Recent zbMATH articles in MSC 39https://www.zbmath.org/atom/cc/392021-03-30T15:24:00+00:00WerkzeugHypertranscendence of solutions of Mahler equations.https://www.zbmath.org/1455.111072021-03-30T15:24:00+00:00"Dreyfus, Thomas"https://www.zbmath.org/authors/?q=ai:dreyfus.thomas"Hardouin, Charlotte"https://www.zbmath.org/authors/?q=ai:hardouin.charlotte"Roques, Julien"https://www.zbmath.org/authors/?q=ai:roques.julienGeneral criteria are proved for the algebraic independence of Mahler functions and their derivatives. The authors give concrete examples for illustrating their criteria, an example of which is the following. Let \(f_{\mathrm BS}\) and \(f_{\mathrm RS}\) be the generating functions of the Baum-Sweet and Rudin-Shapiro sequences. Then the series \(f_{\mathrm BS}(z)\), \(f_{\mathrm BS}(z^2)\), \(f_{\mathrm RS}(z)\), \(f_{\mathrm RS}(-z)\) and all their derivatives are algebraically independent over \({\mathbb{C}}(z)\). The proofs rely on the parametrized difference Galois theory developped by \textit{C. Hardouin} and \textit{M. F. Singer} [Math. Ann. 342, No. 2, 333--377 (2008; Zbl 1163.12002)]
and involve an unpublished contribution by M.~Singer where he proved the hypertranscendence of the Mahler series \(\sum_{n\ge 0} z^{p^n}\).
Let us quote the following so-called user friendly hypertranscendence criterion. Let \(p\ge 2\), let
\({\mathbb{K}}=\bigcup_{j\ge 1}{\mathbb{C}}(z^{1/j})\) be endowed with the structure of \(\phi\)-field given by \(\phi(f(z)):=f(z^p)\), let \(\delta\) be the derivation \(\delta=z\log(z)\frac{\mathrm{d}}{\mathrm{d}z}\) over \({\mathbb{K}}(\log z)\), let \(\widehat{\mathbb{C}}\) be a differential closure of \(({\mathbb{C}},\delta)\) and let \({\mathbb{L}}=\bigcup_{j\ge 1}\widehat{\mathbb{C}}(z^{1/j})(\log(z))\).
Consider a Mahler system \((\star)\, \phi(Y)=AY\)
with \(A\in{\mathrm{GL}}_n({\mathbb{C}}(z))\). Assume that the difference Galois group of the Mahler system \((\star)\) over the \(\phi\)--field \({\mathbb{K}}\) contains \({\mathrm{SL}}_n({\mathbb{C}})\) and that \({\mathrm{det}} A(z)\) is a monomial. Then the following properties hold.
(i) The parametrized difference Galois group of the Mahler system \((\star)\) over \({\mathbb{L}}\) is a subgroup of \({\mathbb{C}}^\times{\mathrm{SL}}_n(\widehat{\mathbb{C}})\) containing \({\mathrm{SL}}_n(\widehat{\mathbb{C}})\).
(ii) Let \(u=(u_1,\dots,u_n)^t\) be a nonzero solution of \((\star)\) with entries in \({\mathbb{C}}((z))\). Then the series \(u_1,\dots,u_n\) and all their derivatives are algebraically independent over \({\mathbb{C}}(z)\). In particular, any \(u_i\) is hypertranscendental over \({\mathbb{C}}(z)\).
Reviewer: Michel Waldschmidt (Paris)Delay reaction-diffusion systems via discrete dynamics.https://www.zbmath.org/1455.350242021-03-30T15:24:00+00:00"Ruiz-Herrera, Alfonso"https://www.zbmath.org/authors/?q=ai:ruiz-herrera.alfonsoA new generalization of Wilson's functional equation.https://www.zbmath.org/1455.390052021-03-30T15:24:00+00:00"Dimou, Hajira"https://www.zbmath.org/authors/?q=ai:dimou.hajira"Chahbi, Abdellatif"https://www.zbmath.org/authors/?q=ai:chahbi.abdellatif"Kabbaj, Samir"https://www.zbmath.org/authors/?q=ai:kabbaj.samirAssume that $G$ is a group, $\sigma:G\to G$ is an involutive automorphism, and $\chi_1,\chi_2:G\to\mathbb{C}^\ast$ are two characters of $G$ such that $\chi_2(x\sigma(x))=1$ for all $x\in G$. The authors study the solutions $f,g:G\to\mathbb{C}$ of the functional equation
\[
\chi_1(y)f(xy)+\chi_2(y)f(\sigma(y)x)=2f(x)g(y),\quad x,y\in G
\]
in terms of characters and additive functions. As an application, they find the complex-valued solutions of the functional equations $f(xy)+f(\sigma(y)x)=2f(x)g(y)$ and $\chi_1(y)f(xy)+f(yx)=2f(x)g(y)$, where $x,y\in G$.
Reviewer: Mohammad Sal Moslehian (Mashhad)Significance and relevances of functional equations in various fields.https://www.zbmath.org/1455.390032021-03-30T15:24:00+00:00"Senthil Kumar, B. V."https://www.zbmath.org/authors/?q=ai:senthil-kumar.b-v"Dutta, Hemen"https://www.zbmath.org/authors/?q=ai:dutta.hemenFrom the summary: This chapter illustrates some examples how functional equations are applied to solve some interesting problems in geometry, finance, information theory, wireless sensor networks, and electric circuits with parallel resistances. It discusses various other multiplicative inverse functional equations and their applications.
For the entire collection see [Zbl 1453.00001].On extensions of the generalized quadratic functions from ``large'' subsets of semigroups.https://www.zbmath.org/1455.390042021-03-30T15:24:00+00:00"Bahyrycz, A."https://www.zbmath.org/authors/?q=ai:bahyrycz.anna"Brzdęk, J."https://www.zbmath.org/authors/?q=ai:brzdek.janusz"Jabłońska, E."https://www.zbmath.org/authors/?q=ai:jablonska.elizaThe set up includes an abelian semigroup \(G\) with an involutive automorphism \(\tau: G \to G\), and an abelian group \(H\) such that \(h \mapsto 2h\) is a bijection of \(H\) onto itself.
Consider for \(D \subseteq G\) functions \(g:D \to H\) such that
\[
g(x+y) + g(x + \tau(y)) = 2g(x) + 2g(y) \text{ whenever }x,y,x+y, x+ \tau(y) \in D.\tag{1}
\]
The authors find \(D\)'s for which each such \(g\) has one and only one extension \(f:G \to H\) satisfying
\[
f(x+y) + f(x + \tau(y)) = 2f(x) + 2f(y) \text{ for all } x,y \in G.\tag{2}
\]
\(D\) is an element of a filter \(\mathcal{L} \subsetneqq 2^G\) of subsets of \(G\) with the property that the sets \(\tfrac{1}{2}B, \tau(B)\) and \(B \pm x\) are elements of \(\mathcal{L}\) whenever \(B\in \mathcal{L}\) and \(x \in G\). In any non-compact, locally compact group \(G\) such that \(x \mapsto 2x\) and \(\tau\) are homeomorphisms, an example is \(\mathcal{L} := \{ A \subseteq G \mid \mu(G \setminus A) < \infty \}\) where \(\mu\) denotes a Haar measure on \(G\).
For \(A \in 2^G\) and \(x \in G\) we introduce the notations
\[
A_x := A \cap (A-x) \ \text{ and } \ A^+ := \{ x \in A \mid x + \tau(x) \in A\}.
\]
Then the main result of the paper is:
Theorem. Let \(D \in \mathcal{L}\) and let \(g:D \to H\) satisfy (1). If there exists \(y_0 \in D\) such that \((D_{y_0})^+ \in \mathcal{L}\), then there exists exactly one solution \(f:G \to H\) of (2) such that \(f = g\) on \(D\).
There are no applications in the paper.
The same technique was applied to other functional equations in [\textit{A. Bahyrycz} et al., Publ. Math. 89, No. 3, 263--275 (2016; Zbl 1399.39043); \textit{J. Brzdęk} and \textit{E. Jabłońska}, Bull. Aust. Math. Soc. 96, No. 1, 110--116 (2017; Zbl 1368.39018)].
Reviewer: Henrik Stetkær (Aarhus)Fractional \(q\)-deformed chaotic maps: a weight function approach.https://www.zbmath.org/1455.390022021-03-30T15:24:00+00:00"Wu, Guo-Cheng"https://www.zbmath.org/authors/?q=ai:wu.guocheng"Niyazi Çankaya, Mehmet"https://www.zbmath.org/authors/?q=ai:cankaya.mehmet-niyazi"Banerjee, Santo"https://www.zbmath.org/authors/?q=ai:banerjee.santoSummary: The fractional derivative holds long-time memory effects or non-locality. It successfully depicts the dynamical systems with long-range interactions. However, it becomes challenging to investigate chaos in the deformed fractional discrete-time systems. This study turns to fractional quantum calculus on the time scale and reports chaos in fractional \(q\)-deformed maps. The discrete memory kernels are used, and a weight function approach is proposed for fractional modeling. Rich \(q\)-deformed dynamics are demonstrated, which shows the methodology's efficiency.
{\copyright 2020 American Institute of Physics}D'Alembert's and Wilson's equations on semigroups.https://www.zbmath.org/1455.390062021-03-30T15:24:00+00:00"Łukasik, Radosław"https://www.zbmath.org/authors/?q=ai:lukasik.radoslawLet $(S,+)$ be an abelian semigroup, let $\mathbb{K}$ be a quadratically closed field with $\mathrm{char}\, \mathbb{K}\neq 2$, and let $f,h:S\to K$ be two functions. The author studies the functional equation $f(x+2y)+f(x)=2f(x+y)f(y),$ $x,y\in S$. This equation is equivalent to d'Alembert's functional equation $f(x+y)+f(x-y)=2f(x)f(y)$ on groups and he shows that its solutions are the same as those of d'Alembert's functional equation. He also characterizes solutions of the equivalent form $h(x+2y)+h(x)=2f(y)h(x+y)$ of Wilson's functional equation.
Reviewer: Mohammad Sal Moslehian (Mashhad)Hyers-Ulam stability of an additive-quadratic functional equation.https://www.zbmath.org/1455.390072021-03-30T15:24:00+00:00"Govindan, Vediyappan"https://www.zbmath.org/authors/?q=ai:govindan.vediyappan"Park, Choonkil"https://www.zbmath.org/authors/?q=ai:park.choonkil"Pinelas, Sandra"https://www.zbmath.org/authors/?q=ai:pinelas.sandra"Rassias, Themistocles M."https://www.zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: In this paper, we introduce the following \((a, b, c)\)-mixed type functional equation of the form
\[ \begin{multlined}
g(ax_1+bx_2+cx_3)-g(-ax_1+bx_2+cx_3) +g(ax_1-bx_2+cx_3)-g(ax_1+bx_2-cx_3) +\\
2a^2[g(x_1) +g(-x_1)] + 2b^2[g(x_2) +g(-x_2)] + 2c^2[g(x_3) +g(-x_3)] +a[g(x_1)-g(-x_1)] +\\
b[g(x_2)-g(-x_2)] +c[g(x_3)-g(-x_3)] = 4g(ax_1+cx_3) + 2g(-bx_2) + 2g(bx_2)
\end{multlined} \]
where \(a\), \(b\), \(c\) are positive integers with \(a >1\), and investigate the solution and the Hyers-Ulam stability of the above functional equation in Banach spaces by using two different methods.Asymptotics and estimates for the discrete spectrum of the Schrödinger operator on a discrete periodic graph.https://www.zbmath.org/1455.351612021-03-30T15:24:00+00:00"Korotyaev, E. L."https://www.zbmath.org/authors/?q=ai:korotyaev.evgeny-l"Sloushch, V. A."https://www.zbmath.org/authors/?q=ai:sloushch.vladimir-anatolevichSummary: The periodic Schrödinger operator \(H\) on a discrete periodic graph is treated. The discrete spectrum is estimated for the perturbed operator \(H_{\pm }(t)=H\pm tV\), \(t>0\), where \(V\ge 0\) is a decaying potential. In the case when the potential has a power asymptotics at infinity, an asymptotics is obtained for the discrete spectrum of the operator \(H_{\pm }(t)\) for a large coupling constant.Normal mode analysis of 3D incompressible viscous fluid flow models.https://www.zbmath.org/1455.651432021-03-30T15:24:00+00:00"Zhang, Guoping"https://www.zbmath.org/authors/?q=ai:zhang.guoping"Cai, Mingchao"https://www.zbmath.org/authors/?q=ai:cai.mingchaoThe focus of the paper is to study the formal mode solutions of incompressible viscous flow models, in particular, the 3D incompressible Navier-Stokes equations (NSEs). Because the increase in dimension leads to theoretical and numerical difficulties, a quasi-one dimensional Stokes model is considered. The stability and accuracy of time-stepping schemes such as Crank-Nicolson scheme and Backward Euler scheme, as well as the splitting method are investigated for the solution of incompressible viscous flow models. By using the normal mode
analysis, it was proved that both the Backward Euler scheme and Crank-Nicolson schemes for the 3D Stokes equations are unconditionally stable. The time errors of the Backward Euler scheme and the Crank-Nicolson scheme are of the order \(\mathcal{O}(\Delta t)\) and \(\mathcal{O}(\Delta t)^2\), respectively. Moreover, using the normal mode analysis, the error orders of each variable, and the intermediate
variables are estimated for the splitting method.
Reviewer: Bülent Karasözen (Ankara)On the symmetrizations of \(\epsilon \)-isometries on Banach spaces.https://www.zbmath.org/1455.460142021-03-30T15:24:00+00:00"Cheng, Lixin"https://www.zbmath.org/authors/?q=ai:cheng.lixin"Sun, Longfa"https://www.zbmath.org/authors/?q=ai:sun.longfaThe authors present a sharp weak stability of the symmetrization of an \(\varepsilon\)-isometry on Banach spaces. As applications, they obtain new stability results for symmetrizations of general \(\varepsilon\)-isometries.
Reviewer: Hark-Mahn Kim (Daejeon)Maximal regularity in \(l_{p}\) spaces for discrete time fractional shifted equations.https://www.zbmath.org/1455.352892021-03-30T15:24:00+00:00"Lizama, Carlos"https://www.zbmath.org/authors/?q=ai:lizama.carlos"Murillo-Arcila, Marina"https://www.zbmath.org/authors/?q=ai:murillo-arcila.marinaThe article at hand is concerned with the following operator equation. Given a UMD-space \(X\), a closed linear operators \(A\) on \(X\) the authors consider finding \(u\colon \mathbb{Z}\times X \to X\) such that given \(f\colon \mathbb{Z}\times \Omega \to X\) we have
\[
\Delta^\alpha u(n) = Au(n) + \sum_{j=1}^k \beta_ju(n-\tau_j) + f(n,u(n)) \text{ in }X
\]for all \(n\in \mathbb{Z}\), where \(\tau_j \in \mathbb{Z}\), \(\beta_j\in \mathbb{R}\), and \(\alpha>0\) are given. Here
\[
(\Delta^\alpha f)(n) := \sum_{j=-\infty}^n k^{-\alpha}(n-j) f(j)
\]
for suitable \(f\), where \(k^\alpha(n) = \alpha(\alpha+1)\cdots(\alpha+n-1)/(n!)\) for \(n\geq 0\).
The main result is a characterisation of maximal \(\ell_p\)-regularity (1<p< \(\infty\) ) of the above problem with \(f(n,u(n))\) replaced by \(f(n)\) (i.e., \(A\) having maximal \(\ell_p\)-regularity) for some \(X\)-valued sequence \(f\) in the following sense:
Assume that \[
\{ (1-e^{-it})^\alpha-\sum_{j=1}^k \beta_j e^{-it\tau_j}; t\in [-\pi,\pi]\}\subseteq \rho(A).
\] Then the following conditions (i), (ii), (iii) are equivalent:
(i) for all \(f\in \ell_p(\mathbb{Z};X)\) there is a unique \(u\in \ell_p(\mathbb{Z};\operatorname{dom}(A))\) such that
\[
\Delta^\alpha u(n) = Au(n) + \sum_{j=1}^k \beta_ju(n-\tau_j) + f(n)\quad (n\in \mathbb{Z}).
\]
(ii) \(M(t):= ((1-e^{-it})^\alpha-\sum_{j=1}^k \beta_j e^{-it\tau_j}-A)^{-1}\) is an \(\ell_p\)-multiplier from \(X\) to \(\operatorname{dom}(A)\).
(iii) \(\{M(t); t \in [-\pi,\pi]\}\) is R-bounded.
This characterisation is then applied to derive \(\ell_p\)-regularity statements for the solution of the above nonlinear problem involving the Nemitskii-type operator given by \(f(n,u(n)) \).
Reviewer: Marcus Waurick (Glasgow)LMS-based variable step-size algorithms: a unified analysis approach.https://www.zbmath.org/1455.652282021-03-30T15:24:00+00:00"Saeed, Muhammad Omer Bin"https://www.zbmath.org/authors/?q=ai:saeed.muhammad-omer-binSummary: Several variable step-size strategies have been suggested in the literature to improve the performance of the least-mean-square (LMS) algorithm. Although they enhance performance, a major drawback is the complexity in the theoretical analysis of these algorithms. Researchers use several assumptions to find closed-form analytical solutions. This work presents a unified approach for the analysis of variable step-size LMS algorithms. The approach is then applied to several variable step-size strategies, and theoretical and simulation results are compared.Independence characterization for Wishart and Kummer random matrices.https://www.zbmath.org/1455.620452021-03-30T15:24:00+00:00"Kolodziejek, Bartosz"https://www.zbmath.org/authors/?q=ai:kolodziejek.bartosz"Piliszek, Agnieszka"https://www.zbmath.org/authors/?q=ai:piliszek.agnieszkaSummary: We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let \(X\) and \(Y\) be independent, non-degenerate random variables valued in \((0,\infty)\), then \(U=Y/(1 +X) \) and \(V=X(1 +U)\) are independent if and only if \(X\) follows the Kummer distribution and \(Y\) follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization of Wishart and matrix-Kummer distributions, which is our second main result.On the Aizerman problem: coefficient conditions for the existence of three- and six-period cycles in a second-order discrete-time system.https://www.zbmath.org/1455.931612021-03-30T15:24:00+00:00"Zvyagintseva, T. E."https://www.zbmath.org/authors/?q=ai:zvyagintseva.tatyana-evgenevnaSummary: In this paper, an automatic control discrete-time system of the second order is studied. The nonlinearity of this system satisfies the generalized Routh-Hurwitz condition. Systems of this type are widely used in solving modern applied problems of the theory of automatic control. This work is a continuation of the results of research presented in our paper
[Vestn. St. Petersbg. Univ., Math. 53, No. 1, 37--44 (2020; Zbl 07310917); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 7(65), No. 1, 50--59 (2020)],
in which systems with two-periodic nonlinearity lying in the Hurwitz angle were studied. In above-mentioned paper, the conditions on the parameters under which a system with two-periodic nonlinearity can possess a family of nonisolated four-period cycles are indicated and a method for constructing such nonlinearity is proposed. In the current paper, we assume that the nonlinearity is three-periodic and lies in a Hurwitz angle. We study a system for all possible parameter values. We explicitly present the conditions for the parameters under which it is possible to construct a three-periodic nonlinearity in such a way that a system with specified nonlinearity is not globally asymptotically stable. We show that a family of three-period cycles and a family of six-period cycles can exist in the system with this nonlinearity. A method for constructing such nonlinearities is proposed. The cycles are nonisolated; any solution of the system with the initial data, which lies on a certain specified ray, is a periodic solution.Hodge-GUE correspondence and the discrete KdV equation.https://www.zbmath.org/1455.370562021-03-30T15:24:00+00:00"Dubrovin, Boris"https://www.zbmath.org/authors/?q=ai:dubrovin.boris-a"Liu, Si-Qi"https://www.zbmath.org/authors/?q=ai:liu.siqi"Yang, Di"https://www.zbmath.org/authors/?q=ai:yang.di"Zhang, Youjin"https://www.zbmath.org/authors/?q=ai:zhang.youjinThe authors prove the conjectural relationship between the certain special cubic Hodge integrals of the Gopkumar-Marino-Vafa type and GUE correlators. They show that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy. Here, they mainly use the Virasoro constraints to show this equivalence. It may be more interesting to consider the equivalence of the bilinear equations.
Reviewer: Jipeng Cheng (Xuzhou)\(q\)-fractional Dirac type systems.https://www.zbmath.org/1455.390012021-03-30T15:24:00+00:00"Allahverdiev, Bilender P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, Hüseyin"https://www.zbmath.org/authors/?q=ai:tuna.huseyinSummary: This paper is devoted to study a regular \(q\)-fractional Dirac type system. We investigate the properties of the eigenvalues and the eigenfunctions of this system. By using a fixed point theorem we give a sufficient condition on eigenvalues for the existence and uniqueness of the associated eigenfunctions.