Recent zbMATH articles in MSC 39https://zbmath.org/atom/cc/392021-11-25T18:46:10.358925ZWerkzeugDiscrete encountershttps://zbmath.org/1472.050032021-11-25T18:46:10.358925Z"Bauer, Craig P."https://zbmath.org/authors/?q=ai:bauer.craig-p``This book provides a refreshing approach to discrete mathematics. The author blends traditional course topics and applications with historical context, pop culture reference, and open problems. It focuses on the historical development of the subject and provides fascinating details of the people behind the mathematics, along with their motivations, deepening readers' appreciation of mathematics. This unique book covers many of the same topics found in traditional textbooks, but does so in an alternative, entertaining style that better captures readers' attention. In addition to standard discrete mathematics material, the author shows the interplay between the discrete and the continuous and includes high-interest topics such as fractals, chaos theory, cellular automata, money-saving financial mathematics, and much more. Not only will readers gain a greater understanding of mathematics and its culture, they will also be encouraged to further explore the subject. Long lists of references at the end of each chapter make this easy.'' (from the presentation of the book).
\par Its chapters are the following: Continuous vs. discrete; Logic; Proof techniques; Practice with proofs; Set theory; Venn diagrams; The functional view of mathematics; The multiplication principle; Permutations; Combinations; Pascal and the arithmetic triangle; Stirling and Bell numbers; The basics of probability; The Fibonacci sequence; The tower of Hanoi; Population models; Financial mathematics (and more); More difference equations; Chaos theory and fractals; Cellular automata; Graph theory; Trees; Relations, partial orderings, and partitions; Index. Each of them ends with a set of exercises and references/further reading. The book also includes many illustrations and portraits from the history of mathematics and ends with a series of 39 colored illustrations. The text's narrative style is that of a popular book, not a dry textbook. Its multidisciplinary approach makes this nice book ideal for liberal arts mathematics classes, leisure reading, or as a reference for professors looking to supplement traditional courses.On the nature of four models of symmetric walks avoiding a quadranthttps://zbmath.org/1472.050062021-11-25T18:46:10.358925Z"Dreyfus, Thomas"https://zbmath.org/authors/?q=ai:dreyfus.thomas"Trotignon, Amélie"https://zbmath.org/authors/?q=ai:trotignon.amelieSummary: We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set is diagonally symmetric (i.e., with no steps in anti-diagonal directions). In that situation, after a transformation of the plane, we derive a quadrant-like functional equation. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially algebraic generating series.Explicit inverse of near Toeplitz pentadiagonal matrices related to higher order difference operatorshttps://zbmath.org/1472.150042021-11-25T18:46:10.358925Z"Kurmanbek, Bakytzhan"https://zbmath.org/authors/?q=ai:kurmanbek.bakytzhan"Erlangga, Yogi"https://zbmath.org/authors/?q=ai:erlangga.yogi-a"Amanbek, Yerlan"https://zbmath.org/authors/?q=ai:amanbek.yerlanSummary: This paper analyzes the inverse of near Toeplitz pentadiagonal matrices, arising from a finite-difference approximation to the fourth-order nonlinear beam equation. Explicit non-recursive inverse matrix formulas and bounds of norms of the inverse matrix are derived for the clamped-free and clamped-clamped boundary conditions. The bound of norms is then used to construct a convergence bound for the fixed-point iteration of the form \(\boldsymbol{u}=f(\boldsymbol{u})\) for solving the nonlinear equation. Numerical computations presented in this paper confirm the theoretical results.On the stability of left \(\delta \)-centralizers on Banach Lie triple systemshttps://zbmath.org/1472.170152021-11-25T18:46:10.358925Z"Ghobadipour, Norouz"https://zbmath.org/authors/?q=ai:ghobadipour.norouz"Sepasian, Ali Reza"https://zbmath.org/authors/?q=ai:sepasian.ali-rezaSummary: In this paper under a condition, we prove that every Jordan left \(\delta \)-centralizer on a Lie triple system is a left \(\delta \)-centralizer. Moreover, we use a fixed point method to prove the generalized Hyers-Ulam-Rassias stability associated with the Pexiderized Cauchy-Jensen type functional equation \[rf\left(\frac{x+y}{r}\right)+sg\left(\frac{x-y}{s}\right)=2h(x),\] for \(r,s \in \mathbb R \setminus \{0\}\) in Banach Lie triple systems.Extending means to several variableshttps://zbmath.org/1472.260162021-11-25T18:46:10.358925Z"Losonczi, Attila"https://zbmath.org/authors/?q=ai:losonczi.attilaAuthor's abstract: We begin the study of how to extend few variable means to several variable ones and how to shrink means of several variables to less variables. With the help of one of the techniques we show that it is enough to check an inequality between two quasi-arithmetic means in 2-variables and that simply implies the inequality in m-variables. The technique has some relation to Markov chains. This method can be applied to symmetrization and compounding means as well.Two variations on \((A_3 \times A_1 \times A_1)^{(1)}\) type discrete Painlevé equationshttps://zbmath.org/1472.341602021-11-25T18:46:10.358925Z"Shi, Yang"https://zbmath.org/authors/?q=ai:shi.yangSummary: By considering the normalizers of reflection subgroups of types \(A^{(1)}_1\) and \(A^{(1)}_3\) in \(\widetilde{W}(D_5^{( 1 )})\), two subgroups: \( \widetilde{W} ( A_3 \times A_1 )^{( 1 )} \ltimes W(A_1^{( 1 )})\) and \(\widetilde{W} ( A_1 \times A_1 )^{( 1 )} \ltimes W(A_3^{( 1 )})\) can be constructed from a \((A_3 \times A_1 \times A_1)^{(1)}\) type subroot system. These two symmetries arose in the studies of discrete Painlevé equations
[\textit{K. Kajiwara} et al., Lett. Math. Phys. 62, No. 3, 259--268 (2002; Zbl 1030.37045);
\textit{T. Takenawa}, Funkc. Ekvacioj, Ser. Int. 46, No. 1, 173--186 (2003; Zbl 1151.34341);
\textit{N. Okubo} and \textit{T. Suzuki}, ``Generalized \(q\)-Painlevé VI systems of type \((A_2n+1 + A_1 + A_1)^{(1)}\) arising from cluster algebra'', Preprint, \url{arXiv:1810.03252}],
where certain non-translational elements of infinite order were shown to give rise to discrete Painlevé equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups [\textit{R. B. Howlett}, J. Lond. Math. Soc., II. Ser. 21, 62--80 (1980; Zbl 0427.20040);
\textit{B. Brink} and \textit{R. B. Howlett}, Invent. Math. 136, No. 2, 323--351 (1999; Zbl 0926.20024)].
This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.Functional difference equations and eigenfunctions of a Schrödinger operator with \(\delta' -\) interaction on a circular conical surfacehttps://zbmath.org/1472.351052021-11-25T18:46:10.358925Z"Lyalinov, Mikhail A."https://zbmath.org/authors/?q=ai:lyalinov.mikhail-anatolievichSummary: Eigenfunctions and their asymptotic behaviour at large distances for the Laplace operator with singular potential, the support of which is on a circular conical surface in three-dimensional space, are studied. Within the framework of incomplete separation of variables an integral representation of the Kontorovich-Lebedev (KL) type for the eigenfunctions is obtained in terms of solution of an auxiliary functional difference equation with a meromorphic potential. Solutions of the functional difference equation are studied by reducing it to an integral equation with a bounded self-adjoint integral operator. To calculate the leading term of the asymptotics of eigenfunctions, the KL integral representation is transformed to a Sommerfeld-type integral which is well adapted to application of the saddle point technique. Outside a small angular vicinity of the so-called singular directions the asymptotic expression takes on an elementary form of exponent decreasing in distance. However, in an asymptotically small neighbourhood of singular directions, the leading term of the asymptotics also depends on a special function closely related to the function of parabolic cylinder (Weber function).Linear encoding of the spatiotemporal cathttps://zbmath.org/1472.370642021-11-25T18:46:10.358925Z"Gutkin, B."https://zbmath.org/authors/?q=ai:gutkin.boris-s"Cvitanović, P."https://zbmath.org/authors/?q=ai:cvitanovic.predrag"Jafari, R."https://zbmath.org/authors/?q=ai:jafari.roya|jafari.rad-nader|jafari.rahim|jafari.reza|jafari.raheleh.1"Saremi, A. K."https://zbmath.org/authors/?q=ai:saremi.a-k"Han, L."https://zbmath.org/authors/?q=ai:han.liu|han.limin|han.lifeng|han.lixia|han.lengyi|han.lili|han.lijia|han.libin|han.liantao|han.lifang|han.lingzhan|han.liguo|khan.lakhkar|han.luyi|han.liwei|han.liming|han.lihua|han.luofeng|han.liwen|han.lingjuan|han.lingxiong|han.lixing|han.lixin|han.le|han.luqing|han.longsheng|han.lu|han.lin|han.lijuan|han.lun|han.lihong|han.litao|han.lingling|han.luhui|han.liting|han.liping|han.lanshan|han.li|han.liangxiu|khan.liyaquat|han.lansheng|han.ling|han.luchang|han.lei|han.linghui|han.lijun|han.lianfang|han.lining|han.lijie|han.long|han.lidong|han.lina|han.laiju|han.liyan|han.longxi|han.lihui|han.leitao|han.lianghao|han.liubin|han.liangyu|khan.ladlay|khan.latifur|khan.laiq|han.liubing|han.liqin|han.longjun|han.liangThe technical novelty of the paper is a working out of a very special model of many-particle dynamics, interpreted as a discretization of a classical \(d\)-dimensional field theory. The main motivation of this endeavour is to describe, by means of the discrete symbolic dynamics, the spatiotemporal chaos (or turbulence) in spatially extended, strongly nonlinear field theories. To exemplify the idea, one may invoke an approximation of turbulent motions of a physical flow, in terms of coupled map lattice models (with discretized spacetime labels). It is the dynamics of finite domains that captures important small-scale spatial structures, being in turn modeled by discrete time maps (Poincaré sections of a single `particle' dynamics) attached to lattice sites. It is presumed that neighboring sites are coupled, in consistency with ranslational and reflection symmetries of the problem. This particular path of reasoning is followed in the present paper by exploiting the coupled cat map lattice built from the Thom-Anosov-Arnol'd-Sinai cat maps (modeling the Hamiltonian dynamics of individual `particles') at sites of a one-dimensional spatial lattice, presumed to be linearly coupled (hence `linear encoding') to their nearest neighbors. The key insight is that the arising two-dimensional spatiotemporal pattern is best described by the corresponding two-dimensional spatiotemporal symbol lattice rather than by a one-dimensional temporal symbol sequence. In case of the spatiotemporal cat its every solution is uniquely encoded by a linear transformation to the corresponding finite alphabet two-dimensional symbol lattice, a spatiotemporal generalization of the linear code for temporal evolution of a cat map, introduced by \textit{I. Percival} and \textit{F. Vivaldi} [Physica D 27, 373--386 (1987; Zbl 0647.58031)]. It is shown that the state of the system over a finite spatiotemporal domain can be described with exponentially increasing precision by a finite pattern of symbols. A systematic, lattice Green function methodology is provided to calculate the frequency (i.e., the measure) of such states. The authors rephrase this result as follows: ``local dynamics, observed through a finite spatiotemporal window, can often be thought of as a visitation sequence of a finite repertoire of finite patterns. To make statistical predictions about the system, one needs to know how often a given pattern occurs.''Continuum limits of pluri-Lagrangian systemshttps://zbmath.org/1472.370682021-11-25T18:46:10.358925Z"Vermeeren, Mats"https://zbmath.org/authors/?q=ai:vermeeren.matsSummary: A pluri-Lagrangian (or Lagrangian multiform) structure is an attribute of integrability that has mainly been studied in the context of multidimensionally consistent lattice equations. It unifies multidimensional consistency with the variational character of the equations. An analogous continuous structure exists for integrable hierarchies of differential equations. We present a continuum limit procedure for pluri-Lagrangian systems. In this procedure, the lattice parameters are interpreted as Miwa variables, describing a particular embedding in continuous multi-time of the mesh on which the discrete system lives. Then, we seek differential equations whose solutions interpolate the embedded discrete solutions. The continuous systems found this way are hierarchies of differential equations. We show that this continuum limit can also be applied to the corresponding pluri-Lagrangian structures. We apply our method to the discrete Toda lattice and to equations H1 and Q1$_{\delta=0}$ from the ABS list.Direct linearization approach to discrete integrable systems associated with \(\mathbb{Z}_\mathcal{N}\) graded Lax pairshttps://zbmath.org/1472.370802021-11-25T18:46:10.358925Z"Fu, Wei"https://zbmath.org/authors/?q=ai:fu.weiSummary: \textit{A. P. Fordy} and \textit{P. Xenitidis} [J. Phys. A, Math. Theor. 50, No. 16, Article ID 165205, 30 p. (2017; Zbl 1367.37055)]
recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of \(\mathbb{Z}_{\mathcal{N}}\) graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy-Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel'fand-Dikii hierarchy.Lectures on pentagram maps and KdV hierarchieshttps://zbmath.org/1472.370812021-11-25T18:46:10.358925Z"Khesin, Boris"https://zbmath.org/authors/?q=ai:khesin.boris-aThe aim of this paper is to survey definitions and integrability properties of pentagram maps on generic plane polygons and their generalizations to higher dimensions. The author also describes the corresponding continuous limits: in dimension \(d\) the pentagram map turns out to be the \((2,d+1)\)-equation of the Korteweg-de Vries hierarchy, thus generalizing the Boussinesq equation in two dimensions. This paper is organized as follows. Section 1 is an introduction to the subject. Section 2 deals with continuous limits of pentagram maps. Section 3 deals with a concrete example concerning an integrable pentagram map in dimension three. Section 4 deals with more general integrable pentagram maps.
For the entire collection see [Zbl 1458.55002].Conservation laws and self-consistent sources for an integrable lattice hierarchy associated with a three-by-three discrete matrix spectral problemhttps://zbmath.org/1472.370822021-11-25T18:46:10.358925Z"Li, Yu-Qing"https://zbmath.org/authors/?q=ai:li.yuqing"Yin, Bao-Shu"https://zbmath.org/authors/?q=ai:yin.baoshuSummary: A lattice hierarchy with self-consistent sources is deduced starting from a three-by-three discrete matrix spectral problem. The Hamiltonian structures are constructed for the resulting hierarchy. Liouville integrability of the resulting equations is demonstrated. Moreover, infinitely many conservation laws of the resulting hierarchy are obtained.Persistence of a discrete-time predator-prey model with stage-structure in the predatorhttps://zbmath.org/1472.370862021-11-25T18:46:10.358925Z"Ackleh, Azmy S."https://zbmath.org/authors/?q=ai:ackleh.azmy-s"Hossain, Md. Istiaq"https://zbmath.org/authors/?q=ai:hossain.md-istiaq"Veprauskas, Amy"https://zbmath.org/authors/?q=ai:veprauskas.amy"Zhang, Aijun"https://zbmath.org/authors/?q=ai:zhang.aijunSummary: We propose and investigate a discrete-time predator-prey model with a structured predator population. We describe the predator population using two stages, juveniles and adults, and assume that only the adult stage consumes the prey species. For this model, we discuss conditions for the existence and global stability of the extinction and predator-free equilibria as well as conditions for the existence and uniqueness of an interior equilibrium. We show that when the predator-free equilibrium destabilizes, the interior equilibrium is stable in a neighborhood of the bifurcation. We also find the conditions for the persistence of both prey and predator populations. Finally, we use numerical simulations to demonstrate various dynamical scenarios. We find that introducing stage-structure into the predator population allows for complicated dynamics that are not possible when the predator is unstructured.
For the entire collection see [Zbl 1467.39001].General theory of the higher-order quaternion linear difference equations via the complex adjoint matrix and the quaternion characteristic polynomialhttps://zbmath.org/1472.390012021-11-25T18:46:10.358925Z"Wang, Chao"https://zbmath.org/authors/?q=ai:wang.chao"Chen, Desu"https://zbmath.org/authors/?q=ai:chen.desu"Li, Zhien"https://zbmath.org/authors/?q=ai:li.zhienSummary: Since the non-commutativity and particular structure of the quaternion algebra, the quternion difference equations (short for QDCEs) have a large difference from the classical theory of difference equations. In this paper, we establish a general theory of the higher-order linear QDCEs including the criteria of the linear independence (or dependence) of the discrete functions in the quaternion space, Liouville formulas, the structure theorems of the general solutions and the particular solutions with the quaternion power and trigonometric form, etc. By introducing the complex adjoint difference equations and the quaternion characteristic polynomial of the higher-order linear QDCEs, some basic results of the homogeneous and non-homogeneous difference equations are obtained. Through the analysis of the complex adjoint matrix and the quaternion eigenvalue, the general solutions of the higher-order linear QDCEs with variable and with constant coefficients are established. Several methods of obtaining the general solutions for the higher-order linear QDCEs are demonstrated and several examples are provided to illustrate the feasibility of our obtained results.Factorization method for solving multipoint problems for second order difference equations with polynomial coefficientshttps://zbmath.org/1472.390022021-11-25T18:46:10.358925Z"Parasidis, I. N."https://zbmath.org/authors/?q=ai:parasidis.ioannis-nestorios|parasidis.ivan-nesterovich"Hahamis, P."https://zbmath.org/authors/?q=ai:hahamis.pSummary: This paper is devoted to the study of second order linear difference equations with polynomial coefficients subject to multipoint boundary conditions. We provide necessary and sufficient conditions for the existence and uniqueness of solutions and find the unique solution in closed form by using factorization techniques.
For the entire collection see [Zbl 07261771].A note on ergodicity for nonautonomous linear difference equationshttps://zbmath.org/1472.390032021-11-25T18:46:10.358925Z"Pituk, Mihály"https://zbmath.org/authors/?q=ai:pituk.mihalySummary: For a class of nonautonomous linear difference equations with bounded, nonnegative and uniformly primitive coefficients it is shown that the normalized positive solutions are asymptotically equivalent to the Perron vectors of the transition matrix at infinity.
For the entire collection see [Zbl 1467.39001].A matrix approach to some second-order difference equations with sign-alternating coefficientshttps://zbmath.org/1472.390042021-11-25T18:46:10.358925Z"Anđelić, Milica"https://zbmath.org/authors/?q=ai:andelic.milica"Du, Zhibin"https://zbmath.org/authors/?q=ai:du.zhibin"da Fonseca, Carlos M."https://zbmath.org/authors/?q=ai:da-fonseca.carlos-martins"Kılıç, Emrah"https://zbmath.org/authors/?q=ai:kilic.emrahSummary: In this paper, we analyse and unify some recent results on the double sequence \(\{y_{n,k}\}\), for \(n, k\geqslant 1\), defined by the second-order difference equation \[y_{n,k}=(-1)^{\lfloor(n-1)/k\rfloor}y_{n-1,k}-y_{n-2,k},\]
with \(y_{1,k}=1\) and \(y_{2,k}=0\), in terms of matrix theory and orthogonal polynomials theory. Moreover, we provide a general solution to
\[z_{n,k}=(-1)^{\lfloor(n-1)/k\rfloor}z_{n-1,k}-(-1)^{\lfloor (n-2)/a\rfloor}z_{n-2,k},\]
using a closely related approach. We discuss briefly other recent problems involving a general recurrence relation of second order and relate them with the existing literature.Analytic invariant curves for an iterative equation related to Ricker-type second-order equationhttps://zbmath.org/1472.390052021-11-25T18:46:10.358925Z"Zhao, Hou Yu"https://zbmath.org/authors/?q=ai:zhao.houyu"Fečkan, Michal"https://zbmath.org/authors/?q=ai:feckan.michalThe authors show the existence of analytic invariant curves of the difference equation
\[
x_{n+1}=x_{n-1}e^{a-x_{n-1}-x_{n}},
\]
or equivalently of the mapping \(T(x,y)=(y,xe^{a-x-y})\). They seek for analytic invariant curves of \(T\) in the form \(y=f(x)\). The considered system can be written in the form of the iterative equation
\[
f(f(x))=xe^{a-x-f(x)},
\]
where \(x\in\mathbb{C}\) and \(a\in\mathbb{R}\) is a fixed number. This equation is reduced, with \(f(x)=g(\alpha g^{-1}(x) )\), to the auxiliary equation
\[
g(\alpha^{2}x)=g(x)e^{a-g(x)-g(\alpha x)}.
\]
Thus, by proving the existence of analytic solutions for this last equation, the analytic invariant curves of original equation can be determined.
Set \(\alpha = \pm e^{a/2}\). The authors distinguish three different cases for \(\alpha\):
\begin{itemize}
\item[(1)] \(0< |\alpha| < 1\);
\item[(2)] \(\alpha = e^{2\pi i\theta}, \theta\in \mathbb{R}\setminus\mathbb{Q}\) and \(\theta\) defines a Brjuno number, i.e.,
\[
\sum_{n=0}^{\infty}\frac{\log q_{n+1}}{q_{n}}<\infty,
\]
where \(\{p_{n}/q_{n}\}\) denotes the sequence of partial fractions of the continued fraction expansion of \(\theta\);
\item[(3)] \(\alpha = e^{2\pi iq/p}\) for some integer \(p\in \mathbb{N}\) with \(p\geq 2\) and \(q\in \mathbb{Z}\setminus\{ 0 \}\) and \(\alpha \neq e^{2\pi i\xi/\upsilon}\) for all \(1\leq \upsilon\leq p-1\) and \(\xi\in \mathbb{Z}\setminus\{ 0 \}\).
\end{itemize}Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequalityhttps://zbmath.org/1472.390062021-11-25T18:46:10.358925Z"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Al-Mdallal, Qasem M."https://zbmath.org/authors/?q=ai:al-mdallal.qasem-mAuthors' abstract: In this article, we studied the Caputo and Riemann-Liouville type discrete fractional difference initial value problems with discrete Mittag-Leffler kernels. The existence and uniqueness of the solution is proved by using Banach contraction principle. The linear type equations are used to prove new discrete fractional versions of the Gronwall's inequality. The nabla discrete Laplace transform is used to obtain solution representations. The proven Gronwall's inequality under a new defined $\alpha$-Lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions. Numerical examples are discussed to demonstrate the reliability of the theoretical results.New estimations for discrete Sturm-Liouville problemshttps://zbmath.org/1472.390072021-11-25T18:46:10.358925Z"Bas, Erdal"https://zbmath.org/authors/?q=ai:bas.erdal"Ozarslan, Ramazan"https://zbmath.org/authors/?q=ai:ozarslan.ramazanThe authors consider the second-order Sturm-Liouville difference equation
\[
\Delta^2x(n-1)+q(n)x(n)=-\lambda x(n), \quad n\in[a,b]_{\mathbb{Z}}, \tag{1}
\]
where \([a,b]_{\mathbb{Z}}:=[a,b]\cap\mathbb{Z}\) is a discrete interval, with the initial conditions
\[
x(a-1)=-h, \quad x(a)=1.
\]
Equation (1) is uniquely solvable by recursion, given the initial values at \(x(a-1)\) and \(x(a)\). Note that the authors use \(x(0)\) and \(x(1)\) at this place without any warning. As a main result the authors present a formula for the solution of (1), which is obtained in the following way. Consider first the homogeneous part of equation (1), i.e., \[ \Delta^2x(n-1)+\lambda x(n)=0, \quad n\in[a,b]_{\mathbb{Z}}, \] whose solution \(x_h(n)=c_1x_1(n)+c_2x_2(n)\) can be obtained explicitly in terms of \(\lambda\) and \(n\). Then they consider the method of variation of constants to incorporate the nonhomogeneous term in the right-hand side of \[ \Delta^2x(n-1)+\lambda x(n)=-q(n)x(n), \quad n\in[a,b]_{\mathbb{Z}}. \] In this way they obtain a formula for \(x(n)\), which of course contains \(x(n)\) on both sides (!) (Equation (13) in Theorem 6). A similar construction (see Equation (33) in Theorem 7) is derived for equation (1) with the initial conditions
\[
x(a-1)=1, \quad x(a)=0.
\]
In addition, certain asymptotic formulas of the form \(x(n)=O(e^{|t|n})\) for \(n\in\mathbb{Z}^+\) are given in Theorems 8 and 9. Here the role of the parameter \(t\) is not specified. The authors also make a comparison of the values of \(x(n)\) with some sample values of \(n\) for the potentials \(q(n)=1/(n+1)\) and \(q(n)=1/\sqrt{n+1}\) and the spectral parameters \(\lambda=1\) and \(\lambda=2\).
Reviewer's remark: The present reviewer finds difficult to understand the usefulness of the above approach. Furthermore, the paper contains inaccuracies in the formulations of the theorems and their proofs.Discrete harmonic analysis associated with ultraspherical expansionshttps://zbmath.org/1472.390082021-11-25T18:46:10.358925Z"Betancor, Jorge J."https://zbmath.org/authors/?q=ai:betancor.jorge-j"Castro, Alejandro J."https://zbmath.org/authors/?q=ai:castro.alejandro-j"Fariña, Juan C."https://zbmath.org/authors/?q=ai:farina.juan-carlos"Rodríguez-Mesa, L."https://zbmath.org/authors/?q=ai:rodriguez-mesa.lourdesAuthors' abstract: In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted \(\ell^p\)-boundedness properties of maximal operators and Littlewood-Paley \(g\)-functions defined by Poisson and heat semigroups generated by the difference operator \[\Delta_{\lambda} f(n):=a_n^{\lambda} f(n+1)-2f(n)+a_{n-1}^{\lambda} f(n-1),\quad n\in \mathbb{N}, \,\lambda >0,\] where \(a_n^{\lambda } :=\{(2\lambda +n)(n+1)/[(n+\lambda )(n+1+\lambda )]\}^{1/2}\), \(n\in \mathbb{N} \), and \(a_{-1}^{\lambda }:=0\). We also prove weighted \(\ell^p \)-boundedness properties of transplantation operators associated with the system \(\{\varphi_n^{\lambda } \}_{n\in \mathbb{N}}\) of ultraspherical functions, a family of eigenfunctions of \(\Delta_\lambda \). In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.Convexity, monotonicity, and positivity results for sequential fractional nabla difference operators with discrete exponential kernelshttps://zbmath.org/1472.390092021-11-25T18:46:10.358925Z"Goodrich, Christopher S."https://zbmath.org/authors/?q=ai:goodrich.christopher-s"Jonnalagadda, Jagan M."https://zbmath.org/authors/?q=ai:jonnalagadda.jagan-m"Lyons, Benjamin"https://zbmath.org/authors/?q=ai:lyons.benjaminSummary: We consider positivity, monotonicity, and convexity results for discrete fractional operators with exponential kernels. Our results cover both the sequential and nonsequential cases, and we demonstrate both similarities and dissimilarities between the exponential kernel case and fractional differences with other types of kernels. This shows that the qualitative information gleaned in the exponential kernel case is not precisely the same as in other cases.Global stability of traveling waves for a spatially discrete diffusion system with time delayhttps://zbmath.org/1472.390102021-11-25T18:46:10.358925Z"Liu, Ting"https://zbmath.org/authors/?q=ai:liu.ting"Zhang, Guo-Bao"https://zbmath.org/authors/?q=ai:zhang.guobao.1|zhang.guobaoSummary: This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in \( L^\infty(\mathbb{R})\times L^\infty(\mathbb{R}) \) with the exponential convergence rate \( e^{-\mu t}\) for some constant \( \mu>0 \).Zeros, growth and Taylor coefficients of entire solutions of linear \(q\)-difference equationshttps://zbmath.org/1472.390112021-11-25T18:46:10.358925Z"Bergweiler, Walter"https://zbmath.org/authors/?q=ai:bergweiler.walterThe author studies the linear \(q\)-difference equation
\[
\sum_{j=0}^m a_j(z)f(q^jz) = b(z),
\]
where \(q\in\mathbb{C}\) with \(0 < |q| < 1\) and \(b\) and the \(a_j\) are polynomials. He determines the asymptotic behavior of the Taylor coefficients of the transcendental entire solutions. He proves that the zeros of the transcendental entire solutions are asymptotic to finitely many geometric progressions under a suitable assumption on the associated Newton-Puiseux diagram. He also sharpens the results on the growth rate of entire solutions due to the author and \textit{W. K. Hayman} [Comput. Methods Funct. Theory 3, No. 1--2, 55--78 (2003; Zbl 1087.39022)].A symmetric quantum calculushttps://zbmath.org/1472.390122021-11-25T18:46:10.358925Z"Brito Da Cruz, Artur M. C."https://zbmath.org/authors/?q=ai:brito-da-cruz.artur-m-c"Martins, Natália"https://zbmath.org/authors/?q=ai:martins.natalia-f"Torres, Delfim F. M."https://zbmath.org/authors/?q=ai:torres.delfim-f-mSummary: We introduce the \(\alpha\), \(\beta\)-symmetric difference derivative and the \(\alpha\), \(\beta\)-symmetric Nörlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generalization of the forward and backward \(h\)-calculus.
For the entire collection see [Zbl 1277.00035].A note on \(q\)-partial differential equations for generalized \(q\)-2D Hermite polynomialshttps://zbmath.org/1472.390132021-11-25T18:46:10.358925Z"Cao, Jian"https://zbmath.org/authors/?q=ai:cao.jian"Cai, Tianxin"https://zbmath.org/authors/?q=ai:cai.tianxin"Cai, Li-Ping"https://zbmath.org/authors/?q=ai:cai.lipingSummary: In this short paper, we generalize Ismail-Zhang's \(q\)-2D Hermite polynomials [\textit{M. E. H. Ismail} and \textit{R. Zhang}, Trans. Am. Math. Soc. 369, No. 10, 6779--6821 (2017; Zbl 1380.33004)] with an extra parameter and prove that if an analytic function in several variables satisfies a set of partial differential equations of second order, then it can be expanded in terms of the product of the generalized \(q\)-2D Hermite polynomials. In addition, we give some generating functions as applications.
For the entire collection see [Zbl 1467.39001].Solving fractional difference equations using the Laplace transform methodhttps://zbmath.org/1472.390142021-11-25T18:46:10.358925Z"Li, Xiao-yan"https://zbmath.org/authors/?q=ai:li.xiaoyan"Jiang, Wei"https://zbmath.org/authors/?q=ai:jiang.wei.1Summary: We discuss the Laplace transform of the Caputo fractional difference and the fractional discrete Mittag-Leffer functions. On these bases, linear and nonlinear fractional initial value problems are solved by the Laplace transform method.Oscillatory and asymptotic behavior of higher order nonlinear difference equationshttps://zbmath.org/1472.390152021-11-25T18:46:10.358925Z"Grace, Said R."https://zbmath.org/authors/?q=ai:grace.said-rSummary: We shall present new oscillation and asymptotic behavior criteria of higher order nonlinear difference equations of the form
\[
\Delta \Big( (a(t)\left( {\Delta^{n-1} x(t)} \right)^\alpha \Big) +{q(t)x}^\beta(t-m+1)=0,
\]
with nonnegative real coefficients.An improved product type oscillation test for partial difference equationshttps://zbmath.org/1472.390162021-11-25T18:46:10.358925Z"Karpuz, Başak"https://zbmath.org/authors/?q=ai:karpuz.basak"Özsavaş, Büşra"https://zbmath.org/authors/?q=ai:ozsavas.busraSummary: We present new oscillation tests for the \(\mathrm{P}\Delta\mathrm{E}\)
\[
a u (m + 1, n) + b u (m, n + 1) - c u (m, n) + p (m, n) u (m - \tau, n - \sigma) = 0,\ m, n = 0, 1, \dots,
\]
where \(a, b, c \in (0, \infty)\) and \(\tau, \sigma \in\{0, 1, \dots \}\), and \(\{p(m, n)\} \subset [0, \infty)\). Our main result improves some of the well-known results in the literature. We also present a numerical example, where all the previous results in the literature fail to deliver an answer.Oscillation and nonoscillation of difference equations with several delayshttps://zbmath.org/1472.390172021-11-25T18:46:10.358925Z"Karpuz, Başak"https://zbmath.org/authors/?q=ai:karpuz.basak"Stavroulakis, Ioannis P."https://zbmath.org/authors/?q=ai:stavroulakis.ioannis-pThe authors investigate the oscillatory and nonoscillatory solutions of the following delay difference equation:
\[
\Delta x\left( n\right) +\sum_{k=1}^{m}p_{k}\left( n\right) x\left( n-\tau_{k}\right) =0, \qquad n=0,1, \dots
\]
where \(\Delta\) is the forward difference operator, i.e., \(\Delta x\left( n\right) :=x\left( n+1\right) -x\left( n\right) \), for \(k=1,2,\dots,m\), \(\tau_{k}\) is a nonnegative integer and \(\left\{ p_{k}\left( n\right) \right\}_{n=0}^{\infty}\) is a nonnegative sequence of reals. Based on both lower and upper limit conditions, they obtain some criteria for the behavior of solutions.Oscillation criteria for higher-order neutral type difference equationshttps://zbmath.org/1472.390182021-11-25T18:46:10.358925Z"Köprübaşi, Turhan"https://zbmath.org/authors/?q=ai:koprubasi.turhan"Ünal, Zafer"https://zbmath.org/authors/?q=ai:unal.zafer"Bolat, Yaşar"https://zbmath.org/authors/?q=ai:bolat.yasarThe authors consider higher-order neutral-type difference equations of the form \(\Delta(r_n\Delta^{k-1}(y_n+p_ny_{\tau_n}))+q_nf(y_{\sigma_n})=0\), where \(\tau_n\ge n\) and \(\sigma_n\le n\). They establish conditions that guarantee that either the equation is oscillatory for even \(k\) or every nonoscillatory solution tends to zero for odd \(k\).Existence and approximation of zeroes of monotone operators by solutions to nonhomogeneous difference inclusionshttps://zbmath.org/1472.390192021-11-25T18:46:10.358925Z"Djafari Rouhani, Behzad"https://zbmath.org/authors/?q=ai:djafari-rouhani.behzad"Jamshidnezhad, Parisa"https://zbmath.org/authors/?q=ai:jamshidnezhad.parisa"Saeidi, Shahram"https://zbmath.org/authors/?q=ai:saeidi.shahramThe authors extend and improve some known results concerning the convergence of solutions of the nonhomogeneous second order inclusion
\[
\begin{cases} u_{n+1}-(1+\theta_n)u_n+\theta_n u_{n-1}\in c_n Au_n+f_n,\;n\geq 1,\\
u_0=x,\;\sup\{\| u_n \|:\, n\geq 0\}<\infty, \end{cases}
\]
where \(A\) is a maximal monotone operator on a real Hilbert space. They also obtain new results about the existence of zeroes of \(A\) and the weighted averages of solutions to a zero of \(A\).Recent results on summations and Volterra difference equations via Lyapunov functionalshttps://zbmath.org/1472.390202021-11-25T18:46:10.358925Z"Raffoul, Youssef"https://zbmath.org/authors/?q=ai:raffoul.youssef-naimSummary: In this research we utilize Lyapunov functionals to obtain boundedness on all solutions, exponential stability and \(l_p\)-stability on the zero solution of summation equations and Volterra difference equations.
For the entire collection see [Zbl 1467.39001].Caputo nabla fractional boundary value problemshttps://zbmath.org/1472.390212021-11-25T18:46:10.358925Z"Peterson, Allan"https://zbmath.org/authors/?q=ai:peterson.allan-c"Hu, Wei"https://zbmath.org/authors/?q=ai:hu.weiSummary: We study boundary value problems with the Caputo nabla difference in the context of discrete fractional nabla calculus, especially when the right boundary condition has a fractional order. We first construct the Green's function for the general case and study the properties of the Green's function in several cases. We then apply the cone theory in a Banach space to show the existence of positive solutions to a nonlinear boundary value problem.
For the entire collection see [Zbl 1467.39001].Bifurcation scenarios under symbolic template iterations of flat top tent mapshttps://zbmath.org/1472.390222021-11-25T18:46:10.358925Z"Silva, Luís"https://zbmath.org/authors/?q=ai:silva.luis-f-p|silva.luis-g|silva.luis-m-a|silva.luis-carlos|silva.luis-nuno|silva.luis-oSummary: The behavior of orbits for iterated flat top maps has been widely studied since the dawn of discrete dynamics as a research field. However, little is known about orbit behavior if the map changes along with the iterations. In this work we consider a family of flat top tent maps and investigate in which ways the iteration pattern (symbolic template) can affect the structure of the bifurcation scenarios.
For the entire collection see [Zbl 1467.39001].Hyers-Ulam stability and best constant for Cayley \(h\)-difference equationshttps://zbmath.org/1472.390232021-11-25T18:46:10.358925Z"Anderson, Douglas R."https://zbmath.org/authors/?q=ai:anderson.douglas-robert"Onitsuka, Masakazu"https://zbmath.org/authors/?q=ai:onitsuka.masakazuThe authors study the Hyers-Ulam stability, compute the best constant and construct radial solutions for Cayley \(h\)-difference equations with complex parameter \(\mu\). This analysis is new and extends known results when \(\mu= 0\) and when \(\mu = 1\).
In particular, they prove Hyers-Ulam instability on the imaginary \(\mu\)-circle, while otherwise Hyers-Ulam stability holds true. They find an explicit best constant for the case of stability in terms of \(\mu\) and show that this constant can also be expressed as a function of the radius.Stabilization and synchronization of discrete-time fractional chaotic systems with non-identical dimensionshttps://zbmath.org/1472.390242021-11-25T18:46:10.358925Z"Bendoukha, Samir"https://zbmath.org/authors/?q=ai:bendoukha.samirSummary: This paper investigates the stabilization and synchronization of two fractional chaotic maps proposed recently, namely the 2D fractional Hénon map and the 3D fractional generalized Hénon map. We show that although these maps have non-identical dimensions, their synchronization is still possible. The proposed controllers are evaluated experimentally in the case of non-identical orders or time-varying orders. Numerical methods are used to illustrate the results.Stability conditions for linear difference system with two delayshttps://zbmath.org/1472.390252021-11-25T18:46:10.358925Z"Deger, S. U."https://zbmath.org/authors/?q=ai:deger.serbun-ufuk"Bolat, Y."https://zbmath.org/authors/?q=ai:bolat.yasarSummary: In this paper, we give new necessary and sufficient conditions for the asymptotic stability of a linear delay difference system with two delays
\[
x_{n+1} - ax_n + A(x_{n-k} + x_{n-l}) = 0,\ n \in \{0,1,2, \dots\},
\]
where \(A\) is a \(2 \times 2\) constant matrix, \(a \in [-1,1] -\{0\}\) is a real number and \(l, k\) are positive integers such that \(1 \leq l < k\).Shadowing for nonautonomous difference equations with infinite delayhttps://zbmath.org/1472.390262021-11-25T18:46:10.358925Z"Dragičević, Davor"https://zbmath.org/authors/?q=ai:dragicevic.davor"Pituk, Mihály"https://zbmath.org/authors/?q=ai:pituk.mihalySummary: We formulate sufficient conditions under which a large class of semilinear nonautonomous difference equations with infinite delay is Hyers-Ulam stable. These conditions require that the nonautonomous linear part admits an exponential dichotomy and the nonlinear perturbations are uniformly Lipschitz continuous with a sufficiently small Lipschitz constant. In the more general case when the linear part admits a shifted exponential dichotomy, we are able to provide sufficient conditions for the existence of a certain weighted form of the shadowing property.A note on non-hyperbolic fixed points of one-dimensional mapshttps://zbmath.org/1472.390272021-11-25T18:46:10.358925Z"Kapçak, Sinan"https://zbmath.org/authors/?q=ai:kapcak.sinanSummary: This paper deals with the local asymptotic stability of non-hyperbolic fixed points of one-dimensional maps. There are, basically, two stability conditions introduced in this study. One of them is for the stability of fixed points of non-oscillatory maps. The second one is a sufficient condition for the stability for oscillatory maps. Some properties and applications are also presented.
For the entire collection see [Zbl 1467.39001].On a second-order rational difference equation with quadratic terms. IIhttps://zbmath.org/1472.390282021-11-25T18:46:10.358925Z"Kostrov, Yevgeniy"https://zbmath.org/authors/?q=ai:kostrov.yevgeniy"Kudlak, Zachary"https://zbmath.org/authors/?q=ai:kudlak.zacharySummary: We give the character of solutions of the following second-order rational difference equation with quadratic denominator
\[ x_{n+1}=\dfrac{\alpha + \beta x_n}{Bx_n + Dx_nx_{n-1} + x_{n-1}}, \]
where the coefficients are positive numbers, and the initial conditions \(x_{-1}\) and \(x_0\) are nonnegative such that the denominator is nonzero. In particular, we show that the unique positive equilibrium is locally asymptotically stable, and we give conditions on the coefficients for which the unique positive equilibrium is globally stable.
For the entire collection see [Zbl 1467.39001].On a class of 2D integrable lattice equationshttps://zbmath.org/1472.390292021-11-25T18:46:10.358925Z"Ferapontov, E. V."https://zbmath.org/authors/?q=ai:ferapontov.evgeny-vladimirovich"Habibullin, I. T."https://zbmath.org/authors/?q=ai:habibullin.ismagil-t"Kuznetsova, M. N."https://zbmath.org/authors/?q=ai:kuznetsova.mariya-nikolaevna"Novikov, V. S."https://zbmath.org/authors/?q=ai:novikov.vladimir-sThe authors develop a new approach to the classification of integrable lattice type equations in dimension three based on the Darboux integrability of suitably reduced equations. The following two-step classification procedure is proposed:
\begin{itemize}
\item[(1)] Require that the dispersionless limit of the equation is integrable, that is, its characteristic variety defines a conformal structure, which is of Einstein-Weyl type on every solution. In this way one gets some candidate equations;
\item[(2)] Apply the test of Darboux integrability of reductions obtained by imposing suitable cutoff conditions.
\end{itemize}
Some explicit examples are considered.Lax matrices for lattice equations which satisfy consistency-around-a-face-centered-cubehttps://zbmath.org/1472.390302021-11-25T18:46:10.358925Z"Kels, Andrew P."https://zbmath.org/authors/?q=ai:kels.andrew-pConsistency around a face-centered cube (CAFCC) is a recently discovered formulation of the multidimensional consistency integrability condition for lattice equations, which is applicable to equations defined on a vertex and its four nearest neighbors on the square lattice. This paper introduces a method for deriving Lax matrices for the equations which satisfy CAFCC. This method gives novel Lax matrices for such equations, which include previously known equations of discrete Toda-, or Laplace-type, as well as newer equations which have only appeared in the context of CAFCC.
The aim of the author is to present a method to derive Lax pairs for the face-centered quad equations from the property of CAFCC, in analogy with the method used to obtain Lax pairs from consistency around a cube (CAC) for regular quad equations. However, because of the differences in the formulations of CAC and CAFCC, the method of deriving Lax pairs from the former does not extend to the latter. The main obstacle is the additional face variables for the face-centered quad equations which have no analogue for CAC, and need to be considered in an evolution around the face-centered cube. To overcome this, a more suitable evolution on the face-centered cube will be chosen, where instead of evolving from a corner vertex to a corner vertex, the Lax matrices are obtained from different evolutions from face vertices to face vertices. Such an alternative approach will be seen to yield the desired Lax matrices for both the type-A and type-B CAFCC equations.
The main results of this paper are both the new method to derive Lax matrices, and the resulting expressions for the Lax matrices themselves.On properties of meromorphic solutions of certain difference Painlevé III equationshttps://zbmath.org/1472.390312021-11-25T18:46:10.358925Z"Lan, Shuang-Ting"https://zbmath.org/authors/?q=ai:lan.shuangting"Chen, Zong-Xuan"https://zbmath.org/authors/?q=ai:chen.zongxuanSummary: We mainly study the exponents of convergence of zeros and poles of difference and divided difference of transcendental meromorphic solutions for certain difference Painlevé III equations.Difference equations related to number theoryhttps://zbmath.org/1472.390322021-11-25T18:46:10.358925Z"Heim, Bernhard"https://zbmath.org/authors/?q=ai:heim.bernhard-ernst"Neuhauser, Markus"https://zbmath.org/authors/?q=ai:neuhauser.markusAs part of a collection on difference equations and applications [Zbl 1467.39001], this article requires a high level of knowledge on the subject.
The authors recall the Dedekind's \(\eta\)-function they already studied in [Res. Math. Sci. 7, No. 1, Paper No. 3, 8 p. (2020; Zbl 1472.11122)], clarifying how its powers are linked to a polynomial defined recursively as follows:
\begin{align*}
P_1(x) &= x , \\
P_n(x) &= \frac{x}{n} \left( \sigma(n) + \sum_{k=1}^{n-1} \sigma(k) P_{n-k}(x) \right) ,
\end{align*}
where \(x \in \mathbb{C}\) and \(\sigma(k)\) is the sum of the divisors of \(k\).
After remarking the importance of \(P_n(x)\) in number theory, the authors acknowledge its irreducibility to a recurrence relation of bounded length and they propose to generalize it through the arithmetic functions \(g,h\):
\begin{align*}
P_1^{g,h}(x) &= x , \\
P_n^{g,h}(x) &= \frac{x}{h(n)} \left( g(n) + \sum_{k=1}^{n-1} g(k) P_{n-k}^{g,h}(x) \right) ,
\end{align*}
with \(g: \mathbb{N} \rightarrow \mathbb{C}\), \(h: \mathbb{N} \rightarrow \mathbb{R}\), and \(g(1)=h(1)=1\).
Beside a quick mention to the classical orthogonal polynomials as solutions of a specific differential equation, the authors distinguish the following subcases:
\begin{align*}
P_n^g (x) &= \frac{x}{n} \left( g(n) + \sum_{k=1}^{n-1} g(k) P_{n-k}^{g}(x) \right) , \\
Q_n^g (x) &= x \left( g(n) + \sum_{k=1}^{n-1} g(k) Q_{n-k}^{g}(x) \right) ,
\end{align*}
related, respectively, to the associated Laguerre polynomials and to the Chebyshev polynomials of the second kind.
The authors employ some of their previous results, obtained in collaboration with \textit{R. Tröger} [J. Difference Equ. Appl. 26, No. 4, 510--531 (2020; Zbl 1456.30010)], in order to analyze the limiting behavior of the main sequence \(\left( P_n^{g,h}(x) \right)_{n \in \mathbb{N}}\) and of the subsequence \(\left( Q_n^g (x) \right)_{n \in \mathbb{N}}\).
Then the authors focus on the recurrence relations of \(P_n^g (x)\) and \(Q_n^g (x)\) for \(g(n)=1\):
\begin{align*}
P_n^1 (x) &= (-1)^n \binom{-x}{n} , \\
Q_n^1 (x) &= (x+1)^{n-1}x ,
\end{align*}
finding a connection between them via a theorem of \textit{H. Poincaré} [Am. J. Math. 7, 203--258 (1885; JFM 17.0290.01)]; they eventually suggest a further investigation under arbitrary conditions.
For the entire collection see [Zbl 1467.39001].Impulse effect on a population model with piecewise constant argumenthttps://zbmath.org/1472.390332021-11-25T18:46:10.358925Z"Karakoç, Fatma"https://zbmath.org/authors/?q=ai:karakoc.fatmaSummary: We consider a population model with piecewise constant argument under impulse effect. First, we deal with the model with impulses. Sufficient conditions for the oscillation of the solutions are obtained. We also investigate asymptotic behavior of the non-oscillatory solutions. Then we obtain similar results for the same model without impulse effect. Finally, we compare the results with non-impulsive case and we give some examples to illustrate our results.
For the entire collection see [Zbl 1467.39001].Population motivated discrete-time disease modelshttps://zbmath.org/1472.390342021-11-25T18:46:10.358925Z"Li, Ye"https://zbmath.org/authors/?q=ai:li.ye.1|li.ye.3|li.ye.4|li.ye"Xu, Jiawei"https://zbmath.org/authors/?q=ai:xu.jiaweiSummary: Infectious diseases are now widely analyzed by compartmental models. This paper introduces a SIR model coupled with a social mobility model (SMM). After discretization by a forward Euler Method, and a mixed type Euler method (structured with both forward and backward Euler elements), we obtained a difference equations model for our social mobility model. We calculate the basic reproduction number \(R_0\) using the next-generation matrix method. When \(R_0<1\), there will be a disease-free equilibrium (DFE), and \(R_0<1\) implies DFE will be locally asymptotically stable, while \(R_0>1\) implies DFE is unstable. When \(R_0=1\), DFE may stable or unstable. Then we obtain a hyperbolic forward Kolmogorov equation corresponding to the SIR epidemic model. We also generate the hyperbolic forward Kolmogorov equations for the SIR model with SMM between 2 locations.
For the entire collection see [Zbl 1467.39001].An alternative delayed population growth difference equation modelhttps://zbmath.org/1472.390352021-11-25T18:46:10.358925Z"Streipert, Sabrina H."https://zbmath.org/authors/?q=ai:streipert.sabrina-h"Wolkowicz, Gail S. K."https://zbmath.org/authors/?q=ai:wolkowicz.gail-s-kThe Beverton-Holt recurrence equation is a discrete-time logistic population growth model which can, in some sense, be thought of as a discretized version of Verhulst logistic growth model used in the modelling of fish populations. The Beverton-Holt model can be defined as
\[
y_{t+1} = \frac{\rho K y_t}{K + (\rho - 1) y_t},
\]
where \(K>0\) is the {carrying capacity} and \(\rho>1\) the {proliferation rate} over a generation. This model assumes that every generation of the population is non-overlapping and that the offspring of a generation instantaneously starts breeding in the next one.
In this article, the authors introduce a delayed version of the Beverton-Holt model in which newborn individuals of the population do not become mature until the age \(\tau\), hence do not contribute to the population growth until that time. The delayed Beverton-Holt model derived in this article is written as
\[
z_{t+1} = \frac{1}{1 + b + c z_t}\left( z_t + \frac{abz_{t-\tau+1}}{b(b+1)^{\tau} + ((b+1)^\tau - 1)c z_{t-\tau+1}} \right),
\]
where \(z_t = y_t/K\) is the fraction of the carrying capacity currently occupied by the population, \(a\) is the {natural growth component}, \(b\) the {natural death component} and \(c\) the {intraspecific competition} of the population. Note that this model corresponds to the classical Beverton-Holt model when \(\tau=0\) and \(a-b = c = \frac{\rho-1}{\rho}\).
The authors then study the asymptotic behaviour of their model, proving that if the delay is larger than the {critical delay} \[\tau_c = \frac{\log\left(\frac{a}{b}\right)}{\log(1+b)},\] i.e., if the maturation period of juvenile individuals is too long, then the population grows extinct. Conversely, as long as the delay is smaller than \(\tau_c\), the population (started from a non-trivial initial condition) stabilizes around a positive equilibrium point.A \(\beta\)-Sturm-Liouville problem associated with the general quantum operatorhttps://zbmath.org/1472.390362021-11-25T18:46:10.358925Z"Cardoso, J. L."https://zbmath.org/authors/?q=ai:cardoso.joao-lopes|cardoso.jose-luis|cardoso-cortes.jose-luisSummary: Let \(I \subseteq \mathbb{R}\) be an interval and \(\beta : I \to I\) a strictly increasing and continuous function with a unique fixed point \(s_0 \in I\) that satisfies \((s_0 - t)(\beta(t)-t)\geq 0\) for all \(t \in I\), where the equality holds only when \(t = s_0\). The general quantum operator defined by \textit{A. E. Hamza} et al. [Adv. Difference Equ. 2015, Paper No. 182, 19 p. (2015; Zbl 1422.39010)], \(D_{\beta}[f](t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t}\) if \(t \neq s_0\) and \(D_{\beta}[f](s_0):=f'(s_0)\) if \(t=s_0\) generalizes the Jackson \(q\)-operator \(D_q\) and also the Hahn (quantum derivative) operator, \(D_{q,\omega}\). Regarding a \(\beta\)-Sturm-Liouville eigenvalue problem associated with the above operator \(D_{\beta}\), we construct the \(\beta\)-Lagrange's identity, show that it is self-adjoint in \(\mathscr{L}_{\beta}^2([a,b])\) and exhibit some properties for the corresponding eigenvalues and eigenfunctions.Periodic one-point rank one commuting difference operatorshttps://zbmath.org/1472.390372021-11-25T18:46:10.358925Z"Dobrogowska, Alina"https://zbmath.org/authors/?q=ai:dobrogowska.alina"Mironov, Andrey E."https://zbmath.org/authors/?q=ai:mironov.andrei-evgenevichSummary: In this paper we study one-point rank one commutative rings of difference operators. We find conditions on spectral data which specify such operators with periodic coefficients.
For the entire collection see [Zbl 1472.53006].A series representation of the discrete fractional Laplace operator of arbitrary orderhttps://zbmath.org/1472.390382021-11-25T18:46:10.358925Z"Frugé Jones, Tiffany"https://zbmath.org/authors/?q=ai:fruge-jones.tiffany"Kostadinova, Evdokiya Georgieva"https://zbmath.org/authors/?q=ai:kostadinova.evdokiya-georgieva"Padgett, Joshua Lee"https://zbmath.org/authors/?q=ai:padgett.joshua-lee"Sheng, Qin"https://zbmath.org/authors/?q=ai:sheng.qinSummary: Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.Positive solution of discrete BVPs involving the mean curvature operator on the set of nonnegative integershttps://zbmath.org/1472.390392021-11-25T18:46:10.358925Z"Wei, Liping"https://zbmath.org/authors/?q=ai:wei.liping"Ma, Ruyun"https://zbmath.org/authors/?q=ai:ma.ruyun"Zhao, Zhongzi"https://zbmath.org/authors/?q=ai:zhao.zhongziSummary: Setting \(\mathbb{N}=\{1,2,\ldots ,\infty\}\), \(\mathbb{N}_0=\{0,1,2,\ldots ,\infty \}\), we show the existence of positive solutions of the quasilinear boundary value problem
\[
\begin{aligned}
&-\nabla \Big (\frac{\Delta u(x)}{\sqrt{1+(\Delta u(x))^2}}\Big)= g(x,u(x),\Delta u(x)), \quad x\in{\mathbb{N}},\\
&\quad u(0)=0, \quad \lim \limits_{x\rightarrow +\infty}\Delta u(x)=0,
\end{aligned}
\]
where \(\Delta\) is the forward difference operator, \(\nabla\) is the backward difference operator, and \(g: \mathbb{N}_0\times [0,+\infty)\times [0,+\infty)\rightarrow [0,+\infty)\) is continuous. Under suitable assumptions on nonlinearity, we obtained the existence of at least one positive solution by a simple application of a Fixed Point Theorem in cones.Linear operators associated with differential and difference systems: what is different?https://zbmath.org/1472.390402021-11-25T18:46:10.358925Z"Zemánek, Petr"https://zbmath.org/authors/?q=ai:zemanek.petrSummary: The existence of a densely defined operator associated with (time-reversed) discrete symplectic systems is discussed and the necessity of the development of the spectral theory for these systems by using linear relations instead of operators is shown. An explanation of this phenomenon is provided by using the time scale calculus. In addition, the density of the domain of the maximal linear relation associated with the system is also investigated.
For the entire collection see [Zbl 1467.39001].Travelling wave solutions in a predator-prey integrodifference systemhttps://zbmath.org/1472.390412021-11-25T18:46:10.358925Z"Wang, Yahui"https://zbmath.org/authors/?q=ai:wang.yahui"Lin, Guo"https://zbmath.org/authors/?q=ai:lin.guo"Niu, Yibin"https://zbmath.org/authors/?q=ai:niu.yibinSummary: This paper studies the minimal wave speed of travelling wave solutions in a predator-prey integrodifference system. Even if the predator vanishes, the corresponding scalar equation of prey may be nonmonotonic. Without the assumption of classical comparison principle in this coupled system, we investigate travelling wave solutions modelling the process that the predator invades the habitat of the prey. By showing the existence and nonexistence of nonconstant travelling wave solutions, the minimal wave speed of travelling wave solutions is confirmed. To achieve the purpose, we utilize recipes of constructing generalized upper and lower solutions, introducing auxiliary equations and applying the propagation theory of scalar equations.Existence of solutions of polynomial-like iterative equation with discontinuous known functionshttps://zbmath.org/1472.390422021-11-25T18:46:10.358925Z"Yu, Zhiheng"https://zbmath.org/authors/?q=ai:yu.zhiheng"Liu, Jinghua"https://zbmath.org/authors/?q=ai:liu.jinghuaThe authors study the existence of solutions of polynomial-like iterative equation \(\lambda_{1}f(x)+\lambda_{2}f^{2}(x)+\dots+\lambda_{n}f^{n}(x)=F(x)\), \(x\in I\), with discontinuous known functions. Let \(I=[a, b]\). Then the authors consider the set
\[
\mathcal{F}=\{f \mid f : I \rightarrow I \text{ is strictly increasing,} \; f(a)=a \text{ and } f(b)=b\},
\]
which contains continuous and discontinuous functions. They define the following three classes of ``good'' functions. For a given \(t\in \operatorname{int} (I)\) define:
\begin{gather*}
\mathcal{C}_{t}(I) :=\{f \mid f \in\mathcal{F}(I) \text{ is continuous on } I \text{ and } f(t)=t\}, \\
\tilde{\mathcal{C}}_{t}(I) :=\{f \mid f \in\mathcal{F}(I) \text{ is continuous on } \operatorname{int} I \text{ and } f(t)=t\}, \\
\mathcal{H}_{t}(I) :=\{f \mid f \in \mathcal{F}(I) \text{ is discontinuous exactly at point } t \text{ and } f(t)=t\}.
\end{gather*}
The authors show that if \(f_{1}, f_{2} \in \mathcal{C}_{t}\), then \(f_{1} \circ f_{2} \in\mathcal{C}_{t}\). Similarly, if \(f_{1}, f_{2} \in \mathcal{H}_{t}\), then \(f_{1} \circ f_{2} \in\mathcal{H}_{t}\).
Further, the authors introduce piecewise bi-Lipschitz functions and construct a functional space consisting of such functions. They denote the subclasses of all functions \(f\) from \(\mathcal{C}_{t}(I)\) and from \(\mathcal{H}_{t}(I)\) by \(\mathcal{A}_{t}(I, m, M)\) and \(\mathcal{B}_{t}(I, m, M)\) respectively. Moreover, \(\mathcal{G}_{t}(I, m, M)=\mathcal{A}_{t}(I, m, M) \cup\mathcal{B}_{t}(I, m, M)\). They show that the set \(\mathcal{G}_{t}(I, m, M)\) endowed with the distance \(\mathcal{D}(f_{1}, f_{2})=\sup\{|f_{1}(x)-f_{2}(x)|, x\in I\}\) is a complete metric space.
Then they define the operator \(\mathcal{T}: \mathcal{G}_{t}(I, m, M) \rightarrow\mathcal{F}(I)\) as \[\mathcal{T}f=\frac{1}{\lambda_{1}}\bigg(F-\sum_{i=2}^{n}\lambda_{i}f^{i}\bigg)\] and prove the existence of solutions by means of the Banach fixed point principle. An example to justify their main results
is discussed.A functional equation originated from the product in a cubic number fieldhttps://zbmath.org/1472.390432021-11-25T18:46:10.358925Z"Mouzoun, A."https://zbmath.org/authors/?q=ai:mouzoun.aziz"Zeglami, D."https://zbmath.org/authors/?q=ai:zeglami.driss"Ayoubi, M."https://zbmath.org/authors/?q=ai:ayoubi.mohamedSummary: Let \(\mathbb{K}\) be either \(\mathbb{R}\) or \(\mathbb{C}\) and \(\alpha \in \mathbb{R}\). We determine the solutions \(f:\mathbb{R}^3\rightarrow \mathbb{K}\) of the following new parametric functional equation:
\[
\begin{multlined} f(x_1x_2+\alpha y_1z_2+\alpha y_2z_1,x_1y_2+x_2y_1+\alpha z_1z_2,x_1z_2+x_2z_1+y_1y_2) \\
=f(x_1,y_1,z_1)f(x_2,y_2,z_2),\; (x_1,y_1,z_1),(x_2,y_2,z_2)\in \mathbb{R}^3,
\end{multlined}
\]
which results from the product of two numbers in a cubic free field. We equip \(\mathbb{R}^3\) with a binary operation to show that the non-zero solutions of this equation are monoid homomorphisms and we investigate our results to introduce and find the solutions of d'Alembert's functional equations with endomorphisms.The archetypal equation and its solutions attaining the global extremumhttps://zbmath.org/1472.390442021-11-25T18:46:10.358925Z"Sudzik, Mariusz"https://zbmath.org/authors/?q=ai:sudzik.mariuszThe paper is devoted to the functional equation \[ \varphi(x)=\int\int_{\mathbb{R}^2}\varphi(a(x-b))\mu(da,db), \tag{1} \] where \(\mu\) is a Borel probability measure on \(\mathbb{R}^2\) and \(\varphi\colon\mathbb{R}\to\mathbb{R}\) is an unknown function. This equation was introduced by \textit{G. Derfel} [Ukr. Math. J. 41, 1137--1141 (1989; Zbl 0713.45003)]. More recently the study of this equation was continued, e.g., by \textit{L.V. Bogachev} et al. [Proc. R. Soc. A471, 1--19 (2015; Zbl 1371.60110)] and the name \textit{archetypal equation} was coined for (\(1\)).
The present paper is devoted to study the bounded continuous solutions of (\(1\)). The main result states that (under some mild assumptions on the measure \(\mu\)) every bounded continuous solution of equation (\(1\)) attaining its global extremum must be a constant mapping. The proof of that takes more than eleven pages and requires a lot of persistence in considering several cases.
Additionally, some description of the behavior of non-constant bounded continuous solutions is given.Holomorphic solutions of some functional equations related to nonlinear second order \(q\)-difference equations which has only one characteristic valuehttps://zbmath.org/1472.390452021-11-25T18:46:10.358925Z"Suzuki, Mami"https://zbmath.org/authors/?q=ai:suzuki.mamiSummary: Here we consider the following functional equation,
\[
\Psi(X(t,x,\Psi(x)))=Y(t,x,\Psi(x)),
\]
where \(X(t,x,y)\) and \(Y(t,x,y)\) are holomorphic functions in \(t< \theta,\vert x\vert< \theta,\vert y\vert < \theta\), and \(x\) is a function of \(t\) such that \(x=x(t)\). If we have a solution of nonlinear second order difference equations, then we can write general solutions by making use of a solution \(\Psi\).
In this present paper, we will prove the existence of a solution of the above functional equation, after we will show a relationship between the above functional equation and nonlinear second order \(q\)-difference equations which has only one characteristic value.The Drygas functional equation on abelian semigroups with endomorphismshttps://zbmath.org/1472.390462021-11-25T18:46:10.358925Z"Akkaoui, Ahmed"https://zbmath.org/authors/?q=ai:akkaoui.ahmed"Fadli, Brahim"https://zbmath.org/authors/?q=ai:fadli.brahim"El Fatini, Mohamed"https://zbmath.org/authors/?q=ai:el-fatini.mohamedFor an abelian semigroup \((S,+)\), a uniquely 2-divisible group \((H,+)\), endomorphisms \(\sigma,\tau\colon S\to S\) and an unknown mapping \(f\colon S\to H\) the authors consider a {Drygas-type functional equation} \[ f(x+\sigma(y))+f(x+\tau(y))=2f(x)+f(\sigma(y))+f(\tau(y)),\qquad x,y\in S. \] The solutions of this equation are determined in the form \[ f(x)=A(x)+Q(x,x),\qquad x\in S, \] where \(A\colon S\to H\) and \(Q\colon S\times S\to H\) are additive and bi-additive mappings, respectively. Further, continuous solutions, if \(S\) is a topological group and \(H\) is a topological vector space, are constructed as well.Generalized sine and cosine addition laws and a Levi-Civita functional equation on monoidshttps://zbmath.org/1472.390472021-11-25T18:46:10.358925Z"Ebanks, Bruce"https://zbmath.org/authors/?q=ai:ebanks.bruce-rFor a semigroup \(S\) and two unknown mappings \(f,g\colon S\to\mathbb{C}\) the functional equations \[ f(xy)=f(x)g(y)+g(x)f(y),\qquad x,y\in S \] and \[ g(xy)=g(x)g(y)-f(x)f(y),\qquad x,y\in S \] are called the \textit{sine} and \textit{cosine additional law}, respectively.
Here some extensions and generalizations of known results for the above equations are obtained. In particular, the paper deals with continuous solutions on topological (semi)groups, pexiderizations \[ f(xy)=f(x)g(y)+h(x)f(y),\qquad x,y\in S, \] \[ g(xy)=g(x)g(y)+h(x)k(y),\qquad x,y\in S,\] with unknown functions \(f,g,h,k\colon S\to\mathbb{C}\) and finally with the Levi-Civita functional equation: \[ f(xy)=g_1(x)h_1(y)+g_2(x)h_2(y),\qquad x,y\in S, \] with unknown functions \(f,g_1,g_2,h_1,h_2\colon S\to\mathbb{C}\).Hypercontractivity of the semigroup of the fractional Laplacian on the \(n\)-spherehttps://zbmath.org/1472.390482021-11-25T18:46:10.358925Z"Frank, Rupert L."https://zbmath.org/authors/?q=ai:frank.rupert-l"Ivanisvili, Paata"https://zbmath.org/authors/?q=ai:ivanisvili.paataThe paper presents a further contribution to a problem concerning the hypercontractivity of the Poisson semigroup of \(e^{-t\sqrt{-\Delta}}\) from \(L^p(\mathbb{S}^n)\) to \(L^q(\mathbb{S}^n)\) for \(t>0\) on the sphere \(\mathbb{S}^n\) of dimension \(n\). This question was posed by \textit{C. E. Mueller} and \textit{F. B. Weissler} [J. Funct. Anal. 48, 252--283 (1982; Zbl 0506.46022)].
The main result of the paper states that for \(1<p\leq q\) the condition \(e^{-t\sqrt{n}}\leq \sqrt{\frac{p-1}{q-1}}\), carrying the smallest nonzero eigenvalue \(\sqrt{n}\) of \(\sqrt{-\Delta}\), is necessary and sufficient for dimension \(n\leq 3\). Noteworthy, in case of \(q>\max\{2,p\}\) the aforementioned condition is not sufficient for dimension \(n\geq 4\).
The reason of the exceptionality for \(n=1,2,3\) is explained in detail in Subsection 2.2. In the remaining part of the paper it is proved, by contradiction, why the sufficient condition does not hold in general.
Summing up, the question of finding an hypercontractivity equivalence for the Poisson semigroup on \(t>0\) for \(n\geq 4\) remains open.On superstability of the Wigner equationhttps://zbmath.org/1472.390492021-11-25T18:46:10.358925Z"Ilišević, Dijana"https://zbmath.org/authors/?q=ai:ilisevic.dijana"Turnšek, Aleksej"https://zbmath.org/authors/?q=ai:turnsek.aleksejLet \(M\) and \(N\) be inner product spaces over \(F\in \{\mathbb{R}, \mathbb{C}\}\). The famous Wigner's theorem says that any mapping \(f:M \to N\) which preserves the transition probability, i.e.,
\[
\| \langle f(x), f(y)\rangle\| = \|\langle x, y\rangle\|, \qquad x,y\in M, \tag{W}
\]
must be phase-equivalent to a linear or a conjugate-linear isometry.
Now suppose that a mapping \(f : M \to N\) satisfies (W) approximately. More precisely,
assume that
\[
\|\langle(x), f(y)\rangle \| - \|\langle x, y\rangle\| \leq \phi(x,y), \qquad x,y\in M,
\]
where \(\phi: M \times M \to [0,\infty)\) is an appropriate control function.
The authors investigate the superstability of above inequality. In particular, they find that if \(f\) is surjective
all solutions of above inequality are in fact solutions of (W).A variant of Wigner's theorem in normed spaceshttps://zbmath.org/1472.390502021-11-25T18:46:10.358925Z"Ilišević, Dijana"https://zbmath.org/authors/?q=ai:ilisevic.dijana"Turnšek, Aleksej"https://zbmath.org/authors/?q=ai:turnsek.aleksejLet \(X\) and \(Y\) be normed spaces over \(\mathbb{F}\) and let \(U:X \rightarrow Y\) be a linear (or a conjugate linear) isometry. If a function \(f:X \rightarrow Y\) has the property \[ f(x)=\sigma(x) Ux, \quad x\in H, \] where \(\sigma\) is a phase function, i.e., \(\sigma\) takes values in modulus one scalars, then a function \(f\) is called phase equivalent to a linear (or a conjugate linear) isometry.
The main result of this paper is given in the following theorem.
Theorem. Let \(X\) and \(Y\) be normed spaces over \(\mathbb{F}\) and \(f:X \rightarrow Y\) a surjective mapping. Suppose that for all semi-inner products on \(X\) and \(Y\), we have \[ | [f(x),f(y)]| = |[x,y]|, \quad x,y \in X.\]
The following statements hold true:
\begin{itemize}
\item[(i)] If dim \(X=1\), then \(f\) is phase equivalent to a linear surjective isometry;
\item[(ii)] If dim \(X\geq 2\) and \(\mathbb{F} = \mathbb{R}\), then \(f\) is phase equivalent to a linear surjective isometry;
\item[(iii)] If dim \(X\geq 2\) and \(\mathbb{F} =\mathbb{C}\), then \(f\) is phase equivalent to a linear or conjugate linear surjective isometry.
\end{itemize}Endpoint regularity of the discrete multisublinear fractional maximal operatorshttps://zbmath.org/1472.420352021-11-25T18:46:10.358925Z"Zhang, Xiao"https://zbmath.org/authors/?q=ai:zhang.xiaoThe main results of this article are about the discrete centered and uncentered \(m\)-sublinear fractional maximal operators, \( \mathfrak{M}_\alpha \) and \( \widetilde{ \mathfrak{M} }_\alpha \) (respectively). First, a theorem on endpoint regularity is obtained for \( \widetilde{ \mathfrak{M} }_\alpha \) from the \(m\)-fold Cartesian product of \( \text{BV}(\mathbb{Z}) \) (\(0 \leq \alpha < 1 \)) or \( \ell^1 (\mathbb{Z}) \) (\( m-1 \leq \alpha < m \)) into the space of functions with bounded \(q\)-variation, where \(q\) depends on the subscript \(\alpha\). Here \( \text{BV}(\mathbb{Z}) \) denotes the space of functions of bounded variation defined on \(\mathbb{Z}\). After that, a theorem showing the boundedness of \(\mathfrak{M}_\alpha \) and \( \widetilde{ \mathfrak{M} }_\alpha \), \( 0 \leq \alpha < m \), from the \(m\)-fold Cartesian product of \( \ell^1 (\mathbb{Z}) \) into \( \text{BV}(\mathbb{Z}) \) is obtained.
Both theorems can be considered as the discrete version of the corresponding results contained in [\textit{F. Liu} and \textit{H. Wu}, Can. Math. Bull. 60, No. 3, 586--603 (2017; Zbl 1372.42015)] and also extend the corresponding already known results of \textit{E. Carneiro} and \textit{J. Madrid} [Trans. Am. Math. Soc. 369, No. 6, 4063--4092 (2017; Zbl 1370.26022)] and \textit{F. Liu} [Bull. Aust. Math. Soc. 95, No. 1, 108--120 (2017; Zbl 1364.42020)].Moment functions on affine groupshttps://zbmath.org/1472.430102021-11-25T18:46:10.358925Z"Fechner, Żywilla"https://zbmath.org/authors/?q=ai:fechner.zywilla"Székelyhidi, László"https://zbmath.org/authors/?q=ai:szekelyhidi.laszloSummary: Moments of probability measures on a hypergroup can be obtained from so-called (generalized) moment functions of a given order. The aim of this paper is to characterize generalized moment functions on a non-commutative affine group. We consider a locally compact group \(G\) and its compact subgroup \(K\). First we recall the notion of the double coset space \(G /\!/ K\) of a locally compact group \(G\) and introduce a hypergroup structure on it. We present the connection between \(K\)-spherical functions on \(G\) and exponentials on the double coset hypergroup \(G /\!/ K\). The definition of the generalized moment functions and their connection to the spherical functions is discussed. We study an important class of double coset hypergroups: specyfing \(K\) as a compact subgroup of the group of invertible linear transformations on a finitely dimensional linear space \(V\) we consider the affine group \(\mathrm{Aff}\, K\). Using the fact that in the finitely dimensional case \((\mathrm{Aff}\, K,K)\) is a Gelfand pair we give a description of exponentials on the double coset hypergroup \({\mathrm{Aff}}\, K/\!/K\) in terms of \(K\)-spherical functions. Moreover, we give a general description of generalized moment functions on \({\mathrm{Aff}}\, K\) and specific examples for \(K=\mathrm{SO}(n)\), and on the so-called \(ax+b\)-group.Riesz means on homogeneous treeshttps://zbmath.org/1472.430112021-11-25T18:46:10.358925Z"Papageorgiou, Effie"https://zbmath.org/authors/?q=ai:papageorgiou.effie-gSummary: Let \(\mathbb{T}\) be a homogeneous tree. We prove that if \(f \in L^p (\mathbb{T})\), \(1 \leq p \leq 2\), then the Riesz means \(S^z_R (f)\) converge to \(f\) everywhere as \(R \rightarrow \infty\), whenever \(\Re z > 0\).Existence of solutions and algorithm for generalized vector quasi-complementarity problems with application to traffic network problemshttps://zbmath.org/1472.490202021-11-25T18:46:10.358925Z"Nguyen Van Hung"https://zbmath.org/authors/?q=ai:nguyen-van-hung."Vo Minh Tam"https://zbmath.org/authors/?q=ai:vo-minh-tam."Köbis, Elisabeth"https://zbmath.org/authors/?q=ai:kobis.elisabeth"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: In this paper, we establish a new class of generalized vector quasi complementarity problems with fuzzy mappings in Hausdorff topological vector spaces. Afterward, we prove that generalized vector quasi-complementarity problems with fuzzy mappings are equivalent to generalized vector quasi-variational inequality problems with fuzzy mappings by using suitable conditions. The existence of solutions for our problems by using the Kakutani-Fan-Glicksberg fixed point theorem is obtained. Further, based on an auxiliary problem, we propose a projection iterative algorithm for generalized vector variational inequality problems with fuzzy mappings. Under suitable conditions, we prove the convergence of this iterative algorithm and solve the generalized vector complementarity problem with fuzzy mappings. As a real-world application, we consider the special case of traffic network problems. A part of the main results obtained in this paper represents an affirmative answer to an open problem posed in [\textit{A. Kılıçman} et al., Fuzzy Sets Syst. 280, 133--141 (2015; Zbl 1378.49009)].Some fixed point theorems for weakly subsequentially continuous and compatible of type (E) mappings with an applicationhttps://zbmath.org/1472.540232021-11-25T18:46:10.358925Z"Beloul, Said"https://zbmath.org/authors/?q=ai:beloul.saidSummary: In this paper, we will establish some fixed point results, for two pairs of self mappings satisfying generalized contractive condition, by using a new concept as weak subsequential continuity, with compatibility of type (E) in metric spaces, as an application the existence of unique solution for a system of functional equations arising in system programming is proved.Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equationshttps://zbmath.org/1472.600072021-11-25T18:46:10.358925Z"Baron, Karol"https://zbmath.org/authors/?q=ai:baron.karolSummary: The paper consists of two parts. At first, assuming that \((\Omega, \mathcal{A}, P)\) is a probability space and \((X, \varrho)\) is a complete and separable metric space with the \(\sigma\)-algebra \(\mathcal{B}\) of all its Borel subsets we consider the set \(\mathcal{R}_c\) of all \(\mathcal{B} \otimes \mathcal{A}\)-measurable and contractive in mean functions \(f: X \times \Omega \rightarrow X\) with finite integral \(\int_\Omega \varrho (f(x, \omega), x) P (d \omega)\) for \(x \in X\), the weak limit \(\pi^f\) of the sequence of \textit{iterates} of \(f \in \mathcal{R}_c\), and investigate continuity-like property of the function \(f \mapsto \pi^f\), \(f \in \mathcal{R}_c\), and Lipschitz solutions \(\varphi\) that take values in a separable Banach space of the equation:
\[
\varphi (x) = \int_\Omega \varphi (f(x,\omega) P ( d\omega) + F(x).
\]
Next, assuming that \(X\) is a real separable Hilbert space, \( \Lambda \): \(X \rightarrow X\) is linear and continuous with \(\Vert \Lambda \Vert < 1\), and \(\mu\) is a probability Borel measure on \(X\) with finite first moment we examine continuous at zero solutions \(\varphi : X \rightarrow \mathbb{C}\) of the equation
\[
\varphi(x) = \hat{\mu}(x)\varphi (\Lambda x)
\]
which characterizes the limit distribution \(\pi^{f}\) for some special \(f \in \mathcal{R}_c\).Stability of stochastic dynamic equations with time-varying delay on time scaleshttps://zbmath.org/1472.600942021-11-25T18:46:10.358925Z"Du, Nguyen Huu"https://zbmath.org/authors/?q=ai:nguyen-huu-du."Tuan, Le Anh"https://zbmath.org/authors/?q=ai:tuan.le-anh"Dieu, Nguyen Thanh"https://zbmath.org/authors/?q=ai:dieu.nguyen-thanhSummary: The aim of this article is to consider the existence, uniqueness and uniformly exponential \(p\)-stability of the solution for \(\nabla\)-delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations.Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein-Gordon equation and numerical comparisonshttps://zbmath.org/1472.810832021-11-25T18:46:10.358925Z"Muda, Y."https://zbmath.org/authors/?q=ai:muda.yuslenita"Akbar, F. T."https://zbmath.org/authors/?q=ai:akbar.fiki-taufik"Kusdiantara, R."https://zbmath.org/authors/?q=ai:kusdiantara.rudy"Gunara, B. E."https://zbmath.org/authors/?q=ai:gunara.bobby-eka"Susanto, H."https://zbmath.org/authors/?q=ai:susanto.hadiSummary: We consider a damped, parametrically driven discrete nonlinear Klein-Gordon equation, that models coupled pendula and micromechanical arrays, among others. To study the equation, one usually uses a small-amplitude wave ansatz, that reduces the equation into a discrete nonlinear Schrödinger equation with damping and parametric drive. Here, we justify the approximation by looking for the error bound with the method of energy estimates. Furthermore, we prove the local and global existence of solutions to the discrete nonlinear Schrödinger equation. To illustrate the main results, we consider numerical simulations showing the dynamics of errors made by the discrete nonlinear equation. We consider two types of initial conditions, with one of them being a discrete soliton of the nonlinear Schrödinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein-Gordon equation.Configurational complexity of nonautonomous discrete one-soliton and rogue waves in Ablowitz-Ladik-Hirota waveguidehttps://zbmath.org/1472.811062021-11-25T18:46:10.358925Z"Thakur, Pooja"https://zbmath.org/authors/?q=ai:thakur.pooja"Gleiser, Marcelo"https://zbmath.org/authors/?q=ai:gleiser.marcelo"Kumar, Anil"https://zbmath.org/authors/?q=ai:kumar.anil"Gupta, Rama"https://zbmath.org/authors/?q=ai:gupta.ramaSummary: We compute the configurational complexity (CC) for discrete soliton and rogue waves traveling along an Ablowitz-Ladik-Hirota (ALH) waveguide and modeled by a discrete nonlinear Schrödinger equation. We show that for a specific range of the soliton transverse direction \(\kappa\) propagating along the parametric time \(\zeta(t)\), CC reaches an evolving series of global minima. These minima represent maximum compressibility of information in the momentum modes along the Ablowitz-Ladik-Hirota waveguide. Computing the CC for rogue waves as a function of background amplitude modulation \(\mu\), we show that it displays two essential features: a maximum representing the optimal value for the rogue wave inception (the ``gradient catastrophe'') and saturation representing the rogue wave dispersion into constituent wave modes. We show that saturation is achieved earlier for higher values of modulation amplitude as the discrete rogue wave evolves along time \(\zeta(t)\).Pharmacokinetics and pharmacodynamics models of tumor growth and anticancer effects in discrete timehttps://zbmath.org/1472.921182021-11-25T18:46:10.358925Z"Atıcı, Ferhan M."https://zbmath.org/authors/?q=ai:merdivenci-atici.ferhan"Nguyen, Ngoc"https://zbmath.org/authors/?q=ai:nguyen.ngoc-hung|nguyen.ngoc-hoai-an|nguyen.ngoc-hieu|nguyen.ngoc-khoat|nguyen.ngoc-anh|nguyen.ngoc-son|nguyen.ngoc-khai|nguyen.ngoc-khanh|nguyen-ngoc-dong-quan.|nguyen-thanh-ngoc.|nguyen.ngoc-thach|nguyen.ngoc-tuan|nguyen.ngoc-cuong|nguyen.ngoc-tuy|nguyen.ngoc-trung|nguyen.ngoc-thinh|nguyen.ngoc-ai-van"Dadashova, Kamala"https://zbmath.org/authors/?q=ai:dadashova.kamala"Pedersen, Sarah E."https://zbmath.org/authors/?q=ai:pedersen.sarah-e"Koch, Gilbert"https://zbmath.org/authors/?q=ai:koch.gilbertSummary: We study the \(h\)-discrete and \(h\)-discrete fractional representation of a pharmacokinetics-pharmacodynamics (PK-PD) model describing tumor growth and anticancer effects in continuous time considering a time scale \(h\mathbb{N}_0\), where \(h > 0\). Since the measurements of the drug concentration in plasma were taken hourly, we consider \(h = 1/24\) and obtain the model in discrete time (i.e. hourly). We then continue with fractionalizing the \(h\)-discrete nabla operator in the \(h\)-discrete model to obtain the model as a system of nabla \(h\)-fractional difference equations. In order to solve the fractional \(h\)-discrete system analytically we state and prove some theorems in the theory of discrete fractional calculus. After estimating and getting confidence intervals of the model parameters, we compare residual squared sum values of the models in one table. Our study shows that the new introduced models provide fitting as good as the existing models in continuous time.