Recent zbMATH articles in MSC 39https://zbmath.org/atom/cc/392024-02-28T19:32:02.718555ZWerkzeugWeakly invariant norms: geometry of spheres in the space of skew-Hermitian matriceshttps://zbmath.org/1527.150182024-02-28T19:32:02.718555Z"Larotonda, Gabriel"https://zbmath.org/authors/?q=ai:larotonda.gabriel"Rey, Ivan"https://zbmath.org/authors/?q=ai:rey.ivanLet \(U_n\) denote the Lie group of all unitary \(n\times n\) complex matrices, and let \({u}_n\) stand for its Lie algebra, the real linear space of skew-Hermitian matrices. The authors study the dual norm of a weakly unitarily invariant norm on \({u}_n\), and show the correspondence among unit spheres of both norms by using the notion of polar duality. They show that faces of the unit sphere can be described by unit norm functionals, or equivalently, by elements of dual unit norm in \({u}_n\). In addition, they obtain a characterization of norming functionals, extreme points and smooth points of the sphere. Furthermore, they establish necessary and sufficient conditions for the inequality \(\varphi([X, [X, V ]]) \leq 0\) to be an equality, where \(\varphi\) is a norming functional for \(V\in {u}_n\).
As an application, they investigate the adjoint action \(V \to V + [X, V]\) of \({u}_n\) on itself. They show that this action always pushes vectors away from the unit sphere, and it only preserves the norm under strict conditions. In particular, these conditions ensure that for a strictly convex norm \(\|\cdot\|\), it holds that \(\|V + [X, V ]\| \geq \|V\|\), with equality if and only if \([X, V ] = 0\).
Reviewer: Mohammad Sal Moslehian (Mashhad)Refinements of generalised Hermite-Hadamard inequalityhttps://zbmath.org/1527.260152024-02-28T19:32:02.718555Z"Ojo, Adefisayo"https://zbmath.org/authors/?q=ai:ojo.adefisayo"Olanipekun, Peter Olamide"https://zbmath.org/authors/?q=ai:olanipekun.peter-olamideSummary: New insights, improvements and refinements of the well known Hermite-Hadamard inequality are established for a general class of convex functions. Inequalities involving products of two harmonically \(\phi_{h-s}\) functions are also obtained.Meromorphic solutions of Fermat type differential and difference equations of certain typeshttps://zbmath.org/1527.300252024-02-28T19:32:02.718555Z"Guo, Yinhao"https://zbmath.org/authors/?q=ai:guo.yinhao"Liu, Kai"https://zbmath.org/authors/?q=ai:liu.kai.6|liu.kai.3|liu.kai.1|liu.kai|liu.kai.4Summary: In this paper, we mainly consider the Fermat type differential equation
\[
f(z)^n+f'(z)^n=\varphi (z),
\]
where \(\varphi (z)=e^{h(z)}\) or \(1-e^{2h(z)}\), and \(h(z)\) is any entire function, and the Fermat type difference equation
\[
f(z)^n+f(z+c)^m=e^{P(z)},
\]
where \(P(z)\) is any entire function and \(c\) is a non-zero constant. We also provide short proofs for some existence results without complicated computations using elliptic functions.Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomialshttps://zbmath.org/1527.330072024-02-28T19:32:02.718555Z"Yafaev, D. R."https://zbmath.org/authors/?q=ai:yafaev.dimitri-rSummary: We find and discuss asymptotic formulas for orthonormal polynomials \(P_n(z)\) with recurrence coefficients \(a_n, b_n\). Our main goal is to consider the case where off-diagonal elements \(a_n\to\infty\) as \(n\to\infty\). Formulas obtained are essentially different for relatively small and large diagonal elements \(b_n\). Our analysis is intimately linked with spectral theory of Jacobi operators \(J\) with coefficients \(a_n, b_n\) and a study of the corresponding second order difference equations. We introduce the Jost solutions \(f_n(z), n\geq - 1\), of such equations by a condition for \(n\to\infty\) and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions \(P_n(z)\) by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for \(P_n(z)\) as \(n\to\infty\) in terms of the Wronskian of the solutions \(P_n(z)\) and \(f_n(z)\). The formulas obtained for \(P_n(z)\) generalize the asymptotic formulas for the classical Hermite polynomials where \(a_n=\sqrt{(n+1)/2}\) and \(b_n=0\). The spectral structure of Jacobi operators \(J\) depends crucially on a rate of growth of the off-diagonal elements \(a_n\) as \(n\to\infty\). If the Carleman condition is satisfied, which, roughly speaking, means that \(a_n=O(n)\), and the diagonal elements \(b_n\) are small compared to \(a_n\), then \(J\) has the absolutely continuous spectrum covering the whole real axis. We obtain an expression for the corresponding spectral measure in terms of the boundary values \(| f_{- 1}(\lambda\pm i0)|\) of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of \(J\) is discrete. We also review the case of stabilizing recurrence coefficients when \(a_n\) tend to a positive constant and \(b_n\to 0\) as \(n\to\infty\). It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way.Multiscale linearization of nonautonomous systemshttps://zbmath.org/1527.340662024-02-28T19:32:02.718555Z"Backes, Lucas"https://zbmath.org/authors/?q=ai:backes.lucas-h"Dragičević, Davor"https://zbmath.org/authors/?q=ai:dragicevic.davorThe authors provide sufficient conditions under which linear nonautonomous systems
\[
x_{n+1}=A_nx_n,\qquad \dot x=A(t) x
\]
and their nonlinear perturbations
\[
x_{n+1}=A_nx_n+F_n(x_n),\qquad \dot x=A(t) x+F(t,x)
\]
are topologically conjugated on the half-line. These conditions assume that the nonlinearities are well-behaved (Lipschitz, bounded), while there are summability conditions on the linear system. Furthermore, the assumptions on the nonlinear perturbations may differ along finitely many mutually complementary directions (explaining the term `Multiscale' in the title).
Reviewer: Christian Pötzsche (Klagenfurt)Riccati equation as topology-based model of computer worms and discrete SIR model with constant infectious periodhttps://zbmath.org/1527.340822024-02-28T19:32:02.718555Z"Satoh, Daisuke"https://zbmath.org/authors/?q=ai:satoh.daisuke"Uchida, Masato"https://zbmath.org/authors/?q=ai:uchida.masatoSummary: We propose discrete and continuous infection models of computer worms via e-mail or social networking site (SNS) messengers that were previously classified as worms spreading through topological neighbors. The discrete model is made on the basis of a new classification of worms as ``permanently'' or ``temporarily'' infectious. A temporary infection means that only the most recently infected nodes are infectious according to a difference equation. The discrete model is reduced to a Riccati differential equation (the continuous model) at the limit of a zero difference interval for the difference equation. The discrete and continuous models well describe actual data and are superior to a linear model in terms of the Akaike information criterion (AIC). Both models overcome the overestimation that is generated by applying a scan-based model to topology-based infection, especially in the early stages. The discrete model gives a condition in which all nodes are infected because the vulnerable nodes of the Riccati difference equation are finite and the solution of the Riccati difference equation plots discrete values on the exact solution of the Riccati differential equation. Also, the discrete model can also be understood as a model for the spread of infections of an epidemic virus with a constant infectious period and is described with a discrete susceptible-infected-recovered (SIR) model. The discrete SIR model has an exact solution. A control to reduce the infection is considered through the discrete SIR model.Delay dynamic equations on isolated time scales and the relevance of one-periodic coefficientshttps://zbmath.org/1527.341362024-02-28T19:32:02.718555Z"Bohner, Martin"https://zbmath.org/authors/?q=ai:bohner.martin-j"Cuchta, Tom"https://zbmath.org/authors/?q=ai:cuchta.tom"Streipert, Sabrina"https://zbmath.org/authors/?q=ai:streipert.sabrina-h(no abstract)The Taylor series method of order \(p\) and Adams-Bashforth method on time scaleshttps://zbmath.org/1527.341372024-02-28T19:32:02.718555Z"Georgiev, Svetlin G."https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Erhan, İnci M."https://zbmath.org/authors/?q=ai:erhan.inci-m(no abstract)Traveling waves for discrete reaction-diffusion equations in the general monostable casehttps://zbmath.org/1527.351182024-02-28T19:32:02.718555Z"Al Haj, M."https://zbmath.org/authors/?q=ai:haj.m-al|al-haj.mohammad"Monneau, R."https://zbmath.org/authors/?q=ai:monneau.regisSummary: We consider general fully nonlinear discrete reaction-diffusion equations \(u_t = F [u]\), described by some function \(F\). In the positively monostable case, we study monotone traveling waves of velocity \(c\), connecting the unstable state 0 to a stable state 1. Under Lipschitz regularity of \(F\), we show that there is a minimal velocity \(c_F^+\) such that there is a branch of traveling waves with velocities \(c \geq c_F^+\) and no traveling waves for \(c < c_F^+\). We also show that the map \(F \mapsto c_F^+\) is not continuous for the \(L^\infty\) norm on \(F\). Assuming more regularity of \(F\) close to the unstable state 0, we show that \(c_F^+ \geq c_F^\ast\), where the velocity \(c_F^\ast\) can be computed from the linearization of the equation around the unstable state 0. In addition, we show that the inequality can be strict for certain nonlinearities \(F\). On the contrary, under a KPP condition on \(F\), we show the equality \(c_F^+ = c_F^\ast\). Finally, we provide an example in which \(c_F^+\) is negative.The delayed Cucker-Smale model with attractive power-law potentialshttps://zbmath.org/1527.351412024-02-28T19:32:02.718555Z"Nie, Long"https://zbmath.org/authors/?q=ai:nie.long"Chen, Zili"https://zbmath.org/authors/?q=ai:chen.ziliSummary: We study the large-time behavior of the delayed Cucker-Smale model with attractive power-law potentials. By making full use of the energy fluctuation and another Lyapunov functional involving the communication function, it is proved that this model achieves consensus, i.e. velocity difference and space diameter converge to zero. More importantly, the precise convergence rate is established.Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domainhttps://zbmath.org/1527.351582024-02-28T19:32:02.718555Z"Guo, Hongpeng"https://zbmath.org/authors/?q=ai:guo.hongpeng"Guo, Zhiming"https://zbmath.org/authors/?q=ai:guo.zhiming"Li, Yijie"https://zbmath.org/authors/?q=ai:li.yijieSummary: This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.Scattering properties and dispersion estimates for a one-dimensional discrete Dirac equationhttps://zbmath.org/1527.353292024-02-28T19:32:02.718555Z"Kopylova, Elena"https://zbmath.org/authors/?q=ai:kopylova.elena-a"Teschl, Gerald"https://zbmath.org/authors/?q=ai:teschl.geraldThe authors consider the 1D discrete Dirac equation \(i\overset{.}{w}(t)= \mathcal{D}w(t)=(\mathcal{D}_{0}+Q)w(t)\), \(w_{n}=(u_{n},v_{n})\in \mathbb{C} ^{2}\), \(n\in \mathbb{Z}\), where the discrete free Dirac operator \(\mathcal{D} _{0}\) is defined by \(\mathcal{D}_{0}=\left( \begin{array}{cc} m & d \\
d^{\ast } & -m \end{array} \right) \), \(m>0\), with \((du)_{n}=u_{n+1}-u_{n}\), and \(Q\) is the bounded operator defined as \(Q=\left( \begin{array}{cc} 0 & q_{n} \\
q_{n} & 0 \end{array} \right) \), with \(q_{n}\neq 1\), \(n\in \mathbb{Z}\). \(\mathcal{D}\) is a bounded self-adjoint operator on \(\mathbf{l}^{2}(\mathbb{Z})=l^{2}(\mathbb{Z})\oplus l^{2}(\mathbb{Z})\). The purpose of the paper is to prove dispersion estimates for solutions to this problem, the authors improving results from their previous paper [J. Math. Anal. Appl. 434, No. 1, 191--208 (2016; Zbl 1330.35366)]. The authors first prove the \(\mathbf{l}^{1}\rightarrow \mathbf{ l}^{\infty }\) decay: \(\left\Vert e^{-it\mathcal{D}}P_{c}\right\Vert _{ \mathbf{l}^{1}\rightarrow \mathbf{l}^{\infty }}=O(t^{-1/3})\), \(t\rightarrow \infty \), assuming that \(d\in l_{1}^{1}\), where \(P_{c}\) is the orthogonal projection in \(\mathbf{l}^{2}\) onto the continuous spectrum of \(\mathcal{D}\) . The second main result proves the decay in \(\mathbf{l}_{\sigma }^{2}\rightarrow \mathbf{l}_{-\sigma }^{2}\): \(\left\Vert e^{-it\mathcal{D} }P_{c}\right\Vert _{\mathbf{l}_{\sigma }^{2}\rightarrow \mathbf{l}_{-\sigma }^{2}}=O(t^{-1/2})\), \(t\rightarrow \infty \), \(\sigma >1/2\). The next results consider the non-resonance case. Assuming that \(q\in l_{2}^{1}\), the authors prove that \(\left\Vert e^{-it\mathcal{D}}P_{c}\right\Vert _{\mathbf{l} _{1}^{1}\rightarrow \mathbf{l}_{-1}^{\infty }}=O(t^{-4/3})\), \(t\rightarrow \infty \), and \(\left\Vert e^{-it\mathcal{D}}P_{c}\right\Vert _{\mathbf{l} _{\sigma }^{2}\rightarrow \mathbf{l}_{-\sigma }^{2}}=O(t^{-3/2})\), \( t\rightarrow \infty \), \(\sigma >3/2\). The weighted space \(l_{\sigma }^{p}\), \( \sigma \in \mathbb{R}\), is equipped with the norm \(\left\Vert u\right\Vert _{l_{\sigma }^{p}}=\left( \sum_{n\in \mathbb{Z}}(1+\left\vert n\right\vert )^{p\sigma }\left\vert u_{n}\right\vert ^{p}\right) ^{1/p}\), \(p\in \lbrack 1,\infty )\), or \(\left\Vert u\right\Vert _{l_{\sigma }^{\infty }}=\sup_{n}(1+\left\vert n\right\vert )^{\sigma }\left\vert u_{n}\right\vert \). The authors recall properties of the Jost solutions \(w=(u,v)\) to the equation \(\mathcal{D}w=\lambda w\), they introduced in their previous paper and defined by the boundary conditions \(w_{n}^{\pm }(\theta )=\left( \begin{array}{c} u_{n}^{\pm }(\theta ) \\
v_{n}^{\pm }(\theta ) \end{array} \right) \rightarrow \left( \begin{array}{c} 1 \\
\alpha _{\mp }(\theta ) \end{array} \right) e^{\pm i\theta _{n}}\), with \(\alpha _{\mp }(\theta )=\frac{e^{\pm i\theta }-1}{m+\lambda }\), where \(\theta \in \overline{\Sigma }=\{-\pi \leq Re(\theta) \leq \pi \); \(Im(\theta) \geq 0\}\) is the solution to \( 2-2\cos \theta =\lambda ^{2}-m^{2}\). They recall the associated Gelfand-Levitan-Marchenko equations. They then prove that the scattering matrix of the operator \(\mathcal{D}\) is in the Wiener algebra defined as the set of all integrable functions whose Fourier coefficients are integrable, if the first moment of the potential is summable. For the proof of the dispersive decays, they use a variant of the van der Corput lemma concerning the oscillatory integral \(I(t)=\int_{a}^{b}eit\phi (\theta )f(\theta )d\theta \), \(-\pi \leq a<b\leq \pi \), where \(\phi (\theta )\) is a real-valued smooth function and \(f\) belongs to the Wiener algebra.
Reviewer: Alain Brillard (Riedisheim)Hamilton-Jacobi flows with nowhere differentiable initial datahttps://zbmath.org/1527.354912024-02-28T19:32:02.718555Z"Fujita, Yasuhiro"https://zbmath.org/authors/?q=ai:fujita.yasuhiro"Siconolfi, Antonio"https://zbmath.org/authors/?q=ai:siconolfi.antonio"Yamaguchi, Norikazu"https://zbmath.org/authors/?q=ai:yamaguchi.norikazuSummary: We deal with a sort of inverse problem. Namely, we aim at deriving features of initial data of an Hamilton-Jacobi flow starting from specific assumptions on the flow. To this scope, we introduce a class of nowhere differentiable functions, and characterize a function in this class through a property of the Hamilton-Jacobi flow issued from this function. This analysis shows that, despite the regularizing effect of the flow, nowhere differentiability of initial data can be nevertheless detected looking at the flow for positive times.Extensions and generalizations of lattice Gelfand-Dickey hierarchyhttps://zbmath.org/1527.370742024-02-28T19:32:02.718555Z"Zhang, Lixiang"https://zbmath.org/authors/?q=ai:zhang.lixiang"Li, Chuanzhong"https://zbmath.org/authors/?q=ai:li.chuanzhongSummary: In this paper, for the extended lattice Gelfand-Dickey hierarchy, we construct its \(n\)-fold Darboux transformation and additional flows. And we prove that these flows are actually symmetries of the extended lattice Gelfand-Dickey hierarchy. Further, we show how the additional flows act on the tau function. On this basis, we generalize the extended lattice Gelfand-Dickey hierarchy to the multicomponent and noncommutative versions, and give the Lax equations, Sato equations, zero-curvature equations and other equivalent expressions of these versions. Moreover, we investigate their Darboux transformations and additional symmetries.Miura type transform between non-abelian Volterra and Toda lattices and inverse spectral problem for band operatorshttps://zbmath.org/1527.370772024-02-28T19:32:02.718555Z"Osipov, A."https://zbmath.org/authors/?q=ai:osipov.alexander-a|osipov.a-yu|osipov.a-v|osipov.alexander-v|osipov.alexandr-v|osipov.a-i|osipov.andrey-v|osipov.andrey-s|osipov.andrei-v|osipov.aleksey-vSummary: We study a discrete Miura-type transformation between the hierarchies of non-Abelian semi-infinite Volterra (Kac-van Moerbeke) and Toda lattices with operator coefficients in terms of the inverse spectral problem for three-diagonal band operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl operator-valued function, can be used in solving initial-boundary value problem for the systems of both these hierarchies. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra hierarchy and the Toda sub-hierarchy which can be characterized via Lax operators corresponding to its lattices.Matrix spectral problems and integrability aspects of the Błaszak-Marciniak lattice equationshttps://zbmath.org/1527.370822024-02-28T19:32:02.718555Z"Wang, Deng-Shan"https://zbmath.org/authors/?q=ai:wang.dengshan"Li, Qian"https://zbmath.org/authors/?q=ai:li.qian.2"Wen, Xiao-Yong"https://zbmath.org/authors/?q=ai:wen.xiaoyong"Liu, Ling"https://zbmath.org/authors/?q=ai:liu.lingSummary: A method to derive the matrix spectral problems of the Błaszak-Marciniak lattice equations is proposed, and the matrix Lax representations of all the three-field and four-field Błaszak-Marciniak lattice equations are given explicitly. The integrability aspects of a three-field Błaszak-Marciniak lattice equation is studied as an example. To be specific, an integrable lattice hierarchy is constructed based on the matrix spectral problem of this lattice equation, the \(N\)-fold Darboux transformation and exact solutions are derived, the solitary wave structures and interaction behaviour of these exact solutions are displayed graphically, and finally the infinitely many conservation laws are listed in a standard wayStrong resonance bifurcations for a discrete-time prey-predator modelhttps://zbmath.org/1527.370912024-02-28T19:32:02.718555Z"Li, Bo"https://zbmath.org/authors/?q=ai:li.bo.12"Eskandari, Zohreh"https://zbmath.org/authors/?q=ai:eskandari.zohreh"Avazzadeh, Zakieh"https://zbmath.org/authors/?q=ai:avazzadeh.zakiehSummary: The aim of this paper is to introduce a two-dimensional discrete-time prey-predator, identify its fixed points, as well as investigate one- and two-parameter bifurcations. Numerical normal forms are used in bifurcation analysis. For this model, the Neimark-Sacker, period doubling and strong resonance bifurcations are observed. Based on the critical coefficients, the bifurcation scenarios can be identified. Based on numerical continuation methods, we use the MATLAB package MatContM to verify the analytical results and observe complex dynamics up to 16- iterate.Dynamics of a novel chaotic maphttps://zbmath.org/1527.370932024-02-28T19:32:02.718555Z"Sriram, Gokulakrishnan"https://zbmath.org/authors/?q=ai:sriram.gokulakrishnan"Ali, Ahmed M. Ali"https://zbmath.org/authors/?q=ai:ali.ahmed-m-ali"Natiq, Hayder"https://zbmath.org/authors/?q=ai:natiq.hayder"Ahmadi, Atefeh"https://zbmath.org/authors/?q=ai:ahmadi.atefeh"Rajagopal, Karthikeyan"https://zbmath.org/authors/?q=ai:rajagopal.karthikeyan"Jafari, Sajad"https://zbmath.org/authors/?q=ai:jafari.sajadSummary: In this study, another one-dimensional sinusoidal chaotic map around the bisector is constructed, which consists of two sine functions with irrational frequency ratios but comparable amplitude and phase. This design is motivated by the process equation. Investigating the unusual dynamics of this map and contrasting them with the process equation are the goals of this work. When two sine functions are included in a map, their interaction results in complex behaviors that have not been seen for the process equation. Like the process equation, this newly created map has an endless number of fixed points, but unlike the process equation, their stability cannot be fully defined. Furthermore, the suggested chaotic map can display a biotic-like time series when escaping regions are produced; however, unlike the process equation, these biotic-like time series only consist of transient components. In other words, the introduced map converges to a fixed point or periodic solution and never becomes unbounded because of the unlimited number of stable and unstable fixed points. The parameters of the map determine how long these biotic-like transient sections last. Additionally, coexisting attractors and multi-stability are seen in this map, much like in the process equation. The significant dynamics of this map are examined with time series, cobweb, bifurcation, and Lyapunov exponent diagrams. Tests are conducted before and after the creation of the escaping regions. Additionally, two-dimensional bifurcation diagrams are used to study the simultaneous impact of many pairs of factors on the map's dynamics.Chaotic dynamics of the fractional order Schnakenberg model and its controlhttps://zbmath.org/1527.371002024-02-28T19:32:02.718555Z"Uddin, Md. Jasim"https://zbmath.org/authors/?q=ai:uddin.md-jasim"Rana, S. M. Sohel"https://zbmath.org/authors/?q=ai:rana.sarker-md-sohelSummary: The Schnakenberg model is thought to be the Caputo fractional derivative. In order to create caputo fractional differential equations for the Schnakenberg model, a discretization process is first used. The fixed points in the model are categorized topologically. Then, we show analytically that, under certain parametric conditions, a Neimark-Sacker (NS) bifurcation and a Flip-bifurcation are supported by a fractional order Schnakenberg model. Using central manifold and bifurcation theory, we demonstrate the presence and direction of NS and Flip bifurcations. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order Schnakenberg model. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, period 2, 4, 7, 8, 10, 16, 20 and 40 orbits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. In order to support the system's chaotic characteristics, we also compute the maximal Lyapunov exponents and fractal dimensions quantitatively. Finally, the chaotic trajectory of the system is stopped using the OGY approach, hybrid control method, and state feedback method.Asymptotic behavior of homogeneous linear recurrent processes and their perturbationshttps://zbmath.org/1527.390012024-02-28T19:32:02.718555Z"Alexandru, Lazari"https://zbmath.org/authors/?q=ai:alexandru.lazariSummary: In this paper the impact of small perturbations on asymptotic evolution of homogeneous linear recurrent processes is investigated. Analytical methods for describing homogeneous linear recurrent systems, from convergence, periodicity and boundedness perspective, are presented. These methods are based on Jury Stability Criterion and the classification of the roots of minimal characteristic polynomial in relation to unit disc.Comparing a singular linear discrete time system to nabla difference equations of fractional orderhttps://zbmath.org/1527.390022024-02-28T19:32:02.718555Z"Ortega, Fernando"https://zbmath.org/authors/?q=ai:ortega.fernando"Cho, Sung"https://zbmath.org/authors/?q=ai:cho.sung"Kontzalis, Charalambos P."https://zbmath.org/authors/?q=ai:kontzalis.charalambos-pSummary: In this article we focus our attention on the relation between a singular linear discrete time system and a singular linear system of fractional nabla difference equations whose coefficients are square constant matrices. By using matrix pencil theory, first we give necessary and sufficient condition to obtain a unique solution for the continuous time model. After by assuming that the input vector changes only at equally space sampling
instants, we shall derive the corresponding discrete time state equation which yield the values of the solutions of the continuous time model which will connect the initial system to the singular linear system of fractional nabla difference equations.Time-step heat problem on the mesh: asymptotic behavior and decay rateshttps://zbmath.org/1527.390032024-02-28T19:32:02.718555Z"Abadias, Luciano"https://zbmath.org/authors/?q=ai:abadias.luciano"González-Camus, Jorge"https://zbmath.org/authors/?q=ai:gonzalez-camus.jorge"Rueda, Silvia"https://zbmath.org/authors/?q=ai:rueda.silviaSummary: In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and \(\ell^p\) estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on \(\ell^p\) spaces.The effects of hyperbolic eigenparameter on spectral analysis of a quantum difference equationshttps://zbmath.org/1527.390042024-02-28T19:32:02.718555Z"Aygar, Y."https://zbmath.org/authors/?q=ai:aygar.yeldaSummary: In this study, second-order nonselfadjoint expression and its associated boundary condition depending on an hyperbolic eigenparameter are discussed. We introduce the sets of eigenvalues and spectral singularities of a boundary value problem (BVP) which is defined with same quantum difference expression and boundary condition. Next, some spectral properties of eigenvalues and spectral singularities are investigated using the Jost solution, green function and resolvent operator of this BVP.Two finite sequences of symmetric \(q\)-orthogonal polynomials generated by two \(q\)-Sturm-Liouville problemshttps://zbmath.org/1527.390052024-02-28T19:32:02.718555Z"Masjed-Jamei, Mohammad"https://zbmath.org/authors/?q=ai:masjed-jamei.mohammad"Soleyman, Fatemeh"https://zbmath.org/authors/?q=ai:soleyman.fatemeh"Koepf, Wolfram"https://zbmath.org/authors/?q=ai:koepf.wolfram-aSummary: By using a symmetric generalization of Sturm-Liouville problems in \(q\)-difference spaces, we introduce two finite sequences of symmetric \(q\)-orthogonal polynomials and obtain their basic properties such as a second-order \(q\)-difference equations, the explicit form of the polynomials in terms of basic hypergeometric series, three-term recurrence relations and norm-square values based on a Ramanujan identity. We also show that one of the introduced sequences is connected with the little \(q\)-Jacobi polynomials.On a system of difference equations defined by the product of separable homogeneous functionshttps://zbmath.org/1527.390062024-02-28T19:32:02.718555Z"Boulouh, Mounira"https://zbmath.org/authors/?q=ai:boulouh.mounira"Touafek, Nouressadat"https://zbmath.org/authors/?q=ai:touafek.nouressadat"Tollu, Durhasan Turgut"https://zbmath.org/authors/?q=ai:tollu.durhasan-turgutSummary: In this work, we present results on the stability, the existence of periodic and oscillatory solutions of a general second order system of difference equations defined by the product of separable homogeneous functions of degree zero. Concrete systems for the obtained results are provided.Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler typehttps://zbmath.org/1527.390072024-02-28T19:32:02.718555Z"Diblík, Josef"https://zbmath.org/authors/?q=ai:diblik.josef"Korobko, Evgeniya"https://zbmath.org/authors/?q=ai:korobko.evgeniyaSummary: The article investigates a second-order nonlinear difference equation of Emden-Fowler type
\[
\Delta^2u(k)\pm k^{\alpha}{u}^m(k)=0,
\]
where \(k\) is the independent variable with values \(k=k_0, k_0+1,\ldots \,\), \(u:\{k_0,k_0+1,\ldots \}\to{\mathbb{R}}\) is the dependent variable, \(k_0\) is a fixed integer, and \({\Delta}^2u(k)\) is its second-order forward difference. New conditions with respect to parameters \(\alpha \in{\mathbb{R}}\) and \(m\in{\mathbb{R}}\), \(m\ne 1\), are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation
\[
y^{\prime\prime}(x)\pm{x}^{\alpha}{y}^m(x)=0.
\]
Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward differences as well. Previously known results are discussed, and illustrative examples are considered.On chaos and projective synchronization of a fractional difference map with no equilibria using a fuzzy-based state feedback controlhttps://zbmath.org/1527.390082024-02-28T19:32:02.718555Z"Zambrano-Serrano, Ernesto"https://zbmath.org/authors/?q=ai:zambrano-serrano.ernesto"Bekiros, Stelios"https://zbmath.org/authors/?q=ai:bekiros.stelios-d"Platas-Garza, Miguel A."https://zbmath.org/authors/?q=ai:platas-garza.miguel-a"Posadas-Castillo, Cornelio"https://zbmath.org/authors/?q=ai:posadas-castillo.cornelio"Agarwal, Praveen"https://zbmath.org/authors/?q=ai:agarwal.praveen"Jahanshahi, Hadi"https://zbmath.org/authors/?q=ai:jahanshahi.hadi"Aly, Ayman A."https://zbmath.org/authors/?q=ai:aly.ayman-aSummary: In this paper, by considering the Caputo-like delta difference definition, a fractional difference order map with chaotic dynamics and with no equilibria is proposed. The complex dynamical behaviors associated with fractional difference order maps are analyzed employing the phase portraits, bifurcations diagrams, and Lyapunov exponents. The complexity of the sequence generated by the chaotic difference map is studied using the permutation entropy approach. Afterwards, projective synchronization of the systems is investigated. Fuzzy logic engines as intelligent schemes are strong tools for control of various systems. However, studies that apply fuzzy logic engines for control of fractional-order discrete-time systems are rare. Hence, in the current study, by taking advantages of fuzzy systems, a new controller is proposed for the fractional-order discrete-time map. The fuzzy logic engine is implemented in order to enhance the performance and agility of the proposed control technique. The stability of the closed-loop systems and asymptotic convergence of the projective synchronization error based on the proposed control scheme are proven. Finally, numerical simulations which clearly confirm that the offered control technique is able to push the states of the fractional-order discrete-time system to the desired value in a short period of time are presented.Uniqueness related to higher order difference operators of entire functionshttps://zbmath.org/1527.390092024-02-28T19:32:02.718555Z"Liu, Xinmei"https://zbmath.org/authors/?q=ai:liu.xinmei"Chen, Junfan"https://zbmath.org/authors/?q=ai:chen.junfanNevanlinna's ``Five values theorem'' states that if two meromorphic functions share five distinct values ignoring multiplicities then the functions are identical. In 1977 \textit{L. A. Rubel} and \textit{C.-C. Yang} published a result in [Lect. Notes Math. 599, 101--103 (1977; Zbl 0362.30026)] showing that if a nonconstant entire function \(f\) shares with its derivative two distinct finite values (counting multiplicities) then \(f(z)\equiv f'(z).\) Modifications and generalisations of these results by various authors contribute to a branch of value distribution theory known as uniqueness theory. More recently, multiple papers are devoted to a variant of this theory concerning difference operators.
Let \(f(z)\) be an entire function and let \(T(r,f)\) be Nevanlinna's characteristic of \(f.\) Let \(\varrho(f)\) denote the order of \(f\) and \(\lambda(f)\) -- the exponent of convergence of zeros of \(f\). Moreover, let \(S(f)\) denote the set of all meromorphic functions \(a(z)\) small with respect to \(f(z)\) in the sense that \(T(r,a)=o(T(r,f))\) apart from, possibly, \(r\) in a set of finite logarithmic measure. We say that the functions \(f\) and \(g\) share a function \(a\in S(f)\cap S(g)\) CM if \(f-a\) and \(g-a\) have zeros at the same points with the same multiplicities.
For a nonzero constant \(\eta,\) the difference operator is defined as follows: \[\Delta_\eta f(z):=f(z+\eta)-f(z),\]
\[\Delta_\eta^{n+1}f(z):=\Delta_\eta^n f(z+\eta)-\Delta_\eta^nf(z).\] The main result in the paper is the following theorem.
Theorem. Let \(f(z)\) be a finite order transcendental entire function such that \(\lambda(f-a)<\varrho(f),\) where \(a(z)\in S(f)\) is an entire function and satisfies \(\varrho(a)<1\) and let \(\eta\in\mathbb{C}\) be a constant such that \(\Delta_\eta^{n+1}f(z)\not\equiv0.\) If \(\Delta_\eta^{n+1}f(z)\) and \(\Delta_\eta^{n}f(z)\) share \(\Delta_\eta^{n}a(z)\)CM, where \(\Delta_\eta^{n}a(z)\in S(\Delta_\eta^{n+1}f(z)),\) then \[f(z)=a(z)+Be^{Az},\] where \(A\) and \(B\) are two nonzero constants and \(a(z)\) reduces to a constant.
Reviewer: Ewa Ciechanowicz (Szczecin)Qualitative properties of nonlocal discrete operatorshttps://zbmath.org/1527.390102024-02-28T19:32:02.718555Z"Bravo, Jennifer"https://zbmath.org/authors/?q=ai:bravo.jennifer"Lizama, Carlos"https://zbmath.org/authors/?q=ai:lizama.carlos"Rueda, Silvia"https://zbmath.org/authors/?q=ai:rueda.silviaSummary: In this article, we prove new convexity results for nonlocal discrete operators across an entire region that covers sequential orders in two parameters. We review and extend current studies on the properties of positivity, monotonicity, and convexity, explore borderline cases, and provide new insights on such properties by means of original examples evidencing the sharpness of the results. Our method is based on the novel principle of transference.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}A variant of Wilson's functional equation on semigroupshttps://zbmath.org/1527.390112024-02-28T19:32:02.718555Z"Aserrar, Youssef"https://zbmath.org/authors/?q=ai:aserrar.youssef"Chahbi, Abdellatif"https://zbmath.org/authors/?q=ai:chahbi.abdellatif"Elqorachi, Elhoucien"https://zbmath.org/authors/?q=ai:elqorachi.elhoucienSummary: Let \(S\) be a semigroup. We determine the complex-valued solutions of the following functional equation
\[
f(xy)+\mu (y)f(\sigma (y)x) = 2f(x)g(y), \quad x,y\in S,
\]
where \(\sigma:S\rightarrow S\) is an automorphism, and \(\mu :S\rightarrow \mathbb{C}\) is a multiplicative function such that \(\mu (x\sigma (x))=1\) for all \(x\in S\).On a new class of two-variable functional equations on semigroups with involutionshttps://zbmath.org/1527.390122024-02-28T19:32:02.718555Z"EL-Fassi, Iz-iddine"https://zbmath.org/authors/?q=ai:el-fassi.iz-iddineSummary: Let \(S\) be a commutative semigroup, \(K\) a quadratically closed commutative field of characteristic different from 2, \(G\) a 2-cancellative abelian group and \(H\) an abelian group uniquely divisible by 2. The goal of this paper is to find the general non-zero solution \(f\colon S^2\to K\) of the d'Alembert type equation
\[
f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z)f(y,w),\quad x,y,z,w\in S,
\]
the general non-zero solution \(f\colon S^2\to G\) of the Jensen type equation
\[
f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z),\quad x,y,z,w\in S,
\]
the general non-zero solution \(f\colon S^2\to H\) of the quadratic type equation
\[
f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z)+2f(y,w),\quad x,y,z,w\in S,
\]
where \(\sigma\), \(\tau\colon S\to S\) are two involutions.Ulam stability of Jensen functional inequality on a class of noncommutative groupshttps://zbmath.org/1527.390132024-02-28T19:32:02.718555Z"Lu, Gang"https://zbmath.org/authors/?q=ai:lu.gang"Sun, Wenlong"https://zbmath.org/authors/?q=ai:sun.wenlong"Jin, Yuanfeng"https://zbmath.org/authors/?q=ai:jin.yuanfeng"Liu, Qi"https://zbmath.org/authors/?q=ai:liu.qiSummary: In the paper, we introduce new \(\rho \)-functional inequalities related to the Jensen functional equation and some properties. The Hyers-Ulam stability of functional inequalities is proved.On the Hyers-Ulam solution and stability problem for general set-valued Euler-Lagrange quadratic functional equationshttps://zbmath.org/1527.390142024-02-28T19:32:02.718555Z"Zhang, Dongwen"https://zbmath.org/authors/?q=ai:zhang.dongwen"Rassias, John Michael"https://zbmath.org/authors/?q=ai:rassias.john-michael"Li, Yongjin"https://zbmath.org/authors/?q=ai:li.yongjinSummary: By established a Banach space with the Hausdorff distance, we introduce the alternative fixed-point theorem to explore the existence and uniqueness of a fixed subset of Y and investigate the stability of set-valued Euler-Lagrange functional equations in this space. Some properties of the Hausdorff distance are furthermore explored by a short and simple way.Quantum oscillator as a minimization problemhttps://zbmath.org/1527.810472024-02-28T19:32:02.718555Z"D'Eliseo, Maurizio M."https://zbmath.org/authors/?q=ai:deliseo.maurizio-m(no abstract)The essential spectrum of a three particle Schrödinger operator on latticeshttps://zbmath.org/1527.810572024-02-28T19:32:02.718555Z"Lakaev, S. N."https://zbmath.org/authors/?q=ai:lakaev.saidakhmat-n"Boltaev, A. T."https://zbmath.org/authors/?q=ai:boltaev.a-tSummary: We consider the Hamiltonian \(\text{H}_{\mu\lambda},\mu,\lambda\in\mathbb{R}\) of a system of three-particles (two identical bosons and one different particle) moving on the lattice \({\mathbb{Z}}^d\), \(d=1,2\) interacting through zero-range pairwise potentials \(\mu\neq 0\) and \(\lambda\neq 0\). The essential spectrum of the three-particle discrete Schrödinger operator \(H_{\mu\lambda}(K)\), \(K\in\mathbb{T}^d \), being the three-particle quasi-momentum, is described by means of the spectrum of non-perturbed three-particle operator \(H_0(K)\) and the two-particle discrete Schrödinger operator \(h_{\mu}(k),h_{\lambda,\gamma}(k),k\in\mathbb{T}^d\), \(\gamma>0\). It is established that the essential spectrum of the three-particle discrete Schrödinger operator \(H_{\mu\lambda}(K)\), \(K\in\mathbb{T}^d\) consists of no more than three bounded closed intervals.On the number and locations of eigenvalues of the discrete Schrödinger operator on a latticehttps://zbmath.org/1527.810632024-02-28T19:32:02.718555Z"Akhmadova, M. O."https://zbmath.org/authors/?q=ai:akhmadova.mukhayyo-o"Alladustova, I. U."https://zbmath.org/authors/?q=ai:alladustova.i-u"Lakaev, S. N."https://zbmath.org/authors/?q=ai:lakaev.saidakhmat-norjigitovich|lakaev.saidakhmat-nSummary: We consider the family of Schrödinger operators \(\hat{H}_{\gamma\lambda\mu}=\hat{H}_0+\hat{V}_{\gamma\lambda\mu}\) on the one-dimensional lattice \(\mathbb{Z} \), where \(\hat{H}_0\) is a convolution operator with a given Hopping matrix \(\hat{\varepsilon} \), and \(\hat{V}_{\gamma\lambda\mu}\) is a multiplication operator by the function \(\hat{v}\) such that \(\hat{v}(0)=\gamma, \hat{v}(x)=\frac{\lambda}{2}\) for \(|x|=1, \hat{v}(x)=\frac{\mu}{2}\) for \(|x|=2\) and \(\hat{v}(x)=0\) for \(|x|>2, \gamma,\lambda,\mu\in\mathbb{R} \). Under certain conditions on the potential, we prove that the discrete Schrödinger operator \(\hat{H}_{\gamma\lambda\mu}\) can have zero, one, two or three eigenvalues outside the essential spectrum. Moreover, we obtain conditions for the existence of three eigenvalues, two of them situated below the bottom of the essential spectrum and the other one above its top.Multistability and stochastic dynamics of Rulkov neurons coupled via a chemical synapsehttps://zbmath.org/1527.920142024-02-28T19:32:02.718555Z"Bashkirtseva, Irina"https://zbmath.org/authors/?q=ai:bashkirtseva.irina-adolfovna"Pisarchik, Alexander N."https://zbmath.org/authors/?q=ai:pisarchik.alexander-n"Ryashko, Lev"https://zbmath.org/authors/?q=ai:ryashko.lev-borisovichSummary: We study complex dynamics of two Rulkov neurons unidirectionally connected via a chemical synapse with respect to three control parameters: (i) a parameter responsible for the type of dynamical behavior of a solitary neuron, (ii) coupling strength, and (iii) noise intensity. The coupled system exhibits various scenarios on the route from a stable equilibrium to chaos with respect to the coupling strength. We observe a variety of dynamical regimes, including mono-, bi- and tri-stability, order-chaos transitions and vice versa, as well as the coexistence of in-phase and anti-phase synchronization. We also study transitions between in-phase and out-of-phase synchronization with statistics on the duration of synchronization intervals and transitions from order to chaos. In addition to numerical simulations, we demonstrate the effectiveness of the analytical confidence ellipses method based on stochastic sensitivity approach.Complex dynamics of a discrete-time seasonally forced SIR epidemic modelhttps://zbmath.org/1527.920622024-02-28T19:32:02.718555Z"Naik, Parvaiz Ahmad"https://zbmath.org/authors/?q=ai:naik.parvaiz-ahmad"Eskandari, Zohreh"https://zbmath.org/authors/?q=ai:eskandari.zohreh"Madzvamuse, Anotida"https://zbmath.org/authors/?q=ai:madzvamuse.anotida"Avazzadeh, Zakieh"https://zbmath.org/authors/?q=ai:avazzadeh.zakieh"Zu, Jian"https://zbmath.org/authors/?q=ai:zu.jianSummary: In this paper, a discrete-time seasonally forced SIR epidemic model with a nonstandard discretization scheme is investigated for different types of bifurcations. Although many researchers have already suggested numerically that this model can exhibit chaotic dynamics, not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Second, the flip, Neimark-Sacker, and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. The model is discretized by a novel technique, namely a nonstandard finite difference discretization scheme (NSFD). Some graphical representations of the model are presented to verify the obtained results.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Model reduction of non-minimal discrete-time linear-time-invariant systemshttps://zbmath.org/1527.930252024-02-28T19:32:02.718555Z"Arif, D. K."https://zbmath.org/authors/?q=ai:arif.didik-khusnul"Adzkiya, D."https://zbmath.org/authors/?q=ai:adzkiya.dieky"Apriliani, E."https://zbmath.org/authors/?q=ai:apriliani.erna"Khasanah, I. N."https://zbmath.org/authors/?q=ai:khasanah.i-nSummary: Model reduction is a method for reducing the order of mathematical models such that the behavior of reduced system is similar with the original system. Many models in real systems are non minimal. However, to the best of our knowledge, there is no literature that discusses the model reduction of non-minimal systems. Therefore in this paper, we propose a procedure for model reduction of non-minimal discrete-time linear-time-invariant systems by using balanced truncation methods. In this paper, we generate an algorithm to reduce non-minimal discrete-time linear-time-invariant systems. From the simulation results, we obtain a reduced system with similar behavior with the original system. Furthermore, we also conclude that the behavior of the reduced system is very close to the original system in high frequency.