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The Thomas model is a system of two reaction–diffusion equations which was originally proposed in the context of enzyme kinetics. It was subsequently realised that it offers a plausible chemical mechanism for the generation of coat markings on mammals. To that end previous investigations have focused on establishing the conditions for the Turing instability and on following the associated patterns as the bifurcation parameter increases through the instability. In this paper we use modern ideas from the theory of dynamical systems to systematically investigate the formation of localised structures in this model. The Turing instability is found to exhibit a degeneracy at which it changes from being subcritical to supercritical (or vice versa). Associated with such degenerate points is a heteroclinic connection which is a crucial requirement for the generation of localised patterns. Localised solutions containing a single ‘spike’ are associated with the so-called Belyakov-Devaney transition. Localised solutions containing either multiple ‘peaks’ or multiple ‘holes’ are associated with homoclinic snaking bifurcations. These structures are investigated using continuation software. The snaking bifurcations are found to collapse when the system crosses the Belyakov-Devaney transition. The isolated spike solutions collapse at a codimension two heteroclinic connection to create the so-called collapsed snaking. We show that the temporal stability of the localised solutions depends upon the presence of Hopf bifurcations which in turn are controlled by the diffusion ratio. For small values of the ratio only solutions in the spike region are destabilised by a Hopf bifurcation. For larger values of the diffusion ratio Hopf bifurcations destabilise most of the localised patterns.

Column generation (CG) algorithms are well known to suffer from convergence issues due, mainly, to the degenerate structure of their master problem and the instability associated with the dual variables involved in the process. In the literature, several strategies have been proposed to overcome this issue. These techniques rely either on the modification of the standard CG algorithm or on some prior information about the set of dual optimal solutions. In this paper, we propose a new stabilization framework, which relies on the dynamic generation of aggregated rows from the CG master problem. To evaluate the performance of our method and its flexibility, we consider instances of three different problems, namely, vehicle routing with time windows (VRPTW), bin packing with conflicts (BPPC), and multiperson pose estimation (MPPEP). When solving the VRPTW, the proposed stabilized CG method yields significant improvements in terms of CPU time and number of iterations with respect to a standard CG algorithm. Huge reductions in CPU time are also achieved when solving the BPPC and the MPPEP. For the latter, our method has shown to be competitive when compared with a tailored method. Summary of Contribution: Column generation (CG) algorithms are among the most important and studied solution methods in operations research. CG algorithms are suitable to cope with large-scale problems arising from several real-life applications. The present paper proposes a generic stabilization framework to address two of the main issues found in a CG method: degeneracy in the master problem and massive instability of the dual variables. The newly devised method, called dynamic separation of aggregated rows (dyn-SAR), relies on an extended master problem that contains redundant constraints obtained by aggregating constraints from the original master problem formulation. This new formulation is solved in a column/row generation fashion. The efficacy of the proposed method is tested through an extensive experimental campaign, where we solve three different problems that differ considerably in terms of their constraints and objective function. Despite being a generic framework, dyn-SAR requires the embedded CG algorithm to be tailored to the application at hand.

Flexible load at the demand-side has been regarded as an effective measure to cope with volatile distributed renewable generations. To unlock the demand-side flexibility, this paper proposes a peer-to-peer energy sharing mechanism that facilitates energy exchange among users while preserving privacy. We prove the existence and partial uniqueness of the energy sharing market equilibrium and provide a centralized optimization to obtain the equilibrium. The centralized optimization is further linearized by a convex combination approach, turning into a multi-parametric linear program (MP-LP) with renewable output deviations being the parameters. The flexibility requirement of individual users is calculated based on this MP-LP. To be specific, an adaptive vertex generation algorithm is established to construct a piecewise linear estimator of the optimal total cost subject to a given error tolerance. Critical regions and optimal strategies are retrieved from the obtained approximate cost function to evaluate the flexibility requirement. The proposed algorithm does not rely on the exact characterization of optimal basis invariant sets and thus is not influenced by model degeneracy, a common difficulty faced by existing approaches. Case studies validate the theoretical results and show that the proposed method is scalable., Comment: 11 pages, 10 figures

This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Holder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L 1 -stable, which is the first frame work on L 1 -wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L 1 -weighted energy estimates combining the compactness analysis, but with some new development.

We address the problem of accelerating column generation (CG) for set-covering formulations via dual optimal inequalities (DOI). DOI use knowledge of the dual solution space to derive inequalities that might be violated by intermediate solutions to a restricted master problem, and as such are efficient at reducing the number of iterations and the oscillations of the dual variables commonly observed in column generation procedures. We study two novel classes of DOI which are referred to as Flexible DOI (F-DOI) and Smooth-DOI (S-DOI), respectively (and jointly as SF-DOI). F-DOI provide rebates for covering items more than necessary. S-DOI describe the payment of a penalty to permit the under-coverage of items in exchange for the over-inclusion of other items. Unlike other classes of DOI from the literature, the S-DOI and F-DOI rely on very little to no problem-specific knowledge, and as such have the potential to be applied to a vast number of problem domains. In particular, we illustrate the efficiency of the new inequalities by embedding them within a column generation solver for the single source capacitated facility location problem (SSCFLP). A speed-up of a factor of up to 130x can be observed as when compared to a non-stabilized variant of the same CG procedure to achieve the linear relaxation lower bound on problems with dense columns and structured assignments costs.

A new approach based on a strong degeneracy state to design oscillators that exhibit low phase noise and low power consumption is presented. A 100-MHz oscillator is designed using a periodic structure based on a degenerate band edge that is augmented with an NMOS cross-coupled pair. The analysis and simulation results show 16 dB phase noise improvement compared to oscillators based on a conventional single-ladder periodic structure. It is also shown that this oscillator dissipates less power than a conventional LC oscillator that requires a buffer to drive a 50Ω load.

In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl equations with low regularity (see Oleinik and Samokhin [9] , P.21, Theorem 2.1.1). Due to the degeneracy of the equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5] . In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).

We consider the estimation of the entropy of a discretely-supported time series through a plug-in estimator. We provide a correction of the bias and we study the asymptotic properties of the estimator. We show that the widely-used correction proposed by Roulston (1999) is incorrect as it does not remove the $O\left ({N^{-1}}\right)$ part of the bias while ours does. We provide the asymptotic distribution and we show that it differs when the values taken by the marginal distribution of the process are equiprobable (a situation that we call degeneracy ) and when they are not. We introduce estimators of the bias, the variance and the distribution under degeneracy and we study the estimation error. Finally, we propose a goodness-of-fit test based on entropy and give two motivations for it. The theoretical results are supported by specific numerical examples.

Efficient magnetic control of electronic conduction is at the heart of spintronic functionality for memory and logic applications1,2. Magnets with topological band crossings serve as a good material platform for such control, because their topological band degeneracy can be readily tuned by spin configurations, dramatically modulating electronic conduction3–10. Here we propose that the topological nodal-line degeneracy of spin-polarized bands in magnetic semiconductors induces an extremely large angular response of magnetotransport. Taking a layered ferrimagnet, Mn3Si2Te6, and its derived compounds as a model system, we show that the topological band degeneracy, driven by chiral molecular orbital states, is lifted depending on spin orientation, which leads to a metal–insulator transition in the same ferrimagnetic phase. The resulting variation of angular magnetoresistance with rotating magnetization exceeds a trillion per cent per radian, which we call colossal angular magnetoresistance. Our findings demonstrate that magnetic nodal-line semiconductors are a promising platform for realizing extremely sensitive spin- and orbital-dependent functionalities. A study reports a colossal angular magnetoresistance in the topological magnet Mn3Si2Te6, resulting from a metal-to-insulator transition caused by controlled lifting of a topological band degeneracy, and discusses the key parameters involved.

Laser cooling exploits the physics of light scattering to cool atomic and molecular gases to close to absolute zero. It is the crucial initial step for essentially all atomic gas experiments in which Bose-Einstein condensation and, more generally, quantum degeneracy is reached. The ongoing development of laser-cooling methods has allowed more elements to be brought to quantum degeneracy, with each additional atomic species offering its own experimental opportunities. Improved methods are opening new avenues, for example, reaching Bose-Einstein condensation purely through laser cooling as well as the realization of continuous Bose-Einstein condensation. Here we review these recent innovations in laser cooling and provide an outlook on methods that may enable new ways of creating quantum gases., 13 pages, 4 figures

An accurate and efficient method is proposed to characterize waveguides with arbitrary cross sections exhibiting discrete rotational symmetries, known in the literature as $C_{N}$ symmetries. This new method is based on a novel tailored 2-D finite element method (FEM) formulation exploiting rotationally periodic boundary conditions involved in the minimum sector that defines the waveguide cross section. While retaining the proven accuracy and convergence rate of FEM, it allows for a reduction in dimensionality of the problem and classifies the modes with respect to their rotational symmetry. Most importantly, it discriminates reliably and unambiguously degenerate modes (no matter how high their order is) with the exact same computed cutoff frequency value. This is especially interesting in modal analysis of devices involving circular polarization, since the formulation inherently provides different mode sets for clockwise and counterclockwise circularly polarized modes. The method has been tested on a wide variety of waveguides and computed higher order modes have been compared against analytical solutions and general-purpose FEM.

The paper studies recoverability of missing values for sequences in a pathwise setting without probabilistic assumptions. Sufficient conditions of recoverability are obtained; it is shown that square summable sequences are recoverable if the Z-transforms vanish at a single point. The recovery is based on a new family of transfer functions defined by locations of missing values. Some robustness of the solution with respect to noise contamination and truncation is established.

Simultaneous localization and mapping (SLAM) is a widely used technology in autonomous mobile robots, where sensors such as Lidar or cameras are typically used. Sensor fusion using multiple sensors has been employed to compensate for the shortcomings of each sensor in SLAM. However, the sensor cost cannot be ignored when considering its practical usage. Therefore, this study aims at realizing a high-precision SLAM using a sensor switching system, combining multiple low-cost sensors. The sensor switching system consists of a low-cost Lidar SLAM and a monocular localization. Since a low-cost Lidar has a short laser range, degeneracy often occurs due to the fact that they cannot capture features while building maps. The proposed system uses localization data from monocular localization to ensure precision in regions where degeneracy occurs. The proposed system was evaluated through the simulation assuming the museum environment where the degeneracy occurred. The accuracy of the robot trajectory and the built map proved the effectiveness of the proposed system.

In this paper, we give a Moser-type theorem for C l -smooth hyperbolic-type degenerate Hamiltonian system with the following Hamiltonian H = 〈 ω , y 〉 + 1 2 v 2 − u 2 d + P ( x , y , u , v ) , ( x , y , u , v ) ∈ T n × R n × R 2 , which is associated with the standard symplectic structure, with d ≥ 1 . Due to the difficulty coming from the degeneracy, our result is quite different from L. Chierchia and D. Qian's work [8] (non-degenerate case). An interesting phenomenon shown in degenerate case is the l-regularity of above Hamiltonian system not only relies on the tori's dimension n but also strongly relies on the degenerate index d. Under arbitrary small perturbation P, we prove that if l ≥ ( 5 d + 2 ) ( 8 τ + 3 ) , where τ > n − 1 , the above hyperbolic-type degenerate Hamiltonian system admits lower dimensional Diophantine tori which are proved to be of class C β for any β ≤ 8 τ + 2 . Our result can be seen a generalization of paper [42] from analytic case to C l -smooth case and can also be seen a generalization of paper [8] from non-degenerate case to degenerate case.

In this paper, we are concerned with the following singular perturbation Kirchhoff equation −(e2a+eb∫R3|∇u|2dy)Δu+V(y)u=|u|p−1u,u∈H1(R3), where constants a,b,e>0 and 1

For the equation $$u_{{xx}}+yu_{{yy}}+\alpha u=0 $$ with a parameter $$\alpha \leq -1/2 $$ in the mixed domain lying in the right half-plane and bounded by the half-axis $$y\geq 0$$ and the characteristic $$ x=2\sqrt {-y}$$ , we consider the Neumann–Tricomi problem in which the values of the normal derivative of the unknown function are given on the half-axis, the values of the function itself are given on the characteristic, and matching conditions are specified on the singular line. The solution is sought in the class of functions having singularities at infinity of order no higher than a given one. Sufficient conditions on the input data under which the problem is uniquely solvable are obtained. The proof is carried out by the integral equation method.

We present a construction of the Kitaev honeycomb lattice model on an arbitrary higher genus surface. We first generalize the exact solution of the model based on the Jordan-Wigner fermionization to a surface with genus $g = 2$, and then use this as a basic module to extend the solution to lattices of arbitrary genus. We demonstrate our method by calculating the ground states of the model in both the Abelian doubled $\mathbb{Z}_2$ phase and the non-Abelian Ising topological phase on lattices with the genus up to $g = 6$. We verify the expected ground state degeneracy of the system in both topological phases and further illuminate the role of fermionic parity in the Abelian phase.

This article proposes a novel sparse kernel ridge regression assisted particle filter (SKRR-PF) based remaining useful life (RUL) estimation method to address the reliability of the cascode gallium nitride field-effect transistors in emerging power electronics systems. The proposed method will overcome three main challenges in this area: first, a large variation in the RUL estimation under the severe noise uncertainty; second, sample degeneracy and sample impoverishment; and third, the low capability of tracing the dynamic and abrupt change in the fault precursor trajectory. The state-of-the-art RUL estimation methods require a significant number of samples to address such issues, including particle degeneracy and sample impoverishment. Also, the state-of-the-art methods mostly fail to apply under a system's dynamic condition changes over a switch's lifetime. The proposed method will significantly enhance the estimation accuracy by introducing SKRR in resampling the posterior probability density function estimation, especially under the dynamically varying system's health condition due to the harsh industrial operation. Thus, the proposed method will offer fast tracing the sudden change in the RDS, ON trajectory, leading to a significant accurate RUL estimation. The performance of the proposed method has been rigorously validated through the purposely designed power cycling testbed.

We consider small perturbations of analytic non-twist area preserving mappings, and prove the existence of invariant curves with prescribed frequency by KAM iteration. Generally speaking, the frequency of invariant curve may undergo some drift, if the twist condition is not satisfied. But in this paper, we deal with a degenerate situation where the unperturbed rotation angle function r → w + r2n+1 is odd order degenerate at r = 0, and prove the existence of invariant curve without any drift in its frequency. Furthermore, we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders.

Delta self-consistent-field methods are widely used in studies of electronically excited states. However, the nonaufbau determinants are generally spin-contaminated. Here, we describe a general approach for spin-coupling interactions of open-shell molecules, making use of multistate density functional theory (MSDFT). In particular, the effective exchange integrals that determine spin coupling are obtained by enforcing the multiplet degeneracy of the S+1 state in the MS = S manifold. Consequently, they are consistent with the energy of the high-spin state that is adequately treated by Kohn-Sham density functional theory (DFT) and, thereby, free of double counting of correlation. The method was applied to core excitations of open-shell molecules and compared with those by spin-adapted time-dependent DFT. An excellent agreement with experiment was found employing the BLYP functional and aug-cc-pCVQZ basis set. Overall, MSDFT provides an effective combination of the strengths of DFT and wave function theory to achieve efficiency and accuracy.

This paper is about traveling wave solutions and asymptotic spreading for a class of degenerate equations with time delay. The influence of delay and degeneracy on the propagation threshold and complete spreading or vanishing is explored. We first study the existence of traveling wave solutions by constructing proper super-solutions and show the monotonicity, asymptotic behavior and uniqueness up to translation of traveling wave solutions. Different from the nondegenerate case, our results reflect that the traveling wave solutions with critical speed decay exponentially, while those with non-critical speed may not decay exponentially. Then some sufficient conditions on the complete spreading or vanishing are given by using suitable super- and sub-solutions, which depend on the degeneracy of nonlinearity as well as the initial value. To illustrate our results, several degenerate equations are investigated, which shows that the large delay could slow down the propagation threshold.

We analyze the long-time behavior of the semigroup S(t) generated by a heat equation with degenerate past history and a nonlinear heat supply posed in a three dimensional bounded domain $$\Omega $$ . Assuming that the degeneracy occurs in a positive measure subset of $$\Omega $$ , we prove the existence and regularity of the global attractor associated to this semigroup.

The relationship between the feature-tuning curve and information transfer profile of individual neurons provides vital insights about neural encoding. However, the relationship between the spatial tuning curve and spatial information transfer of hippocampal place cells remains unexplored. Here, employing a stochastic search procedure spanning thousands of models, we arrived at 127 conductance-based place-cell models that exhibited signature electrophysiological characteristics and sharp spatial tuning, with parametric values that exhibited neither clustering nor strong pairwise correlations. We introduced trial-to-trial variability in responses and computed model tuning curves and information transfer profiles, using stimulus-specific (SSI) and mutual (MI) information metrics, across locations within the place field. We found spatial information transfer to be heterogeneous across models, but to reduce consistently with increasing degrees of variability. Importantly, whereas reliable low-variability responses implied that maximal information transfer occurred at high-slope regions of the tuning curve, increase in variability resulted in maximal transfer occurring at the peak-firing location in a subset of models. Moreover, experience-dependent asymmetry in place-field firing introduced asymmetries in the information transfer computed through MI, but not SSI, and the impact of activity-dependent variability on information transfer was minimal compared to activity-independent variability. Biophysically, we unveiled a many-to-one relationship between different ion channels and information transfer, and demonstrated critical roles for N-methyl-D-aspartate receptors, transient potassium and dendritic sodium channels in regulating information transfer. Our results emphasize the need to account for trial-to-trial variability, tuning-curve shape and biological heterogeneities while assessing information transfer, and demonstrate ion-channel degeneracy in the regulation of spatial information transfer.

In this paper, motivated by recent papers on the stabilization of evolution problems with nonlocal degenerate damping terms, we address an extensible beam model with degenerate nonlocal damping of Balakrishnan-Taylor type. We discuss initially on the well-posedness with respect to weak and regular solutions. Then we show for the first time how hard is to guarantee the stability of the energy solution (related to regular solutions) in the scenarios of constant and non-constant coefficient of extensibility. The degeneracy (in time) of the single nonlocal damping coefficient and the methodology employed in the stability approach are the main novelty for this kind of beam models with degenerate damping.

The paper studies predicting problems for continuous time signals with the Fourier transforms vanishing with a certain rate at a single point. It shows that, with some linear causal predictors, anticausal convolution integrals over future times for these processes can be approximated by causal convolutions over past times. The corresponding predicting kernels are time invariant, and they are presented explicitly in the frequency domain via their transfer functions. These predictors are “universal” meaning that they do not require to know details of the spectrum of the underlying signals; the same predictor can be used for the entire class of signals with a single point spectrum degeneracy. The predictors feature some robustness with respect to noise contamination.

A dynamic geometry system, as an important application in the field of geometric constraint solving, is widely used in elementary mathematics education; moreover, the dynamic geometry system is also a fundamental environment for automated theorem proving in geometry. In a geometric constraint solving process, a situation involving a critical point is often encountered, and geometric element degeneracy may occur at this point. Usually, the degeneracy situation must be substantively focused on during the learning and exploration process. However, many degeneracy situations cannot be completely presented even by the well-known dynamic geometry software. In this paper, the mechanisms causing the degeneracy of a geometric element are analyzed, and relevant definitions and formalized descriptions for the problem are provided according to the relevant modern Euclidean geometry theories. To solve the problem, the data structure is optimized, and a domain model design for the geometric element and the constraint relationships thereof in the dynamic geometry system are formed; furthermore, an update algorithm for the element is proposed based on the novel domain model. In addition, instances show that the proposed domain model and the update algorithm can effectively cope with the geometric element degeneracy situations in the geometric constraint solving process, thereby achieving unification of the dynamic geometry drawing and the geometric intuition of the user.

With the wide application of distributed energy technology in active distribution networks (ADNs), virtual power plants (VPPs) are introduced as a promising integration and management technology. To coordinate their operation schedule and ensure confidentiality, we establish a coordinated optimal power flow (OPF) model and propose an efficient decentralized algorithm based on multi-parametric quadratic programming and Benders decomposition. In feasible sub-problems, the quadratic exchanging functions are built, which greatly accelerates the convergence process. In infeasible sub-problems, feasible cut sets can be calculated based on benders decomposition. Besides, an effective relaxation method is developed to address the degeneracy problem. To eliminate redundant feasible cut sets, a predetermined feasible region is constructed according to the characteristics of each VPP. The effectiveness of the proposed method is demonstrated via numerical tests using two cases of different scales. The results show that the proposed method converges much faster than some prevailing methods. Furthermore, the coordinated OPF has a better fuel economy than the conventional OPF scheme.

In this paper, we derive some $$\partial \overline \partial $$ -Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, and some rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. non-negative) l-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results [5, 6] proved recently by L. Ni on Kahler manifolds to Hermitian manifolds. We also derive an integral inequality for a holomorphic map between Hermitian manifolds.

Intermetallic semiconductors with the cubic Half-Heusler structure (XYZ) have excellent thermoelectric properties. This has been attributed to the high degeneracy of the carrier pockets in the band structure, but large differences are found between different material compositions. Half-Heuslers are often interpreted within Zintl chemistry, making a clear distinction between an electropositive cation ($X^{n+}$) and an extended polyanion ($YZ^{n-}$). Based on quantitative real space chemical bonding analysis, we unravel large degrees of covalent bonding between the formal cation and anion, making the Zintl distinction clearly invalid. This covalence is shown to strongly affect the band structure, thermoelectric properties and response properties in the materials, with improved thermoelectric properties observed for those materials that least follow the Zintl concept. This expands our knowledge of the chemical bonding motifs governing physical properties, and gives a critical view on the simplistic chemical concepts too often applied for design of complex materials., Comment: 29 pages, 12 figures

In this paper, we develop novel local discontinuous Galerkin (LDG) methods for fractional diffusion equations with non-smooth solutions. We consider such problems, for which the solutions are not smooth at boundary, and therefore the traditional LDG methods with piecewise polynomial solutions suffer accuracy degeneracy. The novel LDG methods utilize a solution information enriched basis, simulate the problem on a paired special mesh, and achieve optimal order of accuracy. We analyze the $$L^2$$ stability and optimal error estimate in $$L^2$$ -norm. Finally, numerical examples are presented for validating the theoretical conclusions.

We are concerned with the shock diffraction configuration for isothermal gas modeled by the conservation laws of nonlinear wave system. We reformulate the shock diffraction problem into a linear degenerate elliptic equation in a fixed bounded domain. The degeneracy is of Keldysh type—the derivative of a solution blows up at the boundary. We establish the global existence of solutions and prove the $${C^{0,{1 \over 2}}}$$ -regularity of solutions near the degenerate boundary. We also compare the difference of solutions between the isothermal gas and the polytropic gas.

For the equation $$(\mathrm {sign}\thinspace y)|y|^{m}u_{{xx}}+u_{{yy}}-(m/2y)u_{y}=0$$ considered in a mixed domain, we prove theorems on the uniqueness and existence of a solution of a problem with a missing Tricomi condition on the boundary characteristic and with an analog of the Frankl condition on an internal characteristic and on the degeneracy segment.

In this paper, we present a new construction of asymmetric quantum codes (AQCs) by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counterparts. Consequently, we construct several families of AQCs with better parameters than known results in the literature. Third, AQCTPCs can be efficiently decoded although they are degenerate, provided that the inner and outer codes are efficiently decodable. In particular, we significantly reduce the inner decoding complexity of TPCs from $\Omega(n_2a^{n_1})(a>1)$ to $O(n_2)$ by considering error degeneracy, where $n_1$ and $n_2$ are the block length of the inner code and the outer code, respectively. Furthermore, we generalize our concatenation scheme by using the generalized CCs and TPCs correspondingly., Comment: 36pages, accepted by IEEE Transactions on Communications

In this communication, based upon a squeezed-state trial wave function, we have performed a simple variational study of the spectral collapse in the anisotropic two-photon Rabi model. Our analysis indicates that the light-matter interaction and the spin-flipping (together with the anisotropy) effectively constitute two competing impacts upon the radiation mode. Whilst the former tries to decrease the radiation mode frequency, the latter may counteract or reinforce it. The light-matter interaction appears to dominate the frequency modulation as its coupling strengths go beyond the critical values, leading to the emergence of the spectral collapse. However, at the critical couplings the dominance of the light-matter interaction is not complete, and incomplete spectral collapse appears. Accordingly, at the critical couplings the eigenenergy spectrum comprises both a set of discrete energy levels and a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in a finite potential well, and the number of bound states available is determined by the energy difference between the two atomic levels. Each of these eigenenergies has a twofold degeneracy corresponding to the spin degree of freedom.

In the present paper, we deal with the Dirichlet problem for a model nonlinear second-order equation with isotropic and degenerate (with respect to independent variables) coefficients, lower term and L1-right-hand side. The degeneracy with respect to independent variables is described by the presence of a weighted function in the equation's higher term. Our main result is the theorem on the uniqueness of entropy solution of the problem under consideration. It is proved under minimal conditions on the involved weighted function. Namely, we need these assumptions on its integrability for introduction of the corresponding weighted isotropic energy Sobolev space.

We prove that a matrix-differential operator A in a Banach space is a Fredholm operator with zero index and construct the corresponding subspaces and projections. We find the spectrum of A. Based on properties of the operator A, we study solutions to the initial-boundary value problem for a second order partial differential equation with a small parameter e at the higher order derivatives and some parameter c at lower order terms. We clarify how the parameter c affects the behavior of the solution as e → 0.

In this paper we versally unfold a homogeneous cubic degenerate center, to which the Poincare normal form is not applicable, in the strong monodromic family. Reducing to a normal form of the strong monodromic class, we prove that the degeneracy of the homogeneous cubic degenerate center with perturbations is of codimension one. Using one unfolding parameter, we display bifurcations at the center in the strong monodromic family, which shows that at most one limit cycle arises if the 3-th order generalized Lyapunov quantity does not vanish.

An advanced characteristic mode tracking method is investigated in this article. The tracking method is based on the eigenvector correlation, and a novel tracking error correction algorithm (TECA) is proposed to solve the challenging issues caused by the mode swapping and the mode degeneracy. As the core of the modal tracking method, the TECA can quickly find out the cross-mode pairs according to the eigenvalue differences among different modes, identify the wrongly mapped modes using the eigenvector correlation, and finally correct all the possible tracking errors. The detailed process of each step is described and discussed. In addition, the characteristic modes of two classical fractal antenna structures are investigated for validation. It has been shown that the modes including the swapped modes and degenerated modes are all tracked correctly, verifying the modal tracking method is reliable and efficient. Compared with the previous approaches, the proposed method is not subject to the number of tracked modes and the magnitude of eigenvalue. Also, it requires less computational time. Moreover, the sensitivity to the sizes of frequency step and mesh is reduced to some extent.

For the stability of the non-Newtonian fluid equation ∂ u ∂ t − div ( a ( x ) | ∇ u | p − 2 ∇ u ) − ∑ i = 1 N b i ( x ) D i u + c ( x , t ) u = f ( x , t ) , where a ( x ) | x ∈ Ω > 0 , a ( x ) | x ∈ ∂ Ω = 0 and b i ( x ) ∈ C 1 ( Ω ‾ ) , we know that the degeneracy of a ( x ) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the stability of weak solutions.

This paper is concerned with the existence and global stability of forced waves for reaction diffusion equations with density-dependent diffusion in a shifting environment, which arises from population dynamics. Our argument is based on the introduction of suitable families of upper and lower solutions and on a comparison result through Holmgren's method. Our analysis yields, for any wave speed c > 0 , the density-dependent dispersal population models with shifting habitats admit forced traveling waves. To overcome the difficulties caused by peculiar structure of forced waves and degeneracy, we develop a novel weighted estimate technique to prove the global and exponential stability of these waves. Different from the spatial homogenous case Huang et al. 2018 [24] , the weight function is introduced near the positive equilibrium r ( + ∞ ) instead of the zero equilibrium.

We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $\rho $ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $\rho =1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $K$-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.

Using the linearized Vlasov–Maxwell model and the polarization tensor components for weakly magnetized ( $$\left| \omega -{\mathbf {k}}\cdot {\mathbf {v}}\right| >\varOmega $$ ) electron plasma the dispersion relations of parallel and perpendicular propagating electromagnetic modes are evaluated under high frequency/long wavelength limit (i.e., $$\omega >{\mathbf{k }}\cdot {\mathbf{v }}$$ ). The propagation characteristics of R–L and ordinary waves are discussed in the presence of isotropic Fermi–Dirac distribution function in a relativistic regime. The Polylog functions that arise due to arbitrary/partial degeneracy are examined in various degeneracy and relativistic limits. The graphical results are also extracted in the presence of quantum non-degenerate, degenerate and fully degenerate regimes due to the variation in relativistic parameter $$(\frac{T}{ m_{0}c^{2}})$$ , Lorentz factor ( $$\gamma $$ ) and degeneracy parameter $$(\frac{\mu }{T})$$ in weakly and strongly relativistic limits. Furthermore, the derived results are found to be in complete agreement with the previous investigations.

In certain pulsar timing experiments, where observations are scheduled approximately periodically (e.g. daily), timing models with significantly different frequencies (including but not limited to glitch models with different frequency increments) return near-equivalent timing residuals. The average scheduling aperiodicity divided by the phase error due to time-of-arrival uncertainties is a useful indicator of when the degeneracy is important. Synthetic data are used to explore the effect of this degeneracy systematically. It is found that phase-coherent tempo2 or temponest-based approaches are biased sometimes toward reporting small glitch sizes regardless of the true glitch size. Local estimates of the spin frequency alleviate this bias. A hidden Markov model is free from bias towards small glitches and announces explicitly the existence of multiple glitch solutions but sometimes fails to recover the correct glitch size. Two glitches in the UTMOST public data release are re-assessed, one in PSR J1709$-$4429 at MJD 58178 and the other in PSR J1452$-$6036 at MJD 58600. The estimated fractional frequency jump in PSR J1709$-$4429 is revised upward from $\Delta f/f = (54.6\pm 1.0) \times 10^{-9}$ to $\Delta f/f = (2432.2 \pm 0.1) \times 10^{-9}$ with the aid of additional data from the Parkes radio telescope. We find that the available UTMOST data for PSR J1452$-$6036 are consistent with $\Delta f/f = 270 \times 10^{-9} + N/(fT)$ with $N = 0,1,2$, where $T \approx 1\,\text{sidereal day}$ is the observation scheduling period. Data from the Parkes radio telescope can be included, and the $N = 0$ case is selected unambiguously with a combined dataset., Comment: Accepted for publication in MNRAS. 16 pages, 12 figures

In this paper, we propose a method to handle the problem of complete degeneracy in the study of lower dimensional invariant tori surviving from destructed resonant torus. As its application, we obtain a different proof of the existence of co-dimension one invariant tori. No other condition is assumed on the perturbation except for the smallness.

Exceptional point degeneracies, occurring in non-Hermitian systems, have challenged many well established concepts and led to the development of remarkable technologies. Here, we propose a family of autonomous motors whose operational principle relies on exceptional points via the opportune implementation of a (pseudo-)PT-symmetry and its spontaneous or explicit violation. These motors demonstrate a parameter domain of coexisting high efficiency and maximum work. In the photonic framework, they can be propelled by thermal radiation from the ambient thermal reservoirs and utilized as autonomous self-powered microrobots, or as micro-pumps for microfluidics in biological environments. The same designs can be also implemented with electromechanical elements for harvesting ambient mechanical (e.g., vibrational) noise for powering a variety of auxiliary systems. We expect that our proposal will contribute to the research agenda of energy harvesting by introducing concepts from mathematical and non-Hermitian wave physics. Fil: Fernández, Lucas Jonatan. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Nordeste. Instituto de Modelado e Innovación Tecnológica. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas Naturales y Agrimensura. Instituto de Modelado e Innovación Tecnológica; Argentina Fil: Kononchuk, Rodion. Ohio Wesleyan University.; Estados Unidos Fil: Kottos, Tsampikos. Ohio Wesleyan University.; Estados Unidos

International audience; In this paper, we consider the numerical approximation of hyperbolic systems of conservation laws with sti source terms and parabolic degeneracy in the asymptotic limit. We are more precisely interested in the design of high-order asymptotic-preserving schemes on unstructured meshes. Our approach is based on a very simple modication of the numerical ux associated with the usual HLL scheme and boils down to a sharp control of the underlying numerical diusion. The strategy allows to capture the correct asymptotic parabolic behavior and to preserve the high-order accuracy also in the asymptotic limit. Numerical experiments are proposed to illustrate these properties.

We consider the problem of sample degeneracy in Approximate Bayesian Computation. It arises when proposed values of the parameters, once given as input to the generative model, rarely lead to simulations resembling the observed data and are hence discarded. Such “poor” parameter proposals do not contribute at all to the representation of the parameter’s posterior distribution. This leads to a very large number of required simulations and/or a waste of computational resources, as well as to distortions in the computed posterior distribution. To mitigate this problem, we propose an algorithm, referred to as the Large Deviations Weighted Approximate Bayesian Computation algorithm, where, via Sanov’s Theorem, strictly positive weights are computed for all proposed parameters, thus avoiding the rejection step altogether. In order to derive a computable asymptotic approximation from Sanov’s result, we adopt the information theoretic “method of types” formulation of the method of Large Deviations, thus restricting our attention to models for i.i.d. discrete random variables. Finally, we experimentally evaluate our method through a proof-of-concept implementation.

The Edmond-Ogston model for phase separation in binary polymer mixtures is based on a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers. The second virial coefficients (B11, B12, B22) are the three parameters of the model. Analytical solutions are presented for the critical point and the spinodal in terms of molar concentrations. The calculation of the binodal is simplified by splitting the problem into a part that can be solved analytically and a (two-dimensional) problem that generally needs to be solved numerically, except in some specific cases. The slope of the tie-lines is identified as a suitable parameter that can be varied between two well-defined limits (close to and far away from the critical point) to perform the numerical part of the calculation systematically. Surprisingly, the analysis reveals a degenerate behavior within the model in the sense that a critical point or tie-line corresponds to an infinite set of triplets of second virial coefficients (B11, B12, B22). Since the Edmond-Ogston model is equivalent to the Flory-Huggins model up to the second order of the expansion in the concentrations, this degeneracy is also present in the Flory-Huggins model. However, as long as the virial coefficients predict the correct critical point, the shape of the binodal is relatively insensitive to the specific choice of the virial coefficients, except in a narrow range of values for the cross-virial coefficient B12.

In this paper we study a boundary value problem with the Poincare–Tricomi condition for a degenerate partial differential equation of elliptic-hyperbolic type of the second kind. In the hyperbolic part of a degenerate mixed differential equation of the second kind the line of degeneracy is a characteristic. For this type of differential equations a class of generalized solutions is introduced in the characteristic triangle. Using the properties of generalized solutions, the modified Cauchy and Dirichlet problems are studied. The solutions of these problems are found in the convenient form for further investigations. A new method has been developed for a differential equation of mixed type of the second kind, based on energy integrals. Using this method, the uniqueness of the considering problem is proved. The existence of a solution of the considering problem reduces to investigation of a singular integral equation and the unique solvability of this problem is proved by the Carleman–Vekua regularization method.