Nondegenerate Case Research Articles

Caratheodory periodic perturbations of degenerate systems

We study the structure of the set of harmonic solutions to T-periodically perturbed coupled differential equations on differentiable manifolds, where the perturbation is allowed to be of Caratheodory-type regularity. Employing degree-theoretic methods, we prove the existence of a noncompact connected set of nontrivial T-periodic solutions that, in a sense, emanates from the set of zeros of the unperturbed vector field. The latter is assumed to be ''degenerate'': Meaning that, contrary to the usual assumptions on the leading vector field, it is not required to be either trivial nor to have a compact set of zeros. In fact, known results in the ``nondegenerate case can be recovered from our ones. We also provide some illustrating examples of Lienard- and \(\phi\)-Laplacian-type perturbed equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/39/abstr.html

Decoupling numerical method based on deep neural network for nonlinear degenerate interface problems

Many practical problems, including modeling composite materials, nuclear waste disposal, oil reservoir simulations, and flows in porous medium, commonly involve interface problems. However, the solution to interface problems with discontinuous coefficients of PDEs using fully decoupled numerical methods is challenging. The main objective is to solve the interface problems with fully decoupled numerical methods. This paper proposes an efficient decoupled numerical method for solving degenerate interface problems with double singularities. First, we divide the whole domain into singular and regular subdomains. Then, we use the Deep Neural Network (DNN) to find the solution on the singular subdomain and approximate the solution on the regular subdomain using the finite difference method. The scheme combines the solutions of singular and regular subdomains, which is an exciting idea. The key to the new approach is to split nonlinear degenerate partial differential equations with an interface into two independent boundary value problems based on deep learning. In this way, the expansion of the solution on the singular domain does not contain undetermined parameters, and two independent boundary value problems can be solved with any well-known traditional numerical methods. The main advantage of the proposed scheme is that we not only get the order of convergence of the degenerate interface problems on the whole domain, but we also can calculate VERY BIG jump ratio (such as 1012:1 or 1:1012) for the interface problems including degenerate and non-degenerate cases. Finally, with examples, we demonstrate the efficiency and accuracy of methods for 1 and 2D problems. It is also interesting that the proposed method is valid for the interface problems with degenerate and non-degenerate cases, we show it with some examples.

The Schur–Potapov Algorithm in the General Matrix Case and Its Application to the Matricial Schur Problem

This paper is a generalization of the topic handled in Bogner et al. (Oper Theory 1(1):55–95, 2007a, Oper Theory 1(2):235–278, 2007b) where the Schur–Potapov algorithm (SP-algorithm) was handled in the context of non-degenerate p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences and non-degenerate p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur functions. In particular, the interplay between both types of algorithms was intensively studied there. This was itself a generalization of the classical Schur algorithm (Schur in J Reine Angew Math 148:122–145, 1918) to the non-degenerate matrix case. In treating the matrix case a result due to Potapov (Potapov in Trudy Moskov Mat Obšč 4:125–236, 1955) concerning particular linear fractional transformations of contractive p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} matrices was used. For this reason, the notation SP-algorithm was already chosen in Dubovoj et al. (Matricial version of the classical Schur problem, volume 129 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992). We are going to introduce both types of SP-algorithms as well for arbitrary p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences as for arbitrary p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur functions. Again we will intensively discuss the interplay between both types of algorithms. Applying the SP-algorithm, a complete treatment of the matricial Schur problem in the most general case is established. A one-step extension problem for finite p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences is considered. Central p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences are studied under the view of SP-parameters.

Description of the set of all possible masses at a fixed point of solutions of a truncated matricial Hamburger moment problem

We describe the set of all possible masses at a fixed point of the real axis in the context of the most general case of a truncated matricial (possibly degenerate) Hamburger moment problem. This set is a matricial interval, which we explicitly construct from the given data in the non-degenerate and degenerate cases simultaneously. In particular, we determine the maximum possible mass.

A Potapov-Type Approach to a Particular Truncated Stieltjes Moment Problem in the Case of an Odd Number of Prescribed Matricial Moments

The paper gives a parametrization of the solution set of a matricial Stieltjes-type truncated power moment problem in the non-degenerate and degenerate cases. The problem will be reformulated as an interpolation problem for a distinguished class of holomorphic matrix-valued functions. An essential role plays a solution of the corresponding system of Potapov’s fundamental matrix inequalities.

A posteriori error estimates for the Richards equation

The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated H 1 ( H − 1 ) H^1(H^{-1}) , L 2 ( L 2 ) L^2(L^2) , and the L 2 ( H 1 ) L^2(H^1) errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space–time efficiency error bounds are then obtained in a standard H 1 ( H − 1 ) ∩ L 2 ( H 1 ) H^1(H^{-1})\cap L^2(H^1) norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as space discretization, time discretization, quadrature, linearization, and data oscillation are identified and separated. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic case and a benchmark case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.

Triad resonance for internal waves in a uniformly stratified fluid: Rogue waves and breathers.

Three-wave (triad) resonance in a uniformly stratified fluid is investigated as a case study of energy transfer among oscillatory modes. The existence of a degenerate triad is demonstrated explicitly, where two components have identical group velocity. An illuminating example is a resonance involving waves from modes 1, 3, 5 families, but many other combinations are possible. The physical applications and nonlinear dynamics of rogue waves derived analytically in the literature are examined. Exact solutions with four free parameters (two related to the amplitudes of the background plane waves, two related to the frequencies of slowly varying envelopes) describe motions localized in both space and time. The differences between rogue waves of the degenerate versus the nondegenerate cases are highlighted. The phase and profile of the degenerate case rogue waves are correlated. The volume or energy of the rogue wave (defined as the total extent or energy contents of the fluid set in motion for the duration of the rogue wave) may change drastically, if the wave envelope parameters vary. Pulsating modes (breathers) have been studied previously by layered-fluid and modified Korteweg-de Vries models. Here we extend the consideration to stratified fluids but for the simpler case of nondegenerate triads. Instabilities of fission and fusion of breathers are confirmed computationally with Floquet analysis. This knowledge should prove useful for energy transfer processes in the oceans.

The quantum Carnot-like heat engine: The level degenerate case

The two-level quantum heat engines have been studied extensively. Before that, the optimization of most heat engines was mainly concentrated in non-degenerate cases. We consider the level degeneracy, then calculate the maximum operation time and the irreversible dissipation coefficient of the two-level system under the scheme of linear tuned levels. In the high-temperature limit, we find that the level degeneracy affects the maximum power of the two-level heat engine. Especially in the case of the same degeneracy of the ground state and the excited state, the influence is the greatest. In addition, we also show that the degeneracy does not affect the efficiency at maximum power and the consistent relation of the two-level Carnot-like heat engine.

Existence of solutions to a critical Kirchhoff equation with logarithmic perturbation

In this paper, the following Kirchhoff type elliptic equation − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u ln ⁡ | u | 2 + | u | 4 u , which involves a power type nonlinearity with critical Sobolev exponent and a logarithmic type perturbation, is investigated. The fact that the logarithmic term satisfies neither the monotonicity condition nor the Ambrosetti-Rabinowitz condition brings some additional difficulties when one is looking for weak solutions to this problem. By the use of the Mountain Pass Lemma and the concentration compactness principle, it is proved that the problem admits a ground state solution for q ∈ ( 4 , 6 ) and for any 0 $ ]]> λ > 0 . It is worth pointing out that the analysis includes both the nondegenerate case (a>0) and the degenerate case (a = 0).

Sum-frequency generation and amplification processes in semiconductor superlattices

Semiconductor superlattices are very well-known structures due to their specific electron transport properties, making them extremely attractive to be employed in electronic or optoelectronic devices. The interest in such structures has been recently additionally stirred up due to the first successful experimental demonstration of parametric gain in GaAs/AlGaAs superlattices, resulting in the generation of harmonics, half-harmonics and fractional harmonics. This invention paves the way for a successful realization of superlattice-based generators and amplifiers up to the terahertz frequency range. Despite the emerging experimental results and decade-long theoretical research, unresolved aspects, related to the physical processes inside the superlattices, persist. Lately, the biasing effect was extensively analysed for the case of degenerate processes in the superlattice; however, the non-degenerate case was left out of frame until now. Within this research, we further expand the boundaries of previous investigation by exploring the differences of non-degenerate processes. The study uncovers the asymmetry appearance of the probe field vs. relative phase dependences as well as the possibility of parametric fractional frequency generation. Finally, the concept of energy reflow between two participating probes is predicted and discussed.

Nondegeneracy implies the existence of parametrized families of free boundaries

In this paper, we introduce the notion of variational free boundary problem. Namely, we say that a free boundary problem is variational if its solutions can be characterized as the critical points of some shape functional. Moreover, we extend the notion of nondegeneracy of a critical point to this setting. As a result, we provide a unified functional-analytical framework that allows us to construct families of solutions to variational free boundary problems whenever the shape functional is nondegenerate at some given solution.As a clarifying example, we apply this machinery to construct families of nontrivial solutions to the two-phase Serrin's overdetermined problem in both the degenerate and nondegenerate case.

The design strain sensitivity of the schenberg spherical resonant antenna for gravitational waves

The main purpose of this study is to review the Schenberg resonant antenna transfer function and to recalculate the antenna design strain sensitivity for gravitational waves. We consider the spherical antenna with six transducers in the semi dodecahedral configuration. When coupled to the antenna, the transducer-sphere system will work as a mass-spring system with three masses. The first one is the antenna effective mass for each quadrupole mode, the second one is the mass of the mechanical structure of the transducer first mechanical mode and the third one is the effective mass of the transducer membrane that makes one of the transducer microwave cavity walls. All the calculations are done for the degenerate (all the sphere quadrupole mode frequencies equal) and non-degenerate sphere cases. We have come to the conclusion that the “ultimate” sensitivity of an advanced version of Schenberg antenna (aSchenberg) is around the standard quantum limit (although the parametric transducers used could, in principle, surpass this limit). However, this sensitivity, in the frequency range where Schenberg operates, has already been achieved by the two aLIGOs in the O3 run, therefore, the only reasonable justification for remounting the Schenberg antenna and trying to place it in the sensitivity of the standard quantum limit would be to detect gravitational waves with another physical principle, different from the one used by laser interferometers. This other physical principle would be the absorption of the gravitational wave energy by a resonant mass like Schenberg.

Flow count data-driven static traffic assignment models through network modularity partitioning

AbstractAccurate static traffic assignment models are important tools for the assessment of strategic transportation policies. In this article we present a novel approach to partition road networks through network modularity to produce data-driven static traffic assignment models from loop detector data on large road systems. The use of partitioning allows the estimation of the key model input of Origin–Destination demand matrices from flow counts alone. Previous network tomography-based demand estimation techniques have been limited by the network size. The amount of partitioning changes the Origin–Destination estimation optimisation problems to different levels of computational difficulty. Different approaches to utilising the partitioning were tested, one which degenerated the road network to the scale of the partitions and others which left the network intact. Applied to a subnetwork of England’s Strategic Road Network and other test networks, our results for the degenerate case showed flow and travel time errors are reasonable with a small amount of degeneration. The results for the non-degenerate cases showed that similar errors in model prediction with lower computation requirements can be obtained when using large partitions compared with the non-partitioned case. This work could be used to improve the effectiveness of national road systems planning and infrastructure models.

Convergence rates to the planar stationary solution to a 2D model of the radiating gas on half space

This paper is concerned with the asymptotic stability of a planar stationary solution to an initial-boundary value problem for a two-dimensional hyperbolic–elliptic coupled system of the radiating gas on half space. We show that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity under small initial perturbation. This result is proved by the standard L2-energy method and the div–curl decomposition. Moreover, we prove that the solution (u, q) converges to the corresponding planar stationary solution at the rate t−α/2−1/4 for the non-degenerate case and t−1/4 for the degenerate case. The proof is based on the time and space weighted energy method.

Nontrivial one-loop recursive reduction relation

In [1], we proposed a universal method to reduce one-loop integrals with both tensor structure and higher-power propagators. But the method is quite redundant as it does not utilize the results of lower rank cases when addressing certain tensor integrals. Recently, we found a remarkable recursion relation [2, 3], where a tensor integral is reduced to lower-rank integrals and lower terms corresponding to integrals with one or more propagators being canceled. However, the expression of the lower terms is unknown. In this paper, we derive this non-trivial recursion relation for non-degenerate and degenerate cases and provides an explicit expression for the lower terms, thus simplifying and speeding up the reduction process.

Asymptotic behavior of solutions for a new general class of parabolic Kirchhoff type equation with variable exponent sources

In this paper, we study the following problem:{ut−M(∫Ω|∇u|p(x)p(x)dx)Δp(x)u=|u|m(x)−2u,(x,t)∈Ω×(0,T),u(x,t)=0,(x,t)∈∂Ω×(0,T),u(x,0)=u0(x),x∈Ω, where Ω⊂RN is a bounded domain with smooth boundary ∂Ω, the functions p,m:Ω¯→R and the Kirchhoff function M:[0,∞)→[0,∞) are specified below. We give a new class of general Kirchhoff function which covers both non-degenerate and degenerate cases, and investigate its effects on the existence and non-existence of global weak solutions to the problem in case the initial energy is subcritical. Moreover, we also give the decay estimate for the energy functional in the former case and an upper bound for the maximal existence time in the latter case. This is a continue works by Nhan et al. (2020) [27] and also an affirmative answer for the comments by Guo et al. (2022) [13].

Scalar dark matter with Z3 symmetry in the type-II seesaw mechanism

We study a simple complex scalar singlet dark matter (DM) model with ${Z}_{3}$ symmetry in the framework of the type-II seesaw mechanism. We use the model to explain the excess of electron-positron flux measured by the AMS-02, DAMPE, and Fermi-LAT Collaborations, which is encouraged by the decay of the triplets produced from dark matter annihilations in the Galactic halo. We focus on the nondegenerate case in which the mass of DM is larger than that of the triplets' and deliberately alleviates the leptophilic properties of the DM, so that the semiannihilation effects are enhanced. With the guarantee of ${Z}_{3}$ symmetry, by fitting the antiproton spectrum observed in the PAMELA and AMS experiments, we find that the DM cubic terms and the couplings between DM and the Higgs boson are strongly constrained, leading to the semiannihilation cross-section fraction less than 11% when DM mass is given at 2 TeV.

Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

Let G be a Lie group and g its Lie algebra. The aim here is to develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in g⊗g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor in the tensor square of the Lie algebra g measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence, in terms of explicit algebraic expressions, between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures arising from the data including the symmetric 2-tensor in g⊗g. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. Along the way, a side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.

Front propagation in a double degenerate equation with delay

Abstract The current article is concerned with the traveling fronts for a class of double degenerate equations with delay. We first show that the traveling fronts decay algebraically at one end, while those may decay exponentially or algebraically at the other end, which depend on the wave speed of traveling fronts. Based on the asymptotical behavior, the uniqueness and stability of traveling fronts are then proved. Of particular interest is the effect of the lower order term and higher order term on the critical speed. We mention that, under the double degenerate case, the nonlinear reaction is less competitive due to the appearance of degeneracy. This yields that the critical speed depends on the lower order term and higher order term, which is different from the nondegenerate case.

Complete Closed-Form and Accurate Solution to Pose Estimation From 3D Correspondences

Computing the pose from 3D data acquired in two different frames is of high importance for several robotic tasks like odometry, SLAM and place recognition. The pose is generally obtained by solving a least-squares problem given points-to-points, points-to-planes or points to lines correspondences. The non-linear least-squares problem can be solved by iterative optimization or, more efficiently, in closed-form by using solvers of polynomial systems. In this letter, a complete and accurate closed-form solution for a weighted least-squares problem is proposed. Adding weights for each correspondence allow to increase robustness to outliers. Contrary to existing methods, the proposed approach is complete since it is able to solve the problem in any non-degenerate case and it is accurate since it is guaranteed to find the global optimal estimate of the weighted least-squares problem. Simulations and experiments on real data demonstrate the superior accuracy and robustness of the proposed algorithm compared to previous approaches.