Controlling laser-beam patterns is indispensable in modern technology, where lasers are typically combined with phase-modulating elements such as diffractive optical elements or spatial light modulators. However, the combination of separate elements is not only a challenge for on-chip miniaturisation but also hinders their integration permitting the switchable control of individual modules. Here, we demonstrate the operation of phase-modulating lasers that emit arbitrarily configurable beam patterns without requiring any optical elements or scanning devices. We introduce a phase-modulating resonator in a semiconductor laser, which allows the concurrent realisation of lasing and phase modulation. The fabricated devices are on-chip-sized, making them suitable for integration. We believe this work will provide a breakthrough in various laser applications such as switchable illumination patterns for bio-medical applications, structured illuminations, and even real three-dimensional or highly realistic displays, which cannot be realised with simple combinations of conventional devices or elements. Several recent laser applications, such as maskless lithography11121314The PCSEL is a surface-emitting semiconductor laser that utilises a two-dimensional periodic structure , namely a photonic crystal111213161920222322162222We here introduce an internal holographic modulation of the lattice arrangement in a PCSEL to obtain freely configurable two-dimensional beam patterns. We employ a phase-modulating resonator in which the in-plane position of air holes is slightly shifted from a square lattice in a holographic manner. The arrangement is thus no longer periodic and the modulation must be imposed carefully so as not to compromise lasing. This novel on-chip-sized device, realising both lasing and phase modulation, is henceforth named an “integrable phase-modulating (iPM) laser”. We demonstrate that the iPM laser, which is small enough for integration, can emit characters, line drawings, multi-spot beams, and even grayscale images.a = 282 nm (see Methods - “Device structure”). There is no air hole at the lattice point of the square lattice (point O). Instead, the centre (point C) of each air hole is on the perimeter of a circle of radius d = 0.06a or 0.10a (see Methods - “Device structure”) centred on O. The rotation angle ϕ is chosen according to the designed phase distribution, described in Methods - “Phase distribution of phase-modulating resonator”. This type of modulation is similar to the “detour-phase” methodWe now focus on the main results. d and rotation angle ϕ. Therefore, the diffracted light wave propagating in the surface-normal direction has a phase that deviates locally at each point, eventually forming a phase distribution. As a result, a finely modulated wave front is emitted and arbitrary beam patterns can be obtained. As the resonance mechanism is similar to that of PCSELs, the operating wavelength of iPM lasers can be scaleda and by choosing an appropriate active medium, the emission wavelength can be controlled. Furthermore, devices for different wavelengths can be integrated into a single module.As demonstrated above, iPM lasers can successfully emit arbitrarily configured light patterns, implying that phase modulation is fully achieved in our devices. We now turn to the lasing mechanism. 29In conclusion, we have proposed a class of new devices referred to as “iPM lasers”, which comprise a phase-modulating light source, suitable for on-chip integration. These devices involve a structure with concurrent phase-modulation and lasing functionality. We demonstrated the operation of semiconductor iPM lasers that emit characters, line drawings, multi-spot beams, and even grayscale images. These results suggest that iPM lasers can emit arbitrary beam patterns. We also discussed the working mechanism of the devices. We believe that iPM lasers will open a new field of applications which ranging from the small scale integration such as switchable illumination patterns for wearable modulated imaginga was 282 nm for the devices used in d. The smaller d, the lower the threshold current. On the other hand, large values of d diminish the zero-order light  and a beam profiler . Each image was acquired by averaging 8 times. The measurements were carried out under pulsed operation to suppress thermal effects at room temperature (25 °C). We used a pulsed current source  and a current probe . The pulse width and frequency were 50 ns and 10 kHz, respectively. The driving current was 4 A. To enhance visibility, the central zero-order spot was saturated.The output power was measured using a photodiode  and measuring software  with no ND filter. Starlab parameter settings for the wavelength, range, and averaging interval were 930 nm, “auto”, and 1 s, respectively. The measurements were carried out under pulsed operation at room temperature (25 °C). We used a pulsed current source  and a current probe . The pulse width and frequency were 50 ns and 10 kHz, respectively. The distance between the device and the photodiode was approximately 3 mm for “Total” and 150 mm for “Zero-order”, respectively. The output power of “Pattern (each)” was calculated astotal and Pzero are the measured output powers for “Total” and “Zero-order”, respectively. We calculated Ppattern because the size of the beam pattern at a distance of 150 mm exceeds the measurement area of the photodiode and cannot be measured directly.where PThe lasing spectra were measured using an optical spectrum analyser  through a single-mode fibre. The resolution, sensitivity, and averaging time were 0.02 nm, “High 2”, and 1, respectively. The measurements were carried out under pulsed operation at room temperature (25 °C). We used a pulsed current source  and a current probe . The pulse width and frequency were 50 ns and 10 kHz, respectively. The driving current was 4 A.θtilt and the azimuthal angle θrot were transformed into the normalised wavevector  in reciprocal space as follows:The measurement setup consisted of a multi-channel spectrometer , multi-mode optical fibre, lens, and rotating arm see . Emissiokx and ky are the wavevectors parallel to the Γ-X and Γ-Y directions, a is the lattice constant, and λ is the wavelength in air. In kx and the vertical axis corresponds to ky. In kx in the Γ-X direction and where kx, ky, λ). Then, a horizontal cross section through this array at a desired wavelength yields the angular dependence of the device. The colour scale indicates the light intensity.To obtain the angular dependence of the device at the lasing wavelength , the meaWe refer to the method of computer-generated hologramsHow to cite this article: Kurosaka, Y. et al. Phase-modulating lasers toward on-chip integration. Sci. Rep.6, 30138; doi: 10.1038/srep30138 (2016).


In other models, the resultant out-degree distribution decays as a poisson or an exponential distribution. However, it has been found that in real complex networks, the out-degree distribution decays as a power-law. In order to obtain out-degree distribution with power-law behavior some models have been proposed. This work introduces a new model that allows to obtain out-degree distributions that decay as a power-law with an exponent in the range from 0 to 1.Many growth models have been published to model the behavior of real complex networks. These models are able to reproduce several of the topological properties of such networks. However, in most of these growth models, the number of outgoing links ( CN) that reproduce some topological properties of real systemsi.e., their out-degree is a constant), as in the model proposed by Barabási-Albertet.alCN. For example, in metabolic networksWWWet.al.et.al.i.e. where the average out-degree of nodes is infinite when N → ∞.In the literature, there are many growth models for complex networks , we solve s = 1, s = 2, and so on as follows:where Q1(N), we need to consider the initial condition•  for n0 with no outgoing links, that is N = 1. For this case the quantity Q1(1) of nodes with out-degree s = 1 is zero (Q1(1) = 0). When the node n1 is added (N = 2), the probability for node n1 to have out-degree s = 1 is Q2(N), we need to consider the initial condition•  for n2, only nodes n0 and n1 are in the network (N = 2) and any of them has s ≥ 2, therefore Q2(2) = 0. When node n2 is added (N = 3), the probability that node n2 has out-degree s = 2 is From the previous results in Normalizing outγ of the out-degree distribution obtained with the proposed model is only determined by the probability p. That is, the out-degree distribution obtained decays as a power-lawoutγ = p.with exponent p (second rule of the proposed model), the average out-degree of the nodes grows with the network size. To validate this hypothesis, we analytically calculate the average out-degree On the other hand, we can deduce that as a consequence of the random out-degree selection by new nodes with probability 1 − N of nodes in the network. On the right-hand side of N by a new node. Thus, the term that describes the increment of the average out-degree Solving N ≫ 1), we can approximate As the total number of nodes in the network increases (N → ∞.From p = 0.1, p = 0.3, p = 0.6, and p = 0.9. In each simulation, we considered the growth of a directed network from 1 to 104 nodes. N for any value of 0 < p < 1 as stated by N → ∞. This contrasts with some large networks that are sparse where the number of edges is much smaller than the maximum possible and the average out-degree increases slowly as the network growsIn order to validate the analytical solutions for the out-degree distribution  and averCN seems to be the result of a set of local processes. We consider that the proposed model in this work can contribute to develop new growth models for directed CN which consider local processes that shape the out-degree of the nodes and, therefore, produce better predictions of the behavior of real CN and thus increases the understanding of these systems.The topological properties of real J.E.G., E.S.N., U.P.N. and J.A.E. designed, performed the research and wrote the manuscript.


However, such structures are rarely known; moreover, predicting them computationally is no rescue, since single sequence structure prediction is highly unreliable.Identifying sequence-structure motifs common to two RNAs can speed up the comparison of structural RNAs substantially. The core algorithm of the existent approach ExpaRNA solves this problem for The novel algorithm ExpaRNA-P computes exactly matching sequence-structure motifs in entire Boltzmann-distributed structure ensembles of two RNAs; thereby we match and fold RNAs simultaneously, analogous to the well-known “simultaneous alignment and folding” of RNAs. While this implies much higher flexibility compared to ExpaRNA, ExpaRNA-P has the same very low complexity (quadratic in time and space), which is enabled by its novel structure ensemble-based sparsification. Furthermore, we devise a generalized chaining algorithm to compute compatible subsets of ExpaRNA-P’s sequence-structure motifs. Resulting in the very fast RNA alignment approach ExpLoc-P, we utilize the best chain as anchor constraints for the sequence-structure alignment tool LocARNA. ExpLoc-P is benchmarked in several variants and versus state-of-the-art approaches. In particular, we formally introduce and evaluate strict and relaxed variants of the problem; the latter makes the approach sensitive to compensatory mutations. Across a benchmark set of typical non-coding RNAs, ExpLoc-P has similar accuracy to LocARNA but is four times faster (in both variants), while it achieves a speed-up over 30-fold for the longest benchmark sequences (≈400nt). Finally, different ExpLoc-P variants enable tailoring of the method to specific application scenarios. ExpaRNA-P and ExpLoc-P are distributed as part of the LocARNA package. The source code is freely available at http://www.bioinf.uni-freiburg.de/Software/ExpaRNA-P.ExpaRNA-P’s novel ensemble-based sparsification reduces its complexity to quadratic time and space. Thereby, ExpaRNA-P significantly speeds up sequence-structure alignment while maintaining the alignment quality. Different ExpaRNA-P variants support a wide range of applications.The online version of this article (doi:10.1186/s12859-014-0404-0) contains supplementary material, which is available to authorized users. Genome-wide high-throughput transcriptomics has revealed evidence for massive transcription of eukaryotic genomes, vastly exceeding translation to proteins -3. Ultimde novo RNA-gene finders qrna  denote the probability of the structure S in the Boltzmann ensemble of A  denotes the base pair probability of , which is defined as i,j)|A] is the probability that a random structure S, drawn from the Boltzmann ensemble of A, contains the base pair .For a sequence ble of A . Pr, Pr[∧|A] denotes the joint probability of the stacked base pairs  and . Such probabilities are computed in slight extension of McCaskill’s algorithm  denotes for i<k<j the joint probability that a structure of X contains the base pair and the unpaired base k such that  is the parent of k.Pr  denotes for i<i′<j′<j the joint probability that a structure of X contains the base pairs and  such that  is the parent of .Pr ≥θ1 and Pr[|B]≥θ1Pr[|A]≥θ2Pr[k∈loop|B]≥θ2,and Pr≥θ3Pr[∈loop|B]≥θ3,and Pr by McCaskill’s algorithm [k ∈ loop|X] and Pr[ ∈ loop|X] for X∈{A,B}. For this purpose, we introduce – on top of the McCaskill matrices – the auxiliary matrix Fundamentally, our novel sparsification technique relies on the joint probabilities of Def. 4. For sequences lgorithm . In thismentclass2pt{minimImportantly, all these probabilities are efficiently precomputed independently for each sequence. Hence, e.g. in clustering scenarios, where all pairs from a set of sequences need to be matched, this preprocessing needs to be done only once for each sequence and not for all quadratically many pairs.Figure Fundamental to our approach, all matrices and evaluations in the recursions are sparse, i.e. only entries and cases are considered where the probabilities of elements pass the respective probability thresholds cf. Def. 5). Corresponding constraints are given in the recursion equations – this is also illustrated in Figure . CorrespD score the best EPM enclosed by each base pair match ij ∼ kl, i.e. D denotes the best score of a significant EPM Ai..j and Bk..l with The matrix entries ij ∼ kl, we determine the (score of the) best or forms a (connected) significant EPM only together with the closing base pair match ij ∼ kl. The first case is covered by the single matrix L, whereas the latter case requires three matrices GA, GAB, and LR. By and large, for deriving one D-entry one starts matching from the left using L. Potentially, one introduces a gap using matrices GA and GAB and continues using matrix LR to match the part that is only connected to the right end of ij ∼ kl.Inside of the base pair match i ∼ k of the base pair match, i.e.  is empty or contains i+1 ∼ k+1. Concretely, Lijkl is such a score, where GA and GAB. In the case of j′ but matches l′; for l′ and potentially does not match j′. Finally, LRijkl is the best sum of scores of two significant EPMs i ∼ k and the second contains j′ ∼ l′. Intuitively, the two EPMs are separated by a gap; formally:  or .In more detail, first we determine the best score of a significant EPM L, we check whether there is a sequence match (second case) or a structure match (third case); otherwise, we assign −∞ (first case). LR is analogous to L, only allowing to close a gap left of the structure or sequence match. For this purpose, we introduce an auxiliary matrix H, which does not need to be stored. The gap itself, computed in GA and GAB, allows skipping an arbitrary number of positions in both sequences. The recursion structure ensures that such a gap is introduced at most once per loop match and sequence. To avoid ambiguity, the recursion enforces to first skip positions in A (using GA) and after that positions in B (using GAB); furthermore we enforce a gap in the matchings computed via LR by its initialization.Our recursion equations Figure  show theD in increasing order with respect to their size so that when computing some D, any D with i<i′<j′<j and k<k′<l′<l is already computed.We compute entries of D), we finally compute the matrix F. The entries F, for 0≤j′≤n and 0≤l′≤m, denote the maximum score of a significant EPM of j′,l′), i.e. with F is almost identical to the recursion for L, except for the first case, which is 0 instead of −∞, since the EPMs in F can start at any point . Also, since the matched base pairs in EPMs of F are external (i.e. they are not enclosed by some other base pair of the EPM), we do not perform checks for the second and third condition of significant EPMs (Def. 5).Since EPMs are not necessarily closed by a base pair match . LRijkl=−∞ (first matrix entry)l′>k (first matrix row)j′>i (first matrix column)LR matrix with −∞, we keep matchings represented by LR and L distinct .By initializing the F is initialized by F=F=0 for all j′,l′.The final matrix GA and GAB matrices.) For strict EPMs this is decided independently of the other traced strict EPMs. It suffices to check whether the strict EPM can be extended into the gap matrices, i.e. whether a sequence or structure match is possible at the borders of the gap matrices.For enumerating only maximal EPMs during suboptimal traceback, we take special care that EPMs cannot be extended at the left or right end of gaps  that satisfy To enumerate all maximal EPMs, we start such tracebacks only from entries  is a maximal strict EPM of A and B, iff Let  is not maximal for A and B . “ ⇐”: Let “ ⇒”: Let  is not maximal for A and B. Then, there is a strict EPM Assume i≤j′<j and k≤l′<l. However in this case, while clearly the parent of j′ ∼ l′ in  is ψ, there is a parent of j′ ∼ l′ in ψ . This contradicts  and Consequently, to satisfy By the same argument, the forward direction holds for relaxed EPMs. Therefore, we enumerate all maximal relaxed EPMss by restricting the traceback in the same way. However, since the backward direction does not hold generally, this procedure can enumerate non-maximal relaxed EPMs. In practice, we observe this very rarely; consequently, while redundant relaxed EPMs could be removed explicitly, we let the chaining procedure handle those EPMs.θ1,θ2, and θ3. Consequently, we compute all DP matrices in only O(n2) time and space. We compute matrices LRijkl only for base pairs  and  that are significant |A]≥θ1 and Pr[|B]≥θ1). Furthermore, we compute only relevant entries of these matrices.ExpaRNA-P’s efficiency depends fundamentally on the sparsity of the DP matrices, which we leverage through fixed thresholds j′ is a candidate of  in sequence A if it is either a significant single-stranded position within , i.e. Pr[j′ ∈ loop|A]≥θ2, or contained in a significant helix of , i.e. Pr[ ∈ loop|A]≥θ3 for some i′. Analogously, we define candidates l′ of  in sequence B. |B]≥θ2 or Pr[ ∈ loop|B]≥θ3 for some k′).This is best illustrated by the notion of candidates; each O(n2) entries LRijkl such that j′ is a candidate of  and l′ is a candidate of . Consequently, ExpaRNA-P has quadratic time and space complexity.There are only Proof sketch: By definition, only candidates j′ or l′ can be part of a significant EPM as defined in Def. 5; otherwise, we assign −∞ to Lijkl and LRijkl. Furthermore, in the latter case, we neither store nor compute the values for θ1 reduces the number of base pairs to a constant number of base pairs per sequence position; in total, quadratically many base pairs pass the filter. The thresholds on joint probabilities guarantee that each sequence position is candidate of only constantly many base pairs. In consequence, each position is considered only a constant number of times during the entire computation; this directly results in quadratic time complexity.  matrix for each hole and takes O(Hn2) time, where H is total number of holes with H≪n2. In contrast to ExpaRNA, there may exist more than one EPM ending at each sequence position pair, i.e. there is no one-to-one correspondence between EPMs and EPM’s end positions. This is why each matrix requires additional steps in the order of the number of input EPMs E in ExpaRNA-P’s chaining; the complexity of the generalized chaining algorithm is O(H·(n2+E)). Since in the most general case, when we enumerate all suboptimal EPMs up to a maximal difference to the optimal score, E∈O(n2) is not guaranteed, we implement in addition several ways to control the number of EPMs. For example, ExpaRNA-P allows setting an ad hoc limit on this number. Furthermore, we suggest a heuristic strategy: for each sequence position pair, keep only the best EPM ending there. Consequently, typical use cases of ExpaRNA-P maintain the chaining complexity of ExpaRNA, i.e. O(Hn2).Chaining selects a non-crossing and non-overlapping subset of EPMs. Our algorithm generalizes the chaining of ExpaRNA . The chai ∼ k, considers only the optimal EPM ending at that match. Our tool supports two ways to control the EPM enumeration by the suboptimal traceback: either by defining the maximum score difference to the optimal score or the maximum number of EPMs.We implemented ExpaRNA-P and the chaining algorithm in C++. In particular, we implemented two versions of the traceback: the suboptimal traceback and a heuristic version that, for each match In order to assess the performance of ExpaRNA-P, we designed the following pipeline: In a first step we compute the significant EPMs with ExpaRNA-P and use the chaining algorithm to extract from these EPMs an optimal non-overlapping and non-crossing subset. Then we compute a sequence structure alignment that includes all matches of the chained EPMs. For this purpose, we utilize the EPMs as anchor constraints for LocARNA. Consequently, LocARNA runs much faster, since each anchor reduces the alignment space. In correspondence with the analogous idea ExpLoc , which uWe performed all benchmarks over the pairwise alignment instances of the BRAliBase 2.1 benchmark set ,45. To mheuristic traceback with exact matchesheuristic traceback with inexact structure matchessuboptimal traceback with exact matchessuboptimal traceback with inexact structure matchessuboptimal traceback with inexact structure matches and the additional second filter stepWe study five ExpLoc-P variants, where we generate anchor constraints respectively by θ1=θ2=θ3=0.01 to predict less false positives. Furthermore, we enumerated EPMs that have a score of at least 90 and fix the maximal number of traced EPMs in the suboptimal traceback to 100. The scoring – as defined in Eq. str_mm to −10 for structure mismatches in inexact modes. Furthermore we set α1=1,α2=5 and α3=5 in order to favor structured regions. In addition to SPS and runtime, we computed the coverage for each benchmark instance – consisting of sequences A and B. For this purpose, we define coverage as the fraction of nucleotides that are matched by the best chain of EPMs In particular, we compare exact modes 1,3), which follow the strict EPM definition, and inexact modes 2,4,5), which allow mismatches at structure positions (relaxed EPMs.) The score parameters were selected ad-hoc without parameter learning; in particular, we set the cutoff probabilities to restrictive values ,3, which,4,5, whiFigure In Table We benchmarked three existing approaches: LocARNA, ExpLoc , and RAFTable The quality drop of RAF alignments at low sequence identities is strongly reminiscent of pure sequence alignment methods. Thus, we conjecture that the specific use of sequence-based heuristics by RAF, while guaranteeing sequence alignment like run-time behavior, compromises RAF’s use for ‘hard’ RNA alignment instances that require structure-based alignment methods.Furthermore, we investigated the dependency of the lengths of the input sequences on the speedup  that – without being more efficient – are much less flexible, since they require a priori known or unreliably predicted structure.Due to its novel ensemble-based sparsification, the algorithm ExpaRNA-P has only a very low (quadratic) time and space complexity, equalling sequence alignment. This sparsification technique is particularly relevant, since similar techniques can likely be applied to other RNA analysis methods.We have developed two major variants of this method; one requires strict matches in all positions of an EPM (strict EPMs), the other relaxes this  to allow mismatches at structural positions. The latter supports compensatory mutations, which are highly relevant in RNA structure analysis in general.Our benchmarks study EPMs as anchor constraints to speed up RNA structure alignments . EPMs from structure ensembles have turned out to be substantially more reliable than EPMs from fixed structures. At comparable speed ups, this results in increased quality. Most importantly, the novel approach keeps up the alignment quality even for sequences of low identity, which is ultimately decisive for structure alignment. In striking contrast, the alignment quality of the similarly fast alignment tool RAF breaks down – very much like pure sequence alignment.We have implemented rigorous suboptimal traceback, which provides extensive control of the set of enumerated EPMs. For example, this level of control is required in the analysis of structural variants common to the RNAs. In addition, we have developed a heuristic traceback, which performs almost indistinguishable in our benchmark. Being much faster than the rigorous method, it offers the best speed-quality balance in such settings.Finally, we conjecture that EPM-based anchor constraints can be combined advantageously with other RNA alignment tools such as RAF. While for LocARNA the constraints yield a considerable speedup, the combination with RAF has the potential to improve RAF’s poor alignment quality for low sequence similarity.


Er) are known inevitable. In this mechanism, whether the first derivative of Er (shear) or the second derivative of Er (curvature) works most is decisive in determining the class of nontrivial solutions (which describe the barrier structure). Here we report the experimental identification of the essential role of the Er-curvature on the ETB formation, for the first time, based on the high-spatiotemporal resolution spectroscopic measurement. We found the decisive importance of Er-curvature on ETB formation during ELM-free phase, but there is only a low correlation with the Er-shear value at the peak of normalized ion temperature gradient. Furthermore, in the ELMing phase, the effect of curvature is also quantified in terms of the relationship between pedestal width and thickness of the layer of inhomogeneous Er. This is the fundamental basis to understand the structure of transport barriers in fusion plasmas.The turbulent structure formation, where strongly-inhomogeneous turbulence and global electromagnetic fields are self-organized, is a fundamental mechanism that governs the evolution of high-temperature plasmas in the universe and laboratory  of the H-mode in the toroidal plasmas). The roles of inhomogeneities of radial electric field ( Er, should play an important role in the mechanism of the L-H transition347910Er and significant reduction in a plasma turbulence level. It is conventionally believed that the shear of E×B drift velocity (i.e. first derivative of Er) is the main parameter that is responsible for suppression of turbulence5121314i.e. second derivative of Er) in suppressing the turbulence and turbulent transport has also been pointed out16171819Er-shear and curvature effects on the ETB formation, simultaneously, due to lack of diagnostic that can withstand up to the second derivative estimation. Whether the first derivative or the second derivative works most in turbulence suppression is decisive in determining of class of nontrivial solutions  by the drift wave (DW) turbulence  the enhanced dissipation of DW fluctuations and (2) the convergence of DW energy into coherent axial vector fields, respectively. It is also important for understanding the formation of transport barriers in toroidal confinement devices. For instance, the “curvature transition”, a kind of the internal transport barriersEr curvature. The trapping of turbulence intensity in the trough of Er 25Er on the ETBs formation, for the first time, based on the high-spatiotemporal resolution spectroscopic measurement2728Er-shear and -curvature) effects on ETBs formation. It is shown that both the shear and curvature of Er work, but the role of curvature is inevitable in the formation of transport barrier. This discovery has a fundamental impact on our understanding of structure formation in magnetized plasmas, in addition to a substantial impact on our predictability of performance of future burning plasma experiments, since the over-all performance of toroidal plasma fusion device critically depends on the property of the ETB. For instance, the uncertainty in the fusion power amplification factor of ITER was estimated as about factor 10, and the main origin for which is the variance in the prediction of the width of the ETBSince the discovery in ASDEX tokamakThe model of turbulence intensity, in the presence of inhomogeneous radial electric field, was assessed based on the model formula for the effect of electric field shear51231where the parameters are as follows:I is the turbulence intensity , ρi is the ion gyro-radius, k is a typical wavenumber for the plasma turbulence, Vd is the diamagnetic velocity (Vd ≡ T/eaB), e: elementary charge, a: characteristic mean-scalelength, B: total magnetic field), Bθ), and I0 is the mean intensity in the limit of Z = 0. The term Z1 and Z2 denote the effects of Er-shear  as Z2 has its sign dependence.Er, the temperature gradient is enhanced. If one employs the gyroBohm dependence of local thermal diffusivity, which is proportional to the temperature gradient and turbulent intensity, the heat flux is proportional to the turbulence intensity multiplied by (∇T)2. For fixed heat flux, one can evaluate the enhancement degree of the ion temperature gradient in terms of Z value as;In association with the turbulence reduction by the inhomogeneous kρi ~ 0.1 – 0.3 1, Z2, and Z are investigated. When the turbulent transport is reduced to the level of neoclassical transport, further reduction of turbulence level is less effective in increasing the ion temperature gradient.Here, the Er on the ETBs formation, Er in the transport barrier, so that their role on ETBs formation can be analyzed separately. This discharge shows a long period of secular increase in the edge ion temperature gradient after a “soft” L-H transition having a longer time-scale (a few hundred milliseconds) in comparison with a conventional “hard” one having shorter time-scale (e.g. a few milliseconds or less) seen in many tokamaks. This is followed by a violent phase, where the edge plasma bifurcates between a strong-Er and a weak-Er state , until it finally settles to the conventional H-mode. The time period of 0 < ΔtL-H < 0.3 s in Er.In examining the effect of non-uniformity of L-H < 0.3 s), we found that theZ is from Z2, and Z1 remains small during the growth of the temperature gradient. An important caveat to this discussion is that there is only a low correlation between Z1 and r ≡ r0 as illustrated in r = r0 becomes large as the Z2 value at r = r0 increases. This shows the relative importance of nonzero second derivative effect r = r0 . It should be noted that the location at which the r = r0) is closely related to that at which the Er (and/or its curvature) has a local peak value as shown in Er-shear has almost zero value. Indeed, this fact is confirmed by a quantitative comparison for various pedestal structures obtained in JT-60U H-mode plasmas with different angular momentum injections (e.g. co- and counter-NBI plus perpendicular-NBI heating)Looking at the slow L-H transition phase (0 < Δtn = 0) seems to be associated with the multi-stage Er-transitions during ELM-free H-phase as shown in fGAM ≈ Cs/2πR) is likely to that of the predicted frequency for Geodesic Acoustic Mode (GAM) known as a high-frequency blanch of Zonal flow (ZF), this observation can also support the hypothesis of ZF (and/or Er-curvature) suppression of turbulence even in the H-mode plasmas.It is interesting that a uniform toroidal MHD oscillation (i.e. ΔtL-H = 0.0–0.32 s) there is a good correlation between the r = r0. The enhancement of the gradient (by the factor of a few times) appears in the range of kρi ~ 0.1–0.3.The quantitative comparison between the temperature gradient and inhomogeneity effect of electric field is illustrated in Z shows a more complex (i.e. non-linear) behaviour after  ΔtL-H = 0.32 s, where ΔtL-H = 0.32–0.36 s, a fast (within milliseconds) and strong drop in ΔtL-H = 0.36 s, the Er structure from strong-Er to weak-Er are visible. These changes are similar to those at ΔtL-H = 0.42 s, but the changes are in opposite direction. However, this does not seem to be the same as the H-L backward transition commonly associated with the L-mode but, rather, this phase (ΔtL-H = 0.36–0.42 s) seems to be similar to the phase ΔtL-H = 0.0–0.32 s. After ΔtL-H = 0.42 s, the plasma changes again, with characteristics similar to the phase ΔtL-H = 0.32–0.36 s. This phase continues until the first ELM onset at ΔtL-H = 0.49 s.The relationship between  be seen , while tr = r0 is likely to be non-linear, especially for the later ELM-free H-phase (during ΔtL-H = 0.32–0.36 s and ΔtL-H = 0.42–0.49 s), requires further studies: Let us explain a hypothesis to understand this stimulating observation. Before the jump of electric field occurs, the suppression of turbulence is strong enough so that the ion thermal transport is reduced to the level of the neoclassical transport. The abrupt bifurcations in the edge Er to larger values , suppress the turbulence as was shown inr = r0 seems to be almost linear for the slow L-H transition phase having weak Er (ΔtL-H = 0.0–0.32 s), including another phase having weak Er (ΔtL-H = 0.36–0.42 s), as expected from The question why the dependence of Er layer. These data are determined from multiple and reproducible Edge Localized Modes (ELMs) cycles for co- and counter-NBI discharges, mapping them onto a single-time basis, as defined by the time of the measurement relative to the ELMs for improved statistics to assess the temporal behavior (in particular inter-ELM phase) of the measurementsExperimental data presented in R-RSEP ≥ −0.08 m for co-NBI case, while it becomes narrower up to R-RSEP ≥ −0.06 m for counter-NBI case. Looking at 1 layer (i.e. impact of the R-RSEP ≥ −0.07 m (co-NBI) and −0.05 m (counter-NBI), respectively, indicating about 1 cm narrower than that of the pedestal width. Furthermore, there is Z2 layer of the negative value (contributing turbulence enhancement) around R-RSEP ≈ −0.04 m in the counter-NBI case.As shown in Er (and/or its curvature) had a local peak values for both co- and counter-NBI cases as shown in solid line on Er-shear had a local peak value shift inwards from the radius where the It should be noted that we found a strong correlation between the location at which the r = r0 defined for discharge E049219) shows a similar nature to that obtained even in the ELMing phase (at the time of ELM-onset) for various locations (discharges E049228 and E049229).As a result, we found the Z-profile had almost non-zero value in the pedestal region  contains both Er-shear and Er-curvature effects, while the Er-curvature effect is dominant contributor for Z value in the former case .This result is not self-evident, since the e.g.e.g. around the pedestal-top) and H-mode pedestal  results in an up-shift of the curve in It is also noted that the improved energy confinement region  in forming the ETBs. Recalling that the non-uniform Er effect presented in this letter should have 2nd order effect of O(Er2), the Z profile is localized strongly around the location, at which the In summary, we revisited the studies of paradigm of shear suppression of turbulence as the mechanism for the ETBs formation with an improved diagnostic for the edge radial electric field from the high resolution measurement on JT-60U tokamak, examining the effects of both shear and curvature, comparing with nd order effect, leading a new prediction of the pedestal width scaling . It should be noted that the thickness in the E × Bshear layer was roughly scaled by a few times of the Indeed, this effect may be able to exude toward the pedestal top (and/or bottom) due to its 2P = 3–3.5 m, the plasma minor radius, aP = 0.6–1.1 m and the maximum toroidal magnetic field, BT ≤ 4 T at R = 3.32 m. We performed the NBI heating experiments by comparing the external momentum input directions between co-, balanced- and counter-NBI cases under a matched plasma shape condition. The plasma current, IP, was 1.6 MA, and the toroidal magnetic field, BT, was 3.9 T. The corresponding safety factor at the 95% flux surfaces, q95, was thus 4.2. The elongation, κ, and triangularity, δ, were 1.47 and 0.36, respectively, and the total plasma volume was 57 m3.The JT-60U tokamak is a single null divertor tokamak device having the plasma major radius, Rr structure at the pedestal region, we measured the pressure gradient, and plasma velocity perpendicular to the magnetic field, and the Er was evaluated by the radial force balance equation.In the study of the JT-60U tokamak for the fiscal year (FY) 2007–2008 experimental campaign, we measured the radial profiles for the density, temperature, and poloidal/toroidal plasma flows of fully stripped carbon impurity ions by means of the Charge eXchange Recombination Spectroscopy (CXRS) diagnostic method with fast time resolution (up to 400 Hz) at 59 spatial points . With regard for determining the EHow to cite this article: Kamiya, K. et al. Experimental validation of non-uniformity effect of the radial electric field on the edge transport barrier formation in JT-60U H-mode plasmas. Sci. Rep.6, 30585; doi: 10.1038/srep30585 (2016).


With the prevalence of Social Networks (SNs) and services, plenty of trust models for Trustworthy Service Recommendation (TSR) in Service-oriented SNs (S-SNs) have been proposed. The reputation-based schemes usually do not contain user preferences and are vulnerable to unfair rating attacks. Meanwhile, the local trust-based schemes generally have low reliability or even fail to work when the trust path is too long or does not exist. Thus it is beneficial to integrate them for TSR in S-SNs. This work improves the state-of-the-art Combining Global and Local Trust (CGLT) scheme and proposes a novel Integrating Reputation and Local Trust (IRLT) model which mainly includes four modules, namely Service Recommendation Interface (SRI) module, Local Trust-based Trust Evaluation (LTTE) module, Reputation-based Trust Evaluation (RTE) module and Aggregation Trust Evaluation (ATE) module. Besides, a synthetic S-SN based on the famous Advogato dataset is deployed and the well-known Discount Cumulative Gain (DCG) metric is employed to measure the service recommendation performance of our IRLT model with comparing to that of the excellent CGLT model. The results illustrate that our IRLT model is slightly superior to the CGLT model in honest environment and significantly outperforms the CGLT model in terms of the robustness against unfair rating attacks. Moreover, the integration of SNs and services has become a trend and formed a new research area known as S-SN . A larger number means a stronger trust relationship and vice versa.TR: The set of trust values among service consumers, i.e. TR = {TR}, where 1 ≤ i, i′ ≤ M and i ≠ i′.TS(i): The set of trustworthy service witnesses in the view of SC(i), i.e. TS(i) = {SC(i′)}, where 1 ≤ i, i′ ≤ M and service witness SC(i′) is trustworthy from the perspective of SC(i).SS(j): The service witness set of SR(j), i.e. SS(j) = {SC(i′)}, where 1 ≤ i′ ≤ M, 1 ≤ j ≤ N and SC(i′) has consumed and rated for SR(j).FS(j): The fair service witness set of SR(j), i.e. FS(j) = {SC(i′)} ⊂ SS(j), where 1 ≤ i′ ≤ M, 1 ≤ j ≤ N and SC(i′) has consumed and given a fair rating to SR(j).RT: The rating value of SC(i) to SR(j) on service aspect a, where 1 ≤ i ≤ M, 1 ≤ j ≤ N and 1 ≤ a ≤ A. It is represented as an integer between 0 and 4, indicating very dissatisfied, dissatisfied, general, satisfied and very satisfied, respectively.RT: The rating vector of SC(i) to SR(j), i.e. RT = , RT, …, RT), where 1 ≤ i ≤ M and 1 ≤ j ≤ N.RT: The set of rating vectors, i.e. RT = {RT}, where 1 ≤ i ≤ M and 1 ≤ j ≤ N.PR: The preference weight value of SC(i) on service aspect a, where 1 ≤ i ≤ M, 1 ≤ a ≤ A and PR(i): The preference weight vector of SC(i), i.e. PR(i) = , PR, …, PR), where 1 ≤ i ≤ M.PR: The set of preference weight vectors, i.e. PR = {PR(i)}, where 1 ≤ i ≤ M.RS: The synthetical rating score of SC(i) to SR(j), i.e. i ≤ M and 1 ≤ j ≤ N.SN: The social network which is composed of all the service consumers and the trust relationships among them, i.e. SN = .BN: The bipartite network which is made up of all the service consumers and services as well as the service ratings and preference weights, i.e. BN = .For illustration purposes, we give the following formal representations of elements in S-SNs:SN and BN. To balance computational complexity and evaluation performance, the trust ) and service rating ) are represented as different forms. Specifically, the former is denoted as a real number in order to reduce the computational complexity of LTTE module, while the latter is denoted as a vector including the rating values on various service aspects with different user preference weights so as to improve the evaluation performance of RTE module.S-SN can be viewed as a combination of As we mentioned earlier, this module mainly takes charge of selecting out all the candidate services which satisfy the functional requirements of the active service requester. To this end, we propose a multi-QoS based filtering method and a simple example is shown in SC(1) submits a request which contains its multi-QoS requirements , every service in SR = {SR(1), SR(2), …, SR(5)} is checked whether meets the multi-QoS needs of SC(1) or not. We can easily find that SR(1), SR(2) and SR(3) do not satisfy the “Function”, “Price” and “Response time” requirements of SC(1), respectively, so they are filtered out. While both SR(4) and SR(5) meet the multi-QoS needs of SC(1), thus they are candidate services and are then sent to RTE and LTTE modules for trust evaluations.When When evaluating the local trust value of certain candidate service, only the service ratings from trustworthy service witnesses are considered, while the opinions of the other service witnesses are ignored. For the purpose of identifying all the trustworthy service witnesses, we propose an efficient MLT algorithm (i.e. Algorithm 1) to obtain the maximum local trust values of all the service witnesses.H) is considered untrustworthy. Suppose SC(p0)→SC(p1)→…→SC(pt) is the optimal trust path from SC(i) to SC(i′) (where SC(p0) = SC(i) and SC(pt) = SC(i′)), then the maximum local trust value MT (i.e. MT[i′] in Algorithm 1) of SC(i′) in the perspective of SC(i) can be derived from  (e.g. 0.1). We can find that both MT and LT range from 0 to 1. From Algorithm 1, Eqs , H, SC(i), α; /* Social network, maximum allowable hop, active service requester and trust decay parameter, respectively. */Output: MT; /* Maximum local trust values of all the service consumers in SC from the perspective of SC(i). */VS ⇐ ⌀; /* The visited service consumer set which is initialized to an empty set. */1: MT, HP; /* The arrays of maximum local trust values and corresponding hops from SC(i), respectively. */2: MT[i] ⇐ 1;3: HP[i] ⇐ 0;4: for each SC(i′) ∈ SC − {SC(i)} do5: ifTR > 0 then6:  MT[i′] ⇐ TR;7:   HP[i′] ⇐ 1;8:   else9:  MT[i′] ⇐ 0;10:   HP[i′] ⇐ ∞;11:   end if12:  end for13: SC(i) to VS;14: Add while |VS| < |SC| do15: SC(i′) ⇐ Find the service consumer with the maximum local trust value in SC − VS;16:  ifHP[i′] < Hthen17:  for each SC(i′′) ∈ SC − VS − {SC(i′)} do18:   ifTR > 0 and then19:    20:     HP[i′′] ⇐ HP[i′] + 1;21:     end if22:    end for23:   end if24:  SC(i′) to VS;25:  Add end while26: returnMT;27: www.tmall.com) for example, it is a well-known e-commerce website and the service rating consists of five grades, namely very dissatisfied, dissatisfied, general, satisfied and very satisfied, which are denoted as 1 ∼ 5, respectively. It also considers various service aspects, namely service performance, service attitude, delivery speed and logistics speed. Though it is intuitive and beneficial to potential service consumers, it is vulnerable to unfair rating attacks and neither user preference nor rating number is taken into account when calculating the reputation value of certain service.Next, we present how to calculate the reputation values of candidate services. In many reputation systems of existing e-commerce websites, the reputation value of centain service is merely a simple average of all the rating values given to it. Taking Tmall  and local trust value LT exceeds system threshold λ (e.g. 0.4), then RS is filtered out as it is considered unfair (i.e. service witness SC(i′)∉FS(j)) and vise versa. Furthermore, we also take the following two weights into consideration:SC(i), the rating score RS from service witness SC(i′) is more credible if SC(i) and SC(i′) have more similar preferences and vice versa. Thus we define the user preference weight GP based on the weighted Euclidean distance between PR(i) and PR(i′)  (e.g. 0.1). We can easily find that GP, GN(j) and RP range from 0 to 1 due to the normalization processing. The above calculation process (i.e. Eqs (Where i.e. Eqs  ∼ 5)) i) iν is aFT of SR(j) in the sight of SC(i) can be gained fromAs mentioned earlier, this module is mainly in charge of integrating the evaluation results of LTTE and RTE modules by weighted summation and obtaining the final trust values of all the candidate services. Specifically, the final trust value ω is a parameter which ranges from 0 to 1 and controls the weights of LTTE and RTE modules in aggregation trust evaluation, thus we can easily find that the range of FT is also . Afterwards, this module can generate the top-k (where k is a parameter) recommendation list according to the final trust values of candidate services and sends it to SC(i) via SRI module. A simple example is shown in Where SR(1)∼SR(5) are five candidate services with diverse local trust and reputation values . Their final trust values (i.e. FT) and top-k recommendation lists (i.e. RL) change with ω as illustrated in ω equals to 0 or 1, FT is equivalent to RP or LT, respectively. In other cases, FT falls between LT and RP. If two candidate services have the same FT, the service with larger LT should be prior to the other one in the top-k recommendation list  is more credible than reputation-based evaluation (RP) in the view of service requester).To illustrate the performance of our IRLT model, we present a series of experiments and analysis in this section. In concrete terms, we deploy a synthetic S-SN based on the famous Advogato dataset, and employ the well-known DCG metric to measure the service recommendation performance of our IRLT model through comparing to the state-of-the-art CGLT model in honest environment and malicious environment with unfair rating attacks.In this work, the comprehensive experiments are implemented through Java language on a Linux server with 2.83GHz × 4 CPU and 8G RAM. Specifically, we deploy a synthetic S-SN which contains a social network SN and a bipartite network BN. In SN part, we adopt Advogato which is a famous trust network dataset consisting of 6541 nodes (i.e. service consumers) and 51127 directed edges (i.e. trust relationships among service consumers) as well as containing three kinds of different trust relationships . In BN part, we assume that there exist 50 candidate services and each service has 100 random service witnesses in SN. Each service witness provides a service rating to corresponding service and the preference weights of service consumers in SN are randomly generated. The parameters in experiments are set as illustrated in k recommendation list in terms of local trust value (i.e. LT), reputation value (i.e. RP) and final trust value (i.e. FT), respectively.In this part, we mainly illustrate the service recommendation performance of our IRLT model in honest environment  through comparing to the state-of-the-art CGLT model. As with the outstanding CGLT scheme, we adopt the well-known DCG metric to measure the service recommendation performance. Specifically, we define three kinds of DCG metrics (i.e. Eqs  ∼ 9)) t) t9)) tok is the size of recommendation list, and LT), RP) and FT) are the local trust value, reputation value and final trust value of the s-th service in the top-k recommendation list (in the opinion of SC(i)), respectively. In each experiment, we randomly choose a service requester from SN and generate the top-k recommendation list for it through utilizing the CGLT and our IRLT schemes, respectively. Afterwards, we can calculate three kinds of DCG metric values in the CGLT and our IRLT schemes when k takes different values (i.e. 1 ∼ 50), respectively. The experiment is repeated 5000 times and the average outputs are illustrated in Where k, and three kinds of DCG metric values in our IRLT model are slightly higher than those in the CGLT model, respectively. This indicates that the top-k services obtained from our IRLT model have higher local trust values, reputation values and final trust values than those derived from the CGLT model, thus our IRLT model has a little better service recommendation quality than the state-of-the-art CGLT model in honest environment.From Ballot stuffing: Malicious service providers collude with service consumers to raise their own trust values.Bad mouthing: Malicious service providers collude with service consumers to slander their honest competitors.In this part, we focus on analyzing and validating the robustness of our IRLT model against the following two kinds of unfair rating attacks , 29:BalAs we know, these two kinds of unfair ratings will bring risk to service requesters, thus an excellent TSR scheme should be able to detect and filter them out. In the outstanding CGLT model, two kinds of trust evaluations are separately made without consideration of filtering unfair ratings. While in our IRLT model, we notice that the strange service witnesses have a higher likelihood of colluding with malicious service providers than the trustworthy ones and the service ratings from the former are more likely to be unfair, thus we filter the service ratings in RTE module by utilizing the results from LTTE module. As a result, our IRLT should have stronger robustness against these two kinds of unfair rating attacks. Next we validate the above analysis through three series of experiments, in which the service ratings from trustworthy service witnesses are assumed to be fair and 50 candidate services are composed of 25 honest services and 25 malicious services.k recommendation list in each case, respectively. The experiment is repeated 5000 times and the average outputs of DCG-FT metric are shown in In this part, we mainly validate the robustness of our IRLT model against ballot stuffing attack through comparing to the outstanding CGLT model. In the ballot stuffing attack, the collusive service consumers provide high rating scores to malicious services in spite of their poor performances. In each experiment, we vary the Percentage of Collusive Service Consumers (PCSC) and calculate the three kinds of metric values of malicious services in the top-In the ideal case (i.e. PCSC = 0%, see k recommendation list is the lower, the better. Therefore, the above experiment and analysis show that our IRLT model greatly outperforms the state-of-the-art CGLT model in terms of the robustness against ballot stuffing attack.It is obvious that malicious services mean risk for service requesters, thus the total DCG-FT metric value of malicious services in the top-k recommendation list in each case, respectively. The experiment is also repeated 5000 times and the average outputs of DCG-FT metric are shown in Next, we compare the robustness of our IRLT model against bad mouthing attack with that of the CGLT model. In the bad mouthing attack, the collusive service consumers provide low rating scores to honest services in spite of their good performances. In each experiment, we vary PCSC and calculate the three kinds of metric values of honest services in the top-In the ideal case (i.e. PCSC = 0%, see k recommendation list is the higher, the better. As a result, the above experiment and analysis demonstrate that our IRLT model is vastly superior to the outstanding CGLT model in terms of the robustness against bad mouthing attack.It is well known that honest services will bring benefit to service requesters, thus the total DCG-FT metric value of honest services in the top-k recommendation list in each case, respectively. The experiment is also repeated 5000 times and the average results of DCG-FT metric are shown in In this part, we verify the recommendation performance of our IRLT model in malicious environment with both ballot stuffing and bad mouthing attacks, in which the collusive service consumers not only provide high rating scores to malicious services but also provide low ones to honest services. In each experiment, we vary PCSC and calculate the three kinds of metric values of honest services and malicious services in the top-In the ideal case (i.e. PCSC = 0%, see k recommendation list is the higher, the better, and that of malicious services is the lower, the better. Therefore, the above experiment and analysis show that our IRLT model has better recommendation performance than the excellent CGLT model in malicious environment with both ballot stuffing and bad mouthing attacks.As we mentioned earlier, honest services will bring benefit and malicious services mean risk for service requesters, so the total DCG-FT metric value of honest services in the top-k recommendation list. Furthermore, we have deployed a synthetic S-SN based on the famous Advogato dataset and adopted the well-known DCG metric to measure the service recommendation performance of our IRLT model through comparing to that of the state-of-the-art CGLT model. The results of experiments and analysis show that our IRLT model has slightly better service recommendation quality than the CGLT model in honest environment and greatly outperforms the CGLT model in terms of the robustness against two kinds of unfair rating attacks.In this work, we have proposed a novel IRLT model, which includes four modules, for TSR in S-SNs. In SRI module, a multi-QoS based filtering method is proposed to select out all the candidate services. In LTTE module, a MLT evaluation algorithm is presented to identify all the trustworthy service witnesses and only their service ratings are considered. In RTE module, the opinions of all the service witnesses are considered through filtering by the results of LTTE module to ease unfair rating attacks. Besides, the preference similarity and rating number are regarded as two important weights. In ATE module, the outputs of RTE and LTTE modules are integrated by weighted summation to calculate the finial trust values of candidate services as well as generate the top-


X be a compact metric space and X endowed with the levelwise topology, we proved that its g-fuzzification is turbulent or erratic if the given system f is turbulent or erratic correspondingly and f is g-fuzzification is Let  X be a compact metric space and f be a continuous self-map. We denote by X endowed with the levelwise topology and we define f:g-fuzzification introduced in Kupka  and its g-fuzzification. See more details in Kupka  be a compact metric space and C(X) denote all continuous functions. X with the Hausdorff metric Let (u on the space X is a function u and u (shortly: supp(u)) .Likewise, LetU, Vbe two subsets ofandThen:Kupka : LetU, VBefore we present the elegant results, we need to introduce some basic definitions of chaotic behavior explored in this paper.f is a turbulent function if there are disjoint nonempty closed subsets X such that:Román-Flores et al. : Let \doRomán-Flores et al. : A map \f is chaotic in the sense of Block and Coppel (in short: B-C chaos) if and only if its iterates is turbulent, i.e., there exists J, K of X such thatRomán-Flores et al. : Let \doIt should be remarkable that the erratic property is stronger than B-C chaos.f is expansive if and only if there exists a real constant Let f is In this case we claim that lco-Cano .e(U) is given here.To prove the turbulent and erratic properties, firstly, a theorem on LetUbe closed subset ofXandandThen,It can be proved by the mathematical induction.When Assume that the theorem is true for in Kupka , we haveLikewise, we can achieve the similar result on LetUbe closed subset ofXandandThen,Here, we make a point that n and Mu  for moref. These conclusions can be listed as follows:f and Román-Flores et al.  exploreds et al.  present Ifis a turbulent function, thenis a turbulent function.f is turbulent, there exists nonempty and closed Since e(U), clearly, e(U) and e(V) are two disjoint nonempty closed subsets of By the definition of entclass1pt{minimaLetbe erratic, then,is a erratic function.f is erratic, there exists a nonempty closed subset e(U) is nonempty closed subset of Since mentclass2pt{minimLetbe erratic, then, is a B-C chaos function.e(U), it can be verified with e(U) at least is perfect for the two theorems to make sense.Because Next, we shall discuss the Letbe a continuous function andThenfisexpansive if and only ifisexpansive.f is  and f and In this paper, exploiting the turbulent and erratic properties, we develop the ideas of Román-Flores et al.  and pres


We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to the bases of the matroids in question, which can also be seen as symmetry preserving graph operations in the original graph.We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2 We think of a bar-and-joint framework as a collection of fixed-length bars (corresponding to the edges of G) which are connected at their ends by universal joints (corresponding to the vertices of G). Frameworks can be used to model various structures with pairwise distance constraints and whose rigidity property is of particular interest in applications ranging from civil engineering  is connected. A connected component of F is the edge set of a connected component of G[F]. C(F) denotes the partition of F into connected components of F, and let F is called a forest if it contains no cycle and called a tree if it is a connected forest. F is called a spanning tree of a graph F is a tree with For simplicity, some properties of edge-induced subgraphs will be associated with the corresponding edge sets as follows. Let walk in G is a sequence W and W and the starting vertex of W and W followed by closed if the starting vertex and the end vertex coincide.Let It is sometimes convenient to regard the empty set as a subgroup of a group. Let In this section we shall review some basic properties of gain graphs. We refer the reader to , 31, 32 gain graphgain functionG is, in some sense, arbitrary, and is used only as a reference orientation: the orientation of each edge may be changed, provided that we also modify g in one direction then it has gain G and the underlying undirected graph and use notations introduced in Sect. Let mentclass2pt{minimW be a walk in gain of W is defined as W, and for a backward edge Let v. Similarly, for v and using only edges of X, where Let X relative to v is defined as Let e, e.g., , p. 88 fswitching operation atvwithg changes the gain function E(G) as follows.E(G) is equivalent to another gain function E(G) if For The following two facts are fundamental  if E(G) is balanced (resp. cyclic), respectively.In the same way, an edge subset X, in analogy with the fact that the cycle space of a graph is spanned by fundamental cycles. For a connected T of the edge induced graph G[X]. By Proposition W by v to T. Then observe Proposition T of G[X], suppose that (i)X is unbalanced if and only if there is an edge in (ii)X is cyclic if and only if all gains of For a connected The following technical lemmas will be used in the proof of our main theorem.X and Y be connected edge subsets such that the graph X and Y are balanced, then If X is balanced and Y is cyclic, then If X, Y are cyclic and If Let T in G[X], G[Y], and Since the graph X and Y are balanced, Lemma If X is balanced, then every label in Y is cyclic. This implies (2).If X, Y are cyclic and X and Y are cyclic, Lemma If X and Y be connected balanced edge subsets. If the number of connected components of the graph Let T of G[X]. Since the number of connected components of Y into three subsets We take a spanning tree X and Y are balanced, we have Y is balanced. Therefore By Proposition E be a finite set. A function submodular if monotone if induces a matroid on E, where Let Dilworth truncation of For a monotone submodular function ids, see  for moreNow we consider the union of two matroids induced by monotone submodular functions G. The 2-dimensional generic rigidity matroid is the one induced by As an example, consider the union of two copies of the graphic matroid of a graph k and l with G is called -sparse if k, l)-count matroid on G. If l times. See e.g. . It is known that, in the generic 2-rigidity matroid, an independent set F with Suppose further that see e.g. ). \documLet Without loss of generality, assume Let Let If =3, then  implies X and Y are connected by Lemma Therefore we assume Y is balanced and, again, Lemma (Case 1) If =2, then  implies ities of  and 15)\document(Case 2) If 0-3c1 by . By \docThis completes the proof. X and Y be Let e are contained in V(X), implying X or Y is not cyclic, then we have Since X and Y are cyclic, and they are connected by Lemma T in G[Y] such that G[X]. By Proposition Y is balanced. Since We hence assume that E. We shall also characterize the rank function of this matroid.We are ready to prove that the family of E. Then E. The rank of a set Let For a partition F. Consider the family J. Since each edge F and Let F is a maximal subset of J with Based on sparsity matroid of The matroid which was introduced and denoted by G has a vertex of degree at most 3 if and only if G is not 4-regular. Thus we shall take special care with 4-regular We now present a constructive characterization of maximum admissible if the resulting graph is Before looking at 4-regular graphs and vertices of degree four, we consider the 0-extension, 1-extension, and loop-1-extension operations. Recall that the corresponding inverse operations are called reductions. A reduction is Let v of degree 2 or 3, the 0-reduction, loop-1-reduction, or some of the 1-reductions at v are admissible if Conversely, for any vertex The proof of the first claim is exactly the same as the proof of Lemma v of degree three, we just need to observe that each circuit of v. To see that some reduction is admissible at a vertex extensions and loop-2-extensions for constructing 4-regular Besides 0-extensions, 1-extensions and loop-1-extensions, we shall introduce 2-extension, we take two existing edges v. More precisely, a 2-extension removes e and f, inserts a new vertex v with four new edges, v for each In a 2-$f=v_3,v_$$\end{doloop-2-extensions, we remove an existing edge v, a new loop l at v and two new edges, v for each e is a loop or not, we have two cases as shown in Fig. In a The following lemma shows that these operations preserve G by a 2-extension or a loop-2-extension is Let e and f and the new edges by e to f to G since Suppose that In the same manner, it can be easily checked that a loop-2-extension also preserves v and two incoming non-loop edges u to w with gain We shall define the inverse moves of these operations. Recall that, for a vertex v be a vertex of degree four, not incident to a loop, and v, assuming that all of them are oriented to v. The 2-reduction (at v) deletes v and adds one of reduction atvthroughandv and adds Let v be a vertex of degree four, incident to a loop l, and v, assuming that all of them are oriented to v. The loop-2-reduction (at v) deletes v and adds Let admissible if the resulting graph is A 2-reduction or a loop-2-reduction is said to be near-cyclic 4-regular graphs.Our main theorem asserts that these operations are sufficient to construct all 4-regular trivial graphs, fancy triangles, and fancy hats. A trivial graph is a The first group consists of three types of special fancy triangle is a A hat is a graph obtained from fancy hat is a A double cyclen vertices by replacing each edge by two parallel edges as shown in Fig. k of the underlying dihedral group The second group consists of near-cyclic graphs, which, intuitively speaking, are the near-cyclic if there is an edge e such that The third group consists of The following lemma shows how to construct near-cyclic graphs.e having a gain in Let F is cyclic, either (i) Suppose that F is connected, and clearly the endvertices of e are contained in V(F). Moreover, every cycle in e has a gain not contained in e is not in By Lemma base graph if it is a trivial graph, a fancy triangle, a fancy hat, or a near-cyclic graph.We are ready to state our constructive characterization of 4-regular Let We have proved that these operations preserve Combining Theorem Let k is odd, in which case every The theorems can be strengthened if Then, at least one of Let r along a line. Suppose that Since r, then consider k has to be even which is a contradiction. Thus If C, then consider rC are non-identity and reflections, if they generate a cyclic group, then k. If Let v be a vertex, and we denote the edges incident to v by v, Let v through G, it is easy to check that We first perform the 2-reduction at G is unbalanced, a and v with Suppose that v. In each reduction, the resulting underlying graph is We now consider the remaining two possible 2-reductions at To see that Combining Theorem Let k. See Fig. k.Lemma In this section we discuss our combinatorial characterization of symmetry-forced infinitesimal rigidity with dihedral symmetry. We begin with a necessary condition based on Lemma H, p) be a generic H, p) is symmetry-forced infinitesimally rigid, then the quotient gain graph contains a spanning maximum Let H and p. By Theorem Let Since k is even.In Sect. Our goal of this subsection is to prove the following characterization of symmetry-forced infinitesimal rigidity.H, p) be a generic H, p) is symmetry-forced infinitesimally rigid if and only if the quotient gain graph contains a spanning maximum Let Necessity follows from Lemma p appropriately. (i) will be solved in Lemma k in these lemmas.By Theorem k-gon, which consists of k-fold rotations around the origin and reflections along (fixed) lines. For a line L through the origin, we denote by L, that is, the line orthogonal to L and through the origin. We first note an elementary fact from geometry.In the rest of this section, we identify g is the reflection along a line L, then If g is a rotation, then If Let Let v be the vertex. Take p(v) does not lie on reflection lines L in (Case 1) Suppose that p randomly and computing the rank of p deterministically).(Case 2) Suppose that ,p) (see  for finde such that (Case 3) Suppose that By Theorem L, in which case there is a m does not satisfy the equation associated with L. Then ⟩=0,  becomes\The next two lemmas show that loop-2-extensions and 2-extensions preserve the independence.Let p is G by a loop-2-extension, by removing an existing edge e, adding a new vertex v with new non-loop edges l incident to v . Suppose that R. Since R is orthogonal to L, we deduce that We take a point h is a rotation, and l is a zero vector in d is a vector linearly independent from L contains the origin, and d. Thus, Case (2). Suppose that d. To see the perturbation more precisely, for each l in l in t. This in turn implies that To avoid the situation where This completes the proof of the lemma. Let p is G by a 2-extension, by removing two existing edges e and f and adding a new vertex v with new non-loop edges v  No three points among L and (ii) If The first claim follows from the proof of Claim L and For the second claim, suppose that p implies that L and In case (i), the C. Moreover, since e and f are loops attached to a vertex  of Fig. G.In case (ii), L is the line through Now, L and (ii-1) If k-gon inscribing C. Since p is generic, if L, k-gon, and hence (ii-2) If L is the line through L can have any slope, by moving L and p, then L and In case (iii), L is the line through L and p is generic.In case (iv), C be the circle whose center is the origin and which passes through Let C. We shall relocate L and q and q or p is generic, L and (iv-1) Suppose that C as shown in Fig. C. We shall relocate L is the tangent of C at q and L and q is the antipodal point of L and through the origin. When p is generic, L is not parallel to any reflection lines in (iv-2) Suppose that see Fig. b. Then, Recall that tradicts . \documeFollowing the statement of Claim L and q be the intersection of L and v. Then, e and f. Consider the rows associated with q is on L, e in L and (Case 1) Suppose that L and q be any point on L with v. Then, the orbit rigidity matrix is described as follows:q is on the line L, L and q is on L, but not on (Case 2) Suppose that Combining Theorem k. Therefore, we obtain the following statement even for a dihedral group k: for a generic H, p) with even k and a free action H, p) is symmetry-forced infinitesimally rigid if the quotient gain graph can be constructed from a disjoint union of base graphs by 0-extensions, 1-extensions, loop-1-extensions, 2-extensions and loop-2-extensions. However, as we have seen in Fig. Notice that all the lemmas given in the last subsection are independent of the parity of i and For r and Then, for any n is odd.Let x- and y-coordinates, respectively. For each i, we subtract the first row from the second row and then multiply the first row by an appropriate scalar. We then have, for each n is even. Let See Fig. essential if it is In this section we prove Theorem Any essential u, v) from u to v is simply denoted by uv, and a For simplicity, in the subsequent discussion we omit gain functions special vertices (C2)G is not cyclic.The following fundamental properties of 4-regular graphs will be frequently used.G is connected, then G is (i)G is not 4-regular and condition (C1) is satisfied, or(ii)G is 4-regular and conditions (C1) and (C2) are satisfied.If G is not connected, then G is Let G is connected, we have G is 4-regular and If the maximum degree is at most 4, almost near-cyclic if there are two incident edges e and f such that Thus, to prove Theorem v be a vertex in G that is not incident to a loop. Let v, and suppose that In particular, G is almost near-cyclic.Let v with G is not cyclic, we must have Let special if it is incident with a loop or two parallel classes of edges with The following technical lemma is one of the key observations. A vertex in a 4-regular graph is called v be a vertex in G that is not special, and v. If G is near-cyclic.v is a cut-vertex in G and Let For simplicity, we denote We claim that there are an equivalent gain function G is neither cyclic nor near-cyclic, we have To see this, first observe that if Note that Let us consider F such that V(F) contains the endvertices of s and t be the endvertices of F contains a path from s to t that does not pass through v is special in G, contradicting the assumption of the lemma.Case 1: If (C1) is violated, then v is not a cut-vertex. Note that, since C, whose gain is included in s and t be the endvertices of P from s to a vertex in V(C). We consider a closed walk P starting at s, then goes around C, and comes back to s through s and t. The concatenation of s with v is a cut-vertex in G.Case 2: We next consider the case when (C2) is violated in G, Suppose that special graph. Special graphs are classified into the following three classes n vertices by replacing each edge by two parallel copies; n vertices, each of which is incident to a loop; n vertices by replacing each edge by two parallel copies and adding one loop to each endvertex of the path.Recall that a vertex is said to be special if it is incident with a loop or two parallel classes of edges. A graph which consists of only special vertices is called a mentclasspt{minimaLet Since l. The resulting graph is G except for the loop l is in G is essential.Suppose that the underlying graph is l. The resulting Suppose that the underlying graph is G except for the loop l is included in a cyclic subgroup G is essential.If G is balanced, and G turns out to be a fancy triangle, contradicting the fact that G is essential. It can be easily observed that The next lemma solves the case when the graph is not 2-connected.G is not 2-connected. Then a 2-reduction is admissible at some vertex.Let G is not equal to G has a cut-vertex v which is not special. We show that a 2-reduction at v is admissible. Note that G. Let v, all of them are directed to v. From the 2-edge-connectivity of G, we can assume, without loss of generality, that the endvertices of By Lemma v through v is not special. Since the endvertices of F such that V(F) contains both endvertices of Consider the 2-reduction at v. Suppose that v through G is not near-cyclic since G is essential). Therefore, if G has a special vertex not incident to a loop.Thus, in the subsequent discussion, we may focus on 2-connected graphs. The next lemma solves the case when G has a special vertex not incident to a loop. Then, G has a vertex at which a 2-reduction is admissible.Let w be a special vertex not incident to a loop. By definition of special vertices, w is incident to two parallel classes of edges. Since G contains two adjacent vertices u and v such that v is not special and u is special and not incident to a loop . Depending on the size of Let u by v to u and u. We perform the 2-reduction at u through u, then Let us denote the edges incident to v through G is not near-cyclic and v is neither a cut-vertex nor a special vertex, Lemma v is admissible. Let G is almost near-cyclic.The next lemma solves the case when G is almost near-cyclic. Then a 2-reduction or a loop-2-reduction is admissible at some vertex in G.Let G is almost near-cyclic, there are two edges v and Since v is not special. Then, since v is not a cut-vertex, a 2-reduction is admissible at v by Lemma v is special. If v is not incident to a loop, then Lemma v is incident to a loop.Suppose that v with e not incident to v. This implies that if we remove v from G, the resulting graph is cyclic. In other words, it suffices to consider the case when Suppose that both v. Let v, where G is essential. Therefore, v adds the edge We hence assume that By using Lemma G is not essentially 4-edge-connected. Then, G has a vertex at which a 2-reduction or a loop-2-reduction is admissible.Let G is 2-edge-connected and is not essentially 4-edge-connected, there exists a subset X of V for which G is not B(X) contains a vertex v not incident to a loop, where B(X) denotes a set of vertices of X adjacent to some vertices of v is not a cut-vertex. Hence, denoting the four edges incident to v by X while Since v is a vertex of degree 3 in v is admissible in v in Note that v. Let F is 2-edge-connected if F for which V(F) contains both endvertices of F such that V(F) contains both endvertices of F, denoted by f, is incident to X and f turns out to be a cut-vertex of G, contradicting the 2-connectivity of G. Thus, We now consider adding G is almost near-cyclic, and we can apply Lemma v. If G has a vertex v with The final special case is when G has a vertex v with Let v is special, Lemma If v is not special, then there are three parallel edges between v and a neighbor of v. By the 4-regularity, if G is not essentially-4-edge-connected, and thus Lemma If G is equal to the graph  and  is not admissible. Then there is an edge subset (i)F is balanced tight with (ii)F is balanced tight with (iii)F is sub-tight with Suppose that the 2-reduction through  and ) if F satisfies (i) or (ii) of Lemma F is called an obstacle of type 2 if F satisfies (iii).Since the first and the second cases of the statement of Lemma v, through  and , through  and , and through  and . By Lemma X, Y, Z. We now investigate properties of these obstacles.As noted above, we have three possible ways for a 2-reduction at We begin with a property of type 2 obstacles.X is an obstacle of type 2 for the 2-reduction through  and . Then, the following holds for X:There is an equivalent gain function Suppose that By definition, v with g if necessary, we may assume that G. Thus, Since In the same manner we also have the following technical lemma.X and Y be obstacles for the 2-reduction through  and  and through  and , respectively. Suppose that X is type 2 and Let X is balanced and v, we may assume that Since By the definition of the obstacles (whether type 1 or type 2), The following lemmas describe different relations among obstacles.X and Y be obstacles for the 2-reduction through  and  and through  and , respectively. If Let Suppose for a contradiction that X and Y are of type 1, G.(Case 1) If both X is type 2. By definition of obstacles (whether type 1 or type 2), (Case 2) Let us consider the case when X and Y be obstacles for the 2-reductions through  and  and through  and , respectively. If Let G, each vertex of N(v) has degree three in X and Y are type 1 with a.Without loss of generality, we assume X is type 2, then Y is type 1, then c or d.If X and Y are type 2, then d. If both X, Y, Z be obstacles for the 2-reductions through  and , through  and , and through  and , respectively. If there is no hat subgraph containing the star of v, then Let X, Y and Z are not singleton sets, then the lemma follows from Lemma Note that a type 2 obstacle consists of more than one edge. If two of X is also a singleton set. Let (Case 1) Let us first consider the case when v is included in a hat subgraph. Indeed, if denoting without loss of generality X, Y and Z are obstacles, we have In case (I), the star of G, as a balanced In case (II), without loss of generality, we assume X is type 1 or type 2.(Case 2) Next, we consider the case when X is type 2, then g. Since Y and Z are obstacles, we have g. In other words, G.If X is type 1. Without loss of generality we assume ac or bd and ad or bc, it suffices to consider the following two cases by symmetry: (i)The remaining case is when a as In subcase (i), E. Therefore, this case does not occur and the proof is complete. In subcase (ii), notice that, obstacles for a 2-reduction at a vertex v with v denotes a vertex with v. Without loss of generality, we assume that there are parallel edges v and a, and we denote In this subsection we shall investigate v. In each case, there exists an obstacle if the operation is not admissible. The proof of the following claim is identical to that of Lemma We again have three possible ways for a 2-reduction at (i)F is balanced tight with (ii)F is balanced tight with (iii)F is sub-tight with Suppose that the 2-reduction through bv, vc), we encounter an even simpler situation.For the 2-reduction through bv, vc) is not admissible. Then there is a balanced tight set Suppose that the 2-reduction through G is almost near-cyclic, which contradicts our assumption on G. If F with Note that the type of an obstacle as in the previous subsection. Lemma bv, vc). The next two lemmas are counterparts of Lemmas According to Lemmas X and Y be obstacles for distinct 2-reductions at v. If X is type 2 and Let X and Y be obstacles for distinct 2-reductions at v. Then, if Let To prove the counterpart of Lemma Z is an obstacle of type 1 for the 2-reduction through bv, vc). Then, there is an equivalent gain function Suppose that v with X be an obstacle of type 2 for the 2-reduction through Let Let a and b, a, b, c. This means that X contains an edge linking from a to c, denoted by Suppose that Therefore, X, Y, Z be obstacles for the 2-reductions through bv, vc), respectively. Then, Let We split the proof into two cases depending on whether a type 1 obstacle exists for the 2-reduction through X is type 2. By Lemma a since b or c.(Case 1) Suppose that there is no type 1 obstacle for the 2-reduction through In a symmetric manner, we are done in the case when a type 1 obstacle does not exist for the 2-reduction through X and Y are type 1. If X or Y is 2-connected, and hence a as a to b. Also, by Lemma a to c or (ii) a to b  We now consider the case when both see Fig. .Fig. 16PE in the underlying In case (i), we have b as Z is type 1 and v, a and b have degree four. So, if c becomes a cut-vertex, contradicting the 2-connectivity of G. On the other hand, if G becomes the graph shown in Fig. G is almost near-cyclic, a contradiction. Let us consider case (ii). If v. Then we have three obstacles X, Y and Z for the three possible 2-reductions at v. Suppose further that the star of v is not contained in a hat subgraph. Then, by Lemmas Suppose that no 2-reduction is admissible at G is essentially 4-edge-connected. We then have G.If Therefore we haveX and Y are connected and balanced, Lemma If the number of connected components in Thus the number of connected components in Hence by  and 22)\documenttight by .X and Y are type 1, then X or Y is type 2, and Lemmas X or Y is type 2) we obtain G, If both  Also by  and \docG is near cyclic, as u is incident to a loop and two non-loop edges by 4-regularity. Observe that removing this loop and one of the two non-loop edges results in a cyclic graph. This means that G is almost near-cyclic.If G turns out to be almost near-cyclic, which contradicts the assumption on G. This completes the proof. In both cases We are now ready to prove Theorem G is 2-connected, essentially 4-edge-connected, not special, and not almost near-cyclic. Also, by Lemma v with By Lemmas G is not special, G has a vertex v that is not incident to a loop. Then v is admissible or the star of v is contained in a hat subgraph H. Suppose the latter holds. We denote the vertices of H by H (and hence v). Since H is balanced, we may assume that all labels in H are identity. Moreover, since G is not a fancy hat, we may assume that Since H.We prove that some 2-reduction at We claim that If G. Consequently, the 2-reduction at Consequently, without loss of generality, we may assume that k. This can be done by computing the rank of the quotient graphs in the corresponding matroids The main results of this paper Theorems  and 8.2 E is independent if and only if (i) G is -sparse and (ii) every nonempty balanced subset k, l)-sparsity for any pair of integers k, l, see e.g. [G satisfies (ii) if and only if every minimally non--sparse graph -circuit or an M-circuit) is unbalanced. Suppose that G satisfies (i) and consider one of its M-components, i.e. a subgraph H of G induced by a connected component of the -sparsity matroid of G (see [M-components). Each -circuit is a subgraph of some M-component, so we may deal with them separately. The key observation is that within H the complements of the -circuits are pairwise edge-disjoint. Since the M-components are pairwise edge-disjoint, this shows that the number of -circuits in G is O(n) and they can easily be enumerated. Then it remains to test whether each of these circuits is unbalanced, which can be done by switching and using Lemma Let entclass1pt{minimasee e.g. , so checentclass1pt{minimaf G (see  for moreE is independent if and only if (i) G is -sparse and (ii) every cyclic subset G satisfies (ii) if and only if every minimally non--sparse graph -circuit) is non-cyclic. Suppose that G satisfies (i). Then it is easy to see that these circuits are edge-disjoint, which shows that we have O(n) circuits to check. As above, they can easily be enumerated, and we can use switching and Lemma G satisfies (ii) as well. As above, it remains to check whether every -circuit is unbalanced. Let H be an M-component of G. It is not hard to see that Next consider


Our study shows that the above results can be used not only to measure the electric-field vector and its spatial distribution in some electrostatic devices, such as the Stark decelerator, Stark velocity filter and electrostatic trap and so on, but also to survey the orientation degree of cold linear molecules in a strong electrostatic field.We calculate the pendular-state spectra of cold linear molecules, and investigated the dependences of “Q-branch” integrated intensity of pendular spectra on both electric-field strength and molecular rotation-temperature. A new multi-peak structure in the “Q-branch” spectrum is appearing when the Stark interaction strength  Particularly, cold/ultracold polar molecules can be used to study their quantum-state controlled pendular spectrum. Currently, they can be prepared and manipulated by using some electrostatic devices, such as Stark decelerator2328−1. Also, the measurements of externally-added the electrostatic field vector and its spatial gradient as well as the orientation degree  of cold heavy-atom molecules (such as YbF303132−1) to a moderate one (∼103 V cm−1) by using electro-optic probe3640−15 T to 103 T353637383940in situ the E-field vector and its spatial distribution in the electrostatic devices2223242526303132The accurate amplitude and direction of the electrostatic fields and their spatial distributions play important roles in the preparation and investigation of cold/ultracold molecules using above electrostatic devices222324252622232425263 with P-, R-branches. The Q-branch transition of (HCCCN)3 is forbidden due to the selection rule. However, a so called “Q-branch” transition of the linear (HCCCN)3 will appearing in an external electric field due to the mixture of the wavefunctions . Generally, polar molecules in a strong electric field will be oriented and converted from pinwheeling rotors into pendular liberators confined to oscillate over a limited angular range along the field direction. In this case, the corresponding rotation eigenstates become the pendular states. Molecules with a low rotational temperature, T, a large dipole moment, and a small molecular constant, B, are very easy to be implemented into the pendular state under a weak electric field. The pendular-state spectra were first investigated experimentally by Miller’s groupω = μE(r)/B was used to govern the extent of hybridization of the rotor states. The Schrödinger equation of a rigid linear molecule in an E-field is given byThe pendular states are linear combinations of the field-free rotor states, J2 is the squared angular momentum operator, M is the projection of the angular momentum J along the E-field vector direction, and θ is the angle between the molecular axis and the E-field vector, and where aJM(ω) is the coefficient, which specifies the eigenfunction as linear combination of all rotational levels in a vibrational state. For any fixed value of the good quantum number M, the range of J involved in this coherent superposition or hybrid wavefunction increases with the ω parameter. The eigenstates are labeled by M and the nominal value of where And the corresponding normalized integrated intensity of the Q-branch spectra is given byv is the vibratioanl transition frequency; J′ and J″ states give rise to non-zero transition probabilities between the states that differ by more than unity in their nominal q  designates the spherical components in the space-fixed frame of the E-field vector of the radiation. q = 0 means that the polarization of the probe beam is parallel to the direction of the E-field vector; q = ±1 means that these two directions are orthogonal each other. The relative strength IQ of the “Q-branch” can be calculated by using 0 is the band origin. ±SQmin and ±Smin are the minimum values of the “Q-branch” and entire ro-vibrational spectra’s both sides. The spectrum line profile was considered as Gaussian function, its full width at half maximum (FWHM) was set as B/4where v1 vibrational mode of C-H stretch of linear cyanoacetylene trimer (HCCCN)3E = 0) were calculated by using 3 molecule are similar to the experimental ones of (HCN)3 3 2H4 The spectra of possible rovibrational transitions of the 3 under the parallel transition (the transitions of ΔMJ = 0) and the perpendicular transition (the transitions of ΔMJ = ±1) are showed in We can see from 3 in different electric fields. In the left side of 3 with a MRT of 1 K and rise with increasing the E-field strength, which is due to the slight difference of the electric dipole moments, μ, and molecular constants B of the upper and lower vibrational ground states, v and v″. To our knowledge, this multi-peak spectral structure is the first finding in the world. It is clear that such a multi-peak spectral structure can be used to precisely measure the electric dipole moments μ and B constants of the selected molecules. In the right side of 3 under different E-field strength were calculated, and the dependence of the normalized intensity of the “Q-branch” of (HCCCN)3 with different MRT on the E-field strength E was studied by using −1, the intensity of the “Q-branch” of (HCCCN)3 with a MRT of 0.25 K will be increased faster than that with a MRT of 5 K, and the low-energy rotational levels will be easily mixed in the E-field. This is because most of cold molecules are populated in lower rotational levels dominantly, thus the colder molecules will reach its saturation in the external E-field quickly, and has stronger relative intensity factor of Q-branch compared with that of a higher molecular temperature. In particular, when T = 0.25 K, the Q-branch integrated intensity of cold (HCCCN)3 will be increased from 0 to about 0.95 when the E-field strength is increased from 0 to 10 kV cm−1; while T = 5 K, it will be increased from 0 to ~0.90 with increasing E-field strength from 0 to 100 kV cm−1.The integrated intensity of the Q-branch of (HCCCN)2 2 will be increased from about 0.3 to 0.95 when the E-field strength is increased from 50 kV cm−1 to 1000 kV cm−1; while T = 10 K, it will be increased from about 0.35 to 0.90 with increasing the E-field strength from 100 kV cm−1 to 1000 kV cm−1. Also, we can see from 2 will be increased from about 0.35 to 0.98 when the E-field strength is increased from 5000 kV cm−1 to 105 kV cm−1; while T = 300 K , it will be increased from about 0.3 to 0.92 with the increase of the E-field strength from 7500 kV cm−1 to 105 kV cm−1. These results show that it is suitable to observe the Q-branch spectrum of cold linear (HCCCN)3, HCN-N2 and the room temperature HCl in a weak, moderate and strong E-field, respectively.We also investigate the dependence of the normalized integrated intensity of the Q-branch on the E-field strength for HCN-N3 Q-branch on the MRT T, and shows that for a certain E-field strength, with the reduction of the MRT, the intensity of Q-branch will be first increased and then slowly reached to a saturation value. Particularly, when T ≤ 1 mK, the E-field measurement value of ultracold (HCCCN)3 can reach ∼ 0.01 V cm−1, even a smaller value. This is because the lower the MRT T is, the higher the population of lower rotational levels will be, and when the MRT T is low enough, the population of the lowest rotational level will reach its highest value, and then the integrated intensity of the Q-branch will reach its saturation value. This result shows that the lower the MRT is, the weaker the needed E-field will be, and ultracold linear molecules in a weaker E-field can be used to obtain a higher sensitivity for the observation of the Q-branch integrated intensity.μ and B constants of different vibration states of cold linear molecules.First, we can see from 3 with a temperature of lower or higher than 1 mK is suitable to measure a weaker E-field strength within 10−5 ∼ 100 kV cm−1, cold HCN-N2 and heat HCl from 77 K to 300 K are suitable to measure a moderate E-field strength (1 ∼ 103 kV cm−1) and a strong one (102 ∼ 105 kV cm−1) by using the pendular Q-branch spectra of the linear molecule, respectively. The measurement principle and method will be introduced in some detail as follows: in our proposed scheme, the space of the measured E-field will be filled with the selected cold linear molecule gas, which can be generated by using buffer-gas cooling and Stark velocity filter23ηcoll, and the quantum efficiency ηDet of the detector were assumed as a constant near the band origin v0, the photoelectric current of the detected fluorescence intensity of the “Q-branch” can be given bySecondly, it is clear from B(v0) = AP0ηcollηdet, A is the system coefficient, P0 and V are the power of the probe beam and the effective probing volume, respectively. From where k is the proportional coefficient, which can be determined by the Q-branch intensity of cold linear molecules in a homogeneous E-field generated by a pair of plate electrodes, that is, it can be determined by the calibration curve of the resulting Q-branch intensity of a plate capacitor. ρ(r) is the spatial density distribution of cold molecules and will be easily affected by the E-field environment, but this problem can be solved by using a CCD camera to obtain the fluorescent intensity distribution from the molecule sample, which is proportional to the spatial density distribution of cold molecules in an entire measurement region. This is similar to the measurement method of cold atomic density used in the MOT or BEC experiments.where EX of the measured E-field vector can be obtained by measuring the photon current iQX from the corresponding LIF intensity IQX of the Q-branch. If the direction of the probe-beam polarization is changed as the Y direction [see EY of the measured E-field vector will be obtained by measuring iQY from the corresponding IQY. Then, the magnitude EZ of the measured E-field vector can be determined by using similar measurement method, and then we can obtain the E-field vector The basic principle for measuring the direction of the E-field vector can be simply described as follows: the magnitude and direction of the E-field vector in a two-dimensional (2D) plane (such as in XOY plane) can be measured by changing the direction of the probe-beam polarization two times. That is, from tion see , the Y cFinally, it is well known that the stronger the electric field is, the higher the orientation degree of cold linear-molecules will beSimilar to the derivation of measurement sensitivity expression in refs ηc is the fluorescence collecting efficiency, n is the density of a supersonic molecular beam, and κ the front factor of E2. The effective detected volume within probe laser beam can be assumed as a spheroid, it’s volume Veff equals to w0 is the waist radius, and f represents the Rayleigh length of the Gaussian beam. When the fluorescence collecting efficiency of detection system is ηc = 10%, the rotational temperature of a supersonic (HCCCN)3 beam is 5 K, and the waist radius of the focused probe beam is w0 = 10 μm, we obtain a static electric-field sensitivity of 2.05 × 10−7 V/cm48. It is clear that when SNR = 1, the relative measurement error  reaches 100%. So in order to obtain a 1% relative measurement error (a relative uncertainty), the SNR should be equal to 100, and the corresponding practical sensitivity is E = 2.05 × 10−5 V/cm, which is far lower than ones (5 V/cm ~ 100 V/cm) of other methods3638where 3, and given reasonable physical explanations. Our study shows that the proposed measurement method can cover an ultrawide range of about 10−5 V cm−1 ~ 105 kV cm−1 theoretically, and the corresponding practical measurement sensitivity is 2.05 10−5 V cm−1. So it is clear that such a high-sensitive and ultrawide-range method can be used to realize the nonintrusive, in-situ measurement of the E-field vector and its spatial distribution in Stark decelerator23303132We have calculated pendular-state spectra of cold linear molecules, and studied the dependences of the integrated intensity of the Q-branch on both the E-field strength and the MRT of cold linear molecules, and found that these dependences can be used to sensitively measure the E-field vector and its spatial distribution by changing the polarization directions of the focused probe-beam as well as the orientation degree of cold linear molecules. Also, we have found some new physical phenomena as follows: (1) the saturation effect of the integrated intensity of the Q-branch in a strong E-field; (2) the similar saturation effect occurred at an ultralow temperature of the linear molecules, and (3) a multi-peak structure in the Q-branch spectrum as well as its appearing condition How to cite this article: Deng, M. et al. Dependences of Q-branch integrated intensity of linear-molecule pendular spectra on electric-field strength and rotational temperature and its potential applications. Sci. Rep.6, 26776; doi: 10.1038/srep26776 (2016).