Such torsional excitation is a form of parametric pumping in the system, which results in the breakage of the time-reversal symmetry. We report that, in quasi-static sense, the system shows topologically non-trivial band-gaps. However, in a dynamic regime where the pumping effect is significant, these band-gaps become asymmetric with respect to the frequency and wavenumber domains in the dispersion relationship. By using numerical simulations, we show that such asymmetry has a direct correspondence to the topological invariant, i.e., Chern number, of the system. We propose that this asymmetry, accompanied by selective inter-band transition, can be utilized for directional isolation of the stress wave propagating along the phononic crystal.We present an active, purely mechanical stress wave isolator that consists of short cylindrical particles arranged in a helical architecture. This phononic structure allows us to change inter-particle stiffness dynamically by controlling the contact angles of the cylinders. We use  A wide variety of researchers have been striving to attain the ability to control the flow of energy in media for centuries. In particular, one-way transmission of electrical energy by semiconductor diodes and transistors has revolutionized the modern era. On the similar lines, one-way propagation of light, heat, and sound have captured the attention of the scientific community, leading to the invention of novel devices, namely photonic5in-situ tunability of wave transmission direction is desired, a linear medium would take the precedence over a non-linear one. This is because the allowed direction of wave propagation can be reversed simply by flipping the external bias, which is not the case with passive settings of non-linearity-based isolators. Moreover, the isolation characteristics could be tuned by adjusting the magnitude of the external bias directly, thereby making such systems active. One could use different types of biases in these active systems. For example, in an acoustic domain, external moving fluid can serve as a bias10121314One of the most common ways to realize one way transmission in a mechanical system is to break the time-reversal symmetrylongitudinal wave by having a direct control over the torsional modes. In particular, we use a torsional travelling wave to induce a specific spatio-temporal modulation of the system’s stiffness, which resembles the sliding lattice potential of the topological Thouless pump in quantum mechanicsbulk wave propagation along the phononic crystal. In this study, we show the topological invariants that we calculate for our one-dimensional system quantify the amount of asymmetry in the dispersion of such bulk waves, in contrast to a two-dimensional system in which those invariants tend to be related to the number of edge modes.In this study, we propose a one-dimensional phononic crystal that consists of a chain of cylinders, to achieve the stress wave isolation in a purely mechanical manner. The array of cylinders has shown extraordinary tunability in terms of tailoring the stress wave propagation through them1617Inspired by the mechanics of the Hertzian contact, the architecture of the helix, and the physics of the Thouless pump, this study claims three-fold novelty as following. First, in terms of design, we propose a purely mechanical system that utilizes a strong coupling between two mechanical wave modes to break the time-reversal symmetry. This implies the control of one mechanical wave mode via another, without resorting to external bias/field. Second, in terms of the fundamental physics of a time-dependent, one-dimensional mechanical system, we show that this system is topologically non-trivial and can be characterized by the topological invariant in two-dimensions, i.e., Chern numberThe phononic crystal to be investigated in this study consists of short cylindrical particles stacked helically as a one-dimensional chain . Each cyet al.i-th cylinder asBased on the previous work by Li α0 is the contact angle between any two cylinders at equilibrium; Aα is the dynamic perturbation in the contact angle caused by a travelling torsional wave; kα = 2π/N represents a wavenumber with N being the spatial period of the contact angle variation; fα denotes the frequency of the travelling torsional wave; and forward and backward direction of the travelling torsional wave, respectively.where αi(t) = αi+ 1(t) − αi(t). Using the Hertz’s contact law, instantaneous contact force between i-th and (i + 1)-th cylinders would be βi(t)δ3/2, with δ as the relative longitudinal displacement between two cylinders. The time-dependent stiffness coefficient βi(t) is of the following form19We can define the contact angle, i.e., relative angle between two cylinders, as ΔY, μ and R denote Young’s modulus, Poisson’s ratio, and the radius of a cylinder. Ke(t) and Ee(t) represent the complete elliptical integrals of the first and second kinds, respectively. The elliptical contact area has eccentricity a and b as its semi-major and semi-minor axes, respectively. b/a is equal to [(1 + cos Δαi(t))/(1 − cos Δαi(t))]−2/3, approximatelyHere, n cylinders of equal masses m. We fix the longitudinal motion and keep the torsional motion free at both ends of the chain. Let xi(t) represent instantaneous longitudinal displacement of the i-th cylinder, and δi be the initial deformation between i-th and (i + 1)-th cylinders under static pre-compression. For a generic scenario, under the excitation forces p1(t) and pn(t) applied to the first and last cylinders of the chain, respectively, we write-Once we have formulated the instantaneous contact stiffness between constituents elements, we can now write the equations of motion for the system consisting of For small instantaneous displacements X(t), stiffness matrix K(t), mass matrix M, and forcing vector F(t) are expressed as-where displacement vector K(t) and a forcing vector F(t), which make the system dynamics fairly complex yet interesting. K(t) imbibes spatio-temporal modulation of contact stiffness in the system, whereas the forcing term F(t) has an external forcing component as well as a component dependent on stiffness modulation. Under no external force p(t) applied to the system, we can easily deduce that a travelling torsional wave as in K(t) = K(t + T) and F(t) = F(t + T), where T = 1/fα. This periodic evolution of K(t) represents parametric pumping in the system caused by torsional wave. The mechanism is analogous to the pumping action performed, for example, when one wants to change the amplitude of oscillation of a swing without exerting an external force. Instead, one changes the center-of-mass and thus vary the effective swing length with timeK(t) acts as such a time varying parameter in this case, which is pumped with frequency fα of the torsional wave.Let us now look at the fα = 0, we see that the second component of F(t) vanishes, and this equation reduces to the familiar standard form: independentK. We can easily analyse it further to obtain eigen-frequencies and Bloch states. However, due to a non-zero fα, simultaneous space-time stiffness modulation is enforced, and a time-periodic set of equations with a period T represents our system. This period is dictated by the pumping frequency fα of the torsional wave. In this way, our system not only supports the Bloch states caused by spatial variation of parameters, but also allows us to tailor them by an additional degree of freedom, i.e., time, hence taking us to the realm of Floquet-Bloch engineeringNote that this parametric pumping has direct consequences on the vibrational modes the system supports. If the perturbation in contact angles were to depend solely on space but not time, i.e., To understand the system dynamics, we start with analysing the normal modes of vibration in the system. We would take the homogenized form of K(t + T) = K(t). This time-periodic system of equation resembles the well studied Mathieu’s equation, and it can be solved using Floquet theoryadiabatic topological quantum pumpquasi-static scenario, and then extend the key characteristics to explain dynamic scenario of pumping. With regard to quasi-static evolution over time, we assume that the variation in the stiffness matrix K(t) with respect to time is significantly small, such that the natural modes of vibration can be calculated directly from the matrix at any time instant. Natural frequencies corresponding to such quasi-static evolution of system are named as static frequencieskα in fα, i.e., the pumping frequency, causes temporal variations in stiffness, and is the measure of how quickly static frequencies are varied over time. If we understand how static frequencies evolve over time, that would give us direct insights into the actual system dynamics for quasi-static pumping. It would also be useful to decipher the complex dynamics in case of rapid pumping  where secondary mode of vibration with frequency offsets, namely frequency cascadeswhere Under the quasi-static assumption, we can convert fL|s denotes the static frequency of the longitudinal waves, which should not be confused with pumping frequency fα of the torsional waves. For the analysis, we choose α0 = 10°, Aα = 2°, N = 6, and n = 300. We take geometrical dimensions and material properties of quartz cylinders from our previous experimental studies fα = 0.1 kHz of the forward propagating torsional wave.Here t = 0. In general, we expect five band-gaps in frequency spectrum for the spatial period (N = 6) of stiffness variation. However, the perturbation Aα in contact angles is small, and therefore we notice only three prominent band-gaps, labelled as 1, 2, and 3 (insets show zoomed view). We then calculate how these frequencies evolve over time due to the quasi-static pumping of torsional wave. Since the pumping is periodic, we expect evolution of the eigen-frequencies to be periodic as well. In T. It is important to note that only frequencies on the edges of band-gaps evolve quite distinctly over time. Frequencies corresponding to global modes do not evolve as distinctly as the edge modes. This is intuitive if we rely on the following argument: spatio-temporal stiffness variation in our system is equivalent to a sliding potential, so at each time instant, stiffness variation is just spatially shifted from the one at last time instant. Quasi-statically speaking, this only reflects as a change in the edge modes of a finite lattice, not the global modes. Hence, the modulation is only evident for the modes on the edges. It is also important to note that all three band-gaps evolve distinctly over time with a different number of degeneracies (crossings). We will now shed light on the same in more detail, and establish their non-trivial topological characteristics.In bulk-boundary correspondence indicates breaking for time-reversal symmetry; and it is the classical analogue of the topological band-gaps of the sliding lattice first proposed by Thouless in 1983Such non-trivial topological nature of band-gaps manifests the quantization of pumped charge in a time-dependent, one-dimensional quantum mechanical system. The average charge/photon flow in one period of pumping cycle can be quantified using a topological invariant in two dimensions, i.e., Chern number2733small pumping frequency, fα = 0.01 kHz. In right end – evolves to a global mode. The other instance  from both left and right ends of the chain separately and see how the pulse propagates in the medium. We send these pulses at the two different time points 1 and 2, and solve We now verify the results obtained through quasi-static analysis, and demonstrate the implications of the bulk-boundary correspondence in the system by carrying out full numerical simulations at a πfαt in It should be also noted that if we flip the sign of the pumping frequency , time-reversal symmetry is broken, and we notice asymmetry in the dispersion curves . However, the most remarkable observation is that each band-gap, on the primary mode branch, shows a unique asymmetry with respect to positive and negative wavenumber axes. Those are not only distinct in terms of the magnitude of frequency and wavenumber offset but also in their directional shifts. For example, band-gap 1 shifts up for positive wavenumbers, whereas band-gap 3, even though showing the same frequency offset, shifts down for the positive wavenumbers. Band-gap 2, however, shows entirely distinct offset with double the magnitude compared to those of band-gap 1 and 3.On the other hand, as we introduce a significant pumping frequency  – which occurs during the time evolution of eigen-frequencies  – essentfα = 2 kHz, we focus on the widest and highest band-gap 3. We choose a longitudinal travelling pulse with frequency fL = 16.78 kHz, such that it lies inside and outside of the band-gap of the primary mode for the right and left branches of dispersion curve, respectively. We use 0.8 ms long Gaussian pulse , and a frequency higher than the band-gap region propagates in the opposite direction (right). These are indicated as the tips of the arrow marked in fα for the current band-gap. Hence, we see the modes travelling in opposite directions lie on different bands of the dispersion curve ; and hence the term inter-band transition is justified.This is indeed a peculiar type of wave dynamics we observe in our system. Though, an input wave can propagate in In this study we have proposed a purely mechanical device based on a helical phononic crystal, which shows non-reciprocity in terms of stress wave propagation. We used the coupling of two mechanical waves – longitudinal and torsional modes – to achieve the breakage of time-reversal symmetry. We related such symmetry breaking to the topological nature of band-gaps induced by spatio-temporal variation of system parameters. In this phononic crystal, the topological characteristics represented by the Chern number directly dictate the level of asymmetry associated with each band-gap in the dispersion curves. Notably, the asymmetry in the dispersion curve can be used to achieve the inter-band transition at a specific frequency range. Towards the practical significance, if the higher frequency mode in such asymmetric wave propagation can be filtered, e.g., by using the fact that it would attenuate more in the presence of material dissipation or by taking advantage of its slower wave propagation speed, the current system would serve the purpose of stress wave isolator, allowing stress waves only in one-direction. Moreover, the allowable direction as well as general characteristics of the longitudinal wave can be simply controlled by fine-tuning the torsional wave parameters, thereby making the proposed system active and tunable.How to cite this article: Chaunsali, R. et al. Stress Wave Isolation by Purely Mechanical Topological Phononic Crystals. Sci. Rep.6, 30662; doi: 10.1038/srep30662 (2016).


The study of various living complex systems by system identification method is important, and the identification of the problem is even more challenging when dealing with a dynamic nonlinear system of discrete time. A well-established model based on kernel functions for input of the maximum length sequence (m-sequence) can be used to estimate nonlinear binary kernel slices using cross-correlation method. In this study, we examine the relevant mathematical properties of kernel slices, particularly their shift-and-product property and overlap distortion problem caused by the irregular shifting of the estimated kernel slices in the cross-correlation function between the input m-sequence and the system output. We then derive the properties of the inverse repeat (IR) m-sequence and propose a method of using IR m-sequence as an input to separately estimate odd- and even-order kernel slices to reduce the chance of kernel-slice overlapping. An instance of third-order Wiener nonlinear model is simulated to justify the proposed method. Living systems usually exhibit complex and nonlinear behaviors –3, whichThe kernel estimation for such nonlinear system usually requires the input signal to be a long Gaussian white noise to completely activate the underlying system. Under such conditions, Lee and Schetzen proposed a convenient cross-correlation method widely used to estimate the kernel functions , 8.In several circumstances, particularly for a variety of living biosystems, input signals are constrained as a series of impulse trains instead of continuous signals, such as the Gaussian white noise , 10. ForUsing the m-sequence approach, Sutter studied the binary kernel for identifying multifocal retinosystem through electroretinography and explained the visual function using the binary kernels . Shi andHowever, the binary kernel slices—derived by making use of the shift-and-product property of the m-sequence—are all laid in the first-order cross-correlation function between the m-sequence input and the system response, that is, the observed output. The specific location of any kernel slice in the cross-correlation function is determined through a complex shift function that cannot be explicitly determined. If the kernel slices are improperly arranged such that overlaps among neighboring slices occur, then the kernel estimation is inevitably distorted. A straightforward approach to solve this problem is to multiply the length of the input m-sequence, which is unfavorable for living systems with more or less time-varying property. Another approach to alleviate the overlap issue is to sparsify the impulse train of the m-sequence at risk of suffering the underestimation caused by the reduced number of available kernel slices .In this study, we addressed the overlap problem through a new strategy using an inverse-repeat (IR) m-sequence. We will drive the estimation equations for the binary kernel slices corresponding to the IR m-sequence, through which the odd- and even-order kernel slices can be separately estimated and thus reduce the chance of slice overlapping. Last, a third-order nonlinear system is simulated to demonstrate the process of the proposed method.b[n] consisting of digit −1 and +1 that pseudorandomly occurred can be generated through the output of a linear circular shift register. The structure of which is determined through a primitive polynomial of degree r, which is also the degree of the periodical m-sequence . The exact value of a shift function depends on the shifting lags of the m-sequences to be multiplied. This property indicates that the product of the m-sequences with different circular shifts is also the same m-sequence circular shifting to a lag determined by a shift function fp that is unknown a priori. A straightforward approach to calculate the specific value of a shift function is to compare bit-by-bit the original m-sequence b[n] and the resulting k-bit shifting version b[n − k] until b[n] = b[n − k], such that fp = k.(i) Shift-and-product propertyj1 = ⋯ = jp−1 = 0, that is, the product of the same j  m-sequences, it yieldsFor p ≥ 3), the m-sequence should be selected with caution in case invalid results occur for the kernel estimation.Equation  providesbnbn−j1⋯bϕbb[n] is a periodical real value function with the same minimal period of L; that is, (ii) The autocorrelation function that is, ϕbbn=1L∑τy[n] of a general Qth-order nonlinear dynamic system in response to the input b[n] can be modeled by a Volterra series expansion  and output y[n] is k = k1 and  li = ki+1 − ki, and then . Therefore, if the shift functions of two neighboring slices satisfys and another kernel slice of order q is less than the length of the prior slice, then a slice overlap occurs. The kernel slices overlapping condition is illustrated in To estimate the Volterra kernel for Gaussian white noise, input is not theoretically feasible for the difficulty of nonorthogonality. Instead, it is preferred to estimate thexpansion . This metimation , 16. A pgiven by wpk1,k2,… y[n] is ϕbpy,ϕbpy is twice as that of b[n], and a[n] satisfies the inverse repeatability, that is, (i) The minimal period of y[n] of a nonlinear system in response to the input a[n] can also be modeled by Volterra series expansion. If we dichotomize y[n] into an odd-order response yo[n] = ∑ℋpb[n],  p = odd, and even-order response ye[n] = ∑ℋpb[n], p is even, they satisfy,(ii) The output a[n] and b[n] is zero, that is, (iii) The cross-correlation function of a[n] isL is odd, then ϕaa[n] = −ϕaa[n + L]; that is, ϕaa[n] is also inversely repeated. The proof of property (iv) is given in (iv) The autocorrelation of a[n] also has the following shift-and-product property, (v) IR m-sequence a[n] be the input of the system, and then the output of the Qth-order nonlinear system expressed in terms of Volterra series expansion is k = k1 and li = ki+1 − ki, and, using IR m-sequence property (v), the Volterra operator on the input a[n] becomesb[n] and odd-order Volterra series term ℋpa[n], (p = odd), is zero and that the cross-correlation function between IR m-sequences a[n] and even-order Volterra series term ℋpa[n] (p = even) is also zero; that is, Based on the above properties of the IR m-sequences, we can derive the binary kernel slices in response to an IR m-sequence. Let ploiting  for the that is, ϕbHp[n]=1pth-order cross-correlation for IR m-sequence input a[n] with the same approach as that of getting  to  to pth-omember isϕapy=12L∑lices as wp, output signal y[n], and cross-correlation ϕby[n]. The estimated kernel slices are plotted in solid trace superimposed on the cross-correlation signal (in dashed trace). The red trace segments indicate the occurrence of slice overlaps. According to the overlap condition defined in  and proposed IR m-sequence (in blue traces) methods are shown in l1 = 4 show a slightly opposite behavior that needs to be accounted for, because no overlap occurs at these affected kernel slices. This inconsistency is only observable at a significantly small scale (see the amplitude scales in A series of third-order kernel slices are shown in The shift-and-product property of the m- and IR m-sequence is crucial in the derivation of new properties to address the overlap problem for short-length m-sequence. It is apparently not beneficial to use a redundantly higher order m-sequence or to sacrifice the number of the estimated slices using sparse m-sequence. In this study, we alternatively propose an approach by introducing the IR m-sequence. We provide and prove several relevant properties of the IR m-sequence allowing the estimation of the binary kernel slices. By examining the special shift-and-product properties of the IR m-sequence derived for odd and even shifts, the odd- and even-order kernel slices can be separately represented in the cross-correlation functions, such that the chance of overlapping significantly decreases. Furthermore, this separation will be useful in some special applications where only odd- or even-order kernel slices might be of significant interest.


Segmentation of injured or unusual anatomic structures in medical imagery is a problem that has continued to elude fully automated solutions. In this paper, the goal of easy-to-use and consistent interactive segmentation is transformed into a control synthesis problem. A nominal level set PDE is assumed to be given; this open-loop system achieves correct segmentation under ideal conditions, but does not agree with a human expert's ideal boundary for real image data. Perturbing the state and dynamics of a level set PDE via the accumulated user input and an observer-like system leads to desirable closed-loop behavior. The input structure is designed such that a user can stabilize the boundary in some desired state without needing to understand any mathematical parameters. Effectiveness of the technique is illustrated with applications to the challenging segmentations of a patellar tendon in MR and a shattered femur in CT. Segmentation is the task of identifying and localizing salient structures in the image volume. Since there is an abundance of raw image data that is not analyzed due to the infeasible time and cost of manual segmentation, automated methods for segmentation are the subject of much recent medical computing literature [Microwave-frequency Magnetic-Resonance-Imaging (MRI)  and X-Raterature –7]. How. Howsegmterature –10]. In. Insegmetime; a partial differential equation (PDE) is used to model the space-time relationship between image data and segmentation boundaries . Gi. Gitime;undaries –20]. It. Ittime;blem; in –23], us, ustime;blem; in , 24],  by byut+ddxents. In , the inv u –34]; e.; e.ut+ddUsing a PDE formulation guarantees that the computational complexity is fixed and that the segmentation result changes continuously over time. By incorporating control-theoretic tools, it is shown that the steady state segmentation can be driven to an ideal reference boundary. Input from the human user indicating locations where the current state does not match the desired reference state is processed and used as feedback in the PDE. An approach similar to backstepping , 35] is is35] isThe remainder of this paper is as follows.Image segmentation based on the narrow-band level-set method is reviewed in Section II. Following an approach similar to back-stepping, a controller is proposed in Section III that stabilizes a labeling-error term, assuming exact knowledge of a reference state. An auxiliary observer-like system is designed in Section IV to process a user input signal and estimate the reference state. Application of the technique to interactive segmentation of CT and MRI volumes is demonstrated in Section V, using images that are difficult to segment with existing methods. The proposed algorithm is compared to related methods in Section VI. Section VII summarizes the results and applicability of the approach. Key components of the final closed-loop system and corresponding paper sections are visualized in level set method, reviewed in Section II-A. Section II-B describes the limitations of open-loop segmentation, thus motivating the feedback control model of interactive image segmentation in Section III-A.The class of open-loop systems considered in this work are PDE-based segmentation algorithms using the popular n to be the spatial domain and x a coordinate in Ω. Labeling assignments are represented with an implicit function ϕ : Ω x . D. D37]. DF in In “variational level set methods” –40], th, th40], artificial time parameter t, which arises solely due to an iterative minimization of t corresponds to a physical time since the human user watches a time-varying visualization of ϕ to decide where and when to provide input.As pointed out in , many imtϕ| > 0 only in a neighborhood of the moving zero level set . Th. Thϕt| > method”  update ϕexterior and interior regions by a regularized version of the Heaviside step function denoted by ε, illustrated in εδ are defined asΩ is divided into This paper considers systems described by narrowband level-set PDEs with the general formG(·) in κ(F), where F is a smooth function with non-vanishing gradient defined on x ∈ Ω.Application-specific goals, such as minimization of a functional, dictate the choice of image-dependent ϕ to a signed distance function  mi miϕ to ap1, p2, and κ0 are fixed constants for a given re-distancing method. These bounds are helpful for control synthesis in Section III.where n-dimensional non-parametric density estimates ..n-dimensI(x) in the interior (ϕ > 0) and exterior (ϕ < 0) regions areNevertheless, a recurring problem is that many objects of interest do not coincide with minima of a proposed functional. As an example, consider segmentation via the “mean-alignment” system . The weiA gradient flow for the functionalgives the following narrowband open-loop system:I(x) in the course of minimizing E(t).ϕ in agreement with a desired segmentation.Rather than having the human user give up on the PDE system and manually outline the desired region of interest, the PDE can be augmented with a user-driven control input. A control solution is sought due to limitations in the efficacy of open-loop system ψ denote the ideal reference segmentation. A human user could manually trace the level set ψ(x) = 0 if given unlimited time. We seek to drive ϕ towards an explicit estimate of ψ, while maintaining closed-loop stability and minimizing burden placed on the user. User-driven control effort should preserve the advantages of the narrowband formulation noted in Section II-A; therefore, an admissible control signal f will act only on the |ϕ| ≤ ε subdomain of Ω. The closed-loop system then becomesLet ψ(x) is assumed known; a control is synthesized to drive ϕ such that it matches ψ. Later in Section IV, a coupled system that estimates ψ from available user input is formulated.In this section, ξ and , respectively:Define the pointwise and total labeling error as ψ is known and available, regulation of (t) is straightforward with the following two theorems in this section. The regulatory control uses known bounds on the image-dependent G term of GM(x), G̅M to be upper bounds such that for any segmentation state ϕ,If Theorem III.1 Using a spatially-varying U(x), a control for 0 > 0, λ1 > 0, ρ ∈ , a sufficient condition for boundedness of  isGiven constants λξ is large in the sense ofFurthermore, when the error the rate of convergence is bounded byProof: Re-arranging terms in ′ and integrating by parts making use of εδ ∘ ϕ = 0 on ∂Ω:L1 guarantees the existence of a constant r such thatThe Poincaré inequality in r is at most half the diameter of Ω  whe whe53] wI(x). To do so, open-loop dynamics are chosen to use the localizing active contours of  be beU(x) fIt is emphasized that the closed-loop control formulation in this paper does not seek to replace existing curve evolution algorithms but rather to augment them. The control-theoretic approach enables a user to reach the desired segmentation at steady-state; running the level set evolution for a longer time will not cause the boundary to “bleed-through” or contract.Two orthopedic images are used as test data to quantitatively compare the method presented in this paper with the popular GrabCut algorithm . The useActuated voxels after initialization are plotted in Y -axis) and the number of newly actuated voxels (X-axis). Each mark corresponds to one iteration when new user input was applied. Linear regression lines are overlaid on the data. The two algorithms have a similar dynamic response in the epiphysis segmentation, shown in Predictability of how the segmentation changes in response to mouse strokes is a criterion for practical ease of use. Two scatterplots quantify the predictability in This paper has presented a modeling approach that enables control-theoretic analysis and design for interactive medical image segmentation. Results shown for a synthetic image (Section IV-D) and real medical volumes (Section V) agree with theoretical expectations of system performance. Section V illustrated two qualitatively different situations: (1) gradual expansion of the boundary to bound the entire femur fragment and (2) prevention of “bleed-through” or over-segmentation with the patellar tendon. In both situations, the user is able to drive the segmentation to a desired steady state and to do so with much less effort in terms of actuated voxels than manual segmentation. In summary, the PDE control formulation enables us to guarantee a user's ability to reach a reference segmentation state while also absolving them of the need to understand mathematical details or use precise mouse movement.Successful use of the closed-loop algorithm by medical students motivates several future extensions. If a single image contains several objects of interest, they would need to be extracted sequentially in the current framework. Such a sequential de-coupled approach does not address natural constraints of the geometry and involves re-editing common boundaries. A coupled formulation using an open-loop system of PDEs such as in  together


In this commentary we note that, where relevant but non-redundant features are binary, they inevitably lead to a classifier capable of capturing only a linear relationship between structural features and activity. If, instead, we were to use relevant but non-redundant real-valued features, the resulting predictive model would be capable of describing a non-linear structure-activity relationship. Hence, we suggest that real-valued features, where available, are to be preferred in this scenario.It is common in cheminformatics to represent the properties of a ligand as a string of 1’s and 0’s, with the intention of elucidating,  One of the major goals of cheminformatics is to predict the relationship between a ligand’s chemical structure and its bioactivity . If thisTo devise an adequate model describing this relationship, the cheminformaticist typically follows a standard approach; starting with a large number of ligand attributes or features considered important for representing the underlying characteristics of the ligand, and relevant to its bioactivity. Then, through feature selection techniques, one selects the ligand attributes deemed to have statistically minimum interdependence among themselves (given the ligand bioactivity), while also showing strong association with the ligand bioactivity –5. With L-dimensional vector k, where k = 1, 2, ..., K with K being the total number of classes of interest.Typically the ligand’s chemical structure is represented by an x without errors is generally impossible. Usually both x and k are treated as random variables such that for a given x we have a distribution x to the class minimising the probability of misclassification is induced from given prototype samples (a training dataset) [Identifying the relevant features mentclass2pt{minim classes . In pracdataset) , 9.p(k)), evidence  and absence (0) of these features—i.e., L relevant features whose relationship with the bioactivity is statistically significant, but whose class-conditional interdependency is statistically insignificant—i.e., retain features that are statistically non-redundant .It is common practice nowadays to assume that the xl∈{0,1} . In pracxl∈{0,1} , 5 that xl∈{0,1} , one migxIn our probabilistic setting, ernoulli , 12, 13,4 of ref ):4\documcument}′ , 12, 13—k in xHowever, the situation can be different if non-redundant real-valued features are utilised to represent the chemical structure of the ligand. In this scenario the entclass1pt{minimaependent . Thus, fIn this commentary it has been noted that, when ligand features are represented by a string of binary numbers, one must end up with a linear model for describing the dependency (if any) between the chemical structure of a ligand and its bioactivity of interest—albeit in a classification setting. Such a linear model may be severely biased and limited in its predictivity. It was also pointed out that, where relevant real-valued features are used, the resulting model can be unbiased as it can adequately capture both linear and non-linear structure-activity relationships.


In consideration of capacity constraints, fuzzy defect ratio, and fuzzy transport loss ratio, this paper attempted to establish an optimized decision model for production planning and distribution of a multiphase, multiproduct reverse supply chain, which addresses defects returned to original manufacturers, and in addition, develops hybrid algorithms such as Particle Swarm Optimization-Genetic Algorithm (PSO-GA), Genetic Algorithm-Simulated Annealing (GA-SA), and Particle Swarm Optimization-Simulated Annealing (PSO-SA) for solving the optimized model. During a case study of a multi-phase, multi-product reverse supply chain network, this paper explained the suitability of the optimized decision model and the applicability of the algorithms. Finally, the hybrid algorithms showed excellent solving capability when compared with original GA and PSO methods. Many scholars have been devoted to the study of positive supply chains, for instance –5, howevShih  discusseThus, this paper focuses on the reverse supply chain and constructs an optimized decision model for the selection of supply chain partners as well as determination of production and distribution quantities. Multiechelon logistics issues could be regarded as a Knapsack of a multiple selection portfolio, and resource distribution as a NP-hard issue ; howeverGAs have been widely used in solving real-world complex optimization problems –16 and MFuzzy theory, first proposed by Zaden in 1965 as an extension of a general set, is a numerical control methodology for imitating human thought and addressing the inaccuracy of all physical systems. According to Fuzzy Theory, the thought logic of human begins as fuzzy and is intended for judgment, even if conditions and data are uncertain, while modern computers feature bipolar logic, that is 0 or 1, different from the logic of humans. However, fuzzy logic theory can represent the degree of fuzzy concepts with values between 0 and 1, namely, “membership function.”Karkowski  indicatef = , where fl, fm, and fu are lower bound, mode, and upper bound values, respectively. f.The triangular fuzzy number can be represented as In addition, it is essential for practical applications that a fuzzy number should be transformed to a numerical value. The transform process is called “defuzzification.” Associated ordinary number (AON) is a simple method in defuzzification and many researches have employed it directly and effectively –43. In tThis paper analyzed the reverse distribution activities of defective products in a complex supply chain network, and explored a method of feeding these products back to the manufacturing partners, with consideration of the capacity constraints of suppliers, and the demands of multi-phase, multi-product production and planning. Moreover, the partners shall be selected based on total objective functions , in addition, the fuzzified production defect ratio and transport loss ratio are taken into account. Given the fact of numerous suppliers in a supply chain network, with different production characteristics and resource allocation capabilities covering their capacity and yield, it is crucial to select proper suppliers in a complex supply chain network.T-treatment for data integration [T-treatment formula  → P(2.1) represents the transport path from the first partner of the first hierarchy to the first partner of the second hierarchy, Gene value is an integral number representing the quantity of partner product transported on the distribution path.romosome  and the Step  2. Generate a random initial population according to the final customer demands of the partners, restrict the population for meeting the demand constraint (nstraint ~12), an, anStep nstraint . Step  3. Use Roulette Wheel method to select the chromosomes. Calculate selection probability of every chromosome, of which the chromosomes of better fitness function have a higher probability. Step  4. Randomly select two chromosomes from the population for crossover, and generate random crossover points, then exchange the genes of the chromosome, as shown in xc, the number of chromosomes is yc, the crossover quantity is xc × yc = nc; a better efficiency could be achieved if crossover rate is 0.75~0.95 [.75~0.95 . Step  5. Randomly select a chromosome for mutation with its position; if the mutation rate is xm, and the number of genes is ym, the mutation number is xm × ym = nm, as shown in Step  6. When the chromosomes of best offspring are superior to those of worst population, this population will be replaced as a new one.Step  7. Repeat Steps  2 to 6 until the stop conditions are satisfied.Step  1. Set the particle number, iteration number, maximum speed, learning factor, and inertia weight. Step  2. Randomly generate the initial speed and position of every particle, with the range specified in the constraint (nstraint ~12). . Step  2Step  3. Substitute the particles of population into the objective function equation , and c1 and c2 are learning factors. where Step  5. Substitute the updated particle position into the constraint (nstraint ~12); re; reStep nstraint ~(12). Step  6. Compare the particle's objective function value with updated particle position and its Pbest. If the objective function value is superior to the Pbest, the Pbest of this particle will be replaced by its objective function value with updated particle position. Step  7. Compare the Pbest and Gbest. If the best value of particle is superior to Gbest, the Gbest will be replaced by the best value of the particle. Step  8. Repeat Steps 2 to 7 until meeting the stop conditions or reaching the set iteration number.Steps  1~3. Perform Steps 1~3 of PSO solving model. Step  4. Perform Step  3 of GA solving model to select the particle positions of better fitness functions stored in the library. With this selection mechanism, there is a higher probability that the particle position of better fitness functions will become Gbest; another new Gbest could be obtained through recombination, and the particles could be evaluated and properly deleted, of which the worst individuals are replaced by optimum individuals, after sequencing of fitness functions. Step  5. Obtain a new group of good populations using the selection mechanism, then perform Steps 1~3 of PSO solving model and obtain the new Pbest and Gbest. Step  6. Perform Step  5 of GA solving model to randomly select mutation particles for single-point mutation and generate a new population. Step  7. Perform Steps 4~7 of PSO solving model to obtain the new Pbest and Gbest of the new population. Step  8. Repeat Steps 2 to 7 until stop conditions are met.Steps  1~6. Perform Steps  1~6 of PSO solving model. Step  7Substep  7.1. Set start temperature (Tstart), end temperature (Tend), cooling rate (α), and length of Markov Chain (M).Substep  7.2. Disturb the updated particle under current temperature T, and generate a neighboring solution k. Substep  7.3. Calculate the objective function, perform disturbance mechanism under current temperature to generate a neighboring solution k, and then substitute it into the objective function equation (f(X) and the difference of the fitness function Δf(n) = f(X′) − f(X). Obtain initial solution X, and through iteration of random disturbances neighboring reasonable solutions, obtain a new solution X′; every particle runs across the length of Markov Chain M several times at start temperature Tstart; a fitness is obtained every time (energy function Δf(n)). equation , and calSubstep  7.4. Judge if this neighboring solution is accepted through probability function, using probability function ; if r ≤ P(n), it will disturb the neighboring solution k; replace the particle and its fitness function; if r > P(n), it will not replace the particle. A new disturbance solution will generate if not accepted, until the termination of set search times.Substep  7.6. Repeat Substeps  7.2–7.5 through Markov Chain (M) until M times, and then perform the cooling steps, indicating that a stable state is reached under this temperature. Substep  7.7. Implement the cooling procedure at the set-cooling rate of α through a cooling mechanism T = T × α. Substep  7.8. Judge if the cycling is finished through the set end temperature Tend, if T ≤ Tend, perform the next step; if T > Tend, repeat Substeps 7.2 to 7.7 until T ≤ Tend. Step  8. Repeat Steps 2 to 7 until reaching the desired iteration number.Steps  1~5. Perform Steps 1 to 5 of GA solving model. Step  6. Perform Substeps 7.1 to 7.8 of PSO-SA solving model. Step  7. Generate new offspring through genetic evolution; if optimum fitness function of offspring is superior to that of the population will be replaced as a new one otherwise, maintain the chromosomes of original population for next-generation evolution. Step  8. Repeat Steps 3 to 7 until the set stop conditions are met.In a complex supply chain network, even a leading manufacturer cannot guarantee 100% yield during the production process, or prevent any defect during the transport process. However, the defect ratio and loss ratio are not fixed; thus, fuzzy defect ratio and fuzzy transport loss ratio are applied in this paper.X products for final customer requirements amount to 400, 350, 450, and Y products amount to 250, 150, 200, and Z products amount to 100, 200, and 250. Q), fuzzy defect ratio (FDR), fuzzy transport loss ratio (FTR), and maximum and minimum capacity (Max. CAP. and Min. CAP.). Since the production defect ratio and transport loss ratio are fuzzy sets with triangular fuzzy number, (P(2.1), its fuzzy number of defect ratio is , then its FDR is 2% according to . Then, the fitness value, execution time, and convergence time of the algorithms were compared using Scheffe's multiple comparison method, with Scheffe's equation as follows:In Tables k is group freedom, Fα(k−1)(n−k) is critical value, MSE is intergroup MS, ni, nj is number of samples.where FitnessμPSO-GA = FitnessμGA-SA = FitnessμPSO-SA < FitnessμPSO = FitnessμGA; that is, PSO-GA, GA-SA, and PSO-SA are all better than PSO and GA, and there are no clear differences in the fitness values of the three algorithms. The comparative result of system execution time is shown in ETμPSO < ETμPSO-GA < ETμPSO-SA < ETμGA-SA < ETμGA is the PSO is superior to other solving models. The convergence times of the algorithms are shown in CTμPSO-GA < CTμPSO < CTμPSO-SA < CTμGA-SA < CTμGA PSO-GA < PSO < PSO-SA < GA-SA < GA; that is, PSO-GA, has faster convergence speed than PSO, PSO-SA, GA-SA, and GA. The results show that PSO-GA performs better in fitness value, execution time, and convergence time. Then, the PSO-GA is employed to make the distribution plan of three periods reverse supply chain and the results are shown in Tables This paper focused on analyzing the issues of returning defective products to original manufacturers in a reverse supply chain system. Fuzzy defect and fuzzy transport loss ratios were considered, and an optimized mathematical model was developed. This model combined cost and quality with T-technology and described how to plan a reverse supply chain on the precondition of meeting customer demands and realizing the capacity of partners. To solve the problems efficiently, three hybrid algorithms were applied to this reverse model, including PSO-GA, PSO-SA, and GA-SA; then, the performances of these algorithms were compared. The experimental results show that if the fitness value, execution time, and convergence time are considered, PSO-GA has the minimal value, which means that PSO-GA has the qualities and capabilities for dealing with the production planning and distribution of a multi-phase, multi-product reverse supply chain.


A long sample-to-detector distance, 160.5 m, can be used at this beamline and a small-angle resolution of 0.25 µm Rubber containing nanoparticles is such a system; the hierarchical structure of the nanoparticles in rubber has a close relationship with reinforced mechanical and viscoelastic properties of filled rubber (reinforcement effect)  which corresponds to a small-angle-scattering resolution of 4.2 µm in real space. There is still a need to measure lower-angle X-ray scattering for clarifying the higher-order structure in a hierarchical structure. In the present study, we used lower-energy X-rays and a smaller beam stop to measure scattering in a much smaller angle. In this paper, we present the details of the USAXS set-up.To overcome the difficulty, we have developed a two-dimensional ultra-small-angle X-ray scattering (USAXS) set-up using a long sample-to-detector distance at BL20XU, SPring-8 . The glancing angle was set to 3.39 mrad, at which the intensity of higher-energy X-rays was reduced as shown in Fig. 2b); in this way, higher harmonic X-rays such as 24 keV X-rays were removed. Parasitic scattering from the mirrors was removed by using another set of guard slits that was placed just downstream of the mirrors. The specimen was placed just behind the guard slits. There were air gaps of approximately 1 m upstream and downstream of the specimen. In the second hutch, a beam stop was set just downstream of the exit window of the vacuum pipe. The diameter of the beam stop was 2 mm. There was another vacuum pipe of 1 m downstream of the beam stop and an X-ray detector was set just behind the exit window of the pipe. An imaging plate (IP)  (Suzuki 3.q-range of 0.25–10 µm−1 is recorded. The crossed pattern originates from the form factor of the rectangular grid hole. The FWHM of each diffraction spot was 0.83 (H) × 0.75 mm (V). There are two factors that broaden the spot size: one is the direct beam size that is determined due to the angular divergence of X-rays and the opening size of the rectangular aperture. As shown in a previous study using 23 keV X-rays, the beam size at the detector position depends on the rectangular aperture size due to the diffraction from the rectangular aperture , where d is the pitch of the periods. In the present case, the FWHM of the spot is larger than the FWHM of the direct beam, and is divided into the contribution from the size of the direct beam and the broadening due to the width of the Laue function. Based on the assumption that the broadening of a spot is expressed as the root-mean square of the two contributions, the maximum transverse length inside which coherent scattering takes place is estimated to be 0.84 × 102 µm (H) × 1.0 × 102 µm (V). These values are consistent with the transverse coherence length estimated from the size of the rectangular aperture, the X-ray wavelength and the distance between the aperture and the sample: 93 µm. Also, the values are equivalent to the beam size. From the above discussion, we conclude that (i) the beam at the sample is nearly full-coherent, and (ii) a reduction in the beam size at the detector position is required for both minimizing the measurable q and Δq values. Further optimization of the mirrors is expected to enhance the performance significantly.Fig. 3q should be combined. Fig. 4et al., 2007et al., 2011q-range 2.5 × 10−4–1.0 nm−1 was recorded: this corresponds to 6 nm–25 µm in real space. The one-dimensional scattering intensity profiles were obtained by azimuthally averaging two-dimensional data. Detailed structural analysis based on the scattering intensity profile is beyond the scope of the present article; analysis of the anisotropic structural distribution of aggregates/nanoparticles in rubber will also be presented elsewhere. The intensity profiles measured using 8 keV X-rays at BL20XU show significant fluctuation. This fluctuation originates from the speckle observed in this setting owing to the coherence of X-rays  approximation. The applicability of the RG approximation has been discussed greatly for small-angle light scattering (Kerker et al., 1963i.e. that δ kaδ k is the wavenumber of the X-ray beam and a is the radius of the sphere. In the present case, where δ is of the order of 10−6 and k is 40.8 nm−1, the condition is expressed as aq-range, it is of great importance to study the applicability of the RG approximations as in studies of the light scattering of aggregates (Farias et al., 1996et al., 2007The q = 0.25 µm−1. An optimization of the mirrors is expected to enhance the performance significantly. The applicability of the RG approximation needs to be carefully examined. Combination of this two-dimensional USAXS technique and an X-ray imaging technique (Kishimoto et al., 2013In conclusion, a small-angle scattering resolution was extended to


This paper establishes the computational complexity status for a problem of deciding on the quality of a committee. Starting with individual preferences over alternatives, we analyse when it can be determined efficiently if a given committee  •We focus on the complexity to decide if a committee satisfies a Condorcet criterion.•For Borda voting this decision problem is •For approval voting, we draw the separation line between hard and easy instances.•It is  Centuries ago, Condorcet committee as a committee that is preferred to every other committee by a majority of voters. In contrast, Condorcet winning set if for each element A natural generalization of a Condorcet winner to committees, i.e., sets of alternatives of a given size, is proposed by Other approaches, however, are taken by In recent papers, the problem of selecting committees is extended by additionally taking into account weight constraints; see, e.g., Similarly to 1.2computationally tractable problems), and into the class of problems for which polynomial algorithms seem unlikely to exist .In line with, e.g., In the literature, multi-winner (or committee) selection problems in proportional representation voting schemes have been considered from a computational point of view, the main attention drawn to the schemes of Monroe and Chamberlin–Courant. Typically, the misrepresentation of a voter is evaluated by means of numerical scores used in positional voting rules, with particular focus on approval and Borda scores  of committee of size -committee. In what follows, the voters’ preferences are assumed to be additively separable. This reflects the idea that, when choosing between two committees, a voter prefers the one for which the sum of the voter’s scoresGiven Definition 2.4Let  With abuse of notation, we abbreviate Definition 2.5Let  Clearly, a strong Condorcet committee is also a weak Condorcet committee. To simplify notation, for Example 1Given the voter preference profile However, 2.3We are now ready to state the weak Condorcet Definition 2.6GIVEN: Set QUESTION: Is  The problem SCC is defined by simply replacing in the above definition “weak Condorcet” by “strong Condorcet”.RemarkNote that CC and SCC belong to the class A natural problem related to these problems is the task of deciding whether there exists a weak respectively strong Condorcet Definition 2.7GIVEN: Set QUESTION: Is there a weak Condorcet  Analogously, in the problem SCC-Existence we ask if there exists a strong Condorcet CC-Existence and SCC-Existence.In the next section, we present a detailed computational complexity study of CC and SCC. In Section  3Min 2-Sat-3 problem, which is known to be In this section, we use the computational complexity status of the Definition 3.1GIVEN: Set QUESTION: Is there a truth assignment  Note that in any instance of Min 2-Sat-3 as defined above the clauses are made up of variables and not of literals. I.e., there are no negated variables in an instance of Min 2-Sat-3.In what follows, we identify a truth assignment Remark 1Min 2-Sat-3, and let satisfying truth assignment for Let  since otherwise Remark 2Min 2-Sat-3 to the decision problem In this paper, we prove the  We begin the analysis of the complexity status of CC and SCC by considering Borda voting. Next, we state results for CC and SCC under approval voting, and conclude Section  3.1First, let us consider the case that the voters’ preference relations are strict orders on Theorem 3.1The CC problem under Borda voting is-complete, even if the preference profile consists of strict orders.ProofMin 2-Sat-3. Let We prove the theorem by a polynomial transformation from an arbitrary instance •for every variable •we introduce •–the two variables that make up –the third to the –the last for every clause king see : –the tw•–for each of these voters, the ranking restricted to the last –putting for all when restricted to the we introduce the voters n2−k see  such tha•we introduce the voters From That is, in instance Claim 1a satisfying truth assignmentforwithIn instancethere is a-committeesuch that, i.e., a majority of voters prefersto.Proof of Claim 1““Let Thus,  Note that Min 2-Sat-3 can be polynomially reduced to the complement of CC under Borda voting, which implies the  An analogous hardness result for SCC follows immediately.Theorem 3.2The SCC problem under Borda voting is-complete, even if the preference profile consists of strict orders.ProofCreate instance  The negative results of Lemma 3.3In instance, eitheris a weak Condorcet-committee or a weak Condorcet-committee does not exist.ProofIt is sufficient to show that for any 3.2In this section, we investigate the complexity of CC and SCC in the case that the voters are allowed to distinguish between “good” and “bad” candidates only .First, let each voter approve of exactly one candidate as it is the case in plurality voting.Proposition 3.4CC under plurality voting can be solved in polynomial time. In addition, a weak Condorcet-committee can be found intime.ProofWe prove the second part of the proposition first. Rank the candidates Let thm see  , the timFor the first part of the proposition, let  With analogous arguments, we get a positive, i.e., computational tractability result if each voter votes against exactly one candidate.Proposition 3.5CC under antiplurality can be solved in polynomial time. In particular, a weak Condorcet-committee can be found intime.Corollary 3.6Under plurality voting and antiplurality voting, SCC can be solved in polynomial time.ProofFollows straightforward from the proof of RemarkIt should be noted that in the cases of plurality and antiplurality voting a weak Condorcet  Therewith, if each voter is allowed to choose only one “good” or one “bad” candidate, CC and SCC are easy to solve. However, this changes drastically if a voter is allowed to choose two “good” or two “bad” candidates, as the following theorems show.Theorem 3.7Under  2-approval, both CC and SCC are-complete.ProofMin 2-Sat-3 with We prove the theorem for CC first. Let •introducing, for every variable •introducing candidates From •for every clause •for each •we introduce •for every pair Thus, the set of candidates is see also : •for evLet Claim 2a satisfying truth assignmentforwithIn instancethere is a committeeof sizesuch that.Proof of Claim 2““•at most •due to the fact that the number of occurrences of each variable is bounded by three, at most Assume prefer  where the fourth inequality follows from Now, Note that Since  Note that in instance Min 2-Sat-3 is polynomial due to Min 2-Sat-3. □For SCC, consider instance  We can derive a lemma to be used in Section  ≠F imply 1≤p≤k−1. Lemma 3.8In instance, eitheris a weak Condorcet-committee or a weak Condorcet-committee does not exist.Proof holds for any For  In addition, from Corollary 3.9Both CC and SCC are-complete in the strong sense. As a consequence of Corollary 3.10Both CC and SCC are-complete under-approval for any fixed.ProofWe prove the corollary for CC . The case For We show that Claim 3such thatinsuch thatin.Proof of Claim 3““I.e., Finally, there cannot exist two or more elements in  Turning our attention to CC and SCC under 2-veto, it turns out that the problems show the same computational complexity behaviour as in the case of 2-approval.Theorem 3.11Under  2-veto, both CC and SCC are-complete.ProofMin 2-Sat-3 with a set Min 2-Sat-3 remains •for every variable •candidates •for each •for each We give a proof for CC first. Let Note that we have Claim 4a satisfying truth assignmentforwithIn instancethere is a committeeof sizesuch that.Proof of Claim 4““Again, let Case 1:  where the second inequality trivially follows from Case 2: (11) Recall that  Because in instance Min 2-Sat-3 is polynomial, which completes the proof.Min 2-Sat-3 iff For SCC, construct instance Lemma 3.12In instance, eitheris a weak Condorcet committee or such a committee does not exist.Proof where the last inequality holds for any Let Corollary 3.13Both CC and SCC are-complete under-veto for any fixed.ProofCeteris paribus, the proof follows from the one of 3.3In the two previous sections, computational complexity results for CC and SCC under Borda voting and approval voting have been presented. In particular, the cases that individual preferences are expressed by linear orders and dichotomous preferences have been considered, and a complete analysis in terms of complexity results of the latter case has been established.However, using ry voter .top class and bottom class respectively.Within this section, for Corollary 3.14Both CC and SCC are-complete if the voters have multichotomous preferences, even if each,, induces the same numberof preference classes for some.ProofWe prove the corollary for CC; the hardness of SCC follows analogously. The case •introduce •let, for Let Let the instance Claim 5The following two decision problems are equivalent:(P1)GIVEN: Instanceofunder  2-approval (as in the proof ofTheorem  3.7)QUESTION: Is there a committeeof sizesuch that, where?(P2)GIVEN: Instanceofunder ruleQUESTION: Is there a committeeof sizesuch that?Proof of Claim 5Let  From the above claim we can conclude the theorem by the proof of  For trichotomous preferences, we may consider the following voting rule.Definition 3.2Let  In case of trichotomous preferences, from (the proof of) 4CC-Existence and SCC-Existence belong to the class CC-Existence is SCC-Existence is In this section, we consider the problems of deciding whether or not a weak respectively strong Condorcet Theorem 4.1SCC-Existence is-hard, even under Borda voting or approval voting. In particular, for approval voting SCC-Existence is Corollary 4.2SCC-Existence isFor approval voting, -complete.ProofSCC-Existence is in SCC-Existence with set SCC-Existence reduces to the problem of deciding if SCC-Existence is in Due to Theorem 4.3CC-Existence is-hard, even under Borda voting or approval voting. In fact, it turns out that CC-Existence is both CC-Existence and SCC-Existence show a different computational complexity behaviour.Theorem 4.4CC-Existence is-hard, even under  2-approval voting.ProofIndependent Set (IS). The IS problem is the following decision problem: Given an undirected graph We provide a reduction from c graphs .CC-Existence as follows: Set For an instance If If  Our results have shown that CC-Existence is both SCC-Existence is CC-Existence under approval voting already–but we do not have a proof.5With respect to classic voting schemes adapted from Social Choice Theory, we have analysed the computational complexity involved in deciding if a given committee is at least as good as (resp. better than) every other committee in a pairwise majority comparison. In particular, we have established the sharp separation line between computationally tractable and intractable instances. As our results show, if the voters vote for or against exactly one candidate it can be determined efficiently if a given committee is a weak or strong Condorcet SCC-Existence is CC-Existence is both As it turns out, also the problem


K, and R a classical maximal K. We study sets of lengths of factorizations of cancellative elements of R into atoms (irreducibles). In a large majority of cases there exists a transfer homomorphism to a monoid of zero-sum sequences over a ray class group of R. If K, we prove that in the remaining cases no such transfer homomorphism can exist and that several invariants dealing with sets of lengths are infinite.Let  Then the monoid of divisorial two-sided ideals is a free abelian monoid. If H is commutative , this gives rise to the construction of a transfer homomorphism G of H. Transfer homomorphisms preserve sets of lengths, and if Sets of lengths  are among the most investigated invariants in factorization theory. So far research has almost been entirely devoted to the commutative setting, and it has focused on commutative noetherian domains, commutative Krull monoids, numerical monoids, and others cf. . Recall Only first hesitant steps were taken so far to study factorization properties in a non-commutative setting . Any approach to study sets of lengths, which runs as described above and involves divisorial two-sided ideals, is restricted to normalizing Krull monoids Let Theorem 1.1Letbe a holomorphy ring in a global field K, A a central simple algebra over K, and R a classical maximal-order of A. Suppose that every stably free left R-ideal is free. Then there exists a transfer homomorphism, whereis a ray class group of, andis the monoid of zero-sum sequences over. In particular,1.The set of distancesis a finite interval, and if it is non-empty, then.2.For every, the union of sets of lengths containing k, denoted by, is a finite interval.3.There is ansuch that for everythe set of lengthsis an AAMP with differenceand bound M.R-ideal is free, we obtain a transfer homomorphism to a monoid of zero-sum sequences over a finite abelian group. Therefore, sets of lengths in R are the same as sets of lengths in a commutative Krull monoid with finite class group.Thus, under the additional hypothesis that every stably free left A satisfies the Eichler condition relative to R-ideal is free by Eichlerʼs Theorem. In particular, if K is a number field and A satisfies the Eichler condition relative to A is a totally definite quaternion algebra. Thus in this setting R-ideal is free is indeed necessary.If Theorem 1.2Letbe the ring of algebraic integers in a number field K, A a central simple algebra over K, and R a classical maximal-order of A. If there exists a stably free left R-ideal that is not free, then there exists no transfer homomorphism, whereis any subset of an abelian group. Moreover,1..2.For every, we have.H is a commutative Krull monoid with an infinite class group such that every class contains a prime divisor, then Kainrath showed that every finite subset of The proof of  lengths , whence 2Let Throughout the paper there will be many statements that can be either formulated “from the left” or “from the right”, and most of the time it is obvious how the symmetric statement should look like. Therefore often just one variant is formulated and it is left to the reader to fill in the symmetric definition or statement if required.2.1H be a small category. In the sequel the objects of H play no role, and therefore we shall identify H with the set of morphisms of H. We denote by Let H is a semigroup if and only if A semigroup may be viewed as a category with a single object (corresponding to its identity element), and elements of the semigroup as morphisms with source and target this unique object. In this way the notion of a small category generalizes the usual notion of a semigroup  with j to be the inclusion map, and we call category of right fractions of H with respect to D. If D also admits a left calculus of fractions, then The notion of semigroups of fractions generalizes to categories of fractions with analogous conditions monoid is a cancellative semigroup satisfying the left and right Ore condition  from Let 2.2G is a small category in which every element is a unit . If A groupoid vertex group or isotropy group of G at e. If a: If G is connected  and satisfies the ACC on divisorial two-sided ideals. A prime Goldie ring R is a Krull ring if it is completely integrally closed and satisfies the ACC on divisorial two-sided ideals  and transfer homomorphisms in the setting of cancellative small categories.Throughout this section, let H be a cancellative small category.H is reduced if atom (or irreducible) if H, and call H atomic if every H is a subset H is defined similarly. A principal left (right) ideal of H is a set of the form Ha (aH) for some H is a commutative monoid, then prime element if Proposition 3.1If H satisfies the ACC on principal left and right ideals, then H is atomic.ProofH is cancellative, this implies Claim AIf, then there existandsuch that.We first note that if Proof of Claim AaH with aH in Ω implies Assume the contrary. Then the setHa with We proceed to show that every rigid factorization) in a cancellative small category. It is modeled after a terminology by Cohn The following definition provides a natural notion of an ordered factorization For all,implies.(b).is called right-saturated if these equivalent conditions are fulfilled.Proof(a) ⇒ (b): Let (b) ⇒ (a): Let Definition 3.6B be a reduced cancellative small category. A homomorphism transfer homomorphism if it has the following properties:(T1)(T2)If Let The notion of a transfer homomorphism plays a central role in studying sets of lengths. It is easily checked that the following still holds in our generalized setting  with differenceand bound M.ProofG. By the previous lemma it is sufficient to show these statements for the monoid of zero-sum sequences The description in 4η, and as the main result in this section, we present a transfer homomorphism to a monoid of zero-sum sequences in In this section we introduce arithmetical groupoids and study the factorization behavior of integral elements. In Section Definition 4.1G together with a relation ⩽ on G such that for all 1.2.3.A lattice-ordered groupoid G, and ∧ and ∨ do not represent meet and join operations on the entire set G in the order-theoretic sense.)If a of a lattice-ordered groupoid is called integral if G.An element Definition 4.2G is called an arithmetical groupoid if it has the following properties for all 1)(PFor 2)(P3)(PIf 4)(PFor every non-empty subset 5)(P6) , 2., henWe summarize some basic properties that follow immediately from the definitions.Lemma 4.3Let.1.holds if eitheror. In particular, forthe following are equivalent: (a) .2.Let. Ifandare integral, thenand.3.If,and, then(i)andif,(ii)andif.4.Letand. Ifexists, then alsoexists, and. Moreover, then alsoexists and. Analogous statements hold forand.5.,,and in particularare conditionally complete as lattices.6.The setof all integral elements forms a reduced wide subcategory of G, andis the groupoid of (left and right) fractions of this subcategory.7.For every, there existandwithand.Proof3, 1 this is equivalent to 3 this is equivalent to 1. Assume first 1. For (a) ⇔ (c) set (a) ⇔ (b) and (c) ⇔ (d) by P3. Similarly, 2. Since 3 implies x from the left gives 3. We show (i), (ii) is similar. Since xc is an upper bound for xM. If xM, then 4. Let For 4, and 5. We show the claim for 4), to ensure that the supremum lies in The proof for G. If a is integral. Since 6. By 2 and the fact that every 5 there exist integral 7. By PFor If Definition & Lemma 4.4Forthe following are equivalent:(a)u is maximal in,(b)u is maximal in,(c).An elementsatisfying these equivalent conditions is called maximal integral.Proofu is maximal in u in the first set, therefore u is maximal in the second set.(a) ⇒ (b): By definition, u in (b) ⇒ (c): Assume v and (c) ⇒ (a): Let Lemma 4.5Let U be an l-group. Forthe following are equivalent:(a)p is maximal integral,(b)p is a prime element in,(c).Proofe be the identity of U.(a) ⇔ (c) is shown as in p be maximal in (a) ⇒ (b): Let p be a prime, (b) ⇒ (c): Let Proposition 4.61.If G is a group , and projectivity is an equivalence relation. Note that in a modular lattice perspective intervals are isomorphic Lemma 4.8If a, b, c are as in the definition of transposability, thenfor someand c is transposable to d.Proofd with Since Lemma 4.9If two lattice intervalsandofare projective, then the integral elementsandare projective to each other in the sense of the previous definition.ProofIt suffices to show that if the lattice interval Definition & Lemma 4.10For every,and we writefor this set. The lower bound is defined by.ProofLet Φ and the notation of Section Lemma 4.11Let.1.Iforwith, then. In particular, if, then.2.If,then. If moreover, then, and if furthermoreandare coprime, then.3.If, thenis prime.4.Letbe projective elements. If, then, and.With the definition of ProofΦ.1. Immediate from the definition of 2. Observe that a, b be integral. Then 1 implies Let now 3. By (e)+) by , 3.. Assv is maximal integral, and may assume that either u is transposable to v or v is transposable to u. Let u is transposable to v via v is maximal integral. If v is transposable to u, one argues along similar lines.4. We first show that u is transposable to v . Let again For the remainder of the claim we may now assume that The converse of Proposition 4.121.The categoryis half-factorial. Explicitly: Everypossesses a rigid factorizationwithandand the number of factors,, is uniquely determined by a. Moreover, ifis another rigid factorization with, then there exists a permutationsuch thatis projective tofor all. In particular,for all.2.Any two rigid factorizations ofcan be transformed into each other by a number of steps, each of which only involves replacing two successive elements by two new ones.3.Transposition.) Ifwithand,,, then there exist uniquely determinedsuch that,and. by  and u′=vUniqueness. If Explicit formulas. Since The equalities 4. Write Corollary 4.13Ifis a subcategory, thenis finite and non-empty for all. If for every primeand all  the setis finite, thenis finite for all.Proof6 that H satisfies the ACC on principal left and right ideals, and hence k is bounded by the length of the factorization of a in Using that H is a commutative monoid and The properties that all sets of lengths, respectively that all sets of factorizations, are finite have been studied a lot in the commutative setting. Note, if Definition & Lemma 4.14There exists a unique groupoid epimorphismsuch thatfor all. We call η the abstract norm of G.Proofa (G is the category of (left and right) fractions of η extends to a unique groupoid homomorphism We need to show existence and uniqueness of such a homomorphism. Let ned by a , it follη is surjective, let first To verify that Φ need not be a homomorphism, but from a the prime factorizations of In general Theorem 4.15Let G be an arithmetical groupoid,the abstract norm, H a right-saturated subcategory of, and. Forset, and. Assume that1.forwith,if and only if,2.for everyand, there exists an elementsuch thatand.Then there exists a transfer homomorphism.Proofa, and not on the particular rigid factorization into maximal integral elements chosen. Let η). Therefore Let θ is a homomorphism: Obviously b into maximal integral elements, then ab. ThusWith this definition θ has properties T1 and T2.We still have to check that a possesses an empty factorization into maximal elements, hence If θ is surjective: a is integral in G and H is right-saturated in θ satisfies T2: Let H is a left-saturated subcategory of θ one then first chooses The theorem remains true if Remark 4.16G is a group, then H is therefore a reduced commutative Krull monoid If is A(G+) . As a saH be a normalizing Krull monoid. Then H, which form the free abelian monoid of integral elements in the free abelian group of divisorial ideals of H. In this way we recover the well-known transfer homomorphism for Krull monoids as given for example in Conversely, let We continue the discussion of normalizing Krull monoids in 5In this section we develop a divisorial one-sided ideal theory in semigroups. This follows again original ideas of Asano and Murata and generalizes the corresponding theory in rings and the theory of divisorial two-sided ideals in cancellative semigroups  fractions of H.A semigroup Lemma 5.1Let Q be a quotient semigroup, anda subsemigroup.1.If H is an order in Q, thenand.2.If, thenand H is an order in Q.3.If H is an order in Q,is a subsemigroup of Q and there existwith, thenis an order in Q.ProofH is a right order in Q, there exist H. Then H is a left order it follows in the same way that a is right-cancellative in 1. It suffices to show 2. It again suffices to show H is an order in Q, there exist 3. It suffices to show that every For the remainder of this section, let Q be a quotient semigroup.H and Q, then H is (Asano-)equivalent to Q. An order H is maximal if it is maximal within its equivalence class with respect to set inclusion.If H, one has to study the ideal theory of all maximal orders in its equivalence class at the same time.A feature of the non-commutative theory is that often there is no unique maximal order in a given equivalence class, and in fact in the most important cases we study there are usually infinitely many, but only finitely many conjugacy classes of them. In studying the divisorial one-sided ideal theory of a maximal order X is a left H-module if right-module if module if it is a left H- and a right H-module is an Let Lemma 5.2Let H,be subsemigroups of Q and let X be an-module.1.andare-modules.2.and.3.andare subsemigroups of Q.4..5.if and only ifif and only if.Proof1. 2. 3. Clearly 4. We show 5. Immediate from the definitions of Definition 5.3For Definition 5.4H and Q.1.fractional left H-ideal is a left H-module I such that A 2.fractional right-ideal is a right I such that A 3.I is a fractional left H-ideal and a fractional right I is a fractional-ideal.If 4.fractional H-ideal is a fractional A 5.I is a left H-ideal if it is a fractional left H-ideal and right-ideal is defined analogously.6.I is a left H-ideal and a right I is an ideal.If 7.H-ideal is an An 8.H-ideal I is integral if A fractional left Let H is a maximal order, then the notions of a left H-ideal and that of an integral fractional left H-ideal coincide  H-ideal, or two-sided to emphasize that it is indeed a fractional H-ideal.If H-ideals and first observe the following.We recall some properties of fractional left Lemma 5.5If H is an order in Q and I is a fractional left H-ideal, thenandare orders. I is a fractional-ideal.ProofH is an order and I is a fractional left I is also a fractional right Let I to be a fractional H-ideal .(b)There exists a fractional-ideal.(c)There exists a fractional-ideal.ProofQ, 1. There exist 2. Using I is an I is a fractional 3. (a) ⇒ (b): By (b) ⇒ (a): By H and (a) ⇔ (c) follows by symmetry, swapping the roles of Lemma 5.8Let H be an order in Q. The following statements are equivalent.(a)H is a maximal order.(b)If I is a fractional left H-ideal, thenand if J is a fractional right H-ideal, then.(c)If I is a fractional H-ideal, then.(d)If I is an H-ideal, then.Proof(a) ⇒ (b): By (b) ⇒ (c) ⇒ (d): Trivial.H and I is an H-ideal, and J is again an H-ideal (we use (d) ⇒ (a): Assume Lemma 5.9Let H be a maximal order in Q, let I and J be fractional left H-ideals, and let K be a fractional left-ideal.1.,is a fractional right H-ideal withandis a fractional left H-ideal withand. Moreover,.2.If, then Ha is a fractional left H-ideal with,and. Ha is integral , if and only if.3.Ifthenand.4.,and.5..6..7.Ifandare also maximal, then.ProofH-ideal. By the symmetric statement of what we just showed for fractional right H-ideals, therefore H-ideal. Applying the first part of the statement to 1. By Ha is a fractional left H-ideal. Certainly Ha is a left H-ideal if and only if it is integral due to maximality of H, and 2. Since H-ideals with H-ideals to obtain 3. If H-ideal with H-ideal with 4. By 5. 6. 7. We use Kv) from . We have⊂IvKv by . By 3, tTo prove the claim it suffices to show −1 using  again. MDefinition 5.10H be an order in Q. A fractional left or right H-ideal I is called divisorial if Let I is a fractional left H-ideal for a maximal order H, then it is not necessarily true that H-ideals with H maximal, already both, v-product, by J is a divisorial fractional left v-invertible, i.e., invertible with respect to this operation.If Proposition 5.11Let H be a maximal order in Q. Let I be a fractional left H-ideal. Then:1.is a maximal order. In particular,is a maximal order.2.and ifis also maximal, then.3.If I is a divisorial fractional left H-ideal, J a divisorial fractional left-ideal and K a divisorial fractional left-ideal, thenProofH is maximal, J is an integral left 1. Because H is maximal, it suffices to show We claim J, Therefore, from the definition of Now 2. We have to show 3. Using Corollary 5.12If H is a maximal order, then every orderwithis contained in a maximal order equivalent to H.ProofI. Then By Corollary 5.13Let α denote an equivalence class of maximal orders of Q. LetandThenis a lattice-ordered groupoid, with identity elements the maximal orders in α. If I, J are inwithor, thenand. Moreover,is the subcategory of integral elements.Proofv-product v-product is associative when it is defined  by , and cleDefinition & Lemma 5.14An order H is bounded if it satisfies the following equivalent conditions:(a)Every fractional left H-ideal and every fractional right H-ideal contains a fractional (two-sided) H-ideal.(b)Every left H-ideal and every right H-ideal contains a (two-sided) H-ideal.(c)For allthere existsuch thatand.(d)For allthere existsuch thatand.(e)For all, HaH is a fractional (two-sided) H-ideal.(f)For allthere exists a fractional (two-sided) H-ideal I with.(g)Ifandwith, then there existwithand.Proof(a) ⇒ (b): Trivial.Hc contains an H-ideal J. If (b) ⇒ (c): Let (c) ⇒ (d): By (c) applied to HaH is an (d) ⇒ (e): (e) ⇒ (f): Trivial.H-ideal, there exists (f) ⇒ (g): I be a fractional left H-ideal and HbH is a fractional H-ideal contained in I  ⇒ (a): Let Lemma 5.151.Let H andbe orders in Q. If H is bounded and, thenis also bounded.2.Let H andbe bounded equivalent maximal orders of Q. Then there exists an-ideal I.ProofH is bounded, there exist 1. Because H and H and 2. We show: If H and Clearly Proposition 5.16Let α be an equivalence class of maximal orders in Q.is an arithmetical groupoid if and only if allequivalently, onesatisfy the following three conditions:1) H-ideals is modular”, as the property for the other side then holds automatically.From the discussion after 2.H be a normalizing monoid. By definition of a monoid, H satisfies the left and right Ore condition, hence it is an order in its quotient group. H-ideal is in fact already a two-sided H-ideal, and thus H is bounded.Let H is a normalizing Krull monoid. Then H-ideals is then modular, even distributive Assume that Definition 5.18H satisfying A1–A3 an arithmetical maximal order. If α is its equivalence class of arithmetical maximal orders, then we denote by We call a maximal order Let from here on H be an arithmetical maximal order in Q, and let α be its equivalence class of arithmetical maximal orders.Ha with H-ideal with inverse v-product coincides with the usual proper product on H determines α) we denote the subcategory of H. Explicitly,H are conjugate. Again the inclusion By H and The following simple lemma gives a correspondence between Lemma 5.19Let.1.Ifwith, thenwith.2.Ifandwith, then there existwith,and.3.Ifwithand, then there existwithand.In particular, forwe haveif and only if.Proof1. The multiplication is defined because 2. Since 3. Since Observe that we may view a rigid factorization Proposition 5.20Let. For everythere is a bijection, given byIfis a transfer homomorphism to a reduced cancellative small category B and having the additional property thatfor alland, then it induces a transfer homomorphismgiven by.ProofThe claimed bijection follows by iterating the previous lemma.θ is a transfer homomorphism and first check that θ is a homomorphism: For We need to verify that Remark 5.21B is necessarily a semigroup.The condition Let Lemma 5.221.Ifand, then.2..ProofI, then 1. It suffices to verify the claim for maximal integral η is a homomorphism)2. The first equality is immediate from 1. For the second equality, note that if H.Applying Theorem 5.23Let Q be a quotient semigroup, H an arithmetical maximal order in Q, and α its equivalence class of arithmetical maximal orders.1.For all,is finite and non-empty. If, for every maximal divisorial H-ideal P, the number of maximal divisorial left H-ideals I withis finite, thenis finite for all.2.Let,, and. Assume:(i)A divisorial fractional left H-ideal I is principal if and only if.(ii)For alland allthere exists a maximal divisorial left-ideal with.Then there exists a transfer homomorphism.ProofBy 1. This follows immediately from I be a fractional left I is principal if and only if the fractional left H-ideal 2. Let Remark 5.241.H be a normalizing Krull monoid. Then H-ideals, while We continue our discussion from C is the divisorial class group of H, and H-ideal. The second condition of the theorem is trivially true by virtue of In the previous theorem we therefore have 2.H is a maximal order satisfying only A1 and A3, then 5 and P6, and still obtain Φ, which can only be defined in the presence of P5). This is possible because P6 is only used to show existence of a rigid factorization of a. A sufficient condition for H satisfies A1, then H) satisfy the ACC, and as in If 5.1R is a ring-theoretic order in Q, then a fractional left R-ideal I in the semigroup-theoretic sense is a fractional left R-ideal in the ring-theoretic sense if and only if Suppose that Let for the remainder of this subsectionbe a quotient ring.Lemma 5.25Let H be an order in the multiplicative semigroupand I a fractional left H-ideal (in the semigroup-theoretic sense). Consider the following statements:(a).(b)is a subring of Q.(c)is a subring of Q.Then (a) ⇒ (b) and (a) ⇒ (c). If H is a maximal order and I is divisorial, then (a) ⇔ (b) ⇔ (c).ProofAssume that (a) holds. We show (b): Let H is maximal, Assume now that Lemma 5.26A ring-theoretic order R in Q is maximal in the ring-theoretic sense if and only if it is maximal in the semigroup-theoretic sense.ProofR is maximal in the ring-theoretic sense, then it is maximal in the semigroup-theoretic sense, as the other direction is trivial. Let I be a fractional left R-ideal in the semigroup-theoretic sense. Then R-ideal in the ring-theoretic sense, and using R in the ring-theoretic sense implies J is a fractional right R-ideal in the ring-theoretic sense then R is maximal in the semigroup-theoretic sense.  □We show that if α be an equivalence class of maximal orders of As before let Lemma 5.27Letand assume that H is a subring of Q .1.Everyis a subring of Q (and therefore an order in Q in the ring-theoretic sense).2.If I is a divisorial fractional left H-ideal and J is a divisorial fractional left-ideal, theni.e., the semigroup-theoretic v-product coincides with the ring-theoretic one.3.If I and J are divisorial fractional left H-ideals, then.ProofI. By maximality of H and 1. By H-ideals (in the semigroup-theoretic sense). Therefore 2. Write H-ideals  R-modules is modular, thenis an arithmetical maximal order inin the semigroup-theoretic sense. In particular, the conclusions ofhold for R.Proofv-product, meet and join coincide, and hence By E, that are prime and satisfy the ACC on divisorial two-sided E-ideals, but do not satisfy the ACC on divisorial left E-ideals or the ACC on divisorial right E-ideals. In fact , the orders E are not even atomic. However, these orders are not Goldie, as they are not of finite left or right uniform dimension, and do not satisfy the ACC on left or right annihilator ideals.In Before going to maximal orders in central simple algebras, we discuss principal ideal rings.Example 5.29R be a bounded order in a quotient ring Q. Assume that every left R-ideal and every right R-ideal is principal. By the characterization in R is then already a maximal order, and it satisfies A1–A3. Thus D is a division ring and Let Q be a quaternion algebra over a field K with X, thereforeK is infinite. , then η of Section But the aim of this subsection is to restrict to the situation where the base ring Lemma 5.31Letand let T be a classical-order in A. The following are equivalent:(a)I is a fractional left T-ideal in the ring-theoretic sense I is a finitely generated-module withand.4If T is maximal, then in addition the following statements are equivalent to the previous ones:(c)I is a divisorial fractional left T-ideal in the semigroup-theoretic sense (.ic sense .(d)I is a divisorial fractional left T-ideal in the ring-theoretic sense  ⇒ (b): Recall that (b) ⇒ (a): Certainly T be maximal. (d) ⇒ (a) is trivial, and (a) ⇒ (d) follows because T is a Dedekind prime ring, and hence every fractional left T-ideal (in the ring-theoretic sense) is invertible , and therefore divisorial.Let now (c) ⇔ (d) follows from T-ideal in T-ideal in the sense of T-ideal in the ring-theoretic sense with T is maximal, then the extra condition A subset v-product coincides with the usual proper product of ideals: A in Reinerʼs terminology  R-modules.To be able to apply our abstract results we still have to check that AWriting Lemma 5.32Ifand, thenand there is a canonical bijectioninducing an isomorphism of free abelian groups. The inverse map is given bywhereis the unique maximal (two-sided) R-ideal lying over. If R is unramified at, then.If all residue fields ofare finite, andis the abstract norm homomorphism, then.ProofM a maximal integral left All but the last statement follow from 6Throughout this section, let K be a global field andbe a holomorphy ring in K.5Furthermore, let A be a central simple algebra over K, and R a classical maximal-order.Setting Lemma 6.1Let r be as in. Then r induces an isomorphismwhere.ProofBy Lemma 6.2For all classical maximal-orders, and all, there exist infinitely many maximal left-ideals I with.ProofLet M with For each )]=[p]=g .  □R-ideals, i.e., In the following equivalent characterizations of the first condition of Lemma 6.3The following are equivalent.(a)A fractional left-ideal withconjugate to R is principal if and only if.(b)A fractional left R-ideal is principal if and only if.(c)Every fractional left R-ideal I withis principal.(d)For the map of finite setsit holds that.(e)Every stably free left R-ideal is free.(f)Every finitely generated projective R-module that is stably free is free.ProofR is hereditary noetherian.The equivalence of (a), (b), (c) and (d) is trivial. The remaining equivalences follow from standard literature: (f) ⇒ (e) is true because R-module. Then R-ideal I and I is stably free and hence free by (e), but then so is M.(e) ⇒ (f): Let and n∈N0 . I is stR-modules of rank one, i.e., the map To see (d) ⇔ (e) it suffices to recall that see e.g.  and thatProof of Theorem 1.1R is an arithmetical maximal order in Q. We verify conditions (i) and (ii) of I be a fractional left R-ideal. By R-ideal is free, this is the case if and only if I is principal, thus condition (i) holds. Condition (ii) holds due to By Remark 6.4K, then there is a transfer homomorphism to either I with If more generally 7Throughout this section, letbe the ring of algebraic integers in a number field K, A a central simple algebra over K, and R a classical maximal-order in A having a stably free left R-ideal that is not free. Furthermore, the discriminant of A is denoted byR-ideal that is not free implies that A is a totally definite quaternion algebra and that K is totally real.  We proceed in three subsections.The aim of this section is to prove 7.1We state two propositions and show how they imply Proposition 7.1There exists a totally positive prime element, a non-empty subsetand for everyan atomsuch thatWe emphasize that E does not depend on l.)Proposition 7.2Ifand, then.Proof of Theorem 1.2 (based on Proposition 7.1 and Proposition 7.2)Claim AIfand, then.We first show that there is no transfer homomorphism ProofLet By p and E be as in In order to show Let 7.2Algebraic number theory. Our notation mainly follows Narkiewicz K, a. Completion at a prime Quaternion algebras. We follow T of A is called a classical Eichler-order if it is the intersection of two classical maximal T takes the form T. Furthermore, because A is totally definite, we have A classical R-ideals, and Proposition 7.3Let. Letwith, and such thatThen, for everywith, there exist at least C maximal left (right) R-ideals of reduced normand class c. Hereis the narrow class number, and w and M are constants depending onsee.onD see.As in the previous section, ProofAlthough not explicitly stated in this way, this is proved by Kirschmer and Voight in Optimal embeddings. Let T a classical Eichler A of squarefree level L, every embedding ι is an optimal embedding if T is still bounded by a constant times h(O) see . Since  cf. . The numQ]−1 cf. , withdL=q with q being locally represented everywhere by the norm form, η small enough, say By Remark 7.7T of A there are infinitely many pairwise non-associated totally positive primes T. This can easily be seen as follows: Let T. If T. But there are infinitely many pairwise non-associated such primes, because every class of the ray class group q of the required form correspond exactly to the trivial class in For any classical Lemma 7.8Let q be a totally positive prime element of. Let I be a non-principal right R-ideal withfor some, and J be a left-ideal withfor somesuch that: as left S-ideals) and I (respectively J) is not contained in any principal left S-ideal except S itself, and not contained in any principal right-ideal  exceptrespectivelyitself.Assume further thatwithand.1.For all,is a principal right R-ideal and an atom of. In particular,for all.2.if it is primitive. In particular ifand, then.ProofI is not contained in any principal right R-ideal, it is in particular not contained in qR, hence primitive. Similarly, J is primitive. Let I and J.Since S-ideals, I, J and a, this is the only rigid factorization of 1. Since I and J are both maximal left S-ideals), then 2. By primitivity,Lemma 7.9Let I be a left R-ideal,, and. Thenwith the left-most and the right-most expressions being proper products ofThe products have to be read in ascending order with “” to the very left.)ProofThe formulas are clear, and so is that the products are proper ones. The key point is that these one-sided ideals are indeed integral. But this is so because Lemma 7.10Let M be a maximal left R-ideal, and N a maximal left-ideal. If, then.ProofSince Proof of Proposition 7.1R If Case 4.Case 4a.J is non-principal. If Case 2. In the case that Case 3 in the same situation.)If Case 4b.If Finally, if If Case 5.Case 2.If Case 6.Case 1.If For Case 1. For Case 4 gives at least one factorization with length in Case 2, Case 3, and Case 4 depends only on the principality of certain one-sided ideals, and not on l. Thus we have shown that there exists a set p a totally positive prime element of R.  □Since there is at least one Remark 7.111.T.In the proof, the classical 2.q with q is a square in a a non-square in If K)→1 cf. , it suffeals cf. .3.a in the proof, we can replace A at all  at some position. Thus, for each ax in a of length l.  □It suffices to prove the claim for