Recent zbMATH articles in MSC 26https://zbmath.org/atom/cc/262023-11-13T18:48:18.785376ZWerkzeugOn \(p\)-adic semi-algebraic continuous selectionshttps://zbmath.org/1521.031032023-11-13T18:48:18.785376Z"Thamrongthanyalak, Athipat"https://zbmath.org/authors/?q=ai:thamrongthanyalak.athipatSummary: Let \(E \subseteq \mathbb{Q}_p^n\) and \(T\) be a set-valued map from \(E\) to \(\mathbb{Q}_p^m\). We prove that if \(T\) is \(p\)-adic semi-algebraic, lower semi-continuous and \(T (x)\) is closed for every \(x \in E\), then \(T\) has a \(p\)-adic semi-algebraic continuous selection. In addition, we include three applications of this result. The first one is related to \textit{C. Fefferman}'s and \textit{J. Kollár}'s [Dev. Math. 28, 233--282 (2013; Zbl 1263.15003)] question on existence of \(p\)-adic semi-algebraic continuous solution of linear equations with polynomial coefficients. The second one is about the existence of \(p\)-adic semi-algebraic continuous extensions of continuous functions. The other application is on the characterization of right invertible \(p\)-adic semi-algebraic continuous functions under the composition.Ideal equal Baire classeshttps://zbmath.org/1521.031502023-11-13T18:48:18.785376Z"Kwela, Adam"https://zbmath.org/authors/?q=ai:kwela.adam"Staniszewski, Marcin"https://zbmath.org/authors/?q=ai:staniszewski.marcinSummary: For any Borel ideal we characterize ideal equal Baire system generated by the families of continuous and quasi-continuous functions, i.e., the families of ideal equal limits of sequences of continuous and quasi-continuous functions.Nordhaus-Gaddum type inequality for the integer \(k\)-matching number of a graphhttps://zbmath.org/1521.051532023-11-13T18:48:18.785376Z"Chen, Qian-Qian"https://zbmath.org/authors/?q=ai:chen.qianqian"Guo, Ji-Ming"https://zbmath.org/authors/?q=ai:guo.jimingSummary: An integer \(k\)-matching of a graph \(G\) is a function \(h : E ( G ) \to \{ 0 , 1 , \ldots , k \}\) such that \(\sum_{e \in \Gamma ( v )} h ( e ) \leq k\) for any \(v \in V ( G )\), where \(\Gamma ( v )\) is the set of edges incident to \(v\). The integer \(k\)-matching number of \(G\), denoted by \(m_k ( G )\), is the maximum number of \(\sum_{e \in E ( G )} h ( e )\) over all integer \(k\)-matching \(h\) of \(G\). In this paper, we establish the following lower bounds on the sum of the integer \(k\)-matching number of a graph \(G\) and its complement by using Gallai-Edmonds Structure Theorem:
(1) \(m_k ( G ) + m_k ( \overline{G} ) \geq \lfloor \frac{ n k}{ 2} \rfloor\) for \(n \geq 2\);
(2) if \(G\) and \(\overline{G}\) are non-empty, then for \(n \geq 25\), \(m_k ( G ) + m_k ( \overline{G} ) \geq \lfloor \frac{ n k + k}{ 2} \rfloor \);
(3) if \(G\) and \(\overline{G}\) have no isolated vertices, then for \(n \geq 25\), \(m_k ( G ) + m_k ( \overline{G} ) \geq \lfloor \frac{ n k}{ 2} \rfloor + 2 k\) .
Furthermore, all extremal graphs attaining the lower bounds are also characterized.Some types of numeral systems and their modelinghttps://zbmath.org/1521.110502023-11-13T18:48:18.785376Z"Serbenyuk, Symon"https://zbmath.org/authors/?q=ai:serbenyuk.symonAuthor's abstract: In this article, the operator approach to modelling numeral systems is introduced. This approach can be useful for coding information and providing computer protection. Certain examples of such numeral systems are considered. In addition, the pseudo-binary representation is investigated. A description of further investigations of the author of this article is given.
Reviewer: Alexey Ustinov (Khabarovsk)On a new measure on the Levi-Civita field \(\mathcal{R}\)https://zbmath.org/1521.120102023-11-13T18:48:18.785376Z"Restrepo Borrero, M."https://zbmath.org/authors/?q=ai:restrepo-borrero.m"Srivastava, Vatsal"https://zbmath.org/authors/?q=ai:srivastava.vatsal"Shamseddine, K."https://zbmath.org/authors/?q=ai:shamseddine.khodrSummary: The Levi-Civita field \(\mathcal{R}\) is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13, Zbl 1130.12301], a measure was defined on \(\mathcal{R}\) in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over \(\mathcal{R}\) in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13, loc. cit.]. Then we will introduce the notion of an outer measure on \(\mathcal{R}\) and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on \(\mathcal{R}\) that proves to be a better generalization of the Lebesgue measure from \(\mathbb{R}\) to \(\mathcal{R}\) and that leads to a family of measurable sets in \(\mathcal{R}\) that strictly contains the family of measurable sets from [13, loc. cit.], and for which most of the classic results for Lebesgue measurable sets in \(\mathbb{R}\) hold.Hardy inequalities for fractional \((k,a)\)-generalized harmonic oscillatorshttps://zbmath.org/1521.220162023-11-13T18:48:18.785376Z"Teng, Wentao"https://zbmath.org/authors/?q=ai:teng.wentaoSummary: We will define a-deformed Laguerre operators \(L_{a,\alpha}\) and a-deformed Laguerre holomorphic semigroups on \(L^2 ((0, \infty), d\mu_{a,\alpha})\). Then we give a spherical harmonic expansion, which reduces to the Bochner-type identity when taking the boundary value \(z = \pi i/2\), of the \((k, a)\)-generalized Laguerre semigroup introduced by Ben Saïd, Kobayashi and Ørsted. We prove a Hardy inequality for fractional powers of the \(a\)-deformed Dunkl harmonic oscillator \(\triangle_{k,a} := |x|^{2-a} \triangle_k - |x|^a\) using this expansion. When \(a = 2\), the fractional Hardy inequality reduces to that of Dunkl-Hermite operators given by Ciaurri, Roncal and Thangavelu. The operators \(L_{a,\alpha}\) also give a tangible characterization of the radial part of the \((k, a)\)-generalized Laguerre semigroup on each \(k\)-spherical component \(\mathcal{H}^m_k (\mathbb{R}^n)\) for
\[
\lambda_{k,a,m} :=\frac{2m + 2 \langle k\rangle + N - 2}{a}\geq -\frac{1}{2}
\]
defined via a decomposition of the unitary representation.Elementary functionshttps://zbmath.org/1521.260012023-11-13T18:48:18.785376Z"Bourchtein, Andrei"https://zbmath.org/authors/?q=ai:bourchtein.andrei"Bourchtein, Ludmila"https://zbmath.org/authors/?q=ai:bourchtein.ludmilaPublisher's description: This textbook focuses on the study of different kinds of elementary functions ubiquitous both in high school Algebra and Calculus. To analyze the functions ranging from polynomial to trigonometric ones, it uses rudimentary techniques available to high school students, and at the same time follows the mathematical rigor appropriate for university level courses.
Contrary to other books of Pre-Calculus, this textbook emphasizes the study of elementary functions with rigor appropriate for university level courses in mathematics, although the exposition is confined to the pre-limit topics and techniques. This makes the book useful, on the one hand, as an introduction to mathematical reasoning and methods of proofs in mathematical analysis, and on the other hand, as a preparatory course on the properties of different kinds of elementary functions.
The textbook is aimed at university freshmen and high-school students interested in learning strict mathematical reasoning and in preparing a solid base for subsequent study of elementary functions at advanced level of Calculus and Analysis. The required prerequisites correspond to the level of the high school Algebra. All the preliminary concepts and results related to the elementary functions are covered in the initial part of the text. This makes the textbook suitable for both classroom use and self-study.The Stieltjes integralhttps://zbmath.org/1521.260022023-11-13T18:48:18.785376Z"Convertito, Gregory"https://zbmath.org/authors/?q=ai:convertito.gregory"Cruz-Uribe, David"https://zbmath.org/authors/?q=ai:cruz-uribe.david-vPublisher's description: The Stieltjes Integral provides a detailed, rigorous treatment of the Stieltjes integral. This integral is a generalization of the Riemann and Darboux integrals of calculus and undergraduate analysis, and can serve as a bridge between classical and modern analysis. It has applications in many areas, including number theory, statistics, physics, and finance. It begins with the Darboux integral, builds the theory of functions of bounded variation, and then develops the Stieltjes integral. It culminates with a proof of the Riesz representation theorem as an application of the Stieltjes integral.
For much of the 20th century the Stjeltjes integral was a standard part of the undergraduate or beginning graduate student sequence in analysis. However, the typical mathematics curriculum has changed at many institutions, and the Stieltjes integral has become less common in undergraduate textbooks and analysis courses. This book seeks to address this by offering an accessible treatment of the subject to students who have had a one semester course in analysis. This book is suitable for a second semester course in analysis, and also for independent study or as the foundation for a senior thesis or Masters project.
Features:
\begin{itemize}
\item Written to be rigorous without sacrificing readability.
\item Accessible to undergraduate students who have taken a one-semester course on real analysis.
\item Contains a large number of exercises from routine to challenging.
\end{itemize}Analysis 1. Focused and colouredhttps://zbmath.org/1521.260032023-11-13T18:48:18.785376Z"Friedl, Stefan"https://zbmath.org/authors/?q=ai:friedl.stefanPublisher's description: Dieses Lehrbuch stellt die Inhalte der Analysis-1-Vorlesung fokussiert auf die Kernaussagen dar. Die Stoffauswahl entspricht genau einer einsemestrigen Vorlesung und ist, zusammen mit den Übungsaufgaben, eine ideale Grundlage für eine Analysis-1-Vorlesung.
In dem Buch werden die Definitionen, Aussagen und Beweise mit zahlreichen Beispielen und farbigen Abbildungen veranschaulicht. Bei den wichtigen Beweisen wird die Grundidee herausgearbeitet und erläutert.
Darüber hinaus wird bewusst viel Farbe im Text und in den Formeln eingesetzt, um den Blick auf die zentralen Punkte zu lenken.
Das Buch ist daher für interessierte Schüler und Studierende auch perfekt zum Selbststudium und für die Prüfungsvorbereitung geeignet.
Das Buch basiert auf einem mehrfach überarbeiteten Skript des Autoren. Durch viele Rückmeldungen von Studierenden und durch intensive Diskussionen mit erfahrenen Dozierenden und
Lehrkräften wurde dem Buch der Feinschliff gegeben.Applied calculus with Rhttps://zbmath.org/1521.260042023-11-13T18:48:18.785376Z"Pfaff, Thomas J."https://zbmath.org/authors/?q=ai:pfaff.thomas-jPublisher's description: This textbook integrates scientific programming with the use of R and uses it both as a tool for applied problems and to aid in learning calculus ideas. Adding R, which is free and used widely outside academia, introduces students to programming and expands the types of problems students can engage. There are no expectations that a student has any coding experience to use this text.
While this is an applied calculus text including real world data sets, a student that decides to go on in mathematics should develop sufficient algebraic skills so that they can be successful in a more traditional second semester calculus course. Hopefully, the applications provide some motivation to learn techniques and theory and to take additional math courses. The book contains chapters in the appendix for algebra review as algebra skills can always be improved. Exercise sets and projects are included throughout with numerous exercises based on graphs.Decomposition of integrals of the some classes in generalized power serieshttps://zbmath.org/1521.260052023-11-13T18:48:18.785376Z"Savina, Svetlana V."https://zbmath.org/authors/?q=ai:savina.svetlana-vSummary: It is investigated the decomposition of the Cauchy integrals the special kind in uniformly and absolutely converging generalized power series which it is possible to ``differentiate'' any number of times of the generalized derivative like the rule of differentiation by \(z\) of usual power series. It is deduced the formulas for a finding of the generalized derivative.Improved matrix inequalities using radical convexityhttps://zbmath.org/1521.260062023-11-13T18:48:18.785376Z"Sababheh, Mohammad"https://zbmath.org/authors/?q=ai:sababheh.mohammad-s"Furuichi, Shigeru"https://zbmath.org/authors/?q=ai:furuichi.shigeru"Moradi, Hamid Reza"https://zbmath.org/authors/?q=ai:moradi.hamid-rezaThe authors present several inequalities for \(2\)-radical convex functions which refine some known inequalities for convex functions.
Let \(f:[0,\infty ) \rightarrow [0,\infty)\) be a continuous function with \(f(0)=0\) and let \(p\geq 1\) be a fixed number. If the function \(g(x)=f(x^{1/p})\) is convex on \([0,\infty)\), we say that \(f\) is \(p\)-radical convex.
Theorem 1. Let \(f\) be a 2-radical convex function.
(i) If \(a,b\geq 0\) and \(t\in [0,1]\), then \[ f((1-t)a+tb)+f\left( \sqrt{\frac{r|1-2t|}{2}}|a-b|\right) + 2r\left( \frac{f(a)+f(b)}{2} -f\left( \frac{a+b}{2}\right) \right) \leq (1-t)f(a) +t f(b), \] where \(r=\min \{ t,1-t\}\).
(ii) If \(a,b> 0\) and \(t\in [0,1]\), then \[ f((1-t)a+tb) \leq \frac{((1-t)a+tb)^2}{(1-t)a^2+tb^2} ((1-t)f(a) +tf(b)).\]
(iii) If (\(0<a<b\) and \(t>1\)) or (\(0<b<a\) and \(t<0\)), then for any \(t\not\in [0,1]\) \[ (1-t)f(a) +tf(b) \leq \frac{(1-t)a^2+tb^2}{((1-t)a+tb)^2} f((1-t)a+tb) .\] \\
Similar inequalities for matrices are also given. If \(X, Y \in \mathcal{M}_n\) are Hermitian matrices, then we say that \(\lambda (X) \leq \lambda(Y) \) if \(\lambda_j(X) \leq \lambda_j(Y)\) for each \(j=1,\ldots ,n\), where \(\lambda_j(X)\) is the \(j\)th largest eigenvalue of \(X\).
Theorem 2. Let \(f\) be a 2-radical convex function. Let \(A,B \in \mathcal{M}_n\) be positive definite matrices, \(m,m',M,M'\in \mathbb{R}\) and \(t\in [0,1]\).
(i) If \(0<m'I \leq A \leq mI \leq MI\leq B\leq M'I\), then \[ \lambda(f((1-t)A+tB))\leq \frac{((1-t)m+tM)^2}{(1-t)m^2+tM^2} \lambda(((1-t)f(A)+tf(B))). \]
(ii) If \(0<m'I \leq B \leq mI \leq MI\leq A\leq M'I\), then \[ \lambda(f((1-t)A+tB))\leq \frac{((1-t)M+tm)^2}{(1-t)M^2+tm^2} \lambda(((1-t)f(A)+tf(B))). \]
At the end of the article, some examples are given to illustrate the main result.
Reviewer: Sanja Varošanec (Zagreb)On the differentiability of symmetric matrix valued functionshttps://zbmath.org/1521.260072023-11-13T18:48:18.785376Z"Shapiro, Alexander"https://zbmath.org/authors/?q=ai:shapiro.alexander.1|shapiro.alexanderSummary: With every real valued function, of a real argument, can be associated a matrix function mapping a linear space of symmetric matrices into itself. In this paper we study directional differentiability properties of such matrix functions associated with directionally differentiable real valued functions. In particular, we show that matrix valued functions inherit semismooth properties of the corresponding real valued functions.Implicit function theorems for continuous mappings and their applicationshttps://zbmath.org/1521.260082023-11-13T18:48:18.785376Z"Arutyunov, A. V."https://zbmath.org/authors/?q=ai:arutyunov.aram-v"Zhukovskiy, S. E."https://zbmath.org/authors/?q=ai:zhukovskiy.s-e|zhukovskiy.sergey-evgenyevich"Mordukhovich, B. Sh."https://zbmath.org/authors/?q=ai:mordukhovich.boris-sThe authors investigate implicit function theorems for continuous mappings and their applications. More precisely, local and nonlocal implicit function theorems are obtained for closed mappings with a parameter from one Asplund space to another.
Reviewer: Savin Treanţă (Bucureşti)A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaceshttps://zbmath.org/1521.260092023-11-13T18:48:18.785376Z"Wachsmuth, Daniel"https://zbmath.org/authors/?q=ai:wachsmuth.daniel"Wachsmuth, Gerd"https://zbmath.org/authors/?q=ai:wachsmuth.gerdSummary: We give a simple proof of the Baillon-Haddad theorem for convex functions defined on open and convex subsets of Hilbert spaces. We also state some generalizations and limitations. In particular, we discuss equivalent characterizations of the Lipschitz continuity of the derivative of convex functions on open and convex subsets of Banach spaces.An inequality for polynomials on the standard simplexhttps://zbmath.org/1521.260102023-11-13T18:48:18.785376Z"Milev, Lozko"https://zbmath.org/authors/?q=ai:milev.lozko-b"Naidenov, Nikola"https://zbmath.org/authors/?q=ai:naidenov.nikolaSummary: Let \(\varDelta := \{ (x,y) \in \mathbb{R}^2: x \ge 0,~ y \ge 0,~ x+y \le 1 \}\) be the standard simplex in \(\mathbb{R}^2\) and \(\partial \varDelta\) be the boundary of \(\varDelta \). We use the notations \(\Vert f \Vert_{\varDelta }\) and \(\Vert f \Vert_{\partial \varDelta }\) for the uniform norm of a continuous function \(f\) on \(\varDelta\) and \(\partial \varDelta \), respectively.
Denote by \(\pi_n\) the set of all real algebraic polynomials of two variables and of total degree not exceeding \(n\). Let \(B_{\varDelta } := \{p \in \pi_2 : \Vert p \Vert_{\varDelta } \le 1\}\) and \(B_{\partial \varDelta } := \{p \in \pi_2 : \Vert p \Vert_{\partial \varDelta } \le 1\}\).
Recently we described the set of all extreme points of \(B_{\varDelta }\). In the present paper we give a full description of the strictly definite extreme points of \(B_{\partial \varDelta }\). As an application we prove the following sharp inequality:
\[\Vert p \Vert_{\varDelta } \le \frac{5}{3} \, \Vert p \Vert_{\partial \varDelta }, \quad \text{for every } p \in \pi_2.\]
We also establish two generalizations of the above inequality, for any triangle in \(\mathbb{R}^2\) and for any dimension \(d \ge 2\).
We hope that our results can be useful in studying numerical problems related with estimates for uniform norms of polynomials and splines.
For the entire collection see [Zbl 1511.65004].Constants in Markov's and Bernstein inequality on a finite interval in \(\mathbb{R}\)https://zbmath.org/1521.260112023-11-13T18:48:18.785376Z"Sroka, Grzegorz"https://zbmath.org/authors/?q=ai:sroka.grzegorzSummary: In this paper we demonstrate the constants in the pointwise Bernstein inequality
\[
|P^{(\alpha)}(x)|\leq\left(\frac{2n}{\sqrt{(x-a)(b-x)}}\right)^\alpha\|P\|_{[a, b]},
\]
for the \(\alpha\)th derivative of an algebraic polynomial in \(L^\infty\)-norms on an interval in \(\mathbb{R}\), where \(\alpha\geq3\). This result was obtained using the tools of theory of pluripotential and we apply it to get the main result which is a new generalization of V. A. Markov's type inequalities
\[
\|P^{(\alpha)}\|_p\leq C^{1/p}\left(\frac{2}{b-a}\right)^\alpha\|T^{(\alpha)}_n\|_{[-1, 1]}n^{2/p}\|P\|_p,
\]
for the \(\alpha\)th derivative of an algebraic polynomial in \(L^p\) norms, where \(p\geq1\). In particular, we show that for any \(\alpha\geq3\) the constant \(C\) in the V. A. Markov inequality satisfies the condition \(C\leq8\left(\frac{32\cdot 3,94741\cdot\pi M\alpha^2}{3\sqrt{3}}\right)^{1/p}\).Remarks on a lemma of Schmidthttps://zbmath.org/1521.260122023-11-13T18:48:18.785376Z"Alzer, Horst"https://zbmath.org/authors/?q=ai:alzer.horst"Kwong, Man Kam"https://zbmath.org/authors/?q=ai:kwong.man-kam"Raşa, Ioan"https://zbmath.org/authors/?q=ai:rasa.ioanSummary: Inspired by a lemma of \textit{W. M. Schmidt} [Proc. Am. Math. Soc. 124, No. 10, 3003--3013 (1996; Zbl 0867.11046)] we present several inequalities involving
\[
\begin{aligned}
& \max_{1\le j\le n} |1-x_j|, \quad \max_{1\le j \le n} |\log ( x_j)| \quad \text{ and } \quad \left( \sum_{i=1}^n |1-x_i|^r \right)^{1/r},\ \\
& \quad \left( \sum_{i=1}^n |\log ( x_i)|^r \right)^{1/r}.
\end{aligned}
\]Post-quantum Hermite-Jensen-Mercer inequalitieshttps://zbmath.org/1521.260132023-11-13T18:48:18.785376Z"Bohner, Martin"https://zbmath.org/authors/?q=ai:bohner.martin-j"Budak, Hüseyin"https://zbmath.org/authors/?q=ai:budak.huseyin"Kara, Hasan"https://zbmath.org/authors/?q=ai:kara.hasan-huseyinLet us present some definitions from \((p,q)\)-calculus which are used in this paper.
Let \(p\) and \(q\) be a positive real numbers such that \(q<p\leq 1\). The \((p,q)\)-integer is defined by:
\[[n]_{p,q} = \frac{p^n - q^n }{p-q}.\]
The definite \((p,q)_a\)-integral of \(f:[a,b] \rightarrow \mathbb{R}\) is given by
\[ \int_a^x f(t)\,_ad_{p,q}t=(p-q)(x-a) \sum_{n=0}^{\infty} \frac{q^n}{p^{n+1}} f\left( \frac{q^n}{p^{n+1}}x + \left( 1-\frac{q^n}{p^{n+1}}\right)a \right)\]
for \(x\in [a,pb+(1-p)a] \).
The definite \((p,q)^b\)-integral of \(f:[a,b] \rightarrow \mathbb{R}\) is given by
\[ \int_x^b f(t)\,^bd_{p,q}t=(p-q)(b-x) \sum_{n=0}^{\infty} \frac{q^n}{p^{n+1}} f\left( \frac{q^n}{p^{n+1}}x + \left( 1-\frac{q^n}{p^{n+1}}\right)b \right)\]
for \(x\in [pa+(1-p)b,b] \).
Obviously, if \(p=1\), then we get the objects from \(q\)-calculus, i.e.,\ \((p,q)\)-integer \([n]_{p,q}\), the definite \((p,q)_a\)-integral of \(f\) and the definite \((p,q)^b\)-integral of \(f\) become \(q\)-integer \([n]_{q}\), the definite \(q_a\)-integral of \(f\) and the definite \(q^b\)-integral of \(f\), respectively.
The Hermite-Jensen-Mercer-type inequalities for \((p,q)\)-integrals are given in the first theorem.
Theorem 1. If \(0<q< p \leq 1\) and \(f:[a,b]\rightarrow \mathbb{R}\) is convex, then
\begin{align*}
f\left( a+b-\frac{x+y}{2}\right) & \leq f(a)+f(b)-\frac{1}{2p(y-x)}\left[ \int_{px+(1-p)y}^y f(t)\,^yd_{p,q}t + \int_x^{py+(1-p)x} f(t)\,_xd_{p,q}t \right] \\
& \leq f(a)+f(b) -\frac 12 \left[ f\left(\frac{px+qy}{[2]_{p,q}}\right) +f\left(\frac{qx+py}{[2]_{p,q}}\right) \right]\\
& \leq f(a)+f(b) -f\left(\frac{x+y}{2}\right)
\end{align*}
for all \(x,y\in [a,b], x< y\).
Also, the following two inequalities are given for a convex \(f:[a,b]\rightarrow \mathbb{R}\):
\begin{align*}
\text{(i)}\qquad f\left(a+b-\frac{px+qy}{[2]_{p,q}}\right) &\leq \frac{1}{p(y-x)}\int_{a+b-y}^{a+b-px-(1-p)y} f(t)\,_{a+b-y}d_{p,q}t \\
&\leq f(a)+f(b) -\frac{pf(x)+qf(y)}{[2]_{p,q}}. \end{align*}
\begin{align*}
\text{(ii)\qquad} f\left(a+b-\frac{qx+py}{[2]_{p,q}}\right) &\leq \frac{1}{p(y-x)}\int_{a+b-py-(1-p)x}^{a+b-x} f(t)\,^{a+b-x}d_{p,q}t \\
&\leq f(a)+f(b) -\frac{qf(x)+pf(y)}{[2]_{p,q}}.
\end{align*}
At the end of the article, some examples are given to illustrate the main result.
Reviewer: Sanja Varošanec (Zagreb)Extensions of recent combinatorial refinements of discrete and integral Jensen inequalitieshttps://zbmath.org/1521.260142023-11-13T18:48:18.785376Z"Horváth, László"https://zbmath.org/authors/?q=ai:horvath.laszloSummary: The main purpose of this work is to present essential extensions of previous results of the author on
Jensen's inequality and apply them to some special situations. Of particular interest is the refinement of the integral Jensen inequality for vector valued integrable functions. The applications related to four topics, namely \(f\)-divergences in information theory (an interesting refinement of the weighted geometric mean-arithmetic mean inequality is obtained as a consequence), norm inequalities, quasi-arithmetic means, Hölder's and Minkowski's inequalities.A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional orderhttps://zbmath.org/1521.260152023-11-13T18:48:18.785376Z"Idczak, Dariusz"https://zbmath.org/authors/?q=ai:idczak.dariuszSummary: In the paper, a new Gronwall lemma for functions of two variables with singular integrals is proved. An application to weak relative compactness of the set of solutions to a fractional partial differential equation is given.Some new Hermite-Hadamard type integral inequalities for twice differentiable \(s\)-convex functionshttps://zbmath.org/1521.260162023-11-13T18:48:18.785376Z"Meftah, Badreddine"https://zbmath.org/authors/?q=ai:meftah.badreddine"Lakhdari, Abdelghani"https://zbmath.org/authors/?q=ai:lakhdari.abdelghani"Benchettah, Djaber Chemseddine"https://zbmath.org/authors/?q=ai:benchettah.djaber-chemseddine(no abstract)Generalizations of some integral inequalities for Riemann-Liouville operatorhttps://zbmath.org/1521.260172023-11-13T18:48:18.785376Z"Sofrani, Mohammed"https://zbmath.org/authors/?q=ai:sofrani.mohammed"Senouci Abdelkader"https://zbmath.org/authors/?q=ai:senouci-abdelkader.Summary: The Chebyshev inquality is one of important inequalities in mathematics. It's a necessary tool in probability theory. The item of Chebyshev's inequality may also refer to Markov's inequality in the context of analysis.
In[6, 7], using the usual Riemann-Liouville fractional integral operator \(I^{\alpha }\), were established and proved some new integral inequalities for the Chebyshev fonctional
\[ T(f,g):=\frac{1}{b-a}\int^b_af(x)g(x)dx-\frac{1}{b-a}\int^b_af(x)dx\frac{1}{b-a}\int^b_ag(x)dx. \]
In this work, we give some generalizations of Chebyshev-type integral inequalities by using Riemann-Liouville fractional integrals of function with respect to another function.The precise representative for the gradient of the Riesz potential of a finite measurehttps://zbmath.org/1521.310092023-11-13T18:48:18.785376Z"Cufí, Julià"https://zbmath.org/authors/?q=ai:cufi.julia"Ponce, Augusto C."https://zbmath.org/authors/?q=ai:ponce.augusto-c"Verdera, Joan"https://zbmath.org/authors/?q=ai:verdera.joanSummary: Given a finite nonnegative Borel measure \(m\) in \(\mathbb{R}^d \), we identify the Lebesgue set \(\mathcal{L}(V_s) \subset \mathbb{R}^d\) of the vector-valued function
\[
V_s(x) = \int_{\mathbb{R}^d}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y),
\]
for any order \(0 < s < d\). We prove that \(a \in \mathcal{L}(V_s)\) if and only if the integral above has a principal value at \(a\) and
\[
\lim_{r \rightarrow 0}{\frac{m(B_r(a))}{r^s}} = 0.
\]
In that case, the precise representative of \(V_s\) at \(a\) coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.Automatic real analyticity and a regal proof of a commutative multivariate Löwner theoremhttps://zbmath.org/1521.320072023-11-13T18:48:18.785376Z"Pascoe, J. E."https://zbmath.org/authors/?q=ai:pascoe.james-eldred"Tully-Doyle, Ryan"https://zbmath.org/authors/?q=ai:tully-doyle.ryanLet \(E\) be an open subset of the space \(\mathbb{R}^n\). Let \(\phi_i\), \(1\leq i \leq d\), be automorphisms of the upper half plane such that the tuple \((\phi_1(t), \dots, \phi_d(t))\) maps \((0,1)\) onto \(E\). A~function \(f:E \rightarrow \mathbb{R}\) is called \textit{matrix monotone lite} if \(f(\phi_1(t), \dots, \phi_d(t))\) is a matrix monotone function of \(t\) whenever \(t \in (0,1)\). It is shown that a function is matrix monotone lite if and only if it analytically continues to the multivariate upper half plane, as a map into the upper half plane. Further, for the case of two variables, it is shown that such functions are locally matrix monotone in the sense of Agler-McCarthy-Young [\textit{J. Agler} et al., Ann. Math. (2) 176, No. 3, 1783--1826 (2012; Zbl 1268.47025)].
Reviewer: K. C. Sivakumar (Chennai)Dynamic equations and almost periodic fuzzy functions on time scaleshttps://zbmath.org/1521.340022023-11-13T18:48:18.785376Z"Wang, Chao"https://zbmath.org/authors/?q=ai:wang.chao"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pThis book establishes an almost periodic theory of multidimensional fuzzy dynamic equations and fuzzy vector-valued functions on a time scale domain. The time scales domain considered here are complete-closed time scales under non-translational shifts. This book consists of six chapters.
The first chapter provides some necessary knowledge of interval and fuzzy arithmetic. A generalization of the Hukuhara difference and its properties are introduced. Applications of Hukuhara difference in solving interval and fuzzy linear equations and fuzzy differential equations. In Chapter 2, authors give an embedding theorem for a fuzzy multidimensional space and new types of multiplication in a fuzzy multidimensional space which are useful for further analysis. In Chapter 3, authors introduce the basic notions of generalized Hukuhara \(\Delta\)-derivatives of fuzzy vector-valued functions on time scales. Also, the \(\Delta\)-integral of fuzzy vector-valued functions is introduces. Further, some fundamental properties of generalized Hukuhara \(\Delta\)-derivatives and \(\Delta\)-integral of fuzzy vector-valued functions are established. Chapter 4 deals with the study of the notion of shift almost periodic fuzzy vector-valued functions on complete-closed time scales under non-translational shifts. Chapter 5 presents basic results of fuzzy multidimensional spaces and the calculus of fuzzy vector-valued functions on time scales. Finally, in Chapter 6, the authors develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales. Several applications of this theory are provided. A study of new type of fuzzy dynamic systems, called fuzzy \(q\)-dynamic systems is also presented and studied. An Appendix of the book introduces the notion of the almost anti-periodic discrete process and provides a new avenue to study the almost anti-periodic process on time scales.
The references in the book are not properly cited. Some reference items are wrong, particularly, in the Preface. However, the text material of the book is presented in a readable format. The book may be a good reference material for researchers working in the related field.
Reviewer: Sanket Tikare (Mumbai)Arbitrary order differential equations with fuzzy parametershttps://zbmath.org/1521.340042023-11-13T18:48:18.785376Z"Allahviranloo, Tofigh"https://zbmath.org/authors/?q=ai:allahviranloo.tofigh"Salahshour, Soheil"https://zbmath.org/authors/?q=ai:salahshour.soheilSummary: In the last decades, some generalization of theory of ordinary differential equations has been considered to the arbitrary order differential equations by many researchers, the so-called theory of arbitrary order differential equations (often called as fractional order differential equations [FDEs]). Because of the ability for modeling real phenomena, arbitrary order differential equations have been applied in various fields such as control systems, biosciences, bioengineering, and references therein. In this chapter, the authors propose arbitrary order differential equations with respect to another function using fuzzy parameters (initial values and the unknown solutions). The generalized fuzzy Laplace transform is applied to obtain the Laplace transform of arbitrary order integral and derivative of fuzzy-valued functions to solve linear FDEs. To obtain the large class of solutions for FDEs, the concept of generalized Hukuhara differentiability is applied.
For the entire collection see [Zbl 1439.74003].On inclusion problems involving Caputo and Hadamard fractional derivativeshttps://zbmath.org/1521.340062023-11-13T18:48:18.785376Z"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2"Ntouyas, Sotiris K."https://zbmath.org/authors/?q=ai:ntouyas.sotiris-k"Tariboon, Jessada"https://zbmath.org/authors/?q=ai:tariboon.jessadaSummary: In this paper, we study the existence of solutions to new inclusion problems involving both Caputo and Hadamard fractional derivatives, and separated boundary conditions. We apply the modern tools of the fixed point theory to study the cases when the multi-valued map (the right hand-side of the inclusions) takes convex as well as non-convex values. Examples illustrating the abstract results are also presented.Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operatorshttps://zbmath.org/1521.340072023-11-13T18:48:18.785376Z"Alharthi, Nadiyah Hussain"https://zbmath.org/authors/?q=ai:alharthi.nadiyah-hussain"Atangana, Abdon"https://zbmath.org/authors/?q=ai:atangana.abdon"Alkahtani, Badr S."https://zbmath.org/authors/?q=ai:alkahtani.badr-saad-tSummary: Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.Existence and H-U stability of a tripled system of sequential fractional differential equations with multipoint boundary conditionshttps://zbmath.org/1521.340112023-11-13T18:48:18.785376Z"Murugesan, Manigandan"https://zbmath.org/authors/?q=ai:murugesan.manigandan"Muthaiah, Subramanian"https://zbmath.org/authors/?q=ai:muthaiah.subramanian"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Nandha Gopal, Thangaraj"https://zbmath.org/authors/?q=ai:gopal.thangaraj-nandhaSummary: In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray-Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers-Ulam type. As an application, we present a few examples to illustrate the main results.On implicit boundary value problems with deformable fractional derivative and delay in \(b\)-metric spaceshttps://zbmath.org/1521.340732023-11-13T18:48:18.785376Z"Salim, Abdelkrim"https://zbmath.org/authors/?q=ai:salim.abdelkrim"Krim, Salim"https://zbmath.org/authors/?q=ai:krim.salim"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffakSummary: We demonstrate various existence and uniqueness results for a class of deformable implicit fractional differential equations with delay in \(b\)-metric spaces with boundary conditions. We base our arguments on some suitable fixed point theorems. In the last section, we provide different examples to illustrate our obtained results.Modification of optimal homotopy asymptotic method for multi-dimensional time-fractional model of Navier-Stokes equationhttps://zbmath.org/1521.351352023-11-13T18:48:18.785376Z"Jan, Himayat Ullah"https://zbmath.org/authors/?q=ai:jan.himayat-ullah"Ullah, Hakeem"https://zbmath.org/authors/?q=ai:ullah.hakeem"Fiza, Mehreen"https://zbmath.org/authors/?q=ai:fiza.mehreen"Khan, Ilyas"https://zbmath.org/authors/?q=ai:khan.ilyas"Mohamed, Abdullah"https://zbmath.org/authors/?q=ai:mohamed.abdullah"Mousa, Abd Allah A."https://zbmath.org/authors/?q=ai:mousa.abd-allah-a(no abstract)\(n\)-soliton, breather, lump solutions and diverse traveling wave solutions of the fractional \((2+1)\)-Dimensional Boussinesq equationhttps://zbmath.org/1521.351422023-11-13T18:48:18.785376Z"Wang, Kang-Jia"https://zbmath.org/authors/?q=ai:wang.kang-jia"Liu, Jing-Hua"https://zbmath.org/authors/?q=ai:liu.jinghua"Si, Jing"https://zbmath.org/authors/?q=ai:si.jing"Shi, Feng"https://zbmath.org/authors/?q=ai:shi.feng"Wang, Guo-Dong"https://zbmath.org/authors/?q=ai:wang.guodong(no abstract)Construction of fractal soliton solutions for the fractional evolution equations with conformable derivativehttps://zbmath.org/1521.351622023-11-13T18:48:18.785376Z"Wang, Kangle"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)New solitary wave solutions of the fractional modified KdV-Kadomtsev-Petviashvili equationhttps://zbmath.org/1521.351632023-11-13T18:48:18.785376Z"Wang, Kang-Le"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)Totally new soliton phenomena in the fractional Zoomeron model for shallow waterhttps://zbmath.org/1521.351642023-11-13T18:48:18.785376Z"Wang, Kang-Le"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)Application of Hosoya polynomial to solve a class of time-fractional diffusion equationshttps://zbmath.org/1521.351872023-11-13T18:48:18.785376Z"Jafari, Hossein"https://zbmath.org/authors/?q=ai:jafari.hossein"Ganji, Roghayeh Moallem"https://zbmath.org/authors/?q=ai:ganji.roghayeh-moallem"Narsale, Sonali Mandar"https://zbmath.org/authors/?q=ai:narsale.sonali-mandar"Kgarose, Maluti"https://zbmath.org/authors/?q=ai:kgarose.maluti"Nguyen, Van Thinh"https://zbmath.org/authors/?q=ai:nguyen.van-thinh(no abstract)Continuous dependence of the weak limit of iterates of some random-valued vector functionshttps://zbmath.org/1521.370512023-11-13T18:48:18.785376Z"Komorek, Dawid"https://zbmath.org/authors/?q=ai:komorek.dawidSummary: Given a probability space \((\Omega ,\mathcal{A},\mathbb{P})\), a complete separable Banach space \(X\) with the \(\sigma \)-algebra \(\mathcal B(X)\) of all its Borel subsets, an operator \(\Lambda :\Omega \rightarrow L(X,X)\) and \(\xi :\Omega \rightarrow X\) we consider the \(\mathcal{B}(X)\otimes \mathcal A\)-measurable function \(f:X\times \Omega \rightarrow X\) given by \(f(x,\omega )=\Lambda (\omega )x+\xi (\omega )\) and investigate the continuous dependence of a weak limit \(\pi^f\) of the sequence of iterates \((f^n(x,\cdot ))_{n\in \mathbb{N}}\) of \(f\), defined by \(f^0(x,\omega )=x\), \(f^{n+1}(x,\omega )=f(f^n(x,\omega ),\omega_{n+1})\) for \(x\in X\) and \(\omega =(\omega_1,\omega_2,\dots )\). Moreover for \(X\) taken as a Hilbert space we characterize \(\pi^f\) via the functional equation
\[\varphi^f(u)=\int_{\Omega }\varphi^f(\Lambda (\omega )u)\varphi^{\xi }(u)\mathbb{P}(d\omega )\]
with the aid of its characteristic function \(\varphi^f\). We also indicate the continuous dependence of a solution of that equation.Well-posedness of fractional stochastic complex Ginzburg-Landau equations driven by regular additive noisehttps://zbmath.org/1521.370922023-11-13T18:48:18.785376Z"Liu, Aili"https://zbmath.org/authors/?q=ai:liu.aili"Zou, Yanyan"https://zbmath.org/authors/?q=ai:zou.yanyan"Ren, Die"https://zbmath.org/authors/?q=ai:ren.die"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.jiSummary: This paper deals with the well-posedness of the solutions of the fractional complex Ginzburg-Landau equation driven by locally Lipschitz nonlinear diffusion terms defined on \(R^n\). We first give the pathwise uniform estimates and uniform estimates on average. Then we prove the existence, uniqueness and measurability of solutions for the equation.On an exponential inequalityhttps://zbmath.org/1521.390242023-11-13T18:48:18.785376Z"Menteshashvili, Marina"https://zbmath.org/authors/?q=ai:menteshashvili.marina"Berikashvilia, Valeri"https://zbmath.org/authors/?q=ai:berikashvilia.valeri"Kvaratskhelia, Vakhtang"https://zbmath.org/authors/?q=ai:kvaratskhelia.vakhtang-vSummary: Exponential function is one of the most important functions in mathematics and is helpful in theoretical investigations and practical applications. For example, exponential functions are the solutions to the simplest types of dynamical systems. In particular, an exponential function arises in simple models of bacterial growth, it can describe growth or decay, etc. The main purpose of this paper is one exponential inequality, which arose in the study of the properties of exponential functions.Strongly barycentrically associative and preassociative functionshttps://zbmath.org/1521.390252023-11-13T18:48:18.785376Z"Marichal, Jean-Luc"https://zbmath.org/authors/?q=ai:marichal.jean-luc"Teheux, Bruno"https://zbmath.org/authors/?q=ai:teheux.brunoSummary: We study the property of strong barycentric associativity, a stronger version of barycentric associativity for functions with indefinite arities. We introduce and discuss the more general property of strong barycentric preassociativity, a generalization of strong barycentric associativity which does not involve any composition of functions. We also provide a generalization of Kolmogoroff-Nagumo's characterization of the quasi-arithmetic mean functions to strongly barycentrically preassociative functions.On the convergence of weighted mean summable improper integrals over \(\mathbb{R}_{\geq 0}\)https://zbmath.org/1521.400022023-11-13T18:48:18.785376Z"Sezer, Sefa Anıl"https://zbmath.org/authors/?q=ai:sezer.sefa-anil"Çanak, İbrahim"https://zbmath.org/authors/?q=ai:canak.ibrahimLet \(w(x)\) be a locally integrable weight function in the sense of Lebesgue over \(\mathbf{R}_{\geq0}:=[0,\infty)\), in symbols: \(w\in L^{1}_{\mathrm{loc}}(\mathbb{R}_{\geq0}).\) The authors assume that \(w(x)\) is positive for almost all \(x\) in \(\mathbb{R}_{\geq0}\) and
\[
W(x):=\int_{0}^{x}w(t)\,dt
\]
satisfies
\[
\liminf_{x\to\infty}\frac{W(\mu x)}{ w(x)}>1, \textrm{ for each }\mu>1. \tag{1}
\]
In particular, (1) implies that \(\lim_{x\to\infty}W(x)=\infty\).
For any real/complex-valued function \(a\in L^{1}_{\mathrm{loc}}(\mathbb{R}_{\geq0})\), they set
\[
s(x):=\int_{0}^{x}a(t) \,dt . \tag{2}
\]
They recall that the weighted means of \(s(x)\) with regard to the weight function \(w(x)\) are defined as
\[
\tau(x):=\frac{1}{w(x)}\int_{0}^{x}s(t)w(t) \,dt
\]
for \(x\in\mathbb{R}_{\geq0}.\) If
\[
\lim_{x\to\infty}\tau(x)=\xi , \tag{3}
\]
then the integral \(\int_{0}^{\infty}a(x)\, dx\) is said to be summable to \(\xi\) by the weighted mean method determined by the weight function \(w(x)\), or in brief, \((\bar N,w)\) summable to \(\xi\), and they write \(\int_{0}^{\infty}a(x)\, dx=\xi (\bar N,w).\)
Note that
\begin{itemize}
\item \(w(x)=1\) for \(x\geq0\) leads to the $(C,1)$ summability method,
\item \(w(x)=\frac{1}{ (x+1)}\) for \(x\geq0\) leads to the $ (l,1)$ summability method.
\end{itemize}
In this paper, the authors give a Tauberian theorem for $(\bar{N}, w)$ summable integrals and then, in the last section, they introduce the $ (\bar{N}, w, k)$ summability method and give some Tauberian theorems for this summability method.
Reviewer: Yilmaz Erdem (Aydin)Generalization of statistical limit-cluster points and the concepts of statistical limit inferior-superior on time scales by using regular integral transformationshttps://zbmath.org/1521.400052023-11-13T18:48:18.785376Z"Yalçin, Ceylan"https://zbmath.org/authors/?q=ai:yalcin.ceylan-turanSummary: With the aid of regular integral operators, we will be able to generalize statistical limit-cluster points and statistical limit inferior-superior ideas on time scales in this work. These two topics, which have previously been researched separately from one another sometimes only in the discrete case and other times in the continuous case, will be studied at in a single study. We will investigate the relations of these concepts with each other and come to a number of new conclusions. On some well-known time scales, we shall analyze these ideas using examples.Asymptotic expansions and complete monotonicity properties for the Barnes \(G\)-functionhttps://zbmath.org/1521.410092023-11-13T18:48:18.785376Z"Han, Xue-Feng"https://zbmath.org/authors/?q=ai:han.xuefeng"Chen, Chao-Ping"https://zbmath.org/authors/?q=ai:chen.chaoping"Srivastava, Hari M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohanSummary: We present several new asymptotic expansions of the logarithm of the Barnes \(G\)-function in terms of the gamma function and the polygamma functions. We also give several explicit formulas and recurrence relations for determining the coefficients in these asymptotic expansions. By using the results obtained, we derive recursion formulas for the Bernoulli numbers and the corresponding results for the Riemann zeta function. Finally, based on the asymptotic expansions which we have presented here, we prove new complete monotonicity properties of some functions involving the Barnes \(G\)-function, the gamma function and the polygamma functions.Boas conjecture on the axis for the Fourier-Dunkl transform and its generalizationhttps://zbmath.org/1521.420052023-11-13T18:48:18.785376Z"Gorbachev, Dmitriĭ Viktorovich"https://zbmath.org/authors/?q=ai:gorbachev.dmitrii-viktorovichSummary: The question of integrability of the Fourier transform and other integral transformations \(\mathcal{F}(f)\) on classes of functions in weighted spaces \(L^p(\mathbb{R}^d)\) is a fundamental problem of harmonic analysis. The classical Hausdorff-Young result says that if a function \(f\) from \(L^p(\mathbb{R}^d)\) with \(p\in [1,2]\), then its Fourier transform \(\mathcal{F}(f)\in L^{p^\prime}(\mathbb{R}^d)\). For \(p>2\) the Fourier transform in the general situation will be a generalized function. The Fourier transform can be defined as an usual function for \(p>2\) by considering the weighted spaces \(L^p(\mathbb{R}^d)\). In particular, the classical Pitt inequality implies that if \(p,q\in (1,\infty)\), \(\delta=d(\frac{1}{q}-\frac{1}{p'})\), \(\gamma\in [(\delta)_+,\frac{d}{q})\) and function \(f\) is integrable in \(L^p(\mathbb{R}^d)\) with power weight \(|x|^{p(\gamma-\delta)} \), then its Fourier transform \(\mathcal{F} (f)\) belongs to the space \(L^q(\mathbb{R}^d)\) with weight \(|x|^{-q\gamma} \). The case \(p=q\) corresponds to the well-known Hardy-Littlewood inequality.
The question arises of extending the conditions for the integrability of the Fourier transform under additional conditions on the functions. In the one-dimensional case, G. Hardy and J. Littlewood proved that if \(f\) is an even nonincreasing function tending to zero and \(f\in L^p(\mathbb{R})\) for \(p\in (1,\infty)\), then \(\mathcal{F}(f)\) belongs to \(L^p(\mathbb{R})\) with weight \(|x|^{p-2} \). \textit{R. P. Boas jun.} [Stud. Math. 44, 365--369 (1972; Zbl 0237.42011)] suggested that for a monotone function \(f\) the membership \(|{ \cdot }|^{\gamma-\delta}f\in L^p(\mathbb{R})\) is equivalent to \(|{ \cdot }|^{-\gamma}\mathcal{F}(f)\in L^p(\mathbb{R})\) if and only if \(\gamma\in (-\frac{1}{p'},\frac{1}{p})\). The one-dimensional Boas conjecture was proved by \textit{Y. Sagher} [J. Math. Anal. Appl. 54, 151--156 (1976; Zbl 0324.46031)].
\textit{D. Gorbachev} et al. [J. Anal. Math. 114, 99--120 (2011; Zbl 1246.42011)] proved the multidimensional Boas conjecture for radial functions, moreover, on a wider class of general monotone non-negative radial functions \(f\): \(\||{ \cdot }|^{-\gamma}\mathcal{F}(f)\|_p\asymp \||{ \cdot }|^{\gamma-\delta}f\|_{p }\) if and only if \(\gamma\in (\frac{d}{p}-\frac{d+1}{2},\frac{d}{p})\), where \(\delta= d(\frac{1}{p}-\frac{1}{p'})\). For radial functions, the Fourier transform is expressed in terms of the Bessel transform of half-integer order, which reduces to the classical Hankel transform and includes the cosine and sine Fourier transforms. For the latter, the Boas conjecture was proved by \textit{E. Liflyand} and \textit{S. Tikhonov} [C. R., Math., Acad. Sci. Paris 346, No. 21--22, 1137--1142 (2008; Zbl 1151.42002)]. For the Bessel-Hankel transform with an arbitrary order, the Boas conjecture was proved by \textit{L. De Carli} et al. [J. Math. Anal. Appl. 408, No. 2, 762--774 (2013; Zbl 1307.44004)]. \textit{D. V. Gorbachev} et al. [Int. Math. Res. Not. 2016, No. 23, 7179--7200 (2016; Zbl 1404.42019)] generalized these results to the case of \((\kappa,a)\)-generalized Fourier transform. \textit{A. Debernardi} [J. Fourier Anal. Appl. 25, No. 6, 3310--3341 (2019; Zbl 1428.42007)] studied the case of the Hankel transform and general monotone alternating functions.
So far, the Boas conjecture has been considered for functions on the semiaxis. In this paper, it is studied on the entire axis. To do this, we consider the integral Dunkl transform, which for even functions reduces to the Bessel-Hankel transform. It is also shown that the Boas conjecture remains valid for the \((\kappa,a)\)-generalized Fourier transform, which gives the Dunkl transform for \(a=2\). As a result, we have \[\||{ \cdot }|^{-\gamma}\mathcal{F}_{\kappa,a}(f)\|_{p,\kappa,a}\asymp \||{ \cdot }|^{\gamma-\delta}f\|_{p,\kappa,a},\]
where \(\gamma\in (\frac{d_{\kappa,a}}{p}-\frac{d_{\kappa,a}+\frac{a}{2}}{2},\frac{d_{\kappa,a} }{p})\), \(\delta=d_{\kappa,a}(\frac{1}{p}-\frac{1}{p^\prime})\), \(d_{\kappa,a}=2\kappa+a-1\).Solvability of quadratic Hadamard-type fractional integral equations in Orlicz spaceshttps://zbmath.org/1521.450032023-11-13T18:48:18.785376Z"Metwali, Mohamed M. A."https://zbmath.org/authors/?q=ai:metwali.mohamed-m-aSummary: We demonstrate some properties of Hadamard fractional operators such as boundedness, monotonicity, continuity, and acting conditions in Orlicz spaces \(L_\varphi\). We apply these properties with a proper measure of noncompactness (MNC) to inspect the existence of monotonic solutions of some general, but abstract, form of quadratic Hadamard-type fractional integral equations in \(L_\varphi\). We also discuss the uniqueness of the solution.Eigenvalue inequalities for \(n\)-tuple of matriceshttps://zbmath.org/1521.470212023-11-13T18:48:18.785376Z"Moradi, Hamid Reza"https://zbmath.org/authors/?q=ai:moradi.hamid-reza"Sababheh, Mohammad"https://zbmath.org/authors/?q=ai:sababheh.mohammad-sSummary: The main goal of this article is to present several inequalities for the eigenvalues of several operators, when filtered through positive linear mappings, convex functions, superquadratic functions and asynchronous functions.Some remarks on substitution and composition operatorshttps://zbmath.org/1521.470452023-11-13T18:48:18.785376Z"Appell, Jürgen"https://zbmath.org/authors/?q=ai:appell.jurgen-m"Brito, Belén López"https://zbmath.org/authors/?q=ai:brito.belen-lopez"Reinwand, Simon"https://zbmath.org/authors/?q=ai:reinwand.simon"Schöller, Kilian"https://zbmath.org/authors/?q=ai:scholler.kilianIn this paper, the authors study the linear substitution operator and nonlinear composition operator, namely,
\[
S_\varphi(f)=f\circ\varphi,\; \varphi:[0,1]\rightarrow [0,1],
\]
and
\[
C_g(f)=g\circ f,\; g:\mathbb{R}\rightarrow \mathbb{R},
\]
respectively. They show that these operators have a very different behavior in the space of continuous functions, Lipschitz functions, functions of bounded variation, and Baire class one functions. Moreover, they give examples and counterexamples which illustrate this behavior.
Reviewer: Mohamed Abdalla Darwish (Damanhour)Corrigendum to: ``Some remarks on substitution and composition operators''https://zbmath.org/1521.470462023-11-13T18:48:18.785376Z"Appell, Jürgen"https://zbmath.org/authors/?q=ai:appell.jurgen-m"Brito, Belén López"https://zbmath.org/authors/?q=ai:brito.belen-lopez"Reinwand, Simon"https://zbmath.org/authors/?q=ai:reinwand.simon"Schöller, Kilian"https://zbmath.org/authors/?q=ai:scholler.kilianThe authors correct an error in Proposition 3.3 (Part (c)) in their paper [ibid. 53, Paper No. 6, 25 p. (2021; Zbl 1521.47045)].
Reviewer: Mohamed Abdalla Darwish (Damanhour)Multi-marginal maximal monotonicity and convex analysishttps://zbmath.org/1521.470842023-11-13T18:48:18.785376Z"Bartz, Sedi"https://zbmath.org/authors/?q=ai:bartz.sedi"Bauschke, Heinz H."https://zbmath.org/authors/?q=ai:bauschke.heinz-h"Phan, Hung M."https://zbmath.org/authors/?q=ai:phan.hung-m"Wang, Xianfu"https://zbmath.org/authors/?q=ai:wang.xianfuThe authors study extensions of classical monotone operator theory and convex analysis to the multi-marginal setting. Multi-marginal \(c\)-monotonicity is characterized in terms of classical monotonicity and firmly nonexpansive mappings. Minty type, continuity and conjugacy criteria for multi-marginal maximal monotonicity are presented. Partition of the identity into a sum of firmly nonexpansive mappings, as well as Moreau's decomposition of the quadratic function into envelopes and proximal mappings, is extended in this framework. Examples and applications are presented. Many open questions are posed.
Reviewer: K. C. Sivakumar (Chennai)On the generalized Jacobian of the inverse of a Lipschitzian mappinghttps://zbmath.org/1521.471002023-11-13T18:48:18.785376Z"Fabian, M."https://zbmath.org/authors/?q=ai:fabian.martin"Hiriart-Urruty, J.-B."https://zbmath.org/authors/?q=ai:hiriart-urruty.jean-baptiste"Pauwels, E."https://zbmath.org/authors/?q=ai:pauwels.eric-j|pauwels.elmar|pauwels.edouard-jean-robertSummary: The objective of this short note is to provide an estimate of the generalized Jacobian of the inverse of a Lipschitzian mapping when Clarke's inverse function theorem applies. Contrary to the classical \(\mathcal{C}^1\) case, inverting matrices of the generalized Jacobian is not enough. Simple counterexamples show that our results are sharp.Numerical solution of distributed-order fractional 2D optimal control problems using the Bernstein polynomialshttps://zbmath.org/1521.490252023-11-13T18:48:18.785376Z"Heydari, Mohammad Hossein"https://zbmath.org/authors/?q=ai:heydari.mohammadhossein"Razzaghi, Mohsen"https://zbmath.org/authors/?q=ai:razzaghi.mohsen"Zhagharian, Shabnam"https://zbmath.org/authors/?q=ai:zhagharian.shabnamSummary: In this work, a new class of two-dimensional optimal control problems is introduced with the help of distributed-order fractional derivative in the Caputo form. The orthonormal Bernstein polynomials are used to make a numerical method to solve these problems. Through this way, some operational matrices are obtained for the classical and fractional derivatives of these polynomials to make effective utilisation of them in constructing the proposed method. The main advantage of the established method is that it turns the solution of the problem under consideration into a system of algebraic equations by approximating the state and control variables by the expressed polynomials and applying the method of Lagrange multipliers. The accuracy and capability of the proposed approach are investigated by solving some numerical examples.The distribution of roots of Ehrhart polynomials for the dual of root polytopes of type \(C\)https://zbmath.org/1521.520062023-11-13T18:48:18.785376Z"Higashitani, Akihiro"https://zbmath.org/authors/?q=ai:higashitani.akihiro"Yamada, Yumi"https://zbmath.org/authors/?q=ai:yamada.yumiConsider an integer polytope of type \(C\) of dimension \(d\) and let \(C^\ast_d\) denotes the dual of the root polytope of such polytope. The authors provide an explicit formula for the values of the Ehrhart polynomial of \(C^\ast_d\) and prove that its roots have the same real part \(-1/2\). They also show that the Ehrhart polynomial of \(C^\ast_d\) satisfies the interlacing property (with respect to dimension \(d\)).
Reviewer: Oleg Karpenkov (Liverpool)Poincaré type inequality for hypersurfaces and rigidity resultshttps://zbmath.org/1521.530292023-11-13T18:48:18.785376Z"Alencar, Hilário"https://zbmath.org/authors/?q=ai:alencar.hilario"Batista, Márcio"https://zbmath.org/authors/?q=ai:batista.marcio"Silva Neto, Gregório"https://zbmath.org/authors/?q=ai:silva-neto.gregorioThe authors prove a general Poincaré type inequality on Riemannian manifolds whose sectional curvature is suitably bounded. They then apply this inequality, combined with some other mild assumptions, to prove the following:
(1) Some isoperimetric inequalities for domains of hypersurfaces;
(2) Minimal hypersurfaces of space forms (satisfying suitable decay properties) are foliated by totally geodesic submanifolds;
(3) Hypersurfaces with a determined constant scalar curvature in Einstein manifolds are totally geodesic;
(4) The hyperplane is rigid, in the sense that is the only homothetic self-similar solution to a class of fully nonlinear curvature flows.
Reviewer: Giorgio Saracco (Firenze)On geodesically convex formulations for the Brascamp-Lieb constanthttps://zbmath.org/1521.650542023-11-13T18:48:18.785376Z"Sra, Suvrit"https://zbmath.org/authors/?q=ai:sra.suvrit"Vishnoi, Nisheeth K."https://zbmath.org/authors/?q=ai:vishnoi.nisheeth-k"Yildiz, Ozan"https://zbmath.org/authors/?q=ai:yildiz.ozanSummary: We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work by \textit{E. H. Lieb} [Invent. Math. 102, No. 1, 179--208 (1990; Zbl 0726.42005)] and the second is new and inspired by the work by \textit{J. Bennett} et al. [Geom. Funct. Anal. 17, No. 5, 1343--1415 (2008; Zbl 1132.26006)]. Recent work by \textit{A. Garg} et al. [STOC 2017, 397--409 (2017; Zbl 1372.65191)] also implies a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends exponentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum.
For the entire collection see [Zbl 1393.68012].Spatiotemporal linear stability of viscoelastic subdiffusive channel flows: a fractional calculus frameworkhttps://zbmath.org/1521.761272023-11-13T18:48:18.785376Z"Chauhan, Tanisha"https://zbmath.org/authors/?q=ai:chauhan.tanisha"Bansal, Diksha"https://zbmath.org/authors/?q=ai:bansal.diksha"Sircar, Sarthok"https://zbmath.org/authors/?q=ai:sircar.sarthokSummary: The temporal and spatiotemporal linear stability analyses of viscoelastic, subdiffusive, plane Poiseuille flow obeying the Fractional Upper Convected Maxwell (FUCM) equation in the limit of low to moderate Reynolds number (Re) and Weissenberg number (We) is reported to identify the regions of topological transition of the advancing flow interface. In particular, we demonstrate how the exponent of the power-law scaling \((t^\alpha\), with \(0<\alpha\le 1)\) in viscoelastic microscale models [\textit{T. G. Mason} and \textit{D. A. Weitz}, Phys. Rev. Lett. 74, No. 5, 1250--1253 (1995; \url{doi:10.1103/PhysRevLett.74.1250})] is related to the fractional order of the time derivative, \(\alpha\), of the corresponding non-linear stress constitutive equation in the continuum. The stability studies are limited to two exponents: monomer diffusion in Rouse chain melts, \(\alpha=1/2\), and in Zimm chain solutions, \(\alpha=2/3\). The temporal stability analysis indicates that with decreasing order of the fractional derivative (a) the most unstable mode decreases with decreasing values of \(\alpha\), (b) the peak of the most unstable mode shifts to lower values of Re, and (c) the peak of the most unstable mode, for the Rouse model precipitates towards the limit \(Re\to 0\). The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilities, and evanescent modes. The spatiotemporal phase diagram indicates an abnormal region of temporal stability at high fluid inertia, revealing the presence of a non-homogeneous environment with hindered flow, thus highlighting the potential of the model to effectively capture certain experimentally observed, flow-instability transition in subdiffusive flows.Two-dimensional nonlinear time fractional reaction-diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous mediahttps://zbmath.org/1521.768242023-11-13T18:48:18.785376Z"Pandey, P."https://zbmath.org/authors/?q=ai:pandey.prashant-k"Das, S."https://zbmath.org/authors/?q=ai:das.subir"Craciun, E.-M."https://zbmath.org/authors/?q=ai:craciun.eduard-marius"Sadowski, T."https://zbmath.org/authors/?q=ai:sadowski.tomaszSummary: In the present article, an efficient operational matrix based on the famous Laguerre polynomials is applied for the numerical solution of two-dimensional non-linear time fractional order reaction-diffusion equation. An operational matrix is constructed for fractional order differentiation and this operational matrix converts our proposed model into a system of non-linear algebraic equations through collocation which can be solved by using the Newton Iteration method. Assuming the surface layers are thermodynamically variant under some specified conditions, many insights and properties are deduced e.g., nonlocal diffusion equations and mass conservation of the binary species which are relevant to many engineering and physical problems. The salient features of present manuscript are finding the convergence analysis of the proposed scheme and also the validation and the exhibitions of effectiveness of the method using the order of convergence through the error analysis between the numerical solutions applying the proposed method and the analytical results for two existing problems. The prominent feature of the present article is the graphical presentations of the effect of reaction term on the behavior of solute profile of the considered model for different particular cases.How long does it take to implement a projective measurement?https://zbmath.org/1521.810142023-11-13T18:48:18.785376Z"Strasberg, Philipp"https://zbmath.org/authors/?q=ai:strasberg.philipp"Modi, Kavan"https://zbmath.org/authors/?q=ai:modi.kavan"Skotiniotis, Michalis"https://zbmath.org/authors/?q=ai:skotiniotis.michalis(no abstract)Newton's identities and positivity of trace class integral operatorshttps://zbmath.org/1521.810162023-11-13T18:48:18.785376Z"Homa, G."https://zbmath.org/authors/?q=ai:homa.gabor"Balka, R."https://zbmath.org/authors/?q=ai:balka.richard"Bernád, J. Z."https://zbmath.org/authors/?q=ai:bernad.j-z"Károly, M."https://zbmath.org/authors/?q=ai:karoly.m"Csordás, A."https://zbmath.org/authors/?q=ai:csordas.andrasSummary: We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton's identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schrödinger's uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy.Quantum complexity of permutationshttps://zbmath.org/1521.810532023-11-13T18:48:18.785376Z"Yu, Andrew"https://zbmath.org/authors/?q=ai:yu.andrew-junfangSummary: Quantum complexity of a unitary measures the runtime of quantum computers. In this article, we study the complexity of a special type of unitaries, permutations. Let \(S_n\) be the symmetric group of all permutations of \(\{1,\cdots,n\}\) with two generators: the transposition and the cyclic permutation (denoted by \(\sigma\) and \(\tau)\). The permutations \(\{\sigma,\tau,\tau^{-1}\}\) serve as logic gates. We give an explicit construction of permutations in \(S_n\) with quadratic quantum complexity lower bound \(\frac{n^2-2n-7}{4}\). We also prove that all permutations in \(S_n\) have quadratic quantum complexity upper bound \(3(n-1)^2\). Finally, we show that almost all permutations in \(S_n\) have quadratic quantum complexity lower bound when \(n\to\infty\). The method described in this paper may shed light on the complexity problem for general unitaries in quantum computation.Characteristic function and operator approach to M-indeterminate probability densitieshttps://zbmath.org/1521.810702023-11-13T18:48:18.785376Z"Loughlin, Patrick"https://zbmath.org/authors/?q=ai:loughlin.patrick-j"Cohen, Leon"https://zbmath.org/authors/?q=ai:cohen.leonSummary: Based on a quantum mechanical approach, we investigate moment- (or M-) indeterminate probability densities by way of the characteristic function and self-adjoint operators. The approach leads to new methods to construct classes of M-indeterminate probability densities.Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivativeshttps://zbmath.org/1521.811202023-11-13T18:48:18.785376Z"Cazacu, Cristian"https://zbmath.org/authors/?q=ai:cazacu.cristian-m"Flynn, Joshua"https://zbmath.org/authors/?q=ai:flynn.joshua"Lam, Nguyen"https://zbmath.org/authors/?q=ai:lam.nguyenSummary: The present work has as a first goal to extend the previous results in [\textit{C. Cazacu} et al., J. Funct. Anal. 283, No. 10, Article ID 109659, 37 p. (2022; Zbl 1513.81089)] to weighted uncertainty principles with nontrivial radially symmetric weights applied to curl-free vector fields. Part of these new inequalities generalize the family of Caffarelli-Kohn-Nirenberg (CKN) inequalities studied by Catrina and Costa in [\textit{F. Catrina} and \textit{D. G. Costa}, J. Differ. Equations 246, No. 1, 164--182 (2009; Zbl 1220.35006)] from scalar fields to curl-free vector fields. We will apply a new representation of curl-free vector fields developed by \textit{N. Hamamoto} and \textit{F. Takahashi} [J. Funct. Anal. 280, No. 1, Article ID 108790, 24 p. (2021; Zbl 1452.26016)]. The newly obtained results are also sharp and minimizers are completely described. Secondly, we prove new sharp second order interpolation functional inequalities for scalar fields with radial weights generalizing the previous results in [Cazacu et al., loc. cit.]. We apply new factorization methods being inspired by our recent work [J. Differ. Equations 302, 533--549 (2021; Zbl 1499.81063)]. The main novelty in this case is that we are able to find a new independent family of minimizers based on the solutions of Kummer's differential equations. We point out that the two types of weighted inequalities under consideration (first order inequalities for curl-free vector fields vs. second order inequalities for scalar fields) represent independent families of inequalities unless the weights are trivial.Loop quantum gravity of a spherically symmetric scalar field coupled to gravity with a clockhttps://zbmath.org/1521.830432023-11-13T18:48:18.785376Z"Gambini, Rodolfo"https://zbmath.org/authors/?q=ai:gambini.rodolfo"Pullin, Jorge"https://zbmath.org/authors/?q=ai:pullin.jorge-aSummary: The inclusion of matter fields in spherically symmetric loop quantum gravity has proved problematic at the level of implementing the constraint algebra including the Hamiltonian constraint. Here we consider the system with the introduction of a clock. Using the abelianizaton technique we introduced in previous papers in the case of gravity coupled to matter, the system can be gauge fixed and rewritten in terms of a restricted set of dynamical variables that satisfy simple Poisson bracket relations. This creates a true Hamiltonian and therefore one bypasses the issue of the constraint algebra. We show how loop quantum gravity techniques may be applied for the vacuum case and show that the Hamiltonian system reproduces previous results for the physical space of states and the observables of a Schwarzchild black hole.High-order matrix method with delimited expansion domainhttps://zbmath.org/1521.831322023-11-13T18:48:18.785376Z"Lin, Kai"https://zbmath.org/authors/?q=ai:lin.kai"Qian, Wei-Liang"https://zbmath.org/authors/?q=ai:qian.wei-liangSummary: Motivated by the substantial instability of the fundamental and high-overtone quasinormal modes (QNMs), recent developments regarding the notion of black hole pseudospectrum call for numerical results with unprecedented precision. This work generalizes and improves the matrix method for black hole QNMs to higher orders, specifically aiming at a class of perturbations to the metric featured by discontinuity intimately associated with the QNM structural instability. The approach is based on the mock-Chebyshev grid, which guarantees its convergence in the degree of the interpolant. In practice, solving for black hole QNMs is a formidable task. The presence of discontinuity poses a further difficulty so that many well-known approaches cannot be employed straightforwardly. Compared with other viable methods, the modified matrix method is competent in speed and accuracy. Therefore, the method serves as a helpful gadget for relevant studies.Seed predation-induced Allee effects, seed Dispersal and masting jointly drive the diversity of seed sources during population expansionhttps://zbmath.org/1521.920682023-11-13T18:48:18.785376Z"Doublet, Violette"https://zbmath.org/authors/?q=ai:doublet.violette"Roques, Lionel"https://zbmath.org/authors/?q=ai:roques.lionel-j"Klein, Etienne K."https://zbmath.org/authors/?q=ai:klein.etienne-k"Lefèvre, François"https://zbmath.org/authors/?q=ai:lefevre.francois"Boivin, Thomas"https://zbmath.org/authors/?q=ai:boivin.thomasSummary: The environmental factors affecting plant reproduction and effective dispersal, in particular biotic interactions, have a strong influence on plant expansion dynamics, but their demographic and genetic consequences remain an understudied body of theory. Here, we use a mathematical model in a one-dimensional space and on a single reproductive period to describe the joint effects of predispersal seed insect predators foraging strategy and plant reproduction strategy (masting) on the spatio-temporal dynamics of seed sources diversity in the colonisation front of expanding plant populations. We show that certain foraging strategies can result in a higher seed predation rate at the colonisation front compared to the core of the population, leading to an Allee effect. This effect promotes the contribution of seed sources from the core to the colonisation front, with long-distance dispersal further increasing this contribution. As a consequence, our study reveals a novel impact of the predispersal seed predation-induced Allee effect, which mitigates the erosion of diversity in expanding populations. We use rearrangement inequalities to show that masting has a buffering role: it mitigates this seed predation-induced Allee effect. This study shows that predispersal seed predation, plant reproductive strategies and seed dispersal patterns can be intermingled drivers of the diversity of seed sources in expanding plant populations, and opens new perspectives concerning the analysis of more complex models such as integro-difference or reaction-diffusion equations.A new seismic control framework of optimal \(\mathrm{PI}^\lambda\mathrm{D}^\mu\) controller series with fuzzy PD controller including soil-structure interactionhttps://zbmath.org/1521.930592023-11-13T18:48:18.785376Z"Etedali, Sadegh"https://zbmath.org/authors/?q=ai:etedali.sadegh"Zamani, Abbas-Ali"https://zbmath.org/authors/?q=ai:zamani.abbas-ali"Akbari, Morteza"https://zbmath.org/authors/?q=ai:akbari.morteza"Seifi, Mohammad"https://zbmath.org/authors/?q=ai:seifi.mohammadSummary: A new framework of optimal fractional order proportional-derivative-integral (FOPID) controller series with fuzzy proportional-derivative (PD) controller, namely OFPD-FOPID controller, is proposed in this study for seismic control of structures equipped with active tuned mass damper (ATMD). Three controllers including optimal PID, optimal FOPID, and fuzzy PID (FPID) controllers are also implemented for comparison purposes. Simulation results carried out on a 15-story building show the FOPID controller than the PID and FPID controllers can remarkably reduce the maximum floor displacement, but they represent a poor performance in mitigation of the maximum floor acceleration in different soil conditions, while the proposed OFPD-FOPID controller tracking the amount of the maximum floor acceleration to estimate the optimal control force of the actuator can provide superior performance. An average reduction of 41\%, 45\%, and 33\% in the maximum floor displacement; 36\%, 33\%, and 20\% in the maximum inter-story drift are given by FOPID in the dense, medium, and soft soils, while it results in an increase of 45\%, 52\% and 24\% in the maximum floor acceleration. Similarly, the proposed OFPD-FOPID controller represents an average reduction of 52\%, 55\%, and 45\% in the maximum floor displacement; 42\%, 44\%, and 28\% in the maximum inter-story drift in the dense, medium, and soft soils, while it also slightly reduces the maximum floor acceleration of the studied structure located on different soil conditions.First- and second-order necessary optimality conditions for a control problem described by nonlinear fractional difference equationshttps://zbmath.org/1521.930862023-11-13T18:48:18.785376Z"Aliyeva, S. T."https://zbmath.org/authors/?q=ai:alieva.s-tSummary: This paper considers an optimal control problem for an object described by a system of nonlinear fractional difference equations. Such problems are a discrete analog of optimal control problems described by fractional ordinary differential equations. The first and second variations of a performance criterion are calculated using a modification of the increment method under the assumption that the control set is open. We establish a first-order necessary optimality condition (an analog of the Euler equation) and a general second-order necessary optimality condition. Adopting the representations of the solution of the linearized fractional difference equations from the general second-order optimality condition, we derive necessary optimality conditions in terms of the original problem parameters. Finally, with a special choice of an admissible variation of control, we formulate a pointwise necessary optimality condition for classical extremals.Event-triggered adaptive fuzzy finite time control of fractional-order non-strict feedback nonlinear systemshttps://zbmath.org/1521.931102023-11-13T18:48:18.785376Z"Xin, Chun"https://zbmath.org/authors/?q=ai:xin.chun"Li, Yuanxin"https://zbmath.org/authors/?q=ai:li.yuanxin"Niu, Ben"https://zbmath.org/authors/?q=ai:niu.benSummary: In this article, the problem of event-triggered adaptive fuzzy finite time control of non-strict feedback fractional order nonlinear systems is investigated. By using the property of fuzzy basis function, the obstacle caused by algebraic loop problems is successfully circumvented. Moreover, a new adaptive event-triggered scheme is designed under the unified framework of backstepping control method, which can largely reduce the amount of communications. The stability of the closed-loop system is ensured through fractional Lyapunov stability analysis. Finally, the effectiveness of the proposed scheme is verified by simulation examples.New stability tests for fractional positive descriptor linear systemshttps://zbmath.org/1521.931312023-11-13T18:48:18.785376Z"Kaczorek, Tadeusz"https://zbmath.org/authors/?q=ai:kaczorek.tadeusz"Ruszewski, Andrzej"https://zbmath.org/authors/?q=ai:ruszewski.andrzejSummary: The asymptotic stability of fractional positive descriptor continuous-time and discrete time linear systems is considered. New sufficient conditions for stability of fractional positive descriptor linear systems are established. The efficiency of the new stability conditions are demonstrated on numerical examples of fractional continuous-time and discrete-time linear systems.Practical Mittag-Leffler stability of quasi-one-sided Lipschitz fractional order systemshttps://zbmath.org/1521.931352023-11-13T18:48:18.785376Z"Basdouri, Imed"https://zbmath.org/authors/?q=ai:basdouri.imed"Kasmi, Souad"https://zbmath.org/authors/?q=ai:kasmi.souad"Lerbet, Jean"https://zbmath.org/authors/?q=ai:lerbet.jeanSummary: This paper focuses on the global practical Mittag-Leffler feedback stabilization problem for a class of uncertain fractional-order systems. This class of systems is a larger class of nonlinearities than the Lipschitz ones. Based on the quasi-one-sided Lipschitz condition, firstly, we provide sufficient conditions for the practical observer design. Then, we exhibit that practical Mittag-Leffler stability of the closed loop system with a linear, state feedback is attained. Finally, a separation principle is established and we prove that the closed loop system is practical Mittag-Leffler stable.Stability analysis of fractional-order neural networks with time-varying delay utilizing free-matrix-based integral inequalitieshttps://zbmath.org/1521.931462023-11-13T18:48:18.785376Z"Cheng, Yali"https://zbmath.org/authors/?q=ai:cheng.yali"Xu, Wenbo"https://zbmath.org/authors/?q=ai:xu.wenbo.1"Jia, Haitao"https://zbmath.org/authors/?q=ai:jia.haitao"Zhong, Shouming"https://zbmath.org/authors/?q=ai:zhong.shou-mingSummary: The stability problem of fractional-order neural networks in consideration of time-varying delay is addressed by utilizing Lyapunov functional approach in this paper. Firstly, a class of novel Lyapunov-Krasovskii functions are designed to deal with the time-varying delay terms, which reduces the conservatism of the stability criteria. In addition, to estimate the fractional-order derivative of the Lyapunov-Krasovskii functions, a novel free-matrix-based fractional-order integral inequality is then established, which encompasses the Wirtinger one and leads to tractable linear matrix inequality criteria. Then, based on the designed adaptive control and the proposed Lyapunov-Krasovskii functions, some stability criteria depending on time-varying delay information and fractional-order \(\alpha\) are deduced. Finally, numerical simulation shows that the presented approach can significantly reduce the conservatism of the existing results and has a broader application prospect.Positivity and stability of fractional-order linear time-delay systemshttps://zbmath.org/1521.931482023-11-13T18:48:18.785376Z"Hao, Yilin"https://zbmath.org/authors/?q=ai:hao.yilin"Huang, Chengdai"https://zbmath.org/authors/?q=ai:huang.chengdai"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jinde"Liu, Heng"https://zbmath.org/authors/?q=ai:liu.hengSummary: This article focuses on the positivity and the asymptotic stability of fractional-order linear time-delay systems (FOLTDSs) which are composed of \(N\) \((N\geq 2)\) subsystems. Firstly, a sufficient and necessary condition is given to ensure the positivity of FOLTDSs. The solutions of the studied systems are obtained by using the Laplace transform method, and it can be observed that the positivity of FOLTDSs is completely determined by the series of matrices and independent of the magnitude of time-delays. Secondly, a theorem is given to prove the asymptotic stability of positive FOLTDSs. By considering the monotonicity and asymptotic properties of systems with constant time-delay, it is further shown that the asymptotic stability of positive FOLTDSs is independent of the time-delay. Next, a state-feedback controller, whose gain matrix is derived by resolving a linear programming question, is designed such that the state variables of the systems are nonnegative and asymptotically convergent. When the order of the FOLTDSs is greater than one, by utilizing a proposed property of Caputo derivative, a sufficient condition for the positivity of FOLTDS is presented. Finally, simulation examples are presented to verify the validity and practicability of the theoretical analysis.Estimating the Hölder exponents based on the \(\epsilon \)-complexity of continuous functions: an experimental analysis of the algorithmhttps://zbmath.org/1521.931852023-11-13T18:48:18.785376Z"Dubnov, Yu. A."https://zbmath.org/authors/?q=ai:dubnov.yuri-a"Popkov, A. Yu."https://zbmath.org/authors/?q=ai:popkov.alexey-yu"Darkhovsky, B. S."https://zbmath.org/authors/?q=ai:darkhovsky.boris-sSummary: This paper describes one method for estimating the Hölder exponent based on the \(\epsilon \)-complexity of continuous functions, a concept formulated lately. Computational experiments are carried out to estimate the Hölder exponent for smooth and fractal functions and study the trajectories of discrete deterministic and stochastic systems. The results of these experiments are presented and discussed.Refined estimations for some types of entropies and divergenceshttps://zbmath.org/1521.940152023-11-13T18:48:18.785376Z"Nikoufar, Ismail"https://zbmath.org/authors/?q=ai:nikoufar.ismail"Kanani Arpatapeh, Mehdi"https://zbmath.org/authors/?q=ai:kanani-arpatapeh.mehdiSummary: In this paper, the improvements in the estimations of some entropies are of our interest. These improvements can also be done sequentially. We find some new and refined bounds for the Tsallis entropy, Tsallis quasilinear entropy, and Tsallis quasilinear divergence. The estimations for the Tsallis entropy are provided with respect to the Shannon entropy and Rényi entropy. We obtain some alternative refined bounds for the Tsallis relative operator entropy and we establish the significant refined relation between the Tsallis relative operator entropy and the generalized relative operator entropy.A curious property of tangentshttps://zbmath.org/1521.970192023-11-13T18:48:18.785376Z"Treeby, David"https://zbmath.org/authors/?q=ai:treeby.david"Ross, Marty"https://zbmath.org/authors/?q=ai:ross.martySummary: We give two proofs of a surprising elementary result concerning tangents drawn to the graphs of convex functions.Do dogs know calculus with early transcendentals?https://zbmath.org/1521.970202023-11-13T18:48:18.785376Z"Weyenberg, Grady"https://zbmath.org/authors/?q=ai:weyenberg.gradySummary: A standard applications problem in differential calculus involves minimizing the travel time needed for a dog on a beach to reach a toy floating in the water, taking into account the different velocities the dog can achieve when running versus swimming. The solution usually presented to such problems are formulated in terms of a Cartesian coordinate. Here, a solution that is based in trigonometry is presented that, while quite natural, appears to be relatively unknown. The solution to the optimization problem in the trigonometric setting has a clear interpretation which is lacking in the Cartesian solution.On that most over skinned of improper integralshttps://zbmath.org/1521.970212023-11-13T18:48:18.785376Z"Stewart, Seán M."https://zbmath.org/authors/?q=ai:stewart.sean-markSummary: Continuing a much discussed topic of the various ways a particular improper integral can be evaluated, we give three further ways its generalization can be evaluated. Using techniques typically encountered immediately after the calculus sequence of courses we show how the improper integral can be evaluated using the beta and gamma functions, by first converting it to a double integral, and using a property of the Laplace transform.A note on teaching the Riemann integralhttps://zbmath.org/1521.970222023-11-13T18:48:18.785376Z"Thomson, Brian S."https://zbmath.org/authors/?q=ai:thomson.brian-sSummary: There are alternative definitions for the Riemann integral, many of which avoid some of the unpleasant computations that arise when using Riemann sums. In this version a simple distance function for step functions is used and the Riemann integral is defined and developed by employing exclusively ``convergent'' sequences of step functions. While only modestly different from the standard presentation it might have some extra intuitive appeal. This is a common device in advanced theories of integration and can be introduced at this elementary level.