Recent zbMATH articles in MSC 39https://zbmath.org/atom/cc/392023-03-23T18:28:47.107421ZWerkzeugTwo methods for the general term formula of a recurrence sequence with order \(m\)https://zbmath.org/1503.110212023-03-23T18:28:47.107421Z"Lu, Guo Xiang"https://zbmath.org/authors/?q=ai:lu.guoxiang(no abstract)Apéry limits for elliptic \(L\)-valueshttps://zbmath.org/1503.110822023-03-23T18:28:47.107421Z"Koutschan, Christoph"https://zbmath.org/authors/?q=ai:koutschan.christoph"Zudilin, Wadim"https://zbmath.org/authors/?q=ai:zudilin.wadimFor an (irreducible) recurrence equation with coefficients in \(\mathbb{Z}[n]\) and two linearly independent solutions \(u_n, v_n\), the limit of \(u_n/v_n\) as \(n\to\infty\) (if exists) is called the Apéry limit.
This paper gives a construction (based on mostly conjectural identities) that realizes certain quotients of \(L\)-values of elliptic curves as Apéry limits. The main tool is the family of certain double integrals and their relations to Mahler's measures and \(L\)-values of certain elliptic curves.
Reviewer: Noriko Yui (Kingston)An eigenvalue problem in fractional \(h\)-discrete calculushttps://zbmath.org/1503.260032023-03-23T18:28:47.107421Z"Atıcı, F. M."https://zbmath.org/authors/?q=ai:merdivenci-atici.ferhan"Jonnalagadda, J. M."https://zbmath.org/authors/?q=ai:jonnalagadda.jaganmohan(no abstract)On generalized strongly modified \(h\)-convex functionshttps://zbmath.org/1503.260202023-03-23T18:28:47.107421Z"Zhao, Taiyin"https://zbmath.org/authors/?q=ai:zhao.taiyin"Saleem, Muhammad Shoaib"https://zbmath.org/authors/?q=ai:saleem.muhammad-shoaib"Nazeer, Waqas"https://zbmath.org/authors/?q=ai:nazeer.waqas"Bashir, Imran"https://zbmath.org/authors/?q=ai:bashir.imran"Hussain, Ijaz"https://zbmath.org/authors/?q=ai:hussain.ijazSummary: We derive some properties and results for a new extended class of convex functions, generalized strongly modified \(h\)-convex functions. Moreover, we discuss Schur-type, Hermite-Hadamard-type, and Fejér-type inequalities for this class. The crucial fact is that this extended class has awesome properties similar to those of convex functions.Instability results for the logarithmic Sobolev inequality and its application to related inequalitieshttps://zbmath.org/1503.260332023-03-23T18:28:47.107421Z"Kim, Daesung"https://zbmath.org/authors/?q=ai:kim.daesungSummary: We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and \(L^p(d\gamma)\) distance for \(p>1\). To this end, we construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not, which leads to instability for the logarithmic Sobolev inequality. As an application, we prove instability results for Talagrand's transportation inequality and the Beckner-Hirschman inequality.Discrete fractional distributed Halanay inequality and applications in discrete fractional order neural network systemshttps://zbmath.org/1503.260352023-03-23T18:28:47.107421Z"Liu, Xiang"https://zbmath.org/authors/?q=ai:liu.xiang"Yu, Yongguang"https://zbmath.org/authors/?q=ai:yu.yongguang(no abstract)There is at most one continuous invariant meanhttps://zbmath.org/1503.260922023-03-23T18:28:47.107421Z"Pasteczka, Paweł"https://zbmath.org/authors/?q=ai:pasteczka.pawelSummary: We show that, for a (not necessarily continuous) weakly contractive mean-type mapping \(\mathbf{M} :I^p\rightarrow I^p\) (where \(I\) is an interval and \(p \in \mathbb{N})\), the functional equation \(K \circ \mathbf{M}=K\) has at most one solution in the family of continuous means \(K :I^p \rightarrow I\). Some general approach to this equation is also given.The weighted discrete dynamic inequalities for 4-convex functions, and its generalization on time scales with constant graininess functionhttps://zbmath.org/1503.260942023-03-23T18:28:47.107421Z"Baig, Hira Ashraf"https://zbmath.org/authors/?q=ai:baig.hira-ashraf"Ahmad, Naveed"https://zbmath.org/authors/?q=ai:ahmad.naveedSummary: Motivated by \textit{J. Pečarić} et al. [J. Math. Inequal. 11, No. 2, 543--550 (2017; Zbl 1370.26026)], we established here weighted discrete dynamic inequalities for the difference of second divided difference of 4-convex functions. Further we extend and unify the two inequalities, by establishing the theory of \(n\)-convexity on time scales having constant graininess function.Uniqueness of a meromorphic function sharing two values CM with its difference operatorhttps://zbmath.org/1503.300732023-03-23T18:28:47.107421Z"Barki, Mahesh"https://zbmath.org/authors/?q=ai:barki.mahesh"Dyavanal, Renukadevi S."https://zbmath.org/authors/?q=ai:dyavanal.renukadevi-sangappa"Bhoosnurmath, Subhas S."https://zbmath.org/authors/?q=ai:bhoosnurmath.subhas-sSummary: This research investigates the uniqueness of meromorphic function \(f(z)\) and its \(n^{th}\) order difference operator with two shared values counting multiplicities. It generalizes the results of \textit{Y. Jiang} and \textit{Z. Chen} [An. Științ. Univ. Al. I. Cuza Iași, Ser. Nouă, Mat. 63, No. 1, 169--175 (2017; Zbl 1399.30124)] and for the case \(n \geq 2\), it admits a very special generalization. Moreover, we give an example to illustrate the validity of our main result.Uniqueness of meromorphic functions concerning \(k\)-th derivatives and difference operatorshttps://zbmath.org/1503.300762023-03-23T18:28:47.107421Z"Haldar, Goutam"https://zbmath.org/authors/?q=ai:haldar.goutam(no abstract)Entire and meromorphic solutions for systems of the differential difference equationshttps://zbmath.org/1503.300812023-03-23T18:28:47.107421Z"Xu, Hong Yan"https://zbmath.org/authors/?q=ai:xu.hongyan"Li, Hong"https://zbmath.org/authors/?q=ai:li.hong.8"Ding, Xin"https://zbmath.org/authors/?q=ai:ding.xinSummary: With the help of the Nevanlinna theory of meromorphic functions, the purpose of this article is to describe the existence and the forms of transcendental entire and meromorphic solutions for several systems of the quadratic trinomial functional equations:
\[
\begin{cases}
f(z)^2 + 2\alpha f(z)g(z+c) + g(z+c)^2=1,\\
g(z)^2 + 2\alpha g(z)f(z+c) + f(z+c)^2=1,
\end{cases}
\]
\[
\begin{cases}
f(z+c)^2 + 2\alpha f(z+c)g^\prime(z) + g^\prime(z)^2=1,\\
g(z+c)^2 + 2\alpha g(z+c)f^\prime(z) + f^\prime(z)^2=1,
\end{cases}
\]
and
\[
\begin{cases}
f(z+c)^2 + 2\alpha f(z+c)g^{\prime\prime}(z) + g^{\prime\prime}(z)^2=1,\\
g(z+c)^2 + 2\alpha g(z+c)f^{\prime\prime}(z) + f^{\prime\prime}(z)^2=1.
\end{cases}
\]
We obtain a series of results on the forms of the entire solutions with finite order for such systems, which are some improvements and generalizations of the previous theorems given by \textit{Z. Gao} et al. [Chin. Ann. Math., Ser. A 14, No. 6, 677--685 (1993; 0801.30029)]. Moreover, we provide some examples to explain the existence and forms of solutions for such systems in each case.Results on certain difference polynomials and shared valueshttps://zbmath.org/1503.300822023-03-23T18:28:47.107421Z"Xu, Hui-Cai"https://zbmath.org/authors/?q=ai:xu.huicai"Li, Xiao-Min"https://zbmath.org/authors/?q=ai:li.xiaomin"Yu, Hui"https://zbmath.org/authors/?q=ai:yu.hui"Zhang, Qing-Cai"https://zbmath.org/authors/?q=ai:zhang.qingcaiSummary: In this paper, we study uniqueness questions for meromorphic functions for which certain difference polynomials share a finite non-zero value, and give mathematical expressions for the meromorphic functions in the conclusions of the main results in the present paper, which are the related to the questions studied in [the second and the third author, Bull. Korean Math. Soc. 55, No. 5, 1529--1561 (2018; Zbl 1403.30011)].Determination of a differential pencil frominterior spectral data on a union of two closed intervalshttps://zbmath.org/1503.310182023-03-23T18:28:47.107421Z"Adalar, İbrahim"https://zbmath.org/authors/?q=ai:adalar.ibrahimSummary: In this paper, we consider a quadratic pencil of Sturm-Liouville operator on closed sets. We study an interior-inverse problem for this kind operator and give a uniqueness theorem with an appropriate example.On multi-step methods for singular fractional \(q\)-integro-differential equationshttps://zbmath.org/1503.340202023-03-23T18:28:47.107421Z"Hajiseyedazizi, Sayyedeh Narges"https://zbmath.org/authors/?q=ai:hajiseyedazizi.sayyedeh-narges"Samei, Mohammad Esmael"https://zbmath.org/authors/?q=ai:samei.mohammad-esmael"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Chu, Yu-ming"https://zbmath.org/authors/?q=ai:chu.yuming(no abstract)Global exponential stability of both continuous-time and discrete-time switched positive time-varying delay systems with interval uncertainties and all unstable subsystemshttps://zbmath.org/1503.341272023-03-23T18:28:47.107421Z"Mouktonglang, Thanasak"https://zbmath.org/authors/?q=ai:mouktonglang.thanasak"Yimnet, Suriyon"https://zbmath.org/authors/?q=ai:yimnet.suriyon(no abstract)Ground state solutions for periodic discrete Schrödinger equations with local super-quadratic conditionshttps://zbmath.org/1503.352152023-03-23T18:28:47.107421Z"Xu, Xionghui"https://zbmath.org/authors/?q=ai:xu.xionghui"Sun, Jijiang"https://zbmath.org/authors/?q=ai:sun.jijiangSummary: In this paper, we consider the following discrete nonlinear Schrödinger equation
\[
\begin{cases}
-\Delta u_n+\epsilon_nu_n-\omega u_n=f_n(u_n), \quad n \in \mathbb{Z}, \\
\lim_{|n| \rightarrow \infty} u_n=0,
\end{cases}
\] with periodic potentials and \(\omega\) belongs to a spectral gap of the operator \(L=-\Delta +\epsilon\) defined by \(Lu_n=-\Delta u_n+\epsilon_n u_n\). In order to investigate the existence of ground state solutions or infinitely many geometrically distinct solutions, it is commonly assumed that \(\lim_{|s| \rightarrow \infty} \frac{\int \limits_0^s f_n(t) \text{d} t}{s^2}=\infty\) uniformly for all \(n \in \mathbb{Z}\) in the existing literature. The purpose of this paper is to prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition \(\lim_{|s| \rightarrow \infty } \frac{\int \limits_0^s f_n(t) \text{d} t}{s^2}=\infty\) for \(n \in G\) just for some set \(G \subset \mathbb{Z}\). Our result sharply extends and improves some existing ones in the literature.Analytic linearization of a generalization of the semi-standard map: radius of convergence and Brjuno sumhttps://zbmath.org/1503.370712023-03-23T18:28:47.107421Z"Chavaudret, Claire"https://zbmath.org/authors/?q=ai:chavaudret.claire"Marmi, Stefano"https://zbmath.org/authors/?q=ai:marmi.stefanoThe celebrated standard map of the cylinder is characterized by a nonlinear term in the form \(\sin x\). It was observed that certain analytic aspects of the system become easier to analyze, albeit in the complex variable, when that nonlinear term is replaced by \(e^{ix}\) and the system is referred to as a semi-standard map.
The present paper includes a further generalization with the nonlinear term in the form of a trigonometric polynomial which contains only positive powers of \(e^{ix}\). The specific problem is: what happens near a smooth invariant curve in the case of irrational rotation numbers? The mapping is expected to be analytically linearizable in a neighborhood of the curve provided that a typical arithmetic condition is satisfied by the rotation number. This is indeed a part of the first theorem of the paper, with the ``typical condition'' being the Brjuno condition, but quite remarkably the theorem also contains a lower estimate of the radius of the linearization domain given in terms of the exponential of a Brjuno sum which involves not only the rotation number, but the common divisor of powers in the trigonometric term. A second theorem provides a similar upper estimate for the trigonometric term in a rather special form, thus establishing the necessity of the Brjuno condition in general. The methods of the proof are classical, starting with a formal power series of the linearization and establishing conditions for its convergence.
The paper makes a valuable contribution to understanding not only analytic linearizations, but also the persistence and breakdown of invariant curves in KAM theory. It should be remarked that the nonlinear term does not need to be small; in this respect the result can be compared to the work of \textit{J.-C. Yoccoz} [Petits diviseurs en dimension 1. Paris: Socièté Math. de France (1995; Zbl 0836.30001)]
on Siegel disks for quadratic polynomials.
Reviewer: Grzegorz Świątek (Warszawa)Snakes on Lieb latticehttps://zbmath.org/1503.370842023-03-23T18:28:47.107421Z"Kusdiantara, R."https://zbmath.org/authors/?q=ai:kusdiantara.rudy"Akbar, F. T."https://zbmath.org/authors/?q=ai:akbar.fiki-taufik"Nuraini, N."https://zbmath.org/authors/?q=ai:nuraini.nuning"Gunara, B. E."https://zbmath.org/authors/?q=ai:gunara.bobby-eka"Susanto, H."https://zbmath.org/authors/?q=ai:susanto.hadiSummary: We consider the discrete Allen-Cahn equation with cubic and quintic nonlinearity on the Lieb lattice. We study localized nonlinear solutions of the system that have linear multistability and hysteresis in their bifurcation diagram. In this work, we investigate the system's homoclinic snaking, i.e. snaking-like structure of the bifurcation diagram, particularly the effect of the lattice type. Numerical continuation using a pseudo-arclength method is used to obtain localized solutions along the bifurcation diagram. We then develop an active-cell approximation to classify the type of solution at the turning points, which gives good agreement with the numerical results when the sites are weakly coupled. Time dynamics of localized solutions inside and outside the pinning region is also discussed.Bifurcation and control of a predator-prey system with unfixed functional responseshttps://zbmath.org/1503.370932023-03-23T18:28:47.107421Z"Fei, Lizhi"https://zbmath.org/authors/?q=ai:fei.lizhi"Chen, Xingwu"https://zbmath.org/authors/?q=ai:chen.xingwuSummary: In this paper we investigate a discrete-time predator-prey system with not only some constant parameters but also unfixed functional responses including growth rate function of prey, conversion factor function and predation probability function. We prove that the maximal number of fixed points is \(3\) and give necessary and sufficient conditions of exactly \(j(j = 1,2,3)\) fixed points, respectively. For transcritical bifurcation and Neimark-Sacker bifurcation, we provide bifurcation conditions depending on these unfixed functional responses. In order to regulate the stability of this biological system, a hybrid control strategy is used to control the Neimark-Sacker bifurcation. Finally, we apply our main results to some examples and carry out numerical simulations for each example to verify the correctness of our theoretical analysis.Intrinsic approach to Galois theory of \(q\)-difference equationshttps://zbmath.org/1503.390012023-03-23T18:28:47.107421Z"Di Vizio, Lucia"https://zbmath.org/authors/?q=ai:di-vizio.lucia"Hardouin, Charlotte"https://zbmath.org/authors/?q=ai:hardouin.charlotte"Granier, Anne"https://zbmath.org/authors/?q=ai:granier.anneSummary: The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of \(q\)-difference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of \(q\)-difference equations and we use it to establish an arithmetical description of some of the Galois groups attached to \(q\)-difference systems.Power series solutions of non-linear \(q\)-difference equations and the Newton-Puiseux polygonhttps://zbmath.org/1503.390022023-03-23T18:28:47.107421Z"Cano, J."https://zbmath.org/authors/?q=ai:cano.jose-maria"Fortuny Ayuso, P."https://zbmath.org/authors/?q=ai:fortuny-ayuso.pedroThe authors study the existence of power series solutions to nonlinear \(q\)-difference equations of any order and degree, by using the Newton-Puiseux polygon process. In addition, they investigate the properties of the set of exponents of the solutions and give a bound for their \(q\)-Gevrey order. An example is given.
Reviewer: Thanin Sitthiwirattham (Bangkok)Asymptotic cycles in fractional maps of arbitrary positive ordershttps://zbmath.org/1503.390032023-03-23T18:28:47.107421Z"Edelman, Mark"https://zbmath.org/authors/?q=ai:edelman.mark"Helman, Avigayil B."https://zbmath.org/authors/?q=ai:helman.avigayil-b(no abstract)Caputo delta weakly fractional difference equationshttps://zbmath.org/1503.390042023-03-23T18:28:47.107421Z"Fečkan, Michal"https://zbmath.org/authors/?q=ai:feckan.michal"Pospíšil, Michal"https://zbmath.org/authors/?q=ai:pospisil.michal"Danca, Marius-F."https://zbmath.org/authors/?q=ai:danca.marius-florin"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrong(no abstract)Lyapunov-type inequalities for higher-order half-linear difference equationshttps://zbmath.org/1503.390052023-03-23T18:28:47.107421Z"Liu, Haidong"https://zbmath.org/authors/?q=ai:liu.haidongSummary: In this paper, we will establish some new Lyapunov-type inequalities for some higher-order superlinear-sublinear difference equations with boundary conditions. Our results not only complement the existing results established in the literature, but also furnish a handy tool for the study of qualitative properties of solutions of some difference equations.Discrete fractional order two-point boundary value problem with some relevant physical applicationshttps://zbmath.org/1503.390062023-03-23T18:28:47.107421Z"Selvam, A. George Maria"https://zbmath.org/authors/?q=ai:selvam.a-george-maria"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Dhineshbabu, R."https://zbmath.org/authors/?q=ai:dhineshbabu.raghupathi"Rashid, S."https://zbmath.org/authors/?q=ai:rashid.sabrina|rashid.sheikh|rashid.sabbir-m|rashid.saima|rashid.suliman|rashid.salim|rashid.shahid"Rehman, M."https://zbmath.org/authors/?q=ai:rehman.mutti-ur|ur-rehman.mujeebSummary: The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann-Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers-Ulam stability and Hyers-Ulam-Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.Discrete boundary value problems as approximate constructionshttps://zbmath.org/1503.390072023-03-23T18:28:47.107421Z"Vasilyev, A. V."https://zbmath.org/authors/?q=ai:vasilev.aleksander-vladimirovich"Vasilyev, V. B."https://zbmath.org/authors/?q=ai:vasilyev.vladimir-b"Tarasova, O. A."https://zbmath.org/authors/?q=ai:tarasova.o-aThe authors investigate some boundary value problems for discrete elliptic pseudo-differential equations in the special domain \(D=\mathbb{R}_+^m=\{x\in\mathbb{R}^m,\,\,x=(x_1,\ldots,x_m),\,\,x_m>0\}\).
Reviewer: Rodica Luca (Iaşi)Boundary value problems for a second-order difference equation involving the mean curvature operatorhttps://zbmath.org/1503.390082023-03-23T18:28:47.107421Z"Wang, Zhenguo"https://zbmath.org/authors/?q=ai:wang.zhenguo"Xie, Qilin"https://zbmath.org/authors/?q=ai:xie.qilinThe authors obtain sufficient conditions for the existence of multiple solutions for the following nonlinear difference equation with the mean curvature operator:
\[
-\Delta(\phi_{c}(\Delta u(t-1)))+q(t)u(t)=\lambda f(t,u(t)),
\]
with \(t\in \mathbb{Z}(1,T), u(0)=u(T+1)=0.\) Here \(q(t)\in \mathbb{R}^{+}\) for each \(t\in \mathbb{Z}(1,T),\) \(\lambda > 0\) is a parameter, \(\Delta\) is the forward difference operator defined by \(\Delta u(t)=u(t+1)-u(t), \Delta^{2}u(t)=\Delta(\Delta u(t))\), \(T\) is a given positive integer, \(f(t,.)\in C(\mathbb{R},\mathbb{R})\) for each \(t \in \mathbb{Z}(1,T)\). For \(a,b \in \mathbb{Z}, \mathbb{Z}(a,b)\) denotes the discrete interval \(\{ a,a+1, \dots, b\}\) if \(a\leq b\); \(\phi_{c}\) is the mean curvature operator defined by
\[
\phi_{c}(\xi)=\frac{\xi}{\sqrt{1+k\xi^{2}}} , \quad k>0.
\]
Here the nonlinear terms do not need any asymptotic and super-linear conditions at 0 or at infinity. For their work, they used Clark's theorem (see [\textit{P. H. Rabinowitz}, Minimax methods in critical point theory with applications to differential equations. Providence, RI: American Mathematical Society (AMS) (1986; Zbl 0609.58002); \textit{D. Bai} and \textit{Y. Xu}, J. Math. Anal. Appl. 326, No. 1, 297--302 (2007; Zbl 1113.39018)]. They also consider the existence of a positive solution by using the strong comparison principle. Examples are given to illustrate the results obtained.
Reviewer: Narahari Parhi (Bhubaneswar)Transcritical bifurcation and flip bifurcation of a new discrete ratio-dependent predator-prey systemhttps://zbmath.org/1503.390092023-03-23T18:28:47.107421Z"Li, Xianyi"https://zbmath.org/authors/?q=ai:li.xianyi"Liu, Yuqing"https://zbmath.org/authors/?q=ai:liu.yuqingThe authors consider the dynamical behavior of a discrete two-species predator-prey system with ratio-dependent functional response. They analyze the stability of all equilibria and derive some sufficient conditions for the occurence of transcritical bifurcations and period-doubling bifurcations.
Reviewer: Fengqin Zhang (Yuncheng)Bifurcation analysis of a piecewise-smooth Ricker map with proportional threshold harvestinghttps://zbmath.org/1503.390102023-03-23T18:28:47.107421Z"Liz, Eduardo"https://zbmath.org/authors/?q=ai:liz.eduardoSummary: Proportional threshold harvesting (PTH) refers to some control rules employed in fishing policies, which specify a biomass level below which no fishing is permitted (the threshold), and a fraction of the surplus above the threshold is removed every year. When these rules are applied to a discrete population model, the resulting map governing the harvesting model is piecewise smooth, so border-collision bifurcations play an essential role in the dynamics. In this paper, we carry out a bifurcation analysis of a PTH model, providing a thorough picture of the 2-parameter bifurcation diagram in the plane \((T,q)\) for a case study. Here, \(T\) is the threshold and \(q\) is the harvest proportion. Our results explain some numerical bifurcation diagrams in previous work for PTH, and uncover new features of the dynamics with interesting consequences for population management.Dynamics in a discrete time model of logistic typehttps://zbmath.org/1503.390112023-03-23T18:28:47.107421Z"Yu, Zhiheng"https://zbmath.org/authors/?q=ai:yu.zhiheng"Zhong, Jiyu"https://zbmath.org/authors/?q=ai:zhong.jiyu"Zeng, Yingying"https://zbmath.org/authors/?q=ai:zeng.yingying"Li, Song"https://zbmath.org/authors/?q=ai:li.song.1|li.songSummary: In this paper, we investigate the qualitative properties and bifurcations of a discrete-time logistic type model for the competitive interaction of two species. Applying \textit{polynomial symbolic algebraic theory} to deal with complex high-order semi-algebraic systems, and using the bifurcation theory, we give not only the topological structure of the orbits near each fixed point but also the parameter conditions such that the model produces transcritical bifurcation, supercritical (or subcritical) flip bifurcation and supercritical (or subcritical) Neimark-Sacker bifurcation, respectively. Besides, the corresponding mapping is proven to be chaotic in the sense of Marotto. At last, we simulate the stable orbits of period 2 produced from the supercritical flip bifurcation, the stable invariant circle resulting from the Neimark-Sacker bifurcation and the chaos in the sense of Marotto to verify our results.Asymptotic solutions of the discrete Painlevé equation of second typehttps://zbmath.org/1503.390122023-03-23T18:28:47.107421Z"Novokshenov, V. Yu."https://zbmath.org/authors/?q=ai:novokshenov.viktor-yurevichSeveral classes of asymptotic solutions of the discrete Painlevé equation of second type (dPII) for large values of the independent variable are found. The cases of complex and real solutions are considered, as well as special solutions related to symmetric group representations.
Reviewer: Mengkun Zhu (Jinan)Stabilizing multiple equilibria and cycles with noisy prediction-based controlhttps://zbmath.org/1503.390132023-03-23T18:28:47.107421Z"Braverman, Elena"https://zbmath.org/authors/?q=ai:braverman.elena"Rodkina, Alexandra"https://zbmath.org/authors/?q=ai:rodkina.alexandraPrediction-based control (PBC) was introduced by \textit{T. Ushio} and \textit{S. Yamamoto} [Phys. Lett., A 264, No. 1, 30--35 (1999; Zbl 0941.93023)] as a technique for controlling unstable or chaotic dynamics in iterative systems. The control itself may be applied at every iteration or periodically, the latter case is referred to as pulsed PBC. In any case, the application of the control at the \(n^{\mathrm{th}}\) step is modulated by a control parameter \(\alpha\) that may be stochastic.
PBC is often studied in the context of population dynamics, and its application to systems governed by a Ricker map, a logistic map, or a Maynard Smyth map, was studied in [\textit{E. Braverman} et al., Chaos 30, No. 9, 093116, 15 p. (2020; Zbl 1454.93286)]. In this article, the authors try to demonstrate how, for a scalar iterative equation governed by a map with several fixed points, PBC can be applied to simultaneously stabilise all odd-indexed fixed points.
Their approach relies on the natural decomposition of the state space of the set of fixed points into intervals, where they show that the controlled system solutions can only converge to one of the equilibrium points or circulate infinitely between these intervals. They then identify how large the control parameter should be in order to suppress the possibility of infinite circulation. The map itself is assumed to obey one-sided Lipschitz-type bounds in each of these intervals.
Finally, the authors introduce a stochastic variation around the mean value of the control parameter that is bounded in magnitude and stepwise independent. They are able to identify the range of noise intensities for which a PBC that is stabilising in the absence of noise, will continue to be stabilising after the introduction of noise, and they also show that the presence of noise can increase the range of possible mean control parameters for which stabilisation is possible.
These results are illustrated with numerical examples that include compound forms of the Ricker map with four or more fixed points.
Reviewer: Conall Kelly (Cork)Asymptotic stability of a stochastic discrete logistic equation with delayhttps://zbmath.org/1503.390142023-03-23T18:28:47.107421Z"Wu, Xiaohua"https://zbmath.org/authors/?q=ai:wu.xiaohua"Liao, Xinyuan"https://zbmath.org/authors/?q=ai:liao.xinyuan"Ge, Lingling"https://zbmath.org/authors/?q=ai:ge.lingling(no abstract)Boundary effects on eigen-problems of discrete Laplacian in latticeshttps://zbmath.org/1503.390152023-03-23T18:28:47.107421Z"Kuo, Yueh-Cheng"https://zbmath.org/authors/?q=ai:kuo.yueh-cheng"Shieh, Shih-Feng"https://zbmath.org/authors/?q=ai:shieh.shih-fengSummary: We consider how distribution of eigenvalues depends on boundary conditions of a discrete Laplacian operator on lattices. We study the Laplacian with boundary conditions given by a linear combination of Dirichlet and Neumann conditions. In particular, we derive a secular equation and investigate the Laplacian operator's eigenvalues with different boundary conditions, including the interlacing property, the first eigenvalue gaps, and the monotonicity property.Taylor theory associated with Hahn difference operatorhttps://zbmath.org/1503.390162023-03-23T18:28:47.107421Z"Oraby, Karima"https://zbmath.org/authors/?q=ai:oraby.karima-m"Hamza, Alaa"https://zbmath.org/authors/?q=ai:hamza.alaa-eSummary: In this paper, we establish Taylor theory based on Hahn's difference operator \(D_{q,\omega}\) which is defined by \(D_{q,\omega}f(t)=\frac{f(qt+\omega)-f(t)}{t(q-1)+\omega}, \ t\neq\frac{\omega}{1-q} \), where \(q\in(0,1)\) and \(\omega\) is a positive number.Transportation inequalities for Markov kernels and their applicationshttps://zbmath.org/1503.390172023-03-23T18:28:47.107421Z"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabrice"Eldredge, Nathaniel"https://zbmath.org/authors/?q=ai:eldredge.nathanielThe aim of the paper is to study the connection between reverse Poincaré inequalities and Hellinger-Kantorovich contraction inequalities for heat semigroups associated with Markovian kernels. One of the main results of the paper (Theorem 3.7) is that given a heat kernel \(P\) on a metric space \(X\) (assumed to be a length space, in particular, path connected, equipped with a strong upper gradient) then the following inequalities are equivalent: the reverse Poincaré inequality
\[
| \nabla P \, f|^2 \le C (P(f^2)-(P(f))^2), \quad \forall \, f \in \mathrm{Lip}_{b}(X),
\]
the Hellinger-Kantorovich contraction inequalities
\[
He_{2}(\mu_0 P, \mu_1 P) \le HK_{4/C}(\mu_0,\mu_1) \le \sqrt{\frac{C}{4}} W_{2}(\mu_0,\mu_1),
\]
where \(\mu_0,\mu_1\) are probability measures on \(X\) and \(He\), \(HK\), \(W_{2}\) are the Hellinger, the Hellinger-Kantorovich and the Wasserstein distances respectively, and the Harnack type inequality
\[
Pf(x) \le P f(y) + \sqrt{C} \, d(x,y) \sqrt{P(f^2)(x)}, \quad\forall \, x,y \in X, \, f \in B_{b}(X), \quad f \ge 0.
\]
This result builds on the following tools:
\begin{itemize}
\item[(1)] A suitable extension of recent results of \textit{G. Luise} and \textit{G. Savaré} [Discrete Contin. Dyn. Syst., Ser. S 14, No. 1, 273--297 (2021; Zbl 1459.49030)] on the contraction properties of heat semigroups in spaces of measures, to general metric spaces;
\item[(2)] The connection of the resulting Hellinger-Kantorovich contraction inequalities to reverse Poincaré inequalities;
\item[(3)] A dynamic dual formulation of the Hellinger-Kantorovich distance to define a new family of divergences on \(P(X)\) which generalize the Rényi divergence.
This allows the authors to prove that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang-Harnack inequalities. \end{itemize}
This important result connects the Hellinger-Kantorovich contraction with well known functional inequalities.
The results of the paper are illustrated by various important applications, including subelliptic diffusions arising in sub-Riemannian geometry, non-symmetric Ornstein-Uhlenbeck operators on Carnot groups, Langevin dynamics driven by Lévy processes, and others.
Reviewer: Athanasios Yannacopoulos (Athína)On some inequalities relative to the Pompeiu-Chebyshev functionalhttps://zbmath.org/1503.390182023-03-23T18:28:47.107421Z"Ianoşi, Daniel"https://zbmath.org/authors/?q=ai:ianosi.daniel"Opriş, Adonia-Augustina"https://zbmath.org/authors/?q=ai:opris.adonia-augustinaSummary: In this paper we study the utility of the functional Pompeiu-Chebyshev in some inequalities. Some results obtained by
\textit{M. W. Alomari} [Result. Math. 74, No. 1, Paper No. 56, 36 p. (2019; Zbl 1408.26018)]
will be generalized regarding inequalities with Pompeiu-Chebyshev type functionals, in which linear and positive functionals intervene. We investigate some new inequalities of Grüss type using Pompeiu's mean value theorem. Improvement of known inequalities is also given.The stability of \(N\)-dimensional quadratic functional inequality in non Archimedean Banach spaceshttps://zbmath.org/1503.390192023-03-23T18:28:47.107421Z"Aribou, Y."https://zbmath.org/authors/?q=ai:aribou.youssef"Kabbaj, S."https://zbmath.org/authors/?q=ai:kabbaj.samirSummary: In this paper, using the direct method, we study the stability of the following inequality:
\[
\begin{multlined}
\left\| f\left( \sum^n_{i=1}x_i\right) +\sum_{1\leq i \prec j\leq n}f(x_i -x_j) -n\sum^n_{i=1} f(x_i)\right\| \\
\qquad \leq \left\| f\left( \frac{\sum^n_{i=1}x_i}{n}\right) +\sum_{1\leq i \prec j\leq n}f\left( \frac{x_i -x_j}{n}\right) -\frac{1}{n}\sum^n_{i=1} f(x_i)\right\|
\end{multlined}
\]
in Banach spaces, and the stability of the following inequality:
\[
\begin{multlined}
\left\| f\left( \frac{\sum^n_{i=1}x_i}{n}\right) +\sum_{1\leq i \prec j\leq n}f\left( \frac{x_i -x_j}{n}\right) -\frac{1}{n}\sum^n_{i=1}f(x_i)\right\|_* \\
\qquad \leq \left\| f\left( \sum^n_{i=1}x_i\right) +\sum_{1\leq i \prec j\leq n}f(x_i -x_j)-n\sum^n_{i=1} f(x_i)\right\|_*
\end{multlined}
\]
in non-Archimedean Banach spaces with \(n\) an integer greater than or equal to \(2\).Multi-quadratic mappings with an involutionhttps://zbmath.org/1503.390202023-03-23T18:28:47.107421Z"Bodaghi, Abasalt"https://zbmath.org/authors/?q=ai:bodaghi.abasaltLet \((G,+)\) be an abelian group, \(V\) be a vector space over \(\mathbb{Q}\), \(\sigma:G\to G\) be a fixed involution of \(G\) and \(I\) be the identity mapping on \(G\). A mapping \(f:G^n \to V\) is called \(n\)-multi-quadratic with the involution \(\sigma\) if \(f\) fulfills the quadratic equation
\[
Q(x+y)+Q(x+\sigma(y))=2Q(x)+2Q(y)
\]
in each variable. This and several related equations were studied for \(Q:G\to \mathbb{C}\) by \textit{H. Stetkær} [Aequationes Math. 54, No. 1--2, 144--172 (1997; Zbl 0899.39007)].
It is shown that \(f:G^n \to V\) is multi-quadratic if and only if it satisfies the equation
\[
\sum\limits_{\sigma_1,\dots,\sigma_n\in\{I,\sigma\}}\!\!\!\!\!f(x_{11}+\sigma_1(x_{21}),\dots, x_{1n}+\sigma_n(x_{2n}))=2^n\!\!\!\!\!\!\!\sum\limits_{j_1,\dots,j_n\in\{1,2\}}\!\!\!\!\!f(x_{j_1,1},\dots,x_{j_n,n})
\]
for all \(x_1=(x_{11},\dots,x_{1n}),x_2=(x_{21},\dots,x_{2n})\in G^n\).
The main result of the paper is about Hyers-Ulam stability of multi-quadratic mappings in normed spaces. Applications and examples of this stability theorem are also given in the setting of normed \(\star\)-algebras.
Reviewer: László Losonczi (Debrecen)Two multi-cubic functional equations and some results on the stability in modular spaceshttps://zbmath.org/1503.390212023-03-23T18:28:47.107421Z"Park, Choonkil"https://zbmath.org/authors/?q=ai:park.choonkil"Bodaghi, Abasalt"https://zbmath.org/authors/?q=ai:bodaghi.abasaltSummary: In this article, we study \(n\)-variable mappings which are cubic in each variable. We also show that such mappings can be described by an equation, say, multi-cubic functional equation. Furthermore, we study the stability of such functional equations in the modular space \(X_{\rho }\) by applying \(\Delta_2\)-condition and the Fatou property (in some cases) on the modular function \(\rho \). Finally, we show that, under some mild conditions, one of these new multi-cubic functional equations can be hyperstable.Stabilities and non-stabilities of a new reciprocal functional equationhttps://zbmath.org/1503.390222023-03-23T18:28:47.107421Z"Senthil Kumar, B. V."https://zbmath.org/authors/?q=ai:senthil-kumar.b-v"Rassias, J. M."https://zbmath.org/authors/?q=ai:rassias.john-michael"Idir, S."https://zbmath.org/authors/?q=ai:idir.sadani"Sabarinathan, S."https://zbmath.org/authors/?q=ai:sabarinathan.sriramuluSummary: The intention of this study is to present some stronger results by investigating Ulam-JRassias product stability and Ulam-JRassias mixed-type sum-product stability of a new reciprocal functional equation. Also, a suitable counter-example is presented to show the failure of stability result for the singular case.Sharp phase transitions for the almost Mathieu operatorhttps://zbmath.org/1503.470412023-03-23T18:28:47.107421Z"Avila, Artur"https://zbmath.org/authors/?q=ai:avila.artur"You, Jiangong"https://zbmath.org/authors/?q=ai:you.jiangong"Zhou, Qi"https://zbmath.org/authors/?q=ai:zhou.qiSummary: It is known that the spectral type of the almost Mathieu operator (AMO) depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya's conjecture [\textit{S. Ya. Jitomirskaya}, in: Proceedings of the XIth international congress on mathematical physics, Paris, France, July 18--23, 1994. Cambridge, MA: International Press. 373--382 (1995; Zbl 1052.82539); Proc. Symp. Pure Math. 76, 613--647 (2007; Zbl 1129.82018)]. Together with [\textit{A. Avila}, ``The absolutely continuous spectrum of the almost Mathieu operator'', Preprint (2008), \url{arXiv:0810.2965}], this gives the sharp description of phase transitions for the AMO for the a.e.\ phase.Riemannian approximation in Carnot groupshttps://zbmath.org/1503.530602023-03-23T18:28:47.107421Z"Domokos, András"https://zbmath.org/authors/?q=ai:domokos.andras.2"Manfredi, Juan J."https://zbmath.org/authors/?q=ai:manfredi.juan-j"Ricciotti, Diego"https://zbmath.org/authors/?q=ai:ricciotti.diegoThis paper concerns left-invariant sub-Riemannian metrics on Carnot groups, also known as stratified Lie groups, and their approximations by Riemannian metrics.
Let \(G\) be a Carnot group of step \(\nu\), whose Lie algebra \(\mathfrak{g}\) has a stratification \(\mathfrak{g} = \bigoplus_{i=1}^\nu V^i\), where \([V^1, V^i] = V^{i+1}\) and \([V^1, V^\nu]=0\). Fix a basis \(X_{ij} : 1 \le j \le n_i= \dim V^i\) for each \(V^i\), which can be identified with left-invariant vector fields on \(G\). Then \(G\) has a natural left-invariant sub-Riemannian metric \(g\), having horizontal distribution \(V^1\) and for which the vector fields \(X_{11}, \dots, X_{1n_1}\) are orthonormal. The sub-Riemannian metric \(g\) can be approximated by a one-parameter family of left-invariant Riemannian metrics \(g^{\varepsilon}\), \(\varepsilon > 0\), for which the vector fields \(\varepsilon^{i-1} X_{ij}\), \(1 \le j \le n_i\), \(1 \le i \le \nu\), are orthonormal.
The main result of this paper is that the volume doubling constants of the Riemannian metrics \(g^{\varepsilon}\) are stable as \(\varepsilon \to 0\). More precisely, let \(B^{\varepsilon}(x,r)\) denote a ball of the Riemannian distance induced by \(g^\varepsilon\), and \(|\cdot|\) the Haar measure on \(G\) (which is identified with Lebesgue measure). The result is then that we have for all \(x,r\) that \(|B^\varepsilon(x, 2r)| \le c_d |B^\varepsilon(x,r)|\), for some constant \(c_d\) which (this is the key point) is independent of \(\varepsilon\).
The authors note that more general results on the stability of volume doubling constants for Riemannian approximations were obtained in [\textit{L. Capogna} et al., Math. Ann. 357, No. 3, 1175--1198 (2013; Zbl 1282.35204); \textit{L. Capogna} and \textit{G. Citti}, Bull. Math. Sci. 6, No. 2, 173--230 (2016; Zbl 1357.35092)], in the setting of general vector fields satisfying the Hörmander bracket generating condition. The fundamental underlying results can be traced back to the seminal paper [\textit{A. Nagel} et al., Acta Math. 155, 103--147 (1985; Zbl 0578.32044)]. However, in the setting of the present paper, the extra structure of the Carnot group allows one the use of the simple and natural family of approximations \(g^\varepsilon\) defined above, and leads to more explicit estimates.
As a corollary of the volume doubling statement, the authors also give a proof that there is a uniform bound on the constants in the Poincaré and Poincaré-Sobolev inequalities for the metrics \(g^{\varepsilon}\). For more on these and other functional inequalities, and their relationship to volume doubling, see [\textit{P. Hajłasz} and \textit{P. Koskela}, Sobolev met Poincaré. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0954.46022); \textit{L. Saloff-Coste}, Aspects of Sobolev-type inequalities. Cambridge: Cambridge University Press (2002; Zbl 0991.35002)].
Reviewer: Nathaniel Eldredge (Storrs)Bands of pure absolutely continuous spectrum for lattice Schrödinger operators with a more general long range conditionhttps://zbmath.org/1503.810272023-03-23T18:28:47.107421Z"Golénia, Sylvain"https://zbmath.org/authors/?q=ai:golenia.sylvain"Mandich, Marc-Adrien"https://zbmath.org/authors/?q=ai:mandich.marc-adrienSummary: Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schrödinger operators, \(\Delta + V\) and \(D + V\) on \(\ell^2(\mathbb{Z}^d)\), with emphasis on \(d = 1, 2, 3\). Considered are electric potentials \(V\) satisfying a long range condition of the following type: \(V - \tau_j^\kappa V\) decays appropriately at infinity for some \(\kappa \in \mathbb{N}\) and all \(1 \leq j \leq d\), where \(\tau_j^\kappa V\) is the potential shifted by \(\kappa\) units on the \(j\) th coordinate. More comprehensive results are obtained for small values of \(\kappa\), e.g., \(\kappa = 1, 2, 3, 4\). We work in a simplified framework in which the main takeaway appears to be the existence of bands where a limiting absorption principle holds, and hence, pure absolutely continuous spectrum exists. Other decay conditions at infinity for \(V\) arise from an isomorphism between \(\Delta\) and \(D\) in dimension 2. Oscillating potentials are examples in application.
{\copyright 2021 American Institute of Physics}A Glazman-Povzner-Wienholtz theorem on graphshttps://zbmath.org/1503.810352023-03-23T18:28:47.107421Z"Kostenko, Aleksey"https://zbmath.org/authors/?q=ai:kostenko.aleksey-s"Malamud, Mark"https://zbmath.org/authors/?q=ai:malamud.mark-m"Nicolussi, Noema"https://zbmath.org/authors/?q=ai:nicolussi.noemaSummary: The Glazman-Povzner-Wienholtz theorem states that the semiboundedness of a Schrödinger operator, when combined with suitable local regularity assumptions on its potential and the completeness of the underlying manifold, guarantees its essential self-adjointness. Our aim is to extend this result to Schrödinger operators on graphs. We first obtain the corresponding theorem for Schrödinger operators on metric graphs, allowing in particular distributional potentials which are locally \(H^{- 1}\). Moreover, we exploit recently discovered connections between Schrödinger operators on metric graphs and weighted graphs in order to prove a discrete version of the Glazman-Povzner-Wienholtz theorem.Self-assembly of geometric space from random graphshttps://zbmath.org/1503.830052023-03-23T18:28:47.107421Z"Kelly, Christy"https://zbmath.org/authors/?q=ai:kelly.christy"Trugenberger, Carlo A."https://zbmath.org/authors/?q=ai:trugenberger.carlo-a"Biancalana, Fabio"https://zbmath.org/authors/?q=ai:biancalana.fabioSummary: We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.Discrete maturity and delay differential-difference model of hematopoietic cell dynamics with applications to acute myelogenous leukemiahttps://zbmath.org/1503.920332023-03-23T18:28:47.107421Z"Adimy, Mostafa"https://zbmath.org/authors/?q=ai:adimy.mostafa"Chekroun, Abdennasser"https://zbmath.org/authors/?q=ai:chekroun.abdennasser"El Abdllaoui, Abderrahim"https://zbmath.org/authors/?q=ai:el-abdllaoui.abderrahim"Marzorati, Arsène"https://zbmath.org/authors/?q=ai:marzorati.arseneSummary: In the last few years, many efforts were oriented towards describing the hematopoiesis phenomenon in normal and pathological situations. This complex biological process is organized as a hierarchical system that begins with primitive hematopoietic stem cells (HSCs) and ends with mature blood cells: red blood cells, white blood cells and platelets. Regarding acute myelogenous leukemia (AML), a cancer of the bone marrow and blood, characterized by a rapid proliferation of immature cells, which eventually invade the bloodstream, there is a consensus about the target cells during the HSCs development which are susceptible to leukemic transformation. We propose and analyze a mathematical model of HSC dynamics taking into account two phases in the cell cycle, a resting and a proliferating one, by allowing just after division a part of HSCs to enter the resting phase and the other part to come back to the proliferating phase to divide again. The resulting mathematical model is a system of nonlinear differential-difference equations. Due to the hierarchical organization of the hematopoiesis, we consider \(n\) stages of HSCs characterized by their maturity levels. We obtain a system of \(2n\) nonlinear differential-difference equations. We study the existence, uniqueness, positivity, boundedness and unboundedness of the solutions. We then investigate the existence of positive and axial steady states for the system, and obtain conditions for their stability. Sufficient conditions for the global asymptotic stability of the trivial steady state as well as conditions for its instability are obtained. Using neutral differential equation associated to the differential-difference system, we also obtain results on the local asymptotic stability of the positive steady state. Numerical simulations are carried out to show the influence of variations of the differentiation rates and self-renewal coefficients of the HSCs on the behavior of the system. In particular, we show that a blocking of differentiation at an early stage of HSC development can lead in an overexpression of very immature cells. Such situation corresponds to the observation in the case of AML.