075)
(0.0008)
(0.0443)
100 0.0550 0.2200
0.0157
0.0088
0.0006
0.0365
(0.0110)
(0.0033)
(0.0004)
(0.0284)
150 0.0556 0.1778
0.0144
0.0063
0.0006
0.0275
(0.0013)
(0.0026)
(0.0006)
(0.0180)
Table 2
2The AECPs and AELs (in parentheses) of four methods with level 90% in Example 1.m Fun
n = 50
n = 100
U(%)
S(%)
D(%)
UD(%)
U(%)
S(%)
D(%)
UD(%)
5
α00
87.63
80.98
80.93
64.08
88.63
83.97
81.05
60.80
(0.9016)
(0.7539)
(0.7671)
(0.5309)
(0.7384)
(0.6467)
(0.6142)
(0.3958)
α01
86.57
79.32
79.22
62.23
86.83
81.52
78.63
57.37
(1.2580)
(1.0501)
(1.0721)
(0.7443)
(1.0919)
(0.9576)
(0.9068)
(0.5814)
α1
86.08
79.00
79.52
64.42
87.82
85.80
81.17
58.14
(0.3500)
(0.2909)
(0.2994)
(0.2092)
(0.3007)
(0.2626)
(0.2508)
(0.6510)
β1
87.53
85.67
81.68
44.95
88.02
86.48
84.88
38.95
(1.0994)
(1.0530)
(0.9188)
(0.3071)
(1.0756)
(1.0301)
(0.9057)
(0.2154)
10
α00
88.40
72.55
85.88
77.98
88.48
74.90
84.88
75.50
(0.7374)
(0.4958)
(0.6884)
(0.5648)
(0.5533)
(0.3873)
(0.5049)
(0.4048)
α01
86.63
68.90
83.28
73.93
87.20
70.90
82.95
72.35
(1.0573)
(0.7118)
(0.9868)
(0.8088)
(0.7926)
(0.5556)
(0.7229)
(0.5788)
α1
88.00
71.70
85.35
76.75
88.15
74.65
84.85
75.08
(0.2770)
(0.1871)
(0.2584)
(0.2114)
(0.2057)
(0.1435)
(0.1880)
(0.1512)
β1
88.25
84.43
81.15
52.58
88.68
83.58
85.25
60.63
(0.8836)
(0.8200)
(0.7146)
(0.3007)
(0.4573)
(0.3999)
(0.4129)
(0.2228)
30
α00
88.90
50.65
88.20
84.35
89.58
53.10
89.25
85.53
(0.6279)
(0.2601)
(0.6218)
(0.5762)
(0.4638)
(0.1965)
(0.4611)
(0.4235)
α01
88.15
47.75
87.75
84.20
88.63
46.37
88.37
84.10
(0.8892)
(0.3691)
(0.8805)
(0.8156)
(0.6545)
(0.2773)
(0.6506)
(0.5975)
α1
88.05
52.70
87.40
85.20
88.82
54.35
88.47
85.75
(0.2374)
(0.0984)
(0.2352)
(0.2180)
(0.1688)
(0.0715)
(0.1678)
(0.1542)
β1
88.50
71.65
85.60
77.85
88.63
72.62
84.85
75.30
(0.4139)
(0.2755)
(0.3775)
(0.3110)
(0.3014)
(0.2089)
(0.2761)
(0.2190)
80
α00
89.01
39.67
88.42
87.16
89.61
38.70
88.88
87.85
(0.5937)
(0.1793)
(0.5851)
(0.5667)
(0.4408)
(0.1309)
(0.4365)
(0.4203)
α01
89.49
38.71
89.07
87.93
89.87
35.09
89.56
89.11
(0.8367)
(0.2523)
(0.8248)
(0.7988)
(0.6178)
(0.1835)
(0.6118)
(0.5891)
α1
88.71
41.89
88.13
87.07
88.90
40.00
88.41
87.63
(0.2198)
(0.0662)
(0.2167)
(0.2098)
(0.1593)
(0.0475)
(0.1578)
(0.1519)
β1
88.91
55.58
88.20
85.93
89.40
53.54
88.70
86.72
(0.3471)
(0.1583)
(0.3315)
(0.3105)
(0.2444)
(0.1080)
(0.2390)
(0.2204)
200
α00
89.50
28.40
89.20
89.05
89.75
32.45
89.55
89.30
(0.5926)
(0.1166)
(0.5849)
(0.5618)
(0.4192)
(0.0979)
(0.4072)
(0.4097)
α01
89.47
34.73
89.25
89.13
89.60
34.80
89.55
89.25
(0.8311)
(0.1636)
(0.8238)
(0.7959)
(0.6050)
(0.1376)
(0.5968)
(0.5539)
α1
89.20
29.93
88.90
88.70
89.65
36.35
89.30
89.05
(0.2069)
(0.0406)
(0.2056)
(0.2032)
(0.1419)
(0.0368)
(0.1412)
(0.1384)
β1
89.60
38.60
89.13
89.04
89.80
43.15
89.35
89.20
(0.3219)
(0.0986)
(0.3163)
(0.3072)
(0.2358)
(0.0917)
(0.2111)
(0.2003)
Table 3
3The AECPs and AELs (in parentheses) of of four methods with level 95% in Example 1.m Fun
n = 50
n = 100
U(%)
S(%)
D(%)
UD(%)
U(%)
S(%)
D(%)
UD(%)
5
α00
93.15
88.93
88.23
72.70
93.68
90.45
88.50
69.23
(1.0818)
(0.9006)
(0.9160)
(0.6467)
(0.8599)
(0.7479)
(0.7303)
(0.4270)
α01
92.68
87.73
87.03
70.88
93.08
88.22
86.12
66.52
(1.5106)
(1.2561)
(1.2804)
(0.9062)
(1.2863)
(1.1242)
(1.0779)
(0.6945)
α1
91.60
87.12
85.92
71.92
93.12
90.10
87.65
70.23
(0.4249)
(0.3520)
(0.3613)
(0.2573)
(0.3540)
(0.3082)
(0.2979)
(0.1945)
β1
92.30
91.00
87.95
52.08
93.30
92.25
90.18
43.37
(1.2933)
(1.2373)
(1.0916)
(0.3655)
(0.9537)
(0.9195)
(0.8595)
(0.2563)
10
α00
93.50
81.28
89.93
82.13
94.13
85.53
89.35
81.13
(0.9012)
(0.6247)
(0.8171)
(0.6706)
(0.6933)
(0.5056)
(0.6030)
(0.4853)
α01
93.50
82.28
89.98
82.40
93.90
86.60
90.65
82.18
(1.2926)
(0.8966)
(1.1718)
(0.9611)
(0.9955)
(0.7274)
(0.8649)
(0.6946)
α1
93.15
82.48
90.40
82.73
93.23
84.05
88.58
80.50
(0.3390)
(0.2357)
(0.3072)
(0.2516)
(0.2561)
(0.1862)
(0.2232)
(0.1801)
β1
93.63
90.15
87.68
60.20
94.33
90.53
92.43
70.48
(0.9368)
(0.8567)
(0.7700)
(0.3599)
(0.5288)
(0.4582)
(0.4908)
(0.2658)
30
α00
93.88
68.25
92.12
89.30
94.15
64.30
94.00
90.75
(0.7657)
(0.3638)
(0.7305)
(0.6761)
(0.5495)
(0.2506)
(0.5376)
(0.4906)
α01
93.90
67.45
92.90
90.20
94.05
59.25
93.05
90.05
(1.0900)
(0.5187)
(1.0402)
(0.9621)
(0.7753)
(0.3535)
(0.7585)
(0.6922)
α1
93.80
72.90
92.05
89.75
94.20
68.75
93.15
89.95
(0.2952)
(0.1404)
(0.2818)
(0.2605)
(0.2012)
(0.0916)
(0.1969)
(0.1798)
β1
94.10
81.00
92.20
86.00
94.55
82.05
92.45
83.85
(0.4793)
(0.3082)
(0.4438)
(0.3708)
(0.3590)
(0.2498)
(0.3256)
(0.2602)
80
α00
94.45
45.35
93.80
92.80
94.60
49.25
93.30
92.05
(0.7298)
(0.2136)
(0.7207)
(0.6933)
(0.5183)
(0.1602)
(0.5107)
(0.4924)
α01
94.25
52.80
94.20
93.80
94.75
51.40
94.35
93.70
(1.0268)
(0.3003)
(1.0141)
(0.9440)
(0.7273)
(0.2250)
(0.7165)
(0.6908)
α1
94.10
51.70
92.75
91.65
94.65
52.90
93.05
91.85
(0.2635)
(0.0769)
(0.2602)
(0.2525)
(0.1966)
(0.0609)
(0.1937)
(0.1867)
β1
94.50
58.15
94.00
92.45
94.80
65.40
94.00
93.15
(0.4093)
(0.1845)
(0.3938)
(0.3674)
(0.2902)
(0.1313)
(0.2805)
(0.2601)
200
α00
94.50
30.95
94.30
94.10
94.85
34.25
94.65
94.45
(0.7169)
(0.1342)
(0.7132)
(0.6843)
(0.5122)
(0.1005)
(0.4998)
(0.4827)
α01
94.80
39.65
94.40
94.25
94.95
40.70
94.85
94.40
(1.0003)
(0.1876)
(0.9950)
(0.9325)
(0.7217)
(0.1407)
(0.7083)
(0.6885)
α1
94.35
35.20
94.15
94.00
94.75
41.45
94.55
94.30
(0.2517)
(0.0507)
(0.2503)
(0.2469)
(0.1842)
(0.0353)
(0.1834)
(0.1809)
β1
94.75
47.40
94.30
94.15
94.85
47.35
94.60
94.35
(0.3838)
(0.1116)
(0.3778)
(0.3580)
(0.2712)
(0.0809)
(0.2683)
(0.2595)
Table 4
4Power of testing (6.1) under confidence level α = 5% and 10%.α
θ
(n, m)
(50,5) (50,10) (100,5) (100,10) (50,30) (30,60) (30,100)
5%
0
0.067
0.055
0.050
0.040
0.055
0.060
0.050
0.2 0.233
0.300
0.265
0.370
0.270
0.225
0.230
0.4 0.500
0.690
0.725
0.885
0.875
0.710
0.770
0.6 0.830
0.955
0.985
1.000
1.000
0.995
1.000
0.8 0.970
1.000
1.000
1.000
1.000
1.000
1.000
1.0 0.997
1.000
1.000
1.000
1.000
1.000
1.000
10%
0
0.113
0.095
0.095
0.085
0.100
0.085
0.090
0.2 0.333
0.410
0.360
0.505
0.345
0.290
0.240
0.4 0.610
0.755
0.815
0.930
0.920
0.765
0.810
0.6 0.857
0.965
0.990
1.000
1.000
0.995
1.000
0.8 0.980
1.000
1.000
1.000
1.000
1.000
1.000
1.0 1.000
1.000
1.000
1.000
1.000
1.000
1.000
Table 5
5Size and power of test (6.2) under confidence level α = 5% and 10%.α
θ
(n, m)
(50,5) (50,10) (100,5) (100,10) (50,30) (30,60) (30,100)
5%
0
0.047
0.057
0.050
0.045
0.040
0.045
0.060
0.2 0.107
0.120
0.173
0.205
0.200
0.200
0.180
0.4 0.203
0.557
0.477
0.825
0.907
0.850
0.807
0.6 0.653
0.983
0.937
1.000
1.000
0.990
0.993
0.8 0.920
1.000
1.000
1.000
1.000
1.000
1.000
1.0 0.997
1.000
1.000
1.000
1.000
1.000
1.000
10%
0
0.080
0.083
0.083
0.090
0.0933
0.090
0.100
0.2 0.157
0.197
0.250
0.350
0.327
0.360
0.260
0.4 0.303
0.670
0.610
0.925
0.933
0.930
0.873
0.6 0.740
0.990
0.967
1.000
1.000
1.000
1.000
0.8 0.947
1.000
1.000
1.000
1.000
1.000
1.000
1.0 0.997
1.000
1.000
1.000
1.000
1.000
1.000
APPENDIX A: APPENDIX SECTIONA.1. Appendix subsection. A function m defined on the interval [a, b] is called to be Lipschitz-continuous, if there exists a fixed constantas the space of all functions m (x) defined on[a, b], such that m is differentiable of r − 1 order, and m (r−1) is Lipschitz-continuous, where m (l) means the l-th order derivative of m.The necessary conditions to validate asymptotic properties are as follows.(A1) The observation time points T ij 's are drawn from an unknown distribution, which has a density f T (t) with the support T , and is continuously differentiable in a neighbourhood of t and is uniformly bounded away from 0 and infinity. (A2) X i 's are independent realizations of stochastic process X(t), and X i 's are independent of T ij 's. The marginal density function f X k (·) of covariates X k is continuously differentiable in a neighbourhood of x and is uniformly bounded away from 0 and infinity. {x ij } i are independent and identically distributed. Moreover, {ν i (·)} i , {x ij } ij and {ε ij } ij are mutually independent. (A6) σ 2 (·) < ∞ is continuously differentiable. γ(t, t ′ ) is continuously differentiable and γ(t, t) = lim t ′ →t γ(t, t ′ ) < ∞. (A7) E{|ν i (·) + σ(·)ε ij | ν } is continuous and bounded from infinity for ν ≤ 4. (A8) k(·) is bounded and symmetric probability density function with a bounded support and a bounded derivative.Remark 4. Assumptions A1 and A2 involve the distributions of time points T ij and k-th covariate X k . Assumption A3 relates to covariates Z, a similar conditions with[4]. Assumption A4 specifies the degree of smoothness of varying-coefficient component functions and additive component functions. Assumptions A5-A7 are necessary for constructing asymptotic distribution, a common conditions with[4]. Assumption A8 is a standard
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With exotic propagation properties, optical Airy beams have been well studied for innovative applications in communications, biomedical imaging, micromachining, and so on. Here we extend those studies to the quantum domain, creating quantum correlated photons in finite-energy Airy transverse modes via spontaneous parametric down conversion and sub-sequential spatial light modulation. Through two-photon coincidence measurements, we verify their Airy spatial wavefunctions, propagation along a parabolic trajectory, and that the spatial modulation does not introduce any observable degradation of quantum correlation between the photons. These results suggest the feasibility of using spatially structured photons for practically advantageous quantum applications.
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1710.11208
|
2ddf75cebc2960a9011e389fa675e8f624e13aaf
|
Quantum Airy Photons
November 1, 2017 30 Oct 2017
Stephanie Maruca
Department of Physics
Stevens Institute of Technology
07030HobokenNJUSA
Center for Quantum Science and Engineering
Stevens Institute of Technology
07030HobokenNJUSA
Santosh Kumar
Department of Physics
Stevens Institute of Technology
07030HobokenNJUSA
Center for Quantum Science and Engineering
Stevens Institute of Technology
07030HobokenNJUSA
Yong Meng Sua
Department of Physics
Stevens Institute of Technology
07030HobokenNJUSA
Center for Quantum Science and Engineering
Stevens Institute of Technology
07030HobokenNJUSA
Jia-Yang Chen
Department of Physics
Stevens Institute of Technology
07030HobokenNJUSA
Center for Quantum Science and Engineering
Stevens Institute of Technology
07030HobokenNJUSA
Amin Shahverdi
Department of Physics
Stevens Institute of Technology
07030HobokenNJUSA
Center for Quantum Science and Engineering
Stevens Institute of Technology
07030HobokenNJUSA
Department of Electrical and Computer Engineering
Stevens Institute of Technology
07030HobokenNJUSA
Yu-Ping Huang *[email protected]
Department of Physics
Stevens Institute of Technology
07030HobokenNJUSA
Center for Quantum Science and Engineering
Stevens Institute of Technology
07030HobokenNJUSA
Quantum Airy Photons
November 1, 2017 30 Oct 201710.1364/ao.XX.XXXXXXLetter Optics Letters 1 † These two authors contributed equally to this work. CompiledOCIS codes: (0706120) Spatial light modulators(0501940) Diffraction(1900190) Nonlinear optics(0305260) Photon counting(2700270) Quan- tum optics
With exotic propagation properties, optical Airy beams have been well studied for innovative applications in communications, biomedical imaging, micromachining, and so on. Here we extend those studies to the quantum domain, creating quantum correlated photons in finite-energy Airy transverse modes via spontaneous parametric down conversion and sub-sequential spatial light modulation. Through two-photon coincidence measurements, we verify their Airy spatial wavefunctions, propagation along a parabolic trajectory, and that the spatial modulation does not introduce any observable degradation of quantum correlation between the photons. These results suggest the feasibility of using spatially structured photons for practically advantageous quantum applications.
http://dx.doi.org/10.1364/ao.XX.XXXXXX There has been growing interest in surpassing the diffraction limit of light beams while engineering their directionality for advanced imaging and remote sensing of chemical and biological agents [1][2][3][4]. In this pursuit, Airy wave packets arose as a unique and promising candidate for their non-diffracting propagation along parabolic trajectories that mimic free acceleration [5]. This extraordinary phenomenon was observed more recently by using finite-energy optical Airy beams, together with self healing where their spatial modes are self reconstructed after being partially blocked [8][9][10]. A common way to create the Airy beams is via spatial modulation of Gaussian beams using a cubic phase mask and a Fourier lens [8][9][10], which allows control of the peakintensity locations during their ballistic propagation by laterally translating the phase mask or by tilting the Fourier lens [10][11][12]. Alternatively, they can also been created directly in microchip lasers [13,14] and through nonlinear optical processes [15]. Besides optical Airy beams, their electronic counterparts have also generated by diffraction of electrons through a nanoscale hologram [16].
Thus far, optical Airy beams have been explored for exotic applications [12] such as micro-particle manipulation [17,18], surface plasmonic bending [19][20][21], directioned filamentation [22][23][24], parabolic plasma channeling [25], micromachining along a curve [26], and superresolution imaging [27]. More recently, Airy beams have been exploited in optical parametric oscillators [28,29] and four-wave mixing in atomic vapor cells [30], with growing applications in medical science [31], defense [32], and optical communications [33][34][35]. Those exciting studies, however, are all in the classical domain. It remains to examine if those distinct advantages found in Airy beams could be exploited in the quantum domain [36][37][38][39][40].
In this paper, we generate quantum correlated photon pairs in finite-energy Airy modes through spontaneous parametric down conversion (SPDC) and subsequentially spatial light modulation using a cubic phase mask followed by a Fourier lens. Through time-correlated photon counting, we verify the Airy spatial wavefunctions of the photons and their propagation along a parabolic trajectory. Importantly, we find that the spatial modulation on the photons does not introduce any observable degradation of quantum correlation between them. Those Airy photons could be deployed in various quantum applications for remote sensing, imaging, communications, etc, where distinct advantages may be expected [41][42][43][44][45][46][47][48][49]. The light passes through an electro-optic modulator to create a pulse train with 100 ps full width at half maximum (FWHM) and 10 MHz repetition rate, synchronized with a radio-frequency source that gates single photon detectors for measurement. The optical pulses are then amplified in an Erbium-doped Fiber Amplifier to obtain high peak power (∼ 1 W), and guided through a fiber collimator, a half-waveplate, and a quarter-waveplate, before coupled into a magnesium-doped periodically poled lithium niobate (PPLN) waveguide through an aspheric lens. The PPLN, about 1 cm long, is temperature stabilized and phase matched to create SPDC pump pulses at 779.83 nm via second harmonic generation. The output pulses are then filtered with three short-pass filters which provide a total >180 dB extinction to remove any residual fundamental light. A 90:10 beamsplitter is used to tap 10% power of the second harmonic light for real-time monitor- ing and the remaining power is guided through an aspheric lens into the second PPLN waveguide with similar phase matching characteristics for photon-pair generation via SPDC (779.83 nm → 1554.7 nm + 1564.7 nm) [50][51][52][53][54]. The created photon pairs are coupled out and collimated by another aspheric lens before passing through a long pass filter with extinction of 50 dB to remove the 779.83-nm light pump pulses. After coupled into a fiber, the signal and idler photon pairs are picked at 1554.7 nm and 1564.7 nm respectively, using wavelength division multiplexers. The idler photons are guided through an optical fiber delay line and detected using an InGaAs avalanche photodiode (InGaAs APD) (ID210, ID Quantique) with 10% quantum efficiency and < 100 dark counts per second. The signal photons are guided through a fiber collimator to the free space Airy-beam setup, as shown in Fig. 1, before they are detected also using a second InGaAs APD. Each detected photon produces a TTL pulse which goes into a multi-channel time-to-digital converter (SENLS, HRM-TDC) for coincidence measurement.
The Airy-beam setup follows the standard approach of first producing a cubic phase modulation on a Gaussian incident beam using a 1.5 cm × 1.1 cm spatial light modulator (SLM) (Santec SLM-100), and then passing it through a Fourier transform lens [8,9]. Here, two-dimensional Airy spatial modes are created for both classical light and the signal photons, whose transverse mode is given by
E(x, y) = Ai x x 0 Ai y y 0 exp a x x 0 + y y 0 ,(1)
where Ai stands for the Airy function, x 0 and y 0 are the scaling factors in x and y transverse directions, respectively, and a is a truncation factor [25]. In this experiment, x 0 = y 0 = 271 µm which corresponds to an Airy spatial mode with a main lobe ∼ 434 µm FWHM. The phase mask is created numerically by discretizing the cubic phase over an 1.04 cm × 1.04 cm area (smaller than the SLM screen) with pixel size ∼ 10.4 µm. A linear phase modulation along the x direction is superposed on the cubic phase mask to deflect the modulated light into its first diffraction order [55] thus separating it from any unmodulated light that will be in the zeroth order. The total loss as the light go through the free-space setup is 22 dB. The spatial profile of the resulting two-dimensional Airy beam is directly measured using a NIR-IR camera (FIND-R-SCOPE Model No. 85700) with pixel resolution of 17.6 µm.
Once the Airy-beam setup is verified and optimized using a CW laser at 1568.7 nm, the signal photons are switched in the same fiber and free-space optical paths. At the focal point of the Fourier lens, they are spatially modulated to be in a twodimensional Airy wavefunction. From there, those Airy photons propagate for 3 meters along a parabolic, self-accelerating trajectory. Both the Airy wavefunction and parabolic trajectory are verified using quantum correlation measurement with the idler photon, by first collecting the Airy photons in a fiber and detecting them using an InGaAs APD.
We first measure the quantum correlation between the signal and idler photons with and without the Airy modulation. For the measurement without modulation, the signal photons bypass the Airy-beam setup and propagate instead through an optical fiber of equivalent path length. Using the standard experimental procedures in [58], the coincident-to-accidentals ratio (CAR) of the photon pairs is measured over an integration time of 60 minutes with 33 mW pump pulse peak power. For the measurement with the modulation, the signal photons are guided through the Airy-beam setup and collected by a single-mode fiber (SMF-28) at the focal point of the Fourier lens using a fiber collimator consisting of an aspheric lens (Thorlabs C220TMD-C). The lens is chosen to have a clear aperture that is much larger than the main lobe of the Airy beam and a numerical aperture that matches the fiber for the maximum coupling efficiency. The measurement results are shown in Fig. 2, where the coincident counts for the unmodulated and Airy signal photons are compared. The CAR values in these two cases are 69±6 and 70±3, respectively, which are within each's statistical error despite substantial loss in the current Airy-beam setup that reduces the photon counts. This result indicates that the phase mod-ulation by the SLM does not have any measurable impact on the quantum states of the photon pairs [44], which is essential for their applications in quantum imaging [45], distant object identification [46], quantum key distribution [48], and so on. We next perform a line scan over the optimized classical Airy beam to examine its transverse mode features, as shown in the Inset of Fig. 3. A pinhole (diameter ∼ 0.25 mm) is placed on a translational stage very close to the fiber coupler. This pinhole is translated across a 2-mm range with a 50 µm step size, covering the Airy beam's main lobe and the first side lobe. The intensity of the higher order lobes is too low for this scanning measurement. The result, which uses a a CW laser at 1568.7 nm and an InGaAs switchable gain amplified detector, is plotted as a blue line in Fig. 3, which measures the main lobe of the Airy beam to be ∼ 500 µm FWHM, close to the ∼ 493 µm value derived from NIR-IR camera image (Fig. 3, Inset) and the theoretical value of ∼ 434 µm. Afterwards, the signal photons are created in the Airy spatial wavefunction, whose profile is similarly scanned and measured by coincident detection with the idler photons. In this measurement, at each scanning point the coincident counts are recorded with pump pulse peak power of 0.2 W for a 600second integration time. The result is plotted as green dots in Fig. 3, which follows closely the profile of the classical Airy beam, thus verifying the Airy wavefunction of the signal photons. Lastly, we verify the accelerating propagation of the Airy photons. To this end, we first construct a classical Airy beam from a CW laser at 1568.7 nm as in Fig. 1, and examine its ability to go around an obstacle. As a reference, a collimated Gaussian beam is introduced to overlap with the Airy beam's path right at the focal point of the Fourier lens and after propagating 3 m, where the two are coupled into the same fiber coupler. Figure 4 (a) shows the measured trajectories of the two beams using the NIR-IR camera, along with the simulated position of the Airy beam's main lobe using the exact phase mask for the SLM. The good agreement between the simulation and measurement validates our entire setup. To further examine the parabolic trajectory, a 1-cm wide block is inserted into the path of Gaussian beam by 1.2 mm in the middle between the iris and fiber collimator. The block impedes the Gaussian beam, causing its power collected in the fiber to drop by 90%. In contrast, it allow the Airy beam to travel around due to the parabolic propagation, causing only a 30% drop. Then we swap in the Airy photons to test their parabolic propagation. Figure 4 (b) shows a typical histogram of the coincidence counts between the Airy and idler photons, comparing the cases with and without the block. In this measurement, the pump pulse peak power is 33 mW and the integration time is 60 minutes. As shown, the coincidence counts for the two cases exhibit similar profiles (with shot-noise fluctuations in each time bin), but with the total coincident counts dropping by 28% when the blocked is inserted, which agrees with the results for the Airy beam. The CAR values without and with the block are 70±3 and 62±4, respectively. This highlights the potential of steering Airy photons on a controlled trajectory, which can be reconfigured in real time by using a computer programmed SLM without any moving optics. This feature could be exploited for robust free-space quantum communications beyond line-of-sight [56].
In conclusion, we have created quantum correlated photons in a spatial Airy-shape wavefunction using spontaneous parametric down conversion and cubic spatial phase modulation. Based on two-photon coincidence measurement, we have found that the spatial modulation on the photons does not introduce any observable degradation to the quantum states of the correlated photon pairs. Contrasting with a collimated Gaussian beam, we have also verified the parabolic propagation trajectory of the Airy photons. Our results suggest the feasibility of using structured photons with exotic spatial and temporal modes [57,58] for practical advantages in long distance quantum communication [42,43], quantum imaging [44][45][46][47], quantum key distribution [48], light-sensitive biological applications [49], and so on. Beside the Airy modes, there exist a multitude of spatiotem-proal modulations [23,[59][60][61] that can be similarly applied to quantum photonic signals to attain distinct advantages for various applications. Finally, the present experiment applies phase modulation directly on the generated photons, which induces loss and significantly reduces the photon production rate. This problem can be solved by using the lossless photon shaping technique demonstrated in [62], which could be a subject of research in the future.
Figure 1
1outlines our experimental setup. A continuous-wave (CW) laser (LaserBlade, Coherent Solutions) generates light at 1559.67 nm with ≤ 100 kHz linewidth.
Fig. 1 .
1Experimental setup for generation and detection of quantum correlated Airy photons. TLS: Tunable Laser Source, FPC: Fiber Polarization Controller, EOM: Electro-optic Modulator, EDFA: Erbium-doped Fiber Amplifier, QWP: Quarter-Waveplate, HWP: Half-Waveplate, OSC: Oscilloscope, BS: Beamsplitter, PBS: Polarizing Beamsplitter, SLM: Spatial Light Modulator, WG: Waveguide, PG: Pulse Generator, AWG: Arbitrary Waveform Generator, RF: Radio-Frequency Source, APD: InGaAs Avalanche Photodiode, TDC: Time-to-Digital Converter, SPF: Short Pass Filter, LPF: Long Pass Filter, WDM: Wavelength Division Multiplexer.
Fig. 2 .
2Two-photon coincident detection for (a) unmodulated photons and (b) Airy photons.
Fig. 3 .
3Line scanning measurement of the Airy beam (blue curve) and Airy photons (green dots). For the latter, the error bars are given as the square root of the coincidence count at each point. Inset shows a CCD image of an Airy Beam, with the red line indicating the scanned area.
Fig. 4 .
4(a) Peak-intensity trajectories of the Gaussian beam (blue dots) and Airy beam (orange dots), where the error bars correspond to the fluctuations in each's CCD images. The blue line is a linear fit to the Gaussian beam's trajectory and green line is the the simulated trajectory. Also shown is the block's position. (b) Histogram of the coincident counting between the idler and Airy signal photons without and without the block.
ACKNOWLEDGMENTSThis research was supported in part by the Office of Naval Research (Award No. N00014-15-1-2393). YPH would like to thank Jianming Wen for initial discussions that motivated this research.
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|
[] |
[
"Supersymmetric Response of Bose-Fermi Mixture to Photoassociation",
"Supersymmetric Response of Bose-Fermi Mixture to Photoassociation"
] |
[
"T Shi \nInstitute of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina\n",
"Yue Yu \nInstitute of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina\n",
"C P Sun \nInstitute of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina\n"
] |
[
"Institute of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina",
"Institute of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina",
"Institute of Theoretical Physics\nChinese Academy of Sciences\nP.O. Box 2735100190BeijingChina"
] |
[] |
We study supersymmetric (SUSY) responses to a photoassociation(PA) process in a mixture of Bose molecules b and Fermi atoms f which turn to mutual superpartners for a set of proper parameters. We consider the molecule b to be a bound state of the atom f and another Fermi atom F with different species. The b-f mixture and a free F atom gas are loaded in an optical lattice. The SUSY nature of the mixture can be signaled in the response to a photon-induced atom-molecule transition: While two new types of fermionic excitations, an individual b particle-f hole pair continuum and the Nambu-Goldstone-fermion-like ( or goldstino-like) collective mode, are concomitant for a generic b-f mixture, the former is completely suppressed in the SUSY b-f mixture and the zero-momentum mode of the latter approaches to an exact eigenstate. This SUSY response can be detected by means of the spectroscopy method, e.g., the PA spectrum which displays the molecular formation rate of F f → b.
|
10.1103/physreva.81.011604
|
[
"https://arxiv.org/pdf/0909.3996v3.pdf"
] | 56,343,685 |
0909.3996
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015c681e294978927e0ef7d9d993a00a4f513e5c
|
Supersymmetric Response of Bose-Fermi Mixture to Photoassociation
26 Jan 2010
T Shi
Institute of Theoretical Physics
Chinese Academy of Sciences
P.O. Box 2735100190BeijingChina
Yue Yu
Institute of Theoretical Physics
Chinese Academy of Sciences
P.O. Box 2735100190BeijingChina
C P Sun
Institute of Theoretical Physics
Chinese Academy of Sciences
P.O. Box 2735100190BeijingChina
Supersymmetric Response of Bose-Fermi Mixture to Photoassociation
26 Jan 2010(Dated: January 26, 2010)numbers: 6785Pq3710Jk1130Pb
We study supersymmetric (SUSY) responses to a photoassociation(PA) process in a mixture of Bose molecules b and Fermi atoms f which turn to mutual superpartners for a set of proper parameters. We consider the molecule b to be a bound state of the atom f and another Fermi atom F with different species. The b-f mixture and a free F atom gas are loaded in an optical lattice. The SUSY nature of the mixture can be signaled in the response to a photon-induced atom-molecule transition: While two new types of fermionic excitations, an individual b particle-f hole pair continuum and the Nambu-Goldstone-fermion-like ( or goldstino-like) collective mode, are concomitant for a generic b-f mixture, the former is completely suppressed in the SUSY b-f mixture and the zero-momentum mode of the latter approaches to an exact eigenstate. This SUSY response can be detected by means of the spectroscopy method, e.g., the PA spectrum which displays the molecular formation rate of F f → b.
Introduction -Recently, studies in the supersymmetry (SUSY) for a mixture of cold Bose and Fermi atoms have made spectacular progress [1][2][3]. In such a cold atomic system, however, a Bose atom never transits to a Fermi atom, its superpartner or vice verse. In addition to the nonrelativity, this is another essential difference of this low-energy SUSY from the SUSY in high-energy physics. For the latter, such SUSY decay processes are always anticipated, e.g., a quark (lepton) may emit or absorb a gaugino and decays to a squark (slepton), the superpartner of the quark (lepton) [4].
To expose the interesting SUSY nature of the mixture, the effective "decay" process must be introduced. For a cold atomic SUSY mixture with Bose-Einstein condensation, there is an effective decay of SUSY generators since they behave as the fermion annihilation and creation operators [3]. Therefore, the SUSY excitations can be simulated by a boson-enhanced fermionic excitation. As a result, a Nambu-Goldstone-fermion-like (or "goldstino-like") mode in the condensation phase of bosons could be observed by means of the single-particle spectroscopy [5,6].
To achieve an exact SUSY mixture, the system parameters have to be fine-tuned, which requires elaborate experimental setups and then loses the generality. In this article, we explore how to observe the SUSY response by means of a spectroscopy measurement, even if the mixture deviates slightly from the SUSY and the bosons do not condense to form a whole ordered phase. This can resolve the fine-tuning restraints in measuring the SUSY response. On the other hand, the explicit breaking of the SUSY may create new excitations, the bosonic particle-fermionic hole individual continuous excitations, other than the collective goldtino-like mode. Although our theory is nonrelativistic, the creation of these new excitations due to SUSY explicit breaking should be quite general . This may be a helpful point to the study of SUSY in relativistic theory. We consider a mixture of Bose molecules b and Fermi atoms f with on-site interaction in a d-dimensional optical lattice (d = 2, 3) (see Fig.1(a)). With properly tuned interactions and hopping amplitudes, this b-f mixture may become SUSY [3]. We are interested in a special kind of molecule b, a bound state of f and another species of Fermi atom F with binding energy E b , and restrict our analysis to the normal phase of the b-f mixture [7]. To probe the SUSY behaviors, we load a free Fermi atom F gas, which does not interact both b and f directly. In a photoassociation(PA) process [8], the transitions between two atoms and one molecule, i.e., F f ↔ b, are induced by two laser beams with frequencies ω 1 and ω 2 . For the SUSY b-f mixture, this resembles a high energy physics process: a 'quark' or a 'lepton' (f ) absorbs a 'fermionic gaugino' ('absorbs' a F and emits a photon) and decays to a 'squark' or a 'slepton' (b) or vice verse. (One can also consider f to be a Fermi molecule formed by the bound state of a Bose atom b and a Fermi atom F , i.e., processes F b ↔ f . We will study these processes separately.)
Bose Fermi mixture − b E 0 δ − F R Π (a) photoassociation process 2 ω 1 ω 0 ∆ 1 R Π 0 Π 0 Π bf U (b) Feynman diagram
For a negative detuning δ 0 = ω 2 − ω 1 − E b , we show that the molecule dissociation process b → F f is forbidden. In the formation process F f → b, two types of new fermionic excitations, an individual (bosonic) particle-(fermionic) hole pair continuum and a collective mode, emerge when the SUSY in the b-f mixture is slightly broken. For a SUSY b-f mixture, the former is completely suppressed while the latter in zero-momentum becomes an exact eigenstate, the Goldstino mode [3]. In this sense, we regard these excitations as the SUSY responses. The PA spectrum is directly related to the the molecular formation rate varying as the detuning and faithfully describes these two types of excitations. The position of peak in the PA spectrum determines the frequency of the collective zero-momentum mode. This molecular formation rate is measured by the number variation of the F atoms in time. Experimentally, the number counting of atoms is much simpler than detecting the single atom spectrum.
Model setup -The system illustrated in Fig. 1
(a) is described by a Hamiltonian
H = H 0 + H ex , where H 0 = H bf +H F with H bf = H b +H f +V .
By means of the Feshbach resonance [9], the scattering lengths between F and the b-f mixture can be adjusted to negligibly small. In the tight-binding approximation, one has
H α = − ij t α a α † i a α j − µ α i a α † i a α i ,(1)V = U bb 2 i n b i (n b i − 1) + U bf i n b i n f i , where a α i = b i , f i , and F i (α = b, f
, and F ) are the annihilation operators of b, f and F at site i; µ α and n α i = a α † i a α i are chemical potentials and the number operators at site i. The definitions of the hopping amplitudes t α and the interaction strengths U αβ by the Wannier function w α (r) can be found in the literature [10]. The spatial inhomogeneity of optical lattices trapping the atoms and molecules has been omitted. For the subsys- [1,3]. In order to prevent the phase separation, the parameters obey, e.g.,
tem b-f mixture, the Hamiltonian H bf is SUSY invariant if t b = t f , U bb = U bf and µ b = µ f4πt f ρ f U bb > U 2 bf in 2-dimensions,
where ρ f is the density of f atoms [11]. We choose the parameters of the system obeying this condition.
The PA processes are realized by simultaneously shining two laser beams with frequencies ω 1 and ω 2 into the lattice (shown in Fig. 1(a)). The ω 1 -beam may turn two free atoms f and F into a higher energy bound state |1 which then may transit to the molecule b by emitting a photon with frequency ω 2 . Meanwhile, the molecule b may also be excited to |1 by the ω 2 -beam and then is unbound with some probability by emitting a photon with frequency ω 1 . For large detuning ∆ 0 , the state |1 can be eliminated adiabatically, so that the PA is modeled by the tight-binding Hamiltonian
H ex = i (g i b † i f i F i e iδ0t + H.c.),(2)
where the detuning
δ 0 = ω c − E b with the effective driven frequency ω c = ω 2 − ω 1 , and g j = g 0 exp(−ik 0 · r j ) with g 0 ∝ d d r exp(−ik 0 · r)w * b (r)w f (r)w F (r)
being the coupling intensity independent of the site.
In the k-space, the Hamiltonian H ex is rewritten as
H ex = g 0 √ ρ( k Q † k−k 0 F k e iδ0t + H.c.),(3)
where
Q † k = p b † p+k f p / √ N and a α k = j a α j exp(−ik · r j )/ √ V (where V stands for the volume and a α k stand for b k , f k , and F k ). ρ = N/V is the total density of b and f with the particle number N = i (n b i + n f i )
. Molecular formation rate -The formation rate of the molecules b can be counted by the PA variation of Ffermion number R = ∂ t ψ(t)| N F |ψ(t) for |ψ(t) being the time evolution from the ground state |G = |g |F of H 0 . It follows from the linear response theory that
R = 2g 2 0 ρ k ImD R (k, −δ 0 ),(4)
where the retarded Green function is given by
D R (k, ω) = ∞ −∞ dxA(k − k 0 , x) n f (x) − n f (ε F k ) x − ε F k − ω − i0 + ,(5)
in terms of one loop calculations ( Fig. 1(b)). The single particle dispersions are ε α k = −2t α d s=1 cos k s − µ α where the lattice spacings are set to be the unit. n f (x) is the Fermi distribution at temperature T and the spectral function A(k, ω) = −ImΠ R (k, ω)/π is defined by the retarded Green functionΠ
R (k, ω) = −i ∞ 0 dt g| {Q k (t), Q † k (0)} |g e iωt .
At sufficiently low temperature, the pole and branch cut in Eq. (5) are not qualitatively affected by T and nor is the molecular formation rate. For simplicity, we take a zero temperature approximation in our calculation. It follows from Eq. (5) that the rate R = R b→F f − R F f →b contains two parts :
R b→F f = k 2πg 2 0 ρA(k − k 0 , ε F k − δ 0 )θ(δ 0 − ε F k )
,
R F f →b = k 2πg 2 0 ρA(k − k 0 , ε F k − δ 0 )θ(−ε F k ).(6)
which respectively are the dissociation rate for b → F f and the formation rate for F f → b.
Collective and individual fermionic modes -In order to obtain R for the weak interactions, we perturbatively calculate Π R (k, ω) = ρ −1 [Π −1 0 (k, ω) + U bf ] −1 , which formally results from the equation of motion of Q k . It then follows from the random phase approximation (RPA) illustrated by the "bubble"in ( Fig. 1(b)) that
Π 0 (k, ω) = d d p (2π) d n f (ε f p ) + n b (ε b k+p ) ω − E kp + i0 + ,(7)
where
E kp = ε b k+p − ε f p + 2ρ b δU + U bf ρ with δU = U bb − U bf and ρ b = N b /V; n b (x)
is the Bose distribution. The isolated pole and branch cut of Π R (k, ω) describe the collective and individual SUSY excitations of Q † k |g . Next we consider the elementary excitations in the two-dimensional lattice with f atoms at half filling, i.e., ρ f = 0.5. For the SUSY b-f mixture, i.e., δU = 0 and δt = t b − t f = 0, the dispersion of the collective modes, E c (k) ≃ ∆µ − α |k| 2 for the small |k| [3], is read out from the poles of the retarded Green function Π R (k, ω) (see Fig. 2(a)), where ∆µ = µ f −µ b . For large |k|, the energy E c (k) = E c (|k| , θ) depends not only on |k| but also on the angle θ = arctan(k y /k x ). The energy E c (|k| , θ) of Q † k |g decreases as |k| increases for a fixed θ. The retarded Green's function Π R (0, ω) = (ω − ∆µ + i0 + ) −1 possesses a pole ω = ∆µ, which corresponds to the goldstino-like excitation Q † 0 |g . This recovers the result in Ref. [3]. The excitation spectrum is schematically shown in Fig. 2(b).
For the b-f mixture deviating slightly from SUSY, the retarded Green function Π R (0, ω) has an isolated pole and a branch cut, which correspond to a collective fermionic mode and individual (bosonic) particle-(fermionic) hole pair continuum modes, respectively. The pole in ω 0 < E 0 − 4δt for δt > 0 (or ω 0 < E 0 for δt < 0) describes the shifted goldstino-like mode. The frequencies ω 0 of collective zero-momentum mode for different δt are shown in Fig. 2(c). For k = 0, the pole of Π R (k, ω) has the form E ′ c (k) ≃ ω 0 − α ′ |k| 2 for small |k|. Remarkably, the branch cut l 0 of Π R (0, ω) emerges, which describes individual zero-momentum modes. Here,
l 0 = [E 0 −4δt, E 0 ] for δt > 0 (or [E 0 , E 0 −4δt] for δt < 0)
, and E 0 = ∆µ+2ρ b δU +U bf ρ. Notice that for the weak interactions U bb and U bf the SUSY breaking from δU does not develop a branch cut but only shifts the positions of the pole and branch cut. The pole and the branch cut can be seen in the spectral function A(k, ω), i.e., the peak and the hump in Fig. 2(d) for k = 0. Note that for the SUSY b-f , the branch cut length l 0 of Π R (0, ω) shrinks to zero so that the individual continuum modes of zero momentum are completely suppressed. On the other hand, as the b-f mixture deviates from the SUSY, the goldstino-like mode is gradually suppressed. We ex- amine the dependence of the spectral function on the interacting strength and find that the hump height may be depressed as the interaction becomes stronger, e.g., the height is lower than 0.5 for U bf = U bb = 0.5 comparing with ∼ 10 in Fig. 2(d) for U bf = U bb = 0.1.
In order to study the PA spectrum of the molecular formation rate, we discuss the excitation spectrum shown in Fig. 2(b). For some momenta k, there is a collective mode (dashed red curve) below the individual continuum. For other momenta k c , the dispersion of the collective mode merges into the continuum. However, for small momentum k, there always exists a collective mode below the individual continuum. For convenience, we define a critical momentum k Q (θ), so that for a fixed θ, when |k| > k Q (θ) the negative frequencies of the mode Q k emerge, i.e., A(k, ω < 0) = 0 when |k| > k Q (θ), and A(k, ω < 0) = 0 when |k| < k Q (θ).
PA spectrum -The rate R varies as detuning δ 0 or the light frequency ω c . Measurement of the b boson formation rate varying as δ 0 is called the PA spectrum S(δ 0 ). For a long wave photon, the coupling g j varies slowly in space and the rates in Eq. (6) are approximately independent of k 0 .
According to Eq. (6), the dissociation rate R b→F f does not vanish only if δ 0 > ε F k and A(k, ε F k −δ 0 ) = 0. Because the spectral function A(k, x < 0) = 0 is defined by the re- Fig. 2. (a) The detuning dependence for δt = 0.05 and −0.05, the solid (red) and dashed (blue) lines, respectively. The unit of S is 2πg 2 0 ρNF and NF /N is taken to be 0.01. (b) The major peak values for different δt. This shows that the peak value is suppressed when the system deviates from SUSY.
tarded Green function, it does not vanish only when the energies for collective modes or individual modes of Q k are negative for the large |k| > k Q (θ). We consider a dilute Fermi gas F with the chemical potential µ F ∼ −4t F , the dispersion relation turns out ε F k = t F |k| 2 − µ eff , where µ eff = µ F + 4t F . In this case, the fermion F possesses a small Fermi momentum k F = µ eff /t F which is much smaller than k Q (θ) for small deviating δt. Therefore, ε F k is always positive when |k| > k Q (see Fig. 2(b)). For the negative detuning δ 0 , the condition δ 0 > ε F k is not satisfied in the regime |k| > k Q (θ). That is, A(k, ε F k − δ 0 ) and θ(δ 0 −ε F k ) can not be non-zero simultaneously for the negative detuning and small Fermi momentum k F . This finishes our proof of R b→F f = 0.
The vanishing of R b→F f for the negative δ 0 and small Fermi momentum k F can be understood in a more straightforward way. The transition b → F f is described by the Hermite conjugate term (H.c.) in the Hamiltonian H ex , which is a high-frequency oscillation term when δ 0 < 0. Hence, the Fermi golden rule results in that R b→F f vanishing under the first-order perturbation (linear response).
For the negative δ 0 and small Fermi momentum k F (µ eff ≪ t F ), the molecule formation rate now is reduced to R = −R F f →b and
R F f →b ≃ Z 0 g 2 0 N/[2(t F + α ′ )], for δ 0 = −ω 0 , 2πg 2 0 ρN F A(0, −δ 0 ), for |δ 0 | ∈ l 0 , 0, otherwise
which leads to our main result: The PA spectrum S(δ 0 ) = −R F f →b (see Fig. 3(a)) displays the spectral function of excitations Q † 0 |g . For the SUSY b-f mixture, the length of branch cut l 0 tends to zero and the individual modes are suppressed. Meanwhile, the residue Z 0 = 1 and the formation rate R F f →b = g 2 0 N/[2(t F + α)] ≡ R 0 ∝ N at δ 0 = −∆µ, and vanishes for the other detunings. There is a sharp peak at δ 0 = −∆µ in the PA spectrum. For a generic b-f mixture deviating from SUSY, the residue Z 0 < 1 decreases as |δt| increases. As a result, the peak height is lowered while its position is shifted to δ 0 = −ω 0 . The ratio R F f →b (ω 0 )/R 0 is shown in Fig. 3(b) for different δt and a small µ eff , where R F f →b (ω 0 ) is the value of R F f →b at δ 0 = −ω 0 . Remarkably, a minor hump develops in the region δ 0 ∈ l 0 due to the emergence of individual modes (See Fig. 3(a)). As the system deviates further from SUSY, the individual modes are enhanced due to the sum rules dωA(0, ω) = 1. The temperature may suppress and broaden the peak and hump. These characters of the PA spectrum in the b-f mixture are experimentally measurable SUSY responses to the light field. Conclusions -We studied how to observe the SUSY nature of the b-f mixture in optical lattices through PA spectra. For the Bose molecules formed with two species of Fermi atoms, we showed that the photon induced atom-molecule transition displays the signal of SUSY. As the response to the PA processes, a fermionic individual continuum and the goldstino-like mode were found. The PA spectrum can explicitly witness the molecular formation rate of F f → b. Because the goldstino-like mode in zero momentum turns to be the exact eigen state for the SUSY mixture, the major peak in the PA spectrum reflects the SUSY response to the light field, even if the mixture is not fine-tuned to a SUSY one.
The authors thank Jinbin Li and Peng Zhang for the useful discussions. YY is grateful to Kun Yang for sharing his idea in the earlier stage of this work. This work is supported in part by National Natural Science Foundation of China, the national program for basic research of MOST of China and a fund from CAS.
FIG. 1 :
1(Color online) (a) Up-panel: The optical lattice with cold particles. The gray (green), black (red), and white dots denote Fermi atoms f , F , and the molecule b, respectively. Low panel: The PA processes of two atoms to one molecule with the binding energy E b . (b) The Feynman diagram for the linear response theory: The Green function of Q is calculated by RPA. Up-panel: The wavy (red) lines and dotted (red) line denote the free Green functions of photon and F , respectively. Low panel: The solid (green) curves and dotted (black) curves denote the free Green functions of f and b, respectively.
online) The dispersions, spectrum and spectral function of the excitations for ∆µ = 3.93 and U bb = U bf = 0.1; t b is taken as the unit. (a) The dispersion of collective mode for the SUSY mixture. (b) The spectrum for the SUSY system: the dashed (red) and dotted (blue) curves denote the dispersions of collective modes for θ = 0 and π/5, respectively. The solid (blue) curve denotes the dispersion of atoms F for a small Fermi momentum kF . (c) The frequencies of Q † 0 |g for different δt. (d) The spectral function of the zero-momentum for δt = −0.1: The sharp peak is the shifted goldstino-like mode.
online) The PA spectra with the same parameters as those in
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If the bosons b are condensed, the elementary excitations are more fruitful. We will leave them in further studies. If the bosons b are condensed, the elementary excitations are more fruitful. We will leave them in further studies.
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|
[] |
[
"The Hubble Deep Field: Observations, Data Reduction, and Galaxy Photometry",
"The Hubble Deep Field: Observations, Data Reduction, and Galaxy Photometry"
] |
[
"Robert E Williams ",
"Brett Blacker ",
"Mark Dickinson ",
"W Van Dyke Dixon \nCurrent address: Space Sciences Laboratory\nUniversity of California\n94720BerkeleyCA\n",
"Henry C Ferguson ",
"Andrew S Fruchter ",
"Mauro Giavalisco \nHubble Fellow; Current address: Carnegie Observatories\n813 Santa Barbara Street Pasadena91101CA\n",
"Ronald L Gilliland ",
"Inge Heyer ",
"Rocio Katsanis ",
"Zolt Levay ",
"Ray A Lucas ",
"Douglas B Mcelroy \nCurrent address: Jet Propulsion Laboratory\nCalifornia Institute of Technology\n4800 Oak Grove Dr91109PasadenaCA\n",
"Larry Petro ",
"Marc Postman ",
"Hans-Martin Adorf ",
"Richard N Hook ",
"\nSpace Telescope European Coordinating Facility, c/o ESO\nSpace Telescope Science Institute\n3700 San Martin Drive Baltimore, Karl-Schwarzschild-Str. 221218, D-85748GarchingMDGermany\n"
] |
[
"Current address: Space Sciences Laboratory\nUniversity of California\n94720BerkeleyCA",
"Hubble Fellow; Current address: Carnegie Observatories\n813 Santa Barbara Street Pasadena91101CA",
"Current address: Jet Propulsion Laboratory\nCalifornia Institute of Technology\n4800 Oak Grove Dr91109PasadenaCA",
"Space Telescope European Coordinating Facility, c/o ESO\nSpace Telescope Science Institute\n3700 San Martin Drive Baltimore, Karl-Schwarzschild-Str. 221218, D-85748GarchingMDGermany"
] |
[] |
The Hubble Deep Field (HDF) is a Director's Discretionary program on HST in Cycle 5 to image an undistinguished field at high Galactic latitude in four passbands as deeply as reasonably possible. These images provide the most detailed view to date of distant field galaxies and are likely to be important for a wide range of studies in galaxy evolution and cosmology. In order to optimize observing in the time available, a field in the northern continuous viewing zone was selected and images were taken for ten consecutive days, or approximately 150 orbits. Shorter 1-2 orbit images were obtained of the fields immediately adjacent to the primary HDF in order to facilitate spectroscopic follow-up by ground-based telescopes. The observations were made from 18 to 30 December 1995, and both raw and reduced data have been put in the public domain as a community service. We present a summary of the criteria for selecting the field, the rationale behind the filter selection and observing times in each band, and the strategies for planning the observations to maximize the exposure time while avoiding earth-scattered light. Data reduction procedures are outlined, and images of the combined frames in each band are presented. Objects detected in these images are listed in a catalog with their basic photometric parameters.
|
10.1086/118105
|
[
"https://export.arxiv.org/pdf/astro-ph/9607174v1.pdf"
] | 17,310,815 |
astro-ph/9607174
|
18270e2283c497cb3c0244566d3a3ac685ce85e3
|
The Hubble Deep Field: Observations, Data Reduction, and Galaxy Photometry
Jul 1996
Robert E Williams
Brett Blacker
Mark Dickinson
W Van Dyke Dixon
Current address: Space Sciences Laboratory
University of California
94720BerkeleyCA
Henry C Ferguson
Andrew S Fruchter
Mauro Giavalisco
Hubble Fellow; Current address: Carnegie Observatories
813 Santa Barbara Street Pasadena91101CA
Ronald L Gilliland
Inge Heyer
Rocio Katsanis
Zolt Levay
Ray A Lucas
Douglas B Mcelroy
Current address: Jet Propulsion Laboratory
California Institute of Technology
4800 Oak Grove Dr91109PasadenaCA
Larry Petro
Marc Postman
Hans-Martin Adorf
Richard N Hook
Space Telescope European Coordinating Facility, c/o ESO
Space Telescope Science Institute
3700 San Martin Drive Baltimore, Karl-Schwarzschild-Str. 221218, D-85748GarchingMDGermany
The Hubble Deep Field: Observations, Data Reduction, and Galaxy Photometry
Jul 1996arXiv:astro-ph/9607174v1 31
The Hubble Deep Field (HDF) is a Director's Discretionary program on HST in Cycle 5 to image an undistinguished field at high Galactic latitude in four passbands as deeply as reasonably possible. These images provide the most detailed view to date of distant field galaxies and are likely to be important for a wide range of studies in galaxy evolution and cosmology. In order to optimize observing in the time available, a field in the northern continuous viewing zone was selected and images were taken for ten consecutive days, or approximately 150 orbits. Shorter 1-2 orbit images were obtained of the fields immediately adjacent to the primary HDF in order to facilitate spectroscopic follow-up by ground-based telescopes. The observations were made from 18 to 30 December 1995, and both raw and reduced data have been put in the public domain as a community service. We present a summary of the criteria for selecting the field, the rationale behind the filter selection and observing times in each band, and the strategies for planning the observations to maximize the exposure time while avoiding earth-scattered light. Data reduction procedures are outlined, and images of the combined frames in each band are presented. Objects detected in these images are listed in a catalog with their basic photometric parameters.
Introduction
The HDF program is an outgrowth of previous, highly successful Hubble Space Telescope imaging projects which have elucidated the evolution of galaxies at high redshift. During Cycles 1 through 5, a variety of HST General Observer and Guaranteed Time Observer programs, as well as the Medium Deep Survey (MDS) key project, imaged distant galaxies in both cluster and field environments, providing (for the first time) kiloparsec-scale morphological data at all redshifts. The MDS used the WFPC-1 and WFPC-2 cameras in parallel mode to image random galaxies near the fields of targeted objects. Analyzing 144 field galaxies having I < 22 from six fields, found from visual classification that early-type spirals, ellipticals, and late-type spirals/irregulars were observed in roughly equal proportions, with the Sd/Irr's having much higher surface density than their counterparts at the current epoch. extended this analysis with a similar study of one very deep field for which they showed that galaxy counts beyond I = 22 continue to be increasingly dominated by Sd/Irr galaxies. Combining ground-based redshift information with HST imaging, Lilly and collaborators obtained B and I images for 32 galaxies from their CFHT survey (17.5 < I < 22.5) with known redshifts in the range 0.5 < z < 1.2 (Schade et al. 1995). They found that the observed galaxy morphologies were similar to those seen locally, but that the B images (rest frame UV) looked far less regular than observed at longer wavelengths. In addition, they determined that the central surface brightnesses of the disks in their sample of late-type spirals were more than 1.2 magnitude brighter than found locally. Also, they found that many of the bluer galaxies were nucleated, and they concluded that both of these effects must be responsible for much of the observed evolution of the luminosity function of blue galaxies.
Other HST programs targeted galaxies with known redshifts based upon their membership in clusters that had been studied from the ground, e.g., 0939+4713 (Dressler et al. 1994a,b) and the cluster(s) associated with the radio galaxy 3C 324 at z = 1.21 (Dickinson 1995a). Both of these programs demonstrated the ability of the refurbished HST to resolve galaxy structure at moderate to high redshift in a way that made morphological classification and a quantitative study of various parameters possible. Cluster 0939+4713 does not look entirely unlike nearby clusters insofar as it is populated largely by spiral and elliptical galaxies. However the disk systems are bluer and more numerous than spiral galaxies in the cores of clusters today, and often show signs of disturbance and tidal interactions. Evidently, these spirals are responsible for the rapidly evolving blue galaxy population first noted in distant clusters by Butcher & Oemler (1978;1984). Looking back to z = 1.21, the cluster associated with 3C 324 includes apparently normal, mature E/S0s, but readily recognizable spiral galaxies appear to be rare, and a large number of irregular, amorphous objects are present (Dickinson 1995a,b). Since the first servicing mission, HST has imaged a number of other distant galaxies at still higher redshifts of 1 < z < 3.5 (c.f. Cowie, Hu, & Songaila 1995;Giavalisco, Steidel, & Macchetto 1996) While much of the information available in these images remains to be interpreted, two things have become clear. First, HST can indeed resolve galaxy-sized systems out to high redshift. Second, the Universe at high redshift looks rather different than it does at the current epoch. The fact that HST can image galaxies back at epochs when they were apparently forming and evolving rapidly is of fundamental importance to our understanding of galaxy evolution, and it is imperative that this capability be fully exploited. Based on the current excellent performance of the telescope, a decision was made to devote a substantial fraction of the Director's Discretionary time in Cycle 5 to the study of distant galaxies. A special Institute Advisory Committee was convened which recommended to the Director that deep imaging of one 'typical' field at high galactic latitude be done with the Wide-Field Planetary Camera 2 (WFPC-2) in several filters, and that the data be made available immediately to the astronomical community for study. Following this recommendation a working group was formed to develop and carry out the project.
It is not our purpose here to interpret the data, but rather to present the images and source catalogs, along with the necessary background to facilitate the use of the HDF in studies of galaxy evolution. As part of the background, we describe the criteria for selecting the field, the scientific rationale for the selection of filters, technical aspects of planning the observations, and details of data reduction and calibration. The images are presented and discussed in §5, and the source catalogs in §6. In view of the wide interest in these observations, we have tried to provide a useful reference work, being reasonably comprehensive in describing the data reduction, and including in printed form the most important photometric parameters for the detected sources. Nevertheless many of the parameters for the individual galaxies (e.g. S/N in the different bands, higher-order image moments, star/galaxy classifications) have been left out of the printed catalog. We have opted to emphasize black & white over color images to provide as much detail as possible in the limited dynamic range of the printed page. The full catalog and color images are maintained on the STScI world-wide web site at http://www.stsci.edu.
Field Selection
Primary Field
One of the suggestions to the Advisory Committee by the Institute was that a field in one of the continuous viewing zones (CVZ) be considered, because the observing efficiency there could be up to a factor of two higher than other locations on the sky. The working group focused its attention on the northern CVZ, thereby constraining the HDF location to a narrow declination range centered around +62 • . In addition to being in the CVZ, the candidate field would have to have low optical extinction (E(B − V ) < 0.01 mag), low HI column density (< 2.5 × 10 20 cm −2 ), and low FIR (λ = 100µm) cirrus emission. Furthermore, to facilitate faint object studies at many wavelengths, the HDF field would need to avoid known bright sources in the x-ray, UV, optical, IR, and radio passbands. This latter criterion, therefore, excluded areas with known nearby (z < 0.3) galaxy clusters. Visual inspection of IPAC's co-added 100µm IRAS maps was used to select regions with no significant Galactic cirrus features. An initial sample of about 20 possible HDF regions was narrowed to 3 optimal candidates, all within the Ursa Major region. Quick VLA snapshots at 3.6 and 21 cm by Kellermann (1995) reduced the choice to two fields due to the presence of a 68 mJy source in the center of one of the fields. Initial optical selection was based on the digitized Palomar sky survey. Eisenhardt (1995) kindly provided KPNO 4m R-band CCD images (2 × 300 second exposures) as further verification that the fields were typical in terms of source counts, and were not affected by scattered light or diffraction spikes from bright stars outside the field A search of the ROSAT data archive (Petre 1995) in the vicinity of the two remaining field candidates placed a conservative upper limit on the flux from any source of 6 × 10 −14 erg cm −2 sec −1 in the 0.1-2.4 keV band.
In June 1995, HST acquired a single orbit F606W observation of each of the two fields to verify guide star acquisition. In order to be conservative in safeguarding the entire sequence of HDF observations, we required an independent pair of back-up guide stars, which are scarce at this high Galactic latitude. The decision between the two remaining candidate fields was thus based on guide star availability. The location and characteristics of the resulting HDF field are given in Table 1, and the KPNO R-band image of the field telescope is shown in Fig. 1 with the WFPC-2 field of view superposed.
Flanking Fields
During the HDF observations, 10 orbits were devoted to short exposures of eight "flanking" fields adjacent to the main survey region. These fields were arranged in a roughly square pattern surrounding the central HDF, as shown in Fig. 1. All exposures in the flanking fields were taken at the same orientation as the central field. The coordinates for these fields are given in Table 2.
Filter Selection
The selection of filters for the HDF observations represents a balance between the desire for depth and the desire for color information and practical considerations involving scattered earth light.
The HDF observations were not aimed at answering one specific question, but are rather intended to be of general use for constraining models of galaxy evolution and cosmology. There was thus no single, well-defined criterion that could be used to develop the optimum observing strategy. For galaxies that are well resolved, color gradients and the dependence of morphology on wavelength are of interest. If such studies were the sole aim of the observations, the best strategy might be to opt for the highest possible S/N in two widely separated bandpasses. On the other hand, for the more numerous faint and barely resolved galaxies, it is the statistical distributions of color vs. magnitude, color vs. angular size, etc. that are of interest. These provide information on both the redshift distribution and the stellar populations of the galaxies. For galaxies fainter than spectroscopic limits, it is important to have at least two colors. A single color (or "spectral index") is less useful because there is no way to separate effects of the intrinsic rest-frame spectral properties from the effect of the k-correction. This argues for observations through at least three filters, preferably with the highest possible efficiency and with minimal overlap in bandpasses. While other options offer slightly better photometric accuracy or slightly less bandpass overlap, the combination of F450W, F606W, and F814W provides a very efficient way to cover the optical portion of the WFPC-2 bandpass (Fig. 2). The addition of observations through the F300W filter greatly improves the leverage for statistical redshifts. These observations cannot reach the limiting depth of the other three filters because of the low detector quantum efficiency (QE) at 3000Å. However, because of the low QE, the F300W data are largely read-noise limited, whereas the other bands are more nearly sky-noise limited. Thus the the F300W observations can make use of the orbital "bright time" (see below) that is not particularly useful for the other filters.
With this four-filter strategy the HDF reaches depths roughly three magnitudes fainter than the deepest ground based images in the red bands, two magnitudes deeper in the B band, and one magnitude deeper in the U band. The 80% completeness limit of current deep spectroscopic surveys is B AB ∼ 24 4 and I AB ∼ 22.5 (Glazebrook et al. 1995;Lilly et al. 1995). With Keck and other large aperture telescopes, these limits may be pushed one or even two magnitudes fainter. Even then, the faintest two or three magnitudes of the HDF survey are beyond current spectroscopic limits. Hence, for at least the next few years, the primary information on the redshift distribution of these very faint galaxies will come from statistical analysis of the color distribution. In the individual F450W, F606W, and F814W bands, the HDF observing strategy provides only a modest improvement over the deepest existing HST images. However, having four passbands provides essential information not available from previous surveys.
Part of the scientific interest in F300W stems from its utility in searches for very high-redshift (z > 3) galaxies (Guhathakurta, Tyson, & Majewski 1990;Steidel & Hamilton 1992,1993 The intrinsic 912Å Lyman break in galaxies, combined with the increasing opacity of the intergalactic medium at high redshifts produces a distinctive feature in the spectra of high-redshift galaxies (Yoshii & Peterson 1994;Madau 1995). Specific color selection criteria for the HDF bandpasses are described by Madau et al. (1996). The F300W filter has a small but significant "redleak" at 8000Å. The transmission curve of this redleak is similar to that of the F814W filter, but the throughput is three orders of magnitude smaller. Thus, a 20th magnitude galaxy with no intrinsic flux in the F300W bandpass will appear as an F300W source of AB magnitude ∼ 27.7. Most of the galaxies detected in the HDF have magnitudes fainter than I 814 = 23, and would thus be below the detection limits of the F300W images unless they have intrinsic emission near 3000Å.
Scattered Light and Observation Scheduling
The choice of the F300W passband was motivated partly by the desire to provide a measure of the surface density of galaxies above redshift z = 3 and partly by the desire to find an observing strategy that avoids contamination from stray earth light. While the zodiacal-light background in the HDF field is extremely low in December, scattered earth light during portions of the CVZ was expected to dominate the background in F450W, F606W, and F814W. The scattered earthshine in the F300W filter is more than 15 times fainter than in the other three filters due to absorption of the solar flux by ozone in that band. HST can in principle observe to within 15.5 degrees of the bright earth limb and 7.6 degrees of the dark limb (these limits are imposed by the Fine Guidance Sensors). However, the effects of scattered light can be detected in archival WFPC-2 images at angles to the bright limb as great as 40 degrees. The amount of scattered light in the HST focal plane varies both with limb angle and the total brightness of the earth. The observations were carefully scheduled to make use of the dark portion of the orbit for observations through the F450W, F606W, and F814W filters, and the bright portion for F300W observations. Efforts over the last two years to understand the sources of background have resulted in the development of a program (SEAM; Bely et al. 1996) that successfully models the WFPC-2 background as a function of wavelength and viewing direction. SEAM has been used to predict the background in various filters considered for the HDF. Figure 3 shows examples of the predicted background as a function of time in several orbits spaced throughout the program. Curves such as these were used to plan the start and stop times of each observation to maximize observing time, while minimizing the total background. To ensure that the timing was correct, the start/stop times were adjusted using the latest available orbit predictions as a last step before the final schedule was frozen several weeks prior to the observations. It should be emphasized that without this careful scheduling of the observations, the extra observing time gained by selecting a field in the CVZ would have been largely wasted. While such micro scheduling is not something that is routinely done for HST observations, it is clearly something that is desirable to implement in a more automated manner for observations in which sky background is the limiting factor in obtaining the best signal-to-noise ratio.
Another decision made in the planning stage was to attempt to obtain observations at nine separate pointing positions through each filter. The 0.1 ′′ pixels of the three wide-field (WF) cameras in the WFPC-2 instrument undersample the point spread function of the telescope, while the planetary camera (PC) provides more nearly optimal sampling. The motions of the telescope were laid out as non-integer multiples of a WF pixel. This "dithering" reduces the photometric errors due to flatfielding uncertainties and also allows reconstruction of a higher-resolution image because sources are sampled in different portions of a pixel at each dither position. The price for these improvements is increased complexity in the data reduction phase. The images must be aligned and resampled to the same pixel scale, correcting for the geometric distortions introduced by the camera optics. We chose to observe at nine dither positions spanning a range of 2.6 arcseconds that would in addition map to nine independent positions within the (WF) pixel scale of 0.1 arcseconds (Fig. 4). Two unplanned dither positions with offsets of ∼ 1 ′′ and a rotation of 4.3 arcminutes resulted from an 11 hour period in which one of the Fine Guidance Sensors locked up on a secondary extremum of the "S-curve." We tried to obtain at least five exposures per filter per dither position that were near enough in time that slow drifts in the telescope pointing (which had been seen in previous CVZ observations) would not complicate the cosmic-ray rejection process.
Data Reduction
Pipeline
The HDF data were reduced three times. Version 1 reflects processing up to January 15, 1996. Version 2 was released February 29, 1996 and is used for this paper. At the time of writing, version 3 was still a month or two from completion.
The HDF data were reduced using the same software as the standard HST pipeline calibration, the STSDAS task calwp2, but with different calibration files and a slightly modified treatment of the darktime. The calibration procedure is fully documented in the HST Data Handbook (Leitherer 1995). For the HDF, several of the calibration files were improved by combining a number of individual calibration frames to produce "super" calibration frames. We discuss these improved calibration files below. Versions 1 and 2 used the same calibration files. These are being modified for version 3, for reasons that will be outlined below.
The superbias frame. To the charge accumulated in each pixel of the WFPC-2 CCDs is added a bias value, designed to keep the output of the analog-to-digital (A/D) conversion consistently above zero. The value of this bias can vary slightly with position across the chip. A bias reference file is therefore subtracted from the data to remove any position-dependent bias pattern. The HDF used a superbias frame constructed from 160 individual bias frames. These frames, in uncalibrated format, were retrieved from the archive in sets of 40 and calibrated (A/D conversion and overscan subtraction) in the standard way. Each set of frames was then combined into a single image with cosmic rays removed. The resulting 40-frame bias images were used to create superdark images (see below) for each epoch. Because of the timing in the WFPC-2 electronics, bias frames are not really zero-length exposures, but have 43.6 seconds (plus the readout time for each chip) of exposure time. To remove this, the superdarks were normalized to exposure times of 43.6 s and subtracted from the superbiases. The dark-subtracted bias frames were then averaged to produce the final superbias frame. To use the file correctly, the DARKTIME header field in the frame to be calibrated is updated to the value
DARKT IME = 60 × int((t + 16.4)/60.) + 43.6,(1)
where t is the integration time of the image in seconds. The standard calwp2 data reduction software will then scale the superdark appropriately, and the dark current remaining in the bias frame will be removed in the superbias subtraction. The version 3 data reduction will use bias frames taken both before and after the observations so any systematic drifts with time will be better averaged out.
The superdark and delta-superdark frames. The dark subtraction is intended to remove spatial structure present in the thermally-induced dark current. In practice, most of this structure is in the form of individual "warm" pixels that are between 2σ and 5σ deviant from the mean, but are relatively stable over long periods of time. The superdark frame used in the HDF is derived from eight sets of 30 individual calibrated dark frames (1800 second exposures taken with the shutter closed). The frames in each set were combined and cosmic rays removed, then the eight sets were combined into a single file. The resulting image was normalized to a darktime of 1.0 second. A significant fraction of the counts seen in WFPC-2 "dark" frames arise from cosmic-ray glow in the MgF 2 faceplates in front of the CCD's. This glow varies with the cosmic ray rate and for version 3 will be subtracted independently of the thermal dark current in the CCD detector. For versions 1 and 2, variations in the amplitude of the MgF 2 glow were ignored (i.e. the superdark was scaled by exposure time and no correction was made for the cosmic ray rate). The result is that individual images, after flatfielding, typically show curvature in the background of a few percent within 200 pixels of the edge of the chip. These variations in background average out to less than 1% when multiple frames are stacked together.
To flag the hot pixels present at the start of the HDF observations, a delta-superdark frame was created. To do this, 11 dark frames (total exposure time 19000 seconds) taken near the start of the HDF observations were averaged together. The superdark frame was subtracted and pixels more than 5σ deviant from the mean were identified. (Note that "cold" pixels -i.e. those less than −5σ deviant -were also identified.) These pixels were added to the superdark frame (so that they would subtract, to first order, from the individual observations), and were also flagged in the data quality file, ensuring that they would be ultimately rejected in the final combinations of the images at the different dither positions. The delta-superdark was the same for versions 1 and 2, but is being remade for version 3.
The flat-field frames. The flat-field frames developed for the HDF have since become the default flats for the HST pipeline calibration. These files are versions of the original Investigation Definition Team (IDT) flats with an improved on-orbit illumination correction applied to scales greater than seven pixels. On the very largest scales, the chip-to-chip normalization of the IDT flats have been preserved. Tests of independent sets of flat-field images suggest that the uncertainties are less 2 %. More information on these files can be found in the WFPC-2 Cycle 4 Calibration Summary (Baggett, Casertano, & Biretta 1995). For version 3 of the HDF data reduction, more recent versions of the flat-field calibration files are being used.
Cosmic Ray Rejection and Initial Image Stacking
The preceding steps in the data reduction were identical for versions 1 and 2 of the HDF data. The versions diverge at the image combination step in two ways: (1) the individual images were weighted differently in the two versions and (2) care was taken in version 2 to remove satellite and space-debris trails from the images.
Cosmic ray (CR) identification was carried out using the stsdas task xcrrej (a modified version of the standard STSDAS crrej task). This program is a more sophisticated implementation of the simple idea of averaging several images of a target while removing cosmic ray events. The process begins by computing sky levels from the histogram of pixel intensities, estimating the best value by parabolic interpolation of the peak and the two adjacent points in the histogram. These sky levels are subtracted and the images are renormalized to the average exposure time to allow CR rejection on frames with differing backgrounds and exposure times. The program then computes the expected RMS deviation in each pixel based on the minimum observed value in that pixel and a model for the noise that includes Poisson noise, read noise, and a noise proportional to the counts (to account for variations in the peak brightness of bright sources, e.g. due to pointing jitter). In the first pass values more than 6σ above minimum are rejected. In the second pass, the average in each pixel is used as the expectation value, and values more than 5σ above and below the average are rejected. In the third pass the average is again used and values more than 4σ above and below are rejected.
Typical cosmic rays seen in WFPC-2 are not single-pixel events. To remove the wings of cosmic rays, pixels adjacent to rejected pixels are rejected if they are deviant by one-half the sigma-threshold applied to the initial rejected pixel.
In version 1 of the HDF processed data, the images were combined with weights proportional to exposure time. This is nearly optimal for F606W, and F814W, but not for the F450W and F300W images, which have a significant read-noise contribution. For version 2, the images were combined with weights proportional to the inverse-variance at the mean background level. The variance, in electrons, at the sky level is computed from the noise model:
σ 2 = bt + dt + r 2 (2)
where t is the exposure time, b is the sky background rate, d is the dark current, and r is the read noise. Pixels affected by cosmic rays were not used in the combination and were set to zero in the output inverse-variance image. These output inverse-variance images were used to compute the weights for the final combined images as described in §4.6 below.
Hot Pixel Removal
The number of hot pixels is minimized by using the delta-superdark frame in the HDF pipeline; however, several thousand remain in each frame after pipeline processing.
"Growing pixels" are hot pixels that have appeared or disappeared since the deltasuperdark. They were identified by combining sets of images taken at different dither positions, subtracting a 5 × 5 pixel median filtered image to remove residual objects, and identifying pixels that deviate from 0 by more than than 5 σ. A growing-pixel mask was created for each day using about 48 hours of data roughly centered at noon UT. The masks for each day were logically OR'ed together and OR'ed with the static mask (the list of constant WFPC-2 bad pixels) and the delta-superdark data quality file (the list of pixels that became hot between the superdark and the delta-superdark). In other words, a pixel flagged as suspect in any one of these files was marked in the final data quality file, and ultimately rejected in the final image combination.
This procedure reduces the number of hot pixels to about a dozen per chip. These are identified with a program that flags individual pixels more than 6 σ deviant from their neighbors in a 3 × 3 box. These pixels are OR'ed into the mask to produce one final mask that flags all pixels that were suspect at any time during the HDF. Altogether roughly 12000 pixels per chip were masked. Because there are eight or nine dither positions per filter, this conservative approach to masking hot pixels has only a modest effect on the final signal-to-noise ratio, reducing it by about 10% in the pixels that were flagged, and this for only about 2% of the total imaging pixels.
Scattered Light Removal
The HDF observations were carefully scheduled to minimize the effect of scattered light. Nevertheless, scattered light from the bright earth produces a visible X pattern in about 25% of the HDF data frames. The X pattern is caused by shadowing of the scattered light by the mirror supports in the WFPC-2 relay optics (Biretta, Ritchie, & Rudloff 1995). Images with a mean background more than a few times higher than the mean for a given bandpass were excluded from the processing. However, most of the affected frames had only a small amount of scattered light, and the S/N was not significantly degraded. These images were used in version 2, and steps were taken to remove the scattered light before combining them with the unaffected frames. Briefly, the scattered light subtraction was done by registering the bright and dark images, subtracting the dark from the bright to remove the sources, smoothing to remove any residual sources left over after subtraction, and subtracting this smoothed "sky" image from the bright frame. This procedure removed the X pattern to within a few percent.
Image Registration and Geometric Distortion Correction
The HDF observations were carried out with nine to eleven different pointing positions per filter, spanning a range of roughly 2.6 arcseconds (Fig 4). After having been fed through the pipeline, the registration of the individual images was checked by comparing the positions of several bright sources. This revealed two unexpected large (0.98 arcsec) shifts, but otherwise suggested that the images at each dither position were registered to within the errors of such a comparison. The images with the large shifts have been treated as separate dither positions.
The roughly five images per dither position were cosmic-ray rejected and stacked into a single image. In order to measure the shifts and rotations between these stacks, these images were first geometrically undistorted and resampled using the "drizzle" task which is described below. The drizzle program, in turn, removes geometric distortion using the polynomial solution described by Trauger et al. (1995). The shifts and rotations between the dither stacks were then measured using a cross-correlation-based technique on each of the three wide-field chips and averaging the results, again using the relative orientations and positions of the chips measured by Trauger et al. (1995).
Except for the two unexpected dither settings with a rotation of 4.3 arcmin, all other exposures align perfectly to an accuracy of less than 0.1 arcmin rotation, the estimated measurement error. Estimated errors in the shifts are a few hundredths of a pixel for F450W, F606W, and F814W. At the time of this writing, the shifts and rotations have not been checked to the ultimate possible precision in each of the input frames. There is thus some possibility of improving the resolution and shape of the final PSF with a more careful combination of the images. However, we anticipate such processing will produce only a few percent gain in S/N at the faintest levels, and a modest additional gain in the ability to distinguish stars from galaxies.
Once the shifts and rotations were measured, the images were aligned and combined using the "drizzle" algorithm, which corrects for geometric distortion and produces an output image that is sampled on a smaller pixel scale than the input images. The final pixel scale chosen for both the PC and WF chips was 0.4 of the original WF pixel or, using the astrometric solution of Trauger et al. (1995) for WF2, 0.03985 arcsec.
Image Combination
The "drizzle" algorithm was developed for the combination of multiple stacks of dithered, geometrically distorted, undersampled image frames. It was used to produce the combined output images of the HDF project from the flat-fielded, cosmic ray and hot pixel cleaned stacked frames corresponding to the different dither positions. The algorithm, which is more formally known as variable-pixel linear reconstruction, will be described in detail in another article (Fruchter & Hook 1996). Here an overview is provided for those interested in understanding the specific processing of the HDF.
The algorithm is conceptually simple. Pixels in the original "input" images are mapped into pixels in the subsampled "output" images, taking into account shifts and rotations between images and the optical distortion of the camera. The final image is built up by averaging together enough images taken at different positions that non-uniformities in exposure time from pixel to pixel in the output image become inconsequential (and are in any case recorded in the output variance map).
If there are enough input images, a simple way to improve the resolution of the output image is to make the area or "footprint" of each pixel in the input frame smaller than the physical pixel size, before mapping it onto the output image. To carry forward the analogy, these shrunken pixels, or "drops", rain down upon the subsampled output image. In the case of the HDF, the drops had linear dimensions one-half that of the input pixelslightly larger than the dimensions of the output subsampled pixels. The flux in each drop is divided up among the overlapping output pixels in proportion to their areas of overlap. In the code this is done by breaking the drops into N × N "droplets" and distributing their flux among the output pixels. The version 1 and version 2 images were made with N = 12. For version 3, the code has been revised to calculate the overlap exactly.
The position of each droplet in the undistorted, subsampled output image is computed and its value is averaged with the previous estimate of that pixel value. This is a weighted average which uses the assigned weight for the droplet and a weight for the previous estimate which is the sum of the weights of all droplets previously averaged into the output pixel. The image value and weight of the output pixel are updated before considering the next droplet. Thus, if a particular drop with value i xy and weight w xy is to be added to an image with value I xy and weight W xy , the resulting value of the image I ′ xy and weight W ′ xy is
I ′ xy = i xy w xy + I xy W xy w xy + W xy (3) W ′ xy = w xy + W xy(4)
This procedure is performed for each of the N 2 droplets in each of the pixels of the input image. In order to preserve resolution, a drop size is used which is smaller than the input pixel size. A particular output pixel may receive no droplets when drizzling an individual input frame. In Figure 6 the top left output pixel represents such a situation. These "zero valleys" are not a concern as long as there are enough input frames with different sub-pixel dither positions to fill in the image. It is, in the end, the placement of the dither positions which determines how small the drop size can be.
This scheme was developed in part as a quick means of implementing an area-weighted interpolation for distorted pixels, similar to the interpolation scheme proposed by Trauger et al. (1995). The weight of a droplet from a particlar pixel is just 1/N 2 times the weight of the pixel, which was computed as described in §4.2. Therefore, to the extent that N is large, the weight of a particular input pixel in a final output pixel is the fraction of the input pixel overlapping with the output pixel times the input pixel weight. This algorithm has the following characteristics. (1) It preserves both surface and absolute photometry; flux density can be measured using an aperture whose size is independent of position on the chip. (2) It handles missing data due to cosmic ray hits and hot pixels. (3) It uses a linear weighting scheme which is statistically optimum when inverse variance maps are used as weights. These weights may vary spatially to accommodate changing signal-to-noise ratios across input frames (e.g. due to variable scattered light). This spatial variation was not included for versions 1 and 2 of the data reduction. (4) It produces an inverse variance map (the weight map) along with the combined output frame. (5) It preserves resolution. (6) It largely eliminates the distortion of absolute photometry produced by the flat-fielding of the geometrically distorted images. In an uncorrected image the total photometry of sources near the corners of the chip is about 3.5% brighter than an equivalent source at the center of the chip (Biretta 1995).
Drizzling does however produce small artifacts in the final image. The interpolation scheme used produces an output image that is optimal for aperture photometry when the output image is weighted by the inverse variance. However, it is not optimal if one is interested primarily in producing a smooth point spread function (PSF). Simulations of the dithering pattern and drizzle parameters used in the HDF show that even in the absence of Poisson errors, drizzling produces a PSF whose FWHM varies by ±5% and whose shape can show noticeable high-frequency "noise". Both of these effects depend upon the pixel phase of the star relative to the dither positions, and thus are hard to predict for any individual object. (This directly limits the possibility of applying deconvolution, a process that amplifies high frequencies, to the drizzled data.) In Fig. 7 we show an example of a PSF from a star in the HDF. Variation about a Gaussian fit is a natural feature of an image convolved with large square pixels. However, because the shifts between images in the HDF do not uniformly sample the plane, the interpolation scheme used in drizzling makes this high-frequency scatter even more apparent.
The observed variations could have been reduced by using a larger final pixel size. However, doing so would have meant either using a different pixel scale for the PC than for the WF or suffering even further degradation of the PC resolution. While the high frequency scatter will affect attempts to fit the PSF, it does not significantly affect aperture photometry. By a radius of two WF pixels (five pixels in the drizzled HDF images), which is an aperture frequently favored by those doing HST photometry in crowded regions, the scatter is essentially gone. Furthermore, the scatter is seen only where the variations in surface brightness of an object are so rapid that they are undersampled in the original image (point sources, for example). Those for whom the scatter is a problem may wish to convolve the image with a narrow Gaussian, and thus trade a little resolution for a smoother PSF.
Drizzling also causes the noise in one pixel to be correlated with the noise in an adjacent one, because a single pixel from an input image typically affects the value in several output pixels, even though most of the power often goes into a single output pixel. The amplitude of this effect is determined by the footprint, or drop size, of the input pixels and by the positions of the sub-pixel dither points. If the data had been combined using the shift-and-add technique, which has a pixel footprint four times larger than that used in the HDF, this effect would have been substantially greater.
The drizzling parameters used in the HDF produce negligible correlations between pixels which are not adjacent. The measured 3 × 3 correlation matrix of adjacent pixels in the F606W frames is shown in Table 3. It includes any correlation in the sky due to inexact flat fielding or true variations in the sky. In the end, this provides a better idea of how sky noise will be reduced as the size of an aperture is increased. The correlation matrix varies both locally and globally due to the changes in sub-pixel placement caused by geometric distortion. The above matrix appears to provide a good indication of the overall noise statistics of the image. Simulated images with the same small-scale noise correlations as the HDF drizzled images can be created by convolving an image which has independent noise in each pixel with the matrix shown in Table 4. The resulting image will have an RMS pixel noise identical to the input image, but because of the correlation introduced between neighboring pixels, the expected standard deviation of an N × N box of pixels (where N is much larger than 1) is 1/N times the sum of the above matrix elements, or about 1.9/N. This procedure only simulates the correlation of pixels with equal intrinsic noise; a more complete simulation would need also to allow for the small pixel-to-pixel variations in the standard deviation of the noise. Figure 8 shows a color composite of the HDF full field from the F450W, F606W, and F814W images. This image was produced from the initial version 1 data reduction, and has slightly lower S/N than the version 2 images. Nevertheless it reveals the striking variety of colors and morphologies of the distant galaxies visible in the field.
The Images
The final, combined images of the HDF which result from the drizzling process in each of the four filters are presented in Figs. 9 -24. In each figure we display the co-added images from each of the four individual WFPC-2 CCD's. The total number of exposures and combined exposure time for each of the images is given in Table 5. Also shown are the sky levels, in units of data numbers (DN) per pixel (one DN corresponds to approximately 7 detected electrons). In the figures, the axes show pixel positions in the same units as the catalog listings. The scale at the bottom shows count levels in DN/1000 sec corresponding to different intensities on the final print. The images for the different bands have been scaled so that the maximum gray level corresponds to a constant AB magnitude of 20.35 magnitudes per square arcsec. The minimum gray level is set at −3 times the RMS of the sky level. For the dark gray levels in the images, therefore, a galaxy with a flat spectrum in f ν will have a constant brightness in the different bands. However for fainter galaxies, the transfer functions diverge for the different bandpasses.
It is clear from visual inspection of the images that detection differ in each filter due the variations in their background noise levels. The higher background noise of the F300W images, for which the individual exposures were all read-noise limited, is clearly evident. Objects that are present above the background noise display a wide variety of morphologies. At brightness levels well above the detection limit there are relatively few stars compared to the number of galaxies. The galaxies have a wide distribution of brightnesses, sizes, and shapes, and the brightest galaxies appear to correspond to the normal Hubble types. Most of the fainter sources also appear to be galaxies, although this must be established carefully because many of them are only marginally spatially resolved, but their morphologies are frequently chaotic and asymmetric. Not surprisingly, the fainter sources have a more compact appearance. To what extent this is an artifact of the (1 + z) 4 diminution in surface brightness produced by increasing distance and the changing metric needs to be ascertained. Some of the fainter objects are undoubtedly bright H II regions and massive star complexes in galaxies, which appear above the background threshold while the remaining lower surface brightness regions of the galaxy do not. A cursory study of the images does not reveal any obvious heretofore unobserved class of objects compared to earlier HST images of moderately distant clusters such as 0939+4713 (Dressler et al. 1994b) and those associated with 3C 324 . There do appear to be a number of the linear structures having the sizes and luminosities of galaxies that have been noted by others (e.g. Cowie, Hu, & Songaila 1995). One of the most distinctive galaxies present in the field is a relatively large, extended, low-surface-brightness galaxy near the NW edge of WF3. This galaxy appears to be relatively nearby, and it might be an outlying galaxy member of one of the Ursa Major clusters. Color differences among the various galaxies are notable in that some of them appear much more prominent relative to neighboring galaxies in one of the filters than in others. Also, different colors in distinct regions of the same galaxy, such as the H II regions, are evident in many of the objects.
Source Detection and Photometry
Overview
The photometric parameters of very faint galaxies are difficult to measure and difficult to interpret in a straightforward way. Systematic errors can arise at every phase of the analysis. During the detection phase, spurious "galaxies" can be identified from noise peaks if the noise properties of the data are not well quantified; galaxies with unusual surface brightness profiles can be missed if their sizes are not well matched by the convolution kernel used to smooth the data. In the photometry phase, systematic errors can arise in converting from isophotal or aperture magnitudes to total magnitudes, in measuring colors based on isophotes defined in only one band, and in measuring moments or radii of galaxies using only light from within a certain radius. In the source counting phase, galaxies may be overcounted if substructures are included as separate objects, or undercounted if overlapping distinct objects get counted as one source.
Such issues have generated considerable discussion in recent years. In constructing a catalog of objects which appear in the HDF, we have tried to take a fairly conservative approach, not digging deeply into the noise, and not adopting complex algorithms for source classification, weighted photometry, or merging and splitting of objects. As the data are public, we anticipate that others will generate their own catalogs. Indeed, comparisons among the various catalogs should be instructive in revealing the strengths and deficiencies of the various algorithms.
For the catalog we used a revised version of FOCAS (Jarvis & Tyson 1981;Valdes 1982). The program revisions are by , and are intended to make the program more useful for HST and ground-based infrared imaging. A specific enhancement useful for our application is the ability to adjust isophotal detection thresholds as a function of position using variance maps. This is important for mosaiced data like the HDF, where the effective exposure time at the image boundaries is less than that at field center due to the telescope dithering process.
We shall limit our discussion here only to the version 2 catalog, which supercedes the initial catalog released with the HDF version 1 data. Source detection and deblending was carried out on the exposure-weighted sum of the F606W and F814W version 2 drizzled images to provide maximum limiting depth. The resulting catalog of objects is then applied to the registered images in each of the four individual bandpasses, so that photometry is carried out through identical apertures at all wavelengths. The F606W+F814W summed image is significantly deeper than any of the individual images, and few normal objects should be missed by using it to define the object catalog for all bands. It is possible, however, that objects with very strong emission lines in the F450W or F300W bands could have escaped detection.
To identify sources, the data are first convolved with a fixed smoothing kernel K ij , shown in Table 6. Pixels with convolved values higher than a fixed threshold above a local sky background are marked as potentially being part of an object. A single value for σ sky was measured for each CCD using a fit to the sky histogram, and then input as a parameter to the FOCAS catalog for object detection. FOCAS sets the detection threshold T to a constant times σ sky :
T = Nσ sky i,j K 2 i,j 1 2 ,(5)
where the sum over the squares of the filter kernel elements accounts for the noise supression expected for Poisson sky noise from the smoothing process. Note, however, that the pixel-to-pixel noise in the drizzled images is correlated (see §4.5), and therefore the measured pixel-to-pixel rms provides an underestimate of the "true" noise level of the image by a factor of 1.9. Smoothing the drizzled data does not suppress the noise as much as FOCAS expects for uncorrelated noise. We have therefore empirically adjusted the scaling factor N until the detection of spurious sources was minimized. In the end, a threshold N = 4 was adopted.
Note also that the sky rms is (relatively) larger on the PC than on the WFC detectors. Therefore the isophotal threshold in physical units (e.g. mag/square arcsecond) on the PC is somewhat higher than that used for the WFC.
After thresholding, regions consisting of more than a certain number of contiguous pixels (including diagonals) are counted as sources. For the HDF data, the source detection threshould was set to 4σ and the minimum area to 25 drizzled pixels, or 0.04 square arcseconds. This area is equivalent to 4 original WF pixels or 19.2 PC pixels. It corresponds to an isophotal diameter limit of roughly 0.2 arcsec, about 1.6 times larger than the FWHM of the PSF on the WF cameras. For a point source on the WFC, this minimum area encompasses roughly 60% of the total object flux. Correspondingly, isophotal fluxes of faint, extended galaxies with sizes close to the minimum size limit must underestimate their total fluxes by at least 40%.
These sources are then examined for sub-components using a variant of the original detection scheme. The detection filter is repeatedly run over the image as the threshold is raised. If, at some threshold, the object breaks into two or more unconnected regions, each fragment which meets the minimum area criterion of 25 pixels is given a separate sub-entry in the catalog, and the process continues.
One additional test was applied to potential "daughter" objects during the splitting process, using the FOCAS-defined "significance" parameter. This parameter is a measure of surface brightness difference between the object's peak and its detection isophote, in units of the locally determined sky rms (see (Valdes 1982) for details). This "significance test" was not applied during the original detection of parent objects, as it largely duplicates the original isophotal thresholding criterion, but we have found empirically that its use helps to suppress the over-splitting of spurious sources from the extended wings of bright stars and large galaxies. Therefore a significance threshold of > 5 was imposed during object splitting.
The splitting process occasionally breaks large clumpy galaxies into many individual sub-components, or produces many spurious sources along the diffraction spikes of bright stars. We have fixed the most egregious cases by manually merging the split objects back into their parent. The cases for which this was done are flagged with an "F" in the catalog below. In general, the only objects for which the splitting was adjusted manually were bright, unmistakably recognizable spiral galaxies, where FOCAS tends to sever off the arms and HII regions into separate objects. These have been re-merged back into the parent. There are many complex, irregular galaxies where we have allowed the original FOCAS splitting to stand, despite the fact that in some cases one's intuition might suggest that the divided object is in fact a single entity. As noted below in §6.2, the catalog presented here includes both parent and daughter objects. The user may therefore adopt or reject the FOCAS splitting as he or she sees fit. In section 7 below, we present one possible algorithm for "re-merging" over-split objects, and apply it to the issue of galaxy number counts.
The Catalog
The source catalog is presented in Table 7. For each object we report the following parameters:
ID:
This is the FOCAS catalog entry number. The numbers after the decimal point indicate the level of splitting. Both parents and daughters are included in the catalog shown here. Thus many objects are included repeatedly in the catalog, both as part of the parent and as a separate daughter entry. For the statistical distributions shown in the next section, we have adopted specific color and separation criteria for merging objects to avoid double counting.
x,y: The x and y pixel positions of each object, as defined by brightest pixel within the 3 × 3 pixel grid with the greatest luminosity within the detection area. For objects with a bright off-center peak, this position can be significantly different from the weighted center of the luminosity distribution within the detection area. For the HDF, such differences are typically less than 0.1 arcsec.
RA,DEC:
Minutes and seconds of the right ascension and declination corresponding to the x,y centers, epoch J2000. For RA these are minutes and seconds of time. To these must be added 12 hours (RA) and 62 degrees (Dec).
The RA and Dec positions given in Table 7 were derived as follows. The geometric distortion of each of the WFPC2 CCDs has been accurately measured (Trauger et al. 1995), and was removed in the drizzling stage of the image reduction process (see §4.5). The relative positions of the CCDs are somewhat less accurately determined, and indeed are known to drift with time. WFPC2 image headers are encoded with a world coordinate system tied to the HST guide star reference frame. Absolute positions in this reference frame may be incorrect by an arcsecond or more, depending on the position on the sky, and on the accuracy of the individual positions for the guide stars used during the HST observations.
We have recalibrated the absolute zeropoint of the HDF coordinates using interferometric radio positions for two sources within the HDF detected by both the Merlin array and the VLA. The radio astrometry for these was kindly provided by Tom Muxlow and Ken Kellerman, and the two independant radio measurements for each source agree with one another within their quoted uncertainties. Here, we have adopted the Merlin positions, whose positional accuracy is of ∼20mas or better. The WFPC2 coordinates for the optical counterparts of these two sources were compared to the Merlin positions, and a simple, mean translational offset (∆α = +0 s .089 ± 0.010 and ∆δ = −1 ′′ .03) was adopted and applied to the "raw" WFPC2 coordinates. A global solution using VLA sources detected in both the central field and the flanking fields (Windhorst 1996) yeilds a mean offset 0.4 arcseconds different from the one adopted above, but consistent to within uncertainties of the measured source coordinates.
The resulting coordinate system used for the HDF catalog should therefore be accurate (within the radio reference frame provided by the Merlin data) to approximately 0 ′′ .4. Relative positions for individual galaxies are also typically accurate to 0 ′′ .1, with the largest uncertainty coming from the ability of FOCAS to determine a peak position for faint, lumpy objects.
m t ,m i :
The magnitudes of the detected sources in the F606W image. These magnitudes are in the AB system, where m = −2.5 log f ν − 48.60. The "isophotal" magnitude m i is determined from the sum of the counts within the detection isophote. The "total" magnitude is computed from the number of counts within a "grown" area. The total area is determined by first filling in any x or y concavities in the isophote shape and then adding by rings around the object until the area exceeds the detection area by at least a factor of two. For daughter objects, the total magnitude is divided between the daughters in proportion to their isophotal luminosities. The isophotal magnitudes correspond to the higher isophotes at which the object broke into multiple components.
u − b, b − v, and v − i:
Colors within the detection area. These are essentially isophotal colors measured to a faint limiting isophote defined from the summed F606W+F814W image. They are expressed in the AB system. (Our preferred notation for these colors is U 300 − B 450 , B 450 − V 606 , and V 606 − I 814 , to avoid confusion with the groundbased Johnson and Strömgren systems. However space in the tables does not allow us to use this convention here.) Galaxies where one band is a non-detection, as defined by having signal-to-noise ratio S/N < 2 within one of the bands, are marked as upper or lower limits (depending on which band drops out). If both bands are upper limits, no color is given.
S/N:
The signal-to-noise ratio of the detection in the summed F606W + F814W image, based on a semi-empirical noise model. If L i is the sky-subtracted number of counts within the detection isophote, the expected variance is
(Γσ(L i )) 2 = ΓN obj + 1.9σ 2 sky A obj + A 2 obj 1.9σ 2 sky /A sky ,(6)
where Γ is the inverse gain (assumed to be 7 for all chips), N obj is the total number of counts in the object aperture, A obj and A sky are the number of pixels within the object and sky apertures, respectively, and σ sky is the measured standard deviation of the level within the sky aperture. The first term accounts for Poisson variations in total counts in the source aperture. The second term accounts for statistical variations in the mean sky level expected from Poisson statistics, and the third term accounts for the random (but not any systematic) uncertainty in determining the mean sky level within the sky aperture. The factor of 1.9 is an empirical correction to the measured pixel-to-pixel standard deviation to account for the fact that the images are in effect smoothed by the subsampling procedure. Therefore the pixel-to-pixel variance underestimates the variance that will be seen on large scales, as described above in §4.6. While this factor is in principle a function of scale, our detection apertures and sky apertures are large enough that a constant value is a good approximation.
A:
Area in pixels within the detection isophote.
r 1 : Intensity-weighted first-moment radius determined from pixels within the detection isophote. The radii are determined relative to the x, y centers listed in the catalog, and the first moment radius is
r 1 = rI(x, y)/ I(x, y),(7)
where I(x, y) is the intensity in each pixel.
PA:
The intensity-weighted position angle defined such that an object pointing North-South has θ = 0, and the position angle increases as the major axis of the object rotates toward the east. In chip coordinates, the position angle is
θ xy = 0.5 tan −1 [2Z/(Y − X)],(8)
where X, Y, and Z are defined from the second moments of the image, as follows:
X = x 2 I(x, y)/ I(x, y),(9)
Y = y 2 I(x, y)/ I(x, y), and (10)
Z = xyI(x, y)/ I(x, y).(11)
The nominal spacecraft roll (PA V3) of 112 • was assumed in converting from the measured position angle to a celestial position angle.
b/a:
The intensity-weighted axial ratio taken from the second moment of the light distribution. Define P = X + Y, and
Q = [(X − Y ) 2 + 4Z 2 ] 1/2 ,(12)
where X, Y , and Z are the moments defined above. Then the axial ratio is
ǫ = [(P − Q)/(P + Q)] 1/2 .(14)
Flags: S indicates that the source is a single object (not split into subcomponents). B indicates that the outer isophote of the source overlaps a chip boundary in one or more bandpasses. F indicates that object components originally detected by FOCAS were manually merged back into their parent as described above.
Galaxy Counts
While it is not our purpose to discuss scientific results in this paper, it is useful to examine some of the basic statistics that will be important for such analyses.
The simplest statistic is the number of galaxies in the image. The counts of galaxies as a function of apparent magnitude are an essential tool of observational cosmology, and obtaining the faintest possible counts was one of the prime motivations behind the HDF project. However, before presenting the counts, it is useful to discuss the issue of object splitting, which has been a source of some interest for the faintest counts (Colley et al. 1996). Figure 25 shows a section of the F606W image on chip 4, with the FOCAS identifications labeled. Examples of objects for which image splitting becomes problematical are sources 858, 774, and 555, which are actually among the most interesting sources in the entire HDF. Source 858 has been spectroscopically identified as a galaxy at z = 3.226 . The spectrum refers to all four components, which have essentially the same color in the HDF image. Source 774 consists of three components which all have the same, very blue color. Source 555 consists of multiple components with different colors. The compact object 555.2 is very red and is probably a moderate-redshift elliptical galaxy. The elongated structure is spectroscopically identified as a galaxy at z = 2.803 . FOCAS splits it into two objects 555.11 and 555.12. These subcomponents have nearly identical colors. In the catalog, we list the parents and all the daughters for these objects. Many objects are therefore double-counted in the catalog. In counting the galaxies, we must decide which components of the merged objects to keep separate, and which to count as a single object. The separations of the components of 858 and 747, for example, are under 100 kpc, even for an open cosmology. They are thus probably more fairly considered as pieces of a larger galaxy, rather than as separate entities. However, a simple separation criterion will often merge objects such as 555.1 and 555.2 that are likely to be chance projections rather than physical associations. We have therefore adopted criteria for merging daughters that combine both color and separation. Specifically, we require that ∆(V 606 − I 814 ) < C and that the separation S ij between two components be
S ij < F × (r i + r j ),(15)
where the radii are determined from the FOCAS isophotal area as r i = A/π and the factor F is a tuneable parameter. The values C = 0.3 and F = 5 were chosen after some experimentation. These values merge all the components of sources 858 and 774, and merge the blue components of 555, leaving the red component as a separate object.
It is clearly a matter of subjective judgement whether to merge such objects. Does it matter? Table 8 compares the counts in three magnitude intervals for our chosen values F = 5, C = 0.3; for values F = 0, C = 0, which provide maximal splitting (keeping only the daughters); and for F = 100, C = 10, which provide maximal merging (keeping only the parents). The change in counts from maximal to minimal merging is less than 20% in any magnitude interval. Thus splitting issues, while interesting and important for studies assessing the sizes, clustering, and morpohology of faint galaxies, have only a small influence on the overall counts.
Incompleteness and uncertainties in the magnitude estimates also influence the counts. These are best addressed by means of simulations, and will be discussed in detail in Ferguson et al. (1996). For the purposes of this paper, it should be noted that FOCAS isophotal, total, and aperture magnitudes are each subject to certain systematic biases. The isophotal magnitudes, in particular, are likely to be approximately 0.2 mag fainter than total magnitudes for V 606 > 28. The bias for total magnitudes is smaller but still non-negligible. The counts presented here have not been corrected for these biases. The counts are presented down to magnitudes at which they are likely to be more than 80% complete (under certain standard assumptions for the intrinsic surface-brightness distribution -see Ferguson et al. 1996 for details). Tables 9 and 10. Figure 30 shows the F450W and F814W band counts together with ground-based data. All the surveys have been corrected to AB magnitudes, but no color corrections have been applied.
The essential shape of the counts is relatively robust. In all bands the slopes flatten at faint magnitudes. The slopes in several magnitude intervals are given in Table 11 (computed from simple least-squares fitting of the binned data over the magnitude intervals shown). The rather sharp flattening in the F300W counts appears to be unlikely to be due to incompleteness , and quite possibly indicates that many of the galaxies at faint magnitudes have redshifts z > 2 such that they drop out of the F300W band due to the combined affects of internal extinction (which can be quite severe in the rest-frame far-UV), internal Lyman continuum absorption, and intergalactic absorption from the Lyman α forest and from Lyman-limit systems Madau et al. 1996). The flattening of the counts in the F450W band may signal the loss of galaxies at z > 3 due to the same effects. Figures 31 -33 show color-magnitude diagrams for galaxies in the field. For comparison, also shown are the colors for template non-evolving galaxies of different types.
The templates for E, Sbc, and Im galaxies are taken from Coleman, Wu, & Weedman (1980), with the extrapolations adopted by Ferguson & McGaugh (1995). We have normalized the E and Sbc spectra to have absolute magnitudes in the F450W band of B 450 = −21.1, roughly L * for H 0 = 50 km s −1 Mpc −1 . The Im spectrum is normalized to an absolute magnitude B 450 = −18, more typical of a Magellanic irregular. The curves are labeled with redshift for galaxies of these assumed absolute magnitudes. It is interesting to note that while the elliptical galaxy template outlines the red envelope of the data reasonably well, there is a large population of galaxies at faint magnitudes that are bluer than the irregular galaxy template at any redshift. Also, as noted for the counts, there is a substantial number of galaxies with U 300 − B 450 and B 450 − V 606 colors and magnitudes consistent with being luminous ( ∼ > L * ) galaxies at high redshift (z > 2).
Conclusions
The Hubble Deep Field observations were taken with the expectation that they will contribute to the resolution of some of the outstanding questions in studies of galaxy formation. This paper describes the motivation, field and filter selection, and data reduction. Data, catalogs, and further information on the project are available on the World-Wide Web at http://www.stsci.edu.
We are extremely grateful to the large number of people who have contributed to the HDF project. In particular, we would like to acknowledge the efforts of the Advisory Committee, who donated a significant amount of effort in advising how best to use the Director's discretionary time, and in offering suggestions on how to carry out the observations. Among the many members of the STScI staff who helped with the observations, we would like to acknowledge especially J. C. Hsu and Ivo Busko for their timely efforts in devoloping software for data reduction. We thank Rogier Windhorst and Tom Muxlow for providing details on the comparison of radio and optical coordinates within the HDF. · · · · · · · · · 246 5.51 5.56 28.75 · · · · · · · · · 248 5.52 5.54 29.25 · · · · · · · · · · · · · · · · · · The projection to WF (100 mas) sub-pixel locations varies significantly from chip to chip due to minor rotations and plate scale differences, as well as with position within a CCD due to differential geometric distortion. The symbols "*", "o", and "+" refer to the first, middle and last third of the exposure set for the F606W filter set plotted. A drift of about 10 mas occurred over multiple days. As noted in the text, dither positions 10 and 11 resulted from anomalous Fine Guidance Sensor tracking, and these frames also have a 4.3 arcminute rotation with respect to all the other data. Table 5. The pixel scale at the borders is provided to allow galaxies in the catalog to be located via their x,y positions. -Galaxy counts as a function of AB magnitude in the F300W band. FOCAS total and isophotal magnitudes are shown for 22 < U 300 < 28. Aperture magnitudes are shown only for galaxies fainter than U 300 = 26, because the brighter galaxies are almost all larger than the 0.5 arcsec radius aperture. . Galaxies detected at more than 5σ are shown as large hexagons. Galaxies detected at less than 5σ are smaller hexagons. Galaxies either undetected in F300W or detected at less than 2σ significance are shown as open triangles at the position of the 2σ limit on the color. Also shown are the colors of fiducial non-evolving spectra of E, Sbc, and Im galaxies (see text), as dotted, solid, and dashed curves, respectively.
i − 2 i − 1 i i + 1 i + 2 j − 2 0 1 2 1 0 j − 1 1 3 3 3 1 j 2 3 4 3 2 j + 1 1 3 3 3 1 j + 2 0 1 2 1 0
Figures 26 -
2629 show the HDF counts in the individual bands. Isophotal, total, and aperture magnitudes are shown separately to illustrate the effects of different magnitude estimates. The counts in isophotal and total magnitudes are tabulated in
Fig. 1 .
1-The HDF and flanking fields, superimposed on a ground-based 300 s R-band image taken with the Mayall 4-m telescope. The small labeled insets show centers of each flanking field image. The naming convention for the fields is given in the last column of table 2.
Fig. 2 .
2-The bandpasses of the four filters chosen for the HDF. The total system throughput is shown, including the contribution from the filter, telescope and camera optics, and the detector.
Fig. 3 .
3-Predicted background as a function of time for representative orbits (indicated as year.day) during early, middle, and late parts of the HDF observing period. The background model includes scattered light from the illuminated earth on the day side of the orbit, which causes the large modulation shown in the figure. The floor at 5 × 10 −7 is from the zodiacal background and the excursion to zero is during earth-occultation.
Fig. 4 .
4-HDF pointing positions. Nine dither positions were planned to cover the large scale (left panel) offsets simultaneously and to provide good sub-pixel sampling (right panel). The offsets are shown for the center of WF2 (after rotating x, y by 180 degrees to match WF4).
Fig. 5 .
5-This figure illustrates the "footprint" of a pixel in the drizzling process. The dark region in the central pixel of this 3 × 3 grid shows the smaller area adopted for the pixel when it is projected onto the subsampled image.
Fig. 6 .
6-This figure illustrates the major operations of drizzling. The location of each input pixel in the output subsampled image is determined from knowledge of the pointing position, rotation, and geometric distortion. The flux within that pixel is then "drizzled" into the overlapping output pixels in quantities proportional to the area of overlap.
Fig. 7 .
7-The point-spread function of a bright star in the final drizzled HDF image.
Fig. 8 .
8-A color composite image of the full HDF field, constructed from the F450W, F606W, and F814W images.
Fig. 9 .
9-PC image in the F300W band. This and the following images are from the version 2 drizzled images. Exposure times are given in
Fig. 10 .
10-PC image in the F450W band.
Fig. 11 .
11-PC image in the F606W band.
Fig. 12 .
12-PC image in the F814W band.
Fig. 13 .
13-WF2 image in the F300W band.
Fig. 14 .
14-WF2 image in the F450W band.
Fig. 15 .
15-WF2 image in the F606W band.
Fig. 16 .
16-WF2 image in the F814W band.
Fig. 17 .
17-WF3 image in the F300W band.
Fig. 18 .
18-WF3 image in the F450W band.
Fig. 19 .
19-WF3 image in the F606W band.
Fig. 20 .
20-WF3 image in the F814W band. -37 -Fig. 21.-WF4 image in the F300W band.
Fig. 22 .
22-WF4 image in the F450W band.
Fig. 23 .
23-WF4 image in the F606W band.
Fig. 24 .
24-WF4 image in the F814W band.
Fig. 25 .
25-A section of the F606W image from WF4, with the FOCAS identifications marked.
Fig. 26 .
26Fig. 26.-Galaxy counts as a function of AB magnitude in the F300W band. FOCAS total and isophotal magnitudes are shown for 22 < U 300 < 28. Aperture magnitudes are shown only for galaxies fainter than U 300 = 26, because the brighter galaxies are almost all larger than the 0.5 arcsec radius aperture.
Fig. 27 .
27-Galaxy counts as a function of AB magnitude in the F450W band. FOCAS total and isophotal magnitudes are shown for 22 < B 450 < 29. Aperture magnitudes are shown only for galaxies fainter than B 450 = 26.Fig. 28.-Galaxy counts as a function of AB magnitude in the F606W band. FOCAS total and isophotal magnitudes are shown for 22 < V 606 < 29.5. Aperture magnitudes are shown only for galaxies fainter than V 606 = 26.
Fig. 29 .
29-Galaxy counts as a function of AB magnitude in the F814W band. FOCAS total and isophotal magnitudes are shown for 22 < I 814 < 29. Aperture magnitudes are shown only for galaxies fainter than I 814 = 26.
Fig. 30 .
30-Galaxy counts as a function of AB magnitude in the F450W and F814W bands, together with a compilation of existing ground-based data. FOCAS total and isophotal magnitudes are shown for 22 < I 814 < 29. No color corrections have been applied to the ground-based data.
Fig. 31 .
31-Color-magnitude diagram U 300 − B 450
Fig. 32 .
32-Color-magnitude diagram B 450 − V 606 . The meanings of the lines and symbols are the same as for the previous figure. Triangles are shown at the position of the F450W 2σ limits.Fig. 33.-Color-magnitude diagram V 606 − I 814 . The meanings of the lines and symbols are the same as for the previous two figures. Triangles are shown at the position of the F606W 2σ limits.
Table 1 .
1Characteristics of the Hubble Deep Field Sky values are total DN for the exposure times listed.Location:
12h 36m 49.4s +62 • 12 ′ 58"
(Epoch J2000.0 / WFPC-2 'WFALL FIX' position)
V3 Position angle = 112 •
E(B − V ):
0.00
HI column density:
1.7 × 10 20 cm −2
DIRBE flux:
< 0.14MJy/ster
Radio sources:
none with flux > 1 mJy at 3.6 cm
IRAS cirrus:
Local minimum in 100µm maps
Bright stars:
None near the field
Galaxy Clusters:
Nearest is 48 armin away
Table 2. HDF Flanking Fields
Equatorial
Total
J2000 Coordinates
UT Date
N exp
T exp (s)
Comments
12 36 35.17 +62 13 38.7
30 Dec 1995
4
5300s
Inner West
12 36 20.93 +62 14 19.3
18 Dec 1995
3
2500s
Outer West
12 37 03.62 +62 12 17.2
30 Dec 1995
4
5300s
Inner East
12 37 17.83 +62 11 36.3
29 Dec 1995
3
3000s
Outer East
12 36 51.03 +62 15 47.6
30 Dec 1995
3
2500s
North West
12 37 05.27 +62 15 06.8
29 Dec 1995
3
2500s
North East
12 36 47.77 +62 10 08.4
29 Dec 1995
3
2500s
South East
12 36 33.56 +62 10 49.1
30 Dec 1995
3
2500s
South West
Table 6 .
6FOCAS Smoothing Kernel
Table 7 .
7HDF Catalog See attached pages.
Table 8 .
8Counts for Various Amounts of SplittingV 606
F = 5
F = 0
F = 100
range
C = 0.3
C = 0
C = 10
23 − 25
137
128
133
25 − 27
491
528
450
27 − 29
1131
1239
1050
Table 10 .
10HDF Galaxy Counts -F606W and F814W bands The F300W limit here is 26-28; it is 26-29 for the other filters.V 606
I 814
isophotal
total
isophotal
total
AB
log(n)
log(n)
log(n)
log(n)
mag
N
mag −1 deg −2
mag −1 deg −2
N
mag −1 deg −2
mag −1 deg −2
22.25
7
3.97
3.97
14
4.27
4.20
22.75
8
4.03
4.12
18
4.38
4.42
23.25
15
4.30
4.33
28
4.57
4.60
23.75
28
4.57
4.64
27
4.55
4.64
24.25
43
4.76
4.72
48
4.80
4.85
24.75
41
4.74
4.78
70
4.97
4.96
25.25
82
5.04
5.08
84
5.05
5.13
25.75
90
5.08
5.10
118
5.20
5.19
26.25
139
5.27
5.26
134
5.25
5.28
26.75
145
5.29
5.33
174
5.36
5.40
27.25
191
5.40
5.43
196
5.42
5.43
27.75
193
5.41
5.47
232
5.49
5.60
28.25
330
5.64
5.68
343
5.66
5.71
28.75
344
5.66
5.76
391
5.71
5.78
29.25
439
5.77
5.83
· · ·
· · ·
· · ·
Table 11. Counts Slope
mag range
U 300
B 450
V 606
I 814
23 − 26
0.40
0.39
0.35
0.31
26 − 29
0.05 a
0.16
0.17
0.18
a
Most magnitudes in this paper are expressed in the AB system(Oke 1974), where m = −2.5 log f ν −48.60.
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|
[] |
[
"Low-error and broadband microwave frequency measurement in a silicon chip",
"Low-error and broadband microwave frequency measurement in a silicon chip"
] |
[
"Mattia Pagani \nInstitute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia\n",
"Blair Morrison \nInstitute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia\n",
"Yanbing Zhang \nInstitute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia\n",
"Alvaro Casas-Bedoya \nInstitute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia\n",
"Timo Aalto \nVTT Technical Research Centre of Finland\n02040EspooFinland\n",
"Mikko Harjanne \nVTT Technical Research Centre of Finland\n02040EspooFinland\n",
"Markku Kapulainen \nVTT Technical Research Centre of Finland\n02040EspooFinland\n",
"Benjamin J Eggleton \nInstitute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia\n",
"David Marpaung \nInstitute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia\n"
] |
[
"Institute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia",
"Institute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia",
"Institute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia",
"Institute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia",
"VTT Technical Research Centre of Finland\n02040EspooFinland",
"VTT Technical Research Centre of Finland\n02040EspooFinland",
"VTT Technical Research Centre of Finland\n02040EspooFinland",
"Institute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia",
"Institute of Photonics and Optical Sciences (IPOS)\nSchool of Physics\nCentre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)\nUniversity of Sydney\n2006NSWAustralia"
] |
[] |
Instantaneous frequency measurement (IFM) of microwave signals is a fundamental functionality for applications ranging from electronic warfare to biomedical technology. Photonic techniques, and nonlinear optical interactions in particular, have the potential to broaden the frequency measurement range beyond the limits of electronic IFM systems. The key lies in efficiently harnessing optical mixing in an integrated nonlinear platform, with low losses. In this work, we exploit the low loss of a 35 cm long, thick silicon waveguide, to efficiently harness Kerr nonlinearity, and demonstrate the first on-chip four-wave mixing (FWM) based IFM system. We achieve a large 40 GHz measurement bandwidth and record-low measurement error. Finally, we discuss the future prospect of integrating the whole IFM system on a silicon chip to enable the first reconfigurable, broadband IFM receiver with low-latency.
|
10.1364/optica.2.000751
|
[
"https://arxiv.org/pdf/1506.04261v1.pdf"
] | 119,228,239 |
1506.04261
|
249832b3afa09fe597a6a7718294c62085efc00f
|
Low-error and broadband microwave frequency measurement in a silicon chip
Mattia Pagani
Institute of Photonics and Optical Sciences (IPOS)
School of Physics
Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)
University of Sydney
2006NSWAustralia
Blair Morrison
Institute of Photonics and Optical Sciences (IPOS)
School of Physics
Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)
University of Sydney
2006NSWAustralia
Yanbing Zhang
Institute of Photonics and Optical Sciences (IPOS)
School of Physics
Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)
University of Sydney
2006NSWAustralia
Alvaro Casas-Bedoya
Institute of Photonics and Optical Sciences (IPOS)
School of Physics
Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)
University of Sydney
2006NSWAustralia
Timo Aalto
VTT Technical Research Centre of Finland
02040EspooFinland
Mikko Harjanne
VTT Technical Research Centre of Finland
02040EspooFinland
Markku Kapulainen
VTT Technical Research Centre of Finland
02040EspooFinland
Benjamin J Eggleton
Institute of Photonics and Optical Sciences (IPOS)
School of Physics
Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)
University of Sydney
2006NSWAustralia
David Marpaung
Institute of Photonics and Optical Sciences (IPOS)
School of Physics
Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS)
University of Sydney
2006NSWAustralia
Low-error and broadband microwave frequency measurement in a silicon chip
Instantaneous frequency measurement (IFM) of microwave signals is a fundamental functionality for applications ranging from electronic warfare to biomedical technology. Photonic techniques, and nonlinear optical interactions in particular, have the potential to broaden the frequency measurement range beyond the limits of electronic IFM systems. The key lies in efficiently harnessing optical mixing in an integrated nonlinear platform, with low losses. In this work, we exploit the low loss of a 35 cm long, thick silicon waveguide, to efficiently harness Kerr nonlinearity, and demonstrate the first on-chip four-wave mixing (FWM) based IFM system. We achieve a large 40 GHz measurement bandwidth and record-low measurement error. Finally, we discuss the future prospect of integrating the whole IFM system on a silicon chip to enable the first reconfigurable, broadband IFM receiver with low-latency.
Introduction
The ability to measure the frequency of an unknown radio frequency (RF) or microwave signal, without relying on expensive spectrum analyzers and mixers, is a basic requirement for the development and testing of wireless systems. In particular, the ability to do so in real-time (< 50 ns) and with high accuracy (< 1% error) is crucial for radar receivers used in electronic warfare (EW) [1] and biomedical technology [2]. This is known as instantaneous frequency measurement (IFM), and it involves mapping the signal frequency to a more easily measurable quantity, such as power.
The use of photonics for microwave IFM is attractive due to the possibility of pushing the frequency measurement range far beyond the capacity of electronicbased IFM systems [3,4]. Since its conception in 2006 [5], research into photonicbased IFM systems has progressed by various means [6,7,8,9,10], and has recently culminated with the demonstration of a number of on-chip IFM systems [11,12,13,14]. This trend towards integration is especially important for IFM receivers, where the IFM delay is directly related to the length of the path traveled by the light. The inherently reduced size of integrated systems can thus lead to sub-nanosecond latency, as well as improved robustness and lower footprint [15].
Nevertheless, most on-chip IFM demonstrations to date achieved measurement bandwidths below 10 GHz [11,12,13], lower than what is currently possible using electronic IFM systems. This limit was due to the trade-off between quality factor and free spectral range of the ring resonators employed by these prior demonstrations. A different technique made use of a highly compact onchip Bragg grating filter with promising results, but a relatively high 2% measurement error [14]. There are not many techniques that can simultaneously achieve wideband operation and low, sub-1% measurement error. One such technique relies on mapping the RF signal frequency to the power of an optical idler generated through efficient nonlinear optical mixing, in a low-loss platform [7]. The stringent requirements on the nonlinear medium have meant that so far, all implementations [7,16] were bound to using hundreds of meters of fiber, introducing high measurement delays. For high-bandwidth IFM, where low latency is key, it is crucial to harness the nonlinearity efficiently in a low-loss, short platform.
In this work, we report the first low-latency IFM system using on-chip fourwave mixing (FWM), and demonstrate 40 GHz measurement bandwidth, limited only by the measurement equipment, with record-low 0.8% error. These ground-breaking results were enabled by the ultra-low loss and efficient FWM in a unique platform: a 35 cm long thick silicon waveguide with a highly compact footprint. This allowed us to generate an idler with enhanced signal-to-noise ratio, thereby significantly reducing the measurement error below the level demonstrated in fiber-based systems [7]. This is a breakthrough for high-performance on-chip IFM systems, and reveals a new and unique material platform for nonlinear optical processing of microwave signals [17]. Finally, we discuss the feasibility of integrating the whole IFM system onto a silicon chip, highlighting the potential for implementing the first, ultra-low latency, fully reconfigurable IFM system.
Principle of operation
The basic structure of our FWM-based IFM system is shown in Fig. 1(a). An RF signal of unknown frequency Ω is received and modulates two CW optical carriers at different frequencies ω 1 , ω 2 . This results in two copies of an optical signal, in two different channels. These two optical signals are then demultiplexed using a coarse wavelength division multiplexer (CWDM). In this way, one signal is delayed by ∆t relative to the other, before both are recombined. The two signals are then sent through a nonlinear medium, where they mix through FWM, generating idlers in adjacent channels.
There are three idler components generated in the vicinity of frequency 2ω 1 − ω 2 . The middle component is an idler "carrier", resulting from degenerate FWM (DFWM) between the two original carriers. The two remaining components, or idler sidebands, occur at frequencies 2ω 1 − ω 2 ± Ω. There are in fact two FWM processes which occur simultaneously, shown in Fig. 1(b), giving rise to two separate pairs of idler sidebands. The first is a DFWM process, which gives rise to a delayed pair of idler sidebands. Neglecting dispersion, the complex field for the upper DFWM idler sideband is
E DFWM (t) ∝ 3e jω1t · e jω1t · e −j(ω2−Ω)(t−∆t) .(1)
The second process is a non-degenerate FWM (NDFWM) process, where the generated idler pair is a copy of the non-delayed optical signal. Neglecting dispersion, the complex field of the upper NDFWM idler sideband is
E NDFWM (t) ∝ 6e jω1t · e j(ω1+Ω)t · e −jω2(t−∆t) .(2)
The total sideband idler field is therefore a coherent sum of two separate idler fields with a differential delay of ∆t, resulting in an interference effect. The total upper idler sideband is given by
E idler (t) = E DFWM (t) + E NDFWM (t) ∝ (3e −jΩ∆t + 6)e j(2ω1−ω2+Ω)t .(3)
A similar expression can be derived for the lower idler sideband.
Using an optical bandpass filter (BPF) the total idler field (idler "carrier" and sidebands) can finally be isolated, and its power measured with an optical power meter (i.e. low-speed photodetector) to be:
P idler = A + B cos(Ω∆t),(4)
for some constants A and B, with A > B. This expression clearly shows that once the system has been characterized, i.e. A, B and ∆t are known, an optical power meter can be used to estimate the unknown RF frequency Ω, without the need for expensive high-speed RF equipment. An inspection of Eq. (4) reveals the inherent trade-off between measurement bandwidth (∝ 1/∆t), and measurement sensitivity, defined as dP idler /dΩ (∝ ∆t). To bypass this trade-off and maximize measurement sensitivity without compromising bandwidth, it is necessary to maximize the idler power, both to operate further above the noise floor, and to increase the slope of Eq. (4). It is then crucial to harness FWM in a low-loss, nonlinear platform.
Experiment
Device Description
In this work, the nonlinear platform used to harness the FWM was a 35 cm long silicon strip waveguide [18]. Silicon is a highly attractive platform for nonlinear optics partly due its large Kerr coefficient (100 times larger than that of silica), and due to its compatibility with CMOS processes [19,20].
The distinguishing feature of this waveguide was its large mode area of 2.7 µm 2 , combined with a small footprint. Ordinarily, waveguides with such large mode areas required mm or even cm-scale bending radii, due to high mode coupling losses, and were thus constrained to lengths of a few centimeters [21]. This particular sample however used Euler bends, where the bending radius continuously varies along the whole bend, ensuring minimum coupling to higher order modes [18]. This resulted in µm-scale bending radii, comparable to that of nanowires. Consequently, the whole 35 cm long spiral, shown in Fig. 2(a), occupied less than 3 mm 2 surface area on an silicon-on-insulator (SOI) chip. The dimensions of the strip cross section were 3×1.875 µm. There are several advantages that follow from this large size: low 0.15 dB/cm linear propagation loss (compared to 1-3 dB/cm for silicon nanowires), a higher nonlinear loss threshold, and efficient coupling to lensed fibers (1.5 dB/facet coupling losses). The coupling was done into a single-mode rib waveguide which then tapered into the strip waveguide, as shown in Fig. 2(c), ensuring excitation of the fundamental mode. The waveguide dispersion at 1550 nm was normal, which led to a FWM 3-dB bandwidth of 6 nm. This is sufficient for most RF photonic applications, where the signal content is in the range of 1-100 GHz, or less than 2 nm.
The nonlinear optical properties of the waveguide were measured through a series of self-phase modulation (SPM) and FWM experiments. By launching picosecond pulses into the waveguide, we observed SPM broadening and were able to estimate the Kerr coefficient as n 2 ∼ 1.2×10 −18 m 2 W −1 . This value was slightly lower than for a typical silicon waveguide, possibly due to the change in crystal orientation that occurs in the spiral bends [22]. The nonlinear parameter was then estimated as γ ∼ 1.8 m −1 W −1 . We note that the maximum FWM conversion efficiency for this waveguide was comparable to that obtainable using a much shorter 5 mm standard silicon nanowire. However, our thick silicon spiral was not optimized for FWM. More importantly, the low coupling losses meant that the total waveguide insertion loss was lower than for a nanowire. This property is central to IFM applications, where the output idler power has a direct effect on the system performance.
IFM Experiment
The experimental setup was based on the structure shown in Fig. 1(a). Two semiconductor laser diodes were used to generate the optical carriers, with wavelengths of 1550.0 and 1551.7 nm. The electro-optic modulator (EOM) used was a Mach-Zehnder modulator, while the ∆t delay was adjusted using a tunable optical delay line. Figure 3 shows the optical spectrum at the output of the silicon spiral, when the microwave input frequency is Ω/2π = 40 GHz. By adjusting the tunable delay line, we were able to vary the product Ω∆t between 0 and π. According to Eq. (4), these two points corresponded to maximum and minimum idler sidebands power, respectively. This is clearly visible in Fig. 3, where there is a 10 dB power contrast for the idler sidebands as the delay is tuned. Figure 3: Optical spectra at the output of the silicon waveguide for two different ∆t values. The 10 dB power difference between the idler sidebands is a manifestation of the interference effect between the idlers generated through DFWM and NDFWM.
Initially, the time delay between the optical signals was set to 8.3 ps. Both optical carriers entered the waveguide with 18.7 dBm power, which resulted in a FWM conversion efficiency of −30.1 dB, here defined as P idler,out /P signal,in . The IFM system was then characterized by sweeping an RF signal generator from 0 to 40 GHz, and measuring the generated idler power. This measurement is shown in Fig. 4(a), together with a theoretical fit. The fit was then used to estimate unknown frequencies in the measurement range, and the results are shown in Fig. 4(b). The root mean square (rms) of the frequency estimation error was 318.9 MHz, which corresponded to 0.8% of the 40 GHz measurement bandwidth. This combination of wide measurement bandwidth (limited only by the range of the RF signal generator), and sub-1% error, is a record for an IMWP IFM system.
As explained in Section 2, increasing the ∆t delay can lead to enhanced measurement sensitivity (i.e. lower error), at the cost of a reduced measurement bandwidth. This trade-off is displayed in Fig. 5 where, as ∆t is increased, a Figure 5: Reconfiguration of the IFM system response between high bandwidth/error state (low ∆t) and low bandwidth/error state (high ∆t).
RF frequency
given idler power measurement no longer maps to a single RF frequency band, but to multiple, narrower bands. Nevertheless, by rapidly reconfiguring ∆t, it is possible to exploit the high accuracy of a high-∆t measurement, combined with the wide bandwidth of a low-∆t measurement. This is a process consisting of two steps. Initially, a small ∆t is chosen to obtain a rough estimate of the frequency band in which the RF signal resides (e.g. Band 4 in Fig. 5). Following this, ∆t is increased, producing a highly accurate, unambiguous estimate of the signal frequency.
To demonstrate this process, we reconfigured the system by increasing the time delay to 69.4 ps. The system characterization and theoretical fit are shown in Fig. 6(a). The decaying sinusoid response is due to the frequency roll-off of the modulator. Each linear region of the system response was then assigned to a particular frequency band. This resulted in six, 7.2 GHz bands, which were then used to estimate the frequency of various RF tones. The result of this measurement, shown in Fig. 6(b), exhibits a low estimation error, with rms value of 40.2 MHz, or 0.56% of the 7.2 GHz measurement bandwidth. These measurements show a unique and important feature of FWM-based IFM: the ability to easily tune the system response to optimize for bandwidth or sensitivity. Therefore, it is crucial to be able to quickly tune the ∆t delay, so as to perform fast consecutive measurements with wide bandwidth and high accuracy. Our technique for implementing the delay constitutes a significant improvement over that used in previous FWM-based IFM [7,16], where a dispersive element introduced a delay between the two channels. This is because a tunable delay line offers a much faster mechanism for tuning ∆t, compared to tuning a laser wavelength. Furthermore, a recent breakthrough in setting tunable time delay has achieved sub-nanosecond settling time [23]. This opens the way to achieving the first IFM system which overcomes the bandwidth/error trade-off through sub-nanosecond reconfiguration.
Future Vision
The ability to harness nonlinear optical processes efficiently, in a low-loss integrated platform, is fundamental for a wide variety of microwave photonic applications [15]. This is particularly true of FWM-based IFM where, as we have seen, the generated idler power has a direct effect on the measurement error. In this work, we have shown that long, thick silicon waveguides exhibit all of these properties, while being able to maintain low footprints and a high nonlinear loss threshold.
Nevertheless, to be able to monolithically integrate all critical components of the IFM system, one will need to combine the thick silicon platform with a more standard SOI technology. One such technology is 220 nm thin SOI, which enables access to a full library of active and passive components.
Our vision of the layout of a future fully-integrated SOI FWM-based IFM system is presented in Fig. 7, featuring a single transition from thin to thick silicon. This transition could be implemented using a section of tapered rib waveguide that minimizes the excitation of higher order modes [24], or even through photonic wire bonding [25].
The thick silicon part of the chip houses the FWM waveguide, BPF, and photodetector. One of the main advantages of the current FWM-based IFM scheme is that it requires only a simple low-speed photodetector. Such a device has already been demonstrated using vertical p-i-n germanium, integrated on a thick SOI waveguide [26]. Bandpass filters composed of cascaded Mach-Zehnder interferometers have also been demonstrated in thick silicon [27]. For these components, Euler bends will need to be implemented so as to maintain a low footprint.
The thin silicon section of the chip hosts the electro-optic modulators, and arrayed waveguide grating. The intensity modulator used by the input RF signal would be implemented using doped thin SOI technology. Such Mach-Zehnder modulators have been shown capable of high performance, with low 1.6 dB insertion loss, and wide 27.8 GHz bandwidth [28]. Demultiplexing of the two channels would occur in an arrayed waveguide grating, which have also been : Future vision of the SOI integrated FWM-based IFM system. The chip is divided into thin and thick silicon sections. The RF input signal is applied to the intensity modulator. The tunable time delay unit consists of an optical delay interferometer and a phase modulator [23]. Tuning of ∆t is achieved by varying the voltage input to the phase modulator. The photodetector outputs a DC voltage proportional to the idler power, which is then mapped to an estimate for the RF input signal frequency. demonstrated in thin SOI with 200 GHz channel spacing [29]. Finally, tunable time delay could be realized using a novel approach [23], employing an optical delay interferometer and a phase modulator. High-speed phase modulators have been implemented using silicon-organic hybrid (SOH) technology [30], and would allow for fast tuning of the ∆t delay. Such a chip, boasting fast tunable delay and compact size will result in the first ultra-low latency and highly accurate IFM system with bandwidth and capabilities beyond what is achievable using state-of-the-art RF technologies.
Conclusion
We have presented the first IFM system using on-chip FWM, capable of extremely high frequency measurements, and low error through easy reconfiguration of the system response. The enabling technology was a low-loss, long silicon waveguide, which was used to generate strong FWM idlers. This allowed us to achieve record-low measurement error over a wide frequency range, greatly surpassing that of previous IMWP IFM systems. The novel setup we have presented consists of components which are all capable of SOI integration, opening the way for the first reconfigurable, monolithically integrated, IFM receiver.
Funding Information
Australian Research Council (ARC) (DE150101535, FL120100029, CE110001018).
Figure 1 :
1(a) Schematic of the on-chip FWM-based IFM system. EOM: electrooptic modulator; CWDM: coarse wavelength division multiplexer; ∆t: tunable delay element; BPF: optical bandpass filter. (b) Degenerate (DFWM) and nondegenerate (NDFWM) mixing processes between the two channels.
Figure 2 :
2(a) 35 cm long, thick silicon spiral top view. (b) Simulation of the fundamental mode for the 3×1.875 µm silicon strip waveguide. (c) Rib-to-strip converter for coupling to the fundamental mode.
Figure 4 :
4(a) IFM RF system response with ∆t = 8.3 ps. (b) Frequency estimation measurement over a single 40 GHz frequency band (inset: histogram of the frequency measurement error, rms value = 318.9 MHz).
Figure 6 :
6(a) IFM system response with ∆t = 69.4 ps. (b) Frequency estimation measurement for six separate 7.2 GHz frequency bands (inset: histogram of the frequency measurement error, rms value = 40.2 MHz).
Figure 7
7Photodetector
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[] |
[
"Higher order Whitehead products and L ∞ structures on the homology of a DGL",
"Higher order Whitehead products and L ∞ structures on the homology of a DGL"
] |
[
"Francisco Belchí ",
"Urtzi Buijs ",
"José M Moreno-Fernández ",
"Aniceto Murillo "
] |
[] |
[] |
We detect higher order Whitehead products on the homology H of a differential graded Lie algebra L in terms of higher brackets in the transferred L ∞ structure on H via a given homotopy retraction of L onto H.
|
10.1016/j.laa.2017.01.008
|
[
"https://arxiv.org/pdf/1604.01478v2.pdf"
] | 59,938,813 |
1604.01478
|
33e025660246a74a0124fb6961cb0902c8ea5847
|
Higher order Whitehead products and L ∞ structures on the homology of a DGL
29 Apr 2016 May 2, 2016
Francisco Belchí
Urtzi Buijs
José M Moreno-Fernández
Aniceto Murillo
Higher order Whitehead products and L ∞ structures on the homology of a DGL
29 Apr 2016 May 2, 2016arXiv:1604.01478v2 [math.AT]
We detect higher order Whitehead products on the homology H of a differential graded Lie algebra L in terms of higher brackets in the transferred L ∞ structure on H via a given homotopy retraction of L onto H.
Introduction
Topological higher order Whitehead products were introduced in [14]: given simply connected spheres S n 1 , . . . , S n k , denote by W = S n 1 ∨ · · · ∨ S n k and T = T (S n 1 , . . . , S n k ) their wedge and fat wedge respectively. Then, there is an attaching map (in what follows we shall not distinguish a map from the homotopy class that it represents) ω : S N −1 → T with N = n 1 + · · · + n k , for which S n 1 × · · · × S n k = T ∪ ω e N .
known that a DGL L is formal is there exists a inherited L ∞ structure on H as above for which ℓ n = 0, n ≥ 3. Moreover, a necessary condition for L to be formal is that the zero class be a higher Whitehead bracket of any order. In this paper, and given L any DGL, our goal will be then to detect Whitehead brackets of order k as kth brackets on the induced L ∞ structure on H. The most general assertion in this direction that we obtain is based in [2, Thm. 4.1]: given x ∈ [x 1 , . . . , x k ] W , and up to a sign, ℓ k (x 1 , . . . , x k ) = x modulo brackets (of the L ∞ structure) of order less than or equal to k − 1, see Proposition 3.1 for a precise statement.
To be more accurate, and in high contrast with the Eckmann-Hilton situation concerning Massey products [11,Theorem 3.1], extra conditions are needed. We define higher Whitehead brackets adapted to a given homotopy retract and prove (see Theorem 3.3
):
Theorem 1. 1. For any homotopy retract of L adapted to a given x ∈ [x 1 , . . . , x k ] W , and up to a sign, ℓ k (x 1 , . . . , x k ) = x.
A similar assertion is obtained for any homotopy retract under the vanishing of brackets of length up to k − 2. That is (see Theorem 3.5
):
Theorem 1. 2. Let ℓ i = 0 for i ≤ k − 2 with k ≥ 3. Then, if [x 1 , . . . , x k ] W is non empty, and also up to a sign, ℓ k (x 1 , . . . , x k ) ∈ [x 1 , . . . , x k ] W .
In particular, if [x 1 , x 2 , x 3 ] W = ∅, then ℓ 3 (x 1 , x 2 , x 3 ) ∈ [x 1 , x 2 , x 3 ] W . Also, ℓ 4 (x 1 , x 2 , x 3 , x 4 ) ∈ [x 1 , x 2 , x 3 , x 4 ] W as long as this is not the empty set and the homology of L is abelian . We finish with an example which shows that adapted retracts are needed and that the above are the best possible results in this direction even for reduced DGL's.
Preliminaries
We assume the reader is familiar with the basics of higher homotopy structures being [12] an excellent reference. We will also rely on some known results from rational homotopy theory for which [3] is now a classic reference. With the aim of fixing notation we give some definitions and sketch some results we will need. Throughout this paper we assume that Q is the base field.
A graded Lie algebra is a Z-graded vector space L = ⊕ p∈Z L p with a bilinear product called the Lie bracket and denoted by [ , ] verifying graded antisymmetry, [x, y] = −(−1) |x||y| [y, x], and graded Jacobi identity,
(−1) |x||z| x, [y, z] + (−1) |y||x| y, [z, x] + (−1) |z||y| z, [x, y] = 0,
where |x| denotes the degree of x.
A differential graded Lie algebra (DGL henceforth) is a graded Lie algebra L endowed with a linear derivation ∂ of degree −1 such that ∂ 2 = 0. It is called free if L is free as a Lie algebra, L = L(V ) for some graded vector space V . We say that L is a reduced DGL if L p = 0 for p ≤ 0.
The Quillen chain functor associates to any differential graded Lie algebra (L, ∂) the differential graded coalgebra, DGC henceforth, C(L) = (ΛsL, δ) which is the cocommutative cofree coalgebra generated by the suspension on L and whose differential is given by δ = δ 1 + δ 2 ,
δ 1 (sx 1 ∧ ... ∧ sx k ) = − k i=1 (−1) n i sx 1 ∧ ... ∧ s∂x i ∧ ... ∧ sx k , δ 2 (sx 1 ∧ ... ∧ sx k ) = − i<j (−1) n ij +|x i | s[x i , x j ] ∧ sx 1 ... sx i ... sx j ... ∧ sx k .
Here, n i = j<i |sx j | and n ij is the sign given by the equality sx 1 ∧. ..∧sx
k = (−1) n ij sx i ∧ sx j ∧ sx 1 ... sx i ... sx j ... ∧ sx k .
In [15], D. Quillen constructed an equivalence Simply connected spaces
λ / / − o o Reduced DGL's
between the homotopy category of simply connected rational complexes and the homotopy category of reduced differential graded Lie algebras. The reduced DGL L is a model of the simply connected complex X if there is a sequence of DGL quasi-isomorphisms
L ≃ → · · · ≃ ← λX.
For any model one has H(L) ∼ = π * (ΩX) ⊗ Q. If L = (L(V ), ∂) is free we say that it is a Quillen model of X. For such a model one has H(V, ∂ 1 ) ∼ = s H(X; Q) where ∂ 1 : V → V denotes the linear part of ∂ and s denotes the suspension operator which is defined for any graded vector space W by (sW ) p = W p−1 Next, we briefly recall from [18, §V] how to read the set of higher order Whitehead products of a simply connected complex X in a given Quillen model L. On the one hand, following the notation in the introduction, the map g is modeled by
ϕ : (L(u 1 , . . . , u k ), 0) −→ L
in which |u j | = n j − 1 for each j = 1, . . . , k, and the class ϕ(u j ) represents the element x j ∈ π n j (X).
On the other hand, arguing cellularly [18, §V.2], the inclusion W ֒→ T is modeled by the DGL inclusion
(L(u 1 , . . . , u k ), 0) ֒→ (L(U), ∂) in which |u j | = n j − 1, j = 1, . . . , k, U = u i 1 ...is , 1 ≤ i 1 < · · · < i s ≤ k, s < k, |u i 1 ...is | = n i 1 + · · · + n is − 1,
and the differential is given by
∂u i 1 ...is = s−1 p=1 σ∈ S(p,s−p) ε(σ) u i σ(1) ...i σ(p) , u i σ(p+1) ...i σ(s) ,
where S(p, s−p) denotes the set of shuffle permutations σ such that σ(1) = 1, and ε(σ) is given by the Koszul convention.
Moreover, a Quillen model for S n 1 × · · · × S n k is obtained by attaching a single generator to L(U) in the same way. That is:
(L(U ⊕ u 1...k ), ∂) with |u i...k | = N − 1 and ∂u 1...k = k−1 p=1 σ∈ S(p,k−p) ε(σ) u i σ(1) ...i σ(p) , u i σ(p+1) ...i σ(k) .
(2.1)
We denote ∂u i 1 ...i k = ω henceforth as it encodes the homotopy class S N −1 ω → T . It follows that there is a bijective set correspondence of homology classes
[x 1 , . . . , x k ] W ∼ = {φ(ω) | φ : (L(U), ∂) → L an extension of ϕ}. (L(u 1 , . . . , u k ), 0) _ ϕ / / L (L(U), ∂) φ 8 8 ♣ ♣ ♣ ♣ ♣ ♣ ♣ (2.2)
At a purely algebraic level, and for any DGL, the above subset of H(L) defines the kth order Whitehead bracket set [x 1 , . . . , x k ] W of given homology classes
x 1 , . . . , x k ∈ H(L), see [18, Def. V.3(2)].
From now on, and for simplicity in the notation, we will omit the symbol ⊗ in any element of a tensor algebra.
An L ∞ -algebra (L, {ℓ k }) is a graded vector space L together with linear maps ℓ k : L ⊗k → L of degree k − 2, for k ≥ 1, satisfying the following two conditions:
(i) For any permutation σ of k elements,
ℓ k (x σ(1) . . . x σ(k) ) = ε σ εℓ k (x 1 . . . x k ),
where ε σ is the signature of the permutation and ε is the sign given by the Koszul convention.
(ii) The generalized Jacobi identity holds, that is,
i+j=n+1 σ∈S(i,n−i) ε σ ε(−1) i(j−1) ℓ n−i (ℓ i (x σ(1) . . . x σ(i) )x σ(i+1) . . . x σ(n) ) = 0, where S(i, n − i) denotes the set of (i, n − i) shuffles.
Each L ∞ structure in L corresponds with a differential δ in the cofree graded cocommutative coalgebra Λ + sL generated by the suspension of L. Indeed, every ℓ k determines a degree −1 linear map
h k = (−1) k(k−1) 2 s • ℓ k • (s −1 ) ⊗k : Λ k sL → sL,(2.3)
which extends to a coderivation δ k : Λ + sL −→ Λ + sL decreasing the word length by k − 1, that is, δ k (Λ p sL) ⊂ Λ p−k+1 sL for any p:
δ k (sx 1 ∧ ... ∧ sx p ) = i 1 <···<i k ε h k (sx i 1 ∧ ... ∧ sx i k ) ∧ sx 1 ∧ ... sx i 1 ... sx i k ... ∧ sx p .
(2.4) Every differential graded Lie algebra (L, ∂) is an L ∞ -algebra by setting ℓ 1 = ∂, ℓ 2 = [ , ] and ℓ k = 0 for k > 2. The corresponding DGC structure is precisely C(L).
An
L ∞ -algebra (L, {ℓ k }) is called minimal if ℓ 1 = 0. An L ∞ -morphism between L and L ′ is a DGC morphism f : (Λ + sL, δ) −→ (Λ + sL ′ , δ ′ ),
often denoted simply by f : L → L ′ , which is encoded by a system of skewsymmetric linear maps f (k) : L ⊗k → L ′ of degree 1 − k, k ≥ 1, satisfying an infinite sequence of equations involving the brackets ℓ k and ℓ ′ k (see for instance [10]).
An
L ∞ -morphism is a quasi-isomorphism if f (1) : (L, ℓ 1 ) → (L ′ , ℓ ′ 1 ) is a quasi-isomorphism of complexes.
Given L a DGL, consider the following diagram
K < < (L, ∂) q / / (H, 0) i o o in which H = H(L),
i is a quasi-isomorphism, qi = id H and K is a chain homotopy between id L and iq, i.e., id L − iq = ∂K + K∂. We encode this data as (L, i, q, K) and call it a homotopy retract of L. In this setting, the classical Homotopy Transfer Theorem reads [12]:
Theorem 2.1.
There exists an L ∞ -algebra structure {ℓ k } on H, unique up to isomorphism, and L ∞ quasi-isomorphisms
(L, ∂) Q / / (H, {ℓ k }) I o o such that I (1) = i and Q (1) = q.
In other words, there are DGC quasiisomorphisms extending i and q
C(L) Q / / (ΛsH, δ) I o o
which make (ΛsH, δ) a quasi-isomorphic retract of the Quillen chains on L.
The transferred higher brackets are given by
ℓ k = T ∈T k q • ℓ T | Aut(T )| . (2.5)
We describe here every item in formula (2.5). Let T k be the set of isomorphism classes of directed planar binary rooted trees with exactly k leaves. For such a tree T label the leaves by i, each internal edge by K, and each internal vertex by [ , ]. This produces a linear map
ℓ T : H ⊗k −→ H
by moving down from the leaves to the root. For example, for k = 4, the following tree
i ✾ ✾ ✾ ✾ ✾ i ✆ ✆ ✆ ✆ ✆ i ✾ ✾ ✾ ✾ ✾ i ✆ ✆ ✆ ✆ ✆ [ , ] K ▲ ▲ ▲ ▲ ▲ ▲ [ , ] K s s s s s s [ , ]
produces the map
[ , ] • ((K • [ , ] • (i ⊗ i)) ⊗ (K • [ , ] • (i ⊗ i))). Then, ℓ T = ℓ T • S k where S k : V ⊗k → V ⊗k , S k (v 1 . . . v k ) = σ∈S k ε σ ε v σ(1) . . . v σ(n)
is the symmetrization map in which ε σ denotes the signature of the permutation and ε is the sign given by the Koszul convention. Finally, Aut(T ) stands for the automorphism group of the tree T . (ii) Another invariant of transferred L ∞ structures on H, in fact on isomorphism classes of minimal L ∞ algebras is the least k for which ℓ k is non trivial, and the bracket ℓ k itself.
Higher order Whitehead products and L ∞ structures
Let (L, i, q, K) be a homotopy retract of a given differential graded Lie algebra L. The most general result relating Whitehead brackets on H and brackets of the transferred L ∞ structure depends heavily on a theorem of C. Allday [2, Thm. 4.1], see also [18, Thm. V.7 (7)], and reads as follows.
Proposition 3. 1. Let x 1 , . . . , x k ∈ H and assume that [x 1 , . . . , x k ] W is non empty. Then, for any homotopy retract of L and for any x ∈ [x 1 , . . . ,
x k ] W , ǫ ℓ k (x 1 , . . . , x k ) = x + Γ, Γ ∈ k−1 j=1 Im ℓ j , where ǫ = (−1) k−1 i=1 (k−i)|x i | . In particular, if ℓ j = 0 for j ≤ k − 1, then up to a sign, ℓ k (x 1 , . . . , x k ) ∈ [x 1 , . . . , x k ] W .
In the remaining of the paper ǫ will always denote the above sign.
Proof.
Recall that the Quillen spectral sequence of L [15] is defined by filtering the chains C(L) by the kernel of the reduced diagonals, F p = Λ ≤p sL. Consider the DGC quasi-isomorphisms of Theorem 2.1
C(L) Q / / (ΛsH, δ) I o o ,
choose the same filtration on ΛsH, and observe that at the E 1 level the induced morphisms of spectral sequences are both the identity on ΛsH. By comparison, all the terms in both spectral sequences are also isomorphic. Now, translating [2, Thm. 4.1] to the spectral sequence on ΛsH we obtain that if [x 1 , . . . , x k ] W is non empty, then the element
sx 1 ∧ . . . ∧ sx k survives to the k − 1 page (E k−1 , δ k−1 ). Moreover, given any x ∈ [x 1 , . . . , x k ] W , one has δ k−1 sx 1 ∧ . . . ∧ sx k k−1 = sx k−1 .
Here (·) k−1 denotes the class in E k−1 . This is to say that there exists Φ ∈
Λ ≤k−1 sH such that δ(sx 1 ∧ . . . ∧ sx k + Φ) = sx. (3.1) Write δ = i≥1 δ i with each δ i as in formula (2.4), and decompose Φ = k−1 i=2 Φ i with Φ i ∈ Λ i sH.
By a word length argument,
δ k (sx 1 ∧ . . . ∧ sx k ) + k−1 i=2 δ i (Φ i ) = sx.
Note also that δ k = h k for elements of word length k, with h k as in (2.3) and (2.4). Therefore,
h k (sx 1 ∧ . . . ∧ sx k ) + k−1 i=2 h i (Φ i ) = sx.
To finish, apply to this equation the identity (2.3) which is equivalent to
ℓ i = s −1 • h i • s ⊗i for any i ≥ 1.
In particular, the sign ε appears when writing
ℓ k (x 1 , . . . , x k ) = s −1 • h k • s ⊗k (x 1 , . . . , x k ) = ε s −1 h k (sx 1 ∧ . . . ∧ sx k ).
Next we find kth order Whitehead brackets that are detected precisely and only by kth brackets of the L ∞ structure.
Recall that, any x ∈ [x 1 , . . . , x k ] W is produced by a DGL morphism φ : (L(U), ∂) → L as in diagram (2.2). Write,
U = u 1 , . . . , u k ⊕ V, that is, V = u i 1 ...is , s ≥ 2.
On the other hand, any homotopy retract can be obtained by decomposing L = A ⊕ ∂A ⊕ C with ∂ : A
x ∈ [x 1 , . . . , x k ] W if φ(V ) ⊂ A. In particular, K∂φ(u i 1 ...is ) = φ(u i 1 ...is ) for any generator u i 1 ...is ∈ V. (3.2)
Theorem 3. 3. Let x ∈ [x 1 , . . . , x k ] W . Then, for any homotopy retract of L adapted to x, ǫ ℓ k (x 1 , . . . , x k ) = x.
Proof. Let φ : (L(U), ∂) → L with φ(ω) = x and consider in H the L ∞ structure induced by a given homotopy retract (L, i, q, K) of L adapted to x. We prove by induction on p, with 2 ≤ p ≤ k, that
φ(∂u i 1 ...ip ) = ǫ T ∈Tp 1 |AutT | ℓ T (x i 1 , . . . , x ip ). (3.3)
The assertion is trivial for p = 2 and assume it is satisfied for p < k.
Write the set T k of isomorphism classes of directed planar binary rooted trees and exactly k leaves as
T k = 1≤p≤⌈ k 2 ⌉ T p,k−p ,
where T p,k−p is the set of (classes of) rooted trees T of the form
F ❄ ❄ ❄ ❄ ❄ ❄ G ⑧ ⑧ ⑧ ⑧ ⑧ ⑧
with F ∈ T p and G ∈ T k−p . Note that, whenever k is even and p = k 2 then, for any pair F, G ∈ Tk 2 with F = G, the trees
F ❄ ❄ ❄ ❄ ❄ ❄ G ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ G ❄ ❄ ❄ ❄ ❄ ❄ F ⑧ ⑧ ⑧ ⑧ ⑧ ⑧
are in the same class.
If T ∈ 1≤p≤⌈ k 2 ⌉ T p,k−p , then |AutT | = |AutF ||AutG| except when p = k 2 and T ∈ Tk 2 , k 2 is such that F = G. In this case, which only occurs whenever k is even, |AutT | = 2|AutF ||AutG|.
In what follows we omit signs to avoid excessive notation. On the one hand, splitting the summation for p = 1, 1 < p < ⌈ k 2 ⌉ and p = k 2 (which only occurs whenever k is even), we have: . . , x νσ(k) ) = ( †)
T ∈T k 1 |AutT | ℓ T (x 1 , . . . , , x k ) = T ∈T k 1 |AutT | ℓ T • S k (x 1 , . . . , x k ) = = T ∈T k σ∈S k 1 |AutT | ℓ T (x σ(1) , . . . , x σ(k) ) = σ∈S(1,k−1) ix σ(1) , K T ∈T k−1 1 |AutT | ℓ T (x σ(2) , . . . , x σ(k) ) + 1<p<⌈ k 2 ⌉ σ∈S(p,k−p) K F ∈Tp τ ∈Sp 1 |AutF | ℓ F (x τ σ(1) , . . . , x τ σ(p) ) , , K G∈T k−p ν∈S k−p 1 |AutG| ℓ G (x νσ(p+1) , . . . , x νσ(k) ) + 1 2 σ∈S( k 2 , k 2 ) K F ∈T k 2 τ ∈S k 2 1 |AutF | ℓ F (x τ σ(1) , . . . , x τ σ( k 2 ) ) , , K G∈T k 2 ν∈S k 2 1 |AutG| ℓ G (x νσ( k 2 +1) , .
Note that the last summand appears only if k is even. The 1 2 coefficient arises from the observation above.
On the other hand, remark that the formula (2.1) can be written alternatively as
∂u 1...k = 1≤p<⌈ k 2 ⌉ σ∈S(p,k−p) u σ(1)...σ(p) , u σ(p+1)...σ(k) + 1 2 σ∈S( k 2 , k 2 ) u σ(1)...σ( k 2 ) , u σ( k modulo signs: φ(∂u 1...k ) = 1≤p<⌈ k 2 ⌉ σ∈S(p,k−p) φ(u σ(1)...σ(p) ), φ(u σ(p+1)...σ(k) ) + 1 2 σ∈S( k 2 , k 2 ) φ(u σ(1)...σ( k 2 ) ), φ(u σ( k 2 +1)...σ(k) ) = σ∈S(1,k−1) φu σ(1) , Kφ∂u σ(2)...σ(k) + 1<p<⌈ k 2 ⌉ σ∈S(p,k−p)
Kφ∂u σ(1). ..σ(p) , Kφ∂u σ(p+1)...σ(k)
+ 1 2 σ∈S( k 2 , k 2 ) Kφ∂u σ(1)...σ( k 2 ) , Kφ∂u σ( k 2 +1)...σ(k) = σ∈S(1,k−1) ix σ(1) , K T ∈T k−1 1 |AutT | ℓ T • S k−1 (x σ(2) , . . . , x σ(k) ) + 1<p<⌈ k 2 ⌉ σ∈S(p,k−p) K F ∈Tp 1 |AutF | ℓ F • S p (x σ(1) , . . . , x σ(p) ) , , K G∈T k−p 1 |AutG| ℓ G • S k−p (x σ(p+1) , . . . , x σ(k) ) + 1 2 σ∈S( k 2 , k 2 ) K F ∈T k 2 1 |AutF | ℓ F • Sk 2 (x σ(1) , . . . , x σ( k 2 ) ) , , K G∈T k 2 1 |AutG| ℓ G • Sk 2 (x σ( k 2 +1) , . . . , x σ(k) ) = ( †)
and the assertion is proved. In particular, by the explicit formula for ℓ k in Theorem 2.1,
qφ(ω) = ǫ q T ∈T k 1 |AutT | ℓ T (x 1 , . . . , x k ) = ǫ ℓ k (x 1 , . . . , x k ).
That is, ǫ ℓ k (x 1 , . . . , x k ) ∈ [x 1 , . . . , x k ] W .
Remark 3. 4. The Higher Massey products set [13] a 1 , . . . , a k M ⊂ H * (A) of order k of classes a 1 , . . . , a k ∈ H * (A) in the cohomology of a given differential graded algebra (A, d) (or simply A) can be thought of as the "Eckmann-Hilton" dual of higher Whitehead brackets of order k. There is also an A ∞ version of Theorem 2.1 for a given retract of A,
K < < (A, d) q / / (H, 0), i o o
which produces an A ∞ structure {m k } on H and A ∞ quasi-isomorphisms (see for instance [6] or [7]):
(A, d) Q / / (H, {m k }). I o o
Recall that an A ∞ -algebra [17] is a graded vector space H endowed with a sequence of maps m k : H ⊗k → H of degree 2 − k, for k ≥ 1, satisfying a series of "associative" identities. Each of these maps is identified, up to suspensions and signs, with a degree 1 map δ k : (sH) ⊗k → sH which produces a differential δ on the graded colgebra T (sH). Filtering this DGC by F p = (sH) ⊗≤p we obtain the Eilenberg-Moore spectral sequence from which, following the argument of In this setting one can go a step further, see [11,Theorem 3.1], and prove that for any homotopy retract of A, if a 1 , . . . , a k M is non empty, then ǫ m k (a 1 , . . . , a k ) ∈ a 1 , . . . , a k M .
Except for the case k = 3 in Corollary 3.6 below, this particular behavior cannot be attained in general in the L ∞ setting and additional conditions are required as the following result shows.
Theorem 3.5. Let L be a DGL such that, on H, ℓ i = 0 for i ≤ k − 2 with k ≥ 3. If [x 1 , . . . , x k ] W = ∅, then ǫ ℓ k (x 1 , . . . , x k ) ∈ [x 1 , . . . , x k ] W .
Observe that, in view of (ii) of Remark 2.2 the assumption on the vanishing of ℓ i for i ≤ k − 2 is independent of the chosen retract of L and hence, the result remains valid for any of them.
Proof. We first observe the following: consider (ΛsH, δ) the DGC equivalent to the L ∞ structure on H given by any homotopy retract of L. The condition ℓ i = 0 for i ≤ k − 2 is clearly equivalent via equations (2.3) and (2.4) to the vanishing of the coderivations δ i of Λ + sH also for i ≤ k − 2. On the other hand, as in equation (3.1) in the proof of Proposition 3.1, for any x ∈ [x 1 , . . . , x k ] W , we have.
δ(sx 1 ∧ . . . ∧ sx k + Φ) = sx with Φ ∈ Λ ≤k−1 sH.
Write again δ = i≥1 δ i with each δ i as in formula (2.4). Thus, a word length argument together with δ i = 0, for i ≤ k − 2, readily implies in particular that δ k−1 (sx 1 ∧ · · · ∧ sx k ) = 0.
But
δ k−1 (sx 1 ∧ . . . ∧ sx k ) = k i=1 ε h k−1 (sx 1 ∧ ... sx i ... ∧ sx k ) ∧ sx i .
Hence, via identity (2.3), . . , x i k−1 ) = 0 for any 1 ≤ i 1 < · · · < i k−1 ≤ k. (3.4) Next, for each p ≤ k, let U p ⊂ U be the subspace generated by
ℓ k−1 (x i 1 , .U s = u i 1 ...is , 1 ≤ i 1 < · · · < i s ≤ k, s < p.
Clearly, (L(U p ), ∂) is a sub DGL of (L(U), ∂) and (L(U k ), ∂) = (L(U), ∂).
We also denote,
V p = u i 1 ...is ∈ U p , s ≥ 2 .
Again U p = V p ⊕ u 1 , . . . , u k and V k = V . Let L = A ⊕ ∂A ⊕ C the decomposition giving rise to the chosen arbitrary homotopy retract. By induction on p, with 3 ≤ p ≤ k, we will construct a DGL morphism φ :
L(U p ) → L for which φ(V p ) ⊂ A. For p = 3, as [x 1 , . . . , x k ] W is non empty, let ψ : (L(U), ∂) → (L, ∂) as in (2.2). We define φ : L(U 3 ) → L by φ(u i ) = ψ(u i ), i = 1, 2, 3. φ(u i 1 i 2 ) = K∂ψ(u i 1 i 2 ), 1 ≤ i 1 < i 2 ≤ k,
Obviously φ(V 3 ) ⊂ A and using the trivial identity for any homotopy retract ∂K∂ = ∂ we also see that φ commutes with the differential:
∂φ(u i 1 i 2 ) = ∂K∂ψ(u i 1 i 2 ) = ∂ψ(u i 1 i 2 ) = ψ∂(u i 1 i 2 ) = φ∂(u i 1 i 2 ).
Assume the assertion true for k − 1. That is, there exists a DGL morphism
φ : L(U k−1 ) −→ L for which φ(V k−1 ) ⊂ A.
In particular, we have,
K∂φ(u i 1 ...is ) = φ(u i 1 ...is ) for any generator u i 1 ...is ∈ V k−1 ,
which is equation (3.2) for φ. Then, the same argument as in the proof of Theorem 3.3 proves the analogous of equation (3.3). In particular,
φ(∂u i 1 ...i k−1 ) = ǫ T ∈T k−1 1 |AutT | ℓ T (x i 1 , . . . , x i k−1 ),
and therefore,
qφ(∂u i 1 ...i k−1 ) = ǫ q T ∈T k−1 1 |AutT | ℓ T (x i 1 , . . . , x i k−1 ) = ǫ ℓ k−1 (x i 1 , . . . , x i k−1 ),
which is zero by the observation (3.4) above. Hence,
φ(∂u i 1 ...i k−1 ) = ∂Ψ i 1 ...i k−1 and we define φ(u i 1 ...i k−1 ) = K∂Ψ i 1 ...i k−1 .
Obviously φ(V k ) = φ(V ) ⊂ A and using again the identity ∂K∂ = ∂ we see that φ commutes with differentials. Therefore φ(ω) is an element in [x 1 , . . . , x k ] W for which we can apply Theorem 3.3 and the proof is finished.
Corollary 3.6. Let x 1 , x 2 , x 3 ∈ H such that [x 1 , x 2 , x 3 ] W = ∅.
Then, for any homotopy retract,
ǫ ℓ 3 (x 1 , x 2 , x 3 ) ∈ [x 1 , x 2 , x 3 ] W .
Corollary 3. 7. Let L be a DGL such that H is abelian. If [x 1 , x 2 , x 3 , x 4 ] W = ∅, then, for any homotopy retract,
ǫ ℓ 4 (x 1 , x 2 , x 3 , x 4 ) ∈ [x 1 , x 2 , x 3 , x 4 ] W .
We finish with an example which shows that Theorem 3.5 is the most general version of its Eckmann-Hilton dual, even for reduced DGL's or equivalently, for simply connected rational complexes.
Example 3. 8. Consider the following DGL,
(L, ∂) = (L(v 1 , v 2 , v 3 , v 4 , v 12 , v 13 , v 14 , v 23 , v 24 , v 34 , z, w 123 , w 124 , v 134 , v 234 ), ∂),
where |v i | = 2, 1 ≤ i ≤ 4, |z| = 5 and the differential is given by:
∂(v i ) = 0, 1 ≤ i ≤ 4; ∂(v ij ) = [v i , v j ], 1 ≤ i < j ≤ 4; ∂(z) = 0; ∂(v ijk ) = [v i , v jk ] − [v ij , v k ] − [v j , v ik ]; ∂(w ijk ) = [v i , v jk ] − [v ij + z, v k ] − [v j , v ik ].
The realization of this DGL is (of the rational homotopy type of) the complex X obtained by removing two 9-cells from the space T (S 3 , S 3 , S 3 , S 3 ) ∨S 6 and attach them again in a twisted way using the sphere S 6 .
We claim that [v 1 , v 2 , v 3 , v 4 ] W is non empty and that, for any homotopy retract of L,
ℓ 4 (v 1 , v 2 , v 3 , v 4 ) / ∈ [v 1 , v 2 , v 3 , v 4 ] W .
For it, define a DGL morphism φ which solves the extension problem (L(u 1 , u 2 , u 3 , u 4 ), 0) Note that H is not an abelian Lie algebra and that, for any decomposition A ⊕ ∂A ⊕ C giving rise to any chosen homotopy retract, the element φ(u 12 ) = v 12 + z / ∈ A as z represents a non zero class. Hence, theorems 3.3 and 3.5 do not apply.
In fact, it is straightforward to check that Φ generates H 10 (L) and it is the only element in [v 1 , v 2 , v 3 , v 4 ] W . Moreover, for any homotopy retract,
ℓ 4 (v 1 , v 2 , v 3 , v 4 ) = Φ.
To help the reader with the computations, we make explicit a particular decomposition of L as A ⊕ ∂A ⊕ C with ∂ : A ∼ = → ∂A and C ∼ = H up to degree 10. Here, the first column denotes degree and the twisted arrow indicates the action of ∂ in the corresponding set.
A ∂A C 2 v 1 , v 2 , v 3 , v 4 3 4 [v 1 , v 2 ], [v 1 , v 3 ], [v 1 , v 4 ], [v 2 , v 3 ], [v 2 , v 4 ], [v 3 , v 4 ] 5 v 12 , v 13 , v 14 , z v 23 , v 24 , v 34 6 v 1 , [v 1 , v 2 ] , v 1 , [v 1 , v 3 ] , v 1 , [v 1 , v 4 ] , v 2 , [v 2 , v 1 ] , v 2 , [v 2 , v 3 ] , v 2 , [v 2 , v 4 ] , v 3 , [v 3 , v 1 ] , v 3 , [v 3 , v 2 ] , v 3 , [v 3 , v 4 ] , v 4 , [v 4 , v 1 ] , v 4 , [v 4 , v 2 ] , v 4 , [v 4 , v 3 ] , v 1 , [v 2 , v 3 ] , v 2 , [v 1 , v 3 ] , v 1 , [v 2 , v 4 ] , v 2 , [v 1 , v 4 ] ,
Remark 2. 2 .
2(i) The uniqueness property is clear. Indeed, different homotopy retracts of L produce quasi-isomorphic L ∞ structures on H. But, since all of them are minimal, they are also isomorphic (see for instance[10, §4]).
→
∂A and C ∼ = H a subspace of cycles. For it define i : H ∼ = C ֒→ L, q : L ։ C ∼ = H and K(A) = K(C) = 0, K : ∂A ∼ = → A.
Definition 3. 2 .
2With the notation above, a homotopy retract of L is adapted to
Proposition 3.1 now based in [18, Thm. V.7(6)], we obtain: If a 1 , . . . , a k M is non empty, then, for any a ∈ a 1 , . . . , a k M , and any homotopy retract of A, ǫ m k (a 1 . . . a k ) = a + Γ, Γ ∈
u 1 ) = v 1 ; φ(u 2 ) = v 2 ; φ(u 3 ) = v 3 ; φ(u 4 ) = v 4 ; φ(u 12 ) = v 12 + z; φ(u 13 ) = v 13 ; φ(u 14 ) = v 14 ; φ(u 23 ) = v 23 ; φ(u 24 ) = v 24 ; φ(u 34 ) = v 34 φ(u 123 ) = w 123 ; φ(u 124 ) = w 124 ; φ(u 134 ) = v 134 ; φ(u 234 ) = v 234 . Then, [v 1 , v 2 , v 3 , v 4 ] W = ∅.More precisely, the morphism φ defines the non zero 4th order Whitehead bracket φ(ω) = Φ whereΦ = [w 123 , v 4 ] − [w 124 , v 3 ] + [v 12 , v 34 ] + [z, v 34 ] + [v 14 , v 23 ] + [v 1 , v 234 ] − [v 13 , v 24 ] + [v 134 , v 2 ].
+1)...σ(k) Thus, in view of equation(3.2) and by induction hypothesis, we have, also
. . . . . , 9 . . . . . . . . .
. . . . , 10 . . . . . .
Let X be a 1-connected CW-complex of finite type and let L be a reduced, finite type DGL model of X. If (ΛsH, δ) is the DGC equivalent to a transferred L ∞ structure on H, then, its dual (ΛsH, δ) ♯ is isomorphic [3, §23] to (Λ(sH) ♯ , d) which is the Sullivan minimal model of X. Φ Remark 3.9. In this case, Proposition 3.1 and theorems 3.3 and 3.5 are in some sense the reciprocal of Theorem 5.4 in [1Φ Remark 3.9. Let X be a 1-connected CW-complex of finite type and let L be a reduced, finite type DGL model of X. If (ΛsH, δ) is the DGC equivalent to a transferred L ∞ structure on H, then, its dual (ΛsH, δ) ♯ is isomorphic [3, §23] to (Λ(sH) ♯ , d) which is the Sullivan minimal model of X. In this case, Proposition 3.1 and theorems 3.3 and 3.5 are in some sense the reciprocal of Theorem 5.4 in [1].
Sullivan's minimal models and higher order Whitehead products. P Andrews, M Arkowitz, Canad. J. Math. 305P. Andrews and M. Arkowitz, Sullivan's minimal models and higher order Whitehead products, Canad. J. Math. 30(5) (1978), 961-982.
Rational Whitehead products and a spectral sequence of Quillen II. C Allday, Houston J. Math. 33C. Allday, Rational Whitehead products and a spectral sequence of Quillen II, Houston J. Math. 3(3) (1977), 301-308.
Y Félix, S Halperin, J.-C Thomas, Rational Homotopy Theory. 205SpingerY. Félix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Spinger (2000).
The homotopy type of the polyhedral product for shifted complexes. J Grbić, S Theriault, Adv. in Math. 245J. Grbić and S. Theriault, The homotopy type of the polyhedral product for shifted complexes, Adv. in Math. 245 (2013), 690-715.
J Grbić, S Theriault, arXiv:1011.2133Higher Whitehead products in toric topology. J. Grbić and S. Theriault, Higher Whitehead products in toric topology, arXiv:1011.2133.
On perturbations and A ∞ -structures. V K A M Gugenheim, J D Stasheff, Bull. Soc. Math. Belg. Sér. A. 38V. K. A. M. Gugenheim and J. D. Stasheff, On perturbations and A ∞ - structures. Bull. Soc. Math. Belg. Sér. A 38 (1986), 237-246
Small models for chain algebras. J Huebschmann, T Kadeishvili, Math. Z. 2072J. Huebschmann and T. Kadeishvili, Small models for chain algebras. Math. Z. 207(2) (1991), 245-280.
Decompositions of polyhedral products for shifted complexes. K Iriye, D Kishimoto, Adv. in Math. 245K.Iriye and D. Kishimoto, Decompositions of polyhedral products for shifted complexes, Adv. in Math. 245 (2013), 716-736.
K Iriye, D Kishimoto, arXiv:1505.04892Polyhedral products for shifted complexes and higher Whitehead products. K. Iriye and D. Kishimoto, Polyhedral products for shifted complexes and higher Whitehead products, arXiv:1505.04892.
Deformation quantization of Poisson manifolds. M Kontsevich, Lett. Math. Phys. 663M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66(3) (2003), 157-216.
A-infinity structure on Ext algebras. D.-M Lu, J H Palmieri, Q.-S Wu, J J Zhang, J. Pure Appl. Algebra. 21311D.-M. Lu, J. H. Palmieri, Q.-S. Wu and J. J. Zhang, A-infinity structure on Ext algebras, J. Pure Appl. Algebra 213(11) (2009), 2017-2037.
Algebraic Operads. J.-L Loday, B Vallette, Grundlehren Math. Wiss. 346SpringerJ.-L. Loday and B. Vallette, Algebraic Operads, Grundlehren Math. Wiss. 346, Springer (2012).
Some higher order cohomology operations. W.-S Massey, Int. Symp. Alg. Top. Mexico. W.-S. Massey, Some higher order cohomology operations, Int. Symp. Alg. Top. Mexico, (1958), 145-154.
Higher order Witehead products. G J Porter, Topology. 3G. J. Porter, Higher order Witehead products, Topology 3 (1965), 123- 135.
Rational homotopy theory. D Quillen, Ann. of Math. 2D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205-295.
Homotopy type of Euclidean configuration spaces. P Salvatore, Rend. Circ. Mat. Palermo. 11SupplP. Salvatore, Homotopy type of Euclidean configuration spaces, Rend. Circ. Mat. Palermo 11 Suppl. 66 (2001), 161-164.
Homotopy associativity of H-spaces. I, II. J D Stasheff, Trans. Amer. Math. Soc. 108ibid. 108J. D. Stasheff, Homotopy associativity of H-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293-312.
. D Tanré, Homotopie Rationnelle, Lecture Notes in Mathematics. Modèles de Chen, Quillen, Sullivan1025SpringerD. Tanré, Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics 1025, Springer (1983).
. Urtzi Buijs, M José, Aniceto Moreno-Fernández, Murillo Departamento Deálgebra, Geometría Y Topología, Ap. 5929080Universidad de MálagaUrtzi Buijs, José M. Moreno-Fernández and Aniceto Murillo Departamento deÁlgebra, Geometría y Topología, Universidad de Málaga, Ap. 59, 29080 Málaga, Spain.
|
[] |
[
"Difficulties in Probing Nuclear Physics: A Study of 44 Ti and 56 Ni",
"Difficulties in Probing Nuclear Physics: A Study of 44 Ti and 56 Ni"
] |
[
"Aimee Hungerford ",
"Christopher L Fryer [email protected] ",
"Francis X Timmes ",
"Patrick Young ",
"Michael Bennett ",
"Steven Diehl ",
"Falk Herwig ",
"Raphael Hirschi ",
"Marco Pignatari ",
"Georgios Magkotsios ",
"Gabriel Rockefeller ",
"\nThe NuGrid Collaboration b Computational Methods (CCS\n\n",
"\nLos Alamos National Laboratory\n87544Los AlamosNMUSA\n",
"\nSchool of Earth and Space Exploration\nArizona State University\n85287TempeAZUSA\n",
"\nAstrophysics Group\nKeele University\nST5 5BGUK\n",
"\nTheoretical Astrophysics Group (T-6)\nLos Alamos National Laboratory\n87544Los AlamosNMUSA\n",
"\nDept. of Physics & Astronomy\nV8W 3P6VictoriaBCCanada\n",
"\nJoint Institute for Nuclear Astrophysics\nUniversity of Notre Dame\n46556INUSA\n",
"\nMackinac Island\nMichiganUSA\n"
] |
[
"The NuGrid Collaboration b Computational Methods (CCS\n",
"Los Alamos National Laboratory\n87544Los AlamosNMUSA",
"School of Earth and Space Exploration\nArizona State University\n85287TempeAZUSA",
"Astrophysics Group\nKeele University\nST5 5BGUK",
"Theoretical Astrophysics Group (T-6)\nLos Alamos National Laboratory\n87544Los AlamosNMUSA",
"Dept. of Physics & Astronomy\nV8W 3P6VictoriaBCCanada",
"Joint Institute for Nuclear Astrophysics\nUniversity of Notre Dame\n46556INUSA",
"Mackinac Island\nMichiganUSA"
] |
[] |
The nucleosynthetic yield from a supernova explosion depends upon a variety of effects: progenitor evolution, explosion process, details of the nuclear network, and nuclear rates. Especially in studies of integrated stellar yields, simplifications reduce these uncertainties. But nature is much more complex, and to actually study nuclear rates, we will have to understand the full, complex set of processes involved in nucleosynthesis. Here we discuss a few of these complexities and detail how the NuGrid collaboration will address them.10th Symposium on Nuclei in the CosmosFigure 1: A cartoon displaying the key effects in nucleosynthesis for shocked elements focusing on the 44 Ti and 56 Ni yields[2]. Note that the position of the particles determines which effects (fast freezeout, slow freezeout, α−rich freezeout) occur and this determines the important rates for these yield calculations. The position of the particle used in the The et al.[1] study (ρ = 10 7 gcm −3 , T = 5.5 × 10 9 K) led them to believe one rate was important, but in reality, the trajectories vary tremendously.
| null |
[
"https://arxiv.org/pdf/0811.4645v1.pdf"
] | 1,686,891 |
0811.4645
|
b9f63e14e9387005ce286bf46131ee70660cf0ae
|
Difficulties in Probing Nuclear Physics: A Study of 44 Ti and 56 Ni
28 Nov 2008 July 27 -August 1 2008
Aimee Hungerford
Christopher L Fryer [email protected]
Francis X Timmes
Patrick Young
Michael Bennett
Steven Diehl
Falk Herwig
Raphael Hirschi
Marco Pignatari
Georgios Magkotsios
Gabriel Rockefeller
The NuGrid Collaboration b Computational Methods (CCS
Los Alamos National Laboratory
87544Los AlamosNMUSA
School of Earth and Space Exploration
Arizona State University
85287TempeAZUSA
Astrophysics Group
Keele University
ST5 5BGUK
Theoretical Astrophysics Group (T-6)
Los Alamos National Laboratory
87544Los AlamosNMUSA
Dept. of Physics & Astronomy
V8W 3P6VictoriaBCCanada
Joint Institute for Nuclear Astrophysics
University of Notre Dame
46556INUSA
Mackinac Island
MichiganUSA
Difficulties in Probing Nuclear Physics: A Study of 44 Ti and 56 Ni
28 Nov 2008 July 27 -August 1 2008* Speaker.
The nucleosynthetic yield from a supernova explosion depends upon a variety of effects: progenitor evolution, explosion process, details of the nuclear network, and nuclear rates. Especially in studies of integrated stellar yields, simplifications reduce these uncertainties. But nature is much more complex, and to actually study nuclear rates, we will have to understand the full, complex set of processes involved in nucleosynthesis. Here we discuss a few of these complexities and detail how the NuGrid collaboration will address them.10th Symposium on Nuclei in the CosmosFigure 1: A cartoon displaying the key effects in nucleosynthesis for shocked elements focusing on the 44 Ti and 56 Ni yields[2]. Note that the position of the particles determines which effects (fast freezeout, slow freezeout, α−rich freezeout) occur and this determines the important rates for these yield calculations. The position of the particle used in the The et al.[1] study (ρ = 10 7 gcm −3 , T = 5.5 × 10 9 K) led them to believe one rate was important, but in reality, the trajectories vary tremendously.
Understanding Key Rates in Astrophysics
The field of nuclear astrophysics is predicated on the belief that astronomers can cull from the tens of thousands of rates a handful of critical rates that define the nuclear yields in astronomy. Although at some level, this is true: rates at some critical weighting points do make a big difference in the yields; for the most part, complications in nucleosynthesis make it very difficult to pick out a single rate. Early astrophysics success in pinpointing specific rates has driven the nuclear physics community to expect that if they solve the rates surrounding a few tens of isotopes, they can solve nuclear astrophysics. But many of these successes that pinpointed specific rates did so because they focused on very specific points in the density/temperature evolution. In nature, the rate pinpointed by these studies may be important for only a small amount of material and, when comparing to observations, they may be completely neglible.
Here we present some of the pitfalls that can occur in determining key rates using our study of the production of 44 Ti and 56 Ni as an example. But the complexity of understanding the role nuclear rates plays on nucleosynthesis spans all discussions of nuclear astrophysics and we present an r-process example as well. Finally, we conclude with the approach that will be taken by the NuGrid team. [3,4], a rotating 2-dimensional explosion [5], and 2 weak-strong models mimicking fallback gamma-ray bursts [6]. Note that the peak values span the entire trajectory space.
Understanding 44 Ti and 56 Ni Production
One example of the complexitiies in understanding nucleosynthesis is the study of α-element production and, in particular, the production of 44 Ti and 56 Ni. Let's make the simplifying assumption that the yield of a piece of matter is determined solely by its peak temperature and density. Figure 1 is a cartoon of the peak density/temperature space generally studied in explosive nucleosynthesis. The et al. [1] did exactly this analysis, focusing on a single peak temperature/peak density: (ρ = 10 7 gcm −3 , T = 5.5 × 10 9 K). This density/temperature pair lies directly on the boundary of two different effects. As such, a single rate might change the yield by a large amount and The et al. [1] found that the triple-α rate changed the yield dramatically. But elsewhere on this diagram, the triple-α rate is unimportant. Unless we can assume all explosions produce elements only at a single point, studying that single point will provide us with a skewed set of important rates.
We have begun a more systematic study of this entire grid. A first step might be to determine what peaks are common in supernova explosions. Figure 2 again shows a plot of our peak density/temperature grid with overlying points for 4 different explosion calculations. As one can see, they span a wide part of this grid. Not only are the points from different explosion models spread in peak temperatures and densities, the points within each explosion model possess a range of electron fractions for the fluid element represented. Magkotsios et al. 2008 provide a more detailed view of the added complication from variation in electron fraction (these proceedings.) It appears that the supernova conditions will not permit a narrowing of the important parameter space and an understanding of the entire space is ultimately needed. , low entropy (middle) and particles that produced reasonable amounts of third r-process peak isotopes (bottom). The zero point on the time axis is set to the time when the density reaches its maximum value (generally corresponding to the peak temperature as well). It is very difficult to distinguish the peak densities from each other and it has not yet been determined what path is required to make the r-process.
With our simplifying assumption that we can determine everything from a single peak density/temperature pair, the problem of determining a yield (and the most crucial rates for that yield) presents us with considerable work studying the entire grid space. But, for some problems, the work doesn't end there. Our simplifying assumption is not true for all nucleosynthetic problems. In the study of nuclear rates of r-process, most scientists have focused on the reactions and trajectories behind wind-driven supernovae. Again, this is a too-narrow view and scientists working outside of this narrow view have discovered an entirely new nucleosynthetic path (or paths) to make r-process [9,7,8]. Unfortunately, these new paths depend on the subsequent evolution of the cooling matter as well as the peak temperature/density. Figure 3 shows density trajectories for matter that did not make the r-proces peak and matter that did [8]. Matter with the same peaks produced very different yields. Even worse, it is not clear what trajectory is required to make r-process isotopes.
NuGrid Plans
With all of these complexities, it would seem impossible to actually understand astrophysical explosions and nuclear networks sufficiently well to actually determine what rates are important. But without trying, we will definitely not solve this problem. In many cases, the peak tempera- Figure 4: 56 Ni and 44 Ti yields as a function of enclosed mass for two different stellar explosions. We compare the yields from the standard post-process network (solid lines) to those inferred using peak densities and temperatures. The good agreement means that we can use these peak density/temperature diagrams to improve our intuition about nuclear network yields. ture/density studies produce results that are very close to studies that follow trajectories (Fig. 4) and we can use these simple studies to develop our intuition. But in the long run, we'll have to approach this from all angles: studies of simplified problems, like the density/temperature peak diagrams and their production tracks, studies of temperature/density evolution tracks to better understand which tracks produce what matter, and finally, integrated yield studies (the more common study) to compare to observations. One approach alone will not work. NuGrid is developing a suite of tools ideally suited for all these studies and our collaboration will approach this problem from all directions.
Figure 2 :
2A plot of 44 Ti yield on a peak density/temperature grid with points from simulated explosions showing where they lie on this grid. From top to bottom, the points correspond to a magnetohydrodynamic explosion of a collapsar model
Figure 3 :
3Density versus time for 3 sets of particles: high entropy (top)
Nuclear Reactions Governing the Nucleosynthesis of 44 Ti. L.-S The, D D Clayton, L Jin, B S Meyer, ApJ. 504500The, L.-S., Clayton, D.D., Jin, L., Meyer, B.S., Nuclear Reactions Governing the Nucleosynthesis of 44 Ti, ApJ, 504, 500
Explosive Nucleosynthesis Trends in Core-Collapse Supernovae. A L Hungerford, C L Fryer, G Magkotsios, F X Timmes, P A Young, in preparationA.L. Hungerford, C.L. Fryer, G. Magkotsios F.X. Timmes, P.A. Young, Explosive Nucleosynthesis Trends in Core-Collapse Supernovae, in preparation
G Rockefeller, C L Fryer, H Li, astro-ph/0608028Collapsars in Three Dimensions. G. Rockefeller, C.L. Fryer, H. Li 2008, Collapsars in Three Dimensions, astro-ph/0608028
G Rockefeller, C L Fryer, P A Young, H Li, Nucleosynthetic Yields from Collapsars. in preparationG. Rockefeller, C.L. Fryer, P.A. Young, H. Li 2008, Nucleosynthetic Yields from Collapsars, in preparation
Core-Collapse Simulations of Rotating Stars. C L Fryer, A Heger, ApJ. 5411033C.L. Fryer, A. Heger 2000, Core-Collapse Simulations of Rotating Stars, ApJ, 541, 1033
Light-Curve Calculations of Supernovae from Fallback Gamma-Ray Bursts. C L Fryer, A L Hungerford, P A Young, ApJ. 66255C.L. Fryer, A.L. Hungerford, P.A. Young 2007, Light-Curve Calculations of Supernovae from Fallback Gamma-Ray Bursts, ApJ, 662, L55
C L Fryer, F Herwig, A L Hungerford, F X Timmes, Supernova Fallback: A Possible Site for the r-Process. 646131C.L. Fryer, F. Herwig, A.L. Hungerford, F.X. Timmes 2007 Supernova Fallback: A Possible Site for the r-Process, ApJ, 646, L131
Predictive r-Process Calculations. C L Fryer, F Herwig, A L Hungerford, F X Timmes, Proc. of the Third ANL. Duguet, Esbensenof the Third ANLNollett, RobertsWorld ScientificC.L. Fryer, F. Herwig, A.L. Hungerford, F.X. Timmes 2007 Predictive r-Process Calculations, Proc. of the Third ANL/MSU/JINA/INT RIA Workshop, eds. Duguet, Esbensen, Nollett, Roberts, World Scientific 2007.
. B S Meyer, PRL. 89231101r-Process Nucleosynthesis without Excess NeutronsB.S. Meyer 2002, r-Process Nucleosynthesis without Excess Neutrons, PRL, 89, 231101
|
[] |
[
"A Truth of Molecular Chaos",
"A Truth of Molecular Chaos"
] |
[
"Yuriy E Kuzovlev \nA.A.Galkin Physics and Technology Institute of NASU\n83114DonetskUkraine\n"
] |
[
"A.A.Galkin Physics and Technology Institute of NASU\n83114DonetskUkraine"
] |
[] |
The BBGKY hierarchy of equations for a particle interacting with an ideal gas is investigated. Principal properties of its solutions are disclosed, as exact identities which connect probability distribution of path of the particle, its derivatives in respect to gas density and irreducible manyparticle correlations between gas molecules and the path. They show that all the correlations always give equally important contributions to evolution of the path distribution, and therefore the exact theory does not reduce to the classical kinetics even at arbitrary small gas density.
| null |
[
"https://arxiv.org/pdf/0902.2855v1.pdf"
] | 118,201,181 |
0902.2855
|
71a51e00376781d920f6e90836562891c44a8df9
|
A Truth of Molecular Chaos
17 Feb 2009
Yuriy E Kuzovlev
A.A.Galkin Physics and Technology Institute of NASU
83114DonetskUkraine
A Truth of Molecular Chaos
17 Feb 2009
The BBGKY hierarchy of equations for a particle interacting with an ideal gas is investigated. Principal properties of its solutions are disclosed, as exact identities which connect probability distribution of path of the particle, its derivatives in respect to gas density and irreducible manyparticle correlations between gas molecules and the path. They show that all the correlations always give equally important contributions to evolution of the path distribution, and therefore the exact theory does not reduce to the classical kinetics even at arbitrary small gas density.
1.
One of creators of the modern probability theory A. Kolmogorov underscored [1] that in it " the concept of independence of experiments fills most important place " and " correspondingly one of most important objectives of philosophy of natural sciences " is " clearingup and refinement of those prerequisites under which one can treat given phenomena as independent ". Recall that in the probability theory some random phenomena or quantities A and B by definition are independent if their probability distributions are independent, that is P (A, B) = P (A) P (B) [1]. However, in natural sciences the independence of phenomena A and B is thought as absence of cause-and-effect relations between them, that is an influence of one to another. Whether independence in this usual sense does mean independence in the sense of the probability theory?
From viewpoints of common sense and philosophy, certainly does not mean. Merely because A and B which do not directly influence one on another nevertheless both can be parts of a same another random event and thus turn out to be indirectly related.
From the scientific point of view, it is natural to bring the same question to the statistical mechanics. One of creators of modern theory of dynamical systems and statistical mechanics N. Krylov thoroughly analyzed it [2] and also came to the negative answer: he concluded that opinions that " phenomena which are "manifestly independent" should have independent probability distributions ", and the like, are nothing but " prejudices " [2]. Especially Krylov pointed [2] to firmness of such prejudices [3]. Only it explains why the molecular chaos hypothesis put forward by Boltzmann many years ago [4] until now dominates kinetics although never was logically substantiated [5]. And why N. Bogolyubov, when he obtained [6] an exact hierarchy of evolution equations for s -particle distribution functions, straight away truncated his equations at s = 2 thus reducing it to the Boltzmann equation.
Undoubtedly, molecules of sufficiently rarefied gas are independent in usual sense since almost surely have noth- * Electronic address: [email protected] ing common in the past. Nevertheless they can be essentially dependent in the sense of the probability theory. This is quite understandable [7] (or you may see [8]). The absence of common causes of colliding particles in the past means, for each of them, absence of any back reaction of the gas to its past collisions. Therefore arbitrary long fluctuations in relative frequency of collisions are allowable [9]. These fluctuations just play the role of aforesaid random events producing indirect statistical interdependencies between pairs (or groups) of molecules capable of being participators of one and the same collision (or a cluster of successive collisions).
As the consequence, P (A, B) = P (A) P (B) where P (A) is probability of finding a molecule at (phase) point A and P (A, B) is probability of finding simultaneously two molecules at points A and B . At that, relaxation of one-particle distribution P (A) is determined by pair correlation P (A, B) − P (A) P (B) . Relaxation of the latter just similarly always (regardless of the gas rarefaction) is determined by three-particle correlation. And so on up to infinity. Since during time interval t a molecular undergoes ∼ t/τ collisions (with τ being characteristic free-flight time), a correct description of gas evolution over this interval requires taking into account s-particle correlations with at least s < ∼ t/τ . Hence, in practice the whole hierarchy of equations deduced by Bogolyubov [6] is necessary.
In work [7] (and additionally or instead in [8]) and in work [10] approximate solutions to this hierarchy or, in other words, the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) equations [11] were suggested for the problem about random wandering of a test molecule, and explanations were expounded why the Boltzmann's hypothesis is wrong. The aim of the present communication is to prove the statements of preceding paragraph without any approximations. At that we will strengthen the proof and besides simplify it due to replacing the usual gas by ideal gas whose molecules interact with the test molecule only but not with each other.
2.
We want to consider thermal random motion of a test molecule (TM) in thermodynamically equilibrium gas, at that specifying its position R(t) at some initial time moment t = 0 : R(0) = R 0 .
Let P and M denote momentum and mass of TM, m , r j and p j ( j = 1, 2, ... ) denote masses, coordinates and momenta of other molecules, Φ(r) is (short-range repulsive) potential of interaction between any of them and TM, and n is gas density (mean concentration of molecules). At arbitrary time t ≥ 0 , full statistical description of this system is presented by the chain of (k + 1)-particle distribution functions ( k = 0, 1, 2, ... ): F 0 (t, R, P| R 0 ; n ) which is normalized (to unit) density of probability distribution of TM's variables, and
F k (t, R, r (k) , P, p (k) | R 0 ; n ) (where r (k) = {r 1 ... r k } , p (k) = {p 1 ... p k } )
which are probability densities of finding TM at point R with momentum P and simultaneously finding out some k molecules at points r j with momenta p j . A rigorous definition of such distribution functions (DF) was done in [6]. In respect to the coordinates r j they are not normalized, but instead (as in [6]) obey the conditions of weakening of inter-particle correlations under spatial separation of particles. Subject to the symmetry of DF in respect to x j = {r j , p j } these conditions can be compactly written as follows:
F k → F k−1 G m (p k ) at r k → ∞ , where G m (p)
is the Maxwell momentum distribution of a particle with mass m . The enumerated DF satisfy a standard chain of the Bogolyubov equations [6]:
∂F k ∂t = [ H k , F k ] + n ∂ ∂P k+1 Φ ′ (R − r k+1 ) F k+1 (1) ( k = 0, 1, ... ) along with obvious initial conditions F k | t= 0 = δ(R − R 0 ) exp (−H k /T ) = = δ(R − R 0 ) G M (P) k j =1 E(r j − R) G m (p j ) ,(2)
where H k is Hamiltonian of subsystem " k molecules + TM ", k ... = ... dr k dp k , [..., ...] means the Poisson brackets, Φ ′ (r) = ∇Φ(r) , and E(r) = exp [−Φ(r)/T ] . Notice that TM can be considered as a molecule of non-uniformly distributed impurity, and equations (1) are identical to the equations of twocomponent gas [6] in the limit of infinitely rare impurity, when the main component is spatially homogeneous and thermodynamically equilibrium.
Equations (1) together with (2) unambiguously determine evolution of F 0 and eventually probability distribution of TM's displacement R − R 0 . These equations will become more clear if we make a linear change of DF F k by new functions V k with the help of recurrent relations as follow:
F 0 (t, R, P| R 0 ; n) = V 0 (t, R, P| R 0 ; n) , F 1 (t, R, r 1 , P, p 1 | R 0 ; n) = = V 0 (t, R, P| R 0 ; n) f (r 1 − R, p 1 ) + + V 1 (t, R, r 1 , P, p 1 | R 0 ; n) ,
where f (r, p) = E(r) G m (p) , F 2 (t, R, r (2) , P, p (2) |R 0 ; n) = = V 0 (t, R, P|R 0 ; n)f (r 1 − R, p 1 )f (r 2 − R, p 2 ) + + V 1 (t, R, r 1 , P, p 1 | R 0 ; n) f (r 2 − R, p 2 ) + + V 1 (t, R, r 2 , P, p 2 | R 0 ; n) f (r 1 − R, p 1 ) + + V 2 (t, R, r (2) , P, p (2) | R 0 ; n) , and so on. Apparently, from viewpoint of the probability theory, V k represent a kind of cumulants (semi-invariants), or cumulant functions (CF). It is important to notice that zero values of these CF would mean that all conditional DF of gas, F k /F 0 , are independent on initial position R 0 of TM and thus on its displacement R − R 0 . This fact makes visible an interesting specificity of CF V k : they are irreducible correlations of not only current dynamic states of TM and k gas molecules but also correlations of all them with total previous TM's displacement.
In terms of the CF the BBGKY hierarchy acquires a more complicated tridiagonal structure (we omit uninteresting algebraic details):
∂V k ∂t = [H k , V k ] + n ∂ ∂P k+1 Φ ′ (R − r k+1 )V k+1 + + T k j =1 P kj G m (p k ) E ′ (r k − R) P M T + ∂ ∂P V k−1 (4)
Here E ′ (r) = ∇E(r) , and P kj symbolizes transposition of the pairs of arguments x j and x k . On the other hand, initial conditions (2) and the above-mentioned conditions of weakening of correlations [6] take very simple form:
V 0 (0 , R, P| R 0 ; n) = δ(R − R 0 ) G M (P) , V k (0 , R, r (k) , P, p (k) | R 0 ; n) = 0 , V k (t, R, r (k) , P, p (k) |R 0 ; n) → 0 at r j → ∞ (5) ( 1 ≤ j ≤ k )
. Thus, as it should be with cumulants, CF V k disappear under removal of even one of molecules.
From these equations and initial conditions (as well as physical reasonings) it is clear that the reduction to zero in (5) realizes in an integrable way, so that integrals V k = k+1 V k+1 are finite. Let us consider them. By applying the operation k to equations (4) one easy obtains equations
∂V k ∂t = [H k , V k ] + n ∂ ∂P k+1 Φ ′ (R − r k+1 ) V k+1 + + ∂ ∂P k+1 Φ ′ (R − r k+1 ) V k+1 +(6)+ T k j =1 P kj G m (p k ) E ′ (r k − R) P M T + ∂ ∂P V k−1
(with k = 0, 1, ... ). Because of (5) initial conditions to these equations are zero: V k (t = 0) = 0 at any k . Now, in addition to V k , let us consider derivatives of CF in respect to the gas density, V ′ k = ∂V k /∂n . It is easy to see that differentiation of (4) in respect to n yields equations for the V ′ k which exactly coincide with (6) after changing there V k by V ′ k . Besides, in view of (5), initial conditions to these equations again all are zero: V ′ k (t = 0) = 0 at any k ≥ 0 . These observations strictly imply exact equalities
V ′ k = V k , or ∂ ∂n V k (t, R, r (k) , P, p (k) | R 0 ; n) = (7) = k+1 V k+1 (t, R, r (k+1) , P, p (k+1) | R 0 ; n)
This is main formal result of the present paper.
3. The result (7) contains the proof promised in Sec.1. Indeed, equalities (7) show, firstly, that all the manyparticle correlations between gas molecules and past displacement of test molecule (TM) really exist, i.e. differ from zero. Secondly, all they have roughly one and the same order of magnitude. For instance, if comparing their integral values, due to (7) we can write, in natural dimensionless units,
n k 1 ... k V k dP = n k V (k) 0 (t, ∆; n) ∼ c k V 0 (t, ∆; n) , where V 0 (t, ∆; n) = V 0 (t, R, P|R 0 ; n) dP is probabil- ity distribution of the TM's displacement ∆ = R − R 0 , V (k)
0 (t, ∆; n) = ∂ k V 0 (t, ∆; n)/∂n k are its derivatives in respect to gas density n , and c k some numeric coefficients. Hence, all the correlations are equally important, and none of them can be neglected if we aim at knowledge about true statistics of TM's random walk.
For more details, let us suppose that (s + 1)-particle correlation is so insignificant that one can assign V s = 0 in (4). At that, according to (4)-(5), all higher-order correlations also will be rejected. Then, obviously, according to (7), distribution V 0 (t, R, P|R 0 ; n) and thus V 0 (t, ∆; n) must depend on n definitely as an (s − 1)order polynomial. But, from the other hand, distribution V 0 what follows from the truncated chain of equations (4) certainly is absolutely non-polynomial function of n . With taking into account that equalities (7) do express exact properties of solutions to (4)-(5) we see that very deep contradiction is on hand.
This contradiction clearly prompts us that truncation of the BBGKY hierarchy leads to qualitative losses in its solution.
Some possible losses already were characterized in [7] and [8] (and firstly even much earlier in [12]) and in part filled up in [7,10]. Therefore here we confine ourselves (continuing 5-th paragraph of Sec.1) by remark that cutting of the (s + 1)-particle correlation means cutting of s-th and higher statistical moments of fluctuations in relative frequency of TM's collisions with gas molecules (in other words, fluctuations in diffusivity of TM [12]). At s = 2 these fluctuations are completely ignored, and such truncated equations (4) yield a closed equation for V 0 (t, R, P|R 0 ; n) which is equivalent to the Boltzmann-Lorentz equation [11].
It is necessary to emphasize that above reasonings, as well as the exact relations (7), are indifferent to a degree of the gas rarefaction. Consequently, one can state that the Boltzmann-Lorentz equation (moreover, all the classical kinetics including the Boltzmann equation and its generalizations) does not represent a (low-density) limit of the exact statistical mechanical theory. The conventional kinetics is only (more or less adequate or caricature) probabilistic model of exact theory. Of course, in the latter also molecular chaos does prevail. But here it is much more rich, even if speaking about rarefied gas, and does not keep within naive probabilistic logics.
Foundations of the theory of probability. A N Kolmogorov, Chelsea, New YorkA. N. Kolmogorov. Foundations of the theory of proba- bility. Chelsea, New York, 1956.
Works on the foundations of statistical physics. N S Krylov, PrincetonN. S. Krylov. Works on the foundations of statistical physics. Princeton, 1979.
These prejudices "are so habitual that even persons who agreed with our argumentation usually automatically go back to them when facing with a new question. These prejudices "are so habitual that even persons who agreed with our argumentation usually automatically go back to them when facing with a new question " [2].
L Boltzmann, Vorlesungen uber Gastheorie. Bd. 1-2. Leipzig. L. Boltzmann. Vorlesungen uber Gastheorie. Bd. 1-2. Leipzig, 1896-1898.
An introduction to chaos in nonequilibrium statistical mechanics. J R Dorfman, CambridgeJ. R. Dorfman. An introduction to chaos in non- equilibrium statistical mechanics. Cambridge, 1999.
Problems of dynamical theory in statistical physics. N N Bogolyubov, North-HollandN. N. Bogolyubov. Problems of dynamical theory in sta- tistical physics. North-Holland, 1962.
. Yu E Kuzovlev, Sov.Phys. -JETP. 67122469Yu. E. Kuzovlev, Sov.Phys. -JETP 67, No. 12, 2469 (1988).
. Yu E Kuzovlev, arXiv:cond-mat/9903350Yu. E. Kuzovlev, arXiv: cond-mat/9903350 .
relative frequencies of some phenomenon along a given phase trajectory, generally speaking. " , in no way are connected to probabilities " [2" ... relative frequencies of some phenomenon along a given phase trajectory, generally speaking, in no way are connected to probabilities " [2].
. Yu E Kuzovlev, arXiv:cond-mat/0609515612325Yu. E. Kuzovlev, arXiv: cond-mat/0609515, 0612325 .
Classical kinetic theory of fluids. P Resibois, M De Leener, WileyNew-YorkP. Resibois and M. de Leener. Classical kinetic theory of fluids. Wiley, New-York, 1977.
. Yu E Kuzovlev, G N Bochkov, Radiophysics and Quantum Electronics. 263228Yu. E. Kuzovlev and G. N. Bochkov, Radiophysics and Quantum Electronics 26, No. 3, 228 (1983); 27, No.9 (1984).
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[] |
[
"17B69 (primary), 81R10, 81T40, 14D21 (secondary)",
"17B69 (primary), 81R10, 81T40, 14D21 (secondary)"
] |
[
"Chiara Damiolini ",
"ANDAngela Gibney ",
"Nicola Tarasca "
] |
[] |
[] |
Representations of vertex operator algebras define sheaves of coinvariants and conformal blocks on moduli of stable pointed curves. Assuming certain finiteness and semisimplicity conditions, we prove that such sheaves satisfy the factorization conjecture and consequently are vector bundles. Factorization is essential to a recursive formulation of invariants, like ranks and Chern classes, and to produce new constructions of rational conformal field theories and cohomological field theories.MSC2010. 14H10, 17B69 (primary), 81R10, 81T40, 14D21 (secondary).
| null |
[
"https://arxiv.org/pdf/1909.04683v4.pdf"
] | 202,558,729 |
1909.04683
|
99165573ba13b44240bf1b385986462bcc08d3cf
|
17B69 (primary), 81R10, 81T40, 14D21 (secondary)
Chiara Damiolini
ANDAngela Gibney
Nicola Tarasca
17B69 (primary), 81R10, 81T40, 14D21 (secondary)
arXiv:1909.04683v4 [math.AG] 5 Jul 2022and phrases Vertex algebrasconformal blocks and coinvariantsvector bundles on moduli of curvesfactorization and sewing
Representations of vertex operator algebras define sheaves of coinvariants and conformal blocks on moduli of stable pointed curves. Assuming certain finiteness and semisimplicity conditions, we prove that such sheaves satisfy the factorization conjecture and consequently are vector bundles. Factorization is essential to a recursive formulation of invariants, like ranks and Chern classes, and to produce new constructions of rational conformal field theories and cohomological field theories.MSC2010. 14H10, 17B69 (primary), 81R10, 81T40, 14D21 (secondary).
By assigning a module over a vertex operator algebra to each marked point on a stable pointed curve, one can construct dual vector spaces of coinvariants and conformal blocks, giving rise to sheaves on moduli spaces of stable pointed curves. The main result of this paper is that these sheaves satisfy the factorization property (Theorem 7.0.1), as conjectured in [Zhu1,FBZ]. Namely, if certain finiteness and semisimplicity conditions hold, vector spaces of coinvariants and conformal blocks at a nodal curve decompose as products of analogous spaces at each component of its normalization.
We also show that sheaves of coinvariants satisfy the sewing property (Theorem 8.5.1), a refined version of factorization at infinitesimal smoothings of nodal curves. From this and their projectively flat connection on families of smooth curves [DGT1], we deduce they are vector bundles (VB corollary).
Our findings generalize a number of results known in special cases and for low genus. A historical account with references is given in § §0.2 and 0.3.
Factorization leads to recursive formulas for ranks and is used to show that the Chern characters define semisimple cohomological field theories, hence the Chern classes lie in the tautological ring ( [DGT2], building on results for affine Lie algebras from [MOP1, MOP + 2]). A study of these tautological classes may lead to progress on open questions: As proposed by Pandharipande [Pan], a computation of such Chern classes independently of the projective flatness of the connection [DGT1] would yield relations in the tautological ring, and could be used to test Pixton's conjecture [Pix, Jan].
For sheaves defined by integrable modules over affine Lie algebras, vector spaces of conformal blocks are canonically isomorphic to generalized theta functions (see [LS] and references therein). When in addition the genus is zero, vector bundles of such coinvariants are globally generated [Fak], hence their Chern classes have positivity properties. For instance, first Chern classes are base-point-free and thus give rise to morphisms, some with images having modular interpretations [GJMS1,GJMS2]. It is natural to expect that the more general vector bundles of coinvariants and conformal blocks studied here can be shown to have analogous properties under appropriate assumptions. This has been supported by a preliminary investigation [DG1].
To outline our results, we set some notation. We refer to a stable pointed coordinatized curve as a triple (C, P • , t • ) where (C, P • ) is a stable n-pointed curve, P • = (P 1 , . . . , P n ), and t • = (t 1 , . . . , t n ) with t i a formal coordinate at the point P i . Let M • = (M 1 , . . . , M n ) be an n-tuple of finitely generated admissible modules over a vertex operator algebra V (see § §1.1-1.2). When C \ P • is affine, the vector space of coinvariants V(V ; M • ) (C,P•,t•) is defined as the largest quotient of the tensor product ⊗ n i=1 M i by the action of a Lie algebra determined by V and (C, P • , t • ), see §4.2. In general, by adding more marked points, one can reduce to the case when C \ P • is affine, see (30). Before now, two Lie algebras have been used to define coinvariants: Zhu's Lie algebra g C\P• (V ) and the (former) chiral Lie algebra L C\P• (V ). Here, we introduce a new chiral Lie algebra L C\P• (V ).
Zhu's Lie algebra ( §A.1) is defined when the vertex algebra V is quasiprimary generated and Z ≥0 -graded with lowest degree space of dimension one, for either fixed smooth curves [Zhu1,AN], or for rational stable pointed curves with coordinates [NT]. To show that g C\P• (V ) is a Lie algebra, Zhu uses that any fixed smooth algebraic curve admits an atlas such that all transition functions are Möbius transformations. Transition functions between charts on families of curves of arbitrary genus are more complicated, and for an arbitrary vertex operator algebra, g C\P• (V ) is not well-defined.
The chiral Lie algebra L C\P• (V ) is available for curves of arbitrary genus and for more general vertex operator algebras V , not necessarily quasiprimary generated. Defined for smooth pointed curves by Frenkel and Ben-Zvi [FBZ,§19.4.14] and shown in [FBZ] to coincide with that studied by Beilinson and Drinfeld [BD], the chiral Lie algebra was extended to nodal curves in [DGT1], enabling the construction of coinvariants on such curves.
In §3, we introduce and study a new chiral Lie algebra L C\P• (V ), whose coinvariants are projectively flat over M g,n , and additionally satisfy factorization under some natural hypotheses. See §3.3 for a description of L C\P• (V ) on nodal curves.
In [NT], Nagatomo and Tsuchiya remark that the coinvariants on rational curves with coordinates using Zhu's Lie algebra are equivalent to those considered by Beilinson and Drinfeld. In Proposition A.2.1, we verify their genus zero statement, and further show that coinvariants from the chiral Lie algebra are isomorphic to those given by g C\P• (V ) whenever Zhu's Lie algebra is defined. For instance, when V is quasi-primary generated and one works over a family of curves admitting an atlas where all transition functions are Möbius transformations, both perspectives are equivalent.
To state our main result, let (C, P • , t • ) be a stable pointed coordinatized curve, as above, with one node Q. Let ‹ C → C be the normalization, Q • := (Q + , Q − ) the pair of preimages of Q in ‹ C, and s • := (s + , s − ) with s ± a formal coordinate at Q ± . Let W be a set of representatives of isomorphism classes of simple V -modules, and for W ∈ W , let W ′ be its contragredient module ( §1.8).
Factorization theorem (Theorem 7.0.1). Let V be a rational, C 2 -cofinite vertex operator algebra with one-dimensional weight zero space, and let M • be an n-tuple of finitely-generated admissible V -modules. One has:
(1) V (V ; M • ) (C,P•,t•) ∼ = W ∈W V V ; M • ⊗ W ⊗ W ′ ( ‹ C,P•⊔Q•,t•⊔s•) .
If ‹ C = C + ⊔ C − is disconnected, with Q ± ∈ C ± , then (1) is isomorphic to
W ∈W V V ; M • + ⊗ W X + ⊗ V V ; M • − ⊗ W ′ X −
where X ± := (C ± , P • | C ± ⊔ Q ± , t • | C ± ⊔ s ± ), and M • ± are the modules at those points P • that are in C ± .
The isomorphism giving the factorization theorem is constructed in §7. Propositions 3.3.1, 5.1.1, and 6.2.1 are at the heart of this work, and arise from the construction in §2 of the sheaf V C globalizing a vertex algebra V over a nodal curve C, integral to the definition of the chiral Lie algebra. The sheaf V C is a slight variant on the sheaf V C that we defined in [DGT1]. The two sheaves coincide for smooth curves, and as we explain here, we recover the results of [DGT1] using V C in place of V C . So while the projectively flat connection can be obtained in both constructions (see Proposition 2.7.3 for V C ), to have factorization, we have used V C (see §2.5, and §2.6). In Proposition 3.3.1, we explicitly describe the chiral Lie algebra on a nodal curve in terms of elements of the chiral Lie algebra on its normalization. This involves stable k-differentials that satisfy an infinite number of identities.
In Proposition 5.1.1, we show that vector spaces of coinvariants defined from the chiral Lie algebra and smooth curves of arbitrary genus are finitedimensional. Known to be true in special cases, this result is a natural generalization of work of Abe and Nagatomo [AN] for coinvariants defined from Zhu's Lie algebra and smooth curves with formal coordinates (see §0.3 for the history of the problem). As in [AN], we assume here that V is C 2 -cofinite, which implies C k -cofiniteness for k ≥ 1 [Buh, KL].
Proposition 6.2.1 reinterprets coinvariants at a nodal curve as coinvariants on the normalization by the action of a Lie subalgebra of the chiral Lie algebra. The proof of this result, for which we assume V is C 1 -cofinite, uses that lowest weight V -modules admit spanning sets of PBW-type, following [KL,Cor. 3.12].
In §8.7, following Tsuchimoto [Tsu] in the case of simple affine vertex algebras, we show the sheaf of coinvariants V (V ; M • ) is coordinate-free and descends to the moduli space M g,n of stable n-pointed curves of genus g. As an application of the factorization theorem, we show:
VB corollary. Let V be a rational, C 2 -cofinite vertex operator algebra with one-dimensional weight zero space, and M • be an n-tuple of finitelygenerated admissible V -modules. Then V (V ; M • ) is a vector bundle of finite rank on M g,n .
The proof of the VB corollary is presented in §8.8. The result on the interior of M g,n , the moduli space M g,n of smooth pointed curves, follows from finite-dimensionality of coinvariants (Proposition 5.1.1) and the existence of a projectively flat connection [DGT1]. Two ingredients are needed to give VB corollary on the whole space M g,n : Theorem 8.4.2, a more general result on finite-dimensionality of coinvariants, and the sewing property (Theorem 8.5.1). Each of these involve evaluation of the sheaf of coinvariants on a family formed by infinitesimally smoothing a nodal curve ( §8.1). The proof of the sewing theorem ( §8) relies on the factorization theorem and a sewing procedure originally found in [TUY,§6.2].
0.1. Results on sheaves of coinvariants from [DGT1]. As stated, the vertex algebra sheaf V C defined in §2 is a slight variant on the sheaf V C defined in [DGT1]. These give rise to sheaves of chiral Lie algebras L C\P• (V ) and L C\P• (V ), respectively, which in turn define two sheaves of coinvariants on the moduli space of curves, potentially different on the boundary. As these sheaves agree on the interior M g,n , i.e., the moduli space of smooth pointed curves, the results of [DGT1] on M g,n hold for the sheaves of coinvariants defined here. In particular, the identification of the Atiyah algebra acting on sheaves of coinvariants on M g,n [DGT1,Theorem], and consequently, the computation of their Chern character on M g,n [DGT1,Corollary] hold here. Combining these results with the factorization theorem, we show in [DGT2] that the sheaves of coinvariants defined here in the case of self-contragredient simple vertex algebras give rise to semisimple cohomological field theories, thus allowing one to determine their Chern character on M g,n in terms of the fusion rules.
A natural question is whether the two sheaves of coinvariants coincide on the whole of M g,n . In particular, it is natural to ask whether sheaves of coinvariants defined by L C\P• (V ) satisfy factorization. 0.2. History of factorization and sewing. Tsuchiya and Kanie used integrable modules at a fixed level over affine Lie algebras to form spaces of coinvariants on smooth pointed rational curves with coordinates [TK]. Generalized by Tsuchiya, Ueno, and Yamada to moduli of stable pointed coordinatized curves of arbitrary genus, these sheaves were shown to satisfy a number of good properties including factorization and sewing [TUY]. Tsuchimoto [Tsu] proved the bundles are independent of coordinates and descend to M g,n . Beilinson, Feigin, and Mazur [BFM] showed that factorization holds for coinvariants defined by modules over Virasoro algebras. Our arguments follow [NT] in the genus zero case after our study of the chiral Lie algebra allows one to replace Zhu's Lie algebra in the general case.
Sewing was proved for g ∈ {0, 1} by Huang [Hua1,Hua2,Hua4,Hua5]. His approach was to prove the operator product expansion and the modular invariance of intertwining operators, both conjectured by Moore and Seiberg [MS]. Huang assumes that (i) V = ⊕ i≥0 V i with V 0 ∼ = C; (ii) Every Ngradable weak V -module is completely reducible; and (iii) V is C 2 -cofinite. Our assumptions are (i); (ii') V is rational; and (iii). Conditions (ii) and (iii) of Huang are equivalent to our conditions (ii') and (iii) (see §1.7 for more details). Huang shows that if one assumes in addition (iv) V ∼ = V ′ , then the modular tensor categories he constructs for g ∈ {0, 1} are rigid and nondegenerate. In case (i)-(iv) hold, V is sometimes called strongly rational.
Codogni in [Cod] proves factorization in case our hypotheses hold, with the additional assumptions (iv') V ∼ = V ′ and (v) V has no nontrivial modules. Like Nagatomo and Tsuchiya, Codogni works with coinvariants defined on the moduli space of curves with formal coordinates. We note that the additional assumption (iv') gives that V is quasi-primary generated and in particular, coinvariants defined from Zhu's Lie algebra and the chiral Lie algebra are both well-defined and agree (see §A).
Here we do not assume condition (iv) nor condition (v). There are examples satisfying (1-3) but not (v), see §9. Furthermore, through private communications with Sven Möller, we have learned the existence of a vertex operator algebra that satisfies our conditions (1-3) but not condition (iv). Therefore, while establishing factorization for all g ≥ 0, our work also covers new examples of factorization for g ∈ {0, 1}. 0.3. History of coherence. In [FBZ,page 3 and §5.5.4] the authors single out rational vertex algebras as good candidates for defining finite-dimensional coinvariants (and hence leading to finite-rank vector bundles). At that time, rationality and C 2 -cofiniteness were conjectured to be equivalent conditions [DLM2,ABD]. This has been disproved, as Abe has given a C 2 -cofinite, non-rational vertex operator algebra [Abe].
Coherence of sheaves of coinvariants was shown previously in special cases: for (1) integrable modules at positive integral levels over affine Lie algebras [TUY]; (2) modules over C 2 -cofinite Virasoro vertex algebras [BFM]; (3) curves of genus zero and modules over C 2 -cofinite vertex operator algebras NT]; (4) fixed smooth curves of positive genus and modules over a quasi-primary generated, C 2 -cofinite vertex operator algebras V = ⊕ i∈N V i such that V 0 ∼ = C [AN]. After the first draft of this paper, [vEH] showed finite-dimensionality of spaces of coinvariants associated with trivial modules over elliptic curves under a weaker assumption than C 2 -cofiniteness. In [DG1] it is shown that if V is generated in degree 1 and g = 0, then coinvariants are finite-dimensional.
V = ⊕ i∈N V i such that V 0 ∼ = C [
1. Background 1.1. Vertex operator algebras. We work with a non-negatively graded vertex operator algebra, that is, a 4-tuple V, 1 V , ω, Y (·, z) , throughout simply denoted V for short, such that:
(i) V = ⊕ i∈Z ≥0 V i is a vector space over C with dim V i < ∞; (ii) 1 V ∈ V 0 (the vacuum vector ), and ω ∈ V 2 (the conformal vector ); (iii) Y (·, z) : V → End(V ) z, z −1 is a linear function assigning to every element A ∈ V the vertex operator Y (A, z) := i∈Z A (i) z −i−1 .
The datum V, 1 V , ω, Y (·, z) must satisfy the following axioms:
(a) (vertex operators are fields) for all A, B ∈ V , A (i) B = 0, for i ≫ 0; (b) (vertex operators of the vacuum) Y (1 V , z) = id V : 1 V (−1) = id V and 1 V (i) = 0, for i = −1, and for all A ∈ V , Y (A, z)1 V ∈ A + zV z : A (−1) 1 V = A and A (i) 1 V = 0, for i ≥ 0; (c) (weak commutativity) for all A, B ∈ V , there exists an N ∈ Z ≥0 such that (z − w) N [Y (A, z), Y (B, w)] = 0 in End(V ) z ±1 , w ±1 ; (d) (conformal structure) for Y (ω, z) = i∈Z ω (i) z −i−1 , ω (p+1) , ω (q+1) = (p − q) ω (p+q+1) + c 12 δ p+q,0 (p 3 − p) id V .
Here c ∈ C is the central charge of V . Moreover:
ω (1) | V i = i · id V , for all i, and Y ω (0) A, z = ∂ z Y (A, z).
1.1.1. Action of Virasoro. As we next explain, the conformal structure encodes an action of the Virasoro (Lie) algebra Vir on V . The Witt (Lie) algebra Der K represents the functor assigning to a C-algebra R the Lie algebra Der K(R) := R((z))∂ z generated over R by the derivations L p := −z p+1 ∂ z , for p ∈ Z, with relations [L p , L q ] = (p − q)L p+q . The Virasoro (Lie) algebra Vir represents the functor assigning to R the Lie algebra generated over R by a formal vector K and the elements L p , for p ∈ Z, with Lie bracket given by
[K, L p ] = 0, [L p , L q ] = (p − q)L p+q + K 12 (p 3 − p)δ p+q,0 . Setting L p = ω (p+1) ∈ End(V ), axiom (d)
gives an action of Vir on V with central charge c ∈ C, that is, K ∈ Vir acts as c · id V . 1.1.2. Degree of A (i) . As a consequence of the axioms, one has A (i) V k ⊆ V k+d−i−1 for homogeneous A ∈ V d (see e.g., [Zhu2]). Thus, we have
(2) deg A (i) := deg(A) − i − 1, for homogeneous A in V .
Axiom (d) implies that L 0 acts as a degree operator on V , and L −1 , called the translation operator, is determined by
L −1 A = A (−2) 1 V , for A ∈ V .
1.2. Modules of vertex operator algebras. Let V, 1 V , ω, Y (·, z) be a vertex operator algebra. A weak V -module is a pair M, Y M (·, z) , where:
(i) M is a vector space over C; (ii) Y M (·, z) : V → End(M ) z, z −1 is a linear function that assigns to A ∈ V an End(M )-valued vertex operator Y M (A, z) := i∈Z A M (i) z −i−1 .
The pair M, Y M (·, z) must satisfy the following axioms:
(a) for all A ∈ V and v ∈ M , one has A M (i) v = 0, for i ≫ 0; (b) Y M 1 V , z = id M ; (c) for all A, B ∈ V , there exists N ∈ Z ≥0 such that for all v ∈ M one has (z − w) N î Y M (A, z), Y M (B, w) ó v = 0; (d) for all A ∈ V and v ∈ M , there exists N ∈ Z ≥0 , such that for all B ∈ V one has (w + z) N Ä Y M (Y (A, w)B, z) − Y M (A, w + z)Y M (B, z) ä v = 0; (e) For Y M (ω, z) = i∈Z ω M (i) z −i−1 , one has î ω M (p+1) , ω M (q+1) ó = (p − q) ω M (p+q+1) + c 12 δ p+q,0 (p 3 − p) id M , where c ∈ C is the central charge of V . We identify ω (p+1) ∈ End(M ) with an action of L p ∈ Vir on M . Moreover Y M (L −1 A, z) = ∂ z Y M (a, z).
In the literature, axiom (c) is referred to as weak commutativity, and axiom (d) as weak associativity. Weak associativity and weak commutativity are known to be equivalent to the Jacobi identity (see e.g., [DL,FHL,LL,Li1]). Moreover, by [DLM2,Lemma 2.2], axiom (e) is redundant.
Throughout, by V -modules we mean admissible V -modules. These are weak V -modules such that
(f) M = ⊕ i∈N M i is N-graded and A M (i) M k ⊆ M k+deg(A)−i−1 for every homo- geneous A ∈ V .
Note that V is a module over itself (see [LL,Thm 3.5.4] or [FBZ,§3.2.1]). We give a description of V -modules in §1.5.
1.3. The Lie algebra ancillary to V . Given a formal variable t, we call
L t (V ) := V ⊗ C((t)) Im ∂ the Lie algebra L(V ) = L t (V ) ancillary to V . Here (3) ∂ := L −1 ⊗ id C((t)) + id V ⊗ ∂ t . The space L(V ) is spanned by series of type i≥i 0 c i A [i] , for A ∈ V , c i ∈ C, and i 0 ∈ Z, where A [i] denotes the projection in L(V ) of A ⊗ t i ∈ V ⊗ C((t)). The Lie bracket of L(V ) is induced by (4) A [i] , B [j] := k≥0 Ç i k å A (k) · B [i+j−k] .
The axiom on the vacuum vector 1 V implies that 1 V [−1] is central. The Lie algebra L(V ) is isomorphic to the current Lie algebra in [NT]. For t we will use a formal coordinate at a point P on a curve, and we will denote L P (V ) = L t (V ). A coordinate-free description of L t (V ) is discussed in §2.8.
1.4. The universal enveloping algebra. For a vertex algebra V , there is a complete topological associative algebra U(V ), defined originally by Frenkel and Zhu [FZ]. We review it here following the presentation in [FBZ]. Consider the universal enveloping algebra U (L(V )) of L(V ), and its completion
‹ U (L(V )) := lim ← − N U (L(V ))/I N ,
where I N is the left ideal generated by A [i] , for A ∈ V , and i > N . The universal enveloping algebra U(V ) of V is defined as the quotient of ‹ U (L(V )) by the two-sided ideal generated by the Fourier coefficients of the series
Y A (−1) B, z − : Y [A, z] Y [B, z] :, for all A, B ∈ V, where Y [A, z] = i∈Z A [i]
z −i−1 and ": :" is the normal ordering (see [FLM2]).
For an affine vertex algebra V = V ℓ (g), one has U(V ) is isomorphic to a completion ‹ U ℓ ( g) of U ℓ ( g), for all ℓ ∈ C. Here, U ℓ ( g) is the quotient of U ( g) by the two-sided ideal generated by K − ℓ, where K ∈ g/g ⊗ C((t)), and
‹ U ℓ ( g) := lim ← − N U ℓ ( g)/U ℓ ( g) · g ⊗ t N C t .
For more detail and other examples see [FBZ,§4.3.2, §5.1.8].
1.5. Action on V -modules. Both L(V ) and U(V ) act on any V -module M via the Lie algebra homomorphism L(V ) → End(M ) obtained by map-
ping A [i] to the Fourier coefficient A (i) of the vertex operator Y M (A, z) = i A (i) z −i−1 . Thus, the series i≥i 0 c i A [i] acts on a V -module M via Res z=0 Y M (A, z) i≥i 0 c i z i dz.
An L(V )-module need not be a V -module. On the other hand, there is an equivalence between the categories of weak V -modules and smooth U(V )modules (see [FBZ,§5.1.6]
). A U(V )-module M is smooth if for any w ∈ M and A ∈ V , one has A [i] w = 0 for i ≫ 0.
A V -module is finitely generated if it is finitely generated as a U (V )module. The modules in [NT] are also finitely generated and admissible.
1.6. Correspondence between V -modules and A(V )-modules. A Vmodule W is irreducible, or simple, if it is non zero and it has no subrepresentation other than the trivial representation 0 and W itself. We review here Zhu's associative algebra A(V ), and the one-to-one correspondence between isomorphism classes of finite-dimensional simple A(V )-modules and isomorphism classes of simple V -modules [Zhu2].
Zhu's algebra is the quotient A(
V ) := V /O(V ), where O(V )
is the subspace of V linearly spanned by elements of the form
Res z=0 (1 + z) deg A z 2 Y (A, z)B, where A is homogeneous in V . The image of an element A ∈ V in A(V ) is denoted by o(A). The product in A(V ) is defined by o(A) * o(B) = Res z=0 (1 + z) deg A z Y (A, z)B,
for homogeneous A in V . Nagatomo and Tsuchiya [NT] consider an isomorphic copy of A(V ), which they refer to as the zero-mode algebra. Given a V -module W = i≥0 W i , one has that W 0 is an A(V )-module [Zhu2,Thm 2.2.2]. The action of A(V ) on W 0 is defined as follows: an
element o(A) ∈ A(V ), image of a homogeneous element A ∈ V , acts on W 0 as the endomorphism A (deg A−1) , a Fourier coefficient of Y W (A, z).
For the other direction, recall the triangular decomposition:
(5) L(V ) = L(V ) <0 ⊕ L(V ) 0 ⊕ L(V ) >0 , determined by the degree deg(A [i] ) := deg(A)−i−1, for homogeneous A ∈ V .
From the Lie bracket (4) of L(V ), one checks that each summand above is a Lie subalgebra of L(V ). This induces a subalgebra
U(V ) ≤0 of U(V ). Given a finite-dimensional A(V )-module E, the generalized Verma U(V )- module is M (E) := U(V ) ⊗ U(V ) ≤0 E.
To make E into an U(V ) ≤0 -module, one lets L(V ) <0 act trivially on E, and L(V ) 0 act by the homomorphism of Lie algebras L
(V ) 0 ։ A(V ) Lie induced by the identity endomorphism of V [Li2, Lemma 3.2.1]. For homogeneous A ∈ V k , the image of the element A [k−1] ∈ L(V ) 0 in A(V ) is o(A). By construction, M (E) is automatically a V -module. Given an irreducible V -module W = i≥0 W i , the space W 0 is a finite- dimensional irreducible A(V )-module; conversely, given a finite-dimensional irreducible A(V )-module E, there is a unique maximal proper V -submodule N (E) of the V -module M (E) with N (E) ∩ E = 0 such that L(E) = M (E)/N (E) is an irreducible V -module [Zhu2].
1.7. Rational vertex algebras. A vertex algebra V is rational if every finitely generated V -module is a direct sum of irreducible V -modules. A rational vertex algebra has only finitely many isomorphism classes of irreducible modules, and an irreducible module M = i≥0 M i over a rational vertex algebra satisfies dim M i < ∞ [DLM3].
An ordinary V -module is a weak V -module which carries a C-grading M = λ∈C M λ such that: L 0 | M λ = λ id M λ ; dim M λ < ∞; and for fixed λ, one has M λ+n = 0, for integers n ≪ 0. Ordinary V -modules are admissible, and when V is rational, every simple admissible V -module is ordinary [DLM3], [DLM2,Rmk 2.4]. It follows that for rational V , finitely generated V -modules are direct sums of simple ordinary V -modules. In particular, a finitely generated admissible module M = i≥0 M i over a rational vertex algebra satisfies dim M i < ∞, for all i, and L 0 acts semisimply on such M .
When V is rational, the associative algebra A(V ) is semisimple [Zhu2]. For a rational vertex algebra V , given a simple module E over Zhu's algebra A(V ), the Verma module M (E) remains simple. In general, Verma modules are not necessarily simple, but they are indecomposable. Hence complete reducibility implies that simple and indecomposable coincide.
1.8. Dual modules. Isomorphism classes of simple V -modules and simple A(V )-modules are in correspondence ( §1.5). Here we describe the V -module (and corresponding A(V )-module) structures on their duals.
1.8.1. Duality for V -modules. Let V be a vertex algebra. Following [FHL,§5.2
], given a V -module M = ⊕ i≥0 M i , Y M (−, z) , its contragredient module is Ä M ′ , Y M ′ (−, z) ä , where M ′ is the graded dual of M , that is, M ′ := ⊕ i≥0 M ∨ i , with M ∨ i := Hom C (M i , C), and Y M ′ (−, z) : V → End M ′ z, z −1
is the unique linear map determined by
(6)¨Y M ′ (A, z)ψ, m ∂ =¨ψ, Y M Ä e zL 1 (−z −2 ) L 0 A, z −1 ä m ∂ for A ∈ V , ψ ∈ M ′ , and m ∈ M .
Here and throughout, ·, · is the natural pairing between a vector space and its graded dual.
Duality for
A(V )-modules. For a V -module M , the A(V )-module struc- ture on M ∨ 0 requires the involution ϑ : L(V ) f → L(V ) f induced from (7) ϑ A [j] := (−1) a−1 i≥0 1 i! L i 1 A [2a−j−i−2]
for a homogeneous element A ∈ V of degree a. Here L(V ) f ⊂ L(V ) denotes the quotient of V ⊗ C C[t, t −1 ] by Im ∂. Observe that since the operator L 1 has negative degree, the above sum is finite, and that ϑ is a Lie algebra homomorphism [NT,Prop 4.1.1] (note the sign difference between ϑ used here and the one in [NT]). This involution appeared in [Bor], and it naturally arises from the action of the vertex operators on the contragredient module V ′ . Since ϑ restricts to an involution of L(V ) 0 leaving O(V ) invariant, it induces an involution on Zhu's algebra A(V ). The following statement is a direct consequence of the definition of contragredient modules.
Lemma 1.8.3. i) The image of ψ ∈ M ∨ 0 under the action of σ ∈ L(V ) 0 is the linear functional σ · ψ = −ψ • ϑ(σ). ii) The image of ψ ∈ M ∨ 0 under the action of o(A) ∈ A(V ) is o(A) · ψ = −ψ • ϑ(o(A)).
For homogeneous A ∈ V of degree a and for m ∈ M 0 , this is
o(A) · ψ, m = (−1) a ∞ ψ, i≥0 1 i! L i 1 A (a−i−1) m ∫ .
See [FBZ,§10.4.8] for a geometric realization of the involution ϑ. In §3, we use ϑ to describe chiral Lie algebras on nodal curves.
1.9. Stable k-differentials. Let (C, P • ) be a stable n-pointed curve with at least one node, and let ω C be the dualizing sheaf on C. We review here (stable) k-differentials on C, that is, sections of ω ⊗k C , for an integer k. When k ≥ 1, by ω ⊗−k C we mean (ω ∨ C ) ⊗k . Let ‹ C → C be the partial normalization of C at a node Q, let Q + , Q − ∈ ‹ C be the two preimages of Q, and set Q • = (Q + , Q − ). Note that the curve ‹ C may not be connected. Let s + and s − be formal coordinates at the points Q + and Q − , respectively. We write Q ± to denote either point, and similarly use s ± to denote either formal coordinate. For a section
µ ∈ H 0 Ä ‹ C \ P • ⊔ Q • , ω ⊗k ‹ C ä =: Γ,
let µ Q ± ∈ C((s ± ))(ds ± ) k be the Laurent series expansion of µ at Q ± , that is, the image of µ under the restriction morphism
H 0 Ä ‹ C \ P • ⊔ Q • , ω ⊗k ‹ C ä → H 0 Ä D × Q ± , ω ⊗k ‹ C ä ≃ s ± C((s ± ))(ds ± ) k .
Here D × Q ± is the punctured formal disk about Q ± , that is, the spectrum of the field of fractions of the completed local ring " O Q ± , and ≃ s ± denotes the isomorphism given by fixing the formal coordinate s ± at Q ± .
For a k-differential µ, define the order ord Q ± (µ) of µ at the point Q ± as the highest integer m such that µ Q ± ∈ s m ± C s ± (ds ± ) k . For a k-differential µ with ord Q ± (µ) ≥ −k, the k-residue Res k Q ± (µ) of µ at the point Q ± is defined as the coefficient of s −k ± (ds ± ) k in µ Q ± . The order and the k-residue at Q ± are independent of the formal coordinate s ± at Q ± . Here we only require the case ord Q ± (µ) ≥ −k. For the definition of the k-residue without the assumption ord Q ± (µ) ≥ −k, see e.g., [BCG + ]. Lemma 1.9.1. Assume that C \ P • is affine. For all integers k, one has
η * H 0 Ä C \ P • , ω ⊗k C ä = µ ∈ Γ ord Q ± (µ) ≥ −k, Res k Q + (µ) = (−1) k Res k Q − (µ) .
Proof. It is enough to prove the statement for k ∈ {−1, 0, 1}: indeed, for negative integers k, sections of ω ⊗k C on the affine C \ P • are tensor products of sections of ω −1 C , and the Laurent series expansions of sections of ω ⊗k ‹ C at Q + and Q − are obtained as tensors of the Laurent series expansions of sections of ω −1 ‹ C at Q + and Q − , respectively. An analogous argument may be made in case k is a positive integer.
When k = 1, the statement is about sections of ω C , and by definition sections of ω C are sections of ω ‹ C with at most simple poles at Q + and Q − such that Res Q + (µ) + Res Q − (µ) = 0. When k = 0, the statement is about sections of O C , and indeed a regular function on C is a regular
function µ on ‹ C such that µ(Q + ) = µ(Q − ). When k = −1, by definition we have ω −1 C = H om O C (ω C , O C )
, and the statement follows from a direct computation using the cases k ∈ {0, 1}. We have used the notation η * to denote that the identification of elements
of H 0 Ä C \ P • , ω ⊗k C ä
with elements of Γ is naturally induced by the map η. We will use the same notation in the statement of Proposition 3.3.1.
1.10. A consequence of Riemann-Roch. We will frequently use the following corollary of the Riemann-Roch theorem. Let C be a smooth curve, possibly disconnected, with two non-empty sets of distinct marked points P • = (P 1 , . . . , P n ) and Q • = (Q 1 , . . . , Q m ). Assume that each irreducible component of C contains at least one of the marked points P
• , so that C \ P • is affine. Let s i be a formal coordinate at the point Q i , for each i. Fix an integer k. For all integers d and N , there exists µ ∈ H 0 Ä C \ P • ⊔ Q • , ω ⊗k C ä
such that its Laurent series expansions at the points Q • satisfy:
µ Q i ≡ s d i (ds i ) k ∈ C((s i ))(ds i ) k /s N i C s i (ds i ) k , for a fixed i, µ Q j ≡ 0 ∈ C((s j ))(ds j ) k /s N j C s j (ds j ) k , for all j = i.
This statement has appeared for instance in [Zhu1].
Sheaves of vertex algebras on curves
We describe the sheaf of vertex algebras V C on a curve C with (at worst) simple nodal singularities in §2.5, and its flat connection in §2.7. This yields a coordinate-free description of the Lie algebra ancillary to V in §2.8.
2.1. The group scheme Aut O. Consider the functor which assigns to a C-algebra R the group:
Aut O(R) = z → ρ(z) = a 1 z + a 2 z 2 + · · · | a i ∈ R, a 1 a unit
of continuous automorphisms of the algebra R z preserving the ideal zR z . The group law is given by composition of series: ρ 1 ·ρ 2 := ρ 2 •ρ 1 . This functor is represented by a group scheme, denoted Aut O.
To construct the sheaf of vertex algebras V C on a stable curve C, we describe below the principal (Aut O)-bundle A ut C → C, and actions of Aut O on the vertex operator algebra V and on A ut C × V .
2.2. Coordinatized curves. Given a flat family C → S of stable curves of genus g, we construct an (Aut O)-torsor A ut C/S → C/S in §2.2.2. This torsor is pulled back from a universal (Aut O)-torsor Ë M g,1 → M g,1 . To define these objects, we begin in §2.2.1 with S = Spec(C).
2.2.1. The principal (Aut O)-bundle A ut C at a fixed curve C. Assume first that C is a smooth curve. Let A ut C be the infinite-dimensional smooth variety whose points consist of pairs (P, t), where P is a point in C, and t is a formal coordinate at P (see [ADCKP]). A formal coordinate t at P is an element of the completed local ring "
O P such that t ∈ m P \ m 2 P , where m P is the maximal ideal of " O P .
There is a natural forgetful map A ut C → C, with fiber at a point P ∈ C equal to the set of formal coordinates at P :
A ut P = ¶ t ∈ " O P | t ∈ m P \ m 2 P © .
The group scheme Aut O acts on fibers of A ut C → C by change of coordinates:
A ut C × Aut O → A ut C , ((P, t), ρ) → (P, t · ρ := ρ(t)) .
This action is simply transitive, thus
A ut C is a principal (Aut O)-bundle on C.
The choice of a formal coordinate t at P gives rise to the trivialization
Aut O ≃t −→ A ut P , ρ → ρ(t).
For a nodal curve C, let ‹ C → C denote the normalization of C. Assume for simplicity that C has only one node Q, and let Q + and Q − be the two preimages in ‹ C of Q. A choice of formal coordinates s + and s − at Q + and Q − , respectively, determines a smoothing of the nodal curve C over Spec(C q ) such that, locally around the point Q in C, the family is defined by s + s − = q (see §8.1 for more details). A principal (Aut O)-bundle on a nodal curve is equivalent to the datum of a principal (Aut O)-bundle on its normalization together with a gluing isomorphism between the fibers over the two preimages of each node. In particular, one constructs the principal (Aut O)-bundle A ut C on C from the principal (Aut O)-bundle A ut ‹ C on ‹ C by identifying the fibers at the two preimages Q + and Q − of the node Q in C by the following gluing isomorphism:
(8) A ut Q + ≃ s + Aut O ∼ = − → Aut O ≃ s − A ut Q − , ρ(s + ) → ρ • γ(s − ), where γ ∈ Aut O is defined as (9) γ(z) := 1 1 + z − 1 = −z + z 2 − z 3 + · · · .
Note that (γ • γ)(z) = z, hence (8) determines an involution of Aut O. The isomorphism (8) is induced from the identification s + = γ(s − ).
2.2.2.
A ut C/S on a family C → S and the moduli space Ë M g,1 . The above construction of A ut C can be carried out in families, and for a flat family C → S of stable curves, one thus obtains a principal (Aut O)-bundle A ut C/S → C/S. This determines a principal (Aut O)-bundle Ê C g → C g , where C g → M g is the universal curve over the moduli space of stable curves of genus g. One has the following diagram made of fiber product squares:
A ut C/S Ê C g C C g S M g . Aut O
It can be convenient to identify the universal curve C g with the moduli space of stable pointed curves M g,1 . This leads us to consider a principal
(Aut O)-bundle Ë M g,1 → M g,1 identified with Ê C g → C g . Namely:
Definition 2.2.3. Let Ë M g,1 be the moduli space of coordinatized stable pointed curves. An object over a scheme S consists of a semistable curve C → S with a section P : S → C mapping to the smooth locus of C, together with a formally unramified thickening S × Spf(C t ) → C of the section P , such that every genus one component has at least one special point and every rational component has at least three special points.
Here a special point is either a marked point or a node, counted with multiplicity. Moreover, Spf(C t ) is the formal spectrum of the complete topological ring C t [FBZ, §A.1.1].
The action of Aut O via change of coordinates identifies Ë M g,1 as a principal (Aut O)-bundle over M g,1 . One has the diagram of Cartesian squares:
A ut P/S A ut C/S Ë M g,1 S C M g,1 S M g . Aut O P
In particular, for a flat family C → S of stable curves of genus g, the space A ut C/S is the pull-back of Ë M g,1 via the moduli map S → M g .
For a nodal curve C, the fiber of A ut C over a node can be described as follows. Assume for simplicity that C has only one node Q. Over the singular point Q ∈ C, the fiber A ut Q can be identified with the space of formal coordinates at the point Q ′ in C ′ , where (C ′ , Q ′ ) is formed by stabilization of the unstable pointed curve (C, Q). Stabilization is carried out by blowing-up. As a result, C ′ consists of the partial normalization ‹ C of C at Q together with a rational exceptional component meeting ‹ C transversally at the two preimages Q + and Q − ∈ ‹ C of the node Q. Such a rational component contains three special points: the two attaching points and a point labelled Q ′ . Up to isomorphism, one can identify this rational component with P 1 , with the points attached to Q + and Q − identified with 0 and ∞ ∈ P 1 , respectively, and the point Q ′ identified with 1 ∈ P 1 . The fiber A ut Q of A ut C over the node Q in C is identified with the space of formal coordinates at the point 1 in P 1 ⊂ C ′ .
Here are more details: Choose formal coordinates s + and s − at Q + and Q − , respectively. As mentioned in the previous section, a choice of such coordinates determines a smoothing of C over Spec(C q ) such that, locally around the point Q in C, the family is defined by s + s − = q. After blowingup such a family at the point Q, locally around the two resulting nodes incident to the exceptional component P 1 , the curve C ′ at q = 0 is given by equations s + s 0 = 0 and s ∞ s − = 0, where s 0 , s ∞ are formal coordinates at 0, ∞ ∈ P 1 , respectively. The formal coordinates s 0 and s ∞ satisfy s 0 s ∞ = 1. A pair of such coordinates s 0 , s ∞ induce formal coordinates s 0 −1 and s ∞ −1 at 1 ∈ P 1 , with change of coordinates given precisely by s 0 − 1 = γ(s ∞ − 1), where γ is as in (9). It follows that for the nodal curve C ′ , one has the identifications s + = γ(s 0 ), s − = γ(s ∞ ), and s 0 − 1 = γ(s ∞ − 1). These identifications determine the identification of the space A ut C with the fiber of the projection Ë M g,1 → M g over the moduli point [C] ∈ M g . More generally, one defines Ë M g,n as the moduli space of objects of type (C, P • , t • ), where (C, P • = (P 1 , . . . , P n )) is a stable n-pointed genus g curve, and t • = (t 1 , . . . , t n ) with t i a formal coordinate at P i , for each i. The action of (Aut O) ⊕n by change of coordinates endows Ë M g,n with the structure of an (Aut O) ⊕n -torsor over M g,n . For further information about Ë M g,n over the locus of smooth curves, see [ADCKP] and [FBZ,§6.5 [DG2,Exp. II §3,4]. This is
Der 0 O(R) = R z z∂ z .
The Lie algebra Der 0 O(R) is generated over R by the Virasoro elements L p , for p ≥ 0, hence Der 0 O is a Lie subalgebra of the Virasoro Lie algebra. The action of the Virasoro Lie algebra on a vertex operator algebra V restricts to an action of Der 0 on V . One can integrate this to obtain a left action of Aut O on V defined as the inductive limit of the finite-dimensional submodules V ≤k := i≤k V i . This follows from the fact that L 0 acts semi-simply with integral eigenvalues, and L p acts locally nilpotently for p > 0 [FBZ,§6.3].
Explicitly, to compute the action on V of an an element ρ ∈ Aut O, one proceeds as follows. The element ρ(z) can be expressed as
ρ(z) = exp Ñ i≥0 a i z i+1 ∂ z é (z)
for some a i ∈ C (see e.g., [FBZ,§6.3.1]). Assuming 0 ≤ Im(a 0 ) < 2π, the coefficients a i are uniquely determined. Hence, ρ acts on V as exp Ä i≥0 −a i L i ä . As an example and for later use, the element γ ∈ Aut O from (9) can be expressed as
(10) γ(z) = e −z 2 ∂z (−1) −z∂z (z).
Thus, γ acts on V as e L 1 (−1) L 0 . This is a special case of the computation in [FBZ,(10.4.3)] and is essential to the gluing isomorphism for V C below.
Action of Aut
O on A ut C × V . The group Aut O has a right equi- variant action on the trivial bundle A ut C × V → A ut C defined by (P, t, A) · ρ = P, ρ(t), ρ −1 · A , for ρ ∈ Aut O and (P, t, A) ∈ A ut C × V .
2.5. The sheaf of vertex algebras. As we next describe, the sheaf V C of vertex algebras on a smooth curve C is constructed by faithfully flat descent of an (Aut O)-equivariant sheaf along an (Aut O)-torsor, in order to remove coordinate dependence. The description of the sheaf V C over a nodal curve is more complex. See §8.7 for the extension to families of stable curves. Assume first that the curve C is smooth. The quotient of A ut C × V by the action of Aut O descends along the map π : A ut C → C to the vertex algebra bundle V C on C:
A ut C × V A ut C × Aut O V =: V C A ut C C. Aut O π
In V C , one has identities
(11) (P, t, A) = P, ρ(t), ρ −1 · A , for ρ ∈ Aut O and (P, t, A) ∈ A ut C × V .
The sheaf of sections of V C is the sheaf of vertex algebras:
V C := (V ⊗ π * O A ut C ) Aut O .
This is a locally-free sheaf of O C -modules on C. The fiber of V C at a point P ∈ C is isomorphic to A ut P × Aut O V . Given a formal coordinate t at P , one has the trivialization
A ut P × Aut O V ≃ t V.
For a nodal curve C, the description of V C is more involved. Assume for simplicity that C has only one node Q. Let η : ‹ C → C be the normalization of C, and let Q + and Q − in ‹ C be the two preimages of Q. Choose formal coordinates s + and s − at Q + and Q − , respectively. Denote by π :
A ut ‹ C → ‹ C the natural projection. The action of Aut O on V ⊗ π * O A ut ‹ C restricts to an action of Aut O on k≥0 V k ⊗ O ‹ C (−kQ + − kQ − ) ⊗ π * O A ut ‹ C .
Consider the sheaf
(12) V := Ñ k≥0 V k ⊗ O ‹ C (−kQ + − kQ − ) ⊗ π * O A ut ‹ C é Aut O . The sheaf V C is realized as a subsheaf of η * ( V) which coincides with η * ( V) on C \ Q.
To describe its fiber over Q, we use the involution γ ∈ Aut O from (9) and (10), hence for homogeneous A ∈ V of degree a, one has
γ(A) = e L 1 (−1) L 0 A = (−1) a i≥0 1 i! L i 1 (A).
The fiber of V C over Q is obtained by identifying the fibers of V at Q + and Q − via the gluing isomorphism induced by γ as in the diagram below:
V V k≥0 V k ⊗ C s k + k≥0 V k ⊗ C s k − A ut Q + × Aut O V A ut Q − × Aut O V. γ ∼ = ∼ = ≃s − ∼ = ≃s +
Equivalently, after trivializing with respect to s ± , the gluing isomorphism is
k≥0 V k ⊗ C s k + → k≥0 V k ⊗ C s k − , A ⊗ C s k + → (−1) k i≥0 1 i! L i 1 (A) ⊗ C s k−i − .
We next describe spaces of sections of V C for the nodal curve C. Over an open set
U ⊂ C with Q / ∈ U , one has V C | U := η * V ‹ C | η −1 (U ) .
Since η is an isomorphism away from Q, sections of V C over U are defined as in the case of smooth curves. To define spaces of sections of V C over an open set containing Q, it is sufficient to do so on the formal neighborhood
(13) D Q = Spec " O Q = Spec(C s + , s − /(s + s − )). The space of sections V C (D Q ) is defined as the subspace of η * V(D Q ) = k≥0 V k ⊗ Ä s k + C s + ⊕ s k − C s − ä consisting of elements in the kernel of the map k≥0 V k ⊗ Ä s k + C s + ⊕ s k − C s − ä −→ V induced by Ä A ⊗ s a + f (s + ), B ⊗ s b − g(s − ) ä → f (0) γ(A) − g(0) B for homogeneous A, B ∈ V with deg(A) = a and deg(B) = b, and for all f (s + ) ∈ C s + and g(s − ) ∈ C s − . Hence, V C (D Q ) is spanned by elements A ⊗ s a + , (−1) a i≥0 1 i! L i 1 (A) ⊗ s a−i − and Ä B ⊗ s b+1 + f (s + ), D ⊗ s d+1 − g(s − ) ä for homogeneous A ∈ V a , B ∈ V b , and D ∈ V d , with f (s + ) ∈ C s + and g(s − ) ∈ C s − . The sheaf V C | D Q is naturally an " O Q -module.
To summarize, the sheaf V C on the nodal curve C can be described as follows. Consider the exact sequence of O ‹ C -modules
0 O ‹ C (−Q + − Q − ) ⊗ V V V Q + ⊕ V Q − 0,
where V Q ± is the skyscraper sheaf supported at Q ± with space of sections isomorphic to V via the choice of the coordinate s ± . Pushing forward this sequence along η, we obtain an exact sequence which fits in the diagram
(14) 0 η * Ä O ‹ C (−Q + − Q − ) ⊗ V ä η * V V ⊕2 Q 0 V Q . q γ • π 1 −π 2 Here, π i : V ⊕2 Q → V Q is the natural projection, for i = 1, 2. The sheaf V C is then defined as ker ((γ • π 1 − π 2 ) • q).
Remark 2.5.1. The sheaf V C defined here over a nodal curve C does not coincide in general with the sheaf V C defined in [DGT1], although they agree when restricted to the complements of the nodes of C. For instance, V C is quasi-coherent, while V C is not quasi-coherent. Indeed, if V C were quasicoherent, then the next Lemma 2.6.3 would hold also over nodal curves. This together with Lemma 2.6.1 would imply that the gluing of V C over nodes would be given by (−1) L 0 , instead of γ = e L 1 (−1) L 0 . The gluing γ allows us to deduce Proposition 3.3.1, a key ingredient in the proof of the factorization theorem.
2.6. The structure of the sheaf V C . We describe here some properties of V C . For a smooth curve C, the sheaf V C is filtered by the sheaves V ≤k defined as the sheaves of sections of the vector bundles of finite rank
A ut C × Aut O V ≤k . While the action of Aut O on V ≤k is well-defined, the action of Aut O on V k is so only modulo V ≤k−1 , for each k. In particular, V ≤k is well-defined, but A ut C × Aut O V k only makes sense as a quotient of A ut C × Aut O V ≤k modulo A ut C × Aut O V ≤k−1 .
Assuming for simplicity that the curve C has only one node Q, the sheaves
k i=0 V i ⊗ O ‹ C (−i Q + − i Q − ) ⊗ π * O A ut ‹ C Aut O
provide an increasing filtration of the sheaf V on the normalization ‹ C from (12). The restriction of the gluing isomorphism in §2.5 gives a gluing isomorphism between the fibers of such sheaves at the two preimages Q + and Q − of the node Q, and hence induces an increasing filtration of V C . We denote the subsheaves of such a filtration as V ≤k , as in the smooth case.
Consider the associated graded sheaf
gr • V C := ⊕ k≥0 gr k V C , where gr k V C := V ≤k /V ≤k−1 .
Lemma 2.6.1. One has
gr • V C ∼ = ⊕ k≥0 Ä ω ⊗−k C ä ⊕ dim V k .
This was proved in [FBZ,§6.5.9] for smooth curves. The argument made there extends to stable curves if one replaces the sheaf of differentials Ω 1 C with the dualizing sheaf ω C . We sketch the proof for the reader's convenience. [FBZ,§6.5.9], the transition functions of A and ω ⊗−k C match, hence one concludes that A ∼ = ω ⊗−k C , and this implies the statement.
Proof. Consider V k as the quotient (Aut O)-representation V ≤k /V ≤k−1 , and let A ∈ V k be nonzero. One has L 0 ·A = kA and L p ·A = 0 in V k = V ≤k /V ≤k−1 for p > 0. Assume first that C is smooth. Then it follows that A := (CA ⊗ π * O A ut C ) Aut O is a line sub-bundle of gr k V C . From
Next, we consider the case when C is nodal. Assume for simplicity that C has only one node Q. Consider the line bundle A constructed from the line bundle
A ′ := Ä CA ⊗ π * O A ut ‹ C ä Aut O ⊗ O ‹ C (−kQ + − kQ − ) on the normalization ‹
C of C by identifying the fibers at the preimages Q + and Q − of the node. From the discussion of the smooth case, one has
A ′ ∼ = ω ⊗−k ‹ C (−kQ + − kQ − ) .
It remains to determine the isomorphisms used to identify the fibers at Q + and Q − . The gluing isomorphism
A → e L 1 (−1) L 0 A of V ≤k from §2.5 induces the gluing A → (−1) L 0 A of gr k V C .
In particular, this is the gluing isomorphism of A , which coincides with the gluing of sections for ω ⊗−k C given by the condition on the residues in Lemma 1.9.1. Hence one has A ∼ = ω ⊗−k C in this case as well. The assertion therefore follows.
Remark 2.6.2. The reader will notice how both sheaves
Ä ω ⊗−k C ä ⊕ dim V k
and V ≤k /V ≤k−1 are defined using diagrams similar to the one in (14). From this point of view, the above proof can be seen as comparing the gluing data encoded by the vertical maps in the corresponding diagrams.
As a consequence of [FBZ,§6.5.9], which is Lemma 2.6.1 for smooth curves, one has the following statement, which will be used throughout.
Lemma 2.6.3. Let (C, P • ) be a smooth n-pointed curve. One has:
H 0 (C \ P • , V C ) ∼ = H 0 (C \ P • , gr • V C ).
Proof. We claim that on the affine open set C \ P • , one has
(15) V ≤k ∼ = ⊕ i≤k V ≤i /V ≤i−1 = gr ≤k V C .
Assuming (15), then one has, for every k ∈ Z ≥0 , an injection
φ k : gr ≤k V C ֒→ V C ,
altogether defining a map φ : gr • V C −→ V C . The map φ gives the isomorphism we seek. To see that φ is injective, note that any element x in gr • V C is in fact a finite sum, and hence x is in gr ≤k V for some k. So if x is mapped to zero by φ, then x is mapped to zero by φ k for some k. Since all maps φ k are injective, x is zero. Surjectivity of φ follows from the fact that V C is filtered by the V ≤k . We prove (15) by induction on k with base case k = 0. Lemma 2.6.1 implies that gr k V C is locally free. On affines, locally free sheaves are projective, and hence on the affine open set C \ P • , the following sequence splits:
0 → V ≤k−1 → V ≤k → gr k V C → 0. In particular, on C \ P • V ≤k ∼ = V ≤k−1 ⊕ gr k V C ∼ = gr ≤k−1 V C ⊕ gr k V C ,
and (15) holds.
Remark 2.6.4. Lemma 2.6.3 does not hold on nodal curves since, as noted in Remark 2.5.1, the sheaf V ≤k is not quasi-coherent on a curve with singularities.
2.7. The logarithmic connection. In this section, we describe the logarithmic connection on V C . For smooth curves, this was defined in [FBZ,§6]. The adjective logarithmic is used here to emphasize that we work with possibly nodal curves. For this purpose, we replace the sheaf Ω 1 C , used in [FBZ,§6], with the dualizing sheaf ω C which arises from logarithmic differentials on the normalization of C. Hence a logarithmic connection on V C is a C- (3)). This canonically defines a connection ∇ : V C → V C ⊗ω C independent of the choice of the coordinate t (as in [FBZ,Thm 6.6.3]). 2.7.2. The connection on nodal curves. Let C be a nodal curve, and let η : ‹ C → C be its normalization.
linear map ∇ : V C → V C ⊗ ω C satisfying ∇(f s) = s ⊗ df + f ∇(s)U → A 1 ), one has a trivialization V C | U ≃ t V ⊗O U . On V C | U , the connection ∇ is given by the endomorphism L −1 ⊗id U +id V ⊗∂ t (compare with
Proposition 2.7.3. The connection on V ‹ C described in §2.7.1 naturally induces a logarithmic connection ∇ :
V C → V C ⊗ ω C .
Proof. For simplicity, we assume that C has only one node Q. Recall that the sheaf V C is defined as a subsheaf of η * ( V), where V is the sheaf in (12), and similarly, ω C is a subsheaf of η * ω ‹ C (Q + + Q − ). Since ‹ C is a smooth curve, restricting the prescription given in §2.
7.1 to V ⊂ V ‹ C defines ‹ ∇ : V −→ V ⊗ O ‹ C (Q + + Q − ) ⊗ ω ‹ C .
Pushing forward to C along η and restricting to V C ⊂ η * ( V), we obtain
(16) ∇ : V C −→ η * Ä V ⊗ ω ‹ C (Q + + Q − ) ä .
We claim that this factors through V C ⊗ ω C , defining in this way the desired logarithmic connection. That this holds away from Q is clear. Therefore, it remains to check the assertion over the formal neighborhood D Q from (13).
The sheaf ω C is locally free over C, and by an argument analogous to the proof of Lemma 1.9.1, we obtain an explicit trivialization over D Q as
ω C (D Q ) ∼ = C s + , s − (s + s − ) Å ds + s + , − ds − s − ã ⊂ C s + ds + s + ⊕ C s − ds − s − .
Using this, the restriction of (16) to D Q can be explicitly written as
(17) ∇ : V C (D Q ) −→ k≥0 V k ⊗ Ä s k−1 + C s + ds + ⊕ s k−1 − C s − ds − ä .
To conclude, we show that this factors through
V C (D Q ) ⊗ Ä ds + s + , − ds − s − ä . By definition, the image via ∇ of an element (A ⊗ f (s + ), B ⊗ g(s − )) ∈ V C (D Q ), with A, B ∈ V , is (L −1 (A) ⊗ f (s + )ds + + A ⊗ f ′ (s + )ds + , L −1 (B) ⊗ g(s − )ds − + B ⊗ g ′ (s − )ds − ). The coefficient of Ä ds + s + , − ds − s − ä is (18) (L −1 (A)⊗s + f (s + )+A⊗s + f ′ (s + ), −L −1 (B)⊗s − g(s − )−B⊗s − g ′ (s − )).
We are then left to check that if (A ⊗ f (s + ), B ⊗ g(s − )) ∈ V C (D Q ), then (18) is also an element of V C (D Q ). To do so, it is enough to check that this is true for the elements of type
A ⊗ s a + , (−1) a i≥0 1 i! L i 1 (A) ⊗ s a−i − ∈ V C (D Q )
for homogeneous A ∈ V of degree a. Checking this amounts to showing that
(−1) a+1 a+1 j=0 1 j! L j 1 (L −1 (A)) ⊗ s a+1−j − + a (−1) a a i=0 1 i! L i 1 (A) ⊗ s a−i − is equal to (−1) a+1 a i=0 1 i! L −1 (L i 1 (A)) ⊗ s a−i+1 − + (−1) a+1 a i=0 a − i i! L i 1 (A) ⊗ s a−i − .
Here the sums are truncated, since the terms with larger values of i and j vanish for degree reasons. Comparing powers of s − , we need to verify for every k ∈ {0, . . . , a} that
(19) (−1) a+1 (k + 1)! L k+1 1 L −1 (A) + a(−1) a k! L k 1 (A) is equal to (20) (−1) a+1 (k + 1)! L −1 L k+1 1 (A) + (a − k)(−1) a+1 k! L k 1 (A) in V . Repeatedly applying the commutator [L 1 , L −1 ] = 2L 0 , one has L k+1 1 L −1 (A) = L −1 L k+1 1 (A) + (2a − k)(k + 1) L k 1 (A)
, which indeed implies that (19) and (20) are equal in V . It follows that (17) factors
through V C (D Q ) ⊗ Ä ds + s + , − ds − s − ä
, hence the statement.
2.7.4. The connection: summary. In conclusion, the connection is given by:
Definition 2.7.5. Let C be a curve which has at worst simple nodal singularities. We define the logarithmic connection ∇ : V C → V C ⊗ ω C as the map induced by the following endomorphisms on formal neighborhoods of points of C: for a smooth point P ∈ C with formal coordinate t at P , the endomorphism of V C (D P ) given by L −1 ⊗ id C t + id V ⊗ ∂ t , and for a node Q ∈ C locally given by s + s − = 0, the endomorphism of V C (D Q ) given by
(21) L −1 ⊗ s + id C s + + id V ⊗ s + ∂ s + , −L −1 ⊗ s − id C s − − id V ⊗ s − ∂ s − .
2.8. The coordinate-free Lie algebra ancillary to V . As a first application, one obtains a coordinate-free version of the Lie algebra ancillary to V . For a punctured disk D × P about a smooth point P on C and a formal coordinate t at P , one has
(22) H 0 D × P , V C ⊗ ω C /Im∇ ≃t −→ L t (V ). A section of V C ⊗ ω C on D × P with respect to the t-trivialization B ⊗ i≥i 0 a i t i dt ∈ V ⊗ C C((t)) ⊗ C((t)) C((t))dt ≃ t H 0 D × P , V C ⊗ ω C maps to Res t=0 Y [B, t] i≥i 0 a i t i dt ∈ L t (V ), where Y [B, t] := i∈Z B [i] t −i−1 .
Sections in Im∇ ⊂ V C ⊗ ω C map to zero, and this defines a linear map from sections of V C ⊗ω C /Im∇ on D × P to L t (V ). The vector space H 0 D × P , V C ⊗ ω C /Im∇ has the structure of a Lie algebra such that (22) is an isomorphism of Lie algebras [FBZ,§ §19.4.14, 6.6.9].
3. The new chiral Lie algebra L C\P• (V ) 3.1. Definition of the chiral Lie algebra. For (C, P • ) a stable n-pointed curve and V a vertex operator algebra, set
L C\P• (V ) := H 0 Å C \ P • , V C ⊗ ω C Im∇ ã .
Here V C and its logarithmic connection ∇ are as in §2.
3.2. The chiral Lie algebra maps to the Lie algebra ancillary to V . For each i, let t i be a formal coordinate at P i , let D × P i be the punctured formal disk about P i on C, and L t i (V ) be the Lie algebra ancillary to V ( §1.3). Consider the linear map obtained as the composition
(23) L C\P• (V ) → H 0 Ä D × P i , V C ⊗ ω C /Im∇ ä ∼ = − → L t i (V ).
The first map is canonical and obtained by restricting sections. The second map is the isomorphism of Lie algebras (22) and depends on the formal coordinates t i . From (23), we construct the linear map
(24) ϕ L : L C\P• (V ) → ⊕ n i=1 H 0 Ä D × P i , V C ⊗ ω C /Im∇ ä ∼ = − → ⊕ n i=1 L t i (V )
. After [FBZ,§19.4.14], when C is smooth, the first map of (23) is a homomorphism of Lie algebras, hence so is ϕ L , next denoted simply ϕ. The map ϕ thus induces an action of L C\P• (V ) on L(V ) ⊕n -modules. This will be used in §4. In Proposition 3.3.2 we show an analogous result for nodal curves.
3.3. A close look at the chiral Lie algebra for nodal curves. Let (C, P • ) be a stable n-pointed curve such that C \ P • is affine. Assume for simplicity that C has exactly one simple node, which we denote by Q. Let η : ‹ C → C be the normalization of C, let Q + and Q − be the two preimages of Q, and set Q • = (Q + , Q − ). Let s + and s − be formal coordinates at Q + and Q − , respectively, such that locally around Q, the curve C is given by the equation s + s − = 0. The chiral Lie algebra for
Ä ‹ C, P • ⊔ Q • ä is L ‹ C\P•⊔Q• (V ) = H 0 Ä ‹ C \ P • ⊔ Q • , V ‹ C ⊗ ω ‹ C /Im ∇ ä ,
and consider the linear map given by restriction:
(25) L ‹ C\P•⊔Q• (V ) → H 0 Ä D × Q + , V ‹ C ⊗ ω ‹ C /Im ∇ ä ≃s + − −− → L Q + (V ).
Recall the triangular decomposition of L Q ± (V ) from (5):
L Q ± (V ) = L Q ± (V ) <0 ⊕ L Q ± (V ) 0 ⊕ L Q ± (V ) >0 .
Let σ Q ± ∈ L Q ± (V ) be the image of σ ∈ L ‹ C\P•⊔Q• (V ), and let σ Q ± 0 be the image of σ Q ± under the projection L Q ± (V ) → L Q ± (V ) 0 .
Recall the involution ϑ of L(V ) f in (7). This restricts to an involution on L(V ) 0 given for homogeneous A ∈ V of degree a by
ϑ A [a−1] = (−1) a−1 i≥0 1 i! L i 1 A [a−i−1] . Proposition 3.3.1. For C \ P • affine, one has η * L C\P• (V ) = σ ∈ L ‹ C\P•⊔Q• (V ) (i) σ Q + , σ Q − ∈ L(V ) ≤0 (ii) σ Q − 0 = ϑ σ Q + 0 .
Proof. Since C \ P • is affine, one has
L C\P• (V ) = H 0 (C \ P • , V C ⊗ ω C ) /∇H 0 (C \ P • , V C )
and similarly, since ‹ C \ P • ⊔ Q • is also affine, one has
L ‹ C\P•⊔Q• (V ) = H 0 Ä ‹ C \ P • ⊔ Q • , V ‹ C ⊗ ω ‹ C ä /∇H 0 Ä ‹ C \ P • ⊔ Q • , V ‹ C ä .
To characterize elements in L C\P• (V ), we can first describe their lifts in the vector space H 0 (C \ P • , V C ⊗ ω C ), and then show that the description descends to the quotient by the image of ∇. By definition of V C over nodal curves via the sheaf V in (12) and by Lemma 1.9.1, we have the inclusion
(26) η * H 0 (C \ P • , V C ⊗ ω C ) ⊆ H 0 Ä ‹ C \ P • , V ⊗ ω ‹ C (Q + + Q − ) ä .
To see that (26) implies (i), consider the composition of linear maps
H 0 Ä ‹ C \ P • , V ⊗ ω ‹ C (Q + + Q − ) ä k≥0 V k ⊗ C s k−1 ± C s ± ds ± H 0 Ä D Q ± , V ⊗ ω ‹ C (Q + + Q − ) ä ≃s ±(27)
where the left vertical map is the restriction, followed by the s ± -trivialization. By §2.8, the projection V ⊗ C((s ± ))ds ± → L Q ± (V ) is given by
B ⊗ µ → Res s ± =0 Y [B, s ± ] µ ∈ L Q ± (V ).
It follows that the image of
(28) k≥0 V k ⊗ C s k−1 ± C s ± ds ± → L Q ± (V ) lies in L Q ± (V ) ≤0 .
Composing (27) with (28), we deduce that for σ in (26),
its image σ Q ± in L Q ± (V ) lies in L Q ± (V ) ≤0 ∼ = L(V ) ≤0 , hence (i).
The assertion (ii) follows from the gluing isomorphisms that define V C and ω C as subsheaves of η * V and η * ω ‹ C (Q + + Q − ), respectively, using diagrams as in (14). Given σ in (26), denote by σ Q ± 0 its image via the composition of (27) with the projection
k≥0 V k ⊗ C s k−1 ± C s ± ds ± ։ k≥0 V k ⊗ C s k−1 ± ds ± .
In view of a diagram as in (14), σ Q ± 0 is the restriction of σ at the fibers at Q ± . For σ to correspond to a section of η * H 0 (C \ P • , V C ⊗ ω C ), the elements σ Q + 0 and σ Q − 0 need to satisfy an identity coming from the gluing isomorphism between the fibers of V ⊗ ω ‹ C (Q + + Q − ) at Q ± . For homogeneous A ∈ V of degree a, the gluing isomorphism on fibers of η * V at Q ± defining V C is given by
A ⊗ s a + → (−1) a i≥0 1 i! L i 1 (A) ⊗ s a−i − .
The gluing on fibers of η * ω ‹ C (Q + + Q − ) at Q ± defining ω C is given by s −1 + ds + → −s −1 − ds + . Combining the two, the induced gluing on fibers of
V ⊗ ω ‹ C (Q + + Q − ) at Q ± is given by (29) A ⊗ s a−1 + ds + → (−1) a−1 i≥0 1 i! L i 1 (A) ⊗ s a−i−1 − ds − .
After mapping to L Q ± (V ) 0 via the restriction of (28), the gluing (29) implies that the images of the elements σ Q ± 0 in L Q ± (V ) 0 , still denoted σ Q ± 0 , satisfy (ii) by definition of ϑ.
To show that the conditions are ∇-equivariant, it is enough to check on a neighborhood of Q.
Since ∇ on V C is constructed from ∇ on V ‹ C , one verifies that η * ∇V C (D Q ) ⊂ ∇V ‹ C (D × Q + ⊔ D × Q − )
, hence the statement. We combine §3.2 with Proposition 3.3.1 to show the following: Proof. For simplicity, we may assume that C has a single node Q. By Proposition 3.3.1, η * L C\P• (V ) can be identified with the subspace of sections in L ‹ C\P•⊔Q• (V ) whose restrictions to Q ± is given by the subspace of
L Q + (V ) ⊕ L Q − (V ) ≃ s ± L(V ) ⊕2 generated by L(V ) ⊕2
<0 and the elements of type (A [a−1] , ϑ(A [a−1] )) ∈ L(V ) ⊕2 0 for homogeneous A ∈ V of degree a. Since L(V ) <0 and L(V ) 0 are Lie subalgebras of L(V ) and ϑ is a Lie algebra morphism, it follows that η * L C\P• (V ) is a Lie subalgebra of L ‹ C\P•⊔Q• (V ). To conclude, the analogue of the morphism (24) for the nodal C \ P • is the composition of the Lie algebra morphisms:
L ‹ C\P•⊔Q• (V ) L C\P• (V ) ⊕ n i=1 L t i (V ), ϕ η * ϕ
where η * is described in Proposition 3.3.1 and ϕ is as in (24). 3.4. A consequence of Riemann-Roch for chiral Lie algebras. We give here a statement parallel to §1.10 for chiral Lie algebras. Let C be a smooth curve, possibly disconnected, with two non-empty sets of distinct marked points P • = (P 1 , . . . , P n ) and Q • = (Q 1 , . . . , Q m ). For i ∈ {1, . . . , m}, let s i be a formal coordinate at Q i . For σ ∈ L C\P•⊔Q• (V ), let σ Q i be the image of σ under the map given by restriction
L C\P•⊔Q• (V ) → H 0 Ä D × Q i , V C ⊗ ω C /Im ∇ ä ≃s i − − → L Q i (V ).
For an integer N , consider
L Q i (V, N Q i ) = V ⊗ s N i C s i /Im ∂.
This is a Lie subalgebra of L Q i (V ).
Proposition 3.4.1. Assume C \ P • is affine, fix E ∈ V homogeneous, and integers d and N . There exists σ ∈ L C\P•⊔Q• (V ) such that:
σ Q i ≡ E [d] ∈ L Q i (V )/L Q i (V, N Q i ), for a fixed i, σ Q j ≡ 0 ∈ L Q j (V )/L Q j (V, N Q j ), for all j = i. Proof. Since C \P • is affine, so is C \P • ⊔Q • . As in Proof of Proposition 3.3.1, elements of L C\P•⊔Q• (V ) can be lifted to sections of V C ⊗ ω C on C \ P • ⊔ Q • ,
and thus, via Lemma 2.6.3, described as sections of ⊕ k≥0
Ä ω ⊗1−k C ä ⊕ dim V k on C \ P • ⊔ Q • .
The statement thus follows from the analogous property of sections of tensor products of ω C , discussed in §1.10.
Spaces of coinvariants
Given a stable pointed curve (C, P • ) and a vertex operator algebra V , we define spaces of coinvariants using representations of the chiral Lie algebra.
4.1.
Representations of the chiral Lie algebra. The chiral Lie algebra L C\P• (V ) acts on the tensor product M • := M 1 ⊗ · · · ⊗ M n of V -modules M 1 , . . . , M n . For each i, let t i be a formal coordinate at P i , and L t i (V ) be the Lie algebra ancillary to V ( §1.3). Each L t i (V ) acts on the V -module M i as in §1.3, and the sum ⊕ n i=1 L t i (V ) acts diagonally on M • . The map (24) thus induces an action of L C\P• (V ) on M • as follows: for σ ∈ L C\P• (V ) and A i ∈ M i , one has σ A 1 ⊗ · · · ⊗ A n = n i=1 A 1 ⊗ · · · ⊗ σ P i A i ⊗ · · · ⊗ A n , where σ P i is the restriction of the section σ to the punctured formal disk D × P i about P i on C.
4.2.
Coinvariants. When C \ P • is affine, the space of coinvariants at
(C, P • , t • ) is V (V ; M • ) (C,P•,t•) := M • L C\P• (V ) = M • L C\P• (V ) · M • .
This is the largest quotient of M • on which L C\P• (V ) acts trivially. In general, when C \ P • is not necessarily affine, the space of coinvariants at (C, P • , t • ) is defined as the direct limit where Q • = (Q 1 , . . . , Q m ) ranges over the set of smooth points of C such that P • ∩ Q • = ∅ and C \ P • ⊔ Q • is affine, and s • = (s 1 , . . . , s m ), with s i a formal coordinate at Q i , for each i. As in the case of affine Lie algebras [Fak, Loo], the above direct limit is well defined thanks to the propagation of vacua theorem, which is discussed in §4.3. The construction of the chiral Lie algebra in §3 extends to families of smooth pointed curves over an arbitrary smooth base, and one obtains sheaves of coinvariants as follows. Let (C → S, P • ) be a family of smooth n-pointed curves. In this case, C \ P • (S) is affine over S. Let t i be formal coordinates at P i (S), for i = 1, . . . , n. Equivalently, fix a formally unramified thickening S × Spf(C t i ) → C of the section P i , for each i. One then obtains sheaves of Lie algebras L C\P• (V ) and of coinvariants
V(V ; M • ) (C/S,P•,t•) := (M • ⊗ O S ) L C\P• (V )
over S, for given V -modules M 1 , . . . , M n .
We will extend sheaves of coinvariants over families of stable pointed curves over an arbitrary smooth base in §8.7. The result was established in the generality we need here by Codogni [Cod,Thm 3.6] (see also [DGT1,Thm 6.2]). Other special cases were previously treated in the literature, including the case of coinvariants defined by quasiprimary generated vertex operator algebras V for which V 0 ∼ = C either at a fixed smooth pointed curve with coordinates [Zhu1,AN], or on stable pointed rational curves [NT]. Moreover, propagation of vacua was proved for conformal blocks defined at a fixed smooth curve in [FBZ,§10.3.1].
To state the theorem, we need the following setup. Let (C → S, P • , t • ) be a family of stable n-pointed curves with coordinates. Let Q : S → C be a section such that Q(S) is contained in the smooth locus of C and is disjoint from P i (S), for each i ∈ {1, . . . , n}, and let r be a formal coordinate at Q(S). Theorem 4.3.1 (Propagation of Vacua [Cod,Thm 3.6]). Let V be a vertex operator algebra with one-dimensional weight zero space. Assume that C \ P • (S) is affine over S. The linear map
M • → M • ⊗ V, u → u ⊗ 1 V induces a canonical O S -module isomorphism V (V ; M • ) (C/S,P•,t•) ∼ = − → V (V ; M • ⊗ V ) (C/S,P•⊔Q,t•⊔r) .
Varying (Q, r), the induced isomorphisms are compatible. Moreover, as 1 V is fixed by the action of Aut O, the isomorphism is equivariant with respect to change of coordinates.
The proof requires two main ingredients: (1) the axiom on the vacuum vector; and (2) the existence of a PBW basis for V [GN].
Finite-dimensionality of coinvariants
Using coinvariants by the action of Zhu's Lie algebra ( §A), Abe and Nagatomo show that spaces of coinvariants at smooth pointed curves of arbitrary genus are finite-dimensional [AN].
We show here that the result of [AN] extends to coinvariants by the action of the chiral Lie algebra. Moreover, we further extend the result in [AN] by allowing the following twist of the chiral Lie algebra: given a smooth npointed curve (C, P • ), and an effective divisor
D = m i=1 n i Q i on C not supported at P • , consider (31) L C\P• (V, D) := H 0 (C \ P • , V C ⊗ ω C (−D)/Im∇) ,
where Im∇ denotes the intersection of ∇(V C ) and V C ⊗ ω C (−D). This is the space of sections in L C\P• (V ) vanishing with order at least n i at Q i , for each i, and gives a Lie subalgebra of L C\P• (V ).
5.1. C 2 -cofiniteness. For k ≥ 2 and a V -module M (e.g., M = V ), set:
C k (M ) := span C A (−k) m : A ∈ V, m ∈ M . One says that M is C k -cofinite if dim C M/C k (M ) < ∞.
For a C 2 -cofinite vertex operator algebra V with one-dimensional weight zero space, any finitely generated V -module is C k -cofinite, for k ≥ 2 [Buh]. As explained in [Ara1], the C 2 -cofiniteness has a natural geometric interpretation which generalizes the concept of lisse modules introduced in [BFM] for the Virasoro algebra.
Proposition 5.1.1. Let V be a C 2 -cofinite vertex operator algebra with one-dimensional weight zero space. Let C be a smooth curve with distinct points P 1 , . . . , P n , and D an effective divisor on C not supported at P • . Fix formal coordinates t i at P i , for each i. For finitely generated V -modules M 1 , . . . , M n , the coinvariants M • L C\P• (V,D) are finite-dimensional.
Proof. Recall the map from (23) obtained by fixing the formal coordinates t i at P i , for each i:
L C\P• (V, D) → L P i (V ), σ → σ P i . For k ∈ N, define F k L C\P• (V, D) := σ ∈ L C\P• (V, D) | deg σ P i ≤ k, for all i ,
ON FACTORIZATION OF CONFORMAL BLOCKS FROM VERTEX ALGEBRAS 31
which gives L C\P• (V, D) the structure of a filtered Lie algebra. Let
F k M • = 0≤d≤k M • d , where M • d := d 1 +···+dn=d M 1 d 1 ⊗ · · · ⊗ M n dn . Since F k L C\P• (V, D) · F l M • ⊂ F k+l M • , the L C\P• (V, D)-module M • is a filtered L C\P• (V, D)-module. One has an induced filtration on M • L C\P• (V,D) : F k Ä M • L C\P• (V,D) ä := F k M • + L C\P• (V, D) · M • L C\P• (V, D) · M • .
Step 1. Let U be a finite-dimensional subspace of V such that V = U ⊕ C 2 (V ). Contrary to [AN], elements of U are not required to be quasi-primary here. Let d U be the maximum of the degree of the homogeneous elements in U . Similar to [AN,Lemma 4.1], by an application of the Riemann-Roch and the Weierstrass gap theorem, there exists an integer N such that
H 0 Ä C, ω ⊗1−k C (lP i − D) ä = 0, for all k ≤ d U , l ≥ N, i ∈ {1, . . . , n}.
Step 2. For a V -module M and with N as in Step 1, define the subset
C N (U, M ) = span C A (−l) m : A ∈ U, m ∈ M, l ≥ N .
We claim that for each i the set M 1 ⊗ · · · ⊗ C N U, M i ⊗ · · · ⊗ M n is in the kernel of the canonical surjective linear map
M • π ։ gr • Ä M • L C\P• (V,D) ä := ⊕ k≥0 F k Ä M • L C\P• (V,D) ä F k−1 Ä M • L C\P• (V,D)
ä .
For this, it is enough to show that π m 1 ⊗ · · · ⊗ A (−l) m i ⊗ · · · ⊗ m n = 0, for homogeneous A ∈ U of degree a, m i ∈ M i d i , and l ≥ N . Note that C \ P i is affine for all i. As in the proof of Proposition 3.3.1, elements of L C\P i (V, D) ⊂ L C\P• (V ) can be lifted to sections of V C ⊗ ω C (−D) on C \ P i . By Lemmas 2.6.3 and 2.6.1, the vector space of such sections is isomorphic to the space of sections of
(32) ⊕ k≥0 V k ⊗ ω ⊗1−k C (−D)
on C \ P i . Following
Step 1, there exists a section σ = A ⊗ µ of (32) on C \ P i such that its image via the map (23) is
L C\P i (V, D) → L P i (V ) fromσ P i = A [−l] + j>−l c j A [j]
, for some c j ∈ C.
One has
A [−l] ·M i d i ⊂ M i d i +a+l−1 and A [j] ·M i d i ⊂ M i d i +a+l−2 for j > −l.
Moreover, since µ is holomorphic at a point P r = P i , one has σ Pr = s≥0 a s A [s] , for some a s ∈ C. It follows that σ Pr · M r dr ⊂ M r dr+a−1 . From the identity σ (m 1 ⊗ · · · ⊗ m n ) = n r=1 m 1 ⊗ · · · ⊗ σ Pr (m r ) ⊗ · · · ⊗ m n , one has
m 1 ⊗ · · · ⊗ A (−l) m i ⊗ · · · ⊗ m n ∈ F r dr+a+l−2 M • + L C\P• (V, D) · M • .
Since the element on the left-hand side is in F r dr+a+l−1 M • , it follows that it maps to zero via π. The claim follows.
Step 3. After Step 2, the map π factors through
(33) M 1 /C N U, M 1 ⊗ · · · ⊗ M n /C N (U, M n ) π ։ gr • Ä M • L C\P• (V,D) ä .
By [AN,Prop. 4.5], there is a positive integer k such that
C k M i ⊂ C N U, M i for all i. In particular, dim M i /C N U, M i < dim M i /C k M i .
These are finite as the M i are all C k -cofinite by [Buh]. It follows that the source in (33) is finite-dimensional, hence so is the target. This implies that the coinvariants are finite-dimensional as well.
The proof of Proposition 5.1.1 extends over families of smooth curves C → S with n disjoint sections P 1 , . . . , P n , and for each i, a formal coordinate t i at P i (S). Hence, we conclude:
Corollary 5.1.2. Let V be a C 2 -cofinite vertex operator algebra with onedimensional weight zero space. For any collection of finitely generated Vmodules M 1 , . . . , M n , the sheaf of coinvariants V(V ;
M • ) (C/S,P•,t•) is a co- herent O S -module.
The modules Z and Z
In service of the proof of the factorization theorem, we consider the modules Z and Z in §6.1, and coinvariants constructed from them in §6.2.
6.1. Definitions and properties. Let V be a vertex operator algebra. Recall the associative algebra U(V ) from §1.4. Consider the U(V ) ⊗2 -module
Z := Ind U(V ) U(V ) ≤0 A(V ) ⊗2 = Ä U(V ) ⊗ U(V ) ≤0 A(V ) ä ⊗2
where U(V ) <0 acts trivially on A(V ), and the action of U(V ) 0 on A(V ) is induced from the projection U(V ) 0 → A(V ). With the notation from §1.6, one has that
Z = M (A(V )) ⊗2 , where M (A(V )) is the generalized Verma U(V )-module induced from the natural representation A(V ) of A(V ).
We will also consider a quotient Z of Z defined as follows. Let P be the
subalgebra of U(V ) ⊗2 generated by U(V ) ⊗ C U(V ) <0 , U(V ) <0 ⊗ C U(V ), and U(V ) 0 ⊗ C U(V ) 0 . Consider the U(V ) ⊗2 -module Z := Ind U(V ) ⊗2 P A(V ) = U(V ) ⊗2 ⊗ P A(V ) where U(V ) ⊗ C U(V ) <0 and U(V ) <0 ⊗ C U(V ) act trivially on A(V ), and the action of U(V ) 0 ⊗ C U(V ) 0 on A(V ) is induced via the natural surjection U(V ) 0 ⊗ C U(V ) 0 → A(V )⊗ C A(V ) from the action of A(V )⊗ C A(V ) given by (a ⊗ b)(c) = a · c · (−ϑ(b)), for a ⊗ b ∈ A(V ) ⊗ A(V ), c ∈ A(V ).
Lemma 6.1.1. Let V be a rational vertex operator algebra. One has U(V ) ⊗2module isomorphisms
Z ∼ = W,Y ∈W W ⊗ W ∨ 0 ⊗ Y ⊗ Y ∨ 0 and Z ∼ = W ∈W W ⊗ W ′ ,
with W the set of representatives of isomorphism classes of simple V -modules.
Proof. Since V is rational, the algebra A(V ) is semisimple [Zhu2]. From Wedderburn's theorem, one has A(V ) = ⊕ E∈E E ⊗ E ∨ , where E is the finite set of representatives of isomorphism classes of simple A(V )-modules.
Using the one-to-one correspondence between simple V -modules and simple A(V )-modules [Zhu2], and rationality of V which implies that the V -module induced from any simple A(V )-module is simple, it follows that each simple
V -module is W = U(V ) ⊗ U(V ) ≤0 E, for some E ∈ E . Moreover, there exists a canonical V -module isomorphism U(V ) ⊗ U(V ) ≤0 E ∨ ∼ = (U(V ) ⊗ U(V ) ≤0 E) ′ , for E ∈ E [NT,(V, {Q + , Q − }) of the chiral Lie algebra L C\P• (V ).
We begin by defining L C\P• (V, {Q + , Q − }). For this, let C be a smooth curve, possibly disconnected, with two nonempty, disjoint sets of distinct marked points P • = (P 1 , . . . , P n ) and Q • = (Q + , Q − ). Assume that C \ P • is affine. After Lemmas 2.6.3 and 2.6.1, one has
H 0 (C \ P • , V C ) ∼ = ⊕ k≥0 H 0 Ä C \ P • , V k ⊗ C ω ⊗−k C ä .
Fixing an isomorphism, consider the following Lie subalgebra of the chiral Lie algebra L C\P• (V ): (34)
L C\P• (V, {Q + , Q − }) := k≥0 H 0 Ä C \ P • , V k ⊗ C ω ⊗1−k C (−kQ + − kQ − ) ä ∇H 0 (C \ P • , V C ) .
As in (31), ∇H 0 (C \ P • , V C ) is the intersection of Im∇ with the subspace
k≥0 H 0 Ä C \ P • , V k ⊗ C ω ⊗1−k C (−kQ + − kQ − ) ä of H 0 (C \ P • , V C ⊗ ω C ). Select formal coordinates t i at P i and s i at Q i . Let L P• (V ) := ⊕ n i=1 L P i (V ) and L Q• (V ) := L Q + (V ) ⊕ L Q − (V ). There are Lie algebra injections (35) L C\P• (V, {Q + , Q − }) → L P• (V ) and L C\P• (V, {Q + , Q − }) → L Q• (V ).
The image of an element of (34) in
L Q• (V ) ∼ = L(V ) ⊕2 via the restriction map in (35) is i≥k a i A [i] ⊕ j≥k b j A [j] ∈ L(V ) ⊕2 <0 ⊂ L Q• (V )
for homogeneous A ∈ V of degree k ≥ 0 and coefficients a i , b j ∈ C.
We use here the assumption that V is C 1 -cofinite, i.e., dim C V /C 1 (V ) < ∞, where C 1 (V ) is the subspace of V linearly spanned by A (−1) B for A, B ∈ V >0 and by L −1 V . If V is C 1 -cofinite and V 0 ∼ = C1 V , then lowest weight Vmodules admit spanning sets of PBW-type [KL]. Also, C 2 -cofinitess implies C 1 -cofinitess [KL], hence the assumption here is weaker.
Proposition 6.2.1. Consider (C, P • ⊔ Q • , t • ⊔ s • ),
i.e., a smooth coordinatized (n + 2)-pointed curve, possibly disconnected, such that C \ P • is affine. Let V be a C 1 -cofinite vertex operator algebra with one-dimensional weight zero space, and let M 1 , . . . , M n be V -modules. The map
M • → M • ⊗ C Z, w → w ⊗ 1 A(V ) ⊗ 1 A(V ) ,
where 1 A(V ) ∈ A(V ) is the unit, induces an isomorphism of vector spaces
h : M • L C\P• (V,{Q + ,Q − }) ∼ = − → (M • ⊗ C Z) L C\P•⊔Q• (V ) .
Proof. We proceed in three steps.
Step 1 . We first show that the map h is well-defined. Observe that Z is naturally equipped with a left action of L C\P•⊔Q• (V ) induced by the Lie algebra homomorphisms
L C\P•⊔Q• (V ) → L Q• (V ) → U(V ) ⊗2 . Given an element σ ∈ L C\P• (V, {Q + , Q − }) ⊂ L C\P•⊔Q• (V )
, let σ P• be the image of σ in L P• (V ) via the restriction map in (35), and similarly let σ Q i be its image in L Q i (V ), for i = 1, 2. Since σ Q i ∈ L(V ) <0 , the elements σ Q + ⊗ 1 and 1 ⊗ σ Q − act trivially on A(V ) ⊗ A(V ) ⊂ Z. This implies
(36) σ P• (w) ⊗ 1 A(V ) ⊗ 1 A(V ) = σ Ä w ⊗ 1 A(V ) ⊗ 1 A(V ) ä − w ⊗ σ Q + Ä 1 A(V ) ä ⊗ 1 A(V ) − w ⊗ 1 A(V ) ⊗ σ Q − Ä 1 A(V ) ä = σ Ä w ⊗ 1 A(V ) ⊗ 1 A(V ) ä
for w ∈ M • . It follows that the image of any element is independent of the equivalence class representative in the quotient, hence the map h between the spaces of coinvariants is well-defined.
Step 2. Next, we show that the map h is surjective:
Given w ⊗ z 1 ⊗ z 2 in M • ⊗ Z, there exists w ′ ∈ M • such that w ⊗ z 1 ⊗ z 2 ≡ w ′ ⊗ 1 A(V ) ⊗ 1 A(V ) mod L C\P•⊔Q• (V ) (M • ⊗ Z).
By linearity, and reordering elements in U(V ), we can reduce to the case
z 1 ⊗ z 2 = D l · · · D 1 1 A(V ) ⊗ E m · · · E 1 1 A(V ) ,
with each D i and E j in L(V ) ≥0 . The surjectivity is clear when l = m = 0. By induction on l (and similarly on m), it is then enough to show that D [d] in L(V ) ≥0 . Each component of the curve C has at least one of the marked points in P • . By Proposition 3.4.1, there exists σ ∈ L C\P•⊔Q• (V ) such that
w ⊗ z 1 ⊗ z 2 ≡ w ′ ⊗ z ′ 1 ⊗ z 2 mod L C\P•⊔Q• (V )(M • ⊗ Z) for some w ′ in M • , when z 1 = D [d] (z ′ 1 ) for some homogeneous D ∈ V andσ Q + ≡ D [d] ∈ L Q + (V )/L Q + (V, N Q + ), σ Q − ≡ 0 ∈ L Q − (V )/L Q − (V, N Q − ),
for N ≫ 0. It is enough to take N such that
D [i] (z ′ 1 ) ⊗ z 2 = z ′ 1 ⊗ D [i] (z 2 ) = 0 in Z for all i ≥ N .
Such N exists because U(V ) acts smoothly on each factor. This implies
σ Q + (z ′ 1 ) ⊗ z 2 + z ′ 1 ⊗ σ Q − (z 2 ) = D [d] · z ′ 1 ⊗ z 2 = z 1 ⊗ z 2 . It follows that w ⊗ z 1 ⊗ z 2 = σ(w ⊗ z ′ 1 ⊗ z 2 ) − σ P• (w) ⊗ z ′ 1 ⊗ z 2 , hence w ⊗ z 1 ⊗ z 2 ≡ −σ P• (w) ⊗ z ′ 1 ⊗ z 2 mod L C\P•⊔Q• (V )(M • ⊗ Z).
Repeating the same argument for z 2 , the surjectivity of h follows.
Step 3. It remains to show that h is injective. Equivalently, we show the surjectivity of the dual map
h ∨ : Ä (M • ⊗ C Z) L C\P•⊔Q• (V ) ä ∨ → Ä M • L C\P• (V,{Q + ,Q − }) ä ∨ given by h ∨ (Φ)(w) = Φ Ä w ⊗ 1 A(V ) ⊗ 1 A(V ) ä , for w ∈ M • .
For this, select an element Φ in the target of h ∨ . We construct an element Φ in the source of h ∨ such that h ∨ (Φ) = Φ. First, we define Φ as a linear functional on M • ⊗ C Z. We use a spanning set for Z obtained as follows. Since V is C 1 -cofinite and V 0 ∼ = C1 V , lowest weight V -modules admit spanning sets of PBW-type [KL,Cor. 3.12]. Since Z is the tensor product of two lowest weight V -modules, we conclude that Z admits a spanning set of PBW-type. Select a spanning set of PBW-type for Z, and consider a generating element
(37) z 1 ⊗ z 2 = D l [i l ] · · · D 1 [i 1 ] 1 A(V ) ⊗ E m [jm] · · · E 1 [j 1 ] 1 A(V ) ∈ Z
with D 1 , . . . , D l , E 1 , . . . , E m ∈ V , and i 1 , . . . , i l , j 1 , . . . , j m ∈ Z such that
D l [i l ] , . . . , D 2 [i 2 ] , E m [jm] , . . . , E 2 [j 2 ] ∈ L(V ) >0 and D 1 [i 1 ] , E 1 [j 1 ] ∈ L(V ) ≥0 .
Note that the elements D 1 [i 1 ] , E 1 [j 1 ] ∈ L(V ) 0 generate the lowest weight space from the vector 1 A(V ) ⊗ 1 A(V ) . We proceed by induction on l and m. We start by defining
Φ Ä w ⊗ 1 A(V ) ⊗ 1 A(V ) ä := Φ(w). Moreover, we define Φ Ä w ⊗ D [i] 1 A(V ) ⊗ E [j] 1 A(V ) ä := 0 when D [i] or E [j] ∈ L(V ) 0 C1 V [−1]
. Next, assume that Φ has been defined on elements w ⊗ z 1 ⊗ z 2 for w ∈ M • and z 1 ⊗ z 2 as in (37) for fixed l and m. Consider an element
w ⊗ D + [i + ] z 1 ⊗ z 2 ∈ M • ⊗ C Z for some w ⊗ z 1 ⊗ z 2 ∈ M • ⊗ C Z, D + ∈ V , and i + ∈ Z. Choose an integer N ≫ 0 such that D + [i] z 1 ⊗ z 2 = z 1 ⊗ D + [i] z 2 = 0 in Z for all i ≥ N .
As in
Step 2, such N exists because U(V ) acts smoothly on each of the two factors of Z. By Proposition 3.4.1, there exists σ ∈ L C\P•⊔Q• (V ) such that
(38) σ Q + ≡ D + [i + ] ∈ L Q + (V )/L Q + (V, N Q + ), σ Q − ≡ 0 ∈ L Q − (V )/L Q − (V, N Q − ). Define Φ Ä w ⊗ D + [i + ] z 1 ⊗ z 2 ä := − n i=1 Φ (w 1 ⊗ · · · ⊗ σ P i (w i ) ⊗ · · · ⊗ w n ⊗ z 1 ⊗ z 2 )
where w = w 1 ⊗ · · · ⊗ w n ∈ M • . Since Φ vanishes on L C\P• (V, {Q + , Q − }), the definition of Φ here is independent of the choice of σ satisfying (38). One proceeds in a similar way to define Φ on elements of type
w ⊗ z 1 ⊗ E + [j + ] z 2 ∈ M • ⊗ C Z for w ⊗z 1 ⊗z 2 ∈ M • ⊗ C Z, E + ∈ V ,
and j + ∈ Z. By induction, this defines Φ on a spanning set of PBW-type for M • ⊗ C Z, hence by linearity, on
M • ⊗ C Z. A direct calculation shows that Φ Ä w ⊗ D a [ia] D b [i b ] z 1 ⊗ z 2 ä − Φ Ä w ⊗ D b [i b ] D a [ia] z 1 ⊗ z 2 ä = Φ Ä w ⊗ î D a [ia] , D b [i b ] ó z 1 ⊗ z 2 ä and similarly Φ Ä w ⊗ z 1 ⊗ E a [ja] E b [j b ] z 2 ä − Φ Ä w ⊗ z 1 ⊗ E b [j b ] E a [ja] z 2 ä = Φ Ä w ⊗ z 1 ⊗ î E a [ja] , E b [j b ] ó z 2 ä ,
hence Φ is independent of the choice of the spanning set.
Finally, we check that Φ vanishes on
L C\P•⊔Q• (V ) (M • ⊗ C Z). Fix τ in L C\P•⊔Q• (V ). By definition of Φ, one has Φ w ⊗ τ Q + (z 1 ) ⊗ z 2 + Φ w ⊗ z 1 ⊗ τ Q − (z 2 ) = −Φ (σ P• (w) ⊗ z 1 ⊗ z 2 )
for any σ ∈ L C\P•⊔Q• (V ) satisfying conditions analogous to (38), hence we can choose σ = τ . It follows that Φ vanishes on τ (M • ⊗ C Z) for all τ ∈ L C\P•⊔Q• (V ), hence Φ is in the source of h ∨ . By construction, one has h ∨ (Φ) = Φ, hence h ∨ is surjective. This ends the proof.
Proof of the factorization theorem
Here we prove our main result, which we state in complete detail below. For this let us first set some notation. Let (C, P • ) be a stable n-pointed curve with exactly one simple node, denoted Q. Let ‹ C → C be the normalization of C, let Q + and Q − ∈ ‹ C be the two preimages of Q, and set Q • = (Q + , Q − ). The curve ‹ C may not be connected. Fix formal coordinates t i at P i , for each i = 1, . . . , n, and s ± at Q ± . Suppose M 1 , . . . , M n are V -modules, set M • = ⊗ n i=1 M i , and let W be the set of representatives of isomorphism classes of simple V -modules. Consider the map
M • → W ∈W M • ⊗ C W ⊗ C W ′ , u → ⊕ W ∈W u ⊗ 1 W 0 (39) where 1 W 0 = id W 0 ∈ End(W 0 ) ∼ = W 0 ⊗ W ∨ 0 .
Here W 0 is the degree zero space of the module W = i≥0 W i . Recall that, by definition, the vector spaces W 0 and W ∨ 0 are finite-dimensional.
Theorem 7.0.1 (Factorization theorem). Let V be a rational, C 2 -cofinite vertex operator algebra with one-dimensional weight zero space. The map (39) gives rise to a canonical isomorphism of vector spaces
V(V ; M • ) (C,P•,t•) ∼ = W ∈W V V ; M • ⊗ C W ⊗ C W ′ ( ‹ C,P•⊔Q•,t•⊔s•) .
This isomorphism is equivariant with respect to change of the coordinates t • .
The proof we give here roughly follows the outline of the proof in [NT,§8.6], with the generalizations to coinvariants defined using the chiral Lie algebra instead of Zhu's Lie algebra, and for curves of arbitrary genus, made possible by Propositions 3.3.1 and 6.2.1.
Proof. By definition (30), due to propagation of vacua (Theorem 4.3.1), we can reduce to the case C \P • affine, after possibly adding more marked points P i and corresponding modules V . Using the formal coordinates t i at P i , for i = 1, . . . , n, and s ± at Q ± , one has Lie algebra homomorphisms
L C\P• (V ) → L P• (V ) and L ‹ C\P•⊔Q• (V ) → L P• (V ) ⊕ L Q• (V ).
In the following, we show that (39) induces a canonical isomorphism
(40) M • L C\P• (V ) ∼ = W ∈W M • ⊗ C W ⊗ C W ′ L ‹ C\P•⊔Q• (V ) .
We will argue that there is a commutative diagram
M • L ‹ C\P• (V,{Q + ,Q − }) (M • ⊗ C Z) L ‹ C\P•⊔Q• (V ) M • L C\P• (V ) M • ⊗ C Z L ‹ C\P•⊔Q• (V ) . h ∼ = f ∼ =
Then after Lemma 6.1.1, the isomorphism f gives (40).
Step 1 . The top horizontal isomorphism h is given by Proposition 6.2.1.
Step 2 . We argue that there is an inclusion (34) is identified as a Lie subalgebra of L C\P• (V ) by pullback along the normalization, since its elements satisfy σ Q ± ∈ L(V ) <0 .
ι : L ‹ C\P• (V, {Q + , Q − }) ֒→ L C\P• (V ). Indeed, by Proposition 3.3.1 an element of L C\P• (V ) can be realized as σ in L ‹ C\P•⊔Q• (V ) such that σ Q ± ∈ L(V ) ≤0 and the restrictions σ Q ± 0 of σ Q ± to L(V ) 0 satisfy σ Q − 0 = ϑ σ Q + 0 . In particular, L ‹ C\P• (V, {Q + , Q − }) from
Step 3 . To show that the bottom horizontal map f is an isomorphism, it remains to verify that the kernel of the left vertical map is identified with the kernel of the right vertical map by the isomorphism h.
Step 3(a). The kernel of the left vertical map is the space
K := L C\P• (V ) M • L ‹ C\P• (V,{Q + ,Q − }) .
Note that for σ in L C\P• (V ), the formula for the bracket (4) gives
î ϕ(σ), ϕ Ä L ‹ C\P• (V, {Q + , Q − }) äó ⊂ ϕ Ä L ‹ C\P• (V, {Q + , Q − }) ä ,
where ϕ is as in (24). It follows that σ acts on the source of h. The left vertical map is thus the quotient by the action of
L C\P• (V )/L ‹ C\P• (V, {Q + , Q − }).
Step 3(b). We conclude the argument by showing that the right vertical map is the quotient by h(K).
Recall that h(w) = w ⊗ 1 A(V ) ⊗ 1 A(V ) , for w ∈ M • . Hence, h(K) is linearly spanned by (41) σ P• (w) ⊗ 1 A(V ) ⊗ 1 A(V ) ≡ − w ⊗ σ Q + Ä 1 A(V ) ä ⊗ 1 A(V ) − w ⊗ 1 A(V ) ⊗ σ Q − Ä 1 A(V ) ä modulo L ‹ C\P•⊔Q• (V ) (M • ⊗ Z), for σ ∈ L C\P• (V ) and w ∈ M • . Since U(V ) <0 acts trivially on A(V ), (41) is congruent to −w ⊗ σ Q + 0 Ä 1 A(V ) ä ⊗ 1 A(V ) − w ⊗ 1 A(V ) ⊗ σ Q − 0 Ä 1 A(V ) ä modulo L ‹ C\P•⊔Q• (V ) (M • ⊗ Z)
. From Proposition 3.3.1, one has σ Q − 0 = ϑ σ Q + 0 . By linearity, it follows that h(K) is linearly spanned by ) . Using this in (42), we conclude that h(K) is linearly spanned by
(42) − w ⊗ B [k−1] Ä 1 A(V ) ä ⊗ 1 A(V ) − w ⊗ 1 A(V ) ⊗ ϑ B [k−1] Ä 1 A(V ) ä modulo L ‹ C\P•⊔Q• (V ) (M • ⊗ Z), for w ∈ M • and homogeneous B ∈ V of degree k. Here, B [k−1] and ϑ B [k−1] are in L(V ) 0 , and recall that the action of L(V ) 0 on A(V ) is induced from the projection L(V ) 0 → A(V ). One has b Ä 1 A(V ) ä = Ä 1 A(V ) ä b for all b ∈ L(V ) 0 , where Ä 1 A(V ) ä b denotes the right action of b ∈ L(V ) 0 on 1 A(V(43) − w ⊗ Ä 1 A(V ) ä B [k−1] ⊗ 1 A(V ) − w ⊗ 1 A(V ) ⊗ Ä 1 A(V ) ä ϑ B [k−1] modulo L ‹ C\P•⊔Q• (V ) (M • ⊗ Z), for w ∈ M • and homogeneous B ∈ V of degree k.
From Lemma 6.1.1, the target of h is isomorphic to
(M • ⊗ Z) L ‹ C\P•⊔Q• (V ) ∼ = W,Y ∈W M • ⊗ W ⊗ W ∨ 0 ⊗ Y ⊗ Y ∨ 0 L ‹ C\P•⊔Q• (V ) ∼ = W,Y ∈W (M • ⊗ W ⊗ Y ) L ‹ C\P•⊔Q• (V ) ⊗ W ∨ 0 ⊗ Y ∨ 0 .(44)
The second isomorphism is due to the fact that L Q + (V ) acts on the left of W ⊗ W ∨ 0 , and similarly, L Q − (V ) acts on the left of Y ⊗ Y ∨ 0 , so that L ‹ C\P•⊔Q• (V ) only acts on M • ⊗W ⊗Y . Expressing (43) via the isomorphisms in (44), h(K) is linearly spanned by
− (M • ⊗ W ⊗ Y ) L ‹ C\P•⊔Q• (V ) ⊗ B [k−1] W ∨ 0 ⊗ Y ∨ 0 − (M • ⊗ W ⊗ Y ) L ‹ C\P•⊔Q• (V ) ⊗ W ∨ 0 ⊗ ϑ B [k−1] Y ∨ 0
for W, Y ∈ W and homogeneous B ∈ V of degree k. Here the right action of B [k−1] and ϑ B [k−1] in L(V ) 0 on W ∨ 0 and Y ∨ 0 is given by Lemma 1.8.3, and recall that ϑ is an involution. It follows that h
(K) in (M • ⊗ Z) L ‹ C\P•⊔Q• (V ) is isomorphic to W,Y ∈W (M • ⊗ W ⊗ Y ) L ‹ C\P•⊔Q• (V ) ⊗ I W ∨ 0 , Y ∨ 0 with I (W ∨ 0 , Y ∨ 0 ) ⊂ W ∨ 0 ⊗ Y ∨ 0 linearly spanned by ψ W • ϑ B [k−1] ⊗ ψ Y + ψ W ⊗ ψ Y • B [k−1] , where ψ W ∈ W ∨ 0 , ψ Y ∈ Y ∨ 0 for W, Y ∈ W , and B ∈ V k , for k ≥ 0. One has W ∨ 0 ⊗ Y ∨ 0 /I W ∨ 0 , Y ∨ 0 = Hom A(V ) W 0 , Y ∨ 0 ,
and by Schur's Lemma, this is isomorphic to C when Y = W ′ and zero otherwise. This and the description of Z from Lemma 6.1.1 imply that, after taking the quotient of (M
• ⊗ C Z) L ‹ C\P•⊔Q• (V ) by h(K), one obtains M • ⊗ C Z L ‹ C\P•⊔Q• (V )
, hence the statement.
Sewing and local freeness
In this section we prove VB corollary. For this, we start with the sewing theorem (Theorem 8.5.1), a refined version of the factorization theorem. This requires the notion of formal smoothings, reviewed below.
8.1. Formal smoothings. For a C-algebra R with smooth Spec(R), let C 0 → S 0 := Spec(R) be a flat family of nodal curves with a single simple node defined by a section Q and with n distinct smooth points given by sections P • = (P 1 , . . . , P n ). Assume that C 0 \ P • (S 0 ) is affine over S 0 . Up to anétale base change of S 0 of degree two, we can normalize C 0 and obtain a smooth family of (n+2)-pointed curves ‹ C 0 → S 0 with sections P • ⊔(Q + , Q − ), where Q ± (S 0 ) ⊂ ‹ C 0 are the preimages of the node in C 0 /S 0 . Fix formal coordinates s + and s − at Q + (S 0 ) and Q − (S 0 ), respectively. Such coordinates determine a smoothing of (C 0 , P • ) over S := Spec(R q ). That is, a flat family C → S = Spec(R q ) with sections P • = (P 1 , . . . , P n ) such that the general fiber is smooth and the special fiber is identified with C 0 → S 0 . The family ‹ C 0 → S 0 trivially extends to a family of smooth curves ‹ C → S with n + 2 sections P • , Q + , and Q − :
‹ C C S = Spec(R q ) . P• P•,Q + ,Q −
The formal coordinate at Q ± (S 0 ) extends to a formal coordinate, still denoted s ± , at Q ± (S) -that is, s ± is a generator of the ideal of the completed local R q -algebra of ‹ C at Q ± (R) -such that locally around the node, the family C is defined by s + s − = q. For more details, see [Loo,p. 457] and [ACG,. We emphasize that the existence of such families holds over the formal base S = Spec(R q ), or equivalently, over the complex open unit disk around S 0 in the analytic category, but fails over a more general base. Moreover, one still has that ‹ C \ P • (S) and C \ P • (S) are affine over S.
8.2.
The sheaf of vertex algebras over formal smoothings. We define here the sheaf of vertex algebras V C over a family C → S as in §8.1. After §2.5, it is enough to describe V C on the formal neighborhood D Q of the node Q. The completed local ring " O Q consists of elements of the form i,j≥0 α i,j s i + s j − for α i,j ∈ R. After identifying s + = (s + , q s − ) and s − = ( q s + , s − ), the ring " O Q is realized as the subring of R((s + )) q ⊕ R((s − )) q consisting of elements of type i,j≥0
α i,j Ä s i−j + q j , s j−i − q i ä with α i,j ∈ R. The space of sections of V C over D Q is generated by (45) i,j≥0 α i,j A ⊗ s a+i−j + q j , (−1) a k≥0 1 k! L k 1 (A) ⊗ s a−k+j−i − q i
for homogeneous A ∈ V of degree a and α i,j ∈ R. For q = 0, one recovers the description given in §2.5 for nodal curves. Similar to the case of a nodal curve, we can identify a logarithmic connection ∇ : V C → V C ⊗ω C /S . The description of ∇ on a smooth open set follows from the description of the connection ∇ over a smooth curve. Specifically, for every smooth point P ∈ C \ Q, there is an open set S ′ ⊂ S and an open set U ⊂ C \ Q over S ′ and containing P such that there exists anétale map U → A 1 S ′ . This implies that U has a global coordinate t, and thus
V| U ≃ t V ⊗ O U .
On the open set U , the connection ∇ is provided by the endomorphism of V| U given by L −1 ⊗ id U + id V ⊗ ∂ t , as in §2.7.1.
We are left to describe ∇ over D Q . Recall that
Ä ds + s + , − ds − s − ä is a generator of ω C /S over " O Q .
Using this trivialization, the connection ∇ is defined by the endomorphism (21) extended over R q by linearity, that is:
L −1 ⊗ s + id R((s + )) q + id V ⊗ s + ∂ s + , −L −1 ⊗ s − id R((s − )) q − id V ⊗ s − ∂ s − .
As in the proof of Proposition 2.7.3, one needs to show that the target of this map lies inside V C (D Q ), and not merely in V ⊗ R((s + )) q ⊕ R((s − )) q . This amounts to a verification as in the proof of Proposition 2.7.3. 8.3. The chiral Lie algebra over formal smoothings. Next, we globalize the Lie algebra L(V ) ancillary to V and the chiral Lie algebra.
Select a formal coordinate t i at P i (S). On the formal neighborhood
D P i = Spec R t i , q , one has V C | D P i ≃ t i (V ⊗ R t i ) q .
Since the restriction of ∇ over D P i is given by L −1 ⊗ id R t i ,q + id V ⊗ ∂ t i , we have an identification as in (22):
(46) H 0 Ä D × P i , V C ⊗ ω C /S /Im∇ ä ≃t i − − → L t i (V ) q .
We define the chiral Lie algebra assigned to V and (C , P • ) as
L C \P• (V ) := H 0 Å C \ P • (S), V C ⊗ ω C /S Im∇ ã .
This extends §3.1. The map (46) is a Lie algebra morphism.
L C \P• (V ) → ⊕ n i=1 H 0 Ä D × P i , V C ⊗ ω C /S /Im∇ ä ∼ = − → ⊕ n i=1 L t i (V ) q induced
by the restriction of sections and
Remark 8.3.1. As in Proposition 3.3.1, L C \P• (V ) can be described as the subspace of L ‹ C \P•⊔Q• (V ) generated by those elements σ whose restrictions σ Q ± to L Q ± (V ) q satisfy
σ Q + = i,j≥0 α i,j A [a+i−j−1] q j , σ Q − = (−1) a−1 i,j≥0 α i,j k≥0 1 k! Ä L k 1 A ä [a−k+j−i−1] q i = i,j≥0 α i,j ϑ A [a+i−j−1] q i
for homogeneous A ∈ V of degree a and integers i, j ∈ Z ≥0 . This uses that sections of V C over D Q are generated by (45)
V(V ; M • ) (C /S,P•,t•) := (M • ⊗ O S ) L C \P• (V ) ,
The factorization theorem holds for the restriction of V(V ; M • ) (C /S,P•,t•) to the special fiber C 0 → S 0 :
Theorem 8.4.1. Let V be a rational, C 2 -cofinite vertex operator algebra with one-dimensional weight zero space. The map (39) induces a canonical
O S 0 -module isomorphism V(V ; M • ) (C 0 /S 0 ,P•,t•) ∼ = W ∈W V V ; M • ⊗ W ⊗ W ′ Ä ‹ C 0 /S 0 ,P•⊔Q•,t•⊔s• ä .
As a consequence of Corollary 5.1.2 and Theorem 8.4.1 we obtain the following result on coherence.
Theorem 8.4.2. Let V be a rational, C 2 -cofinite vertex operator algebra with one-dimensional weight zero space. For finitely generated V -modules M 1 , . . . , M n , the sheaf V(V ; M • ) (C /S,P•,t•) is a coherent O S -module.
Proof. Corollary 5.1.2 implies that the restriction of V(V ; M • ) (C /S,P•,t•) to S\S 0 is coherent. We are left to show that the restriction of V(V ; M • ) (C /S,P•,t•) to S 0 , i.e. V(V ; M • ) (C 0 /S 0 ,P•,t•) is coherent. This follows from Theorem 8.4.1 and Corollary 5.
1.2 applied to V (V ; M • ⊗ W ⊗ W ′ ) Ä ‹ C 0 /S 0 ,P•⊔Q•,t•⊔s• ä . 8.5. Sewing. Given a simple V -module W = i≥0 W i , define 1 W := i≥0 1 W i q i ∈ (W ⊗ W ′ ) q , where 1 W i := id W i ∈ End(W i ) ∼ = W i ⊗ W ∨ i . Consider the map (47) M • −→ ⊕ W ∈W M • ⊗ (W ⊗ W ′ ) q , u → ⊕ W ∈W u ⊗ 1 W .
The following result extends [NT,Thm 8.4.6] to curves of arbitrary genus.
Theorem 8.5.1 (Sewing Theorem). Let V be a rational, C 2 -cofinite vertex operator algebra with one-dimensional weight zero space, and set M
• = n i=1 M i for V -modules M i . The map (47) induces a canonical O S 0 q - module isomorphism Ψ such that the following diagram commutes V(V ; M • ) (C /S,P•,t•) W ∈W V (V ; M • ⊗ W ⊗ W ′ ) Ä ‹ C 0 /S 0 ,P•⊔Q•,t•⊔s• ä ⊗ O S 0 O S 0 q V(V ; M • ) (C 0 /S 0 ,P•,t•) W ∈W V (V ; M • ⊗ W ⊗ W ′ ) Ä ‹ C 0 /S 0 ,P•⊔Q•,t•⊔s• ä . Ψ ∼ =
This isomorphism is equivariant with respect to change of the coordinates t • .
Remark 8.5.2. Theorems 8.4.1 and 8.5.1 give a canonical isomorphism
V(V ; M • ) (C /S,P•,t•) ∼ = V (V ; M • ) (C 0 /S 0 ,P•,t•) q .
In particular, this means that to the non-trivial deformation C of C 0 , there corresponds a trivial deformation of the space of conformal blocks.
Proof of Theorem 8.5.1. As in the proof of the factorization theorem, we can reduce to the case C \ P • affine over S, and show that (47) induces a canonical R q -module isomorphism, still denoted Ψ, such that the following diagram commutes
(M • ⊗ O S ) L C \P• (V ) W ∈W (M • ⊗ W ⊗ W ′ ⊗ O S 0 ) L C 0 \P•⊔Q• (V ) ⊗ O S 0 O S 0 q (M • ⊗ O S 0 ) L C 0 \P• (V ) W ∈W (M • ⊗ W ⊗ W ′ ⊗ O S 0 ) L C 0 \P•⊔Q• (V ) . Ψ ∼ =
Here, the vertical maps are obtained by specializing at q = 0.
Step 1 . We show that (47) induces a well-defined map Ψ between spaces of coinvariants. For this, it is enough to show that for each σ ∈ L C \P• (V ) and W ∈ W , one has σ P• (M • ) ⊗ 1 W = σ M • ⊗ 1 W , or equivalently
(48) σ Q + ⊗ 1 + 1 ⊗ σ Q − Ä 1 W ä = 0.
From Remark 8.3.1 and by linearity, we can reduce to the case when
σ Q + = A [a+i−j−1] q j and σ Q − = ϑ A [a+i−j−1] q i
for homogeneous A ∈ V of degree a and integers i, j ≥ 0. The vanishing of (48) follows from the identity [NT,Lemma 8.7.1] (there is a sign difference between the involution ϑ used here and the involution used in [NT]). Thus we conclude that the map Ψ is well-defined and makes the diagram above commute.
(49) A [a+i−j−1] ⊗ 1 + 1 ⊗ ϑ A [a+i−j−1] q i−j Ä 1 W ä = 0 established in
Step 2 . Since (i) the target of Ψ is a free O S 0 q -module of finite rank, (ii) the source is finitely generated (Theorem 8.4.2), and (iii) Φ is an isomorphism modulo q (Theorem 8.4.1), Nakayama's lemma implies that Ψ is an isomorphism (this is as in [TUY, Loo, NT]).
8.6. The sheaf of coinvariants on Ë M g,n . Let M g,n be the restriction of Ë M g,n over the locus M g,n of smooth pointed curves (see §2.2.2 for definitions). The vertex algebra bundle and the chiral Lie algebra defined on smooth curves in § §2 and 3, respectively, give the vertex algebra bundle and the sheaf of chiral Lie algebras on M g,n . In § §8.2 and 8.3, the vertex algebra bundle and the sheaf of chiral Lie algebras are defined on formal smoothings of families of nodal curves. Gluing as in [BL], one obtains the vertex algebra bundle and the sheaf of chiral Lie algebras on Ë M g,n . Similarly, the sheaf of coinvariants defined on families of smooth curves in §4 gives the sheaf of coinvariants V(V ; M • ) on M g,n . In §8.4, the sheaf of coinvariants is defined on formal smoothings of families of nodal curves. By gluing as in [BL], one obtains the sheaf of coinvariants V(V ; M • ) on Ë M g,n .
8.7. The sheaf of coinvariants on M g,n . Throughout this section, we require every V -module M to further satisfy the following property: there exists a complex number c M called the conformal dimension (or conformal weight) of M such that for every homogeneous v ∈ M one has L 0 (v) = (deg(v) + c M )v. This condition holds whenever M = V or, for instance, when M is a simple V -module. Furthermore, if V is C 2 -cofinite, then c M is a rational number [Miy1].
The sheaf of coinvariants on M g,n is obtained from a two-step process [DGT1,§6.3], reviewed next. Consider the group scheme Aut + O which represents the functor assigning to a C-algebra R the group:
Aut + O(R) = z → ρ(z) = z + a 2 z 2 + · · · | a i ∈ R .
This is a subgroup scheme of the group scheme Aut O. In particular, one has Aut O = G m ⋉ Aut + O. The (Aut O) ⊕n -torsor Ë M g,n → M g,n factors as the composition of an (Aut + O) ⊕n -torsor and a G ⊕n m -torsor:
(50) Ë M g,n J × g,n M g,n . (Aut + O) ⊕n (Aut O) ⊕n G ⊕n m
Here J × g,n is the space of tuples (C, P • , τ • ), where τ • = (τ 1 , . . . , τ n ) with τ i a non-zero 1-jet of a formal coordinate at P i for each i. The idea for the descent of V J (V ; M • ) to M g,n is inspired by Tsuchimoto [Tsu] (and used to prove [DGT1,Theorem 8.1]): first, one tensors V J (V ; M • ) with an appropriate line bundle to obtain a new sheaf on which G ⊕n m acts; after descending the new sheaf, one then tensors back with the dual of the line bundle. Next, we detail this argument using root stacks.
The case n > 1 can be treated by iterating the procedure used for n = 1, hence we discuss only this latter case and set M 1 =: M . We will restrict to the case in which the conformal dimension c M of M is a rational number, and we write c M = a d for a ∈ Z and d ∈ N. We consider line bundles L M = (Ψ ∨ ) ⊗a on M := M g,1 , and L J = π * L M on J := J × g,1 , where π : J → M is the map forgetting the 1-jets and Ψ is the cotangent line bundle corresponding to the marked point. 8.7.1. Root stacks. We briefly review some properties of the root stacks d L M /M and d L J /J . Our primary reference is [Lie1], and more information can be found in [Lie2,§ §3.3 and 4], [Ols,§10.3], and [AGV, App. B]. The root stack d L M /M is the stack parametrizing d-th roots of the line bundle L M . In other words, d L M /M represents the functor which associates to every scheme φ : Y → M the groupoid of pairs (N , f ) where N is a line bundle on Y and f : N ⊗d → φ * L M is an isomorphism. An isomorphism between (N 1 , f 1 ) and (N 2 , f 2 ) is an isomorphism of line bundles g : N 1 → N 2 such that f 2 • g ⊗d = f 1 . One defines d L J /J similarly.
Since L J is the pullback of L M along π, one has a Cartesian diagram
(51) d L J /J d L M /M J M. d √ π p J p M π In particular, d L J /J → d L M /M is a G m -torsor. The stacks d L M /M and d L J /J have universal line bundles U M and U J = d √ π * U M such that U ⊗d M = p * M L M and U ⊗d J = p * J L J .
The key property that we will use about d L J /J (and d L M /M as well) is that its category of quasi-coherent sheaves has an eigendecomposition with respect to the action of the inertial group µ d of d-th roots of unity, and the degree zero component on d L J /J (resp., d L M /M) consists of pullbacks of quasi-coherent sheaves on J (resp., M). By [Lie1, Lemma 3.1.1.7], the eigensheaves for the trivial character on the root stack are identified with sheaves on the base stack M. This allows one to conclude that the pullback via p J is fully faithful, i.e., two quasi-coherent sheaves on J are isomorphic if and only if they are isomorphic when pulled back to d L J /J . (ii) For simplicity, we have given the details of the argument when the conformal dimensions are rational. This is indeed the case for simple modules over a C 2 -cofinite V , and allows for the use of a root stack defined by a line bundle. While not needed for this paper, an analogous argument can be made for complex, irrational conformal dimensions, using a gerbe that is a generalization of the root stack. 8.8. Proof of the VB corollary. By means of Theorems 8.4.2 and 8.5.1, one concludes that the sheaf of coinvariants V(V ; M • ) is a vector bundle of finite rank on Ë M g,n , and this gives rise to a vector bundle of finite rank on M g,n , as in [TUY], [Sor,§2.7], [Loo], [NT]. We sketch the argument for completeness.
First, we argue that V(V ; M • ) is a vector bundle of finite rank on Ë M g,n . The sheaf V(V ; M • ) on M g,n is coherent (Corollary 5.1.2) and is equipped with a projectively flat connection [DGT1]. As in [TUY], see also [Sor,§2.7], it follows that V(V ; M • ) is locally free of finite rank on M g,n . After Theorem 8.4.2 and gluing the sheaf as in [BL], it follows that the sheaf V(V ; M • ) is also coherent on Ë M g,n . It remains to show that V(V ; M • ) is locally free on Ë M g,n . For this, consider a stable family of n-pointed nodal curves (C 0 → Spec(R), P • ), and for simplicity, assume that it has only one simple node. Consider its formal smoothing (C → Spec(R q ), P • ) as described in §8.1. For each i, fix a formal coordinate t i at P i (S). The sewing theorem (Theorem 8.5.1) implies that V(V ; M • ) (C /S,P•,t•) is locally free of finite rank, hence we conclude the argument. For families of curves with more nodes, one proceeds similarly. It follows that V(V ; M • ) is a vector bundle of finite rank on Ë M g,n .
Finally, since V is rational, by §8.7 and Remark 8.7.3(i), we can descend V(V ; M • ) to a sheaf of coinvariants on M g,n . As the descent of a vector bundle is a vector bundle, this concludes the proof of the VB corollary.
Examples
Here we list examples of vertex operator algebras V satisfying the hypotheses of our theorems, namely: (1) V = i∈Z ≥0 V i with V 0 ∼ = C; (2) V is rational; and (3) V is C 2 -cofinite. 9.1. Virasoro VOAs. Given the Lie subalgebra Vir ≥0 := CK ⊕ zC z ∂ z of the Virasoro Lie algebra Vir, and c, h ∈ C, let M c,h := U (Vir) ⊗ U (Vir≥0) C1, where C1 inherits the structure of a Vir ≥0 -module by setting L p>0 1 = 0, L 0 1 = h1, and K1 = c1. There is a unique maximal proper submodule J c,h ⊂ M c,h . For h = 0, J c,0 contains a submodule generated by the singular vector L −1 1 ∈ M c,0 [FF]. Set If c = c p,q := 1 − 6(p−q) 2 pq , with relatively prime p, q ∈ N such that 1 < p < q, then M c ∼ = V ir c , that is, J c,0 = L −1 1 , while for c = c p,q , the submodule J c,0 is generated by two singular vectors [FF]. By [FZ,Thm 4.3], M c and V ir c are VOAs. Since A(M c ) ∼ = C[x] [FZ,Thm 4.6], M c is not rational. However, V ir c is rational if and only if c = c p,q [Wan,Thm 4.2] if and only if V ir c is C 2 -cofinite [DLM4,Lemma 12.3] (see also [Ara1,Prop. 3.4.1]). In this case, A(V ir c ) is a quotient of C[x], and simple V ir c -modules are the L c,h , for h = (np−mq) 2 −(p−q) 2 4pq with 0 < m < p and 0 < n < q. 9.2. Simple affine VOAs. One may associate to a finite-dimensional complex simple Lie algebra g and ℓ ∈ Z >0 , a simple vertex operator algebra L ℓ (g), described in [FZ,§2], [LL,§6.2] (see also [NT,§A.1.1]). This is rational by [FZ,Thm 3.1.3] and C 2 -cofinite by [Zhu2] (see also [DLM4,Prop. 12.6], [Ara1,Prop. 3.5.1]). 9.3. The moonshine module V ♮ . A rational vertex operator algebra V with no nontrivial simple V -modules is called holomorphic, and if (1) and (3) also hold, then V must have central charge divisible by 8 [DM1]. One example is given by the moonshine module V ♮ of central charge 24, whose automorphism group is the monster group [FLM1]. 9.4. Even lattice vertex algebras. Vertex operator algebras V L given by positive-definite even lattices L of finite rank [Bor] are rational [Don] and C 2 -cofinite [DLM4]. Zhu's algebra is described in [DLM1,Thm 3.4]. 9.5. Exceptional W -algebras. Arakawa [Ara3] has shown that a large class of simple W -algebras are C 2 -cofinite, including the minimal series principal W -algebras [FKW] and the exceptional W -algebras of Kac-Wakimoto [KW]. Moreover, the minimal series principal W -algebras and a large subclass of exceptional affine W -algebras are rational [Ara2,Ara4,AvE]. 9.6. Orbifolds, commutants, and tensor products. More vertex operator algebras can be obtained from the examples discussed above through standard constructions resulting in orbifold algebras, commutants, and tensor products. These constructions often preserve our desired properties. For instance, for a finite subgroup G of the automorphism group Aut(V ), the orbifold algebra V G is conjecturally C 2 -cofinite and rational when so is V . This holds for G solvable and V simple [Miy2,CM]. For a subalgebra A ⊂ V , the commutant Com(A, V ) [FZ] is conjecturally C 2 -cofinite and rational if so are both A and V . This holds for parafermions [DR]. The VOAs V 1 , . . . , V m are rational if and only if V = m i=1 V i is rational [DMZ]; in this case, if the V i are C 2 -cofinite, then so is V [DLM2].
Appendix A. Zhu's Lie algebra and isomorphic coinvariants For a smooth curve C and a quasi-primary generated vertex algebra V with V 0 ∼ = C, in addition to the chiral Lie algebra L C\P• (V ) ( §3.1), one also has Zhu's Lie algebra g C\P• (V ), reviewed in §A.1.
In Proposition A.2.1, we show that when defined, g C\P• (V ) is isomorphic to the image of L C\P• (V ) under the Lie algebra homomorphism ϕ L (Proposition A.2.1). Nagatomo and Tsuchiya extend the definition of g C\P• (V ) to stable pointed rational curves [NT], and they indicate that their coinvariants are equivalent to those studied by Beilinson and Drinfeld in [BD], suggesting they knew that Proposition A.2.1 holds in that case.
A quasi-primary vector is an element A ∈ V such that L 1 A = 0, and V is quasi-primary generated if and only if L 1 V 1 = 0 [DLM5]. A vertex algebra V = i≥0 V i with V 0 ∼ = C satisfies L 1 V 1 = 0 if and only if V ∼ = V ′ (see [FHL,§5.3] and [DM2,§2]). In particular, in the results of Huang [Hua3] and Codogni [Cod], the vertex algebras studied are quasi-primary generated.
A.1. The Lie algebra g C\P• (V ). In [Zhu2], given a smooth pointed curve (C, P • ) and a quasi-primary generated vertex operator algebra V for which V 0 ∼ = C, Zhu defines a Lie algebra g C\P• (V ), generalizing the construction of Tsuchiya, Ueno, and Yamada for affine Lie algebras. Namely, consider
g C\P• (V ) := ϕ g Ä ⊕ k≥0 V k ⊗ H 0 Ä C \ P • , ω ⊗1−k C ää where (53) ϕ g : ⊕ k≥0 V k ⊗ H 0 Ä C \ P • , ω ⊗1−k C ä → ⊕ n i=1 L t i (V ) is the map induced by B ⊗ µ → Ä Res t i =0 Y [B, t i ]µ P i (dt i ) k ä i=1,...,n .
Here t i is a formal coordinate at the point P i , Y [B, t i ] := k∈Z B [k] t −k−1 i , and µ P i is the Laurent series expansion of µ at P i , the image of µ via
H 0 Ä C \ P • , ω ⊗1−k C ä → H 0 Ä D × P i , ω ⊗1−k C ä ≃ t i C((t i ))(dt i ) 1−k .
When V is assumed to be quasi-primary generated with V 0 ∼ = C, Zhu shows that g C\P• (V ) is a Lie subalgebra of L(V ) ⊕n . The argument uses that any fixed smooth algebraic curve admits an atlas such that all transition functions are Möbius transformations. Transition functions between charts on families of curves of arbitrary genus are more general, hence the need to consider the more involved construction for the chiral Lie algebra based on the (Aut O)-twist of V in §2.
A.2. Isomorphism of coinvariants. When g C\P• (V ) is well-defined and C \ P • is affine, one can define the space of coinvariants M • g C\P• (V ) as the quotient of the L(V ) ⊕n -module M • by the action of the Lie subalgebra g C\P• (V ) of L(V ) ⊕n . These spaces were introduced in [Zhu1], and studied also in [AN, NT]. Recall the homomorphisms ϕ L from (24) and ϕ g from (53).
Proposition A.2.1. When g C\P• (V ) is well-defined ( §A.1), one has Im(ϕ L ) ∼ = Im(ϕ g ).
It follows that there exists an isomorphism of vector spaces
M • g C\P• (V ) ∼ = M • L C\P• (V ) .
Proof. One has
(54) ⊕ k≥0 V k ⊗ H 0 Ä C \ P • , ω ⊗1−k C ä ∼ = H 0 Ä C \ P • , ⊕ k≥0 V k ⊗ ω ⊗1−k C ä ∼ = H 0 C \ P • , ⊕ k≥0 Ä ω ⊗1−k C ä ⊕ dim V k .
From Lemma 2.6.1, one has gr
• V C ∼ = ⊕ k≥0 Ä ω ⊗−k C ä ⊕ dim V k . It follows that H 0 C \ P • , ⊕ k≥0 Ä ω ⊗1−k C ä ⊕ dim V k ∼ = H 0 C \ P • , gr • V C ⊗ ω C .
Now by Lemma 2.6.3,
H 0 (C \ P • , gr • V C ⊗ ω C ) ∼ = H 0 (C \ P • , V C ⊗ ω C ) .
On the other hand, as C \ P • is assumed to be affine, one has
L C\P• (V ) ∼ = H 0 (C \ P • , V C ⊗ ω C ) /∇H 0 (C \ P • , V C ) .
The map ϕ L is induced from the composition
(56) H 0 (C \ P • , V C ⊗ ω C ) → ⊕ n i=1 H 0 Ä D × P i , V C ⊗ ω C ä → ⊕ n i=1 L t i (V )
. The first map is canonical and obtained by restricting sections; the second is (22). By [FBZ,§6.6.9], sections in ∇H 0 Ä D × P i , V C ä act trivially. Hence (56) induces a map from the Lie algebra L C\P• (V ) to ⊕ n i=1 L t i (V ). It follows that the image of ϕ L coincides with the image of H 0 (C \ P (56). Composing (54) and (55), and by the definition of ϕ g in (53), the image of the map in (56) coincides with the image of ϕ g .
• , V C ⊗ ω C ) in ⊕ n i=1 L t i (V ) via
for all local sections f of O C and s of V C . We will use that on open sets U ⊂ C where ω C is trivial, one can describe a connection as an endomorphism of V C | U . 2.7.1. The connection on smooth curves. Let C be a smooth curve. On an open subset U of C admitting a global coordinate t (for instance, on an open U admitting anétale map
The normalization map η : ‹ C → C identifies L C\P• (V ) with a Lie subalgebra of L ‹ C\P•⊔Q• (V ).Moreover, this induces an action of L C\P• (V ) on L(V ) ⊕n -modules.
V
(V ; M • ⊔ (V, . . . , V )) (C,P•⊔Q•,t•⊔s•)
4. 3 .
3Propagation of vacua. The propagation of vacua theorem, first proved by Tsuchiya, Ueno, and Yamada for spaces of coinvariants constructed from representations of affine Lie algebras [TUY, Prop 2.2.3, Cor 2.2.4], says that spaces of coinvariants associated to a stable n-pointed curve with coordinates remain invariant when adding a new marked point and the trivial module.
, sections of ω C /S over D Q are generated by Ä ds + s + , − ds − s − ä over " O Q , and the definition of ϑ from (7). 8.4. Sheaf of coinvariants over formal smoothings. Let (C /S, P • ) be as in §8.1. Fix formal coordinates t i at P i (S), for i = 1, . . . , n. Given V -modules M 1 , . . . , M n , let M • := ⊗ n i=1 M i . One defines the sheaf of coinvariants V(V ; M • ) (C /S,P•,t•) as in §4, that is:
As in [DGT1, §6.3.1], (Aut + O) ⊕n acts on V(V ; M • ), and descending along the (Aut + O) ⊕n -torsor in (50), one obtains a sheaf of coinvariants V J (V ; M • ) on J × g,n .
8. 7 . 2 .
72The final descent. Let V J := V J (V ; M ). The quasi-coherent sheaf p * J V J ⊗ U J on d L J /J has an action of G m as in [DGT1, § §4.2.1, 6.3.2] 1 . Descending along the G m -torsor d L J /J → d L M /M, we obtain a sheaf F on d L M /M for which (i) When V is C 2 -cofinite and rational, we can descend V J (V ; M ) to a sheaf V(V ; M ) over M g,n for every finitely-generated V -module M . Indeed, since the category of V -modules is semisimple, we can decompose M = ⊕ ℓ∈I M ℓ with M ℓ a simple V -module with rational conformal dimension and I a finite index set. This induces the decomposition V J (V ; M ) = ⊕ ℓ∈I V J (V ; M ℓ ), and we can apply the descent argument of §8.7.2 to each component V J (V ; M ℓ ).
(iii) When M i = V for all i, the action of G ⊕n m on V J (V ; M • ) from [DGT1, § §4.2.1, 6.3.2] is compatible with the restriction of the action of (Aut O) ⊕n . In this case, the above construction simplifies, since the two descents along the (Aut + O) ⊕n -torsor and G ⊕n m -torsor in (50) are equivalent to the descent along the (Aut O) ⊕n -torsor. (iv) The method used to construct the sheaf V(V ; M • ) on M g,n from V J (V ; M • ) guarantees that, when V(V ; M • ) is of finite rank on M g,n , the Chern character of V(V ; M • ) on M g,n is given by [DGT1, Cor 9.1].
L c,h := M c,h /J c,h , M c := M c,0 / L −1 1 ,and V ir c := L c,0 .
]; over stable curves with singularities, see [DGT1, §3.2]. 2.3. Action of Aut O on V . Let Der 0 O be the Lie algebra functor attached to the group functor Aut O in the sense of
Prop. 7.2.1]. The statement follows by linearity. 6.2. Replacing coinvariants with Z. The main result of this section is Proposition 6.2.1, which generalizes [NT, Prop. 7.2.2, Cor. 8.6.2] to curves of arbitrary genus. The statement describes coinvariants of the action of a Lie subalgebra L C\P•
For this action, one uses the filtration on V J (V ; M ) induced by the Z ≥0 -grading of the module M .
AcknowledgementsWe thank Dennis Gaitsgory for questions which led to the need to replace V C with V C . We thank Igor Frenkel, Bin Gui, Yi-Zhi Huang, Danny Krashen, Jim Lepowsky, and Sven Möller for helpful discussions. Thanks also to Yi-Zhi Huang and Giulio Codogni for comments on a preliminary version of the manuscript, and to André Henriques, on a later version. We are indebted to [FBZ] for their work on smooth curves, and to [NT] for the arguments on the factorization and sewing for coinvariants by Zhu's Lie algebra on rational curves. Finally, we wish to thank the referees for carefully reading our work, and one generous referee in particular, who has provided many valuable comments. Gibney was supported by NSF DMS-1902237.By its construction and the commutativity of (51), one hasSince the pullback of sheaves to a root stack is fully faithful, we deduce that V J = π * V(V ; M ). In particular, V J descends to a sheaf V(V ; M ), which is therefore well-defined on M g,1 .Remark 8.7.3. Here we list some observations.
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[] |
[
"LARGE, GLOBAL SOLUTIONS TO THE THREE-DIMENSIONAL THE NAVIER-STOKES EQUATIONS WITHOUT VERTICAL VISCOSITY",
"LARGE, GLOBAL SOLUTIONS TO THE THREE-DIMENSIONAL THE NAVIER-STOKES EQUATIONS WITHOUT VERTICAL VISCOSITY"
] |
[
"Isabelle Gallagher ",
"Alexandre Yotopoulos "
] |
[] |
[] |
The three-dimensional, homogeneous, incompressible Navier-Stokes equations are studied in the absence of viscosity in one direction. It is shown that there are arbitrarily large initial data generating a unique global solution, the main feature of which is that they are slowly varying in the direction where viscosity is missing. The difficulty arises from the complete absence of a regularising effect in this direction. The special structure of the nonlinear term, joint with the divergence-free condition on the velocity field, is crucial in obtaining the result.
| null |
[
"https://arxiv.org/pdf/2202.11536v1.pdf"
] | 247,058,780 |
2202.11536
|
d018f86640b0bc8f5fdd24a6ea5e2ac85e2d452b
|
LARGE, GLOBAL SOLUTIONS TO THE THREE-DIMENSIONAL THE NAVIER-STOKES EQUATIONS WITHOUT VERTICAL VISCOSITY
23 Feb 2022
Isabelle Gallagher
Alexandre Yotopoulos
LARGE, GLOBAL SOLUTIONS TO THE THREE-DIMENSIONAL THE NAVIER-STOKES EQUATIONS WITHOUT VERTICAL VISCOSITY
23 Feb 2022
The three-dimensional, homogeneous, incompressible Navier-Stokes equations are studied in the absence of viscosity in one direction. It is shown that there are arbitrarily large initial data generating a unique global solution, the main feature of which is that they are slowly varying in the direction where viscosity is missing. The difficulty arises from the complete absence of a regularising effect in this direction. The special structure of the nonlinear term, joint with the divergence-free condition on the velocity field, is crucial in obtaining the result.
Introduction and énoncé du résultat
The incompressible homogeneous Navier-Stokes equations in d space dimensions are written as
(NS) ∂ t u + u · ∇u − ∆u = −∇p div u = 0 u |t=0 = u 0 ,
where p = p(t, x) and u = (u 1 , . . . , u d )(t, x) are respectively the pressure and the velocity field of a viscous incompressible fluid. The viscosity has been set to 1 for simplicity. We are interested here in the case where the viscosity of the fluid is strongly anisotropic: the Laplacian acts only in the horizontal coordinates. By defining ∆ h := ∂ 2 1 + ∂ 2 2 , the system of equations writes
(NS) h ∂ t u + u · ∇u − ∆ h u = −∇p in R + × Ω div u = 0 u |t=0 = u 0 .
The spatial domain Ω will be in the following T 2 × R or T 3 , where T d := (R /Z ) d is the ddimensional torus. The assumption of zero vertical viscosity originates from the study of geophysical fluids, notably the oceans where the viscosity, known as "turbulent", is often much weaker in the vertical variable [9].
Before presenting the results obtained in this article concerning (NS) h , let us recall some known results on these equations starting with the case of (NS), where the viscosity is isotropic (we refer for example to [2,25,26] for details and more references). In the case where the Laplacian acts in the three directions of space it is well known since the work of J. Leray [27] that for any initial data u 0 in the space L 2 (Ω) there exists a global distributional solution u to (NS), of finite energy in the sense that
∀t ≥ 0 , 1 2 u(t) 2 L 2 (Ω) + t 0 ∇u(t ′ ) 2 L 2 (Ω) dt ′ ≤ 1 2 u 0 2 L 2 (Ω) , .
Let us recall that in dimension two space J. Leray shows in [28] the uniqueness of this finite energy solution (and it verifies the energy equality). Uniqueness in the three dimensional
case is an open problem, although recent results (see [20] for numerical evidence, [6] for distributional solutions and [1] in the presence of forcing) tend to indicate that the Leray solution to (NS) may well be non unique.
In the anisotropic context of (NS) h , on the other hand, the absence of compactness in the vertical direction makes the proof of [27] inoperative and the global existence of weak solutions is not known.
Concerning the existence of unique solutions in the isotropic framework, the Cauchy theory is, as often for evolution PDEs, related to the scale invariance of the equation: for all λ > 0, if u is a solution of (NS) associated to the data u 0 then λu(λ 2 t, λx) is a solution of (NS) associated to the data λu 0 (λx). This scale invariance remains for (NS) h . Let us give some examples of scale invariant spaces for the initial data: first we recall the definition of homogeneous Sobolev spaces H s (R d ), given by the norm (for s < d/2) 1 2 where f is the Fourier transform of f . In the case of periodic or hybrid boundary conditions considered in this article, the definition becomes
f H s (R d ) := R d | f (ξ)| 2 |ξ| 2s dξf H s (T 3 ) := n∈N 3 | f (n)| 2 |n| 2s 1 2 and f H s (T 2 ×R) := n∈N 2 R | f (n, ξ)| 2 (|n| 2 +|ξ| 2 ) s dξ 1 2 .
The spaces H 1 2 (Ω) and L 3 (Ω) are scale invariant. The existence of unique solutions to (NS), in short time (globally in time under a smallness condition on the initial data), is known for an initial data in the space H 1 2 since H. Fujita and T. Kato [16], in L 3 since [18,22,33]. In the framework of Besov spaces B −1+ 3 p p,∞ for p < ∞ (see Definition 1.1 below) we know that a global solution exists for small data since [32]. The best result in this context is due to H. Koch and D. Tataru [23]: these authors prove by a fixed point argument (as it is the case for all the uniqueness results mentioned above) that (NS) is globally wellposed under a smallness condition on
u 0 BMO −1 := sup t>0 t 1 2 e t∆ u 0 L ∞ + sup x∈R 3 R>0 1 R 3 2 [0,R 2 ]×B(x,R) |(e t∆ u 0 )(t, y)| 2 dydt 1 2 ,
where B(x, R) is the ball centred at x with radius R. The space BMO −1 , like the other spaces mentioned above, is invariant by the change of scale of the equation. Note that the norm sup t>0 t 1 2 e t∆ u 0 L ∞ which appears above is equivalent to the Besov norm u 0 B −1 ∞,∞ (see (1.1) for the equivalence). This space B −1 ∞,∞ is in fact the space into which any Banach space of scale-invariant tempered distributions is embedded, see [29] -if we want to define a notion of "large" initial data for (NS), it is thus in B −1 ∞,∞ that it should be measured; we refer the reader to the Appendix for more information on these spaces, whose definition we recall below -as well as that of the anisotropic Besov spaces which are used in this article and which are modelled on them. We note ξ = (ξ h , ξ 3 ) = (ξ 1 , ξ 2 , ξ 3 ) with ξ h ∈ Z 2 and ξ 3 ∈ R the Fourier variables on T 2 × R (and ξ 3 ∈ Z if Ω = T 3 ). Definition 1.1 (Isotropic Besov spaces). Let χ be a radial function in D(R) such that χ(t) = 1 for |t| ≤ 1 and χ(t) = 0 for |t| > 2. For all q ∈ Z we define the frequency truncation operators
S q f (ξ) := χ 2 −q |ξ| f (ξ) and ∆ q := S q+1 − S q .
For any p, r in [1, ∞] and any s in R, with s < 3/p (or s ≤ 3/p if r = 1), the homogeneous Besov space B s p,r is the space of tempered distributions f such that f B s p,r := 2 qs ∆ q f L p ℓ r < ∞ . The Sobolev space H s corresponds to the choice p = r = 2. It is well-known (see [2] for instance) that the Besov norm has an equivalent formulation via the heat flow
(1.1) ∀s < 0 , f ∈ f B s p,r ⇐⇒ t − s 2 e t∆ f L p L r ( dt t ) < ∞ .
The anisotropic Besov spaces used in this text are of two types, defined below.
Definition 1.2 (Anisotropic Besov spaces).
With the notations of Definition ??, the horizontal frequency truncation operators are defined for j ∈ Z by
S h j f (ξ) := χ 2 −j |ξ h | f (ξ) and ∆ h j := S h j+1 − S h j , and the vertical frequency truncations for q ∈ Z by S v q f (ξ) := χ(2 −q |ξ 3 |) f (ξ) and ∆ v q := S v q+1 − S v q . For all s ≤ 2/p and s ′ ≤ 1/p, the Besov space B s,s ′ is the space of tempered distributions f such that f B s,s ′ := j,q 2 js+qs ′ ∆ h j ∆ v q f L 2 < ∞ , and the Besov space B 0,s is the space of tempered distributions f such that f B 0,s := q 2 qs ∆ v q f L 2 < ∞ .
Let us notice that B 0,s is continuously embedded in B 0,s , since
(1.2) q 2 qs ∆ v q f L 2 ≤ j,q 2 qs ∆ h j ∆ v q f L 2 .
Note that the anisotropic Besov space B 0, 1 2 appears naturally here because it is modelled on the space L 2 (T 2 ;Ḣ 1 2 (R)) -which is a natural space in the context of (NS) h since L 2 (T 2 ) is associated with the two-dimensional equation andḢ 1 2 is scale-invariant in dimension three -while being a Banach space (unlikeḢ 1 2 (R) for example, and thus L 2 (T 2 ;Ḣ 1 2 (R))). This space was introduced in this context by M. Paicu in [30], who showed the global existence in time of solutions to (NS) h for small data (local in time for any data) in B 0, 1 2 . He also showed the uniqueness of solutions in L ∞ (R + , B 0, 1 2 ) whose horizontal gradient is in L 2 (R + , B 0, 1 2 ). In the present work we seek, in the spirit of the works [10,11,12] for example, to exhibit initial data which may be arbitrarily large but for which there is a unique associate global solution to (NS) h . The natural context, following these works, is to consider initial data varying slowly in one direction, and the specificity of this paper is to assume that this direction is the same as the one in which there is no viscosity (the vertical one for instance). This work thus follows a series of works concerning either the slowly varying case or the anisotropic equation (see for example [3,4,8,10,11,12,14,15,19,21,30,31]). To our knowledge, this is the first time that the slowly variable character in one direction, which allows to obtain global solutions without any smallness assumption on the initial data, is mixed with the absence of vertical viscosity in the equation. This leads to be particularly careful in the estimates since no regularising effect is possible in the vertical direction. In particular the special structure of the nonlinear term, joint with the condition that the velocity field is divergence free, will be crucial in the analysis. The result is as follows. 1 2 ) and such that ∇ h u ε belongs to L 2 (R + , B 0, 1 2 ).
Theorem 1. Let u h 0 = (u 1 0 , u 2 0 ) and w 0 = (w 1 0 , w 2 0 , w 3 0 ) be two divergence free vector fields with (u h 0 , w 3 0 ) belonging to B 0, 1 2 ∩ B −1, 5 2 . Let, for all ε ∈ (0, 1), (1.3) u ε 0 (x) := (u h 0 + εw h 0 , w 3 0 )(x 1 , x 2 , εx 3 ) . For ε small enough, there is a unique global solution u ε to (NS) h associated to the initial data (1.3), in the space L ∞ (R + , B 0,
Remark 1.3. It is shown in [10] that a function of the form
h ε (x) = f (x h )g(εx 3 )
with f and g in the Schwartz class, verifies
h ε B −1 ∞,∞ ≥ 1 4 f Ḃ −1 ∞,∞ g L ∞ so can be as large as desired in B −1 ∞,∞ . Remark 1.4.
The periodic character of the horizontal variable allows us to obtain good regularity estimates on w h from estimates on w 3 via the identity
w h = −∇ h (∆ h ) −1 ∂ 3 w 3 , ∇ h := (∂ 1 , ∂ 2 ) ,
because ∂ 3 w 3 has a zero horizontal mean (see Section 3). In the case where w 0 ≡ 0, the proof of Theorem 1 shows that we can assume indifferently that the horizontal variable is in R 2 or in T 2 .
From now on we note
f ε (x) := f (x h , εx 3 )
. The method of proving the Theorem 1 consists in looking for the solution u ε , and the associated pressure p ε , in the form
(1.4) u ε = u ε app + R ε , p ε = p ε app + q ε with u ε app = u h + εw h , w 3 ε , p ε app = p 0 + εp 1 ε where for all y 3 , u h (·, y 3 )
is a solution of the two-dimensional Navier-Stokes equations with initial data u h 0 (·, y 3 ):
(NS2D) y 3 ∂ t u h + u h · ∇ h u h − ∆ h u h = −∇ h p 0 in R + × T 2 div h u h = 0 u h |t=0 = u h 0 (·, y 3 )
, and w is a solution of the linear equation
(T) ∂ t w + u h · ∇ h w − ∆ h w = −(∇ h p 1 , 0) in R + × Ω div w = 0 w |t=0 = w 0 .
Since u h 0 (·, y 3 ) belongs to L 2 (T 2 ), then given y 3 there is a unique global solution u h to (NS2D) y 3 , in the energy space L ∞ (R + ; L 2 (T 2 )) ∩ L 2 (R + ;Ḣ 1 (T 2 )). The vector field w also exists uniquely for all times (this will be clarified in Section 3), and the main part of the work consists therefore in solving globally in time, for ε sufficiently small, the (perturbed anisotropic Navier-Stokes) equation verified by R ε .
The plan of the article is as follows. In Section 2 we show that for ε sufficiently small, the equation verified by R ε has a global solution: this proof relies on a priori estimates on the approximate solution u ε app which are derived in Section 3 from estimates on (NS2D) y 3 and on (T). The Appendix is devoted to the recollection of classical results concerning the functional spaces appearing in the present work, as well as two important trilinear estimates which can be found in the literature.
Throughout this article and unless otherwise stated, we will note by C a universal constant, in particular independent of ε, which can change from one line to another. We will sometimes note a b if a ≤ Cb. We will note q ∼ q ′ if q ∈ [q ′ − C, q ′ + C]. We will note generically by (s q ) q∈Z a sequence of positive real numbers such that q∈Z s 1 2 q ≤ 1 and by (d q ) q∈Z a sequence of positive real numbers such that
q∈Z d q ≤ 1. Finally, if X is a function space on T 2 and Y a function space on R, we write X h Y v := X(T 2 ; Y (R)) and similarly Y v X h := Y (R; X(T 2 )).
2. Proof of the theorem 2.1. Main steps of the proof. Recalling the notation (1.4), let us write the system of equations verified by R ε . We have
(2.1) ∂ t R ε + R ε · ∇R ε + u ε app · ∇R ε + R ε · ∇u ε app − ∆ h R ε = −∇q ε + εF ε with F ε := ε w h · ∇ h (w h , 0) + w 3 ∂ 3 (w h , 0) ε + w · ∇(u h , w 3 ) ε + 0, [∂ 3 (p 0 + εp 1 )] ε ,
and R ε |t=0 = 0. To prove the global existence of R ε we write an a priori estimate on R ε
L∞ (R + ;B 0, 1 2 ) and ∇ h R ε L2 (R + ;B 0, 1 2 )
(the definition of these spaces is recalled in the Appendix) and omit the classical step of regularization of the system to justify the estimates. Moreover, as recalled in the introduction, only the global existence of solutions has to be proved since the uniqueness for (NS) h in our functional framework is a consequence of [30].
In order to absorb the linear terms in (2.1) we use a Gronwall-type argument, but in the context ofL p spaces in time (see the Appendix for a definition of these spaces). One strategy (see [17] for example) is to write a partition of R + into time intervals
(2.2) R + = K−1 k=0 [t k , t k+1 [
as in Proposition 3.1 below, which depends on a constantC which will be fixed at the end. We then write, following [30], an energy estimate in L 2 on ∆ v q R ε and it comes after integration on a time interval [t k , t k+1 ]
(2.3) 1 2 ∆ v q R ε (t k+1 ) 2 L 2 + t k+1 t k ∇ h ∆ v q R ε (t) 2 L 2 dt ≤ 1 2 ∆ v q R ε (t k ) 2 L 2 + t k+1 t k (∆ v q (R ε · ∇R ε )|∆ v q R ε ) L 2 (t) dt + t k+1 t k (∆ v q (u ε app · ∇R ε )|∆ v q R ε ) L 2 (t) + (∆ v q (R ε · ∇u ε app )|∆ v q R ε ) L 2 (t) dt ′ + ε t k+1 t k (∆ v q F ε |∆ v q R ε ) L 2 (t) dt .
We note that R ε (t 0 ) = R ε (0) = 0. Let us introduce the notatioñ
L r k X :=L r ([t k , t k+1 ]; X) .
From (A.7) we know that Thanks to Young's inequality
(2.4) t k+1 t k (∆ v q (R ε · ∇R ε )|∆ v q R ε ) L 2 (t) dt 2 −q s q ∇ h R ε 2 L 2 k B 0, 1 2 R ε L∞ k B 0, 1 2 and from (A.8) t k+1 t k (∆ v q (u ε app · ∇R ε )|∆ v q R ε ) L 2 (t) dt 2 −q s q R ε 1 2 L ∞ k B 0, 1 2 ∇ h R ε L2 k B 0, 1 2 × ∇ h R ε 1 2 L 2 k B 0, 1 2 u ε app 1 2 L ∞ k B 0, 1 2 ∇ h u ε app 1 2 L 2 k B 0, 1 2 + ∇ h u ε app L2 k B 0, 1 2 R ε 1 2 L ∞ k B 0(2.5) ab a p + b p ′ , 1 p + 1 p ′ = 1 this last inequality can also be written (2.6) t k+1 t k (∆ v q (u ε app · ∇R ε )|∆ v q R ε ) L 2 (t) dt ≤ 2 −q s q 1 4 ∇ h R ε 2 L 2 k B 0, 1 2 + C R ε 2 L ∞ k B 0, 1 2 ∇ h u ε app 2 L 2 k B 0, 1 2 (1 + u ε app 2 L ∞ k B 0, 1 2 ) .
The end of the proof of the theorem relies on the following two lemmas, which are proved respectively in Paragraph 2.2 and in Section 3. q ≤ 1 and such that
(2.7) t k+1 t k (∆ v q (R ε · ∇u ε app )|∆ v q R ε ) L 2 (t) dt ≤ 2 −q s q 1 4 ∇ h R ε 2 L 2 k B 0, 1 2 + C R ε 2 L ∞ k B 0, 1 2 u ε app 2 L 2 k B 1, 1 2 (1 + u ε app 2 L ∞ k B 0, 1 2 ) + ∂ 3 u ε app L1 k B 1,(2.8) 2 q t k+1 t k (∆ v q F ε |∆ v q R ε ) L 2 (t) dt ≤ Cs q R ε L∞ k B 0, 1 2 .
Let us return to (2.3). By gathering (2.4), (2.6), (2.7) and (2.8) we get By taking the square root of the two sides of the equation and summing over q ∈ Z we find that R ε
2 q 2 ∆ v q R ε (t k+1 ) 2 L 2 + 2 q t k+1 t k ∇ h ∆ v q R ε (t) 2 L 2 dt ≤ 2 q 2 ∆ v q R ε (t k ) 2 L 2 + Cs q ∇ h R ε 2 L 2 k B 0, 1 2 R ε L∞ k B 0, 1 2 + Cεs q R ε L∞ k B 0, 1 2 + s q 2 ∇ h R ε 2 L 2 k B 0, 1 2 + Cs q R ε 2 L ∞ k B 0, 1 2 u ε app 2 L 2 k B 1, 1 2 (1 + u ε app 2 L ∞ k B 0, 1 2 ) + ∂ 3 u ε app L1 k B 1,L∞ k B 0, 1 2 + ∇ h R ε L2 k B 0, 1 2 ≤ R ε (t k ) B 0, 1 2 + R ε 1 2 L ∞ k B 0, 1 2 ∇ h R ε L2 k B 0, 1 2 + C √ ε + C R ε L∞ k B 0, 1 2 u ε app L2 k B 1, 1 2 (1 + u ε app L∞ k B 0, 1 2 ) + ∂ 3 u ε app 1 2 L 1 k B 1, 1 2 .
It is then sufficient to choose the partition (2.2) thanks to Proposition 3.1 so that
u ε app L2 k B 1, 1 2 (1 + u ε app |L ∞ k B 0, 1 2 ) ≤ 1 4C
=: 1 C and to choose, still thanks to Proposition 3.1, ε small enough so that
∂ 3 u ε app 1 2 L 1 k B 1, 1 2 ≤ 1 4C ·
Then the above inequality becomes
1 2 R ε L∞ k B 0, 1 2 + ∇ h R ε L2 k B 0, 1 2 ≤ C R ε (t k ) B 0, 1 2 + C R ε 1 2 L ∞ k B 0, 1 2 ∇ h R ε L2 k B 0, 1 2 + Cε . Let now T ε be the maximum time for which ∀t ≤ T ε , R ε 1 2 L ∞ ([0,t];B 0, 1 2 ) ≤ 1 2C ·
Then as long as t k+1 ≤ T ε we have
1 2 R ε L∞ k B 0, 1 2 + 1 2 ∇ h R ε L2 k B 0, 1 2 ≤ C R ε (t k )
B 0, 1 2 + Cε and as R ε (t 0 ) B 0, 1 2 = 0, by iterating K times this inequality we find (see [17,4] for example) that T ε = ∞ and that there exists a constant C 0 (depending on u h 0 and w 0 via Proposition 3.1) such that R ε L∞ (R + ;B 0, 1 2 )
+ 1 2 ∇ h R ε L2 (R + ;B 0, 1 2 ) ε exp C 0 .
This concludes the proof of Theorem 1.
Proof of Lemma 2.1.
We begin by noting that since the divergence of R ε is zero
− ∆ v q (R ε · ∇u ε app )|∆ v q R ε L 2 = 2 ℓ=1 ∆ v q (R ε,ℓ u ε app )|∂ ℓ ∆ v q R ε L 2 + ∆ v q (R ε,3 u ε app )|∂ 3 ∆ v q R ε L 2 .
We set
I q := 2 ℓ=1 ∆ v q (R ε,ℓ u ε app )|∂ ℓ ∆ v q R ε L 2 and J q := ∆ v q (R ε,3 u ε app )|∂ 3 ∆ v q R ε L 2 .
Let us start by studying the contribution of I q . There holds
(2.9) 2 q t k+1 t k I q (t) dt 2 q 2 ∆ v q (R ε,h u ε app ) L 2 k L 2 2 q 2 ∆ v q ∇ h R ε L 2 k L 2 2 q 2 ∆ v q (R ε,h u ε app ) L 2 k L 2 d q ∇ h R ε L2 k B 0, 1 2
with the generic notation presented in the introduction: (d q ) q∈Z is a sequence of positive real numbers such that q∈Z d q ≤ 1 .
We also recall the notationL r k X =L r ([t k , t k+1 ]; X). We then use the Bony decomposition into paraproduct and remainder (A.5) which allows us to write (2.10)
2 q 2 ∆ v q (R ε,ℓ u ε app ) L 2 k L 2 2 q 2 q ′ ∼q S v q ′ −1 R ε L 4 k L 4 h L ∞ v ∆ v q ′ u ε app L 4 k L 4 h L 2 v + 2 q 2 q ′ ∼q S v q ′ −1 u ε app L 4 k L 4 h L ∞ v ∆ v q ′ R ε L 4 k L 4 h L 2 v + 2 q 2 q ′ 2 q q ′′ ∼q ′ ∆ v q ′′ u ε app L 4 k L 4 h L 2 v ∆ v q ′ R ε L 4 k L 4 h L 2 v =: T 1 qh + T 2 qh + R qh ,
where we used Bernstein's inequality (A.1) in the last inequality:
∆ v q (R ε,ℓ u ε app ) L 2 k L 2 2 q 2 ∆ v q (R ε,ℓ u ε app ) L 2 k L 2 h L 1 v .
Let us estimate each of the terms in succession. For T 1 qh we start by noting that for any function a and any x 3 ∈ R we have thanks to the Sobolev embedding H
1 2 (T 2 ) ⊂ L 4 (T 2 ), a(·, x 3 ) L 4 (T 2 ) a(·, x 3 ) H 1 2 (T 2 ) a(·, x 3 ) 1 2 L 2 (T 2 ) ∇ h a(·, x 3 ) 1 2 L 2 (T 2 )
so by the Cauchy-Schwarz inequality in x 3 there holds
a L 2 v L 4 h a 1 2 L 2 ∇ h a 1 2 L 2 . Using again Bernstein's inequality (A.1) ∆ v q ′′ R ε L 4 k L 4 h L ∞ v 2 q ′′ 2 ∆ v q ′′ R ε L 4 k L 4 h L 2 v
and then Minkowski's inequality
∆ v q ′′ R ε L 4 k L 4 h L 2 v ≤ ∆ v q ′′ R ε L 4 k L 2 v L 4 h we gather S v q ′ −1 R ε L 4 k L 4 h L ∞ v 2 q ′′ 2 q ′ 2 q ′′ 2 ∆ v q ′′ R ε L 4 k L 2 v L 4 h 2 q ′′ 2 q ′ 2 q ′′ 2 ∆ v q ′′ R ε 1 2 L ∞ k L 2 ∆ v q ′′ ∇ h R ε 1 2 L 2 k L 2 R ε 1 2 L ∞ k B 0, 1 2 ∇ h R ε 1 2 L 2 k B 0, 1 2 by the Cauchy-Schwarz inequality. Similarly ∆ v q ′ u ε app L 4 k L 4 h L ∞ v 2 q ′ 2 ∆ v q ′ u ε app 1 2 L ∞ k L 2 ∆ v q ′ ∇ h u ε app 1 2 L 2 k L 2 d q ′ u ε app 1 2 L ∞ k B 0, 1 2 ∇ h u ε app 1 2 L 2 k B 0, 1 2 hence finally T 1 qh d q R ε 1 2 L ∞ k B 0, 1 2 ∇ h R ε 1 2 L 2 k B 0, 1 2 u ε app 1 2 L ∞ k B 0, 1 2 ∇ h u ε app 1 2 L 2 k B 0, 1 2 ,
and symmetrically
T 2 qh d q R ε 1 2 L ∞ k B 0, 1 2 ∇ h R ε 1 2 L 2 k B 0, 1 2 u ε app 1 2 L ∞ k B 0, 1 2 ∇ h u ε app 1 2 L 2 k B 0, 1 2 .
Finally R qh can be estimated by an analogous argument. One has indeed
R qh = 2 q 2 q ′ 2 q q ′′ ∼q ′ ∆ v q ′′ u ε app L 4 k L 4 h L 2 v ∆ v q ′ R ε L 4 k L 4 h L 2 v 2 q ′ 2 q q ′′ ∼q ′ 2 q−q ′′ 2 q ′′ 2 ∆ v q ′′ u ε app 1 2 L ∞ k L 2 ∆ v q ′′ ∇ h u ε app 1 2 L ∞ k L 2 2 q ′′ 2 ∆ v q ′ R ε 1 2 L ∞ k L 2 ∆ v q ′ ∇ h R ε 1 2 L ∞ k L 2
hence by Young's inequality for series
R qh d q R ε 1 2 L ∞ k B 0, 1 2 ∇ h R ε 1 2 L 2 k B 0, 1 2 u ε app 1 2 L ∞ k B 0, 1 2 ∇ h u ε app 1 2 L 2 k B 0, 1 2 .
In conclusion, by returning to (2.9) we obtain
2 q t k+1 t k I q (t) dt s q R ε 1 2 L ∞ k B 0, 1 2 ∇ h R ε 3 2 L 2 k B 0, 1 2 u ε app 1 2 L ∞ k B 0, 1 2 ∇ h u ε app 1 2 L 2 k B 02 q t k+1 t k I q (t) dt ≤ s q 100 ∇ h R ε 2 L 2 k B 0, 1 2 + Cs q R ε 2 L ∞ k B 0, 1 2 u ε app 2 L 2 k B 1, 1 2 u ε app 2 L ∞ k B 0, 1 2 .
Let us now study the contribution of J q . We notice that by (A.1)
∂ 3 ∆ v q R ε L 2 2 q ∆ v q R ε L 2 hence (2.12) 2 q t k+1 t k J q (t)dt 2 2q ∆ v q (R ε,3 u ε app ) L 1 k L 2 ∆ v q R ε L ∞ k L 2 2 3q 2 ∆ v q (R ε,3 u ε app ) L 1 k L 2 d q R ε L∞ k B 0, 1
2 , so we proceed as above by decomposing the first term into paraproduct and remainder :
(2.13) 2 3q 2 ∆ v q (R ε,3 u ε app ) L 1 k L 2 2 3q 2 q ′ ∼q S v q ′ −1 R ε,3 L ∞ k L 2 h L ∞ v ∆ v q ′ u ε app L 1 k L ∞ h L 2 v + 2 3q 2 q ′ ∼q S v q ′ −1 u ε app L 2 k L ∞ ∆ v q ′ R ε,3 L 2 k L 2 + 2 2q 2 q ′ 2 q q ′′ ∼q ′ ∆ v q ′′ u ε app L 1 k L ∞ h L 2 v ∆ v q ′ R ε,3 L ∞ k L 2 =: T 1 q3 + T 2 q3 + R q3
. Note on the one hand that
T 1 q3 2 3q 2 2 q ′′ 2 q ′ q ′ ∼q 2 q ′′ 2 ∆ v q ′′ R ε L ∞ k L 2 ∆ v q ′ u ε app L 1 k L ∞ h L 2 v R ε L∞ k B 0, 1 2 q ′ ∼q 2 3q ′ 2 ∆ v q ′ u ε app L 1 k L ∞ h L 2 v
and thanks to (A.4) we therefore have (2.14)
T 1 q3 d q R ε L∞ k B 0, 1 2 ∂ 3 u ε app L 1 k B 1, 1
2 . The second term of the decomposition can thus be estimated by (A.2):
T 2 q3 2 3q 2 2 q ′′ 2 q ′ q ′ ∼q 2 q ′′ 2 ∆ v q ′′ u ε app L 2 k L ∞ h L 2 v ∆ v q ′ R ε,3 L 2 k L 2 2 q 2 2 q ′′ 2 q ′ q ′ ∼q 2 q ′′ 2 ∆ v q ′′ u ε app L 2 k L ∞ h L 2 v ∆ v q ′ ∂ 3 R ε,3 L 2 k L 2 u ε app L2 k B 1, 1 2 2 q 2 q ′ ∼q ∆ v q ′ ∇ h R ε L 2 k L 2
as above and thanks to (A.4) and to the fact that ∂ 3 R ε,3 = −div h R ε,h . It follows that
(2.15) T 2 q3 d q u ε app L2 k B 1, 1 2 ∇ h R ε L2 k B 0, 1 2 .
Finally for the remainder term we write, again by (A.2),
(2.16) R q3 2 2q 2 q ′ 2 q q ′′ ∼q ′ 2 −q ′′ ∆ v q ′′ ∂ 3 u ε app L 1 k L ∞ h L 2 v ∆ v q ′ R ε L ∞ k L 2 2 q ′ 2 q q ′′ ∼q ′ 2 2(q−q ′′ ) 2 q ′′ 2 ∆ v q ′′ ∂ 3 u ε app L 1 k L ∞ h L 2 v 2 q ′ 2 ∆ v q ′ R ε L ∞ k L 2 d q R ε L∞ k B 0, 1 2 ∂ 3 u ε app L 1 k B 1, 1
2 . Inserting (2.14)-(2.16) into (2.13) it follows that
2 3q 2 ∆ v q (R ε,3 u ε app ) L 1 k L 2 d q R ε L∞ k B 0, 1 2 ∂ 3 u ε app L 1 k B 1, 1 2 + ∇ h R ε L2 k B 0, 1 2 u ε app L2 k B 1, 1 2
hence returning to (2.12)
2 q t k+1 t k J q (t)dt s q R ε L∞ k B 0, 1 2 ∂ 3 u ε app L 1 k B 1, 1 2 + ∇ h R ε L2 k B 0, 1 2 u ε app L2 k B 1, 1 2 R ε L∞ k B 0, 1 2 . Finally we find (2.17) 2 q t k+1 t k J q (t) dt ≤ s q 100 ∇ h R ε 2 L 2 k B 0, 1 2 + Cs q R ε 2 L ∞ k B 0, 1 2 u ε app 2 L 2 k B 1, 1 2 + ∂ 3 u ε app L 1 k B 1, 1 2 .
Gathering (2.11) and (2.17), Lemma 2.1 is proved.
3. Proof of Lemma 2.2
Estimates on the approximate solution.
In this section we prove some a priori estimates on u ε app , whose definition we recall: u ε app = u h + εw h , w 3 ε , with u h solution of (NS2D) y 3 and w solution of (T) as defined in the introduction. These estimates were used in Section 2 to prove Theorem 1.
+ ∂ 3 u ε app L2 (R + ;B 0, 1 2 ) + u ε app L2 (R + ;B 0, 1 2 ) ≤ C 1 and ∂ 3 u ε app L 1 (R + ;B 1, 1 2 ) ≤ ε C 2 .
Finally for any constantC > 0 there is a constant K > 1 depending on (u h 0 , w 0 ) B 0, 1 2 and on the times 0 = t 0 < t 1 < · · · < t K = ∞ such that
R + = K−1 k=0 [t k , t k+1 [ and ∀ε ∈]0, 1[ , u ε app L2 (R + ;B 1, 1 2 ) (1 + u ε app L∞ (R + ;B 0, 1 2 ) ) ≤ 1 C · Proof.
C ,
where C depends on the norms of
u h 0 et w 3 0 in B s,s ′ ∩ B 0, 1 2 . Concerning w h we note that w h = −∇ h (∆ h ) −1 ∂ 3 w 3 .
As the horizontal average of where C depends on the norms of u h 0 and w 3 0 in B s,s ′ +1 ∩ B 0, 1 2 . We used above that if a function f defined on T 2 has zero horizontal mean, then
s 1 ≥ s 2 =⇒ f B s 1 ,s ′ ≤ f B s 2 ,s ′ .
The first estimate of Proposition 3.1 comes then simply from the fact that by definition of u ε app , for any σ ∈ R, u ε app Lr (R + ;B σ, 1 2 )
= (u h + εw h , w 3 ) Lr (R + ;B σ,1 2 )
, along with the continuous embedding of B σ, 1 2 into B σ, 1 2 recalled in (1.2). For the second estimate of the Proposition 3.1 we apply (3.1)-(3.3) to s = −1, s ′ = 3/2 and r = 1. From the definition of u ε app we have indeed that
(3.4) ∂ 3 u ε app L1 (R + ;B 1, 1 2 ) = ε ∂ 3 (u h + εw h , w 3 ) L1 (R + ;B 1, 1 2 )
by the same calculations as above, which completes the proof thanks to (3.1)-(3.3).
Finally, the last result of the proposition is simply that and so the time integration interval can be sliced to make the time norms arbitrarily small, regardless of ε. The proposition is proved.
3.2.
Estimates on the pressure.
[∂ 3 p 0 ] ε L 1 (R + ;B 0, 1 2 ) ≤ C 3 ,
and p 1 can be written under the form
p 1 = p 1,h + p 1,3 with [∂ 3 p 1,h ] ε L 1 (R + ;B 0, 1 2 ) + [∇ h p 1,3 ] ε L 1 (R + ;B 0, 1 2 ) ≤ C 4 .
Proof. By definition
∂ 3 p 0 = ∂ 3 2 i,j=1 ∂ i ∂ j (−∆ h ) −1 (u i u j ) .
We recall the product law (A.6)
ab L 1 (R + ;B 0, 1 2 ) a L2 (R + ;B 1 2 , 1 2 ) b L2 (R + ;B 1 2 ,1 2 )
as well as the fact recalled in (1.2) that B 0, 1 2 embeds continuously in B 0, 1 2 . Since the operator ∂ i ∂ j (−∆ h ) −1 is a Fourier multiplier of order 0 if i, j ∈ {1, 2}, it follows from (3.1) that
[∂ 3 p 0 ] ε L 1 (R + ;B 0, 1 2 ) = ∂ 3 p 0 L 1 (R + ;B 0, 1 2 ) ∂ 3 u h L2 (R + ;B 1 2 , 1 2 ) u h L2 (R + ;B 1 2 , 1 2 ) ≤ C , where C depends on u h 0 B − 1 2 , 3 2 ∩B − 1 2 , 1
2 . Furthermore by definition
p 1 = 2 i=1 3 j=1 ∂ i ∂ j (−∆ h ) −1 (u i w j )
and one sets
p 1,h := 2 i,j=1 ∂ i ∂ j (−∆ h ) −1 (u i w j ) and p 1,3 := 2 i=1 ∂ i ∂ 3 (−∆ h ) −1 (u i w 3 ) .∂ j p 1,3 = ∂ 3 2 i=1 ∂ i ∂ j (u i w 3 )
and using again (3.1)-(3.3). Proposition 3.2 is proved.
(∆ v q ∇ h [p 1,3 ] ε |∆ v q R ε,h ) L 2 dt ≤ Cs q R ε L∞ k B 0, 1 2 ∇ h [p 1,3 ] ε L 1 (R + ;B 0, 1 2 ) ≤ Cs q R ε L∞ k B 0, 1
2 . It remains to study the bilinear terms. Here again, the product laws recalled in (A.6) give the result easily since for any function a, thanks to the continuous embedding of B 0, 1 2
into B 0, 1 2 , we have w h · ∇ h a L 1 (R + ;B 0, 1 2 ) ≤ w h · ∇ h a L 1 (R + ;B 0, 1 2 ) w h L2 (R + ;B 1, 1 2 ) ∇ h a L2 (R + ;B 0,1 2 )
and
w 3 ∂ 3 a L 1 (R + ;B 0, 1 2 ) ≤ w 3 ∂ 3 a L 1 (R + ;B 0, 1 2 ) w 3 L2 (R + ;B 1, 1 2 ) a L2 (R + ;B 0,3 2 )
.
We conclude thanks to (3.1)-(3.3). Lemma 2.2 is proved.
Appendix A. Some technical tools A.1. Anisotropic Besov spaces. In this Appendix we recall some useful properties on anisotropic Besov spaces, the definition of which is given in the introduction (see Definitions 1.1 and 1.2).
Let us first recall the anisotropic Bernstein inequalities (see [14,30]).
-If the support of the Fourier transform of a function a defined on R is included in 2 q B where B is a ball of R then for all 1 ≤ p 2 ≤ p 1 ≤ ∞,
(A.1)
∂ α x 3 a L p 1 (R) 2 q(|α|+(1/p 2 −1/p 1 )) a L p 2 (R) .
-If the support of the Fourier transform of a function a defined on R is included in 2 q C where C is a ring of R centered at 0 then (A.2) a L p 1 (R) 2 −q ∂ 3 a L p 1 (R) .
-If the support of the Fourier transform of a function a defined on T 2 is included in 2 j B where B s a ball of R 2 , then for all 1 ≤ p 2 ≤ p 1 ≤ ∞,
(A.3) a L p 1 (R 2 ) 2 2j(1/p 2 −1/p 1 ) a L p 2 (R 2 ) .
It is then not difficult to show, using (A.3), that
(A.4) q∈Z 2 qs ∇ h ∆ v q a L 2 + q∈Z 2 qs ∆ v q a L ∞ h L 2 v a B 1,s .
The spaces given by the following norm, introduced in [13], are used consistently in this text: Finally, let us present the paraproduct algorithm of J.-M. Bony [5] (in the vertical direction): the product of two distributions a, b, when defined, can decompose into
ab = S v q−1 a ∆ v q b + S v q−1 b ∆ v q a + q∼q ′ ∆ v q a ∆ v q ′ b
and thus in particular
(A.5) ∆ v q (ab) = q ′ ∼q S v q ′ −1 a ∆ v q ′ b + q ′ ∼q S v q ′ −1 b ∆ v q ′ a + 2 q ′ 2 q q ′′ ∼q ′ ∆ v q ′′ a ∆ v q ′ b .
This decomposition, with (A.1), makes it possible to prove the following product laws (see for example [3]
a (∆ v q (u · ∇u)|∆ v q u) L 2 (t) dt 2 −q s q ∇ h u 2 L 2 k B 0, 1 2 u L∞ k B 0t k+1 t k (∆ v q (u · ∇v)|∆ v q R ε ) L 2 (t) dt 2 −q s q v 1 2 L ∞ k B 0, 1 2 ∇ h v L2 k B 0, 1 2 × ∇ h v 1 2 L 2 k B 0, 1 2 u 1 2 L ∞ k B 0, 1 2 ∇ h u 1 2 L 2 k B 0, 1 2 + ∇ h u L2 k B 0, 1 2 v 1 2 L ∞ k B 0, 1 2
where (s q ) q∈Z is any sequence of positive real numbers satisfying q∈Z s 1 2 q ≤ 1 .
that (s q ) q∈Z denotes generically a sequence of positive real numbers verifying
Lemma 2. 1 .
1There exists a constant C > 0 such that under the hypotheses of Theorem 1, there exists a sequence (s q ) q∈Z of positive real numbers verifying
1 2 .
2We used the fact, recalled in (A.4), that ∇ h a
∂ 3 w 3 is zero, for all s ∈] − 2, 0[, all s ′ ≥ − 1 2 and all r ∈ [1, R + ;B s+ 2 r ,s ′ ) ∂ 3 w 3 Lr (R + ;B s+ 2 r +1,s ′ ) ∂ 3 w 3 Lr (R + ;B s+ 2 r ,s ′ ) ≤ C ,
u ε app L2 (R + ;B 1, 1 2 ) = (u h + εw h , w 3 ) L2 (R + ;B 1,1 2 )
a
Lr ([0;T ];B σ,s ) := j,q2 js+qs ′ ∆ h j ∆ v q f L r ([0;T ];L 2 ) a Lr ([0;T ];B 0,s ) := q 2 qs ∆ h j ∆ v q f L r ([0;T ];L 2 ) .
1 2 .
2Lemma 2.2.There exists a constant C > 0 such that under the hypotheses of Theorem 1,there exists a sequence (s q ) q∈Z of positive real numbers verifying
q∈Z
s
1
2
q ≤ 1 and such that
Proposition 3.2.There are two constants, C 3 depending on u h0 B − 1
2 , 3
2 ∩B 0, 1
2 and C 4 depend-
ing on (u h
0 , w 3
0 )
B − 1
2 , 5
2 ∩B 0, 1
2 such that the following holds: p 0 satisfies
The term [∂ 3 p 1,h ] ε can be estimated exactly as [∂ 3 p 0 ] ε above thanks to (3.2) and (3.3), and similarly for [∇ h p 1,3 ] ε once noticed that for all j ∈ {1, 2},
ab B 0,s a B 1,s b B 0,s .A.2. Some useful trilinear estimates. We recall here for the convenience of the reader some estimates which were used in the course of the proofs.First from [30, Section 4.1, Corollary 3] we have for any divergence free vector field u):
(A.6)
∀s ≥ 1/2 ,
ab B 1,s
a B 1,s b B 1,s
ab B 0,s
a
B
1
2 ,s b
B
1
2 ,s
(A.7)
t k+1
, 1 2
1and from [30, Lemma from Section 5.1], for all v and for u divergence free (A.8)
Proof of Lemma 2.2. recall thatand let us start with the pressure terms, which are estimated by Proposition 3.2. One has indeed on the one handand on the other hand we notice thatAfter an integration by parts we find thereforeSince R ε is divergence free, another integration by parts givesand we conclude as above thanks to Proposition 3.2 that
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. ) Gallagher, Dma, Cnrs École Normale Supérieure, Research University, 4575005UlmGallagher) DMA, École normale supérieure, CNRS, PSL Research University, 45 rue d'Ulm, 75005
Paris Université De, Paris , CNRS, IMJ-PRG, 75013. Paris, France; Paris, France75006Université de Paris and Sorbonne UniversitéParis, and Université de Paris, 75006 Paris, France (A. Yotopoulos) Université de Paris and Sorbonne Université, CNRS, IMJ-PRG, 75013 Paris, France
|
[] |
[
"Federated Bayesian Neural Regression: A Scalable Global Federated Gaussian Process",
"Federated Bayesian Neural Regression: A Scalable Global Federated Gaussian Process"
] |
[
"Haolin Yu ",
"Kaiyang Guo [email protected] ",
"Mahdi Karami [email protected] ",
"Xi Chen [email protected] ",
"Guojun Zhang [email protected] ",
"Pascal Poupart [email protected] "
] |
[] |
[] |
In typical scenarios where the Federated Learning (FL) framework applies, it is common for clients to have insufficient training data to produce an accurate model. Thus, models that provide not only point estimations, but also some notion of confidence are beneficial. Gaussian Process (GP) is a powerful Bayesian model that comes with naturally well-calibrated variance estimations. However, it is challenging to learn a stand-alone global GP since merging local kernels leads to privacy leakage. To preserve privacy, previous works that consider federated GPs avoid learning a global model by focusing on the personalized setting or learning an ensemble of local models. We present Federated Bayesian Neural Regression (FedBNR), an algorithm that learns a scalable stand-alone global federated GP that respects clients' privacy. We incorporate deep kernel learning and random features for scalability by defining a unifying random kernel. We show this random kernel can recover any stationary kernel and many non-stationary kernels. We then derive a principled approach of learning a global predictive model as if all client data is centralized. We also learn global kernels with knowledge distillation methods for non-identically and independently distributed (non-i.i.d.) clients. Experiments are conducted on real-world regression datasets and show statistically significant improvements compared to other federated GP models.
| null |
[
"https://arxiv.org/pdf/2206.06357v1.pdf"
] | 249,626,069 |
2206.06357
|
07cfe61ee957f50ae681d29935a6fb07e008c4d9
|
Federated Bayesian Neural Regression: A Scalable Global Federated Gaussian Process
Haolin Yu
Kaiyang Guo [email protected]
Mahdi Karami [email protected]
Xi Chen [email protected]
Guojun Zhang [email protected]
Pascal Poupart [email protected]
Federated Bayesian Neural Regression: A Scalable Global Federated Gaussian Process
In typical scenarios where the Federated Learning (FL) framework applies, it is common for clients to have insufficient training data to produce an accurate model. Thus, models that provide not only point estimations, but also some notion of confidence are beneficial. Gaussian Process (GP) is a powerful Bayesian model that comes with naturally well-calibrated variance estimations. However, it is challenging to learn a stand-alone global GP since merging local kernels leads to privacy leakage. To preserve privacy, previous works that consider federated GPs avoid learning a global model by focusing on the personalized setting or learning an ensemble of local models. We present Federated Bayesian Neural Regression (FedBNR), an algorithm that learns a scalable stand-alone global federated GP that respects clients' privacy. We incorporate deep kernel learning and random features for scalability by defining a unifying random kernel. We show this random kernel can recover any stationary kernel and many non-stationary kernels. We then derive a principled approach of learning a global predictive model as if all client data is centralized. We also learn global kernels with knowledge distillation methods for non-identically and independently distributed (non-i.i.d.) clients. Experiments are conducted on real-world regression datasets and show statistically significant improvements compared to other federated GP models.
Introduction
In Federated Learning (FL) [1], we seek to train a model in a distributed way across several clients without any data leaving the clients to preserve privacy. This is particularly attractive in application domains where each client has insufficient data to train a strong model by itself and therefore could benefit from additional information from other clients. A trusted server is often used to aggregate the client models into a global model that improves upon the local models. Since each client has limited data, its local model is uncertain and therefore there is value in representing this uncertainty to improve the aggregation at the server. Intuitively, uncertain models should be given less importance in the aggregation. Furthermore, uncertainty modeling can be used to derive confidence estimates with respect to predictions.
Gaussian Processes (GPs) with deep kernel learning [2,3] provide a good balance between expressiveness and complexity to represent model uncertainty. At one end of the spectrum, most models such as traditional neural networks do not capture any uncertainty, but are simple and scalable. At the other end of the spectrum, Bayesian neural networks express a full distribution over all weights of neural networks, but inference tends to be intractable. In between, a GP with a deep kernel consists of a neural feature extractor (also known as deep kernel) with a distribution over the last layer that facilitates exact inference. Since the weights of the last layer are the most important for prediction, representing their uncertainty is often sufficient to capture most of the uncertainty of a model. Several works have explored distributed GPs [4,5] and federated GPs [6]. While distributed GPs are designed to improve scalability, they pose an important privacy risk since sharing kernels either implies sharing data or sharing pairwise data similarity. In contrast, pFedGP [6] shares only the hyperparameters of deep kernels while learning local GPs that are never shared. This personalized approach reduces privacy risks, but the local GPs do not benefit from other client information (beyond the shared kernel hyperparameters). We propose a new federated GP technique called Federated Bayesian neural regression (FedBNR) that can learn a global GP with reduced privacy risks. We avoid kernel sharing by working directly in the feature space and sharing scatter matrices (instead of kernels). We also propose a unifying random kernel (URK) that leverages random features and deep kernels to approximate any stationary kernel and some non-stationary kernels, including infinite kernels. The approach is demonstrated on real world regression datasets where we achieve statistically significant improvements over prior techniques both in terms of predictions and expected calibration error. The contributions of the paper can be summarized as follows:
• New federated GP technique with deep kernel learning called federated Bayesian neural regression (FedBNR). To our knowledge, this is the first federated GP technique that learns a global GP. We describe an exact aggregation technique of the linear layer that allows inference in a way that is mathematically equivalent to inference with all the data centralized.
• New unifying random kernel (URK) that provides a unifying definition for deep random kernels. URK can approximate any stationary kernel and many non-stationary kernels, including infinite kernels. This kernel has finitely many features and therefore allows us to work directly in the feature space (instead of the dual space).
Related work
Deep kernel learning and GP approximations. There have been many works that committed to increase the model capacity of GPs by incorporating deep neural networks (DNNs). [7,8] either pretrains a deep belief network or directly trains it with a GP to extract first-step features before sending the data into the GP with conventional kernels. [2,3], building on top of [9,10,11,12,13], extends this idea with approximations for scalability and stochastic variational inference for classification tasks. Then [14] studies the variance estimations of deep kernel learning models and propose to use Monte-Carlo Dropout [15] for better calibration, and [16] proposes to use Bayesian Neural Networks instead of deterministic DNNs to prevent over-fitting. Besides, [17] designed a special architecture that makes it possible for DNNs to simulate GP behaviors. [18] forms GPs into a deep architecture that corresponds to a deep belief network based on GP mappings. To make GPs practical, one popular method is the inducing point approximation, where the joint GP prior of training points and inducing points are approximated [19]. Variants includes SoD [19], SoR [20], DTC [21], FITC [22], and PITC [23]. Later, KISS-GP [9] gave another interpretation that inducing point approximations are equivalent to global GP interpolation, and it can exploit Kronecker structures [24]. Another approximation method is random features [25,26,27] that use randomized basis functions to approximate kernels. Details about random features will be covered in Section 3.2, and more information about scalable GPs can be found in this survey [28].
Distributed and Federated Gaussian Processes (GPs). Closely related to our work is the literature on distributed and federated GPs. Distributed GPs [4,5] were initially proposed to improve scalability by partitioning the data into several machines since GPs that operate in the dual space scale cubically with the amount of data in the worst case. The product-of-experts framework has emerged as a popular technique to aggregate local GPs, including generalized product of experts [29] and robust Bayesian committee machines [4]. Distributed optimization of hyperparameters in GPs has also been explored [30]. While those techniques do not ensure data privacy, recent work about federated GPs reduce privacy risks while training in a distributed way. This includes pFedGP [6], which optimizes the hyperparameters of a global deep kernel, while training local GPs. In another line of work, GPs have also been used to estimate correlations between clients in FL in order to Figure 1: FedBNR learns a global federated GP in two phases: kernel learning with FL optimization and last layer updating with exact Bayesian inference. Though we work in the primal space, URK allows us to approximate an infinite kernel in the dual space with finite features in the primal space.
actively select independent clients for aggregation [31]. Our work differs from previous distributed and federated GPs by learning a global GP while reducing privacy risks.
Bayesian FL. Beyond federated GPs, other Bayesian models have been explored to represent distributions over models and predictions in FL. The challenge is in the aggregation of the local posteriors. Various techniques have been proposed including posterior averaging [32], online Laplace approximation [33], Thompson sampling [34], MCMC [35].
Background
Notation. We will use the following notations throughout the paper. X, X ∈ R p×n , x, x ∈ R p are input matrices or vectors. y ∈ R n is the target vector. A * subscript indicates the vectors have not been seen by the models. σ ∈ R is the noise level of Bayesian models. I is the identity matrix and 0 is the zero vector. Their dimensions can be inferred from the context. φ : R p×ā → R p ×ā denotes some basis function, and Φ = φ(X) is the corresponding features of X. k : R p×ā × R p×ā → Rā ×ā denotes a kernel function, and K = k(X, X) is the corresponding kernel matrix.ā is a placeholder that indicates the function can take in matrices or vectors of any dimension. w ∈ R p is the weights of the last linear layer in a Bayesian linear regression model. λ ∈ R is the prior standard deviation. E[·], Cov(·, ·), and tr(·) are the expectation, covariance, and trace function respectively. Any symbol with a c subscript is a local version of the the original symbol, held by some client c.
Gaussian process
A GP can be informally viewed as an infinite dimensional Gaussian distribution over functions f (·).
With a finite set of points of interest X on the support, a GP boils down to a multi-dimensional Gaussian distribution f (X), providing mean and variance estimates at these places. Formally, the noisy version of a GP model is established as:
y = f (x) + ,where ∼ N (0, σ 2 )(1)
After a prior over f (·) is specified, likelihood, posterior, and prediction can be computed as follows:
Prior:
Pr(f (·)) = N (0, k(·, ·))(2)
Likelihood:
Pr(y|X, f (·)) = N (f (X), σ 2 I)(3)
Posterior:
Pr(f (·)|X, y) = N (m(·), k (·, ·))(4)
wherem(·) = k(·, X)(K + σ 2 I) −1 y,
k (·, ·) = k(·, ·) − k(·, X)(K + σ 2 I) −1 k(X, ·)
Prediction:
Pr(y * |x * , X, y) = N (m(x * ), k (x * , x * ))(7)
The complexity of GP is cubic in the amount of data due to the inversion of K. Thus in practice, full GPs are often infeasible and approximations are needed for scalability. The performance of GP models highly depends on the kernel function k(·, ·), and hyperparameters of this function can be learnt by maximizing the log marginal likelihood log Pr GP (y|X) = −y (K + σ 2 I) −1 y − log |K + σ 2 I|.
A valid kernel function is any positive definite function and can always be decomposed into the outer product of some basis functions k(x, x ) = φ(x) φ(x ). An important advantage of working in the dual space is that one can have a kernel corresponding to infinite features without paying a price in terms of complexity. Popular kernels such as the Gaussian kernel tend to have infinite features, providing significant model capacity. However, inference in the dual space relies on evaluating the kernel distance between data points, making it inevitably violates the privacy when data come from different sources.
If the kernel has finite features, a GP degenerates into Bayesian linear regression in the primal space:
f (·) = w φ(·)(8)
Given a prior, likelihood, posterior, and prediction are computed as follows:
Prior:
Pr(w) = N (0, λ 2 I)(9)
Likelihood:
Pr(y|X, w) = N (w X, σ 2 I)(10)
Posterior:
Pr(w|X, y) = N (w, A −1 ) (11) wherew = σ −2 A −1 Φy and A = σ −2 ΦΦ + λ −2 I(12)
Prediction:
Pr(y * |x * , X, y) = N (σ −2 φ(x * )A −1 Φy, σ 2 + φ(x * ) A −1 φ(x * ))(13)
The complexity of Bayesian linear regression is linear in the amount of data, but cubic in the number of features due to the inversion of A. Hyperparameters of φ can be learnt similarly by maximizing the log marginal likelihood log Pr BLR (y|X) = −n log σ 2 − log |λ 2 I| − log |A| − y y/σ 2 + w Aw.
Random features
Random features [25] is a kind of approximation that allows working in the primal space despite the full GP having infinitely many features. The idea is to find randomized basis functions z such that:
k(x, x ) = φ(x) φ(x ) = E[z(x) z(x )] ≈ s m (x) √ m s m (x ) √ m(14)
When s m (x) √ m , the normalized concatenation of m samples of z(x), has much lower dimensionality than φ(x) and the amount of data, the cubic cost in the number of features becomes negligible.
The most renowned random feature approach is random Fourier features (RFF) [25] that can approximate any stationary kernel, based on Bochner's theorem:
Theorem 1 (Bochner [36]). A continuous kernel k(x, x ) = k(x − x ) = k(δ) on R p is positive definite if and only if k(δ) is the Fourier transform of a non-negative measure.
If the kernel is real-valued and properly scaled, its inverse Fourier transform p(ω) is also real-valued and is a proper probability distribution. Then a valid mapping is
z ω (x) = [cos(ω x), sin(ω x)] , since k(δ) = R p p(ω) cos(ω δ)dω = E ω [cos(ω δ)] = E ω [z ω (x) z ω (x )](15)
The true expectation is then approximated by the empirical mean of multiple samples from z ω (x), which makes it possible to recover an infinite kernel with a finite set of features, and enables working directly in the primal space.
FedBNR: a scalable global federated GP
We now describe our approach. First we extend RFF to non-stationary kernels. Then we show how we can learn a stand-alone global GP in a principled way by updating the model in two phases.
Unifying random kernel
Although conventional stationary kernels have been specifically popular due to their distance awareness property, deep kernel learning [2,3] pointed out that incorporating DNNs with stationary kernels further increases the model capacity and makes it more suitable for modern machine learning tasks. However, the common architecture that a DNN is plugged in before the kernel to extract first-step features usually results in non-stationary kernels and also constrained the architecture of the combined kernel. Thus, we wish to extend RFF to non-stationary kernels and provide a unifying definition for random kernels with DNNs, with which people can design any architecture freely.
Let ω ∈ R d be any random variable or vector, and g : R d × R p×ā − → R d×ā be any function that extracts d features out of each input with some random weights ω. Then we construct the random basis functions z as z ω (x) = g(ω, x). We define the true underlying kernel and its approximation, the unifying random kernel (URK) as:
k(x, x ) = E ω [z ω (x) z ω (x )] = tr(Cov ω (z ω (x), z ω (x ))) + E ω [z ω (x)] E ω [z ω (x )](16)≈ URK ω,g (x, x ) = s m ω (x) √ m − 1 s m ω (x ) √ m − 1 ,(17)
where s m ω (x) is the concatenation of m samples of z ω (x). URK can also recover any stationary kernel since RFF is a special case of it: Theorem 2. Given any properly scaled stationary kernel k(x, x ) on R p and its inverse Fourier
transform p(ω), ∃ω ∼ p(ω), g(ω, x) = [cos(ω x), sin(ω x)] s.t. lim m− →∞ URK ω,g (x, x ) = k(x, x ).
Proof. Following Theorem 1 and Equation 15, the construction of ω and z ω is equivalent to RFF,
so k(x, x ) = E ω [z ω (x) z ω (x )]. By Equation 17, URK ω,g (x, x ) is an unbiased estimator of E ω [z ω (x) z ω (x )], so lim m− →∞ URK ω,g (x, x ) = k(x, x ).
However, note that in the definition of URK we do not rely on the inverse Fourier transformation or the Bochner's theorem to find a valid distribution for ω. Instead, any ω and g can give us a valid kernel: Theorem 3. Given any proper probability distribution p(ω) on R d and function g on R d × R p×ā − → R d×ā , the corresponding kernel matrix k(X, X ) = lim m− →∞ URK ω,g (X, X ) is positive definite.
Proof. Following Equation 16, we have
k(X, X ) = d i=1 (Cov ω (z ω (X) i , z ω (X ) i )) + E ω [z ω (X)] E ω [z ω (X )].
z ω (X) i denotes the i th row of the d × n matrix z ω (X). The first term is the addition of d covariance matrices and is always positive definite; the second term is symmetrical and essentially adds one feature to the first term. Thus k(X, X ) is always positive definite.
Since g is an arbitrary function, we can assign it any DNN with any architecture. We call it a Kernel Neural Network (KNN) since its weights, θ, are essentially the kernel hyperparameters. Theorem 3 allows us to train kernels with optimization methods similarly to training DNNs from a much richer hypothesis set than conventional kernels. To provide some insights into possible non-stationary kernels expressed by URK, we give an example construction of a kernel with infinite features below.
Let ω ∼ N (0, I), g(ω, x) = exp(ω x). By definition, k(x, x ) = E ω [exp(ω (x + x ))] = M ω (x + x ) = exp((x + x ) (x + x )/2),
where M ω is the moment generating function of Gaussian distributions. This kernel contains features of all polynomial kernels with c = 0, times exp(x x/2) and a constant. The proof can be found in Appendix A.
Two-phase update
The training procedure of FedBNR can be divided into 2 phases, as illustrated in Fig. 1. In the first phase, we train the KNN by optimization methods. In the second phase, we calculate the weights Algorithm 1 Federated Bayesian Neural Regression (θ: the global KNN, σ: noise level, λ: prior covariance of the linear layer, ω: a set of random numbers, ζ: local learning step, ζ : knowledge distillation step)
Phase 1: Kernel Learning Initialize shared kernel θ ← θ 0 , hyperparameters σ ← σ 0 , λ ← λ 0 for each aggregation round t ← 0, 1, 2, · · · do for each client c ∈ S do θ t c , σ t c , λ t c ← θ t , σ t , λ t for each local update round k ← 0, 1, 2, · · · do θ t c , σ t c , λ t c ← −ζ∇L M L c according to Equation 18 if FedAvg then θ t+1 , σ t+1 , λ t+1 ← mean(θ t S ), mean(λ t S ), mean(λ t S ) else if Knowledge Distillation then
for each knowledge distillation round k ← 0, 1, 2, · · · do θ t+1 , σ t+1 , λ t+1 ← −ζ ∇L KD according to Equation 19 Phase 2: Update the Global Linear Layer Server: send θ, σ, λ, ω to all clients for each Client c ∈ S do compute the random features Φ c ← θ(ω, X c ) and send Φ c Φ c to the server Server: In phase 1, we follow a standard training procedure under the FL framework. We assume there is a central server holding the shared global KNN weights θ and global hyperparameters σ, λ that denote the noise level and the prior variance respectively. We assume there is a set of clients c ∈ S holding local KNNs weights θ c , local hyperparameters σ c , λ c , local inputs X c , and local targets y c . We use θ(x) to denote the result of sending x through the KNN. In the beginning of each aggregation round, the server first sends a copy of aggregated or initialized θ, σ, λ to all the clients. Then all the clients c ∈ S first update their local model for a fixed number of iterations. Then they send θ c , σ c , and λ c back to the server for aggregation and starts another aggregation round. The local loss function is the local log marginal likelihood:
send A −1 ← (σ −2 c (Φ c Φ c ) + λ −2 I) −1 to all clients for each Client c ∈ S dō w c ← σ −2 A −1 ΦL M L c = log Pr BLR (y c |θ c (X c ); σ c , λ c ) = −n c log σ 2 c −log(|λ 2 c I||A c |)−y c y c /σ 2 c +w c A c w c(
18) One commonly used method for aggregation is the FedAvg [37] heuristic, where the new global model parameters are assigned the average of all client model parameters θ = c∈S θ c /|S|. However, multiple works [38,39,40] have pointed out the quality and the convergence rate of this heuristic can suffer from non-i.i.d. clients. To account for this, we propose to adapt kernel knowledge distillation [41] to aggregate the KNNs. We assume the server holds a relatively small dataset X kd , y kd and tries to minimize the following knowledge distillation loss with respect to this dataset:
L KD = L M L kd + α * MSE θ(X kd ) θ(X kd ) − c∈S θ c (X kd ) θ c (X kd )/|S|(19)
Here, L M L kd is the global log marginal likelihood loss L M L with respect to the knowledge distillation dataset X kd , y kd . α is a common hyperparameter in knowledge distillation methods to adjust the ratio between the log marginal likelihood loss and the mean squared error (MSE) loss. The MSE loss factor forces the global kernel θ(X kd ) θ(X kd ) to simulate the mean of all client kernels c∈S θ c (X kd ) θ c (X kd )/|S|, which is akin to concatenating all the features of the client kernels. Ideally, if the global kernel successfully learns to do so, it should not perform worse on any of the clients, while the FedAvg heuristic has no similar guarantees.
In phase 2, we fix the kernel hyperparameters and learn A −1 , the matrix for covariance prediction, andw, the weights of the last linear layer, in an exact way as if all client data are centralized. To understand the procedure, first notice that we can decompose A andw as follows:
A = σ −2 ΦΦ + λ −2 I = σ −2 c∈S Φ c Φ c + λ −2 I (20) w = σ −2 A −1 Φy = c∈S σ −2 A −1 Φ c y c(21)
Here Φ c = θ(X c ) denotes the random features of local inputs extracted by the global KNN. The server first broadcasts the global model to all the clients, and asks them to return the scatter matrices Φ c Φ c , and then the server can calculate A following Equation 20. Next, the server broadcasts A −1 and asks clients for the intermediate weights σ −2 A −1 Φ c y c . Finally, the server can calculatew according to Equation 21 and broadcasts the whole model again to all the clients.
We claim that FedBNR protects privacy of clients at least as well as FedAvg and other federated learning algorithms that send client models to the server. In phase 1, the aggregation only requires client model parameters. In phase 2, we send information twice outside each client: the scatter matrices and the intermediate weights. Sending these matrices and vectors are safer than sending the features Φ c directly since they have limited sizes that are completely independent of the training data size n c , meaning they must only contain limited information about the raw data. Specifically, the scatter matrices are of size md × md, and the intermediate weights are of size md, where m is the number of samples from z and d is the output dimension of the KNN.
Experiments
Synthetic experiment
Although personalized federated learning (PFL) is usually viewed as an advanced version of the plain FL framework since it learns a fine-tuned local model for each client and can automatically handle preference distribution skew [42] (i.e. Pr(y|X) varies for clients), it is noteworthy that if all the clients can agree on a single global model in the hypothesis set, PFL with no global model may not be the best choice due to the trade-off between generalization and personalization. For example, a hospital that only collected data for cancer may also want their model to help diagnosing COVID-19, but personalization would prevent the model from generalizing to other ranges. Moreover, when a new client comes in with few data points, the quality of prediction will suffer compared to other clients, even only querying its own range.
We designed the following synthetic experiments to support these arguments. We first decide a true underlying function, sample 200 points uniformly from the range [−5, 5], and add Gaussian noise with σ = 0.5. We then learn the kernel hyperparameters in a centralized fashion to eliminate any impact from imperfect kernels later on. Details about the kernel sizes and architectures can be found in Appendix B. We then assign the first 100 points in range [−5, 0] to client 1, and the rest to client 2. We train pFedGP and FedBNR with the learnt kernel hyperparameters fixed and query for prediction in range [−5, 5]. The top left graph of Figure 2 shows the result of pFedGP, and the top right one shows the result of FedBNR. The blue curve shows the true underlying function, and the green curve shows the predictions. The light red area is a 95% confidence interval based on variance.
Since both algorithms share the same kernel hyperparameters, the only difference lies in whether they leverage data of both clients for the predictive model. As shown in the pFedGP graph, the personalized model of client 1 only sees its own data (red dots), so it does not generalize well to the range of client 2, while the global model learnt by FedBNR can leverage all the data (blue dots) and generalizes to the full range.
Next, we introduce two new clients into the system. The first client holds data uniformly sampled from range [−5, 5], and the second from range [5,15]. We again fix the kernel hyperparameters learnt centrally and train pFedGP, FedLoc, and FedBNR on these new clients seperated. For testing, we still query the range [− 5,5]. The bottom graphs of Figure 2 shows the MSE loss as the size of data of both new clients grows. As expected, for pFedGP, the quality of prediction is massively impacted when there are few points in the first case, and for FedBNR the loss remains approximately a straight line. For FedLoc, the loss is also impacted due to zero-out effects of non-overlapping client models. Even worse, when training data and testing data are not in the same range for the second new client, the MSE loss of both pFedGP and FedLoc never drops back to the level of previous clients.
UCI regression datasets
We conducted comprehensive experiments on five UCI regression datasets under ten cases. Results of two variants are reported: i) FedBNR that performs the FedAvg heuristic at aggregation; and ii) FedBNR-KD that performs the knowledge distillation method at aggregation.
Two groups of baselines are compared to our method for RMSE error: i) ablation study that contains local+local, local+global, avg+local, and kd+global, in the format of kernel learning method + last linear weight learning method, where we remove the global aggregation of either phase 1 or phase 2 from our methods; ii) previous works that contains (1) FedAvg [37], a standard non-Bayesian FL algorithm that has a global model; (2) FedProx [43], a non-Bayesian FL algorithm that adds a proximal loss to FedAvg to prevent client models from getting too far from the global model; (3) pFedGP [6], a Bayesian PFL method that learns a local GP with a shared deep kernel for each client; and (4) pFedGP [5], a Bayesian FL method that directly applies distributed GP methods [30] without deep kernel learning. We also compare to pFedGP and FedLoc for calibration errors since they are Bayesian models that have a notion of confidence. Besides, we report results of a centralized GP equipped with URK as a casual reference of the testing error lower-bound.
We used fully connected neural networks to extract first-step features for FedAvg, FedProx, and pFedGP. We used Gaussian kernels for the GPs in pFedGP and FedLoc. For fairness, KNNs used by our methods have similar architectures to the combined kernel of pFedGP. For scalability, FITC [22] approximations are implemented for the other GPs as described in pFedGP. We used 50 random samples for KNN and 50 inducing points for pFedGP and FedLoc. Further details are in Appendix B.
Each dataset is uniformly divided into 8:1:1 training:testing:validation sets globally. The training data is sorted by the feature that has the largest absolute correlation coefficient with the output, and divided into multiple chunks. Each client randomly takes two chunks so that their data distributions are heterogeneous. The larger the absolute correlation coefficient is, the more significant the distribution skew is. Before training, we tune hyperparameters that cannot be learnt by gradient descent with grid searching on the validation set. Specially, FedBNR-KD uses 80% of the validation set for knowledge distillation, and the rest 20% for validation. All the datasets are then ran for at most 50 local epochs times 100 aggregation rounds in full batches. Validation and testing error are recorded for each aggregation round. For methods that contain only local models, these errors are defined as the mean error of all local models with respect to the testing/validation set. If the validation error has not improved for 5 rounds, the training process is terminated. We report the average minimum testing RMSE for 10 random seeds for each case in Table 1. We also measured the statistical significance of the results compared to FedBNR with one-tailed Wilcoxon signed-rank tests [44]. We then report expected calibration errors in Table 2 and perform the Wilcoxon test compared to FedBNR-KD. The maximum calibration error, the Brier score, and further details of Table 1 are included in Appendix C.
The results show: i) in terms of RMSE, our methods are statistically better than the Bayesian models pFedGP and FedLoc in all the cases and better than the non-Bayesian models FedAvg and FedProx in most of the cases, especially when the training set at each client is significant smaller than the whole set; ii) compared with the ablation study methods, our methods always perform better, so the global aggregation at both phases are essential; iii) FedLoc without deep kernel learning has a especially smaller model capacity; iv) FedBNR-KD only outperforms FedBNR in 40% of cases in terms of RMSE, which is probably due to the small size of data (8% of all) used for knowledge distillation. However, FedBNR-KD is clearly more stable when client heterogeneity gets worse as the number of clients increases. It is also better calibrated than FedBNR and other Bayesian models in most cases.
Conclusion
In this work, we proposed FedBNR, a novel Bayesian federated learning algorithm that learns a global federated GP without privacy leakage and introduced URK, a unifying definition for deep random features, to approximate kernels with randomized basis functions in the primal space. FedBNR learns a kernel represented by a DNN under the URK definition, and share scatter matrices instead of direct features to achieve the exact global optimum of the last layer. We derived two variants based on the FedAvg heuristic and the knowledge distillation. Both variants shows empirically statistically significant improvements in terms of point estimation and calibration than other federated GP models. Proof.
k(x, x ) = exp((x + x ) (x + x )/2) = exp((x x + x x )/2) exp(x x ) = exp((x x + x x )/2) ∞ l=0 (x x ) l l! , by the Maclaurin series of exp(x) = ∞ l=0 exp(x x/2) √ l! (x x ) l exp(x x /2) √ l! = φ(x) φ(x )
Additionally, we show URK can recover the popular polynomial kernel, a non-stationary kernel beyond RFF's capability.
Theorem A.2. Let c poly , n poly ∈ R. Define p poly = [ 1 2 , 1 2p , 1 2p , · · · , 1 2p ] ∈ R p+1 , ω ∼ M ulti(n poly , p poly ), the multinomial distribution,x = [ 2c poly , √ 2px ] ∈ R p+1 , g(ω, x) = exp(ω logx), then k(x, x ) = lim m− →∞ URK ω,g (x, x ) = (x x + c poly ) n poly Proof. In the following proof, any subscript i means the i th entry of the vector. By definition,
k(x, x ) = E ω [z ω (x) z ω (x )] = E ω [exp ω (logx + logx ) ] = M ω (logx + logx ), the moment generating function of ω = p+1 i=1 p poly,i exp(logx i + logx i ) n poly = 1 2 exp(2 log 2c poly ) + p i=1 1 2p exp(log 2px i + 2p log x i ) n poly = c poly + p i=1
x i x i n poly = (x x + c poly ) n poly
A.2 Greater expressiveness with URK
We expand on new architectures of deep random kernels enabled by the definition of URK in this section to show that URK is more flexible than common heuristics in deep kernel learning. Minimal arguments and evidence are provided below since the ultimate goal of this paper is still to propose a Bayesian FL algorithm that can learn a global GP, not a random feature algorithm that provides better GP approximation.
As illustrated in Figure A.1, URK can recover any standard deep kernel combined with conventional stationary GPs easily. Further more, we can exploit the flexibility of URK and define a distribution shifter h that transforms ω. We can start from a standard normal distribution, which is very easy to sample from, and send the samples through h to simulate a much more complex distribution with minimal computation resources required. If we choose h carefully so that the identity function is in its hypothesis set, we will presumably learn a kernel at least as good as the Gaussian kernel.
If we take a step further beyond DNNs, the function g in URK can be assigned some replication policy that creates randomized versions of x given different ω such as multiplying or adding random Gaussian noise to the input. Combined with the idea of being distance-aware in some latent space, we present another architecture as the leftmost diagram in Figure A.2. We train a GP with URK of these two architectures on a step function and show their predictions in Figure A.2, where f is a very small DNN. The green curve shows the predictions. The light red area is a 95% confidence interval based on variance. The blue points are the training data. The replicate policy (the upper right one), although introduces no additional parameters, further increases the model capacity, and its prediction is more reasonable than just using a DNN and a stationary kernel. We run each random seed of each dataset on 1 CPU and 1 NVIDIA T4 GPU with 16GB RAM. Some important hyperparameters are listed in Table A.1. These hyperparameters are selected through grid searching, as suggested by FedProx.
B Experiment details
C Other metrics
We include the maximum calibration error (MCE) and the Brier score (BRI) of the UCI experiments in Table A.2 intervals and p % test points falling into these intervals. BRI measures the mean squared difference between the confidence and observations, where a test point falling in the CI counts as 1, otherwise 0. We also include the standard error of the mean (SEM) in Table A.4. Our methods still perform better in most of the cases.
c y c and sendw c to the server Server:w ← c∈Sw c of the last linear layer that maps the random features to the output space, by a closed-form formula inferred from Equation 8 -13. The pseudocode of FedBNR is summarized in Algorithm 1.
Figure 2 :
2Top graphs: (i) left: prediction of a pFedGP model trained on two non-overlapping clients; (ii) right: global prediction of FedBNR. Blue curve: underlying truth; green curve: point estimation; red range: 95% confidence interval; dots: training data leveraged by the model. Bottom graphs: MSE of predictions tested on the same range v.s. number of training data from a new client; (i) left: client with training range [-5, 5]; (ii) right: client with training range[5,15]
Figure A. 1 :
1From left to right: a standard deep kernel learning algorithm with conventional stationary kernel GPs; the corresponding URK architecture; a more general architecture enabled by URK for convenient latent stationary kernel learning.A Unifying random kernelA.1 Example construction of a non-stationary kernelTheorem A.1. Let ω ∼ N (0, I), g(ω, x) = exp(ω x), then k(x, x ) = lim m− →∞ URK ω,g (x, x )has infinite features.
Figure A. 3
3shows the architecture of DNNs used in the UCI experiment. For FedAvg, FedProx, and pFedGP, we used the left DNN as their feature extractor. For our methods, we used the rightmost architecture inFigure A.1 with the same feature extractor f , a very small distribution shifter h (the right DNN inFigure A.3), and ω ∼ N (0, I 5 ).
and A.3. MCE measures the (estimated) worst difference between p% confidence Figure A.2: Left: another URK architecture. Right: Train a GP with URK on a step function. Top right: results of the rightmost architecture in Figure A.1. Bottom right: results of the leftmost architecture in Figure A.2. Blue points: training data; green curve: point estimation; red range: 95% confidence interval.
Figure A. 3 :
3Architecture of DNNs used in UCI experiments. Left: the feature extractor for x; right: the distribution shifter for ω. FC is short for "fully connected". The layer size s varies for different datasets.
Table 1 :
1UCI regression datasets, RMSE reported. ⇑ and ↑ denote significantly worse results with p < 0.01 and p < 0.05 respectively; ⇓ and ↓ denote significantly better results similarly.Skillcraft [45]
SML [46]
Parkinsons [47]
Bike [48]
CCPP [49]
train/test size
2670 334
3309 414
4700 587
7008 876
7654 957
corr-coef
-0.660
0.783
0.410
0.539
-0.948
#clients
10
100
10
100
10
100
10
100
10
100
Central GP
0.95
0.21
3.58
0.37
4.02
local+local
1.26 ⇑ 1.48 ⇑ 1.49 ⇑ 2.32 ⇑ 10.9 ⇑ 10.6 ⇑
0.79 ⇑ 0.92 ⇑ 14.6 ⇑ 19.3 ⇑
local+global
1.08 ⇑ 1.22 ⇑ 1.00 ⇑ 1.65 ⇑ 6.42 ⇑ 7.42 ⇑
0.59 ⇑ 0.73 ⇑ 5.62 ⇑ 7.03 ⇑
avg+local
1.05 ⇑ 1.06 ⇑ 0.61 ⇑ 0.81 ⇑ 9.84 ⇑ 8.80 ⇑
0.45 ⇑ 0.54 ⇑ 8.32 ⇑ 13.4 ⇑
kd+local
1.04 ⇑ 1.28 ⇑ 0.86 ⇑ 1.34 ⇑ 6.15 ⇑ 6.97 ⇑
0.51 ⇑ 0.61 ⇑ 10.0 ⇑ 17.1 ⇑
FedAvg [37]
1.00 ↑ 1.03 ⇑ 0.36 ⇑ 0.71 ⇑ 6.83 ⇑ 7.38 ⇑
0.38 ↓ 0.42
4.45
4.44
FedProx [43] 0.98
1.05 ⇑ 0.34 ↑ 0.62 ⇑ 6.32 ⇑ 7.38 ⇑
0.39
0.42
4.43
4.47
pFedGP [6]
0.99 ↑ 1.15 ⇑ 0.75 ⇑ 1.34 ⇑ 9.36 ⇑ 8.91 ⇑
0.45 ⇑ 0.46 ⇑ 14.2 ⇑ 18.2 ⇑
FedLoc [5]
1.08 ⇑ 4.15 ⇑ 2.83 ⇑ 5.43 ⇑ 8.40 ⇑ 11.5 ⇑
0.64 ⇑ 0.75 ⇑ 20.6 ⇑ 44.1 ⇑
ours
FedBNR
0.98
0.97
0.25
0.44
3.10
5.42
0.39
0.42
4.40
4.51
FedBNR-KD 0.96 ⇓ 0.98 ↑ 0.55 ⇑ 0.55 ⇑ 4.58 ⇑ 4.67 ↓
0.43 ⇑ 0.48 ⇑ 4.38
4.38 ↓
Table 2 :
2Expected calibration error (ECE) reported. ⇑ and ↑ denote significantly worse results with p < 0.01 and p < 0.05 respectively; ⇓ and ↓ denote significantly better results similarly.Skillcraft
SML
Parkinsons
Bike
CCPP
#clients
10
100
10
100
10
100
10
100
10
100
Central GP
0.02
0.09
0.27
0.11
0.24
pFedGP [6]
0.43 ⇑ 0.38 ⇑ 0.45 ⇑ 0.45 ⇑ 0.49 ⇑ 0.48 ⇑ 0.42 ⇑ 0.41 ⇑ 0.49 ⇑ 0.49 ⇑
FedLoc [5]
0.32 ⇑ 0.44 ⇑ 0.12 ⇓ 0.27 ⇑ 0.32 ⇑ 0.43 ⇑ 0.26 ⇑ 0.16 ⇑ 0.23 ⇓ 0.50 ⇑
ours
FedBNR
0.05
0.20 ⇑ 0.39 ⇑ 0.37 ⇑ 0.36 ↑ 0.40 ⇑ 0.07
0.04 ⇑ 0.24 ⇓ 0.20 ⇓
FedBNR-KD 0.05
0.06
0.20
0.21
0.29
0.30
0.08
0.09
0.30
0.31
Table A .
A1: Hyperparameters used in the UCI experiment. For FedLoc L = 10ρ to run the Proximal ADMM algorithm.Skillcraft
SML
Parkinsons
Bike
CCPP
#clients
10
100 10
100 10
100 10
100 10
100
size s
200
2000
2000
5000
5000
Table A .
A2: Maximum calibration error (MCE) reported. ⇑ and ↑ denote significantly worse results with p < 0.01 and p < 0.05 respectively; ⇓ and ↓ denote significantly better results similarly. 77 ⇑ 0.65 ⇑ 0.82 ⇑ 0.83 ⇑ 0.93 ⇑ 0.91 ⇑ 0.78 ⇑ 0.73 ⇑ 0.94 ⇑ 0.93 ⇑ FedLoc 0.51 ⇑ 0.81 ⇑ 0.20 ⇓ 0.44 ⇑ 0.54 ⇑ 0.76 ⇑ 0.42 ⇑ 0.28 ⇑ 0.39 ⇓ 0.95Skillcraft
SML
Parkinsons
Bike
CCPP
#clients
10
100
10
100
10
100
10
100
10
100
Central GP
0.03
0.16
0.42
0.16
0.40
pFedGP
0.ours
FedBNR
0.09
0.32 ⇑ 0.67 ⇑ 0.62 ⇑ 0.62 ⇑ 0.72 ⇑ 0.15
0.64 ⇑ 0.39 ⇓ 0.31 ⇓
FedBNR-KD 0.10
0.11
0.33
0.35
0.47
0.48
0.15
0.17
0.49
0.50
Table A .
A3: Brier score (BRI) reported. ⇑ and ↑ denote significantly worse results with p < 0.01 and p < 0.05 respectively; ⇓ and ↓ denote significantly better results similarly.30 ⇑ 0.28 ⇑ 0.31 ⇑ 0.31 ⇑ 0.33 ⇑ 0.33 ⇑ 0.30 ⇑ 0.30 ⇑ 0.33 ⇑ 0.33 ⇑ FedLoc 0.21 0.31 ⇑ 0.18 ⇓ 0.25 ⇑ 0.26 ⇑ 0.30 ⇑ 0.19 ↓ 0.16 ⇓ 0.24 ⇓ 0.33 ⇑Skillcraft
SML
Parkinsons
Bike
CCPP
#clients
10
100
10
100
10
100
10
100
10
100
Central GP
0.16
0.20
0.24
0.18
0.24
pFedGP
0.ours
FedBNR
0.18
0.22 ⇑ 0.29 ⇑ 0.28 ⇑ 0.28 ⇑ 0.30 ⇑ 0.20 ⇑ 0.29 ⇑ 0.24 ⇓ 0.22 ⇓
FedBNR-KD 0.18
0.18
0.23
0.23
0.25
0.26
0.19
0.20
0.25
0.26
Table A .
A4: UCI regression datasets, RMSE ± standard error of the mean (SEM) reported. 26±0.03 1.48±0.01 1.49±0.06 2.32±0.03 10.9±0.26 10.6±0.05 0.79±0.02 0.91±0.01 14.6±0.31 19.3±2.08 local+global 1.08±0.02 1.22±0.01 1.00±0.02 1.65±0.01 6.42±0.10 7.42±0.01 0.59±0.01 0.73±0.01 5.62±0.19 7.03±0.16 avg+local 1.05±0.02 1.06±0.02 0.61±0.06 0.81±0.05 9.84±0.31 8.80±0.11 0.45±0.01 0.54±0.01 8.32±0.43 13.4±0.43 kd+local 1.04±0.01 1.28±0.02 0.86±0.06 1.34±0.12 6.15±0.30 6.97±0.27 0.51±0.01 0.61±0.01 10.0±0.61 17.1±0.63 FedAvg 1.00±0.01 1.03±0.01 0.36±0.02 0.71±0.02 6.83±0.16 7.38±0.21 0.38±0.01 0.42±0.01 4.45±0.06 4.44±0.04 FedProx 0.98±0.01 1.05±0.01 0.34±0.01 0.62±0.02 6.32±0.27 7.38±0.25 0.39±0.01 0.42±0.01 4.43±0.04 4.47±0.02 pFedGP 0.99±0.01 1.15±0.01 0.75±0.05 1.34±0.04 9.36±0.16 8.91±0.09 0.45±0.02 0.46±0.01 14.2±0.61 18.2±0.23 FedLoc 1.08±0.01 4.15±0.26 2.83±0.07 5.43±0.04 8.40±0.02 11.5±0.03 0.64±0.01 0.75±0.01 20.6±0.74 44.1±0.61 ours FedBNR 0.98±0.01 0.97±0.01 0.25±0.01 0.44±0.02 3.10±0.23 5.42±0.20 0.39±0.01 0.42±0.01 4.40±0.01 4.51±0.03 FedBNR-KD 0.96±0.01 0.98±0.01 0.55±0.01 0.55±0.01 4.58±0.05 4.67±0.04 0.43±0.01 0.48±0.01 4.38±0.01 4.38±0.01Skillcraft
SML
Parkinsons
Bike
CCPP
#clients
10
100
10
100
10
100
10
100
10
100
local+local
1.
Acknowledgments and Disclosure of FundingResources used in preparing this research at the University of Waterloo were provided by Huawei Canada, the province of Ontario and the government of Canada through CIFAR and companies sponsoring the Vector Institute.
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"Emily Ryu [email protected]é \nCornell University\nCornell University\nCornell University\n\n",
"Va Tardos [email protected] \nCornell University\nCornell University\nCornell University\n\n"
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A fundamental task underlying many important optimization problems, from influence maximization to sensor placement to content recommendation, is to select the optimal group of k items from a larger set. Submodularity has been very effective in allowing approximation algorithms for such subset selection problems. However, in several applications, we are interested not only in the elements of a set, but also the order in which they appear, breaking the assumption that all selected items receive equal consideration. One such category of applications involves the presentation of search results, product recommendations, news articles, and other content, due to the well-documented phenomenon that humans pay greater attention to higher-ranked items. As a result, optimization in content presentation for diversity, user coverage, calibration, or other objectives more accurately represents a sequence selection problem, to which traditional submodularity approximation results no longer apply. Although extensions of submodularity to sequences have been proposed, none is designed to model settings where items contribute based on their position in a ranked list, and hence they are not able to express these types of optimization problems. In this paper, we aim to address this modeling gap.Here, we propose a new formalism of ordered submodularity that captures these ordering problems in content presentation, and more generally a category of optimization problems over ranked sequences in which different list positions contribute differently to the objective function. We analyze the natural ordered analogue of the greedy algorithm and show that it provides a 2-approximation. We also show that this bound is tight, establishing that our new framework is conceptually and quantitatively distinct from previous formalisms of set and sequence submodularity.
|
10.48550/arxiv.2203.00233
|
[
"https://arxiv.org/pdf/2203.00233v1.pdf"
] | 247,188,021 |
2203.00233
|
a88a05e118584932e85dcaf4d5d50c519dbafb07
|
Ordered Submodularity and its Applications to Diversifying Recommendations
1 Mar 2022 March 2, 2022
Jon Kleinberg [email protected]
Cornell University
Cornell University
Cornell University
Emily Ryu [email protected]é
Cornell University
Cornell University
Cornell University
Va Tardos [email protected]
Cornell University
Cornell University
Cornell University
Ordered Submodularity and its Applications to Diversifying Recommendations
1 Mar 2022 March 2, 2022
A fundamental task underlying many important optimization problems, from influence maximization to sensor placement to content recommendation, is to select the optimal group of k items from a larger set. Submodularity has been very effective in allowing approximation algorithms for such subset selection problems. However, in several applications, we are interested not only in the elements of a set, but also the order in which they appear, breaking the assumption that all selected items receive equal consideration. One such category of applications involves the presentation of search results, product recommendations, news articles, and other content, due to the well-documented phenomenon that humans pay greater attention to higher-ranked items. As a result, optimization in content presentation for diversity, user coverage, calibration, or other objectives more accurately represents a sequence selection problem, to which traditional submodularity approximation results no longer apply. Although extensions of submodularity to sequences have been proposed, none is designed to model settings where items contribute based on their position in a ranked list, and hence they are not able to express these types of optimization problems. In this paper, we aim to address this modeling gap.Here, we propose a new formalism of ordered submodularity that captures these ordering problems in content presentation, and more generally a category of optimization problems over ranked sequences in which different list positions contribute differently to the objective function. We analyze the natural ordered analogue of the greedy algorithm and show that it provides a 2-approximation. We also show that this bound is tight, establishing that our new framework is conceptually and quantitatively distinct from previous formalisms of set and sequence submodularity.
Introduction
Many important optimization problems involve selecting a subset of items from a larger set. Examples of such tasks include influence maximization in social networks [KKT03], sensor placement and experimental design [KMGG08], and recommendation systems [YG11, GKW + 13]. In domains in which the goal is this type of subset selection, submodularity has been widely used to express the notion of "diminishing marginal returns." Submodularity is a powerful framework for approximate optimization; in particular, there is a rich literature on approximation algorithms for selecting subsets achieving near-maximum value with respect to a submodular function [NWF78,CCPV11,FW13,Von13,KG14].
An implicit modeling assumption in the use of submodularity is that the order of the selected elements does not matter; this is crucial, since submodularity is a property of functions that operate on unordered sets. However, in many applications, we are interested not only in the elements of a set, but also the order in which the elements appear. A broad category of such applications, in both on-line and off-line settings, is the presentation of content to an audience -for example, search results, product or entertainment recommendations, news articles, social media posts, and many other instances. Content presentation crucially depends on sequential effects due to well-documented phenomena in human behavior -specifically, that human cognition is generally limited to serially processing information one piece at a time, rather than processing all elements of a list in parallel. Moreover, people tend to have limited attention span and patience, meaning that when items of content are presented in a ranked list, the higher-ranked items are likely to receive significantly greater attention [PHJ + 07]. This results in several empirical observations, such as inverse power law relationships in number of clicks on search results [Wil12] and sharp decreases in webpage viewing time "below the fold" (content that does not fit on the first screen and must be scrolled down to reach) [Fes18].
The use of optimization frameworks for content presentation suggests some of the fundamental limits in the application of submodularity for problem domains where sequential effects are important. In particular, for a number of basic problems in ranking and recommendation, standard formalisms model them as the problem of selecting a subset of items to present to a user, then showing that the resulting objective function over selected subsets is submodular, and thus deriving guarantees for approximating this objective function. But if the value of a set of items to a user is strongly dependent on the order in which it is presented, then the optimization is in fact taking place over sequences rather than sets, and in this richer formalism submodularity would not be applicable.
Our goal in this paper is to propose a formalism that can address these types of ordering issues in optimization problems generally, and for a collection of basic content presentation problems in particular. We begin by observing (and demonstrating in Section 2) that while other generalizations of submodularity to sequences have been formulated, they fundamentally make assumptions that are not well-suited to modeling the sequential effects that arise from phenomena like the diminshing attention of a user reading a ranked list. Hence, a new notion of submodularity for sequences is required. Here, we present such a generalization of a combined monotonicity-submodularity property, which we term ordered submodularity. We provide approximation guarantees for functions of this type, and we show how they capture the sequential effects in a range of standard content presentation problems.
Motivating applications. Throughout our work, it is useful to keep in mind the following two standard problems in ranking and recommendation that help motivate our work. The first is a coverage problem that is used for creating diversity in ranked lists of items as follows [AGHI09,AKBW15]. Suppose we want to produce a list of k recommendations (say of movies) to show to a group of users. Each movie can satisfy only some subset of the users, and we would like to choose the k movies so to maximize the number of users who like at least one item on the list. (In this way, we seek to cover their preferences as completely as possible with k items.) We can view the number of users satisfied as an objective function on the set of k items chosen; in [AGHI09] it is shown that this function is monotone and submodular, and hence greedy maximization provides a (1 − 1 e )-approximation. But as the authors of [AGHI09] observe, in the real application users will have declining attention as they process the list of items, and different users will stop reading the list at different points. This basic addition to the model -that users have differential patience -means that the order of the list is crucial for evaluating the number of users that it satisfies; and once we introduce ordering into the problem, the results from the large body of work on submodular optimization no longer hold in this setting. Is there still a way to find good approximations to the optimal ranked list?
The second problem we draw on for motivation is the task of calibrating recommendations [Ste18]. In this problem, we present a list of k recommendations to a single user (again, suppose they are movies); and we assume that each movie represents a distribution over genres. (For example, a documentary in Italian about the national soccer team is a multi-genre mixture of a movie about sports, an Italian language film, and a documentary.) The list of k items thus induces an average distribution over genres. Now, the user has a target distribution over genres that reflect the extent to which they want to consume each genre in the long run. A natural goal is that the average distribution induced by the list of recommendations should be "close" (in a distributional sense) to the target genre distribution of the user; when these two distributions are close, we say that the set of recommendations is calibrated to the user. (For example, a user who likes both Italian language films and movies about sports might well be dissatisfied with recommendations consisting only of sports movies in English; this set of recommendations would be badly calibrated to the user's target distribution of genres.) For natural measures of distributional similarity, the selection of a set of k items to match the user's target distribution can be formulated as the maximization of a submodular set function. But here too, the work introducing this problem observed that since user attention diminshes over the course of a ranked list, the list of k items is really producing a weighted average over the genres of these items, with the earlier items in the list weighted more highly than the later ones [Ste18]. Once we introduce this natural addition to the problem, based on ordering, it again becomes unclear whether there are good algorithms to find provably well-calibrated lists of recommendations.
A new definition of ordered submodularity. In this paper we introduce a property called ordered submodularity that can be viewed as an analogue of monotonicity and submodularity for functions defined on sequences. It captures both of the motivating applications described above, and more generally captures a category of optimization problems which search over lists, and in which different list positions contribute differently to the objective function.
We define the property as follows. Let f be a function defined on a sequences of elements from some ground set; we say that f ordered-submodular if for all sequences of elements s 1 s 2 . . . s k , the following property holds for all i ∈ [k] and all other elementss i :
f (s 1 . . . s i ) − f (s 1 . . . s i−1 ) ≥ f (s 1 . . . s i . . . s k ) − f (s 1 . . . s i−1si s i+1 . . . s k ).
Notice that if f is an ordered-submodular function that takes sequences as input but does not depend on their order (that is, it produces the same value for all permutations of a given sequence), then it follows immediately from the definition that f is a monotone submodular set function. In this way, monotone submodular set functions are a special case of our class of functions.
We prove that for any ordered-submodular function f , the natural greedy algorithm for maximizing f -building a sequence by always appending the item that produces the largest marginal gain -is a 2-approximation, and there are simple examples of ordered-submodular functions for which the greedy algorithm does no better than a factor of 2. This highlights a key distinction from the unordered case of monotone submodular set functions: there the corresponding greedy algorithm produces the strictly better approximation guarantee of (1 − 1/e). Hence the move to ordered submodularity changes the approximability of the maximization problem in a qualitative way: it still admits a small constant-factor bound, but a different constant.
In the coverage problem described above with users of differential patience, we show directly that the objective function is ordered-submodular, and this provides the first non-trivial approximation guarantee for this problem. (This problem provides some of the simple examples in which the factor of 2 is tight for the performance of the greedy algorithm.) For the calibrated recommendation problem with ranked lists described above, we need to specify how the distance between distributions will be measured; we show that that natural ways of measuring distance (such as the classical family of f -divergences from the statistics and information theory literature) give rise to ordered-submodular functions. We thus obtain the first non-trivial approximation guarantee for this ordered problem as well. As noted above, we find it interesting that existing formalisms extending submodularity to sequences do not capture the objective functions arising from problems such as these two, and the way in which items in these problems contribute based on their position in a ranked list. 1 In the next section, we provide some detail for why these alternative formalisms differ from our proposal and do not capture the objective functions we consider in the paper; following this, we establish our approximation results and their application to the problems discussed here.
Related work
First, we cover general theories of submodularity in sequences and explain how they cannot model the types of problems that our definition does. Then, we discuss applications in the specific context of recommender systems.
Existing frameworks for submodularity in sequences
Alaei, Makhdoumi, and Malekian (2010) introduce the first generalizations of sequence-submodularity and sequence-monotonicity in the context of online ad allocation, and show that the greedy algorithm for sequence-submodular maximization achieves a (1 − 1 e )-approximation to the optimal solution [AMM19]. However, a major limitation of their model is that their definition of sequence-monotonicity is extremely strong. Their result requires that f (A) ≤ f (B) for any sequences A and B such that A is a subsequence of B, which in many settings is too restrictive to be useful. For instance, if an element s 1 only contributes to the value of the objective function when included as the first element of the input sequence but not as the second, it is possible to have f (s 1 ) > f (s 2 s 1 ), violating sequence-monotonicity.
Similarly, Zhang et al. (2013) study the maximization of string submodular functions of strings (or sequences) of actions chosen from a set, a notion similar to sequence-submodularity but only requiring monotonicity and diminishing returns with respect to prefixes, not all subsequences [ZCPM16]. When the function also satisfies monotonicity with respect to postfixes and not only prefixes, then they, too, establish a (1 − 1 e )-approximation ratio for the greedy algorithm, and provide improved guarantees when additional curvature constraints are satisfied. More formally, prefix/postfix monotonicity requires that for any sequences A and B and their concatenation A||B, it must hold that f (A||B) ≥ f (A) (prefix monotonicity) and f (A||B) ≥ f (B) (postfix monotonicity), properties which were both previously suggested by Streeter and Golovin (2008), who considered sequences in the context of an online submodular selection problem [SG08]. As seen above, postfix monotonicity is not a natural property when modeling attention drop-off, since it would imply that prepending a "bad" movie that interests nobody at the front of a ranked list would capture more users, which clearly is not the case.
In another direction, Tschiatschek, Singla, and Krause (2017) approach the selection of maximizing sequences using submodularity by encoding sequential dependencies in a directed acyclic graph [TSK17], and Mitrovic et al. (2018) generalize this concept from DAGs to hypergraphs [MFKK18]. They place an edge between two nodes (u, v) of the graph if there is additional utility in selecting element u before element v, and then consider submodular functions on the edge set of the graph. However, this approach is only able to represent sequential dependencies inherent to the identity of a set of elements (for example, watching a prequel before the sequel), but it cannot represent decreasing attention or other complex dependencies dependencies that may vary with the objective function, or with the position and identity of other elements in the input sequence.
Most recently, Bernardini, Fagnani, and Piacentini (2021) propose a framework in which the set of all elements is equipped with some property g, according to which it has a total ordering. Their objective function is defined recursively as the sum of the marginal increase of appending each element σ to the list of earlier elements, weighted by g(σ). Denoting the subsequence of the first i elements in the list as S i , for any function g and any monotone submodular set function h, they study sequence functions of the form
f (s 1 . . . s k ) = k i=1 g(s i ) · [h(S i ) − h(S i−1 )].
Phrased this way, the sequential nature of the problem results from considering the marginal increase due to each element with respect to the set of elements before it, but the weight assigned to each marginal increase depends solely on the identity of the element, not its rank. While this is a valid assumption in a number of applications, it does not hold in our particular use case of modeling sequential attention drop-off. In contrast, our framework encompasses functions of the form
f (s 1 . . . s k ) = k i=1 g i · [h(S i ) − h(S i−1 )],
where g i can be thought of as the weight assigned to rank i. This key difference allows us to avoid imposing a total g-ordering on the set of all elements (even if such an ordering does exist, this information may not be known to a system designer). Perhaps more significantly, it also introduces an additional sequential aspect that further differentiates our approach from traditional set submodularity.
Applications to diversifying and calibrating recommendations
One important topic in content presentation is the problem of curating search results that are useful to a diverse population of users. Agrawal et al. (2009) establish a mathematical formalization of this user coverage problem, which they study through the lens of submodularity [AGHI09]. They suppose that each item has some probability of satisfying every user type. Then, they seek to display a diverse set of search results to maximize the number of users who find at least one satisfactory document. That is, given a query, they seek to select k search results to maximize the probability that a randomly chosen user drawn from a heterogeneous group likes at least one item in the set. The authors show that this objective is a monotone submodular set function, and consequently observe that there exists a (1 − 1 e )-approximation algorithm for the problem. While mathematically elegant, a key limitation of this formulation is that it assumes that all users are equally patient and give equal consideration to all search results. Acknowledging that this is not an accurate representation of human patience and attention in the real world, the authors suggest as a direction for future work the formulation of an objective function that accounts for the distribution of users who stop at different points in the search results. Our work does exactly this. In doing so, the presentation order of the search results becomes important, and the objective function becomes a sequence function that must be studied using our new definition of ordered submodularity. Ashkan et al. (2015) also study diversification for user coverage in recommender systems, this time using a modular function subject to a submodular constraint [AKBW15]. They maintain the consideration that recommendations should not be only diverse, but also still broadly relevant and useful, by maximizing a weighted sum of a diversification metric and the sum of all the utilities of the recommended items. In their setup, the greedy approach to maximization is optimal. But again, their formulation assumes that all users have equal patience and consider all recommendations equally, so their optimality result does not hold when users have differential patience values. Steck (2018) also considers the question of creating diversity in lists of recommendations, but with the different goal of creating recommendations that are calibrated to the user's interests [Ste18]. (We note that in the literature, "diversification" has historically been used to refer to variants of the coverage problem previously discussed, but we find it more useful to think of "diversity" as a general concept describing lists that include a mixture of categories. The coverage objective is one way to achieve diversity by including as heterogeneous a mixture as possible; the calibration objective is another way that includes categories in a proportional mixture. Section 5.1 of [Ste18] discusses the relationship between diversity, calibration, and other metrics in more detail.) Steck proposes as a heuristic for calibration a modified version of the KL divergence from the recommended distribution to the user's preference distribution. When all the recommended items are assigned equal weight, this induces a submodular set function, which can be used for approximate maximization via the greedy algorithm. But in the case when the recommended items have unequal weights, such as when accounting for attention dropoff, his approximation results do not apply. We discuss more of the technical details of Steck's formalism in Section 6 and describe extensions to our ordered-submodular optimization framework for sequences.
Lastly, another setting in which some notion of weights appears in submodular optimization is the context of knapsack constraints or budgets [Svi04,AGT12,SKIK14]. Here, we note that despite the initial similarities in terminology, the use of weights as capacities in this line of work is quite different from the attenuation of attention and impact that we intend our weights to represent.
Definition of ordered submodularity
In this section we define our extension of submodularity to ordered sets. To simplify notation, for two sequences A and B we will use A||B to denote their concatenation. For a single element s we will use A||s to denote s added at the end of the list A.
Definition 1 (Ordered submodularity). A sequence function f is ordered-submodular if for all sequences A and B, the following property holds for all elements s ands:
f (A||s) − f (A) ≥ f (A||s||B) − f (A||s||B).
Ordered submodularity can be viewed as a generalization of monotonicity and submodularity for set functions. For functions f that depend only on the set of elements in the input sequence and not their order, settings = s implies f (A||s) − f (A) ≥ 0, corresponding to monotonicity, and settings to the "null" element implies
f (A||s) − f (A) ≥ f (A||s||B) − f (A||B),
corresponding to submodularity.
On the other hand, any monotone submodular set function f when viewed as a function on sequences, that does not depend on the order of the elements satisfies
f (A||s) − f (A) ≥ f (A||s||B) − f (A||B) ≥ f (A||s||B)
where the first inequality is due to submodularity and the second inequality is due to monotonicity. This is exactly ordered submodularity when f is interpreted as a sequence function, so we see that ordered submodularity is indeed a very natural and well-motivated property in the sequential setting.
We now demonstrate a few basic ways of constructing ordered-submodular functions from other submodular and ordered-submodular functions.
Lemma 2. If f and g are ordered-submodular, then αf +βg is also ordered-submodular for any α, β ≥ 0.
Proof. We simply multiply and add the two inequalities from the definition of ordered submodularity:
α [f (A||s) − f (A)] ≥ α [f (A||s||B) − f (A||s||B)] β [g(A||s) − g(A)] ≥ β [g(A||s||B) − g(A||s||B)] =⇒ (αf + βg)(A||s) − (αf + βg)(A) ≥ (αf + βg)f (A||s||B) − (αf + βg)f (A||s i ||B).
Lemma 3. Suppose h is a monotone submodular set function. Then the function f constructed by evaluating h on the set of the first t elements of S, that is,
f (S) = h(S) if |S| ≤ t h(S t ) if |S| > t is ordered-submodular.
Here, it is useful to think of t as a threshold beyond which additional elements contribute nothing to the value of f . Once again, S i denotes the sequence of the first i elements of the sequence S, and for a sequence S we use h(S) to denote the value of the submodular function on the set of elements in S, independent of order.
Proof. We seek to show that for all sequences A and B and elements s ands,
f (A||s) − f (A) ≥ f (A||s||B) − f (A||s||B).
We take two cases based on |A|.
Case 1: |A| ≥ t. Then f (A||s) = f (A) = f (A||s||B) = f (A||s||B) = h(A t ), so f (A||s) − f (A) = 0 = f (A||s||B) − f (A||s||B).
Case 2: |A| < t.
Let j = t − |A| − 1. Now, observe that we have
f (A||s) − f (A) = h(A||s) − h(A) ≥ h(A||s||B j ) − h(A||B j ) ≥ h(A||s||B j ) − h(A||s||B j ) = f (A||s||B) − f (A||s||B),
where the first inequality is due to submodularity of h and the second is due to monotonicity of h.
Lemma 4. Suppose h is a monotone submodular set function and {g i } is a sequence of monotonically decreasing weights (i.e., g i ≥ g j if i < j). Then the sequence function defined by
f (S) = k i=1 g i · [h(S i ) − h(S i−1 )],
where k = |S|, is ordered-submodular.
Here we use S i to denote the sequence of the first i elements of the sequence S, and for a sequence S we use h(S) to denote the value of the submodular function on the set of elements in S, independent of the order of the sequence.
Proof. Define g ′ i = g i − g i+1 (where we use an additional term, g k+1 = 0, for notational convenience) and the sequence functions h i (S) = h(S i ), so that we can write f (S) = k i=1 g ′ i · h i (S). By monotonicity, g ′ i ≥ 0 for all i, so by Lemma 2 it suffices to show that each h i (S) is ordered-submodular. But h i (S) is just a monotone submodular set function h evaluated on a threshold of the first i elements of S, so it is ordered-submodular by Lemma 3. Thus we conclude that f is ordered-submodular.
Analysis of simple greedy algorithm
The simple greedy algorithm for cardinality-constrained nonnegative ordered-submodular maximization works as follows: It initializes A 0 = ∅ (the empty sequence), and for ℓ = 1, 2, . . . , k, it selects A ℓ to be the sequence that maximizes f (A) over all sequences obtained by appending an element to the end of A ℓ−1 . In other words, it iteratively appends elements to the sequence A one by one, each time choosing the element that leads to the greatest marginal increase in the value of f .
Proposition 5. The greedy algorithm for nonnegative ordered-submodular function maximization over sets of cardinality k outputs a solution whose value is at least 1 2 times that of the optimum solution. Proof. Denote the sequence of length k maximizing f as S = s 1 s 2 . . . s k and the sequence of length k maximizing the marginal increase at each step as A = a 1 a 2 . . . a k . We write S j = s j s j+1 . . . s k to denote the suffix of S starting at element s j .
Let OP T (k) = f (S), ALG(k) = f (A), so that we seek to show that ALG(k) ≥ 1 2 OP T (k) for all k. We must bound the performance of the greedy algorithm by comparing it to the optimal solution. The key insight is to ask the following question at each step: if we must remain committed to all the greedily chosen elements so far, but make the same choices as the optimum for the rest of the elements, how much have we lost?
To answer this question, we show via induction that for all i,
f (A i ||S i+1 ) ≥ OP T (k) − f (A i ).
The base case of i = 0 is trivial, as
f (A 0 ||S 1 ) = f (S) = OP T (k) ≥ OP T (k) − f (A 0 )
. So suppose the claim is true for some i, and observe that by ordered submodularity we have
f (A i ||s i+1 ) − f (A i ) ≥ f (A i ||s i+1 ||S i+2 ) − f (A i ||a i+1 ||S i+2 ) = f (A i ||S i+1 ) − f (A i+1 ||S i+2 ), f (A i+1 ||S i+2 ) ≥ f (A i ||S i+1 ) + f (A i ) − f (A i ||s i+1 ).
Applying first the induction hypothesis, then the fact that f (A i+1 ) ≥ f (A i ||s i+1 ) by definition of the greedy algorithm, yields
f (A i+1 ||S i+2 ) ≥ (OP T (k) − f (A i )) + f (A i ) − f (A i ||s i+1 ) = OP T (k) − f (A i ||s i+1 ) ≥ OP T (k) − f (A i+1 ),
completing the induction. Finally, taking i = k in the claim gives
f (A) ≥ OP T (k) − f (A) =⇒ f (A) = ALG(k) ≥ 1 2 OP T (k).
Application 1: Diversification for user coverage
Suppose we are designing a movie recommender system which produces a single list of recommendations for a large number of users. Every user has some amount of patience, representing the fact that users are only willing to scroll down so far before deciding that the list is unsatisfactory. We say that the system covers a user if the user is able to find a movie that interests them before their patience expires; otherwise the user gives up on the system and simply walks away. The goal of the designer is to diversify the list of recommendations in order to maximize the number of users covered by the system. In this section, we formally define an objective function for this problem and show that it is ordered-submodular, allowing us to conclude that the greedy algorithm gives a factor of 2 approximation for the coverage problem.
Mathematical formulation
In a realistic application, we may not expect to exactly know each individual user that will ever use the recommendation system; instead, we may only know a probability distribution over the types of users who will use the system. We may also not know with complete certainty that a movie will or will not interest a given user; we may only have an estimated probability that a movie interests a user of a certain type. To generalize our model to this randomized setting, we seek to maximize the expected number of users covered by the system, or equivalently, the probability that a randomly chosen user is covered. Let π represent the probability distribution over user types (so that π u is the probability that a random user has type u). Denote the probability that movie m interests user type u by p m,u . Define θ u , the patience of type u, as the number of recommendations that a user of type u will consider before leaving the system (e.g., if θ u = 2, the system will cover u only if they are interested by the first or second movie in the list). Then, the probability that the recommendation list S = s 1 s 2 . . . s k covers a randomly chosen user from π is
f (S) = u π u 1 − min{θu,|S|} j=1 (1 − p sj ,u ) ,
where the inner expression is obtained as the complement of the probability that a user of type u is not satisfied before their patience expires or they reach the end of the list, whichever comes first. This is the objective function that we now seek to maximize.
Demonstration of ordered submodularity
The objective function is of the form f (S) = u π u f u (S), where
f u (S) = 1 − min{θu,|S|} j=1 (1 − p sj ,u ).
Thus by Lemma 2, to show ordered submodularity, it suffices to fix u and show that f u is orderedsubmodular. But now observe that f u is a function of the set of the first θ u elements only (since multiplication is commutative, and any elements indexed above θ u are not included in the product). Further, the coverage expression on the right hand side is a submodular set function of the type studied by [AGHI09]. So f u is a sequence function defined by imposing a threshold θ u on a submodular set function h, which is ordered-submodular by Lemma 3. Therefore, we conclude that the overall function f is ordered-submodular.
Theorem 6. The user coverage function parametrized by user probability distribution π, movie satisfaction probabilities {p m,u }, and patience values {θ u },
f (S) = u π u 1 − min{θu,|S|} j=1 (1 − p sj ,u ) ,
is ordered-submodular. Thus, the greedy algorithm produces a ranked list covering at least 1 2 as many users as the optimal ranked list.
Greedy approximation ratio of 2 is tight
A simple example in this setting shows that we can do no better than a factor of 2 approximation using the greedy algorithm.
Suppose there are two user types, 1 and 2, with (π 1 , π 2 ) = 1 2 , 1 2 , θ 1 = 1, and θ 2 = 2. There are also two movies, s 1 and s 2 , with p s1,1 = p s2,2 = 1, p s1,2 = p s2,1 = 0. We seek to generate a recommendation list of length 2 (i.e., to rank the two movies in order).
Since f (s 1 ) = π 1 = 1 2 , f (s 2 ) = π 2 = 1 2 , the greedy algorithm may choose arbitrarily between s 1 and s 2 ; suppose it chooses s 2 first. 2 It then chooses s 1 in the second step, but obtains no additional value since s 1 only interests user type 1, but user type 1 will not look at the second movie in the list. Then ALG = f (s 2 s 1 ) = π 1 · 0 + π 2 · 1 = 1 2 .
But the optimal list would place s 1 ahead of s 2 , which first covers user type 1 before their patience expires, then covers user type 2, giving
OP T = f (s 1 s 2 ) = π 1 · 1 + π 2 · 1 = 1, so ALG/OP T = 1 2 exactly. This example can be extended to a recommendation list of arbitrary length k by defining k user types with π i = 1 k , θ i = i (for i = 1, 2, . . . , k) and k movies s j (for j = 1, 2, . . . , k) with p sj ,i = 1 if j = i and p sj ,i = 0 otherwise.
The optimal list is s 1 s 2 . . . s k , which covers each user type exactly before their patience expires, giving OP T = 1. Meanwhile, via induction on the iterations we see that the greedy algorithm can choose the movies in reverse order, producing the list s k s k−1 . . . s 1 . Then only movies s k through s k/2+1 will be able to interest their corresponding user type (for simplicity suppose k is even); for movies s k/2 through s 1 , their corresponding user type will walk away before they are covered. So we have
ALG = k/2 i=1 π i · 0 + k i=k/2+1 π i · 1 = k 2 · 1 k = 1 2 .
Again, ALG/OP T = 1 2 exactly, establishing that the greedy approximation ratio of 2 is tight. Theorem 7. There exist instances of ordered-submodular optimization problems on which the greedy algorithm achieves exactly 1 2 of the optimal value. Thus, the 2-approximation performance bound is tight.
Application 2: Calibration in personalized recommendations
We now consider the setting of personalized recommendations, which generates a tailored list of recommendations for each individual user based on their historical preferences. Much research on personalized recommender systems has worked toward improving prediction accuracy (e.g., how many of the recommended items are indeed relevant to the user), but training solely toward accuracy metrics can actually be detrimental to the performance of the system. For instance, recommendation lists focused only on accuracy may suffer from a lack of diversity or novelty [MRK06]. Another important metric in machine learning is calibration, the degree to which the predicted proportions of the various classes align with the true proportions of the classes in the existing data. From the user's perspective, a recommendation list is calibrated if it closely reflects their various interests in appropriate proportions. This a desirable additional objective when optimizing the user experience; for instance, a user would likely want the system to preserve their minor interests, rather than entirely "crowding them out" in favor of major interests only. Steck (2018) considers the problem of creating calibrated recommendations using the language of movies as the items with which users interact, and genres as the classes of items [Ste18]. Each user has a preference distribution over genres that can be inferred from their previous activity, and the goal is to recommend a list of movies whose genres reflect these preferences (possibly also incorporating a "quality" score for each movie, representing its general utility or relevance). In our work, we adopt Steck's formulation of distributions over genres, which we describe below.
Suppose that each movie i has a distribution over genres g, given by p(g|i). For a user u, we consider two induced distributions: one from the list of movies H that user u has played in the past, and one from the list of movies I that the system recommends to user u:
• p(g|u), the distribution over genres g played by user u in the past:
p(g|u) = i∈H w u,i · p(g|i) i∈H w u,i ,
where w u,i is the weight of movie i (e.g., how recently it was played by user u),
• q(g|u), the distribution over genres g recommended to user u:
q(g|u) = i∈I w r(i) · p(g|i) i∈I w r(i) ,
where w r(i) is the weight of movie i due to its rank r(i).
In general, Steck does not provide much guidance on how the weights are intended to be chosen and interpreted in the context of the greedy algorithm. For our purposes, we suppose the weights are weakly decreasing in rank (i.e., w a ≥ w b if a < b). We also suppose that the desired length of the recommendation list is a fixed constant k (i.e., |I| = k) and k j=1 w i = 1. This assumption is without loss, even with the more typical cardinality constraint that the list may have length at most k -we simply linearly consider each possible length ℓ ∈ [1, k], renormalize so that the first ℓ weights sum to 1, and perform the optimization. We then take the maximally calibrated list over all k length-optimal lists.
The goal of the calibrated recommendations problem is to choose I such that q is "close" to p. To quantify this concept of closeness between distributions, we introduce the formalism of overlap measures.
Overlap measures
For the discussion that follows, we restrict to finite discrete probability spaces Ω for simplicity, although the concepts can be generalized to continuous probability measures.
A common tool for quantitatively comparing distributions is statistical divergences, which measure the "distance" from one distribution to another. A divergence D has the property that D(p, q) ≥ 0 for any two distributions p, q, with equality attained if and only if p = q. This means that divergences cannot directly be used to measure calibration, which we think of as a non-negative metric that is uniquely maximized when p = q. Instead, we define a new but closely related tool that we call overlap, which exactly satisfies the desired properties.
Since divergences have a number of well-studied properties and applications, it is useful to consider overlap measures derived from divergences. We note that [Ste18] does a version of this, modifying the KL divergence into a maximum marginal relevance objective function. However, this proposed objective function may be either positive or negative (see Appendix A.1 for an example), meaning that it cannot be used in our greedy algorithm-and in fact, the concept of approximation guarantees is not even well-specified for functions of variable sign. In contrast, our abstraction of overlap measures satisfies non-negativity for all pairs of distributions p, q.
Our definition is also more general in two important ways. First, we do not limit ourselves to the KL divergence, so that other divergences and distances with useful properties may be used (one such example is the Hellinger distance, H(p, q) = 1 √ 2 || √ p − √ q|| 2 , which forms a bounded metric and has a convenient geometric interpretation using Euclidean distance). Second, in our definition q may be any subdistribution, a term we use to denote a vector of probabilities summing to at most 1. This is useful because our greedy algorithm incrementally constructs q from the 0 vector by adding a new movie (weighted by its rank), so in each iteration we must compute the overlap between the true distribution p and the partially constructed subdistribution q.
With these considerations in mind, we now proceed to define overlap measures. Many classical distances are originally defined on pairs of distributions (p, q), but admit explicit functional forms that can be evaluated using the values of p(x) and q(x) for all x ∈ Ω. This allows us to compute d(p, q), and consequently G d (p, q), when q is a subdistribution. Now, it is clear that G d indeed satisfies both properties of an overlap measure: property (i) follows from the definition of d * , and property (ii) follows from the unique minimization of d at q = p.
Families of ordered-submodular overlap measures
Any overlap measure produces a version of the calibrated recommendations problem (and the corresponding approximation problem), since for the user's target distribution p(g|u), we seek to calibrate the recommended genre distribution q(g|u) to equal p(g|u), which maximizes the overlap G(p, q). But to execute our greedy algorithm, we are interested in overlap measures that give rise to an ordered-submodular optimization problem in particular. As discussed in Section 3, it suffices to study conditions under which the resulting calibration heuristic exhibits diminishing marginal returns and monotonicity (with respect to filling in any position in the sequence that is currently the formal "null" with a new item).
As before, we are also interested in divergence-based overlap measures. One of the main classes of divergences is the family of f -divergences, which are generated from functions f (t) that are convex on t ≥ 0 with f (1) = 0. Given such a function f , the f -divergence of p from q (alternatively, "from q to p") is defined as
D f (p, q) := x∈Ω f p(x) q(x) q(x).
We show that in general, f -divergences yield ordered-submodular problems to which we can apply our greedy algorithm. Recall that p is a given as fixed, while q is constructed incrementally as q(g|u) = i∈I w r(i) · p(g|i). For any genre g, since the p(g|i) are non-negative probabilities, adding a movie i to the sequence always (weakly) increases q(g|u). Then, in the execution of the greedy algorithm, adding i later in the sequence adds onto a larger accumulated value of q(g|u). If g(y) := f c y y is convex for y > 0, then D f (p, q) = y=q(x) g(y) displays "increasing marginal returns" with respect to the sequence of movies, and G D f (p, q) will display decreasing marginal returns.
Indeed, we can verify that we have
g ′ (y) = −f ′ c y · c y 2 · y + f c y · 1 = f c y − f ′ c y c y , g ′′ (y) = −f ′′ c y · c y 2 + f ′′ c y · c y 2 · c y + f ′ c y · c y 2 = f ′′ c y c y 3 ≥ 0,
since c, y, f ′′ c y ≥ 0 (by definition of convexity of f ). Consequently, any bounded f -divergence results in a D f -overlap measure with the diminishing returns property.
Further, we note that many D f -overlap measures based on commonly used f -divergences satisfy the monotonicity property. As a concrete example, consider the squared Hellinger distance (obtained by choosing f (t) = ( √ t − 1) 2 or f (t) = 2(1 − √ t)), which is of the form
H 2 (p, q) = 1 2 x∈Ω ( p(x) − q(x)) 2 = 1 − x∈Ω p(x) · q(x).
The resulting H 2 -overlap measure is
G H 2 (p, q) = x∈Ω p(x) · q(x).
Clearly, if subdistribution q coordinate-wise dominates subdistribution q ′ , then G H 2 (p, q) ≥ G H 2 (p, q ′ ) for all p, which establishes the monotonicity property. Taking these two desired properties together, we have demonstrated that our notion of overlap measures works well with f -divergences to create a general family of ordered-submodular calibration problems.
Theorem 10. Given any bounded f -divergence D f with maximum value d * = max (p,q) d(p, q), the corresponding D f -overlap measure
G D f (p, q) = d * − D f (p, q)
is ordered-submodular. Thus, the greedy algorithm provides a 2-approximation for calibration heuristics using overlap measures of this form.
Inspired by the squared Hellinger-based overlap measure, we also consider the construction of another general family of overlap measures of the form G(p, q) = x∈Ω g 1 (p(x)) · g 2 (q(x)), for nonnegative functions g 1 and g 2 . Here, G is ordered-submodular with respect to the recommendation list as long as g 2 is any non-decreasing (corresponding to monotonicity) and concave (corresponding to diminishing returns) function. Given such a g 2 , we fully specify the overlap measure by choosing g 1 such that G is uniquely maximized when q = p.
That is, we consider the constrained maximization of g i=1 g 1 (p i ) · g 2 (q i ), subject to g i=1 q i ≤ 1. By placing a Lagrange multiplier of λ on the constraint, we see that the maximum occurs when
g 1 (p i ) · g ′ 2 (q i ) = λ
for all i. Since we would like this to be satisfied if q i = p i for all i, and we can scale the overlap measure by a multiplicative constant without loss, it suffices to set g 1 identically to 1
g ′ 2 .
This gives rise to a second family of ordered-submodular overlap measures that includes a wide range of general forms, and also remains easy to compute.
Theorem 11. Given any nonnegative non-decreasing concave function f , the overlap measure
G(p, q) = x∈Ω f (q(x)) f ′ (p(x))
is ordered-submodular. Thus, the greedy algorithm provides a 2-approximation for calibration heuristics using overlap measures of this form.
As a concrete example, taking f (x) = x α for α ∈ (0, 1) gives 1 f ′ (x) = αx 1−α , which produces the (scaled) overlap measure
G(p, q) = x∈Ω p(x) 1−α q(x) α .
Observe that the natural special case of α = 1 2 gives f (x) = 1 f ′ (x) = √ x, providing an alternate construction that recovers the squared Hellinger-based overlap measure.
Finally, we note that until this point our discussion has focused only on calibration, but in practice the recommendations should also have high quality (utility, relevance, etc.). To address this, we can model the overall quality of a list as the sum of quality scores of the individual movies (as in [AKBW15,Ste18]), and then optimize for a weighted sum of quality and calibration. As a straightforward sum of scores, the quality metric is modular (and hence ordered-submodular). Then by Lemma 4 the combination of quality and calibration remains ordered-submodular, and our approximation results still hold.
Corollary 12. The calibrated recommendations problem with combined quality and calibration metrics is ordered-submodular, and thus admits a 2-approximation via the greedy algorithm.
Remark. Since we have established in Theorems 10 and 11 that the calibration heuristic can be formulated as ordered-submodular, our result in Proposition 5 implies that the greedy algorithm gives us a 2-approximation to the calibrated recommendations problem. But unlike in Section 6, where we show that this factor of 2 is tight for the coverage problem, we do not have a corresponding tight instance for calibration.
In simulation, the greedy algorithm performs nearly optimally on most instances of the calibration problem; however, we leave stronger theoretical guarantees as an open problem.
Discussion
In this paper, we have presented a new definition of ordered submodularity, which extends the traditional notion of set submodularity to a class of optimization problems in which the order of elements matters, because elements contribute differently based on their position in the sequence. In particular, our formalism models coverage and calibration of ranked lists, two standard problems in the design of content recommendation systems. We have also shown that greedy ordered-submodular maximization gives a 2-approximation and that this bound is tight on simple instances of the coverage problem. This quantitative result establishes our framework as qualitatively distinct from previous formalisms of set and sequence submodularity, and thus our work has provided the first performance guarantee for approximate optimization of this type.
It is interesting to consider the greedy algorithm in the calibration problem and ask whether the factor of 2 is tight here too, or if the greedy algorithm always performs better in this specific context. Another potential direction for further investigation is parametrizing worst-case instances of the calibration problem, since we found the greedy solution to be very close to optimal across many randomly generated instances. More generally, we pose the natural open question: Does there exist a polynomial time approximation algorithm for ordered submodular maximization achieving a constant factor better than 2? Or does the analogy to set submodularity continue to hold, in that the greedy algorithm provides the best approximation guarantee possible? Further understanding the approximability of this class of problems is a key next step in the continued development and application of our framework.
Since w 1 > 1, we have 2w1+1 w1+2 < 2, thus f (i 1 ) − f (i 2 ) > p 3 1 − ε 2ε + (p 4 − p 2 ) log 2w 1 + 1 w 1 + 2 = 0 =⇒ f (i 1 ) > f (i 2 ).
That is, the greedy algorithm will first choose i 1 instead of i 2 , thereby constructing a suboptimal list. Now, we compute ALG = f (i 1 i 2 ) and OP T = f (i 2 i 1 ) for the following set of parameters: p 1 = 0.05, p 2 = 0.9, p 3 = p 4 = 0.025, ε = 10 −10 , varying w 1 > 1. We now observe that the function does not have consistent sign; ALG and OP T are negative for lower values of w 1 and positive for higher values of w 1 . This is because theq(g|i)'s represent a probability distribution and are thus less than 1, so when the weights are small we take the logarithm of a number less than 1, so the function is negative; when the weights are sufficiently large, then the inner summand exceeds 1 and the function becomes positive.
It is unclear how we should think about approximation when the value of a function is not always positive or negative -for instance, the approximation ratio ALG/OP T is meaningless, especially considering that ALG and OP T may have opposite signs (such as when w 1 = 3.5). So if the simple greedy algorithm is not always optimal, but we have no consistent way of comparing its performance with the optimal solution, then it becomes very difficult to understand the maximization (or approximate maximization) of this specific form of the calibration heuristic.
A.2 Varying sequential dependencies in calibration
In Section 2, we described earlier formalisms of sequential submodularity that rely on postfix monotonicity and argued that many natural ordering problems, including the coverage objective function, are not postfix monotone. A different line of papers encodes sequences using DAGs and hypergraphs. Now, we show that this formalism also does not capture the rank-based sequential dependencies that we desire.
We present a simple instance of the calibration problem which hints at the potential intricacies of sequential dependencies. Suppose there are just 2 genres (g 1 and g 2 ), 4 movies (i 1 , i 2 , i 3 , i 4 ), and 1 user (u). Say that the target distribution is p(g 1 |u) = p(g 2 |u) = 0.5, and the weights of the recommended items are w 1 = 0.5, w 2 = 0.3, w 3 = 0.2. Suppose further that the movies have genre distributions as follows:
p(g 1 |i 1 ) = 0.4, p(g 2 |i 1 ) = 0.6 p(g 1 |i 2 ) = 0.8, p(g 2 |i 2 ) = 0.2 p(g 1 |i 3 ) = 1, p(g 2 |i 3 ) = 0 p(g 1 |i 4 ) = 0, p(g 2 |i 4 ) = 1.
Our heuristic for measuring calibration is the overlap measure G(p, q) = g p(g|u) · q(g|u). We now consider a few different recommended lists as input to the overlap measure:
Definition 8 (
8Overlap measure). An overlap measure G is a function on pairs of distributions and subdistributions (p, q) with the properties that (i) G(p, q) ≥ 0 for all distributions p and subdistributions q,(ii) For any fixed p, G(p, q) is uniquely maximized at q = p.Further, we observe that overlap measures can be constructed based on distance functions.Definition 9 (Distance-based overlap measure). Let d(p, q) be a bounded distance function on the space of distributions p and subdistributions q with the property that d(p, q) ≥ 0 with d(p, q) = 0 if and only if p = q. Denote by d * the maximum value attained by d over allpairs (p, q).Then, the d-overlap measure G d is defined as G d (p, q) := d * − d(p, q).
f
(i 3 i 1 i 2 ) = G(p, (0.78, 0.22)) ≈ 0.956 f (i 3 i 2 i 1 ) = G(p, (0.82, 0.18)) ≈ 0.940 f (i 4 i 1 i 2 ) = G(p, (0.28, 0.72)) ≈ 0.974 f (i 4 i 2 i 1 ) = G(p, (0.32, 0.68)) ≈ 0.983
As one indication of the differences at work, these earlier formalisms for submodularity over sequences have the property that the greedy algorithm continues to be a (1 − 1/e)-approximation for the corresponding maximization problem. But for the ordered coverage problem we have described here, the greedy algorithm can differ from the optimum by a factor of 2; this is the tight bound on its approximation performance, and it suggests that the problem has a qualitatively different type of objective function.
We may also perturb the probabilities by an arbitrarily small amount ε so that π 1 < π 2 , but we make the standard assumption of arbitrary tiebreaking for a cleaner proof of the same result.
A AppendixA.1 Greedy algorithm on variants of the KL divergence A natural hope might be to use the KL divergence as a calibration heuristic, as it is perhaps the most commonly used statistical divergence. Unfortunately, the KL divergence cannot be used directly because it is unbounded; our translation to the distance-based overlap measure is also not well-defined on the KL divergence for the same reason. In[Ste18]an alternative transformation is proposed, yielding the following calibration heuristic:However, this objective function has inconsistent sign, depending on how the recommendation weights are chosen (and we note that Steck does not set any constraints on the weights), and consequently the greedy choice can be far from optimal. In fact, we show that the greedy solution can be negative, while the optimum is positive. So the KL divergence (and variants of it) are not conducive to multiplicative approximation guarantees for the calibration problem.Suppose there are 4 genres (g k for k = 1, 2, 3, 4), 2 movies (i ℓ for ℓ = 1, 2), and 1 user (u), and that we seek a recommendation list of length 2 with weights w 1 > w 2 = 1. For simplicity of notation, we denote p(g k |u) as p k . Suppose further that the movies have the following distributions over genres for some ε ∈ (0, 1 3 ):qFinally, suppose the parameters are such thatThen, observe thatWe can verify that for ε < 1 3 , we have w1(1−ε)+2ε 2w1ε+1−ε < 1−ε 2ε , thusThat is, the optimal recommendation list ranks i 2 first, then i 1 second. However, we also havew1ε 2 + p 4 log w 1 ε w1ε 2 = p 2 log 1 2 + p 3 1 − ε 2ε + p 4 log (2) = p 3 1 − ε 2ε + (p 4 − p 2 ) log (2) .
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this very natural problem setting cannot be satisfactorily encoded by the DAG or hypergraph models of [TSK17] and [MFKK18], providing further motivation for our framework of ordered submodularity. Zhenliang Zhang, K P Edwin, Ali Chong, William Pezeshki, Moran, IEEE Transactions on Automatic Control. 613String submodular functions with curvature constraintsZhenliang Zhang, Edwin K. P. Chong, Ali Pezeshki, and William Moran. String submodular functions with curvature constraints. IEEE Transactions on Automatic Control, 61(3):601- 616, 2016. this very natural problem setting cannot be satisfactorily encoded by the DAG or hypergraph models of [TSK17] and [MFKK18], providing further motivation for our framework of ordered submodularity.
|
[] |
[
"ON SPATIAL ENTROPY AND PERIODIC ENTROPIES OF TWO-DIMENSIONAL SHIFTS OF FINITE TYPE",
"ON SPATIAL ENTROPY AND PERIODIC ENTROPIES OF TWO-DIMENSIONAL SHIFTS OF FINITE TYPE"
] |
[
"Wen-Guei Hu ",
"ANDGuan-Yu Lai ",
"Song-Sun Lin "
] |
[] |
[] |
Topological entropy or spatial entropy is a way to measure the complexity of shift spaces. This study investigates the relationships between the spatial entropy and the various periodic entropies which are computed by skew-coordinated systems γ ∈ GL 2 (Z) on two dimensional shifts of finite type. It is known that there are some aperiodic two dimensional shifts of finite types with positive spatial entropy without any periodic patterns. Hence, the spatial entropy is strictly greater than periodic entropy which is zero. On the other hand, when the shift spaces have some mixing properties then these two entropies are equal. In this paper, we show that some periodic mixing properties imply all of these entropies are equal. Indeed, for two dimensional shift of finite type Σ(B) which is generated by an admissible local patterns B ⊆ {0, 1, ..., r} Z m×m , r ≥ 1 and m ≥ 2. The horizontal periodic transition matrix Tm(B) is a 0-1 matrix which stores horizontal periodic patterns with period m ≥ 1 and height 2 with the maximum eigenvalue ρ(Tm(B)).) k with 1 mk log c(m, k) tends to zero as (m, k) tends to infinte. Uniform dominanting follows if the associated graphs of transition matrices {Tm(B)} ∞ m=1 have uniformly bounded diameters or the shift space Σ(B) has periodic-block gluing properties. The explicit formulas for the transformations between the Hermite normal forms of the finite-index subgroups of Z 2 in different γ coordinated system are obtained. Therefore, it can be proved that uniformly dominant conditions (0.1) and (0.2) implies the spatial entropy of the shift space Σ(B) is equal to the periodic entropy which is computed by γ-coordinated system for all γ ∈ GL 2 (Z). In these case, the spatial entropy can also be computed as lim sup q→∞ log ρ(T γq ,1 ) where γq = 1 q 0 1 .
| null |
[
"https://export.arxiv.org/pdf/2207.11381v1.pdf"
] | 251,040,480 |
2207.11381
|
14434df61fb2afaa464aef775c1f5169f7f7dbb3
|
ON SPATIAL ENTROPY AND PERIODIC ENTROPIES OF TWO-DIMENSIONAL SHIFTS OF FINITE TYPE
Wen-Guei Hu
ANDGuan-Yu Lai
Song-Sun Lin
ON SPATIAL ENTROPY AND PERIODIC ENTROPIES OF TWO-DIMENSIONAL SHIFTS OF FINITE TYPE
Topological entropy or spatial entropy is a way to measure the complexity of shift spaces. This study investigates the relationships between the spatial entropy and the various periodic entropies which are computed by skew-coordinated systems γ ∈ GL 2 (Z) on two dimensional shifts of finite type. It is known that there are some aperiodic two dimensional shifts of finite types with positive spatial entropy without any periodic patterns. Hence, the spatial entropy is strictly greater than periodic entropy which is zero. On the other hand, when the shift spaces have some mixing properties then these two entropies are equal. In this paper, we show that some periodic mixing properties imply all of these entropies are equal. Indeed, for two dimensional shift of finite type Σ(B) which is generated by an admissible local patterns B ⊆ {0, 1, ..., r} Z m×m , r ≥ 1 and m ≥ 2. The horizontal periodic transition matrix Tm(B) is a 0-1 matrix which stores horizontal periodic patterns with period m ≥ 1 and height 2 with the maximum eigenvalue ρ(Tm(B)).) k with 1 mk log c(m, k) tends to zero as (m, k) tends to infinte. Uniform dominanting follows if the associated graphs of transition matrices {Tm(B)} ∞ m=1 have uniformly bounded diameters or the shift space Σ(B) has periodic-block gluing properties. The explicit formulas for the transformations between the Hermite normal forms of the finite-index subgroups of Z 2 in different γ coordinated system are obtained. Therefore, it can be proved that uniformly dominant conditions (0.1) and (0.2) implies the spatial entropy of the shift space Σ(B) is equal to the periodic entropy which is computed by γ-coordinated system for all γ ∈ GL 2 (Z). In these case, the spatial entropy can also be computed as lim sup q→∞ log ρ(T γq ,1 ) where γq = 1 q 0 1 .
Introduction
Topological entropy or spatial entropy is a way to measure the complexity of shift spaces X which has been studied extensively for many years, see [1,3,5,6,7,8,10,11,12,14,16,19,20,21,22,25,26,27,30,31,32,33,34,41,42,44]. The periodic entropy only counts the subset X p of periodic patterns of X. For multi-dimensional symbolic dynamical systems X (or Z d , d ≥ 2 actions). Apart from the rectangularperiodicity, the skew-periodicity can also be considered in each coordinated systems on Z d [2]. This study investigates the relationships between the spatial entropy and various periodic entropies in skew-coordinated systems.
For simplicity, this study presents only the case of two-dimensional shifts of finite type. Let Z m×m be the m×m square lattice in Z 2 and S be the finite set of symbols S = {0, 1, ..., r − 1}, r ≥ 2. Denote by S Zm×m be the set of all local patterns on Z m×m . A given subset B ⊆ S Zm×m is called a basic set of admissible local patterns. Σ (B) is the set of all admissible (global) patterns generated by B on Z 2 . Σ p (B) is the subset of all rectangular periodic patterns of Σ (B). The spatial entropy of Σ (B) is defined by
(1.1) h(B) = lim (n,k)→∞ 1 nk log |Σ n×k (B)| ,
where |Σ n×k (B)| is the cardinal number of Σ n×k (B) which is the set of all admissible patterns generated by B on sublattice Z n×k . The skew-coordinated system can be represented by unimodular transformation γ ∈ GL 2 (Z). The modular group is defined by GL 2 (Z) = a b c d : a, b, c, d ∈ Z and |ad − bc| = 1 .
γ 0 = 1 0 0 1 ∈ GL 2 (Z) is the standard rectangular system. The finite-index subgroup of Z 2 can be parameterized in Hermite normal form [37,39] as L 2 = n 0 k Z 2 : n ≥ 1, k ≥ 1 and 0 ≤ ≤ n − 1 .
In the γ-system, L 2 = n 0 k γ Z 2 : n ≥ 1, k ≥ 1 and 0 ≤ ≤ n − 1 .
The spatial entropy h γ (B) computed in γ system is defined by When γ = γ 0 , h γ0,p (B) is also denoted by h p (B) for simplicity. According to the result of Hu and Lin [24], the spatial entropy h γ (B) computed by any skew-system γ is equal to the standard coordinated system h (B), i.e., (1.4) h γ (B) = h (B) for all γ ∈ GL 2 (Z).
We begin with the study by considering when h (B) = h p (B) in the standard coordinated system γ 0 = 1 0 0 1 .
It is known when r ≥ 5, there are some basic sets B * ⊆ S Z2×2 r with h (B * ) > 0 and Σ p (B * ) = ∅. Therefore (1.5) h (B * ) > h p (B * ) , see [17,28] and related works [9,13,43,48]. On the other hand, when Σ (B) has some mixing property then
(1.6) h (B) = h p (B) ,
see [10,49] and related works [29,35,36,40,41,42,44,45,46,47]. In this paper, we introduce a kind of periodic-mixing property to ensure (1.6) hold. In the following, we briefly introduce the relevant notations and the notation of the periodic-mixing property.
Based on our earlier studies [2,5], we introduce the horizontal-periodic transition matrix T m (B) for all m ≥ 1. Our first result is the following theorem. Furthermore, the uniform connectedness can be expressed in terms of periodicmixing conditions as follows.
The shift space Σ (B) is called horizontal-periodic block gluing if there is an integer K ≥ 1 such that for any pair of m-periodic patterns U m ∈ Σ ∞×k1 (B) and V m ∈ Σ ∞×k2 (B); then U m , V m can be glued together when vertical distance k ≥ K for all m ≥ 1, see the details of Definition 3.21. Then, we have the following result for horizontal-periodic block gluing and uniform connectedness.
Theorem 1.3. If {T m (B)} ∞ m=1 is irreducible and Σ (B) is horizontal-periodic block gluing, then {T m (B)} ∞ m=1 is uniform connected. Furthermore if {T m (B)} ∞ m=1
is uniform connected and for any m ≥ 1, there exists an index i m such that
(1.11) (T m (B)) (im,im) = 1, then Σ (B) is horizontal-periodic block gluing.
To study the problems of h (B) = h γ,p (B), we need to the study the transformation of Hermite normal forms M L 0 K γ between m 0 k γ0 . Indeed, we have the following result.
Theorem 1.4. Given γ = a b c d ∈ GL 2 (Z), ∆ = det γ, with b = 0. Let k = gcd(bM, bL + dK) and k = b 1 (bM ) + b 2 (bL + dK). Let bM = m k and bL + dK = k with b 1 m + b 2 = 1.
Then
(1.12) m k b −dK+ k b 0 K γ ∼ = m K b ak−∆b2K b 0 k γ0 ,
where all entries in (1.12) are integers.
Note that two integer 2 × 2 matrices A and A is called equivalent and is denoted by
A ∼ = A if (1.13) A Z 2 = AZ 2 .
Then we have the following result.
Theorem 1.5. If h(B) = h * (B), then h γ,p (B) = h(B) for all γ ∈ GL 2 (Z). In particular, if {T m (B)} ∞ m=1 is uniformly dominated by {ρ (T m (B))} ∞ m=1 , then h γ,p (B) = h(B) for all γ ∈ GL 2 (Z).
The results also hold when
T m (B) is not irreducible. Indeed, if T m (B) is a maximum irreducible component of T m (B), and T m (B) ∞ m=1 is uniformly domi- nated by {ρ (T m (B))} ∞ m=1
, the results of Theorems 1.1 and 1.5 hold. Furthermore, the horizontal periodicity can also be replaced by vertical-periodicity which will produced the analogous results. In this case, we consider the vertical-periodic transition matricesT m (B) which is obtained by considering the conjugate coordinated
systemγ 0 = 0 1 1 0 of γ 0 = 1 0 0 1 .
Finally, a sequence of skew-coordinated system γ q = 1 q 0 1 , q ≥ 1, is useful in computing the entropy. Indeed, the transformation of Hermite normal form
(1.14) 1 0 0 mq γq ∼ = m 1 0 q γ0
holds. Then we can obtain the following results:
Theorem 1.6. If {T m (B)} ∞ m=1 is irreducible and uniformly dominated by {ρ (T m (B))} ∞ m=1 . Then h (B) = lim sup q→∞ log ρ T γq,1 .
The rest of this paper is organized as follows. Section 2 recalls some useful properties of periodic transition matrices T m (B). In addition, Section 3 recalls the Perron-Frobenius Theorem and introduces uniform domination, uniform connectedness and periodic block gluing and proves Theorems 1.1, 1.2 and 1.3. Finally, Section 4 proves the transformations of Hermite normal forms on different skewcoordinates, then proves Theorems 1.4, 1.5 and 1.6.
Entropy and periodic entropy
In this section, we briefly introduce some properties which concern the entropy of the two-dimensional shift of finite type, see [1,2,3,4,5]. Let Z m×m be the m × m square lattice in Z 2 and S be the finite set of symbols (alphabets or colors). The set of all local patterns (or configurations) on Z m×m is denoted by S Zm×m . A given subset B ⊆ S Zm×m is called a basic set of admissible(or allowable) local patterns.
Σ n×k (B) is the set of all admissible patterns generated by B on Z n×k and Σ(B) is the set of all admissible patterns generated by B on Z 2 . The entropy (spatial entropy) of Σ(B) is defined by
(2.1) h(B) = lim (n,k)→∞ 1 nk log |Σ n×k (B)| .
Due to the subadditive property of log |Σ n×k (B)| in n and k, it can be shown the limit (2.1) always exists, see [15].
As for periodic entropy, the set of finite-index subgroup of Z 2 is denoted by L 2 . L 2 can be parameterized in Hermite normal forms
L 2 = n 0 k Z 2 : n ≥ 1, k ≥ 1 and 0 ≤ ≤ n − 1 ,
see [37,39]. Let P B n 0 k be the set of all n 0 k -periodic and B-addmissible
patterns on Z 2 . The number of P B n 0 k is denoted by Γ B n 0 k , while
the set of all periodic patterns on Z 2 , i.e.,
P(B) = n,k≥1 0≤ ≤n−1 P B n 0 k
is denoted by P(B). Then the periodic entropy h p (B) of P(B) is defined by
(2.2) h p (B) = lim sup (n,k)→∞ sup 0≤ ≤n−1 1 nk log Γ B n 0 k .
Unlike spatial entropy, the limit of (2.2) does not exist in general and lim sup is required. It is clear that
(2.3) P(B) ⊆ Σ(B).
Therefore,
(2.4) h p (B) ≤ h(B).
Let S r = {0, 1, ..., r − 1} be the set of symbols. It is known, when r ≥ 5, for some B ⊆ S
(2.5) h p (B) < h(B),
see [17,28]. On the other hand, when Σ(B) has some mixing property, then
(2.6) h p (B) = h(B),
see [10,49].
In the study of the entropy in [1,2,5], a sequence of matrices of patterns X 2×n and Y m×2 are introduced to store all patterns in Σ 2×n and Σ m×2 , respectively. In the following, the two symbols S 2 = {0, 1} case is introduced. The general cases S r = {0, 1, ..., r − 1} with r ≥ 3 can also be introduced, see [1,5,6,4]. In the case of two symbols, for Σ 2×2 , a 4 × 4 matrix X 2×2 = [x i1i2 ] with 2 × 2 patterns x i1i2 as its entries is defined by
X 2×2 = , and Y 2×2 = [y j1j2 ] is defined by Y 2×2 = .
X 2×2 is called the horizontal pattern matrix with height 2 and Y 2×2 is called the vertical pattern matrix with width 2. Both of them store all 2 × 2 patterns and the arrangements of these patterns are closely related as follows. Indeed, X 2×2 can be presented by y j1j2 as X 2×2 = X 2;1 X 2;2 X 2;3 X 2;4 , with X 2;k = y k1 y k2 y k3 y k4 .
X 2×2 =
Similarly,
(2.8) Y 2×2 = x 11 x 12 x 21 x 22 x 13 x 14 x 23 x 24 x 31 x 32 x 41 x 42 x 33 x 34 x 43 x 44 = Y 2;1 Y 2;2 Y 2;3 Y 2;4 .
Furthermore, all 2 × n patterns in Σ 2×n can be stored in pattern matrix X 2×n with height n which can be defined inductively from X 2×2 . Indeed, denote X 2×n = X n;1 X n;2 X n;3 X n;4 , then X 2×(n+1) = X 2×n • E 2 n−1 ⊗ X 2;1 X 2;2 X 2;3 X 2;4 (2.9) = X 2×2 • E 2 ⊗ X n;1 X n;2 X n;3 X n;4 . (2.10)
Similarly, for Σ m×2 , Y m×2 can be defined recursively with
Y (m+1)×2 = Y m×2 • E 2 m−1 ⊗ Y 2;1 Y 2;2 Y 2;3 Y 2;4 (2.11) = Y 2×2 • E 2 ⊗ Y m;1 Y m;2 Y m;3 Y m;4 , (2.12)
where E 2 n−1 is the 2 n−1 × 2 n−1 matrices and entries are 1. ⊗ is the Kronecker product (tensor product) and • is the Hadamard product i.e., if A = [a i,j ] N ×N and
B = [b i,j ] N ×N then A • B = [a i,j b i,j ] N ×N .
When both a i,j and b i,j are numbers, or matrices of numbers, then a i,j b i,j is the product. When a i,j and b i,j are patterns, it will be understood in the context. Now, given a basic set of admissible patterns B ⊆ S Z2×2 2
, then the horizontal transition matrix H n (B) can be defined for Σ 2×n (B) and vertical transition matrix V n for Σ n×2 , respectively. Indeed, H 2 and V 2 are defined by For n ≥ 3, H n and V n can be defined recursively by applying (2.9) to (2.12) respectively. Indeed, and
H 2 = [h i1,i2 ] 4×4 and V 2 = [v j1,j2 ] 4×4 ,H n+1 = H n • E 2 n−1 ⊗HV m+1 = V m • E 2 m−1 ⊗ V 2;1 V 2;2 V 2;3 V 2;4 (2.15) = V 2 • E 2 ⊗ V m;1 V m;2 V m;3 V m;4 ,(2.V 2 = V 2;1 V 2;2 V 2;3 V 2;4 .
Therefore, the entropy of Σ(B) can be computed by
h(B) = lim (n,k)→∞ 1 nk log H k−1 n (2.19) = lim (m, )→∞ 1 m log V −1 m . (2.20)
Furthermore, by Perron-Frobenius theorem, [18,23,38],
h(B) = lim sup n→∞ 1 n log ρ(H n ) (2.21) = lim sup m→∞ 1 m log ρ(V m ), (2.22)
where ρ(A) is the largest (maximum) eigenvalue of matrix A.
Next, to study the periodic entropy, we need to introduce the matrix of horizontal cylindrical patterns, see [2,5]. Indeed, the x-periodic patterns with periodic 2 in Σ 2×2 can be stored in cylindrical pattern matrix C 2×2 as follows.
(2.23) C 2×2 = .
C 2×2 can also be represented in terms of x i1i2 or y j1j2 . We first introduce the column matrices of patternsX 2×2 of X 2×2 andỸ 2×2 of Y 2×2 , respectively,
(2.24)X 2×2 = x 11 x 21 x 12 x 22 x 31 x 41 x 32 x 42 x 13 x 23 x 14 x 24 x 33 x 43 x 34 x 44 = X 2;1X2;2 X 2;3X2;4 , whereX 2;α = , α = 1 + 2α 1 + α 2 and (2.25)Ỹ 2×2 =
y 11 y 21 y 12 y 22 y 31 y 41 y 32 y 42 y 13 y 23 y 14 y 24 y 33 y 43 y 34 y 44
= Ỹ 2;1Ỹ2;2 Y 2;3Ỹ2;4 , whereỸ 2;β = , β = 1 + 2β 1 + β 2 . Then (2.26) C 2×2 = x 11 x 11 x 12 x 21 x 21 x 12 x 22 x 22 x 13 x 31 x 14 x 41 x 23 x 32 x 24 x 42 x 31 x 13 x 32 x 23 x 41 x 14 x 42 x 24 x 33 x 33 x 34 x 43 x 43 x 34 x 44 x 44 = Y 2×2 •X 2×2 .
Similarly, the y-periodic patterns with period 2 in Σ 2×2 can be stored in vertical cylindrical pattern matrixĈ 2×2 with
(2.27)Ĉ 2×2 = X 2×2 •Ỹ 2×2 .
The horizontal cylindrical pattern matrix which consists of patterns of x-periodic with period n and high 2, defined by
(2.28) C n×2 = Y n×2 • E 2 n−2 ⊗X 2;1 E 2 n−2 ⊗X 2;2 E 2 n−2 ⊗X 2;3 E 2 n−2 ⊗X 2;4 ,
and the vertical cylindrical matrix which consists of patterns of y-periodic with period n and wide 2 can be defined by
(2.29)Ĉ 2×n = X 2×n • E 2 n−2 ⊗Ỹ 2;1 E 2 n−2 ⊗Ỹ 2;2 E 2 n−2 ⊗Ỹ 2;3 E 2 n−2 ⊗Ỹ 2;4 .
Given a basic set of admissible patterns B, then the horizontal cylindrical (or horizontal-periodic) transition matrix T n (B) of C n×2 and vertical cylindrical (or vertical-periodic) matrixT n (B) ofĈ n×2 can be defined by
(2.30) T 2 (B) = V 2 •H 2 and (2.31)T 2 (B) = H 2 •Ṽ 2 ,
whereH 2 andṼ 2 are column matrices of H 2 and V 2 respectively. Furthermore, from (2.28) and (2.29), we have
(2.32) T n = V n • E 2 n−2 ⊗H 2;1 E 2 n−2 ⊗H 2;2 E 2 n−2 ⊗H 2;3 E 2 n−2 ⊗H 2;4 , and (2.33)T n = H n • E 2 n−2 ⊗Ṽ 2;1 E 2 n−2 ⊗Ṽ 2;2 E 2 n−2 ⊗Ṽ 2;3 E 2 n−2 ⊗Ṽ 2;4 .
Next, we recall the periodic patterns in
Z 2 . A global pattern U = [U i,j ] on Z 2 is called n 0 k -periodic if it satisfies (2.34) U i+rn+s ,j+sk = U i,j
for all (i, j) ∈ Z 2 and r, s ∈ Z. Note that n 0 0 k -periodic is just the rectangular (n, k)-periodic. To study these periodic patterns, we need to recall the rotational matrix R n , n ≥ 1, see [2]. The shift (to the left) σ of any n-sequence (u 1 u 2 · · · u n ) ∈ {0, 1} n is defined by
(2.35) σ(u 1 u 2 · · · u n ) = (u 2 u 3 · · · u n u 1 ).
Notably, the shift of any one-dimensional periodic sequence (u 1 u 2 · · · u n ) ∞ = (u 1 u 2 · · · u n u 1 u 2 · · · ) with period n becomes (u 2 u 3 · · · u n u 1 ) ∞ = (u 2 u 3 · · · u n u 1 u 2 u 3 · · · ). The sequence (u 1 u 2 · · · u n ) and its shift (u 2 · · · u n u 1 ) can also be related by their counting number χ(u 1 u 2 · · · u n ). Define
i = χ(u 1 u 2 · · · u n ) = 1 + n j=1 u j 2 n−j ,
and n-shift map
(2.36) σ(i) ≡ σ n (i) = χ(u 2 u 3 · · · u n u 1 ).
It is easy to see that σ is a bijection map on {1, 2, ..., 2 n }. For any n ≥ 1, the 2 n × 2 n rotational matrix R n = [R n;i,j ] , R n;i,j ∈ {0, 1}, is defined by
(2.37) R n;i,j = 1 if and only if j = σ(i).
R n is a permutation matrix. Indeed, R n can be written explicitly as follows.
Proposition 2.1. For any integer n ≥ 2,
(2.38) R n;i,2i−1 = 1 and R n;2 n−1 +i,2i = 1 , if 1 ≤ i ≤ 2 n−1 , R n;i,j = 0 , otherwise.
Equivalently,
(2.39) σ(i) = 2i − 1 , if 1 ≤ i ≤ 2 n−1 , 2(i − 2 n−1 ) , if 2 n−1 < i ≤ 2 n .
Furthermore, (R j n ) i,σ j (i) = 1 for any 1 ≤ j ≤ n − 1 and R n n = I 2 n where I N is the N × N identity matrix.
We recall the following results in [2].
Proposition 2.2. For n ≥ 2, T n = [T n;i,j ] is R n -symmetric, i.e., T n;σ (i),σ (j) = T n;i,j for all 1 ≤ i, j ≤ 2 n and 0 ≤ ≤ n − 1. Proposition 2.3. For n, k ≥ 1 and 0 ≤ ≤ n − 1, (2.40) Γ B n 0 k = tr T k n R n , and (2.41) h p (B) = lim sup (n,k)→∞ sup 0≤ ≤n−1 1 nk log tr T k n R n .
It is worth investigating the periodic patterns in more detail. Indeed, for any integer , define the -shift periodic entropy h (B) by
(2.42) h (B) = lim sup (n,k)→∞ 1 nk log Γ B n 0 k .
In particular, when = 0, h 0 (B) is the rectangular periodic entropy, i.e.,
h 0 (B) = lim sup (n,k)→∞ 1 nk log Γ B n 0 0 k . For = 1, h 1 (B)
is also very useful which will be studied later by using the skewcoordinated system. It is clear that
h (B) ≤ h p (B)
for all . The inequalities may be hold for some . Indeed, consider the 3 symbols basic set B 3,0 defined by
B 3,0 = .
It is easy to see
Γ B3,0 2m 2 0 k = 2 mk , and Γ B3,0 2m 2 + 1 0 k = 0.
Therefore,
h 2 (B 3,0 ) = 1 2 log 2, h 2 +1 (B 3,0 ) = 0, and h p (B 3,0 ) = 1 2 log 2.
The problem of h (B) = h p (B) will be studied in Section 4 by using the skewcoordinated system.
Uniform Controls and Connects
In this section, we will study entropy and periodic entropy and introduce some conditions on B which implies h p (B) = h(B).
We begin with recalling some results from the matrix theory, in particular, from the Perron-Frobenius Theorem and its applications, see [18,23].
A N × N matrix A = [a i,j ] N ×N is non-negative if a i,j ≥ 0 for all 1 ≤ i, j ≤ N and integral if all its entries are integers. λ is an eigenvalue of A with right (column) eigenvector v = (v 1 , ..., v N ) t if (3.1) Av = λv, and left (row) eigenvector u = (u 1 , ..., u N ) of A with eigenvalue λ if (3.2) uA = λu. A matrix A = [a i,j ] is called irreducible if for any 1 ≤ i, j ≤ N , there is some positive integer k such that (A k ) i,j > 0. A is called primitive (irreducible and aperiodic) if there is some positive integer k such that (A k ) i,j > 0 for all 1 ≤ i, j ≤ N . It is clear that if A is primitive then A is irreducible.
Now, we state the Perron-Frobenius Theorem.
Theorem 3.1 (Perron-Frobenius Theorem). (i) Let A be an N × N , N ≥ 2, irreducible matrix.
Then there is the positive maximum eigenvalue ρ = ρ(A) which is algebraic simple and the corresponding eigenvectors v and u are positive (i.e., v i > 0 and u i > 0 for all i.) Any other eigenvalue λ of A, |λ| ≤ ρ, ρ is the only eigenvalue with a non-negative eigenvector.
(ii) If A is primitive then for each 1 ≤ i, j ≤ N , (3.3) lim k→∞ (A k ) i,j ρ k = v i u j , and (3.4) |λ| < ρ
for any other eigenvalue λ of A.
The following results of the structures of irreducible and reducible matrices are very useful, see [23].
Theorem 3.2. (i) If A is an irreducible N × N matrix,
then there is positive integer p ≥ 1 and a permutation P such that
(3.5) P AP t = O B 1 O · · · O O . . . . . . O O B p−1 B p O and (3.6) P A kp P t = A k 1 O . . . O A k p , where A i = B i · · · B p B 1 · · · B i−1 and A i is primitive for any 1 ≤ i ≤ p. (ii) If A is a reducible N × N matrix
then under a permutation, A can be reduced to a normal reducible form :
(3.7) A = A 1 O · · · O O A 2 O · · · O . . . . . . O · · · O A g O · · · O A g+1,1 · · · A g+1,g A g+1 O · · · O A s,1 · · · A s,g · · · A s,s−1 A s , where 1 ≤ g ≤ s, A i , 1 ≤ i ≤ g are irreducible matrices, and (3.8) ρ(A) = max 1≤i≤s ρ(A i ).
From Theorems 3.1 and 3.2, we can obtain the asymptotic result for (A k ) i,j as k → ∞.
≤ i, j ≤ N , there exists a k = k(i, j) such that (3.9) c 1 ρ αp ≤ (A αp+k ) i,j ≤ d 1 ρ αp for large α ≥ 1 where 0 < c 1 ≤ d 1 are positive constants only depend on A. Furthermore, (3.10) lim sup k→∞ 1 k log A k = log ρ(A)
and
(3.11) lim sup k→∞ 1 k log tr(A k ) = log ρ(A).
Proof. From (3.6),
A αp+k = P (A) α P t A k where A = A 1 O . . . O A p , P = [P i,j ] N ×N , P t = [P t i,j ] N ×N ,
and A k is primitive and ρ(A k ) = ρ(A) for all 1 ≤ k ≤ p.
For each 1 ≤ k ≤ p, let v k = (v k,1 , ..., v k,N k ) and u k = (u k,1 , ..., u k,N k ) are right and left positive eigenvectors of A k . Note that p k=1 N k = N .
ON SPATIAL ENTROPY AND PERIODIC ENTROPIES OF TWO-DIMENSIONAL SHIFTS OF FINITE TYPE 15
Then (3.3) implies (3.12) lim α→∞ (A α k ) s,t ρ αp = v k,s u k,t , for 1 ≤ s, t ≤ N k . Then (3.13) (A αp+β ) i,j = N i2,i3=1 P ii2 (A α ) i2,i2 P t i2,i3 (A β ) i3,j .
Since P is a permutation, there is a unique i 2 such that (3.14)
P i,i2 = 1.
For large α, (3.12) implies
(3.15) lim α→∞ (A α ) i2,i2 ρ αp = v k,s u k,s for some k ≥ 1 and s ≥ 1. Since P t is permutation, there exists a unique i 3 such that P t i2,i3 = 1. Therefore, choose k ≥ 1 such that (3.16) (A k ) ij ≥ 1,
where k only depends on i and j. By (3.13), (3.14), (3.15) and (3.16), there are constants 0 < c 1 ≤ d 1 only depend on A k , 1 ≤ k ≤ p (and so A), such that
c 1 ρ αp ≤ (A αp+k ) i,j ≤ d 1 ρ αp .
The proof of (3.9) is complete. (3.10) and (3.11) follow easily from (3.9).
The following lemma concerning the double limit and iterated limits of double sequences are useful. In the case that the double limit exists, then the equalities in (3.17) and (3.18) hold.
The following results for H n and T n (or V m andT m ) are useful in the study of entropy and periodic entropy. Proof. Both sides of (3.19) are the cardinal numbers of m×n B-admissible patterns which are horizontal periodic with period m and height n. The proof is complete.
Furthermore, we have the following results. Then we have the following results. As for (3.24),
h p (B) = lim sup (m,k)→∞ sup 0≤ ≤m−1 1 mk log Γ B m 0 k = lim sup (m,k)→∞ sup 0≤ ≤m−1 1 mk log tr T k m (B)R m ≥ lim sup (m,k)→∞ 1 mk log tr T k m (B) ≥ lim m→∞ 1 m lim sup k→∞ 1 k log tr T k m (B) = lim sup m→∞ 1 m log ρ(T m (B)).
The proof is complete.
The equality of (3.24) does not hold for some B as mentioned in [17,28]. However, for some classes of basic set B, it holds. In [5], using (3.20) it has been proved that when H 2 (B) is symmetric, then
h(B) ≤ 1 2m log ρ(T 2m )
for all m ≥ 1. Hence, h(B) = h * (B). In viewing (3.20), we can introduce another classes of basic sets. We need the following notion. To study the uniformly dominant condition (3.26), we will introduce some notion and results. We first recall the following lemma, see [38].
v = (v 1 , ..., v N ) t . Then (3.30) c d N ρ(A) k ≤ A k ≤ d c N ρ(A) k , where (3.31) c = min i v i and d = max i v i . Proof. Since Av = ρv, A k v = ρ k v. Therefore, N j=1 (A k ) i,j v j = ρ k v i for all i. Hence, (3.32) cρ k ≤ N j=1 (A k ) i,j v j ≤ dρ k
for all 1 ≤ i ≤ N . From the right hand side of (3.32)
N j=1 (A k ) i,j ≤ d c ρ k . Hence A k ≤ d c N ρ(A) k .
From the left hand side of (3.32), we have c
d ρ k ≤ N j=1 (A k ) i,j . Hence c d N ρ(A) k ≤ A k .
The proof is complete.
The following lemma estimates the ratio of d c .
≤ i, j ≤ N , there is 1 ≤ k ≤ K with k = k(i, j) such that (3.33) (A k ) i,j ≥ 1. Then (3.34) d c := max i v i min i v i ≤ ρ K . Proof. For any 1 ≤ i, j ≤ N , (A k ) i,j ≥ 1 and N =1 (A k ) i v = ρ k v i imply v j ≤ (A k ) i,j v j ≤ ρ k v i . Therefore, v j ≤ ρ k v i , and then v j ≤ ρ K v i . Hence (3.34) holds.
We introduce some concepts from graph theory. For any matrix A = [a i,j ] N ×N , with a i,j ∈ {0, 1}, then the associated graph G = G(A) is defined by the vertex set V = {1, 2, ..., N } and for i, j ∈ V there is an edge from i to j if a i,j = 1. If there is k such that (A k ) i,j ≥ 1, then the distance d(i, j) of vertices i and j is defined by
(3.35) d(i, j) = min k : (A k ) i,j ≥ 1 .
Otherwise, if vertices i and j cannot be connected, then there is no k such that (A k ) i,j ≥ 1, as denoted by d(i, j) = ∞. If A is irreducible, then for any vertices i, j ∈ V can be connected in G, that is, there is a k ≥ 1 such that (A k ) i,j ≥ 1. The diameter D(G) of G is the maximum of distances of all i, j ∈ V , that is,
(3.36) D(G) = max {d(i, j) : i, j ∈ V } .
We now introduce the following notion of the uniform connectedness property of {T m } ∞ m=1 . Uniform connectedness of {T m } ∞ m=1 is equivalent to there is a finite bound K of the diameters of associated graphs G(T m ), that is, Since ρ m ≤ |S| m , hence
(3.38) max {D(G(T m )) : m ∈ N} ≤ K.(3.41) T k m ≤ r Km r m ρ k m = r K+1 m ρ k m . (3.26) holds with C = r m(K+1) . Hence {T m } ∞ m=1 is uniformly dominated by {ρ(T m )} ∞ m=1 .
The proof is complete.
Therefore we have the following result. In the case of T m being reducible, then Theorem 3.2 (ii) implies there is a maximum irreducible submatrix T m = [t m;α,β ] Im×Im where t m;α,β ≤ t m;α,β and either t m;α,β = t m;α,β > 0 or t m;α,β = 0 and I m ⊆ [1, r m ] is the set of indices such that for any indices pair α and β in I m then there is k ≥ 1 such that
(3.42) (T k m ) α,β ≥ 1. Since T m is maximum, (3.43) ρ(T m ) = ρ(T m ).
We have the following definition. By using the recursive formula, it can be shown that the normal form V 3 of matrix V 3 is
V 3 = 1 0 1 1 ⊗ 1 0 1 1 ⊗ 1 1 1 1 .
For general m ≥ 3,
V m = ⊗ m−1 1 0 1 1 ⊗ 1 1 1 1 . Therefore, V k m = (k + 2) m−1 2 k+1 . Hence T k m ≤ c 0 (m, k)2 k+1 with c 0 (m, k) = (k + 2) m−1 , which satisfies (3.45). Note that ρ(T m ) = ρ(V m ) = 2 for all m ≥ 2. Hence, h(B) = h p (B) = 0.
It may happen that T m (B) has zero rows or zero columns. In these case, we can reduce T m (B) by deleting its zero rows and columns. When matrix A has zero rows or zero columns, we introduce the following notation, see [6].
Definition 3.17. A non-negative matrix A is called weakly primitive if there exists K ≥ 1 such that each entry of A k is positive except in positions of A where a zero row or zero column is present for all k ≥ K. That is to say after deleting the zero row or zero column in A, the remaining matrix A is primitive. Similarly, A is called weakly irreducible if the remaining matrix A is irreducible after deleting zero rows and zero columns from A. Definition 3.12 can be extended as follows.
Definition 3.18. The maximum irreducible component {T m (B)} ∞ m=1 of {T m (B)} ∞ m=1
is called uniformly connected if there is a positive constant K ≥ 1 such that for any m ≥ 1 and any indices pair i, j ∈ I m , there is 1 ≤ k ≤ K such that
(3.46) (T k m ) i,j ≥ 1.
Therefore, we have the following result. (b) Hard-Hexagon model [26]:
H 2 = 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 .
(c) Strict Golden-Mean shift (SGM):
H 2 = 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 .
All of their H n , V n and T m can be proved to be weakly primitive and {T m } ∞ m=1 are uniformly connected. Therefore, Theorem 3.19 is applicable to them.
It is worth to mention that the concept of uniform connectedness of
{T m (B)} ∞ m=1
can also be interpreted in terms of the mixing properties of a two-dimensional symbolic dynamic system. Indeed, it is a periodic version of block gluing which has been studied in [4,10]. We can now introduce the concept of horizontal-periodic block gluing as follows. For m ≥ 1 and a pair of two horizontal m-periodic pattern U m ∈ Σ ∞×k1 (B) and V m ∈ Σ ∞×k2 (B), U m , V m can be glued together with vertical distance k ≥ 0 if there exists a m-periodic patterns W m ∈ Σ ∞×(k1+k+k2) (B) such that
(3.48) W m α,β = U m α,β , for 1 − k 1 ≤ β ≤ 0 and (3.49) W m α,β = V m α,β , for k ≤ β ≤ k + k 2 − 1.
We can now give the definition of horizontal-periodic block gluing.
Definition 3.21. The shift space Σ (B) is called horizontal-periodic block gluing if there exists an integer K ≥ 1 such that for any two horizontal m-periodic patterns U m ∈ Σ ∞×k1 (B) and V m ∈ Σ ∞×k2 (B), U m , V m can be glued together with vertical distance k for any k ≥ K for all m ≥ 1.
Now we can prove the following theorem which is concerning the uniform connectness and horizontal-periodic block gluing.
T K m (B) i,j ≥ 1. Hence (3.52) T k m (B) i,j ≥ 1 for some 1 ≤ k ≤ K. Therefore, {T m (B)} ∞ m=1 is uniformly connected. On the other hand, if {T m (B)} ∞ m=1
is uniformly connected with finite bound K of diameters of associated graphs G m (B) and satisfying condition (3.50), then for m ≥ 1 and any index pair
1 ≤ i, j ≤ r m , if k ≥ 2K + 1 then T k1 m (B) i,im ≥ 1 and T k2 m (B) im,j ≥ 1 with 1 ≤ k 1 , k 2 ≤ K, and T 2K+1−(k1+k2) m (B) im,im ≥ 1 implies T k m (B) i,j ≥ 1. Hence Σ (B)
is horizontal-periodic block gluing. The proof is complete.
By combining with Theorems 3.14, 3.19 and 3.22, we have the following result.
Entropy studied by skew-coordinated system
In this section, we use a skew-coordinated system to study the entropies. We first recall some properties of skew-coordinated systems [2,37,39]. For a skewcoordinated system γ ∈ GL 2 (Z), GL 2 (Z) is the modular group
γ 1 = (1, 0) t γ = (a, b) t and γ 2 = (0, 1) t γ = (c, d) t .
The height h(γ) of γ is defined by
h = h(γ) = |a| + |b|,
and the width of γ is
w = w(γ) = |c| + |d|. γ 0 = 1 0 0 1 ∈ GL 2 (Z)
is the standard rectangular system. The smallest parallelogram lattices in the γ-coordinate that contain exactly one unit square lattice in γ 0 -coordinates are determined as follows, see [2]. After Proposition 4.1, the cylindrical transition matrices T γ,n (B) which determine the γ 1 -periodic with period n and width 2 in γ 2 -direction can be obtained.
Finally, as in γ 0 -coordinates, the rotational matrix R γ,n for γ-coordinates can be introduced. Then we can have same results as Porposition 2.3 for γ-coordinates.
(4.1) Γ M L 0 K γ = tr T K γ,M R L γ,M .
It is worth investigating the entropies computed by various skew-coordinated systems. Therefore, we will study the transformations of Hermite normal forms among the different skew-coordinated systems.
We first recall the method for transforming a 2×2 integer matrix into its Hermite normal form, see [39]. We denote by A = a 11 a 12 a 21 a 22 an integer matrix with det A = 0 and a 21 = 0. If a 21 = 0, then A is already in a normal form. Let Then k = 0. When a 22 = 0, then k = a 21 and b 1 = 1. Let
(4.4) U = a22 k b 1 − a21 k b 2 .
Then by (4.3), det U = 1, i.e., U is a unitary matrix. Then Hence,
(4.8) AZ 2 = m 0 k Z 2 .
Therefore, m 0 k is the Hermite normal form of A.
Note that, two integer 2×2 matrices A and A are call equivalent and are denoted by
A ∼ = A if (4.9) A Z 2 = AZ 2 ,
i.e., they determine the same sublattices in Z 2 . Hence A and its normal form m 0 k are equivalent in (4.8) which determine the same sublattices of Z 2 . After introducing the procedure to transform the 2 × 2 integer matrix to its normal form,
we are going to study the transformation of normal form M L 0 K γ in γ-coordinates to its normal form m 0 k γ0 in γ 0 -coordinates.
Firstly, M γ = M L 0 K γ with respect to γ-coordinates can be rewritten as in
γ 0 -coordinates by (4.10) γ t M γ = a c b d M L 0 K = aM aL + cK bM bL + dK γ0
It remains to transform (4.10) to its normal form. Indeed, assume b = 0, let k = gcd(bM, bL + dK) (4.11) = b 1 (bM ) + b 2 (bL + dK), (4.12) (4.13) ∆ = det γ, and (4.14)
U γ = ∆ k (bL + dK) b 1 − ∆ k (bM ) b 2 .
Then, it is straightforward to verify that
(4.15) γ t M γ U γ = m 0 k γ0 with (4.16) m = M K k and (4.17) = b 1 (aM ) + b 2 (aL + cK).
That is,
(4.18) m 0 k γ0 = M K k b 1 (aM ) + b 2 (aL + cK) 0 k γ0 .
Therefore, we have the following theorem. Then
(4.22) m k b −dK+ k b 0 K γ ∼ = m K b ak−∆b2K b 0 k γ0 .
Proof. By
= 1 b {a(b 1 bM + b 2 bL + b 2 dK) + (bc − ad)b 2 K} = 1 b (ak − ∆b 2 K).
The proof is complete.
Also, (4.22) in Theorem 4.3 can be stated as the following theorem.
K = µb + ν, where (4.25) 0 ≤ ν 0 ≤ b − 1 and 0 ≤ ν ≤ b − 1.
Then
(4.26) m k b µ 0 − dµ + ν0+dν b 0 µb + ν γ ∼ = m µ + m ν b −∆b 2 µ + ak−∆b2ν b 0 k γ0 .
Furthermore, Then
(4.31) m K a b−ab K −(a d−b c)K a b−ab 0 K γ ∼ = m K a b−ab K (ad −bc ) a b−ab − b 2 K ∆∆ a b−ab 0 K γ . Proof. For γ, γ ∈ GL 2 (Z), if M L 0 K γ and M L 0 K γ are equivalent in γ 0 - coordinates, then (4.32) a c b d M L 0 K U ∼ = a c b d M L 0 K U ,
where U and U are unitary matrices. Then (4.32) implies
(4.33) ∆ M L 0 K ∼ = d −c −b a aM aL + cK bM bL + dK U (U ) −1 ∼ = (ad − bc )M (ad − bc )L + (cd − c d)K (a b − ab )M (a b − ab )L + (ad − b c)K U (U ) −1 .
Since ∆ = ±1,
(4.34) ∆ M L 0 K ∼ = M L 0 K ∼ = ∆ M L 0 K .
Combining (4.33) and (4.34), we obtain
(4.35) M L 0 K ∼ = (ad − bc )M (ad − bc )L + (cd − c d)K (a b − ab )M (a b − ab )L + (ad − b c)K U (U ) −1 .
In the case a b − ab = 0. Since
U (U ) −1 = (a b−ab )L+(a d−b c)K K b 1 (a b−ab )M K b 2 where K = gcd((a b − ab )M, (a b − ab )L + (ad − b c)K) = b 1 (a b − ab )M + b 2 [(a b − ab )L + (ad − b c)K)] = b 1 m K + b 2 K . Then M = m K a b−ab , L = K −(a d−b c)K a b−ab , M = (ad − bc )M (a b − ab )L + (a d − b c)K K − (a b − ab )M K [(ad − bc )L + (cd − c d)K] = M K K [(ad − bc )(a d − b c) − (a b − ab )(cd − c d)] = M K K ∆∆ = m K a b − ab ∆∆ , and L = b 1 (ad − bc )M + b 2 [(ad − bc )L + (cd − c d)K] = b 1 (ad − bc ) m K a b − ab + b 2 ad − bc a b − ab [ K − (a d − b c)K] + (a b − ab )(cd − c d)K a b − ab = b 1 (ad − bc )m K + b 2 K (ad − bc ) a b − ab − b 2 K (ad − bc )(a d − b c) − (a b − ab )(cd − c d) a b − ab = K (ad − bc ) a b − ab − b 2 K ∆∆ a b − ab . Thus, m K a b−ab K −(a d−b c)K a b−ab 0 K γ ∼ = ∆∆ m K a b−ab K (ad −bc ) a b−ab − b 2 K ∆∆ a b−ab 0 K γ ∼ = m K a b−ab K (ad −bc ) a b−ab − b 2 K ∆∆ a b−ab 0 K γ ,
the last ∼ = due to ∆∆ = ±1 and (4.34).
= a b c d ∈ GL 2 (Z) with a = 0, mk aˆ k +cK a 0 K γ ∼ = mK a −bk+∆b2K a 0k γ0
, whereb 1m +b 2ˆ = 1.
Proof. The proof is easily from Theorem 4.5.
In particular, if γ = γ 0 = 1 0 0 1 and γ =γ 0 = 0 1 1 0 . Then we have
Theorem 4.8. m k k 0 k γ0 ∼ = m k b 2 k 0k γ0 , where b 1 m + b 2 = 1.
For completeness, a similar expression forγ 0 as Theorem 4.4 can also be written in the following theorem. The proof is omitted. By the results in [24],
h γ (B) = h(B)
for all basic set B and all γ ∈ GL 2 (Z).
The periodic entropy can also be defined in skew-coordinated systems. Indeed,
for any γ = a b c d ∈ GL 2 (Z),1 M K log Γ M L 0 K γ .
By the Theorem 4.3, we can prove the following lemma. Proof. We only treat the case b = 0. The case b = 0 can also be studied analogously. By Theorem 4.3,
(4.41) M L 0 K γ ∼ = m 0 k γ0 ,
where M, L, K, m, , k satisfy the relations in (4.22). Then, The proof is complete.
h γ,p (B) = lim sup (M,K)→∞ sup L 1 M K log Γ B M L 0 K γ ≥ lim sup K→∞ lim sup M →∞ sup L 1 M K log Γ B M L 0 K γ = lim sup
Hence we have the following theorem. We have the following results.
(4.48) c m ρ(T m ) αpm ≤ (T αpm+βm m ) i,j ≤ d m ρ(T m ) αpm for α ≥ α m ≥ 1, where 0 < c m ≤ d m . Hence (4.48) implies lim sup k→∞ 1 k log tr(T k m R m ) ≥ lim sup α→∞ 1 αp m + β m log r m i=1 (T αpm+βm m ) i,σ − (i) ≥ lim sup α→∞ log ρ(T m ).
Therefore,
h (B) = lim sup (m,k)→∞ 1 mk log tr(T k m R m ) ≥ lim sup m→∞ 1 m lim sup k→∞ 1 k log tr(T k m R m ) ≥ lim sup m→∞ 1 m lim sup k→∞ 1 k log r m i=1 (T k m ) i,σ − (i) ≥ lim sup m→∞ 1 m log ρ(T m ).
Next, if T m is not irreducible, but T m ≤ T m is a maximum irreducible submatrix with cycle p m ≥ 1, i and σ − (i) ∈ I m , the index set of T m . Then, as (4.48), the following inequality holds.
(4.49) c m ρ(T m ) αpm ≤ (T αpm+βm m ) i,σ − (i) ≤ d m ρ(T m ) αpm
for all α ≥ α m ≥ 1 and some β m ≥ 0. Hence (4.46) holds. The proof is completed.
Combing with Theorems 4.12, 3.9 and 3.14 we have the following results. for m ≥ 1. Similarly, the above results for ML 0K γq and mˆ 0k γ0 can also be obtained.
Proof. The proof is easily from Theorem 4.5.
In the following, we will use a sequence of skew-coordinates systems γ q = 1 q 0 1
, q ≥ 1, to compute the periodic entropy. By (4.53), we have (4.54) 1 0 0 mq γq ∼ = m 1 0 q γ0 , for m ≥ 1. The relation of (4.54) enable us to compute the 1-shift entropy h 1 (B) by computing the ρ(T γq,1 ). We draw the pictures to illustrate T γq,1 for q = 1, 2 and 3. The others can also be obtained inductively.
= = (4.57) = x 11 x 21 × × × × × × × × x 31 x 41 × × × × × × × × x 12 x 22 × × × × × × × × x 32 x 42 x 13 x 23 × × × × × × × × x 33 x 43 × × × × × × × × x 14 x 24 × × × × × × × × x 34 x 44 2 3 ×2 3 .
(4.58)
For q ≥ 3, (4.59)
T γq,1 = I 2 q−2 ⊗ x 11 x 21 × × × × x 31 x 41 I 2 q−2 ⊗ x 12 x 22 × × × × x 32 x 42 I 2 q−2 ⊗ x 13 x 23 × × × × x 33 x 43 I 2 q−2 ⊗ x 14 x 24 × × × × x 34 x 44 2 q+1 ×2 q+1 .
Note that in (4.58) to (4.59), the jth column vector (x 1j , x 2j , x 3j , x 4j ) t in X 2×2 has been arranged as
A j = x 1j x 2j × × × × x 3j x 4j .
For r ≥ 3 and γ ∈ GL 2 (Z), T γ,n can also be introduced. For example, for r = 3, We need the following lemma to study (4.62). Proof. Let {λ j } N j=1 be the eigenvalues of A. Then |λ j | ≤ ρ, for all j.
X 2×2 = [x i,j ] 3 2 ×3 2 ,
Hence
tr(A k ) = N j=1 λ k j ≤ N j=1 |λ j | k ≤ N ρ k .
Then, (4.63) follows.
Then we have the following result. From Theorem 4.19, h(B) can be studied by ρ(T γq,1 ) which are easier than computing ρ (T m ).
log |Σ γ;n×k (B)| .The periodic entropy h γ,p (B) of basic set of admissible patterns B is defined by
T m (B) stores all horizontal periodic patterns which are generated by B on Z m×2 , m ≥ 1. Denoted by ρ (T m (B)) to be the maximum eigenvalue of matrix T m (B). The sequence of horizontal periodic transition matrices {T m (B)} ∞ m=1 is called uniformly dominated by {ρ (T m (B)
Theorem 1. 1 .
1If (1.7) and (1.8) hold, then h (B) = h p (B) = h * (B) , where (1.9) h * (B) = lim sup m→∞ 1 m log ρ (T m (B)) .Conditions (1.7) and (1.8) can be obtained by checking the following connectedness conditions.{T m (B)} ∞ m=1 is called uniformly connected if T m (B)is irreducible for all m ≥ 1 and there is a positive integer K such that for any m ≥ 1 and any indices pair(i, j), 1 ≤ i, j ≤ r m where there is 1 ≤ k ≤ K such that (1.10) T k m (B) i,j ≥ 1. Uniform connectiveness of {T m (B)}∞ m=1 is equivalent to the existence of finite upper bound K of the diameters of associated graphs of {T m (B)} ∞ m=1 , see Section 3 for details. By Perron-Frobenius Theorem, uniform connectedness implies uniform domination as follows: Theorem 1.2. If {T m (B)} ∞ m=1 is uniformly connected then {T m (B)} ∞ m=1 is uniformly dominated by {ρ (T m (B))} ∞ m=1 .
r have P(B) = ∅ and Σ(B) = ∅ with h(B) > 0. Therefore,
y
11 y 12 y 21 y 22 y 13 y 14 y 23 y 24 y 31 y 32 y 41 y 42 y 33 y 34 y 43 y
where h i1,i2 = 1 if and only if x i1i2 ∈ B, and v j1,j2 = 1 if and only if y j1j2 ∈ B.
Theorem 3 . 3 .
33Let A = [a i,j ] N ×N be an irreducible non-negative integral matrix with maximum eigenvalue ρ = ρ(A) > 0 and cycle p ≥ 1. Then for any 1
Lemma 3. 4 .
4For any double sequence a m,n , we have
Lemma 3. 5 .
5For any basic set of admissible local patterns B,
( 3 .
319) tr(H m n (B)) = T n−1 m (B) .
Lemma 3. 6 .
6For any basic set of admissible local patterns B, Since the double limit of h(B) holds in(2.19) and equals to iterated limits, log ρ (H n (B)) .On the other hand, by(3.19) of Lemma 3.log |T n m (B)| .By(2.19) and(3.25), the equality holds in(3.22). That is,(3.21) holds. The proof is complete.For any basic set of admissible patterns B, denote by(3.23) h * (B) = lim sup m→∞ 1 m log ρ(T m (B)).
Lemma 3. 7 .
7For any basic set of admissible local patterns B,(3.24) h * (B) ≤ h p (B) ≤ h(B).Proof. From (3.19), we have(3.25) T n−1 m (B) ≤ |H m n (B)| .
Definition 3. 8 .
8For basic set B, the sequence of cylindrical transition matrices {T m (B)} ∞ m=1 is called uniformly dominated (or uniformly controlled or uniformly bounded) by their maximum eigenvalues {ρ (T m (B))} ∞ m=1 if there is a positive function c(m, k) such that (3.26) T k m (B) ≤ c(m, k)ρ(T m ) m ≥ 1 and k ≥ 1.We show that the equality holds in (3.24) when basic set B satisfies the uniformly dominant properties(3.26) and(3.27).
Theorem 3. 9 .
9If (3.26) and (3.27) hold, then
( 3 .
328) h(B) = h p (B) = h * (B). ρ(T m (B)).Therefore, (3.29) holds by Lemma 3.6 and then (3.28) follows. The proof is complete.
Lemma 3 . 10 .
310If the non-negative N × N matrix A = [a i,j ] has positive eigenvalue ρ(A) with positive eigenvector
Lemma 3 . 11 .
311Let N × N non-negative integral matrix A = [a i,j ] have positive eigenvalue ρ ≥ 1 with positive eigenvector v = (v 1 ,..., v N ). If there is a positive constant K ≥ 1 such that for any 1
Definition 3 . 12 .
312For a basic set of admissible patterns B, T m (B) is called uniformly connected if T m (B) is irreducible for any m ≥ 1, and there is a positive integer K such that for any m ≥ 1 and any 1 ≤ i, j ≤ r m , there exists a 1 ≤ k ≤ K such that(3.37) (T k m ) i,j ≥ 1.
Theorem 3. 13 .
13If {T m } ∞ m=1 is irreducible and uniformly connected then {T m } ∞ m=1 is uniformly dominated by {ρ(T m )} ∞ m=1 . Proof. Let K ≥ 1 such that (3.37) holds. Let v m = (v m,1 , ..., v m,N (m) ) be the associated positive eigenvectors and ρ m = ρ(T m ) be the maximum positive eigenvalue of T m . Then Lemma 3.10 implies (3.39) max i v m,i min i v m,i ≤ ρ K m . By (3.30), (3.40) T k m ≤ ρ K m r m ρ k m for all m ≥ 1 and k ≥ 1.
Theorem 3 . 14 .
314If {T m (B)} ∞ m=1 is irreducible and uniformly connected then h(B) = h p (B) = h * (B).
Definition 3 . 15 .
315For any basic set of admissible patterns B ⊆ S r , T k m (B) is called uniformly dominated by T k m (B) if there is a positive function c 0 (m, k) such that (3.44) T k m (B) ≤ c 0 (m, k) log c 0 (m, k) = 0.To illustrate Definition 3.15, we introduce the following example.
Theorem 3 . 19 .
319If {T m (B)} ∞ m=1 is uniformly dominated by the maximum irreducible submatrices {T m (B)} ∞ m=1 and {T m (B)} ∞ m=1 is uniformly connected, then {T m (B)} ∞ m=1 is uniformly dominated by {ρ(T m )} ∞ m=1 .In theses cases,(3.47) h(B) = h p (B) = h * (B).We can provide some examples which have been studied widely as follows.
Theorem 3.22. If {T m (B)} ∞ m=1 is irreducible and Σ (B) is horizontal-periodic block gluing, then {T m (B)} ∞ m=1 is uniformly connected. Furthermore, if {T m (B)} ∞ m=1 is uniformly connected and for any m ≥ 1, there exists an index i m such that (3.50) (T m (B)) im,im = 1, then Σ (B) is horizontal-periodic block gluing. Proof. Denoted by the indices U m α,0 , 0 ≤ α ≤ m−1 and V m α,0 , 0 ≤ α ≤ m−1, by i and j, respectively. Then (3.48) and (3.49) is equivalent to (3.51) T k m (B) i,j ≥ 1. If Σ (B) is horizontal-periodic block gluing, then for any i, j in I m (Σ (B))
Theorem 3 . 23 .
323If {T m (B)} ∞ m=1 is weakly irreducible and Σ (B) is horizontalperiodic block gluing, then h (B) = h p (B) = h * (B) .
a, b, c, d ∈ Z and |ad − bc| = 1 , and its subgroup SL 2 (Z) = a b c d : a, b, c, d ∈ Z and ad − bc = 1 . GL 2 (Z), Z 2 = {(ra+sc, rb+sd) : r, s ∈ Z} holds. Therefore γ is a unimodular transformation on Z 2 and induces a skew-coordinated system on Z 2 . Indeed, the unit lattice points in γ-coordinates are (1, 0) γ = (a, b) and (0, 1) γ = (c, d), and the unit vectors are
Proposition 4. 1 .
1For any γ = a b c d ∈ GL 2 (Z), there exists exactly one unit square lattice in γ 0 -coordinates in the parallelogram lattices with vertices (0, 0) γ , (w, 0) γ , (0, h) γ and (w, h) γ . The unit square lattice has either vertices (0, h) γ and (w, 0) γ or (0, 0) γ and (w, h) γ .
GL 2 (Z), let M, K ≥ 1 and 0 ≤ L ≤ M − 1,
( 4 .
42) k = gcd(a 21 , a 22 ), be the greatest common divisor of a 21 and a 22 with(4.3) k = b 1 a 21 + b 2 a 22 .
1 a 11 + b 2 a 12 .
Theorem 4. 3 .
3Given γ = a b c d ∈ GL 2 (Z), ∆ = det γ, with b = 0. Let k = gcd(bM, bL + dK) and k = b 1 (bM ) + b 2 (bL + dK). Let
1 (aM ) + b 2 (aL + cK)
( 4 .
427) if b|m k then b|m ν, and (4.28) if b|(ν 0 − dν) then b|(ak − ∆b 2 ν).Proof. By (4.23) and (4.24),k−dK b = µ 0 − dµ + ν0−dν b. Given 0 ≤ ν 0 ≤ b − 1, since gcd(b, d) = 1, then there exist m, n ∈ Z such that bν 0 m + dν 0 n = ν 0 . This implies b(ν 0 m − dk) + d(ν 0 n − bk) = ν 0 for all k ∈ Z, which gives a unique 0 ≤ ν ≤ b − 1 such that b|ν 0 − dν. Furthermore, ak − ∆b 2 K = −b∆b 2 µ + ak − ∆b 2 ν. Hence = −∆b 2 µ + ak−∆b2ν b . Finally, (4.27) and (4.28) can be verified directly and the details are omitted here. Theorems 4.3 and 4.4 can also be generalized to any two γ GL 2 (Z) as in the following theorem.
Theorem 4. 5 .
5Given
m + b 2 = 1, ∆ = det γ and ∆ = det γ .
Remark 4. 6 .. 7 .
67If a b − ab = 0, then b a = b a . Now, gcd(a , b ) = 1 and gcd(a, b) = 1 imply either a = a and b = b or a = −a and b = −b. Indeed, we have M = (ad − bc )M , L = (ad − bc )L + (cd − c d)K and K = (a b − ab )L + (ad − b c)K.It is easy to see when γ = γ Given γ
Theorem 4. 9 .
9Given γ = a b c d ∈ GL 2 (Z) with a = 0. Then
m + b 2 = 1, k =μ 0 a +ν 0 , and (4.38)K =μa +ν, 0 ≤ν 0 ≤ a − 1, 0 ≤ν ≤ a − 1.
if a|m k then a|m ν, and (4.40) if a|(ν 0 + cν) then a|(bk − ∆b 2ν ).The entropy can be studied in skew-coordinated systems in any γ = a b c d ∈GL 2 (Z). Indeed, the parallelogram Z γ,m×k in γ-system is defined as m units in γ 1 direction and k units in γ 2 direction. Denoted by |Σ γ;m×k (B)| the total number of admissible patterns is determined by B. Then the entropy h γ (B) computed in γ-coordinated system is defined by h γ (B) = lim sup (n,k)→∞ 1 nk log |Σ γ;m×k (B)| .
defining the periodic entropy h γ,p (
Lemma 4 . 10 .
410For any admissible basic set B and γ ∈ GL 2 (Z),h γ,p (B) ≥ h * (B).
By choosing m = 1, b 1 = 1, b 2 = 0 and k = bm, we have h γ,p (B)
Theorem 4 . 11 .
411If h(B) = h * (B), then h γ,p (B) = h(B) for all γ ∈ GL 2 (Z). In particular, if {T m (B)} ∞ m=1 is uniformly dominated by {ρ(T m )} ∞ m=1 which satisfies(3.26) and (3.27), then h γ,p (B) = h(B) for all γ ∈ GL 2 (Z).
Theorem 4 . 12 .
412If T m (B) is irreducible for all m ≥ 1, then (4.47) h (B) ≥ h * (B). Furthermore, if T m (B) is not irreducible and T m (B) ≤ T m (B) isa maximum irreducible submatrix with index set I m and there is an index pair i and σ − (i) ∈ I m . Then (4.47) holds. Proof. By Theorem 3.3, if T m is irreducible then for any m ≥ 1 and any pair i and j there is β m = β m (i, j) ≥ 0 and cycle p m ≥ 1 and α m ≥ 1 such that
Theorem 4 . 13 .
413If {T m (B)} ∞ m=1 is irreducible and uniformly dominated by {ρ (T m (B))} ∞ m=1 , then
h (B) = h p (B) = h(B) = h * (B) for all integer . Furthermore, if {T m (B)} ∞ m=1 is reducible and {T m (B)} ∞ m=1 is a sequence of maximum irreducible submatrices with an indices pair i and σ − (i) ∈ I m , and {T m (B)} ∞ m=1 is uniformly dominated by {ρ (T m (B)(B) = h(B) = h * (B). Now, Theorem 4.12 implies (4.50). The proof is complete. By combining Theorems 4.13 and 3.22, we have the following result.
Theorem 4 . 14 .
414If {T m (B)} ∞ m=1 is irreducible or weakly irreducible and Σ (B) is horizontal-periodic block gluing, then h (B) = h p (B) = h (B) = h * (B) , for all .
Example 4 . 15 .where b 1 m + b 2 = 1 .
4151The GM, SGM and Hard-Hexagon model in Example 3.20 satisfy the assumption of Theorem 3.14 and then (4.50) holds.The following theorem derives the transformation between When m k = q, then
for any α ≥ 1 .= 1
11In particular, when m
log tr T mq γq,1 .
Lemma 4. 17 .
17Let A = [a i,j ] N ×N be a non-negative matrix with eigenvalues {λ j } N j=1 and the maximum eigenvalue ρ. Then
Theorem 4 . 18 .
418For any basic set B of admissible patterns,
ρ(T γq,1 ),where ρ q = ρ(T γq,1 ) and N q = r q+1 , here r is the number of symbols.Then(4.65) follows from h 1 (B) ≥ lim sup q→∞ lim sup m→∞ 1 mq log tr(T mq γq,1 ) = lim sup q→∞ log ρ(T γq,1 ).On the other hand, (4.63) also implies tr(T mq γq,1 ) ≤ N q ρ(T γq,1 ) mq . Hence (4.64) follows fromh 1 (B) ≤ lim sup (m,q)→∞ 1 mq log N q ρ(T γq,1 ) mq = lim sup q→∞ log ρ(T γq,1 ).Combining Theorems 3.9, 4.13 and 4.18, we haveTheorem 4.19. If {T m } ∞ m=1 is irreducible and uniformly dominated by {ρ(T m )} ∞ m=1 , or if {T m } ∞ m=1 is reducible and the maximum irreducible submatrices {T m } ∞ m=1 with indices pairs i and σ −1 (i) ∈ I m and {T m } ∞ m=1 is uniformly dominated by {ρ(T m )} ∞ m=1 . Then h(B) = lim sup q→∞ log ρ(T γq,1 ).
t m;i1,i2 · · · t m;i k ,σ − (i1) .
In fact, for q ≥ 2is a rearrangement of ith column vector in X 2×2 . Since h 1 (B) = lim sup (m,q)→∞
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. Wen-Guei Hu, Chengdu, 610064College of Mathematics, Sichuan UniversityChina Email address: [email protected](Wen-Guei Hu) College of Mathematics, Sichuan University, Chengdu, 610064, China Email address: [email protected]
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[] |
[
"COVID-19 societal response captured by seismic noise in China and Italy",
"COVID-19 societal response captured by seismic noise in China and Italy"
] |
[
"Han Xiao \nDepartment of Earth Science and Earth Research Institute\nUniversity of California\nSanta BarbaraCAUSA\n",
"Zachary Cohen Eilon \nDepartment of Earth Science and Earth Research Institute\nUniversity of California\nSanta BarbaraCAUSA\n",
"Chen Ji \nDepartment of Earth Science and Earth Research Institute\nUniversity of California\nSanta BarbaraCAUSA\n",
"Toshiro Tanimoto \nDepartment of Earth Science and Earth Research Institute\nUniversity of California\nSanta BarbaraCAUSA\n"
] |
[
"Department of Earth Science and Earth Research Institute\nUniversity of California\nSanta BarbaraCAUSA",
"Department of Earth Science and Earth Research Institute\nUniversity of California\nSanta BarbaraCAUSA",
"Department of Earth Science and Earth Research Institute\nUniversity of California\nSanta BarbaraCAUSA",
"Department of Earth Science and Earth Research Institute\nUniversity of California\nSanta BarbaraCAUSA"
] |
[] |
Seismic noise with frequencies above 1 Hz is often called 'cultural noise' and is generally correlated quite well with human activities. Recently, cities in mainland China and Italy imposed lockdown restrictions in response to COVID-19, which gave us an unprecedented opportunity to study the relationship between seismic noise above 1 Hz and human activities. Using seismic records from stations in China and Italy, we show that seismic noise above 1 Hz was primarily generated by the local transportation systems. The lockdown of the cities and the imposition of travel restrictions led to a ~4-12 dB energy decrease in seismic noise in mainland China. Data also show that different Chinese cities experienced distinct periods of diminished cultural noise, related to differences in local response to the epidemic. In contrast, there was only ~1-6 dB energy decrease of seismic noise in Italy, after the country was put under a lockdown. The noise data indicate that traffic flow did not decrease as much in Italy, but show how different cities reacted distinctly to the lockdown conditions.
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10.1785/0220200147
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[
"https://arxiv.org/pdf/2005.00131v1.pdf"
] | 218,470,589 |
2005.00131
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bac8f4241cf1c9a32c27a012cfe6a0251f103348
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COVID-19 societal response captured by seismic noise in China and Italy
Han Xiao
Department of Earth Science and Earth Research Institute
University of California
Santa BarbaraCAUSA
Zachary Cohen Eilon
Department of Earth Science and Earth Research Institute
University of California
Santa BarbaraCAUSA
Chen Ji
Department of Earth Science and Earth Research Institute
University of California
Santa BarbaraCAUSA
Toshiro Tanimoto
Department of Earth Science and Earth Research Institute
University of California
Santa BarbaraCAUSA
COVID-19 societal response captured by seismic noise in China and Italy
1
Seismic noise with frequencies above 1 Hz is often called 'cultural noise' and is generally correlated quite well with human activities. Recently, cities in mainland China and Italy imposed lockdown restrictions in response to COVID-19, which gave us an unprecedented opportunity to study the relationship between seismic noise above 1 Hz and human activities. Using seismic records from stations in China and Italy, we show that seismic noise above 1 Hz was primarily generated by the local transportation systems. The lockdown of the cities and the imposition of travel restrictions led to a ~4-12 dB energy decrease in seismic noise in mainland China. Data also show that different Chinese cities experienced distinct periods of diminished cultural noise, related to differences in local response to the epidemic. In contrast, there was only ~1-6 dB energy decrease of seismic noise in Italy, after the country was put under a lockdown. The noise data indicate that traffic flow did not decrease as much in Italy, but show how different cities reacted distinctly to the lockdown conditions.
Introduction
Seismic background noise observed at frequencies above 1 Hz primarily consists of cultural noise, which is generated by human activities (Stutzmann et al., 2000;McNamara and Buland, 2004;Groos and Ritter, 2009;Green et al., 2017) such as trains (Chen et al., 2004;Sheen et al., 2009;Ribes-Llario et al., 2017), road traffic (Hao and Ang, 1998;Coward et al., 2003) and airports (Meng and Ben-Zion, 2018). These noise sources are now known to be useful for studying the subsurface structures (Nakata et al., 2011;Riahi and Gerstoft, 2015;Quiros et al., 2016;Ajo-Franklin et al., 2019).
The outbreak of the novel coronavirus SARS-CoV-2 disease (hereafter: COVID-19) was first reported in Wuhan, Hubei, China, in December 2019 (Andersen et al., 2020). In early to mid-January 2020, the virus spread to other Chinese provinces, facilitated by increased travel during the Chinese Lunar New Year. With Wuhan being a major rail transport hub in China, the virus quickly spread throughout the country. On 23 January 2020, Wuhan and other cities of Hubei province imposed a lockdown in an effort to quarantine the epicenter of the COVID-19 outbreak (Lu, 2020). All public transportation except for emergency and supply vehicles were suspended. A total of 12 other counties to prefecture-level cities in Hubei, including Enshi, the location of one of the seismic stations used in this study, were placed on travel restrictions by the end of 24 January 2020 (Table 1). The World Health Organization (WHO) declared the outbreak to be a Public Health Emergency of International Concern on 30 January 2020. By chance, this outbreak closely coincided with the Chinese Lunar New year of 2020, and the Chinese government utilized this coincidence to facilitate lockdown logistics, essentially extending the traditional week-long national holiday for several months. In Italy, following the rapid expansion of an outbreak of COVID-19 cases in the north of the country in late February 2020, the Italian government imposed a lockdown on many of its Northern provinces on 8 March 2020. The lockdown restricted all movement except for work, health circumstances and essential activities. On the evening of 9 March, quarantine measures were extended to the entire nation, becoming effective the next day. This quarantine included some important differences from the restrictions in China. For example, the lockdown did not apply to the public transportation system, including buses, railways, flights, and ferry services. People with self-declared travel exemptions were permitted to travel. On 11 March 2020, the WHO declared the outbreak a pandemic. This sudden declaration of a public health emergency provides an unusual dataset in which we can compare changes in seismic noise to the known timings of social orders.
In this study, we analyze continuous seismic time series from seismic stations in China and Italy (Fig. 1). We particularly focus on understanding the characteristics of cultural noise before and after the lockdowns. We show that the lockdown of cities in China led to a ~4-12 dB reduction in cultural noise. In contrast, there was only ~1-6 dB energy decrease of cultural noise in Italy, possibly due to ongoing traffic.
Data and Methods
We utilized data from the New China Digital Seismograph Network (NCDSN), in operation since 1992, with the network code IC. Data from IC were obtained from the Incorporated Research Institutions for Seismology (IRIS) Data Management Center (DMC; www.iris.edu/dms) from 1 January 2000 to 15 April 2020. Those stations are: station IC.ENH, located in Enshi, Hubei province; station IC.MDJ, located in Mudanjiang, Heilongjiang province in the northeast region of China; station IC.BJT which is located in the northwest of Beijing, the capital of China; and station IC.QIZ which is located on Hainan island in Qiongzhou (Fig. 1a). We also analyzed data from the Italian National We analyzed broadband high-gain vertical seismograms (BHZ) and high-sample-rate highgain broadband three-component seismograms (HHN, HHE, and HHZ), with sample rates of 20 Hz and 100 Hz, respectively. Ground acceleration records were retrieved by deconvolving the instrumental response from the original seismograms. All seismic data were divided into one-hour segments with overlapping time intervals of 50% (30 minutes).
Each one-hour segment was detrended, tapered with a Hanning window, and the power spectral densities (PSD) were calculated. We did not remove earthquake signals from the time series, because the effects of earthquakes are limited to short time intervals and predominantly contain their most characteristic signals at lower frequencies. Thus they have extremely small overall effects on the estimation of cultural noise. We show noise power levels in units of decibels (dB) with respect to 10log &' ( ) * + , -. ) . We also performed a frequency-dependent polarization analysis for the high broadband three-component seismic data to determine the source directions of the cultural noise (Samson, 1983;Park et al., 1987;Koper and Hawley, 2010).
Results
Baseline seismic noise patterns at ENH
We use a twenty-year-long seismic record from station IC.ENH, located in Enshi, Hubei province, China, to establish important baselines and patterns in the cultural noise, illustrating the detailed ways in which this signal is related to societal behavior. The vertical PSD for this station is shown in Figure show that the noise level started to drop 5-6 days before the holiday and did not fully recover until another two weeks after (Fig. 7). This frequency band also exhibits seasonal variations ( Fig. 5a) which is longer strong noise signal in the summer than the winter corresponding to variation in daylight hours. We infer that 1.5-8 Hz noise is probably generated by the local factories, pedestrians (Alyamkin and Eremenko, 2011), and lowspeed urban road traffic (Green et al., 2017).
By contrast, the noise in the frequency band 10-20 Hz is relatively stable during the Chinese Lunar New Year. This persistence allows the noise source to be identified. Long (1971) found that moving vehicles on the freeway generate noise with peak frequency at about 10-20 Hz, and that this signal can be detected within 5-8 km. The station IC.ENH is located ~3 km from the freeway G50 to its southeast (Fig. 1a). The G50 is a 1900 km long and east-west bound expressway connecting Shanghai, China to the east and Chongqing, China to the west. While we are not yet able to retrieve the record of daily traffic-flow volume on the G50 near city Enshi, traffic data for a segment of this freeway named as Huanghuang east of Wuhan can be accessed (http://www.hhgs.org.cn/) and we use it as the representative of the general traffic flow on this road within Hubei province. The average daily traffic-flow volume on the freeway during the 2018 Chinese Lunar New Year was only 31% less than the daily volume on the regular days (Table 2). This suggests that the cultural noise peak in the frequency band 10-20 Hz can be explained by the high-speed cars on the freeway. This is further supported by the back azimuths of the polarization ellipsoids in Figure 3b. The source direction of cultural noise in the 10-20 Hz frequency band is from the southeast (back azimuths are ~ 120° -150°) during the daytime, indicating the noise arrives from the direction of the freeway. The northwest noise source during the night ( Fig. 3c) is probably from the downtown area, which lies in that direction. This dataset also shows a period of freeway traffic control between 13 September 2018 and 26
October 2018, when freeway access was limited or cut off due to road repairs. Station IC.ENH exhibits clear 10-20 Hz noise energy reduction during this period (Fig. 5b).
Our findings agree with previous analyses of frequency bands typical for road traffic (Groos and Ritter, 2009;Boese et al., 2015;Chang et al., 2016;Green et al., 2017) and consistent with a correlation between the cars' noise frequency content and their speed: the peak frequency band for the low-speed urban road traffic is 1.5-8 Hz, and for the high-speed cars traveling on the freeway it is 10-20 Hz (Long, 1971). Overground rail transportation could generate the seismic noise above 30 Hz in the seismograms (Chen et al., 2004;Boese et al., 2015;Riahi and Gerstoft, 2015;Green et al., 2017), and the Chinese Lunar New Year is the peak travel season for the railway stations. However, according to our observations, there is a 5 dB reduction in the frequency band 30-40 Hz during this period, suggesting that the noise generated by the rail does not dominate our seismic record, perhaps due to the high attenuation and relatively far distance (8 km).
Comparison of living habits between China and Italy.
These data also reveal the various living habits of people in different cities (Fig. 5). For station IC.ENH located in Enshi ( Fig. 5a and 5b), China, as we mentioned before, we can see the lull caused by the Chinese Lunar New Year in January or February (Fig. 2). In general, people in Enshi show most activity from 8 a.m. to 5 p.m. during the winter. In the summer, people tend to be active for longer, from 7:00 a.m. to 6:00 p.m. There is a lunchtime lull year-round at noon. However, for the frequency band 10-20 Hz generated by the traveling cars on the freeway (Fig. 5b), the lunch-time lull is longer and continues for about two hours during the summer. This may reflect hot noon-time weather in the summer discouraging people from going outside. Interestingly, the Chinese data do not show clear weekly cycles; work and activity continue week-round. In Milan, Italy ( Fig. 5c and 5d (Fig. 8b). Interestingly, there are four distinct sources in frequency bands 1-2 Hz, 3-8 Hz, 10-20 Hz, and 20-40 Hz for station IV.RMP which located in the ~20 km southeast of Rome (Fig. 8e). We estimate the first peak in 1-2 Hz ( Fig. 9a), with a ~5 dB reduction during the 2019 Christmas period, is generated by pedestrians (Alyamkin and Eremenko, 2011), since we expect fewer pedestrians on the street during the holiday of Christmas. By comparison to the Chinese data, we believe the second peak at 3-8 Hz (Fig. 9b) is generated by the local factories and low-speed cars on the street; this band a ~3dB deduction at Christmas. Finally, the high-speed cars on the freeway and the trains are responsible for the frequency bands at 10-20 Hz and 20-40 Hz respectively, which exhibit a ~2 dB reduction during Christmas (Fig. 8f). However, there is no obvious energy diminishing during the Christmas holiday for station IV.MONC (Fig. 8d), in which the noise is mainly from the freeway nearby (Fig. 1b) time shift in the frequency band 10-40 Hz (Fig. 5d) at the time of clock changes is reflective of the fact that this energy is generated by public transportation, such as trains, and buses, which have a fixed schedule. We highlight how cultural noise data reflects nuances in societal behavior in order to illustrate how these data can provide a detailed proxy account of the societal COVID-19 response.
Cultural noise changes in China
In 2020 we observe a sharp decrease in cultural noise in (Figs 4a and 4d) which coincides with the time when the city of Enshi went under lockdown due to COVID-19 (Table 1), and the extended new year holiday. By comparison to historical data (Fig. 2), in the 1-8 Hz band the ~12 dB decrease was equivalent to the abrupt cessation of roughly 20 years' worth of urbanization and development activity. In Figure 4a, a weak peak still appears during the lockdown time in the frequency band around 10-20 Hz. This could be caused by the official vehicles and the supply vehicles on the road (Groos and Ritter, 2009;Boese et al., 2015;Chang et al., 2016;Green et al., 2017). The energy of this peak increases steadily as more vehicles appear on the road from approximately Julian day 70 onwards. Directional analysis of this noise reveals that it mainly comes from the southeast even at night (Fig. 4b and 4c) where the national freeway is located (Fig. 1a). The systematic increase in traffic as the lockdown eased serves as a natural experiment that we can leverage to better understand the relationship between traffic and seismic noise. Seismic noise generated by pedestrians and local industry is usually in the frequency band 1-5 Hz (Alyamkin and Eremenko, 2011). On the one hand, the marked decrease of seismic energy in this frequency band during the lockdown period (Fig. 4a) reflects the many fewer pedestrians and cars on the street. On the other hand, the observation that 1-5 Hz social noise increased ~5 dB in a two week period from Julian days 71 to 85 suggests that Enshi started to gradually reopen much earlier than the official lift of the lockdown. Note that the increase in social noise at Enshi is correlated well with the traffic flow volume at the highway segment 600 km away (Fig. 4d). It suggests that this gradual reopening is a province-wide activity.
The lockdown of Hubei province came one day before the 2020 Chinese Lunar New Year (January 25), the most important festival in the country. To quantify the energy reduction in cultural noise caused by the coronavirus alone, we compared the daily cultural noise energy variation between 2018 and 2020 using the day of Chinese Lunar New Year as the reference time in Figure 7. It is worth noting that they show a similar pattern before the lockdown of the cities in the Hubei province. However, after the lockdown, the cultural noise energy in 2020 is much lower than in 2018. The average reduction was ~10 dB in the frequency band 1.5-8 Hz and ~12 dB in 10-20 Hz for the station IC.ENH (Fig. 7c).
We conducted a similar analysis at several other stations located within urban centers in China ( Fig. 6 and Fig. 7). None of these regions came under direct lockdown. We again compared the vertical component power spectral densities between the years 2018 ( Fig. 6, left) and 2020 (Fig. 6, right). We found that the peak frequencies of cultural noise appear to be different for different cities. This is probably due to the relative distances to the noise sources and the installation (environmental) conditions at different station sites (Trnkoczy et al., 2012).
For station IC.MDJ, located in the northeast region of China, the cultural noise is seen in mainly two peaks, similarly to at IC.ENH. One spans the frequency band 5-10 Hz and the other the frequency band 10-30 Hz. The first peak seems to be consistent with the local road traffic, and the second peak is consistent with the noise by the nearby railway that is at a distance of about 3 km. A substantial change in noise is observed coincident with the lockdown of Hubei and Lunar New Year. The duration and magnitude of this noise change, when compared to the 2018 record, demonstrates that this change substantially exceeded the 'normal' variation due to the new year holiday, indicating that industry and civilians altered behavior in Mudanjiang in response to COVID-19 despite the lack of formal local lockdown. There was a ~3 dB reduction in the frequency band 5-10 Hz and ~4 dB in the frequency band 10-30 Hz (Fig. 7a). The lowest noise conditions persisted for ~20 days, followed by a slow return to normal noise levels over a further ~60 day period.
Station IC.BJT, located in Beijing, shows only the lower frequency cultural noise, in this station mostly at 2-5 Hz (Figs 6c and 6d). This may be related to site installation: this seismograph was installed in a deep tunnel, which might suppress high-frequency noise (McNamara and Buland, 2004). This 2-5 Hz cultural noise is likely to be generated by both the road traffic and by pedestrians (Alyamkin and Eremenko, 2011;Boese et al., 2015;Green et al., 2017). As with other stations, the COVID-19 effects produced a protracted lull in the cultural noise, with a reduction ~4 dB in 2-5 Hz (Fig. 7b). At this station the duration of the noise reduction was longer, more than 81 days. The return to 'normal' cultural noise levels was much substantially more gradual than at other stations, with a slow increase in amplitudes from Julian day 30 and recovery to early-January noise levels Station IC.QIZ, located on Hainan island, which is famous for its tourism industry during the winter, shows a similar pattern to IC.ENH (Figs 6e and 6f), with a dominant cultural noise peak in the range 2-20 Hz that seems to include distinct sources in bands 2-8 Hz and 8-20 Hz. This is probably because both stations are closer to the freeways (unlike station IC.MDJ) (Fig. 1a). IC.QIZ manifests a similar noise variation to IC.MDJ, with a ~30 day lull, followed by a gradual return to 'normal' but still less than the background levels in 2018 over a further ~51 day period (Fig. 7d). The average reduction is ~10 dB in the frequency band 2-8 Hz and ~8 dB in the frequency band 10-20 Hz. Interestingly, the higher frequency (10-20 Hz) noise at this station seems to recover faster than the lower frequency (2-8 Hz) noise. If the former reflects high-speed vehicular traffic and the latter reflects pedestrian traffic, this staggered recovery may result from civilians feeling safe travelling in their own cars earlier than they feel comfortable walking around.
Cultural noise changes in Italy during lockdown
Italy was put under a dramatic lockdown (Table 1) as the coronavirus continued to spread in the country. Although it was one of the toughest responses implemented outside of China, their lockdown policy was less strict than China. As a result, we might expect traffic noise not to have decreased as sharply as we found in mainland China. We find only ~1 dB of energy decrease of cultural noise in the frequency band 10-40 Hz in IV.MILN and ~5 dB of decrease in IV.MONC after Italy declared its lockdown (Fig. 8). For station IV.RMP, the energy reduction was ~6 dB in 1-2 Hz (Fig. 9a), indicating many fewer pedestrians on the street. There is a ~4 dB reduction in the frequency band 3-8 Hz (Fig. 9b) and ~5 dB reduction in the frequency band 10-40 Hz (Fig. 8f), which implies the decrease in trafficvolume was less than the reduction in foot-traffic. Our observations are consistent with the local lockdown policies. The Italian authorities required that their schools, universities, theaters, cinemas, bars, and nightclubs must be closed. Religious gatherings, including funerals and weddings, and sporting events were suspended or postponed. Restaurants and bars were allowed to be open from 6 a.m. to 6 p.m., and shopping malls and markets could open on weekdays with a decreased density of patrons. Under such conditions, the cultural noise should be primarily generated by transportation systems. Lack of any decreasing seismic noise across the lockdown timing seems to corroborate the inference that the primary noise source was public transportation, which was not impacted by the lockdown (Pepe et al., 2020). It appears that the continuous operation of the public transportation system maintained the persistently high level of cultural noise.
Despite this, a modest but significant decrease in noise level is observed at all Italian stations, from a period beginning at the official lockdown until at least Julian day 106 (time of writing). At stations with higher overall cultural noise (IV. MILN and IV.RMP), the pattern of noisy weekdays and less noisy weekends continues after the lockdown, although both shift to lower-noise than their pre-lockdown counterparts. In fact, for IV.RMP, near
Rome, the post-lockdown week days are less noisy than even the quiet pre-lockdown weekends.
For station IV.MILN, the lowest noise energy appears in the first weekend after lockdown of the country, with the lowest noise conditions persisting just for one day. We also note that even the quietest post-lockdown day is not as quiet as the 2019 Christmas day. Since
Julian day 74 we observe a slow increase in noise over a further 40 day period, perhaps indicating that civilians are increasingly willing to go outside in Milan. However, the noise levels have not yet reached pre-lockdown levels. For station IV. MONC, seismic noise reduced ~5dB following the lockdown over a period of 5 days. There is no clear trend of noise increase at this station, perhaps indicating a more strict maintenance of social distancing and stay-at-home behavior. Station IV.RMP, by contrast, recorded a nearimmediate reduction in seismic noise over the few days following the lockdown, and actually manifests a gradually decreasing trend for the entire cultural noise frequency band 1-40 Hz. The decrease is particularly evident in the 1-2 Hz and 3-8 Hz period bands associated with pedestrians and local urban traffic (Fig. 9). This trend may imply that people in Rome are increasingly concerned by the COVID-19 pandemic and are adjusting their behavior to be more conservative.
Conclusions
Seismic records provide unique signals that can elucidate human activities on a large scale.
In this paper, we examined variations in seismic noise between 1 Hz and 40 Hz, which provide proxy information on cultural behavior. In particular, we focused on the effects of governmental lockdowns and self-imposed behavioral alterations due to the outbreak of
COVID-19 in mainland China and Italy. Using seismic records from stations in China and
Italy, we show that the cultural noise in the range of about 2-40 Hz was primarily generated by the local transportation and population sources and study the living habits of local people by using seismic data. The lockdown of the cities and imposition of travel restrictions led to a ~4-12 dB energy decrease in cultural noise on the background of the noise energy in mainland China. According to our observations, different Chinese cities experienced distinct periods of diminished cultural noise, related to the differing local responses to the epidemic. A marked noise change was found even in cities that did not come under government-mandated quarantine. In contrast, there was only ~1-6 dB energy decrease of cultural noise after Italy was put under a total lockdown, due to continuous public transport. Italian cities seem to be responding differently in terms of social behavior as the lockdown continues.
Data and Resources
The data used in this study were collected from the Incorporated Research Institutions for Seismology (IRIS) Data Management Center (DMC; www.iris.edu/dms) and the Italian National Institute of Geophysics and Volcanology (INGV; http://webservices.ingv.it) in Italy using ObsPy Python package (Beyreuther et al., 2010). We used GMT (Wessel and Smith, 1991) to make many of the figures in this paper. Our data for seismic PSD, polarization results and traffic-flow volume on the freeway can be obtained from https://zenodo.org/record/3740214#.XojbDC2ZNE6. The grey line is plotted in half-hour bins and the red line is plotted in one-day average bins.
Seismic
Network (INSN), with the network code IV. These data, including seismic stations IV.MILN in Milan, IV.MONC close to Torino, and IV.RMP ~20 km southeast of Rome, were obtained from the Italian National Institute of Geophysics and Volcanology (INGV; http://webservices.ingv.it) from 15 December 2019 to 15 April 2020 (Fig. 1b).
://www.macrotrends.net/cities/23612/enshi/population). Typical secondary microseism peaks are distinct in this figure with an approximate frequency band of 0.15-0.5 Hz. Another distinct peak can be identified at high frequencies, approximately in the 1.5-8 Hz frequency band. As we will show, this peak is caused by cultural noise.There is a clear increasing trend in seismic noise in Enshi between 2000 and 2020 whichshows a good correlation with the local economic growth and the number of civil motor vehicles(Figs 2b and 2c). Black arrows inFigure 2ashow the timings of the Chinese LunarNew Year since 2000, correlating well with an annual lull in cultural noise due to decreased traffic flow and closure of factories. The red arrow inFigure 2aindicates the time thatEnshi came under lockdown, resulting in a sudden decrease of cultural noise. We explore the effects of this lockdown by examining vertical component PSDs for the first three months of the year, comparing vertical PSDs between 2018(Fig. 3a)and 2020(Fig. 4a).The 2018 data at this site provide information on various aspects of social life in a regularyear, including diurnal variations, a holiday effect, and seasonal variations (Figs 3a, 5a and 5b
at approximately Julian day 100 (76 days after the Hubei lockdown started). At time of writing (Julian day 102) noise levels at IC.BJT are still lower than the 'normal' background noise levels of 2018, implying a persistent alteration in traffic and social patterns from 'business as usual'.
Figure 1 .
1(a) Map of stations in China. The blue triangles indicate seismometer locations. The base image shows the cumulative cases of COVID-19 for different provinces in mainland China as of 12 March 2020, based on data from the World Health Organization. The insets show the seismometer locations in each urban environment. IC.MDJ is located close to a railway line and a major road. IC.BJT is also close to the freeway and railway line but the seismometer is deployed in the deep tunnel. IC.ENH is located north west of a major freeway and south east of the local urban center. IC.QIZ is on the Hainan island and is also close to the freeway to its north. (b) Same as (a) but for Italy, highlighting seismic stations IV.MILN in Milan, IV.MONC in the area of Torino, and IV.RMC near Rome. Inset maps show local maps: station IV.MILN was deployed in the urban area along the railway line and freeway. Station IV.MONC is in the mountainous area but close to the freeway. Station IV.RMP is in the suburbs 20 km southeast of Rome, Italy's capital city.
Figure 2 .
2(a) Twenty years of power spectral density (PSDs) analysis for IC.ENH (Enshi, Hubei province) which has been operational since 20 September 1997. PSDs from the vertical component are shown in decibels relative to the ground acceleration with units of Black arrows indicate the timing of the Chinese Lunar New Year; the red arrow (the top-right location) indicates the time when the city went under lockdown due to COVID-19. (b) Twenty-year variation of cultural noise in the frequency band 1-8 Hz. The red line indicates the timing that Enshi went under lockdown due to COVID-19. The black line denotes the metro area population of Enshi during this period (c) GDP and the number of civil motor vehicles for the period 2000-2018 in Hubei province. Data are from the China National Bureau of Statistics (http://www.stats.gov.cn/).
Figure 3 .Figure 4 .
34(a) Power spectral density of vertical component (HHZ) for the station IC.ENH in Enshi, Hubei province from Julian days 1 to 86 in 2018. PSDs from the vertical component are shown in decibels relative to the ground acceleration with units of The diurnal variations of cultural noise are obvious in the PSDs; seismic noise is higher during the day than the night. The black box indicates the period of the Chinese Lunar New Year (Julian days 46 to 52). Typically, the low cultural noise period is found during the Chinese Lunar New Year for a duration of about one week. (b) (a) The vertical component noise PSDs for the station IC.ENH in Enshi, Hubei province from Julian days 1 to 86 in 2020. The black line indicates the timing the city went under lockdown due to COVID-19. (b) The back azimuths of the polarization ellipsoids for the different frequencies as a function of time from Julian days 1 to 86 in 2020. (b) Distribution of back azimuths for the frequency band 10-20 Hz at daytime (7:00 a.m. to 7:00 p.m., local time) estimated from Julian days 24 to 86 when Enshi was under lockdown. (c) Same with (b) but at nighttime daytime (7:00 p.m. to 7:00 a.m., local time). (d) Comparison of cultural noise in the frequency band 10-20 Hz (red line) and 1.5-8 Hz (blue line) with the daily traffic-flow volume on the freeway, the traffic data is from the website http://www.hhgs.org.cn/.
Figure 5 .
5(a, b) The noise variations for the frequency band 1.5-8 Hz and 10-40 Hz in halfhour bins across the year of 2018 for the station IC.ENH in Enshi, Hubei province, China. The power is measured from the vertical component in decibels relative to the ground acceleration with units of 10log &Black lines in (b) show the repairing period of the local freeway which is located in the southeast of seismic station IC.ENH. Note there is a constant lull at 12:00 local time. (c, d) Same with (a, b) but for the station IV.MILN in Milan, Lombardy province, Italy, in 2019. The black lines show the summer time in Italy, which is from March 31 to October 27 in 2019.
Figure 6 .
6Comparison of the noise PSDs between 2018 and 2020 for the station (a, b) IC.MDJ (Mudanjiang), (c, d) IC.BJT (Beijing) and (e, f) IC.QIZ (Qiongzhou). The power is measured from the vertical component in decibels relative to the ground acceleration with units of 10log &
Figure 7 .
7Comparison of the cultural noise energy daily variation between 2018 and 2020 for the station (a) IC.MDJ (Mudanjiang), (b) IC.BJT (Beijing), (c) IC.ENH (Enshi) and (d) IC.QIZ (Qiongzhou). The power is measured from the vertical component in decibels relative to the ground acceleration with units of 10log &The time is aligned with the day of the Chinese Lunar New Year which is indicated by the red line.
Figure 8 .
8The energy variations of vertical-component power spectral densities at stations (a, b) IV.MILN in Milan, Italy, and (c, d) IV.MONC in the area of Torino, Italy and (e, f) IV.RMP located in ~20 km southeast of Rome, Italy. The power is measured from the vertical component in decibels relative to the ground acceleration with units of The left panels show noise energy as a function of time and frequency. The black lines indicate the times that the cities where the seismic stations are located went under lockdown. The right plots show the noise variations in the frequency band 10-40 Hz.
Figure 9 .
9The energy variations of vertical-component power spectral densities at station IV.RMP (Rome, Italy) for (a) 1-2 Hz, and (b) 3-8 Hz. The power is measured from the vertical component in decibels relative to the ground acceleration with units of
) ,
)people work longer during weekdays (from 5 a.m. to 11 p.m.) and show clear differences in behavior on weekdays compared to the weekend. We can also see the cultural noise lull caused by local holidays, such as Easter Monday (April 22), Liberation Day (April 25), Ferragosto (August 15) and Christmas (December 25). The cultural noise in the frequency band 10-40 Hz lull caused by the 2019 Christmas holiday for station IV.MILN is ~1 dB at the daytime and ~2 dB at nighttime
. These observations are consistent with our observations from China that there are still lots of traveling cars on the freeway during the Chinese Lunar New Year holiday. The peak frequency of cultural noise for IV.MONC generated by the traveling cars on the freeway is ~20-30 Hz rather than 10-20 Hz. We estimate that this difference may relate to the different local geological conditions or simply the different regulations of highway speed limit (80 km/hr on G50 in Enshi, Hubei province in China vs. 130 km/hr in Italy). We note that there is a remarkable difference in the level of cultural noise (1-40 Hz) between IV.MILN, IV.RMP and IV.MONC, as stations IV.MILN and IV.RMP exhibit much higher noise. We believe the reason is that stations IV.MILN and IV.RMP is in an urban area, whereas IV.MONC is in a mountainous area(Fig. 1b). Italy follows the European Summer Time annual Daylight Saving Time procedure setting the clocks forward one hour from standard time during the summer months. In 2019, summer time was from March 31 to October 27. Figures 5c and 5d show a clear time shift of cultural noise energy due to these clock changes. The abrupt
Table 1
1City
Seismic station
The timing of the
lockdown
The timing of the
lifting lockdown
Enshi
(Hubei
province, China)
IC.ENH
January 24, 2020
(Julian days 24 in
2020)
March 25, 2020
(Julian days 85 in
2020)
Beijing (China)
IC.BJT
No
official
declared lockdown
Mudanjiang
(Heilongjiang
province, China)
IC.MDJ
No
official
declared lockdown
Qiongzhou (Hainan
province, China)
IC.QIZ
No
official
declared lockdown
Milan (Lombardy
province, Italy)
IV.MILN
March 8, 2020
(Julian days 68 in
2020)
Still in place at
time of writing
(April 19 th )
Torino (Piedmont
province, Italy)
IV.MONC
March 10, 2020
(Julian days 70 in
2020)
Still in place at
time of writing
(April 19 th )
Rome
(Lazio
province, Italy)
IV.RMP
March 10, 2020
(Julian days 70 in
2020)
Still in place at
time of writing
(April 19 th )
Table 2
2Average in 2018During 2018 Chinese Lunar New Year (Julian days 46-52) Daily traffic-flow volume on freeway18950 vehicles
13139 vehicles
AcknowledgmentsThe authors thank Mohan Pan, Kaelynn Rose, Scott Condon, and Brennan Brunsvik for comments which improved the manuscript substantially. This work was supported by grants from the Southern California Earthquake Center (SCEC #19037, #20072). SCEC is funded by NSF Cooperative Agreement EAR-1600087 & USGS Cooperative Agreement G17AC00047.
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The source direction of cultural noise in the 10-20 Hz frequency band is from the southeast at the daytime, and from the northwest at the nighttime. 10log &' ( ) * + , -. ). The grey line is. Distribution of back azimuths for the frequency band 10-20 Hz at daytime (7:00 a.m. to 7:00 p.m., local time) estimated from one-year data in 2018 for IC.ENH. (c) Same with (b) but at nighttime (7:00 p.m. to 7:00 a.m., local time). plotted in half-hour bins and the red line is plotted in oneday average binsDistribution of back azimuths for the frequency band 10-20 Hz at daytime (7:00 a.m. to 7:00 p.m., local time) estimated from one-year data in 2018 for IC.ENH. (c) Same with (b) but at nighttime (7:00 p.m. to 7:00 a.m., local time). The source direction of cultural noise in the 10-20 Hz frequency band is from the southeast at the daytime, and from the northwest at the nighttime. 10log &' ( ) * + , -. ). The grey line is plotted in half-hour bins and the red line is plotted in one- day average bins.
|
[] |
[
"On Single Variable Transformation Approach to Markov Chain Monte Carlo",
"On Single Variable Transformation Approach to Markov Chain Monte Carlo"
] |
[
"Kushal K Dey ",
"†+ ",
"Sourabh Bhattacharya ",
"\nUniversity of Chicago\nIL * Indian Statistical Institute\nKolkata\n",
"\nIntroduction\n\n"
] |
[
"University of Chicago\nIL * Indian Statistical Institute\nKolkata",
"Introduction\n"
] |
[] |
In today's times, Markov Chain Monte Carlo (MCMC) methods have everyday use in Statistics and other disciplines like Computer Science, Systems Biology and Astronomy. This technique of generating random samples even from very high dimensional spaces involving very complicated data likelihoods and posterior distributions has simplified many pressing real life problems in recent times. In particular, Bayesian computation, simulation from complex posterior distribution and asymptotics of Bayesian algorithms have benefited a lot from this mechanism (see Gelfand and Smith [GS90], Tierney [Tie94], Gilks et al [GS96]). A very standard approach of simulating from multivariate distributions is to use the Metropolis-Hastings (MH) algorithm [Has70][MRR53] using the random walk proposal. We refer to such algorithm as the Random Walk Metropolis Hastings (RWMH) algorithm. The convergence and optimal scaling of this algorithm has been extensively studied[RGG97]. However, despite the advances, there are certain glaring problems that one may encounter while using RWMH. For very high dimensional, non-standard target distributions, choosing the scales optimally is not feasible in practice, and hence, attempts of jointly updating the parameters using RWMH face serious drop in the acceptance rate, which, in turn, leads to poor convergence. Methods of adpatively selecting the scales usually take very large number of iterations to even converge to the optimal scales; particularly in complex and very high-dimensional situations, this exercise is computationally burdensome in the extreme. The alternative method of updating the parameters sequentially is not only computationally burdensome in high-dimensional problems, high posterior correlation among the parameters usually cause very slow convergence. These issues are discussed in much detail in[DB13b].The TMCMC methodology proposed in Dutta and Bhattacharya [DB11] tries to address these problems. The methodology uses simple deterministic transformations using (typically) a single random variable having an appropriately chosen proposal density. In this paper, we primarily study one version, termed as the Additive TMCMC (ATMCMC) method, and deal with the ergodic behavior of the chain in high dimensions. Our aim is to present a comparative study of ATMCMC and the standard RWMH algorithm with respect to their ergodic behaviors. This paper is organized as follows. In Section 2, we present the ATMCMC algorithm and discuss the intuition behind this algorithm. In Section 3, we discuss some theoretical results regarding the ergodic behavior of the chain. Section 4 focuses on how to optimally select the proposal density for the chain when the target density has a product structure. In Section 5, we present the comparative simulation study of ATMCMC and RWMH and analyze the results.1
| null |
[
"https://arxiv.org/pdf/1408.6667v1.pdf"
] | 88,513,002 |
1408.6667
|
65f60d072ebff5ce1641d0c79f3f422ebd9363a9
|
On Single Variable Transformation Approach to Markov Chain Monte Carlo
Kushal K Dey
†+
Sourabh Bhattacharya
University of Chicago
IL * Indian Statistical Institute
Kolkata
Introduction
On Single Variable Transformation Approach to Markov Chain Monte Carlo
In today's times, Markov Chain Monte Carlo (MCMC) methods have everyday use in Statistics and other disciplines like Computer Science, Systems Biology and Astronomy. This technique of generating random samples even from very high dimensional spaces involving very complicated data likelihoods and posterior distributions has simplified many pressing real life problems in recent times. In particular, Bayesian computation, simulation from complex posterior distribution and asymptotics of Bayesian algorithms have benefited a lot from this mechanism (see Gelfand and Smith [GS90], Tierney [Tie94], Gilks et al [GS96]). A very standard approach of simulating from multivariate distributions is to use the Metropolis-Hastings (MH) algorithm [Has70][MRR53] using the random walk proposal. We refer to such algorithm as the Random Walk Metropolis Hastings (RWMH) algorithm. The convergence and optimal scaling of this algorithm has been extensively studied[RGG97]. However, despite the advances, there are certain glaring problems that one may encounter while using RWMH. For very high dimensional, non-standard target distributions, choosing the scales optimally is not feasible in practice, and hence, attempts of jointly updating the parameters using RWMH face serious drop in the acceptance rate, which, in turn, leads to poor convergence. Methods of adpatively selecting the scales usually take very large number of iterations to even converge to the optimal scales; particularly in complex and very high-dimensional situations, this exercise is computationally burdensome in the extreme. The alternative method of updating the parameters sequentially is not only computationally burdensome in high-dimensional problems, high posterior correlation among the parameters usually cause very slow convergence. These issues are discussed in much detail in[DB13b].The TMCMC methodology proposed in Dutta and Bhattacharya [DB11] tries to address these problems. The methodology uses simple deterministic transformations using (typically) a single random variable having an appropriately chosen proposal density. In this paper, we primarily study one version, termed as the Additive TMCMC (ATMCMC) method, and deal with the ergodic behavior of the chain in high dimensions. Our aim is to present a comparative study of ATMCMC and the standard RWMH algorithm with respect to their ergodic behaviors. This paper is organized as follows. In Section 2, we present the ATMCMC algorithm and discuss the intuition behind this algorithm. In Section 3, we discuss some theoretical results regarding the ergodic behavior of the chain. Section 4 focuses on how to optimally select the proposal density for the chain when the target density has a product structure. In Section 5, we present the comparative simulation study of ATMCMC and RWMH and analyze the results.1
Algorithm
We first briefly describe how additive TMCMC (ATMCMC) works. We explain it for the bivariate case -the multivariate extension would analogously follow. Suppose we start at a point (x 1 , x 2 ). We generate an > 0 from some pre-specified proposal distribution q defined on R + . Then in additive TMCMC we have the following four possible "move-types":
(x 1 , x 2 ) → (x 1 + , x 2 + ) (x 1 , x 2 ) → (x 1 + , x 2 − ) (x 1 , x 2 ) → (x 1 − , x 2 + ) (x 1 , x 2 ) → (x 1 − , x 2 − )(1)
This means we are moving along two lines in each transition from the point (x 1 , x 2 ), one parallel to the line y = x and the other parallel to the direction y = −x. Each of the four transitions described above are indexed as I k for the kth transition, where k varies from 1 to 4 in the bivariate case, and in general from 1 to 2 d in R d . For simplicity we assume that the move-types are chosen with equal probability; see Dutta and Bhattacharya [DB11] for the general case. As with the standard RWMH case, we do attach some probabilities with accepting/rejecting the proposed move such that the reversibility condition is satisfied thereby guaranteeing convergence. Formally, the algorithm may be presented as follows.
Algorithm 2.1. Suppose we are at x n = (x 1 , x 2 , · · · , x d ) at the nth iteration.
1. Generate ∼ g(·) on R + .
2. Select randomly one move type and define b 1 , b 2 , · · · , b d iid ∼ DiscrU nif {−1, 1} y = (x 1 + b 1 , x 2 + b 2 , · · · , b d )
(2)
α(x, ) = min 1, π(y) π(x n )
3. Set x n+1 = y with prob. α(x n , )
x n with prob. 1 − α(x n , )
Now we intuitively discuss why ATMCMC is a better option compared to the RWMH algorithm. Firstly, we tested using simulation experiments (all conducted in MATLAB R2013b) that our algorithm requires less computational time to run compared to RWMH (see Fig 1).
Secondly, and more importantly, ATMCMC is expected to have much higher acceptance rate than RWMH. We discuss this as follows.
In a standard RWMH algorithm in d dimensions, we need to generate d many i 's, for i ∈ {1, 2, · · · , d}. For simpliicty of illustration, assume that the target density π is the product density, π = d i=1 f () of iid components f . Then the acceptance rule for RWMH comprises the ratio TMCMC algorithms corresponding to dimensions varying from 2 to 50 with target density being product of N (0, 5) and the proposal density for additive TMCMC being T N >0 (0, 1) (truncated N (0, 1) left truncated at 0) and for RWMH proposal, every component has N (0, 1) distribution. It is observed that TMCMC has consistently less computation time compared to RWM, specially for higher dimensions.
π(x + ) π(x) = d i=1 f (x i + i ) f (x i ) .
If d is very large, then, by chance, we may obtain some very small or large values of i ∼ q(·) (note that 5% observations are expected to lie outside the 95% confidence region and these are the points that are problematic). This would result in certain very small values of f (x i + i ) for some i and thereby drastically reduce the above ratio. So, the chain has the problem of remaining stuck at a point for a long time. Note that ATMCMC uses only one to update all the co-ordinates using sign change and this counters the above problem. So, we can expect a much higher acceptance rate for ATMCMC over the RWMH algorithm. But there are two pertinent questions here. Firstly, how much can we improve on the RWMH algorithm in terms of the acceptance rate? Secondly, how would the sample we get using the ATMCMC method compare to the RWMH algorithm in terms of the convergence of the iterates to the target density and the mixing among the iterates once the target is attained? We address the first issue in Section 4 and the second in Section 5.
Ergodic Properties of ATMCMC
In case of Markov chains on discrete spaces, there is a well-established notion of irreducibility. However, on general state spaces, such a notion no longer works. This is why we define ψ irreducibility. A Markov chain is said to be ψ-irreducible if there exists a measure ψ such that
ψ(A) > 0 =⇒ ∃n with P n (x, A) > 0 ∀x ∈ χ (4)
where χ is the state space of the Markov chain (in our case, it would most often be R d for some d). For convergence of the process, we must ensure that it is µ-irreducible, where µ is the Lebesgue measure. We also need additional concepts of aperiodicity and small sets. A set E is said to be small if there exists n > 0 , δ > 0 and some measure ν such that P n (x, ·) > δν(·)
x ∈ E
A chain is called aperiodic if the g.c.d of all such n for Eqn 5 holds, is 1. All these concepts of µ-irreducibility, aperiodicity and small sets are very important for laying the basic foundations of stability. The following theorem due to Dutta and Bhattacharya [DB11] establishes these properties for the ATMCMC chain.
Result 3.1. Let π be a continuous target density which is bounded away from 0 on R d . Also, let the proposal density q be positive on all compact sets on R + . Then, every non-empty bounded set in R d is small, and this can be used to show that the chain is both λ-irreducible and aperiodic.
A proof of this result can be found in Dutta and Bhattacharya [DB11], along with a graphical interpretation; see also Dey and Bhattacharya [DB13a]. In fact, in Dutta and Bhattacharya [DB11], a stronger result has been proved that for any n > d (d represents the dimensionality of the state space), the minorization condition is satisfied. From the monorization condition, λ irreducibility follows trivially. Aperiodicity follows because the above result is true for all n > d and the g.c.d of such n is 1.
Let P be the transition kernel of a ψ-irreducible, aperiodic Markov chain with the stationary distribution π. Then the chain is geometrically ergodic if ∃ a function V ≥ 1, which is finite at least one point, and also constants ρ ∈ (0, 1) and M (< ∞), such that
||P n (x, ·) − π(·)|| T V ≤ M V (x)ρ n ∀n ≥ 1,(6)
where ||ν|| T V denotes the total variation norm, defined as
||ν|| T V = sup g:|g|≤V ν(g)
Apart from ensuring geometric rate of convergence of the Markov chain, another utility of geometric ergodicity is that one can apply Central Limit Theorem to a wide class of functions of the Markov chain, and hence, one can also investigate stability of these ergodic estimates (see Roberts, Gelman and Gilks [RGG97]). A very standard way of checking geometric ergodicity is a result that involves the Foster-Lyapunov drift criteria. P is said to have a geometric drift to a set E if there is a function V ≥ 1, finite for at least one point and constants λ < 1 and c < ∞ such that
P V (x) ≤ λV (x) + c1 E (x),(7)
where P V (x) = V (y)P (x, y)dy is the expectation of V after one transition given that one starts at the point x. Theorems 14.0.1 and 15.0.1 in Meyn and Tweedie [MT93] establish the fact that if P has a geometric drift to a small set E, then under certain regularity conditions, P is π-almost everywhere geometric ergodic and the converse is also true. The first result we present is basically adaptation of a result due to Mengersen and Tweedie [MT96]. We now show a sufficient condition that would ensure that Eqn 7 holds.
Lemma 3.1. If ∃ V such that V ≥ 1 and finite on bounded support, such that the following hold:
lim sup |x|→∞ P V (x) V (x) < 1 (8) P V (x) V (x) < ∞ ∀x.(9)
Then this V satisfies the geometric drift condition in Eqn 7, and hence the chain must be geometrically ergodic. Also, if for some V finite, the geometric drift condition is satisfied, then the above condition must also hold true.
Result 3.2. If π, the target density, is sub-exponential and has contours that are nowhere piecewise parallel to {x : |x 1 | = |x 2 | = · · · = |x d |}, then the additive TMCMC chain satisfies geometric drift if and only if
lim inf x →∞ Q(x, A(x)) > 0,(10)
where A(x) denotes the acceptance region when x is updated, and Q(x, A(x)) denotes the probability of the acceptance region under the ATMCMC proposal distribution associated with the density q(·) of .
A proof of this result is given in Dey and Bhattacharya [DB13a]. A similar result holds true for the RWMH algorithm as well (see Jarner and Hansen [JH00] and Roberts and Tweedie [RT96]) except that there we do not need the constraint that the contours are not piecewise parallel to {x : |x 1 | = |x 2 | = · · · = |x d |}, but this is true for most densities we commonly encounter. Even if this condition is not satisfied, we can still show geometric ergodicity for a modified TMCMC chain with moves from (
x 1 , x 2 , · · · , x d ) to (x 1 + b 1 c 1 1 , x 2 + b 2 c 2 2 , · · · , x d + b d c d d )
where c i 's are some positive scalars not all equal.
Optimal Scaling of Additive TMCMC
In this section, we shall restrict our focus on target densities that are products of iid components π = d i=1 f and the proposal density for is given by T N >0 (0, l 2 d ), where l is called the scaling term of the proposal. This section will be dedicated to obtaining the optimal value of this scaling l and determining the limiting expected acceptance rate of ATMCMC under the optimal scaling scenario. If the variance of the proposal density is very small, then the jumps will be of smaller magnitude and this would mean the Markov chain would take very many iterations to traverse the entire state space, and in the process, the convergence rate would be very small. On the other hand, if the variance is very large, then our algorithm will reject too many of the moves. An instance of this argument is depicted in There is an extensive theory on optimal scaling of RWMH chains (see Beskos, Roberts and Stuart [BRS09], Bedard [Bed09] [Bed07], Neal and Roberts [NR06], Roberts, Gelman and Gilks [RGG97]). The magic number for RWMH has been the optimal acceptance rate value of 0.234, which has been achieved through maximization of speed of the process for a wide range of distributions -iid set up, some special class of independent but non-identical set up, as well as a dependent set-up. For our purpose, we have developed an optimal scaling theory for ATMCMC where we have optimized the diffusion speed of our process to obtain optimal acceptance rate for ATMCMC. We present a rough sketch of our approach here, for detailed analysis we refer the reader to Dey and Bhattacharya [DB13b].
We assume that f is Lipschitz continuous and satisfies the following conditions:
(C1) E f (X) f (X) 8 = M 1 < ∞. (11) (C2) E f (X) f (X) 4 = M 2 < ∞.(12)
We define U t d = X d [dt],1 , the sped up first component of the actual Markov chain. Note that this process makes a transition at an interval of 1 d . As we set d → ∞, meaning that as the dimension of the space blows to ∞, the sped up ATMCMC process essentially converges to a continuous time diffusion process.
For our purpose, we define the discrete time generator of the TMCMC approach, as
G d V (x) = d 2 d b i ∈ {−1, +1} ∀i = 1, . . . , d ∞ 0 V (x 1 + b 1 , . . . , x d + b d ) − V (x 1 , . . . , x d ) × min 1, π(x 1 + b 1 , . . . , x d + b d ) π(x 1 , x 2 , . . . , x d ) q( )d .(13)
In the above equation, we may assume that V belongs to the space of inifinitely differentiable functions on compact support (see, for example, [Bed07]) for further details). Note that this function is measurable with respect to the Skorokhod topology and we can treat G d as a continuous time generator that has jumps at the rate d −1 . Given our restricted focus on a one dimensional component of the actual process, we assume V to be a function of the first co-ordinate only. Under this assumption, the generator defined in (13) is a function of only and b 1 , and can be rephrased as
G d V (x) = d 2 ∞ 0 b 1 ∈{−1,+1} V (x 1 + b 1 ) − V (x 1 ) ×E b 2 ,...,b d min 1, π(x 1 + b 1 , . . . , x d + b d ) π(x 1 , . . . , x d ) q( )d ,(14)
where E b 2 ,...,b d is the expectation taken conditional on b 1 and .
First we show that the quantity G d V (x) is a bounded quantity.
G d V (x) ≤ dE {b 1 , } [V (x 1 + b 1 ) − V (x 1 )] = dV (x 1 )E {b 1 , } (b 1 ) + d 2 V (x * 1 )E {b 1 , } ( 2 ) ≤ l 2 M V ,(15)
where x * 1 lies between x 1 and x 1 + b 1 and M V is the maximum value of V . We derive the limit of G d V (x) as d → ∞ that will give us the infinitesimal generator of the associated diffusion process for the ATMCMC chain. It can be shown that
Proposition 4.1. If X ∼ N (µ, σ 2 ), then E min 1, e X = Φ µ σ + e µ+ σ 2 2 Φ −σ − µ σ ,(16)
where Φ is the standard Gaussian cdf.
Using this proposition, we can write
E b 1 min 1, π(x 1 + b 1 , . . . , x d + b d ) π(x 1 , . . . , x d ) = Φ η(x 1 , b 1 , ) − (d−1) 2 2 I (d − 1) 2 I + e η(x 1 ,b 1 , ) Φ − (d − 1) 2 I − η(x 1 , b 1 , ) − (d−1) 2 2 I (d − 1) 2 I = W(b 1 , x 1 ).(17)
Note that using Taylor series expansion around x 1 , we can represent η(x 1 , b 1 , ) as
η(x 1 , b 1 , ) = b 1 [log f (x 1 )] + 2 2 [log f (x 1 )] + b 1 3 3! [log f (ξ 1 )] ,(18)
where ξ 1 lies between x 1 and x 1 + b 1 . Again re-writing b 1 as l √ d z * 1 , where z * 1 follows a N (0, 1) distribution, η and W can be expressed in terms of l and z * 1 as
η(x 1 , z * 1 , d) = lz * 1 √ d [log f (x 1 )] + l 2 z * 1 2 2!d [log f (x 1 )] + l 3 z * 1 3 3!d 3 2 [log f (ξ 1 )](19)
and
W(z * 1 , x 1 , d) = Φ η(x 1 , z * 1 , d) − z * 1 2 l 2 2 I z * 1 2 l 2 I + e η(x 1 ,z * 1 ,d) Φ − z * 1 2 l 2 I 2 − η(x 1 , z * 1 , d) z * 1 2 l 2 I .(20)
The last line follows as the expression η(x 1 , b 1 , ) depends on b 1 and only through the product b 1 . (ATMCMC) implying that once stationarity is reached, there will be faster mixing among the iterates in RWMH compared to ATMCMC. However, an interesting observation is that if l deviates slightly from l opt , the diffusion speed of RWMH drops much faster compared to that of ATMCMC. Thus, ATMCMC is much more robust compared to RWMH with respect to the scaling. This is very important in complex and high-dimensional practical situations where achieving the optimal scaling usually turns out to be infeasible; recall the discussion regarding this in Section 1. Although our above analysis holds true only for the case when all the components of the product density are iid, however, this condition can be relaxed to include independent components with appropriate scaling and inherent regularization properties as in Bedard (2009) [Bed09] and Dey and Bhattacharya (2013) [DB13a] and also to non-regular component densities in Dey and Bhattacharya [DB14]. Also, in all the calculations we have done so far and in the consideration of the diffusion speed and its implications, we must keep in mind our inherent assumption that the process is in stationarity. The major question to address now is that which chain has faster convergence to stationarity. We address this in the next section via simulation studies.
Simulation study comparison
In this section, we compare RWMH and additive TMCMC methods using two parameters, one being the acceptance rate and the other, the Kolmogorov-Smirnov (KS) distance between the empirical distribution at each time point and the target density. For the first measure, we observed the acceptance rates of the two algorithms for varying dimesnions and scaling factors l. The results are reported in Table 1. Table 1 validates that for higher dimensions, under optimal scaling, the acceptance rates of RWMH and additive TMCMC are indeed 0.234 and 0.439 respectively, as the observed values are very close to the theoretical ones. Also, we see that for fixed dimensions, as scaling increases away from the optimal value, the acceptance rate falls drastically for RWMH and this worsens with increase in dimensionality. For dimensions 100 and 200, we skipped providing the acceptance rates for scaling l = 10 as it was understandably very small for RWMH. Comparatively, additive TMCMC is much more stable with change of scaling even for high dimensions. This validates the robustness of the diffusion speed with respect to scaling l in For the second measure of KS distance comparison, we run a number of chains, say L, starting from one fixed point for both RWMH and ATMCMC adaptations. Corresponding to each time point t, we thus get L many iterates. The notion is that, as time t increases (specially after burn-in), these L many iterates should be close to an independently drawn random sample from the target distribution π. So, if we observe the KS statistic for the empirical distribution of these iterates along any particular dimension with respect to the marginal of π along that dimension, we expect the test statistic to be decreasing with time and finally being very close to 0 after a certain time point. Now the question of interest is, of the two approaches, ATMCMC and RWMH, for which method the graph decays faster to 0? Corresponding to two different dimensions d = 10 and d = 100, and two scalings l = 2.4 (optimal given that I = 1 for the target density product of N (0, 1) components) and l = 4, we present the two graphs of additive TMCMC and RWMH simultaneously in Fig 4 and Fig 5. Both the figures, but particularly the latter, clearly indicate faster convergence of ATMCMC to the stationary distribution. Therefore in conclusion it can be stated that • ATMCMC is simple to interpret and does not depend heavily on the target density, and additionally has much lesser computational burden and time complexity.
• Under sub-exponential target density with some regularity constraints on the target density, the ATMCMC algorithm is geometrically ergodic.
• ATMCMC has a higher acceptance rate of 0.439 corresponding to 0.234 for the RWMH algorithm. As observed, our algorithm is more robust to change of scale and across dimensions. But the mixing or diffusion speed of RWMH is higher, meaning that once stationarity is attained RWMH will provide better samples than ATMCMC. Figure 4: The KS distance graph for RWMH and ATMCMC chains for a 30 dimensional target density, which is the product of iid N (0, 1) components. The scalings for the two graphs are l = 2.4 and l = 4. Notice that the KS graph for ATMCMC seems to be lower compared to that of RWMH implying faster rate of convergence for ATMCMC. Here the KS graph for ATMCMC is clearly lower compared to that of RWMH implying faster rate of convergence for ATMCMC.
• The KS test comparison in the simulation study shows that for high dimensions, ATM-CMC has lower KS statistic value compared to RWMH when the chain is not stationary. This also suggests that ATMCMC reaches burn-in faster than RWMH for higher dimensions. But once burn-in is reached, ideally the two methods should both yield KS values close to 0 and that is why we see that the KS graphs stabilize with time for both the approaches.
Figure 1 :
1Computation time (in MATLAB R2013b) of one run of 100,000 iterations with RWM and
Figure 2 :
2The graphical representation of a co-ordinate for a 5-dimensional chain with target density being product of N (0, 1) densities and the values of the scaling factor l for the two cases are taken to be l = 0.8 and l = 8 respectively for the two scenarios a) and b) depicted in the graph
Figure 3 :
3The plot of the diffusion speed with respect to the scaling factor l for RWMH and ATMCMC chains.
Figure 5 :
5The KS distance graph for RWMH and ATMCMC chains for a 100 dimensional target density, which is the product of iid N (0, 1) components. The scalings for the two graphs are l = 2.4 and l = 4.
Now we consider the Taylor series expansion around x 1 of the termFrom (20) it is clear that W(z * 1 , x 1 , d) is continuous but not differentiable at the point 0. Using Taylor series expansion of the terms ΦThe infinitesimal generator GV (x) obtained as the limit of the GV d (x) has therefore a simpler formThis is the form of the generator for a Langevin diffusion process withThe function h is called the diffusion speed and we maximize this quantity with respect to l to derive the optimal scaling. For our case, l opt = 2.4 √ I and we plug this value in the formula for asymptotic expected acceptance rate to obtainFor RWMH too, the diffuion process is Langevin but the form of the diffusion speed is somewhat different (see Roberts, Gelman and Gilks[RGG97]):It was noted in[RGG97]that the limiting expected acceptance rate corresponding to optimal scaling in RWMH is 0.234, while for that for the optimal scaling in additive TMCMC is 0.439 which is almost twice as that of RWMH. It is to be noted that the optimal scale of RWMH is l opt = 2.4 √ I , which, up to the first decimal place, is the same as that of ATMCMC. The graphs of the diffusion speeds over different l for ATMCMC and for standard RWMH are presented inFig 3.Note that the diffusion speed at l opt is higher for RWMH compared to additive TMCMC
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[] |
[
"Optimal Transmit Filters for ISI Channels under Channel Shortening Detection",
"Optimal Transmit Filters for ISI Channels under Channel Shortening Detection"
] |
[
"Student Member, IEEEAndrea Modenini ",
"Fredrik Rusek ",
"Senior Member, IEEEGiulio Colavolpe "
] |
[] |
[
"IEEE TRANSACTIONS ON COMMUNICATIONS"
] |
We consider channels affected by intersymbol interference with reduced-complexity, mutual information optimized, channel-shortening detection. For such settings, we optimize the transmit filter, taking into consideration the reduced receiver complexity constraint. As figure of merit, we consider the achievable information rate of the entire system and with functional analysis, we establish a general form of the optimal transmit filter, which can then be optimized by standard numerical methods. As a corollary to our main result, we obtain some insight of the behavior of the standard waterfilling algorithm for intersymbol interference channels. With only some minor changes, the general form we derive can be applied to multipleinput multiple-output channels with intersymbol interference. To illuminate the practical use of our results, we provide applications of our theoretical results by deriving the optimal shaping pulse of a linear modulation transmitted over a bandlimited additive white Gaussian noise channel which has possible applications in the faster-than-Nyquist/time packing technique.
|
10.1109/tcomm.2013.110813.130385
|
[
"https://arxiv.org/pdf/1310.6265v1.pdf"
] | 18,798,679 |
1310.6265
|
0ec989c23dd4df3c27931ba3d10e094699c04f14
|
Optimal Transmit Filters for ISI Channels under Channel Shortening Detection
23 Oct 2013
Student Member, IEEEAndrea Modenini
Fredrik Rusek
Senior Member, IEEEGiulio Colavolpe
Optimal Transmit Filters for ISI Channels under Channel Shortening Detection
IEEE TRANSACTIONS ON COMMUNICATIONS
123 Oct 2013arXiv:1310.6265v1 [cs.IT]Index Terms-ISI channelschannel shorteningwaterfilling algorithmsreduced complexity detectionmismatched receiversMIMO-ISIfaster-than-Nyquisttime packing
We consider channels affected by intersymbol interference with reduced-complexity, mutual information optimized, channel-shortening detection. For such settings, we optimize the transmit filter, taking into consideration the reduced receiver complexity constraint. As figure of merit, we consider the achievable information rate of the entire system and with functional analysis, we establish a general form of the optimal transmit filter, which can then be optimized by standard numerical methods. As a corollary to our main result, we obtain some insight of the behavior of the standard waterfilling algorithm for intersymbol interference channels. With only some minor changes, the general form we derive can be applied to multipleinput multiple-output channels with intersymbol interference. To illuminate the practical use of our results, we provide applications of our theoretical results by deriving the optimal shaping pulse of a linear modulation transmitted over a bandlimited additive white Gaussian noise channel which has possible applications in the faster-than-Nyquist/time packing technique.
I. INTRODUCTION
The intersymbol interference (ISI) channel has played a central role in communication theory for several decades. It has been heavily researched, and today most of its fundamental properties are known. The capacity of the ISI channel was for example derived by Hirt back in 1988 in [1], and it was shown that Gaussian inputs in combination with the classical waterfilling algorithm achieves capacity. In practice, Gaussian channel inputs are not very common and discrete inputs are typically preferred. In this case the ultimate communication limit was found in the early 2000s through a series of papers [2]- [6]. Further results on capacity properties of ISI channels include Kavcic's elegant method [7] to achieve the capacity of the ISI channel with discrete inputs through a generalized version of the Arimoto-Blahut algorithm, and also Soriaga et al.'s evaluation of the low-rate Shannon limit of ISI channels [8].
However, all of the above mentioned papers study ISI channels under the assumption that the receiver can perform optimal maximum-likelihood (ML) or maximum-a-posteriori Submitted (MAP) detection. Let L H + 1 denote the number of taps in the channel impulse response. Forney showed in 1972 [9] that optimal ML/MAP-detection can be performed by searching a trellis whose number of states is U LH , where U is the cardinality of the employed constellation. The number of trellis states will be considered in the following has a measure of the receiver complexity. In many practical scenarios L H is far too long for practical implementation of optimal ML/MAP detection. This observation spurred significant research efforts to reduce the computational complexity of the MAP/ML algorithm (e.g., see [10], [11] and references therein) or to investigate when a properly designed linear equalizer has the same diversity order of the optimal detector (e.g. see [12], [13] and references therein). An alternative promising approach was channel shortening pioneered by Falconer and Magee in 1973 [14] and further investigated by several researchers (e.g., see [15]- [24]). Traditionally, channel shortening detectors were optimized from a minimum mean-square-error (MMSE) perspective. However, minimizing the mean-square-error does not directly correspond to achieving the highest information rate (in the Shannon sense) that can be supported by a shortening detector. Recently, the achievable rate of channelshortening detectors was optimized in [25] by utilizing the framework of mismatched mutual information [26], [27]. The result of [25] is a closed-form expression of the achievable information rate (AIR) of an ISI channel with Gaussian inputs and an optimized channel-shortening detector that considers the channel memory to be L < L H taps long, where L is a user-defined parameter.
In this paper, we extend [25] by designing a proper transmit filter to be employed jointly with a channel-shortening detector 1 with the aim of further improving the achievable information rate. In other words, we consider to adopt, at the receiver side, a channel-shortening detector and then solve for the optimal transmit filter to be used jointly with it. When the use of the optimal full-complexity receiver is allowed, the answer to this question is the classical waterfilling processing. We are generalizing the waterfilling concept to the case of reduced-complexity channel-shortening detectors, i.e., we essentially redo Hirt's derivations, but this time with the practical constraint of a given receiver complexity.
Our results are not as conclusive as in the unconstrained 1 As in [25], with the term "channel-shortening detector" we mean a detector based on a proper linear filter (the channel shortener) plus a suboptimal reduced-complexity trellis-based detector with proper branch metrics designed for a target channel response of length L < L H . With "optimal" channelshortening detector we mean that proposed in [25] which is optimal from the point of view of the maximization of the achievable information rate. receiver complexity case. With functional analysis, we can prove that, for real channels, the optimal transmit filter has a frequency response described by L + 1 real-scalar values. In general, for complex channels, the optimal transmit filter is described by L + 1 complex scalar values. The transmit filter optimization thereby becomes a problem of finite dimensionality, and a numerical optimization provides the optimal spectrum. Note that, in practice, L is limited to rather small values and L = 1 is an appealing choice from a complexity perspective. This essentially leads to very effective numerical optimizations.
The rest of the paper is organized as follows. In Section II, we lay down the system model and formulates the problem that we intend to solve. In Section III, we derive a general form of the frequency response of the optimal transmit filter. In Section V-B, we derive, by using the same framework, the optimal transmit filter for multiple-input multiple-output channel (MIMO) affected by ISI (MIMO-ISI), and the optimal shaping pulse for a transmission over a bandlimited additive white Gaussian noise (AWGN) channel. Numerical examples and properties of the numerical optimization are given in Section VI. Finally, Section VII concludes the paper.
II. PRELIMINARIES
In this section we give the system model, lay down the fundamentals of channel shortening receivers and their optimization, and formulates the problem that will be solved.
A. System Model
Let us consider the transmission of the sequence of symbols a = {a k } over a discrete-time channel with model 2
y k = LH ℓ=0 a k−ℓ h ℓ + w k ,(1)
where h = {h k } LH k=0 is the channel impulse response, assumed time-invariant and of finite length, and w = {w k } are independent and identically distributed complex Gaussian random variables, with mean zero and variance N 0 -note that bold letters are used for vectors. This system is studied under the assumption of ideal channel state information (CSI) at both transmit and receive side, that is, perfect knowledge of the coefficients and the noise variance. The symbol vector a is a precoded version of the information symbols u = {u k },
a = u ⋆ p ,(2)
where "⋆" denotes convolution and p is a transmit filter subject to the power constraint k |p k | 2 = 1 and with continuous spectrum |P (ω)| 2 , where P (ω) is the discrete time Fourier transform (DTFT) of the vector p. Taken together, the received signal can be expressed as
y = v ⋆ u + w,(3)
where v = h ⋆ p. It is convenient to assembly the presentation on matrix notation, so that (3) becomes
y = V u + w,
where V is a convolutional matrix formed from the vector v, and y, u and w are now column vectors of appropriate sizes. Assume that the combined channel-precoder response v has K+1 non-zero taps. The complexity of MAP sequence (implemented through the Viterbi algorithm) and symbol detection (implemented through the BCJR algorithm) is O(U K ) per symbol, where U is the cardinality of the employed alphabet. Falconer and Magee's idea was to reduce this complexity by a linear filtering
r = y ⋆ q = (v ⋆ q) ⋆ u + (w ⋆ q).
Then, a Viterbi/BCJR algorithm follows assuming a target response t of L+1 taps (L ≤ K), and working on a trellis with U L states. Presumably, the target response t roughly equals the L + 1 strongest taps of (v ⋆ q), but there must not be an exact match if it turns out that it is not optimal to do so. In matrix notation, this procedure can be viewed as if the receiver decodes on the basis of a mismatched conditional probability distribution (pdf) 3
p(y|u) ∝ exp − Qy − T u 2 N 0(4)
instead of the actual conditional pdf
p(y|u) ∝ exp − y − V u 2 N 0 .
Two questions now emerge: (1) For a given target response t, how should the linear filter q be selected? And (2) how should the target response t be selected? These two questions kept researchers busy for several decades, see [14]- [23]. However, in all of those papers, the optimizations of t and q was done with an MMSE cost function, which does not directly correspond to the achievable information rate of the overall system. 4 The optimization for achievable information rate was completely solved in [25] under the assumption of Gaussian input symbols and by using a slightly more general model for channel shortening. This generalization is now described. By expansion of the exponent in (4) we get
p(y|u) ∝ exp − Qy − T u 2 N 0 ∝ exp 2R{u † T † Qy} − u † T † T u N 0 ,(5)
where all terms independent of u have been left out. A MAP sequence detector based on (5) was proposed by Ungerboeck in 1974 [28] and an algorithm for MAP symbol detection in 2005 by Colavolpe and Barbieri [29]. In [25], a reduced complexity channel shortening detector is obtained by substituting in (5) T † Q with (H r ) † and T † T with G r . In addition, the noise density N 0 is also absorbed into H r and G r . This results in a mismatched conditional pdf of the form
p(y|u) = exp 2R{u † (H r ) † y} − u † G r u .
While the front-end H r is unconstrained, the matrix G r must satisfy G r ℓk = 0, |ℓ − k| > L (6) in order to satisfy the reduced-complexity constraint. The matrix T † T in (5) must be positive semi-definite, while no such constraint applies to the matrix G r . Hence, a more general model than (4) for channel shortening is obtained. The AIR of a general mismatched receiver is derived in [26], [27] and equals
I AIR = lim N →∞ 1 N [−E y [log 2 (p(y))] + E y,u [log 2 (p(y|u))]] ,
where N is the number of input symbols (i.e., the length of the vector u), E y denotes the expectation operator with respect to the random variable y and
p(y) up (y|u)p u (u).
The rate I AIR is directly impacted by the choices of G r and H r . The optimization problem reads
I OPT = max G r ,H r I AIR ,(7)
under the constraints specified in (6). Problem (7) for a discrete alphabet is a hard task. On the other hand, it can be solved in closed form under the assumption that transmitted symbols are independent Gaussian random variables [25]. In this case of Gaussian inputs, closed-form expressions for G r , H r can be found with the following algorithm:
• Compute the sequence {b k } L k=−L as b k = 1 2π π −π N 0 |V (ω)| 2 + N 0 e jkω dω = 1 2π π −π N 0 |H(ω)| 2 |P (ω)| 2 + N 0 e jkω dω.
where H(ω) and V (ω) are the DTFT of h and v. • Compute the real-valued scalar
c = b 0 − bB −1 b † ,(8)where b = [b 1 , b 2 , . . . , b L ], and B is L × L Toeplitz with entries B ij = b j−i . • Define the vector u = 1 √ c [1, −bB −1 ]
and find the optimal target response as
G r (ω) = |U (ω)| 2 − 1 .
• Finally, the optimal channel shortener is found as
H r (ω) = V (ω) |V (ω)| 2 + N 0 (G r (ω) + 1) .
By using the optimal channel shortener and the target response I OPT results to be
I OPT = − log 2 (c) .
B. Problem Formulation
The problem we aim at solving is to maximize I OPT over the transmit filter P (ω), i.e., the DTFT of p. Thus, we have the following optimization problem at hand
min P (ω) c[P (ω)]
such that (9) π −π |P (ω)| 2 dω = 2π In (9), we have made explicit the dependency of c on P (ω), but not on N 0 and H(ω), since these are not subject to optimization. Since the starting point is the expression of the AIR when the optimal channel-shortening detector is employed, we are thus jointly optimizing the channel shortening filter, the target response, and the transmit filter, although for Gaussian inputs only. However, as shown in the numerical results, when a low-cardinality discrete alphabet is employed, a significant performance improvement is still observed (see also [25]).
III. GENERAL FORM OF THE OPTIMAL TRANSMIT FILTER
The optimization problem (9) is an instance of calculus of variations. We have not been able to solve it in closed form, but we can reduce the optimization problem into an L + 1 dimensional problem, which can then efficiently be solved by standard numerical methods. The main result of the paper is the following theorem.
Theorem 1: The optimal transmit filter with continuous spectrum for the channel H(ω) with a memory L channelshortening detector satisfies
|P (ω)| 2 = max 0, N 0 |H(ω)| 2 L ℓ=−L A ℓ e jℓω − N 0 |H(ω)| 2 ,(10)
where {A ℓ } are complex-valued scalar constants with Hermitian symmetry, i.e., A ℓ = A * −ℓ . For a proof see the Appendix A.
IV. INTERLUDE: FULL COMPLEXITY DETECTORS
Theorem 1 gives a general form of the optimal transmit filter to be used for a memory L channel shortening detector. By definition, it becomes the classical waterfilling filter when L = K. Hence, it also provides an insight to the behavior of the transmit filter for the classical waterfilling algorithm. We remind the reader that L H + 1 denotes the duration of the channel impulse response and K+1 denotes the duration of the combined transmit filter and channel response. We summarize our finding in the following Theorem 2: Let P (ω) be the transmit filter found through the waterfilling algorithm. Then,
K ≥ L H .
For a proof, see the Appendix B. Whereas the statement is trivial when the transmit filter and the channel have a finite impulse response (FIR), the theorem proves that this fact holds also when they have infinite impulse responses (IIR). Thus, for a FIR channel response, the waterfilling solution cannot contain any pole that cancels a zero of the channel, while, for IIR channels, the waterfilling solution cannot contain any zero that cancels a pole. Thus, the overall channel cannot be with memory shorter than the original one.
Theorem 2 reveals the interesting fact that the waterfilling algorithm trades a rate gain for detection complexity. By using the optimal transmit filter, a capacity gain is achieved, but the associated decoding complexity (of a full complexity detector) must inherently increase. Thus, with waterfilling, it is not possible to achieve both a rate gain and a decoding complexity reduction at the same time.
V. OTHER PRACTICAL APPLICATIONS OF THE OPTIMAL TRANSMIT FILTER
Although we restricted our attention on the discrete-time ISI channel (1), the same framework can be used to derive the optimal precoder for other channels.
A. MIMO-ISI Channels with perfect CSI
Consider the MIMO-ISI channel
y k = LH ℓ=0 H ℓ a k−ℓ + w k .
Without loss of generality, we assume that the channel is N × N , i.e., matrices {H ℓ } LH ℓ=0 have dimension N × N and {y k },{a k },{w k } are column vectors N × 1. In case N × M channels, they can be converted in a equivalent N ×N channel by means of the QR decomposition [25]. Channel shortening receivers for MIMO-ISI channels have been studied before, e.g., in [19], but here we optimize the receiver with respect to mutual information rather than an MMSE cost function as in [19].
The DTFT of {H ℓ }, defined as H(ω) = LH ℓ=0 H ℓ e −jℓω , can be factorized by means of singular value decomposition (SVD) as
H(ω) = U H (ω)Σ(ω)V † H (ω) ,
where U H (ω) and V H (ω) are unitary matrices and Σ(ω) is a diagonal matrix with elements Σ n (ω). By adopting the MIMO filter V H (ω) at the transmitter and the filter U † H (ω) at the receiver, without any information loss we obtain N independent parallel channels with channel responses {Σ n (ω)} N n=1 . The transceiver block diagram is as shown in Fig. 1a for the case N = 2. The objective function to be maximized is
I OPT = N n=1 − log 2 (c n ) under the constraint N n=1 |P n (ω)| 2 dω = 2πN
where c n is given in (8) and P n (ω) is the precoder for the channel Σ n (ω). By solving the Euler-Lagrange equation, the optimal precoders have spectra of the form (10).
H(ω) U † H (ω) p * (−t) information soft H r 1 (ω) soft information P2(ω) G r 2 (ω) G r 1 (ω) y k uk V H (ω) ak r(t) y k G r (ω) H r (ω)
B. Optimal Shaping Pulse for Bandlimited AWGN Channels
We now consider a linearly-modulated transmission over a continuous-time, time-invariant, bandlimited AWGN channel, under the assumptions of ideal synchronization, and we show how to design the optimal shaping pulse for this scenario. The received signal can be expressed as
r(t) = k u kp (t − kT ) + w(t) ,
wherep(t) is the received pulse, taking into account the transmitted pulse and the channel impulse response, symbols u k are independent, zero-mean, and properly normalized such that E{|u k | 2 } = 1, T is the symbol time, and w(t) is a zeromean, circularly symmetric, white Gaussian noise process with two-sided power spectral density N 0 /2.
As before, the channel is assumed perfectly known at the receiver and time-invariant. The shaping pulse, assumed to be of unit energy, has a spectrum with support over a bandwidth W and the channel frequency response is assumed flat over W , although the generalization to the case of a frequencyselective channel is straightforward.
A set of sufficient statistics for detection is given by the samples at the output of a whitened matched filter (WMF) [9], whose output has the expression (3) where the sequence {v k } has power spectral density
|V (ω)| 2 = 1 T k P ω 2πT − k T
2 P (f ) being the Fourier transform ofp(t). Clearly, this discrete-time model will depend on the adopted shaping pulse, its bandwidth, and the employed symbol time.
The corresponding channel shortening detector is shown in Fig. 1b. Since the WMF can be implemented as a cascade of a continuous-time matched filter followed by a discrete-time whitening filter, this latter filter can be "combined" with the channel shortening filter obtaining a single discrete-time filter with frequency response [25] H r (ω) = G r (ω) + 1 |V (ω)| 2 + N 0 .
The power spectral density of {v k } is
|V (ω)| 2 = |P (ω)| 2 |H(ω)| 2 where H(ω) = 1 |ω| ≤ 2W T π 0 otherwise , ω ∈ [−π, π] .
Thus the optimization problem is still given by (9) where the optimal shaping pulse is such that
|P (f )| 2 = T |P (2πT f )| 2
with |P (ω)| 2 given in (10).
Clearly, when 2W T ≥ 1, the optimal solution is trivial and |P (ω)| 2 is flat. Thus, for 2W T = 1 thep(t) is a sinc function, whereas for 2W T > 1 thep(t) can be a pulse whose spectrum has vestigial symmetry (e.g., pulses with a root raised cosine (RRC) spectrum). For 2W T < 1, the symbol time is such that the Nyquist condition for the absence of ISI cannot be satisfied. Thus, we are working in the domain of the faster-than-Nyquist (FTN) paradigm [30]- [32] or its extension represented by time packing [33], [34]. Note that, as said before, the discrete-time channel model, will depend on the values of W and T . When changing the values of W and/or T , the corresponding optimal pulse will change and so the maximum value of the AIR for the given allowed complexity. In general, when reducing the value of W T , the maximum AIR value will decrease. However, the spectral efficiency, defined as the ratio between the AIR and the product W T could, in principle, increase [30]- [35]. This is the rationale behind FTN/time packing that allows to improve the spectral efficiency by accepting interference. The optimal value of T is, in that case, properly optimized to maximize the spectral efficiency. This optimization can be now performed by also using, for each value of T , the corresponding optimal shaping pulse. In other words, we can find the optimal pulse for a constrained complexity detector when FTN/time packing is adopted.
We point out that, for this scenario, the numerical computation of the optimal shaping pulse in the time-domain can require the adoption of some windowing technique or the use of Parks-McClellan algorithm [36] to obtain a practical pulse since H(ω) can have a spectrum with an ideal frequency cut.
VI. NUMERICAL OPTIMIZATION AND EXAMPLES
Theorem 1 provides a general form of the optimal transmit filter for channel shortening detection of ISI channels. What remains to be optimized is the L+1 complex-valued constants {A ℓ }. A closed form optimization seems out of reach since the constraint in (9) has no simple analytical form in {A ℓ }.
We have applied a straightforward numerical optimization of the variables {A ℓ } under the constraints in (9). With a standard workstation and any randomly generated channel impulse response, the optimization is stable, converges to the same solution no matter the starting position as long as the signal-to-noise-ratio (SNR) is not very high or very low, and is altogether a matter of fractions of a second.
We now describe some illuminating examples. In all cases, the transmit power is the same both in the absence and presence of the optimal transmit filter. We first consider the complex channel h = [0.5, 0.5, −0.5, −0.5j] with memory L H = 3. 5 Fig. 2 shows the AIR I OPT for Gaussian inputs when the transmit filter is optimized for different values of the memory L considered by the receiver. For comparison, 5 Other examples can be found in [37]. the figure also gives I OPT for a flat transmit power spectrum (i.e., no transmit filter at all) and the channel capacity (i.e., when using the spectrum obtained by means of the waterfilling algorithm and assuming a receiver with unconstrained complexity). It can be seen that using an optimized transmit filter for each L, significant gains are achieved w.r.t. the flat power spectrum at all SNRs. The flat spectrum reaches its maximum information rate when L = L H but suffers a loss to the channel capacity. On the other hand, we can see that the optimized transmit filter when L = L H achieves an achievable rate which is close to the channel capacity. However, there is not an exact match. This loss is due to the fact that L H must be lower than the combined channel-precoder memory K as stated by Theorem 2. This behavior is clearly illustrated by Fig. 3, which plots the information rate when the transmit filter is found through the waterfilling algorithm and the receiver complexity is constrained with values of the memory L. It can be seen that when the memory L is increased more and more, even above L H , the information rate becomes closer and closer to the channel capacity. Moreover, it is important to notice that if, naïvely, a transmit filter found through the waterfilling algorithm is used when the receiver complexity is constrained, a loss w.r.t. the optimized case occurs and it may even be better to not have any transmit filter at all for high SNR values.
Although the results of this paper were so far presented only for Gaussian symbols, we now show that when the optimized transmit filter and detector for Gaussian inputs are used for low-cardinality discrete alphabets, the ensuing I AIR is still excellent. 6 Fig. 4 shows the AIR for a binary phase shift keying (BPSK) modulation. It can be noticed that the behavior among the curves for BPSK reflects the behavior for Gaussian symbols. The AIR can be approached in practice with proper modulation and coding formats. Fig. 5 shows the bit error rate (BER) of a BPSK-based system using the DVB-S2 lowdensity parity-check code with rate 1/2. In all cases, 10 internal iterations within the LDPC decoder and 10 global iterations were carried out. It can be noticed that the performance are in accordance with the AIR results. All simulations that we have presented were also carried out for other channels (e.g., EPR4, Proakis B and C). However, we have not presented any result for these channels since our findings for those channels are in principle identical to those for the channel presented in the paper.
||h|| 2 /N 0 [dB] flat L=1 flat L=2 flat L=3 opt. L=1 opt. L=2 opt. L=3
A. MIMO-ISI Channels with perfect CSI
We now considered a 2 × 2 MIMO-ISI channel, with L H = 3. Fig. 6 shows the AIR I OPT for Gaussian inputs as a function of E H /N 0 , being E H = ℓ tr(H ℓ H † ℓ ). The transmit filters are optimized for the equivalent channels Σ 1 (ω) and Σ 2 (ω) for different values of the memory L considered by the receiver. For comparison, the figure also gives I OPT for flat transmit power spectra (i.e., E{a k a † k+m } = Iδ m , where I is the identity matrix and δ m is the Kronecker delta) and the channel capacity (i.e., when using the spectra obtained by means of the waterfilling algorithm and assuming a receiver with unconstrained complexity). It can be seen that conclusions for scalar ISI channels also hold for MIMO-ISI. However, we found that, for MIMO-ISI channel, the objective function seems to have some local maxima, and thus the optimization can depend on the starting position. This problem can be easily solved by running the optimization more times (three times were always enough in all our tests) and keeping the maximum value.
B. Bandlimited AWGN channels
We computed the optimal shaping pulse on a bandlimited AWGN channel with 2W T = 0.48. Hence, we are in the realm of FTN/time packing and the considered ISI is only due to the adoption of such a technique. Fig. 7 shows the achievable spectral efficiency (ASE) η = I AIR /W T for a BPSK modulation on the continuous-time AWGN channel as a function of the ratio E b /N 0 , E b being the received signal energy per information bit. Two values of the memory, namely L = 1 and L = 2 are considered at the detector. For comparison, the figure also gives the ASE for pulses with RRC spectrum and roll-off α = 0.1 or α = 0.2, and the unconstrained capacity for the AWGN channel. It can be seen that the optimized pulse outperforms the other pulses.
VII. CONCLUSION
We have studied ISI channels with channel shortening detection. The channel shortening detector that we used is optimized from a mutual information perspective and allows for the highest possible data rate. We then optimized the transmit filter for a given receiver complexity and ISI channel. This is an optimization problem of infinite dimensionality, but we managed to reduce it through functional analysis into an optimization problem of a dimension that equals the memory of the receiver plus one. A standard numerical optimization procedure then follows. Since the memory L of the receiver is in practice typically set to a small value, such as L = 1, the numerical optimization can be easily carried out.
As a side result, we also show that the classical waterfilling algorithm for ISI channels can never result in a shorter channel response at the receiver than the length of the channel response itself. From our numerical experiments, we have found that it is crucial to take the receiver complexity into account when designing the transmit filter, since if the transmit filter found through the waterfilling algorithm is used, then a loss can occur compared with a flat transmit filter.
We have finally shown that the same framework can be used to derive the optimal shaping pulse on a bandlimited AWGN channel.
APPENDIX A: PROOF OF THEOREM 1
We first note that P (ω) only enters the optimization through its square magnitude, and we therefore make the variable substitution S p (ω) = |P (ω)| 2 and optimize over S p (ω) instead.
The proof will consist of three steps • A formula for stationary points.
• The observation that some of these do not have strictly positive spectrum. • Fixing the problem identified in the previous bullet. Let us now start with the first bullet.
From Cramer's rule [38], we get that
B −1 = 1 det(B) [C ij ],
where C ij is the cofactor of entry (i, j) in B. This implies that we can express bB
−1 b † as M m=1 α m b φm,0 0 b φm,1 1 (b * 1 ) φm,2 · · · b φm,2L−1 L (b * L ) φm,2L N n=1 β n b ψn,0 0 b ψn,1 1 (b * 1 ) φm,2 · · · b ψn,2L−3 L−1 (b * L−1 ) ψn,2L−2 ,
where M and N are finite constants that depend on L, α m , β m ∈ {±1}, and both φ m,ℓ and ψ n,ℓ are non-negative integers which satisfy 2L ℓ=0 φ m,ℓ = L + 1 and 2L−2 ℓ=0 ψ n,ℓ = L .
We next introduce the variable substitution
y(ω) = N 0 |H(ω)| 2 S p (ω) + N 0 , S p (ω) = N 0 |H(ω)| 2 1 y(ω) − 1 .
The constraint S p (ω)dω = 2π translates into
e[y(ω)] = π −π 1 y(ω)|H(ω)| 2 dω = π −π 1 |H(ω)| 2 dω + 2π N 0 . Furthermore, we have b k = 1 2π π −π y(ω)e jkω dω.
The constrained Euler-Lagrange equation [39] becomes δc δy = λ δe δy = − λ |H(ω)| 2 y 2 (ω) .
The functional derivative δb s k /δy equals = sb s−1 k e jkω . We now note that b k , raised to any power, is a constant that depends explicitly on y. Therefore, by an application on the quotient rule for the derivative and the chain rule to (8)
where the A ℓ must have Hermitian symmetry.
We have now found a general form for any stationary point. Unfortunately, for a given H(ω), this stationary point may lie outside of the domain of the optimization. The optimal spectrum S p (ω) must therefore lie on the boundary of the optimization domain, which in this case implies that S p (ω) = 0 for ω ∈ I 0 ⊂ [−π, π]. Let us define I + as the subset [−π, π] where S p (ω) > 0 except for the endpoints of I + where S p (ω) = 0 due to the assumption of a continuous spectrum. Note that I + may be the union of several disjoint sub-intervals of [−π, π]. We can now rewrite the constraint and the expressions of b k as e[y(ω)] = I+ 1 |H(ω)| 2 dω + 2π N 0 and b k = 1 2π I+ y(ω)e jkω dω.
From the first part of the proof, i.e., identifying a necessary condition for stationary points, we have that (11) must hold within the interval I + , and the constants {A ℓ } must be such that S opt p (ω) = 0 at the end-points of each sub-interval within I + . Hence, no matter what I + is, we can express the optimal S opt p (ω) as in (10).
APPENDIX B: PROOF OF THEOREM 2
The waterfilling algorithm provides a transmit filter that satisfies [1] |P (ω)| 2 = max 0, θ − N 0 |H(ω)| 2 ,
for some power constant θ. In view of Theorem 1, |P (ω)| 2 in (12) must also satisfy (10). Equating (12) and (10)
Fig. 1 .
1Block diagrams of a) the transceiver for 2×2 MIMO-ISI channels and of b) the channel shortening detector for continuous-time AWGN channels.
Fig. 3 .I
3AIRs for Gaussian inputs with the waterfilling-solution power spectrum, when different values of the memory L are considered at receiver. AIR [bit/ch.use]
Fig. 4 .
4AIRs for BPSK modulation when different values of the memory L are considered at receiver.
Fig. 6 .
6AIRs for Gaussian inputs over a MIMO-ISI channel with N = 2 and L H = 3, when different values of the memory L are considered at receiver.
.
L A ℓ [y]e jℓω C[y] , where the constants A ℓ [y] and C[y] explicitly depend on y, e.g., By manipulation of the Euler-Lagrange equation and by introducing a new set of constants {B ℓ [y]}, we obtainy(ω) = 1 |H(ω)| 2 [ L ℓ=−L B ℓ [y]e jℓω ].This translates into a general form of the optimal S p (ω) which readsS opt p (ω) = N 0 |H(ω)| 2 L ℓ=−L A ℓ e jℓω − N 0 |H(ω)| 2
A
e jℓω = γ|H(ω)| 2 ,for some constant γ. However, ℓ e jℓω = γ LH ℓ=−LH g ℓ e −jℓω , K must at least equal L H .
: May 23, 2013. Revised: October 24, 2013. A. Modenini and G. Colavolpe are with Università di Parma, Dipartimento di Ingegneria dell'Informazione, Parco Area delle Scienze, 181A, 43124 Parma, Italy, email: [email protected], [email protected]. F. Rusek is with Lund University, box 118, 22100 Lund, Sweden, e-mail: [email protected]. The work of F. Rusek was supported by SSF through the distributed antenna project. The paper was presented in part at the IEEE Intern. Conf. Commun. (ICC'13), Budapest, Hungary, June 2013.
Fig. 7. ASE for a BPSK modulation by using the optimized pulse for two values of the memory L considered at receiver.0.5
1
1.5
2
2.5
3
3.5
4
-1
1
3
5
7
9
11
13
η [bit/s/Hz]
E b /N 0 [dB]
L=1
L=2
RRC α=0.2
RRC α=0.1
opt. pulse
AWGN Capacity
For simplicity of exposition, we refer here to this discrete-time model of a channel with finite ISI. We will discuss later the case of a continuous-time, bandlimited AWGN channel.
By T and Q we mean the convolutional matrices formed from the vectors t and q, respectively.4 With "overall system", we mean the chain: prefilter-channel-reduced complexity receiver.
We remind the reader that I OPT refers to an optimized detector while I AIR refers to the achievable rate for a non optimized detector. Since the filters have been optimized for Gaussian inputs, but we are using here lowcardinality constellations, the filters could be further optimized and for these reason we use the notation I AIR .
Andrea Modenini(S'12)was born in Parma, Italy, in 1986. He received the Dr. Eng. degree in telecommunications engineering (cum laude) in december 2010 from the University of Parma, Italy, where he is currently Ph.D. Student at the Dipartimento di Ingegneria dell'Informazione (DII). His main research interests include information theory and digital transmission theory, with particular emphasis on the optimization of detection algorithm from an information theoretic point of view. He participates in several research projects funded by the European Space Agency (ESA-ESTEC) and important telecommunications companies. In the spring 2012 he was a visiting PhD student at the University of Lund, Sweden, for reasearch on channel shortening detection for spectrally efficient modulations.
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