Recent zbMATH articles in MSC 26https://www.zbmath.org/atom/cc/262022-01-14T13:23:02.489162ZWerkzeugIntegral calculus made easy!https://www.zbmath.org/1475.000032022-01-14T13:23:02.489162Z"Balla, Jochen"https://www.zbmath.org/authors/?q=ai:balla.jochenPublisher's description: Das vorliegende Buch bietet eine leicht lesbare und verständliche Darstellung der Kerninhalte der Integralrechnung. Es richtet sich an Studierende der Natur- und Ingenieurwissenschaften, der Wirtschaftswissenschaften und allgemein aller Fachgebiete, in denen Integrale eine Rolle spielen. Auch Mathematikstudierenden, die einen leicht verständlichen Zugang suchen, sollte es gute Dienste leisten.
Die ersten Kapitel behandeln die Integration gewöhnlicher Funktionen einer Veränderlichen, anschließend werden Mehrfachintegrale besprochen. Auch das dazu notwendige Grundwissen über mehrdimensionale Funktionen wird bereitgestellt. Die Theorie wird mit zahlreichen, teilweise auch weiterführenden Beispielen eingeübt und angewendet.
Das Lehrbuch bietet verschiedene Hilfestellungen, die den Zugang erleichtern:
\begin{itemize}
\item
152 Lesehilfen helfen über schwierige Stellen hinweg
\item
40 Zwischenfragen mit Antworten regen zum Nachdenken an
\item
45 Übungsaufgaben mit ausführlichen Lösungen unterstützen das vertiefende Studium
\item
``Das Wichtigste in Kürze'' und eine Formelsammlung fassen am Ende eines jeden Kapitels den Stoff zusammen.
\end{itemize}The continuum, the infinitely small, and the law of continuity in Leibnizhttps://www.zbmath.org/1475.010112022-01-14T13:23:02.489162Z"Levey, Samuel"https://www.zbmath.org/authors/?q=ai:levey.samuelLeibniz's differential calculus went beyond the standards of rigor established by ancient authors like Euclid and Archimedes. The problem of its justification was closely related to a multitude of other questions pertinent to philosophy since antiquity: the composition of the continuum, the existence of minimal and infinitesimal elements, the notion of continuity in time and space and how to understand motion. For example: Is a line a mere aggregate of its points? When dealing with these questions, Leibniz distinguished between the reality of matter and the ideal spaces of geometry. With the aim of presenting Leibniz's ideas on these subjects, the author focusses on two important texts from 1676 while also drawing on other sources: the dialogue \textit{Pacidius Philalethi}, which discusses motion, and the treatise \textit{De quadratura arithmetica circuli ellipseos et hyperbolae}, whose Prop. 6 provides a justification for the use of infinitesimals in geometry. Here, the continuity of the curves is tacitely presupposed. Later in his life, Leibniz subsumed continuity arguments under a general principle, the so-called law of continuity, which he applied to mathematics as well as to physics. One of its incarnations can be summarized as follows: When given quantities approach each other, then those resulting from them also do. Leibniz based one of his attempts to justify the differential calculus on this law. But since he only vaguely formulated and motivated the principle itself, the soundness of this attempt remains questionable (p.~153 ff.).
For the entire collection see [Zbl 1454.01002].
Reviewer: Charlotte Wahl (Hannover)Johann Bernoulli's first lecture from the first integral calculus textbook ever written: an annotated translationhttps://www.zbmath.org/1475.010152022-01-14T13:23:02.489162Z"Scarpello, Giovanni Mingari"https://www.zbmath.org/authors/?q=ai:mingari-scarpello.giovanni"Ritelli, Daniele"https://www.zbmath.org/authors/?q=ai:ritelli.danieleSummary: When Johann Bernoulli published his lectures on integrals in 1742, integral calculus had become very advanced since the time of their composition in 1692. Nevertheless, these lectures are of excellent clarity and simplicity even when the book deals with major problems of Mathematical Physics. Just to pique some interest, we offer a commented translation of the first lecture and some general information about the whole treatise. In his introductory lecture, Johann Bernoulli develops many examples of how the integration of polynomials could lead to the integration of radicals and then provides some relevant changes of variables. When studying the integration of
\[
\frac{a\mathrm{d} x}{\sqrt{2ax+x^2}} + \frac{ x\mathrm{d} x}{\sqrt{2ax+x^2}}
\]
the integrable combinations appear for the first time; several developments will stem from them in the future.A continuity principle equivalent to the monotone \(\Pi^0_1\) fan theoremhttps://www.zbmath.org/1475.031002022-01-14T13:23:02.489162Z"Kawai, Tatsuji"https://www.zbmath.org/authors/?q=ai:kawai.tatsujiThe author's strong continuity principle (SC) says that every pointwise continuous function from a complete separable metric space to a metric space is strongly continuous in the sense of [\textit{D.S.~Bridges}, Constructive functional analysis. London etc.: Pitman (1979; Zbl 0401.03027)]: that is, uniformly continuous near every compact image. Working in Bishop-style constructive reverse mathematics, the author shows that SC is tantamount to the fan theorem for monotone \(\Pi^0_1\) bars, and that the latter is equivalent to a variant of the fan theorem, called sc-FAN, related to but stronger than the principle c-FAN from [\textit{J. Berger}, Lect. Notes Comput. Sci. 3988, 35--39 (2006; Zbl 1145.03339)].
Reviewer: Peter M. Schuster (Verona)Generalized difference sets and autocorrelation integralshttps://www.zbmath.org/1475.050152022-01-14T13:23:02.489162Z"Kravitz, Noah"https://www.zbmath.org/authors/?q=ai:kravitz.noahSummary: \textit{J. Cilleruelo} et al. [Adv. Math. 225, No. 5, 2786--2807 (2010; Zbl 1293.11019)] established a surprising connection between the maximum possible size of a generalized Sidon set in the first \(N\) natural numbers and the optimal constant in an ``analogous'' problem concerning nonnegative-valued functions on \([0,1]\) with autoconvolution integral uniformly bounded above. Answering a recent question of \textit{R. C. Barnard} and \textit{S. Steinerberger} [J. Number Theory 207, 42--55 (2020; Zbl 1447.11008)], we prove the corresponding dual result about the minimum size of a so-called generalized difference set that covers the first \(N\) natural numbers and the optimal constant in an analogous problem concerning nonnegative-valued functions on \(\mathbb{R}\) with autocorrelation integral bounded below on \([0,1]\). These results show that the correspondence of Cilleruelo et al. [loc. cit.] is representative of a more general phenomenon relating discrete problems in additive combinatorics to questions in the continuous world.A note on M-convex functions on jump systemshttps://www.zbmath.org/1475.050242022-01-14T13:23:02.489162Z"Murota, Kazuo"https://www.zbmath.org/authors/?q=ai:murota.kazuoLet \(x,y\in \mathbb{Z}^{n}\). A vector \(s\in \mathbb{Z}^{n}\) is said to be an \((x,y)\)-increment if either \(s\) or \(-s\) is one of the \(n\) unit vectors and \(x\wedge y\leq x+s\leq x\vee y\).
Let \(f:\mathbb{Z}^{n}\rightarrow \mathbb{R\cup \{+\infty \}}\). One says that \(f\) is jump M-convex if for any \(x,y\in \mathrm{dom}~f\) and any \((x,y)\)-increment \(s\), there exists an \((x+s,y)\)-increment \(t\) such that \(x+s+t,~y-s-t\in \mathrm{dom}~f\) and \(f(x)+f(y)\geq f(x+s+t)+f(y-s-t)\). One says that \(f\) is jump M\(^{\sharp}\)-convex if for any \(x,y\in \mathrm{dom}~f\) and any \((x,y)\)-increment \(s\), one of the following two conditions holds:
(i) \(x+s,~y-s\in \mathrm{dom}~f\), and \(f(x)+f(y)\geq f(x+s)+f(y-s)\),
(ii) there exists an \((x+s,y)\)-increment \(t\) such that \(x+s+t,~y-s-t\in \mathrm{dom}~f\) and \(f(x)+f(y)\geq f(x+s+t)+f(y-s-t)\).
For \(x\in \mathbb{Z}^{n}\), define \(\pi (x)=0\) if the component sum of \(x\) is even and \(\pi (x)=1\) otherwise.
The main result states that \(f\) is jump M\(^{\sharp }\)-convex if and only if the function \(\widetilde{f}:\mathbb{Z}^{n+1}\rightarrow \mathbb{R\cup \{+\infty \}}\) defined by \(\widetilde{f}(x_{0},x)=f(x)\) if \(x_{0}=\pi (x)\) and \(\widetilde{f}(x_{0},x)=+\infty \) otherwise is jump M-convex.
Reviewer: Juan-Enrique Martínez-Legaz (Barcelona)Fractional calculus, zeta functions and Shannon entropyhttps://www.zbmath.org/1475.111512022-01-14T13:23:02.489162Z"Guariglia, Emanuel"https://www.zbmath.org/authors/?q=ai:guariglia.emanuelSummary: This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz \(\zeta\) function. All the results are based on the complex generalization of the Grünwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.Strictly real fundamental theorem of algebra using polynomial interlacinghttps://www.zbmath.org/1475.120012022-01-14T13:23:02.489162Z"Basu, Soham"https://www.zbmath.org/authors/?q=ai:basu.sohamThis publication presents a full proof of the fundamental theorem of algebra in the real case, using elementary concepts only. It is quite interesting for it is an ingenious combination of inequalities and some algebra; I found it in connection with the historical spirit of the Italian school -- while solving the 3rd and 4th polynomial in the XVI-th century. However the current consensus of what the field of real numbers is, after works of the XIX-th and XX-th centuries, is essential in this proof, simply since it is related to our actual understanding of infinite sets (use of the least upper bound property).
Reviewer: Gabriel Thomas (Grenoble)Constructing separable Arnold snakes of Morse polynomialshttps://www.zbmath.org/1475.141152022-01-14T13:23:02.489162Z"Sorea, Miruna-Ştefana"https://www.zbmath.org/authors/?q=ai:sorea.miruna-stefanaIn general, any snake can be associated with Morse polynomials in one variable, and the number of snakes is equal to the number of topologically nonequivalent Morse polynomials (see [\textit{V. I. Arnol'd}, Russ. Math. Surv. 47, No. 1, 1 (1992; Zbl 0791.05001); translation from Usp. Mat. Nauk 47, No. 1, 3--45 (1992)]). The author considers snakes associated with alternating permutations, given by the relative positions of critical values of a Morse polynomial (cf. [\textit{S. K. Lando}, Lectures on generating functions. Transl. from the Russian by the author. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1032.05001)]), and calls them Arnold snakes. Then using the notion of separable permutation (see [\textit{P. Bose} et al., Inf. Process. Lett. 65, No. 5, 277--283 (1998; Zbl 1338.68304); \textit{S. Kitaev}, Patterns in permutations and words. Berlin: Springer (2011; Zbl 1257.68007)]) and some other combinatorial objects and considerations, she shows how to construct explicitly polynomials in one variable with preassigned critical values configurations for a special class of Arnold snakes associated with separable permutations.
The author emphasizes also that the paper is based on the first chapter of her PhD thesis [The shapes of level curves of real polynomials near strict local minima. Université de Lille (2018), \url{https://hal.archives-ouvertes.fr/tel-01909028v1}], defended at Paul Painlevé Laboratory in Lille.
Reviewer: Aleksandr G. Aleksandrov (Moskva)Multivariable dynamic calculus on time scaleshttps://www.zbmath.org/1475.260012022-01-14T13:23:02.489162Z"Bohner, Martin"https://www.zbmath.org/authors/?q=ai:bohner.martin-j"Georgiev, Svetlin G."https://www.zbmath.org/authors/?q=ai:georgiev.svetlin-georgievThe present book provides a comprehensive treatise on the multivariable calculus on time scales. It is a very natural and nice sequel of the pair of monographs concerning the fundamental theory of the time scale calculus and dynamic equations [the first author and \textit{A. Peterson}, Dynamic equations on time scales. An introduction with applications. Basel: Birkhäuser (2001; Zbl 0978.39001); the first author (ed.) and \textit{A. Peterson} (ed.), Advances in dynamic equations on time scales. Boston, MA: Birkhäuser (2003; Zbl 1025.34001)]. Let us recall that a time scale \(\mathbb{T}\) is any closed subset of the real numbers \(\mathbb{R}\) and its development was motivated by the effort to study simultaneously the continuous (when \(\mathbb{T}=\mathbb{R}\)) and discrete (when \(\mathbb{T}=\mathbb{Z}\)) calculi and many cases ``in between''. The origin of this field goes back to 1988 and the dissertation of \textit{S. Hilger} [Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Würzburg: Univ. Würzburg (1988; Zbl 0695.34001)], see also [\textit{S. Hilger}, Result. Math. 18, No. 1--2, 18--56 (1990; Zbl 0722.39001)].
The book includes nine chapters:
\begin{itemize}
\item Time scales
\item Differential calculus of functions of one variable
\item Integral calculus of functions of one variable
\item Sequences and series of functions
\item Parameter-dependent integrals
\item Partial differentiation on time scales
\item Multiple integration on time scales
\item Line integrals
\item Surface integrals
\end{itemize}
In the first three chapters, the results of the single-variable time scale calculus are summarized and extended (convexity, extreme values, complete \(\Delta\)-differentiability). Although almost all results are taken from the monographs mentioned above, their presentation seems to be more compact, all proofs are included, and the theory is supplemented by numerous solved examples and exercises (with results).
In the next chapter, necessary and sufficient conditions of the uniform convergence of sequences and series of functions on time scales are investigated. This chapter is based on [\textit{L. Pang} and \textit{K. Wang}, Comput. Math. Appl. 62, No. 9, 3427--3437 (2011; Zbl 1236.41030)].
In the fifth chapter, several results on the ``normal'' and improper parameter-dependent integrals are given, i.e., for integrals of the type
\[
I(t_2)=\int_{a_1}^{b_1} f(t_1,t_2)\,\Delta_1\,t_1,\quad t_2\in[a_2,b_2],
\]
where \([a_1,b_1]\subset\mathbb{T}_1\) and \([a_2,b_2]\subset\mathbb{T}_2\) are two time scale intervals and the function \(f\) is defined on the rectangle \([a_1,b_1]\times[a_2,b_2]\). This chapter is based on [\textit{C. Zhang} and \textit{K. Wang}, Indian J. Pure Appl. Math. 45, No. 2, 139--164 (2014; Zbl 1318.26056)].
Finally, the last four chapters are focused on multivariable calculus. It starts with the multidimensional extension of the notion of the jump and backward jump operators and graininess function, which are subsequently used in the definition of the partial derivative as the number \(f_{t_i}^{\Delta_i}(t)\) (provided it exists) with the property that for any \(\epsilon_i>0\), there exists a neighborhood
\[
U_i=(t_i-\delta,t_i+\delta)\cap\mathbb{T}_i,
\]
for some \(\delta_i>0\), such that
\begin{align*}
\big|f(t_1,\dots,t_{i-1},\sigma_i(t_i),t_{i+1},\dots,t_n)-f(t_1,\dots,t_{i-1},s_i,t_{i+1},\dots,t_n) -f_{t_i}^{\Delta_i}(t)\,(\sigma_i(t_i)-s_i)\big|\leq \epsilon_i\left|\sigma_i(t_i)-s_i\right|,
\end{align*}
where \(f\) is a function on the \(n\)-dimensional time scale \(\mathbb{T}_1\times\mathbb{T}_2\times\dots\times\mathbb{T}_n\) and \(t\in\mathbb{T}_1\times\mathbb{T}_2\times\dots\mathbb{T}_{i-1}\times\mathbb{T}_{i}^\kappa\times\mathbb{T}_{i+1} \times\dots\times\mathbb{T}_n\). Furthermore, the differentiability and complete differentiability are introduced, the equality of mixed partial derivatives is discussed, the chain rule is derived, the directional derivative is defined, and implicit functions are studied. In the seventh chapter, we find the construction and several properties of the multiple Riemann \(\Delta\)-integral over rectangles and, more generally, over Jordan \(\Delta\)-measurable sets. In the eighth and ninth chapters, the attention is focused on a time scale extension of the line and surface integrals (including Green's formula). These chapters are based on the works of the first author and \textit{G. Sh. Guseinov}, see [Dyn. Syst. Appl. 13, No. 3--4, 351--379 (2004; Zbl 1090.26004); Dyn. Syst. Appl. 14, No. 3--4, 579--606 (2005; Zbl 1095.26006); J. Math. Anal. Appl. 326, No. 2, 1124--1141 (2007; Zbl 1118.26009); Dyn. Syst. Appl. 19, No. 3--4, 435--453 (2010; Zbl 1216.26018)].
Overall, the book is well-written and well-organized. I recommend this book to everybody who is interested in time scale calculus. The total number of 275 solved examples and 239 exercises including advanced practical problems make the book also very suitable for beginners such as, e.g., undergraduate students.
The book is dedicated to the memory of Professor Gusein Shirin Guseinov (1951--2015).
Reviewer's remark: The reader should be careful with the concept of convexity as given in Definition~2.94. In fact, the inequality should be required only for all \(t_1,t_2\in\mathbb{T}\) and \(\lambda\in[0,1]\) such that \(\lambda\,t_1+(1-\lambda)\,t_2\in\mathbb{T}\), see also [\textit{C. Dinu}, An. Univ. Craiova, Ser. Mat. Inf. 35, 87--96 (2008; Zbl 1199.26073)].
Reviewer: Petr Zemánek (Brno)Calculus set free. Infinitesimals to the rescuehttps://www.zbmath.org/1475.260022022-01-14T13:23:02.489162Z"Dawson, C. Bryan"https://www.zbmath.org/authors/?q=ai:dawson.c-bryanPublisher's description: Calculus Set Free: Infinitesimals to the Rescue is a single-variable calculus textbook that incorporates the use of infinitesimal methods. The procedures used throughout make many of the calculations simpler and the concepts clearer for undergraduate students, heightening success and easing a significant burden of entry into STEM disciplines.
This text features a student-friendly exposition with ample marginal notes, examples, illustrations, and more. The exercises include a wide range of difficulty levels, stretching from very simple ``rapid response'' questions to the occasional exercise meant to test knowledge. While some exercises require the use of technology to work through, none are dependent on any specific software. The answers to odd-numbered exercises in the back of the book include both simplified and non-simplified answers, hints, or alternative answers.
Throughout the text, notes in the margins include comments meant to supplement understanding, sometimes including line-by-line commentary for worked examples. Without sacrificing academic rigor, Calculus Set Free offers an engaging style that helps students to solidify their understanding on difficult theoretical calculus.Explorations in analysis, topology, and dynamics. An introduction to abstract mathematicshttps://www.zbmath.org/1475.260032022-01-14T13:23:02.489162Z"Uribe A., Alejandro"https://www.zbmath.org/authors/?q=ai:uribe-a.alejandro"Visscher, Daniel A."https://www.zbmath.org/authors/?q=ai:visscher.daniel-aThis book explores the theoretical fundamental concepts of calculus, together with some additional topics, from a novel and engaging perspective. It guides the reader through the basic notions of analysis presenting them by means of stimulating exercises and investigations which make the reader think, explore, formulate conjectures, ask questions and find answers. This idea is so well organized and carried out that the development of the different topics becomes totally natural. The authors do not give complete proofs of the results, but give instead sketchs of proofs which have to be completed by the reader, who is also asked sometimes to give or complete definitions. As it is said by the authors in the preface ``One of the aims of the book is to introduce students to rigorous mathematical language and argumentation'', and I think that this book does this job wonderfully. The whole idea of the book, which is designed to be implemented in a first course of analysis with the guidance of a professor or tutor, is very interesting and is likely to make the students get involved to a great extent in their learning process.
The plan of the book is put into action from the very beginning, since in the first section of the first chapter a seemingly elementary warm-up review of rational numbers challenges the reader to write some definitions and check that they express what they ought to. The authors then give the axioms of an ordered field, the axiom of completeness (in the form of nested intervals) and the Archimedean axiom. In the subsequent sections of the first chapter of the book, the authors introduce sequences of real numbers and the concept of convergence of those sequences, where after a suitable guidance, the reader is once again expected to come up with the definition -- which, due to its major importance, is included in one of the appendices of the book for comparison purposes. The first chapter continues with the study of series of real numbers and some of its properties, and the introduction of the notion of subsequences and the Bolzano-Weierstrass theorem. The last section of the first chapter deals with the study of sequences in \(\mathbb{R}^n\) and the definition of convergence of those type of sequences, together with some related results.
In the second chapter, the authors address several topological notions in \(\mathbb{R}^n\), which are carried out in a not so standard way, defining first the closure of a subset of \(\mathbb{R}^n\) by means of convergent sequences, defining then closed sets, and finally defining open sets as the complements of closed sets. Although this might seem a bit awkward at first glance, this approach is fully justified since sequences play a major role in this book. The authors then define continuous functions using convergent sequences once again, while the well-known \(\varepsilon\)-\(\delta\) definition is given later as an additional material. In this chapter, the reader is also guided through the notion of compact sets in \(\mathbb{R}^n\), the intermediate value theorem, the definition of continuous maps with codomain \(\mathbb{R}^m\), the notion of homeomorphism and the Brouwer fixed point theorem.
In the third chapter, the authors introduce the notion of the derivative of a function and immediately guide the reader towards the concept of linearization of a map, that is, the Taylor polynomial of first order. Then, classical properties of the derivatives are given together with Rolle's theorem, the mean value theorem -- also known as Lagrange's theorem -- and the generalized mean value theorem -- also known as Cauchy's mean value theorem. After that, the theory of Taylor polynomials is developed, generalizing the previously given linearization idea so as to obtain a better approximation of the given map. The importance of estimating the error of the approximation is addressed, and finally Taylor series are explained.
In the fourth chapter, the Riemann integral is introduced in a standard way by means of Riemann sums and approximation of areas. After giving some classical properties of Riemann sums and definite integrals, the authors present the second part of the fundamental theorem of calculus and the mean value theorem for integrals, from which they then derive the first part of the fundamental theorem of calculus. Finally, the substitution and the integration by parts methods are briefly addressed.
The fifth chapter deals with discrete dynamical systems, that is, the study of the iteration of a function from a set \(X\) to itself. The notions of attracting and repelling fixed points are given, and several examples of discrete dynamical systems are discussed and studied in great detail.
The last chapter of the book deals with representation of real numbers in base 10, in base 2 and by continued fractions, by means of iterating suitable algorithms, and thus these are regarded as discrete dynamical systems.
Finally, several brief appendices complete the book. Appendix A deals with writing mathematics and proofs and propositional logic, Appendix B deals with sets and functions between sets, Appendix C contains some graphs of maps which are meant to help with the corresponding iteration exercises given in Chapter 5 and Appendix D gives hints to selected problems.
To sum up, this book develops a very interesting non-standard approach for a first course of analysis, introducing the students to mathematical thinking, writing and exploration.
Reviewer: Miguel Ottina (Mendoza)Oscillation of functions in the Hölder classhttps://www.zbmath.org/1475.260042022-01-14T13:23:02.489162Z"Mozolyako, Pavel"https://www.zbmath.org/authors/?q=ai:mozolyako.pavel-a"Nicolau, Artur"https://www.zbmath.org/authors/?q=ai:nicolau.arturSummary: We study the size of the set of points where the \(\alpha \)-divided difference of a function in the Hölder class \(\Lambda_\alpha\) is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which can be stated in the language of dyadic martingales. Our main technical result in this setting is a sharp estimate of the Hausdorff measure of the set of points where a dyadic martingale with bounded increments has maximal growth.Caputo fractional derivative Hadamard inequalities for strongly \(m\)-convex functionshttps://www.zbmath.org/1475.260052022-01-14T13:23:02.489162Z"Feng, Xue"https://www.zbmath.org/authors/?q=ai:feng.xue"Feng, Baolin"https://www.zbmath.org/authors/?q=ai:feng.baolin"Farid, Ghulam"https://www.zbmath.org/authors/?q=ai:farid.ghulam"Bibi, Sidra"https://www.zbmath.org/authors/?q=ai:bibi.sidra"Xiaoyan, Qi"https://www.zbmath.org/authors/?q=ai:xiaoyan.qi"Wu, Ze"https://www.zbmath.org/authors/?q=ai:wu.zeThe authors derive and prove two versions of Hadamard-type inequalities using the concept of Caputo fractional derivatives and strongly \(m\)-convex functions. Several consequences of their results are pointed out. Furthermore, error bounds of fractional Hadamard inequalities are also derived, proved and discussed.
Reviewer: James Adedayo Oguntuase (Abeokuta)Some new Hermite-Hadamard type inequalities via \(k\)-fractional integrals pertaining differentiable generalized relative semi-\(\mathbf{m}\)-\((r; h_1, h_2)\)-preinvex mappings and their applicationshttps://www.zbmath.org/1475.260062022-01-14T13:23:02.489162Z"Kashuri, Artion"https://www.zbmath.org/authors/?q=ai:kashuri.artion"Awan, Muhammad Uzair"https://www.zbmath.org/authors/?q=ai:awan.muhammad-uzair"Noor, Muhammad Aslam"https://www.zbmath.org/authors/?q=ai:noor.muhammad-aslam"Mihai, Marcela V."https://www.zbmath.org/authors/?q=ai:mihai.marcela-v"Liko, Rozana"https://www.zbmath.org/authors/?q=ai:liko.rozanaIn this paper, the authors define a generalized relative semi-\(\mathbf{m}\)-\((r,h_1,h_2)\)-preinvex mapping and prove several inequalities of Hermite-Hadamard type for this class of functions.
Let \(K \subseteq \mathbb{R}\) be an open \(\mathbf{m}\)-invex set with respect to the mapping \(\eta: K \times K \rightarrow\mathbb{R}\). Further, let \( I\) be a real interval. Suppose \(h_1,h_2: [0,1] \rightarrow [0,\infty \rangle\), \(\varphi :I \rightarrow K\) are continuous functions and \(\mathbf{m} : [0,1] \rightarrow \langle 0,1]\). A mapping \(f:K\rightarrow \langle 0, \infty \rangle\) is said to be a generalized relative semi-\(\mathbf{ m}\)-\((r,h_1,h_2)\)-preinvex mapping if \[ f(\mathbf{m}(t)\varphi(x) + \xi \eta ( \varphi(y), \mathbf{m}(t)\varphi(x))) \leq \Big[ \mathbf{m}(t) h_1(\xi) f^r(x) + h_2(\xi) f^r (y)\Big]^{\frac 1r} \] holds for all \(x,y\in I\) and \(t,\xi \in [0,1]\), where \(r\not= 0 \).
If \(r=1\), then the above definition includes several classes of functions such as: generalized relative
semi-\((\mathbf{m}, P)\)-preinvex mappings, generalized relative semi-\((\mathbf{m}, s)\)-Breckner-preinvex mappings, generalized relative semi-\((\mathbf{m}, s)\)-Godunova-Levin-Dragomir-preinvex mappings, generalized relative semi-\((\mathbf{m}, h)\)-preinvex mappings, generalized relative semi-\((\mathbf{m}, tgs)\)-preinvex mappings, and generalized relative semi-\((\mathbf{m}, MT)\)-preinvex mappings.
Using the Hölder and the Minkowski inequalities together with the definition of the generalized relative
semi-\(\mathbf{m}\)-\((r,h_1,h_2)\)-preinvex functions the following Hermite-Hadamard-type inequality is proved.
Theorem. Let \(\alpha, k >0\) and \(0< r \leq 1\). Suppose \(h_1,h_2: [0,1] \rightarrow [0,\infty \rangle\), \(\varphi :I \rightarrow K\) are continuous functions and \(\mathbf{m} : [0,1] \rightarrow \langle 0,1]\). Let \(K=[\mathbf{m}(t) \varphi(a), \mathbf{m}(t) \varphi(a) + \eta(\varphi(b), \mathbf{m}(t)\varphi(a))] \subseteq \mathbb{R}\) be an \(\mathbf{m}\)-invex subset with respect to \(\eta: K \times K \rightarrow \mathbb{R}\) and let \(\eta(\varphi(b), \mathbf{m}(t)\varphi(a)) >0\) for all \(t\in [0,1]\). Assume that \(f : K \rightarrow \langle 0, \infty \rangle\) be a differentiable mapping on \(K^0\).
If \((f')^q\) is a generalized relative semi-\(\mathbf{m}\)-\((r,h_1,h_2)\)-preinvex mapping on \(K\), \(q>1\), \(\displaystyle \frac 1p + \frac 1q =1\), then for any \(\lambda, \mu \in [0,1]\) and \(r_1 \geq 0\) the following inequality for \(k\)-fractional integrals holds: \[ | I^{\alpha, k}_{f,\eta,\varphi,\mathbf{m}} (\lambda,\mu; r_1,a,b)| \leq \left( \frac{\eta(\varphi(b), \mathbf{m}(t)\varphi(a))}{r_1+1}\right)^{\frac{\alpha}{k}+1}\] \[ \times \left\{ \rho^{\frac 1p} \left(\frac{\alpha}{k}, \lambda, p \right)\Big[ (f'(a))^{rq} I_1^r(h_1(\xi);\mathbf{m}(\xi),r,r_1) + (f'(b))^{rq} I_2^r(h_2(\xi);r,r_1)\Big]^{\frac{1}{rq}} \right. \] \[ + \left. \rho^{\frac 1p} \left(\frac{\alpha}{k}, \mu, p \right)\Big[ (f'(a))^{rq} \overline{I_1}^r(h_1(\xi);\mathbf{m}(\xi),r,r_1) + (f'(b))^{rq} \overline{I_2}^r(h_2(\xi);r,r_1)\Big]^{\frac{1}{rq}} \right\} , \] where \[ I_1^r(h_1(\xi);\mathbf{m}(\xi),r,r_1) := \int_0^1 \mathbf{m}^{\frac 1r}(\xi) h_1^{\frac 1r}\left( \frac{r_1+\xi}{r_1+1}\right) d\xi, \quad I_2^r(h_2(\xi);r,r_1) := \int_0^1 h_2^{\frac 1r}\left( \frac{r_1+\xi}{r_1+1}\right) d\xi, \] \[ \overline{I_1}^r(h_1(\xi);\mathbf{m}(\xi),r,r_1) := \int_0^1 \mathbf{m}^{\frac 1r}(\xi) h_1^{\frac 1r}\left( \frac{1-\xi}{r_1+1}\right) d\xi, \quad \overline{I_2}^r(h_2(\xi);r,r_1) := \int_0^1 h_2^{\frac 1r}\left( \frac{1-\xi}{r_1+1}\right) d\xi, \] \[ \rho \left(\frac{\alpha}{k}, \xi, p \right) := \left\{ \begin{array}{ll} \frac{1}{p\alpha +1}, & \xi=0, \\
\frac{\xi^{p+\frac{1}{\alpha}}}{\alpha} \beta(\frac{1}{\alpha}, p+1) + \frac{(1-\xi)^{p+1}}{\alpha(p+1)} \phantom{.}_2F_1(1-\frac{1}{\alpha}, 1;p+2,1-\xi) & 0<\xi <1,\\
\frac{1}{\alpha} \beta(p+1, \frac{1}{\alpha}) & \xi =1, \end{array} \right. \] and \[ I^{\alpha, k}_{f,\eta,\varphi,\mathbf{m}} (\lambda,\mu; r,a,b)\] \[ := \left( \frac{\eta(\varphi(b), \mathbf{m}(t)\varphi(a))}{r+1}\right)^{\frac{\alpha}{k}+1} \left( \int_0^1 (x^{\frac{\alpha}{k}}-\lambda) f'\left( \mathbf{m}(t) \varphi(a) + \frac{r+x}{r+1} \eta(\varphi(b), \mathbf{m}(t)\varphi(a))\right) dx \right.\] \[ + \left. \int_0^1 (\mu - x^{\frac{\alpha}{k}}) f' \left( \mathbf{m}(t) \varphi(a) + \frac{1-x}{r+1} \eta(\varphi(b), \mathbf{m}(t)\varphi(a))\right) dx \right).\]
By applying the power mean inequality a similar estimation for \(|I^{\alpha, k}_{f,\eta,\varphi,\mathbf{m}} (\lambda,\mu; r,a,b)|\) is given.
Reviewer: Sanja Varošanec (Zagreb)On the exact pairs of classes for the Stieltjes integralhttps://www.zbmath.org/1475.260072022-01-14T13:23:02.489162Z"Derr, V."https://www.zbmath.org/authors/?q=ai:derr.vasilii-yakovlevichSummary: The exact pairs of classes for the Riemann-Stieltjes integral are studied. More precisely, a pair of classes (the integrand and the integrator) is said to be exact if the Riemann-Stieltjes integral exists, and one of the classes may be extended only at the cost of another class contraction. In particular, it is shown that the classes of \(\sigma\)-continuous functions and functions of bounded variation form an exact pair.A direct computation of a certain family of integralshttps://www.zbmath.org/1475.260082022-01-14T13:23:02.489162Z"Fornari, Lorenzo"https://www.zbmath.org/authors/?q=ai:fornari.lorenzo"Laeng, Enrico"https://www.zbmath.org/authors/?q=ai:laeng.enrico"Pata, Vittorino"https://www.zbmath.org/authors/?q=ai:pata.vittorinoSummary: The authors propose a rather elementary method to compute a family of integrals on the half line, involving positive powers of \(\sin x\) and negative powers of \(x\), depending on the integer parameters \(n\geq q\geq 1\).BV continuity for the uncentered Hardy-Littlewood maximal operatorhttps://www.zbmath.org/1475.260092022-01-14T13:23:02.489162Z"González-Riquelme, Cristian"https://www.zbmath.org/authors/?q=ai:gonzalez-riquelme.cristian"Kosz, Dariusz"https://www.zbmath.org/authors/?q=ai:kosz.dariuszThe questions of boundedness and continuity of the classical Hardy-Littlewood operator in the Sobolev space \(W^{1,p}(\mathbb{R}^d)\) for \(1<p<\infty\) have been solved by \textit{J. Kinnunen} [Isr. J. Math. 100, 117--124 (1997; Zbl 0882.43003)] and \textit{H. Luiro} [Proc. Am. Math. Soc. 135, No. 1, 243--251 (2007; Zbl 1136.42018)], respectively. The much more subtle case of \(p=1\) has been the subject of the so called endpoint continuity programme proposed in [\textit{E. Carneiro} et al., J. Funct. Anal. 273, No. 10, 3262--3294 (2017; Zbl 1402.42024)], where some partial answers were given. The aim of this paper is to add to the programme by proving the continuity of the noncentered maximal operator \(\widetilde M(f)=\sup\{|B|^{-1}\int_B |f(x)|dx: B\subset\mathbb{R}^d\ \text{is a ball},\ x\in B\}\) as mapping from \(BV(\mathbb{R})\) to \(BV(\mathbb{R})\). The main tools in the proof are the variation convergence and the pointwise derivative analysis.
Reviewer: Jiří Rákosník (Praha)On Opial-type inequalities via a new generalized integral operatorhttps://www.zbmath.org/1475.260102022-01-14T13:23:02.489162Z"Farid, Ghulam"https://www.zbmath.org/authors/?q=ai:farid.ghulam"Mehboob, Yasir"https://www.zbmath.org/authors/?q=ai:mehboob.yasirSummary: Opial inequality and its consequences are useful in establishing existence and uniqueness of solutions of initial and boundary value problems for differential and difference equations. In this paper we analyze Opial-type inequalities for convex functions. We have studied different versions of these inequalities for a generalized integral operator. Further difference of Opial-type inequalities are utilized to obtain generalized mean value theorems, which further produce various interesting derivations for fractional and conformable integral operators.Valuations on log-concave functionshttps://www.zbmath.org/1475.260112022-01-14T13:23:02.489162Z"Mussnig, Fabian"https://www.zbmath.org/authors/?q=ai:mussnig.fabianThe authors prove a new classification of \(\mathrm{SL}(n)\) and translation covariant Minkowski valuations on logconcave functions. To reach it, characterizations of the moment vector and of the level set body of a log-concave function are provided. Last but not least analogs of the Euler characteristic and volume are characterized as \(\mathrm{SL}(n)\) and translation invariant valuations on log-concave functions as well.
Reviewer: Sorin-Mihai Grad (Paris)Inequalities of Landau-Adamar type for vector functionshttps://www.zbmath.org/1475.260122022-01-14T13:23:02.489162Z"Perov, Anatoly I."https://www.zbmath.org/authors/?q=ai:perov.anatolij-iSummary: Known classical inequalities of Landau-Adamar type for scalari functions transfer on vector functions with meaning in Banach's space.On some new quantum midpoint-type inequalities for twice quantum differentiable convex functionshttps://www.zbmath.org/1475.260132022-01-14T13:23:02.489162Z"Ali, Muhammad Aamir"https://www.zbmath.org/authors/?q=ai:ali.muhammad-aamir"Alp, Necmettin"https://www.zbmath.org/authors/?q=ai:alp.necmettin"Budak, Hüseyin"https://www.zbmath.org/authors/?q=ai:budak.huseyin"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yuming"Zhang, Zhiyue"https://www.zbmath.org/authors/?q=ai:zhang.zhiyueSummary: The present paper aims to find some new midpoint-type inequalities for twice quantum differentiable convex functions. The consequences derived in this paper are unification and generalization of the comparable consequences in the literature on midpoint inequalities.Quantum Ostrowski-type integral inequalities for functions of two variableshttps://www.zbmath.org/1475.260142022-01-14T13:23:02.489162Z"Budak, Hüseyin"https://www.zbmath.org/authors/?q=ai:budak.huseyin"Ali, Muhammad Aamir"https://www.zbmath.org/authors/?q=ai:ali.muhammad-aamir"Tunç, Tuba"https://www.zbmath.org/authors/?q=ai:tunc.tubaSummary: In this study, we established some new inequalities of Ostrowski type for the functions of two variables by using the concept of newly defined double quantum integrals. We also revealed that the results presented in this paper are the consolidation and generalization of some existing results on the literature of Ostrowski inequalities.More new results on integral inequalities for generalized \(\mathcal{K}\)-fractional conformable integral operatorshttps://www.zbmath.org/1475.260152022-01-14T13:23:02.489162Z"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yuming"Rashid, Saima"https://www.zbmath.org/authors/?q=ai:rashid.saima"Jarad, Fahd"https://www.zbmath.org/authors/?q=ai:jarad.fahd"Aslam Noor, Muhammad"https://www.zbmath.org/authors/?q=ai:noor.muhammad-aslam"Kalsoom, Humaira"https://www.zbmath.org/authors/?q=ai:kalsoom.humairaSummary: This paper aims to investigate the several generalizations by newly proposed generalized \( \mathcal{K} \)-fractional conformable integral operator. Based on these novel ideas, we derived a novel framework to study for Čebyšev and Pólya-Szegő type inequalities by generalized \(\mathcal{K}\)-fractional conformable integral operator. Several special cases are apprehended in the light of generalized fractional conformable integral. This novel strategy captures several existing results in the relative literature. We also aim at showing important connections of the results here with those including Riemann-Liouville fractional integral operator.Classical inequalities for \((p, q)\)-calculus on finite intervalshttps://www.zbmath.org/1475.260162022-01-14T13:23:02.489162Z"Jain, Pankaj"https://www.zbmath.org/authors/?q=ai:jain.pankaj"Manglik, Rohit"https://www.zbmath.org/authors/?q=ai:manglik.rohitSummary: In this paper, certain classical inequalities, namely, trapezoidal inequality (first as well as second order), generalized weighted Hölder's inequality, Minkowski's inequality and Grüss type inequalities have been investigated in the framework of \((p, q)\)-calculus. These inequalities extend the corresponding known inequalities in \(q\)-calculus. Moreover, in the case of trapezoidal inequality, we improve upon the constant as well. To prove \((p, q)\)-Grüss inequalities, we first derive \((p, q)\)-Andreief's identity which, in particular, contains \((p, q)\)-Korkine identity.Kolmogorov-type inequalities for the norms of fractional derivatives of functions defined on the positive half linehttps://www.zbmath.org/1475.260172022-01-14T13:23:02.489162Z"Kozynenko, O."https://www.zbmath.org/authors/?q=ai:kozynenko.oleksandr|kozynenko.o-v"Skorokhodov, D."https://www.zbmath.org/authors/?q=ai:skorokhodov.dmytro-s|skorokhodov.dmitriyIn this paper, the authors derive and prove new Kolmogorov-type sharp inequalities to estimate the norm of the Marchaud fractional derivative \(\| D^{k}_{f} \|_{\infty}\) of a function \(f\) defined on the positive half line in terms of \(\| f \|_{p}, 1 < p < \infty\), and \(\| f^{\prime\prime}\|_1\). In particular, the following result
\[
\| D^{k}_{f} \|_{L_{q}(G)} \leq K\| f \|^{1-\lambda}_{L_{p}(G)}\| f^{(r)}\|^{\lambda}_{L_{p}(G)}, \quad \lambda=\frac{k-1/q+1/p}{r-1/s +1/p} \tag{\(*\)}
\]
where \(G\) be either the real axis \(\mathbb{R}\) or the semi axis \(\mathbb{R_{+}} = [0,+1)\), \(k \in \mathbb{R}_{+} \backslash \mathbb{Z}_{+}\), \(D^k\) is the operator of fractional differentiation of order \(k\), \(f\in L^{p}_{p,s}(G)\) and \(K>0\) is a constant independent of \(f\). A sharp new inequality of the form \((*)\) is derived and proved for the case where \(G= \mathbb{R}_{+}\), \(1<p<\infty\), \(q= \infty\), \(s=1\), \(r=2\) and \(k \in (0,1)\) in Theorem 1 and Corollary 1 in the main result. Finally, the Stechkin problem of the best approximation of the operator \(D^{k}_{-}\) by linear bounded operators and the problem of the best possible recovery of the operator \(D^{k}_{-}\) on a class of elements given with errors are also established.
Reviewer: James Adedayo Oguntuase (Abeokuta)On post quantum integral inequalitieshttps://www.zbmath.org/1475.260182022-01-14T13:23:02.489162Z"Awan, Muhammad Uzair"https://www.zbmath.org/authors/?q=ai:awan.muhammad-uzair"Talib, Sadia"https://www.zbmath.org/authors/?q=ai:talib.sadia"Noor, Muhammad Aslam"https://www.zbmath.org/authors/?q=ai:noor.muhammad-aslam"Noor, Khalida Inayat"https://www.zbmath.org/authors/?q=ai:noor.khalida-inayat"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yumingThe theory of quantum calculus has attracted the attention of many authors in recent years due to its many applications in various branches of pure and applied sciences. Several inequalities have been provided in the literature using quantum derivatives and integrals by many authors. Recently, the concept of post quantum derivatives and integrals, which are generalizations of the quantum derivatives and integrals, were introduced in the literature.
In this paper, the authors establish some inequalities of Hermite-Hadamard type using the post quantum calculus for functions whose post quantum derivatives in absolute value to certain powers are h-preinvex. The results thus extend some existing results in the literature.
The results in this paper are well established and quite easy to follow. I believe it will inspire experts in this field to conduct further research on the topic.
Reviewer: Seth Kermausuor (Montgomery)Some Hardy-type integral inequalities involving functions of two independent variableshttps://www.zbmath.org/1475.260192022-01-14T13:23:02.489162Z"Benaissa, Bouharket"https://www.zbmath.org/authors/?q=ai:benaissa.bouharket"Sarikaya, Mehmet Zeki"https://www.zbmath.org/authors/?q=ai:sarikaya.mehmet-zekiSummary: In this paper, we give some new generalizations to the Hardy-type integral inequalities for functions of two variables by using weighted mean operators \(S_1:=S_1^wf\) and \(S_2:=S_2^wf\) defined by
\[
S_1(x,y)=\frac{1}{W(x)W(y)}\int_{\frac{x}{2}}^x\int_{\frac{y }{2}}^yw(t)w(s)f(t,s)\,dsdt,
\]
and
\[
S_2(x,y)= \int_{\frac{x}{2}}^x\int_{\frac{y}{2}}^y\frac{ w(t)w(s)}{W(t)W(s)}f(t,s)\,dsdt,
\]
with
\[
W(z)=\int_0^zw(r)\,dr\quad \text{for }z\in (0,+\infty ),
\]
where \(w\) is a weight function.Some the weighted generalizations the integral inequalities for convex mappingshttps://www.zbmath.org/1475.260202022-01-14T13:23:02.489162Z"Erden, Samet"https://www.zbmath.org/authors/?q=ai:erden.samet"Sarikaya, M. Zeki"https://www.zbmath.org/authors/?q=ai:sarikaya.mehmet-zekiSummary: We establish an important integral identity and new Hermite-Hadamard-Fejér type integral inequalities. Then, it is extended some estimates of the right hand and left hand side of a Hermite-Hadamard-Fejér type inequality for functions whose first derivatives absolute values are convex. In addition, new Hermite-Hadamard-type inequalities involving fractional integral are given.The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applicationshttps://www.zbmath.org/1475.260212022-01-14T13:23:02.489162Z"He, Bing"https://www.zbmath.org/authors/?q=ai:he.bing.1"Hong, Yong"https://www.zbmath.org/authors/?q=ai:hong.yong"Chen, Qiang"https://www.zbmath.org/authors/?q=ai:chen.qiang.1Summary: In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel \(G(n^{\lambda_{1}}\|x\|_{m,\rho}^{\lambda_{2}})\) (\(\lambda_1, \lambda_2 >0\)) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.Improvement of the Hardy inequality involving \(k\)-fractional calculushttps://www.zbmath.org/1475.260222022-01-14T13:23:02.489162Z"Iqbal, Sajid"https://www.zbmath.org/authors/?q=ai:iqbal.sajid"Samraiz, Muhammad"https://www.zbmath.org/authors/?q=ai:samraiz.muhammadSummary: The major idea of this paper is to establish some new improvements of the Hardy inequality by using \(k\)-fractional integral of Riemann-type, Caputo \(k\)-fractional derivative, Hilfer \(k\)-fractional derivative and Riemann-Liouville \((k,r)\)-fractional integral. We discuss the \(\log \)-convexity of the related linear functionals. We also deduce some known results from our general results.New \((p, q)\)-estimates for different types of integral inequalities via \((\alpha, m)\)-convex mappingshttps://www.zbmath.org/1475.260232022-01-14T13:23:02.489162Z"Kalsoom, Humaira"https://www.zbmath.org/authors/?q=ai:kalsoom.humaira"Latif, Muhammad Amer"https://www.zbmath.org/authors/?q=ai:latif.muhammad-amer"Rashid, Saima"https://www.zbmath.org/authors/?q=ai:rashid.saima"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-i"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yumingSummary: In the article, we present a new \((p,q)\)-integral identity for the first-order (p,q)-differentiable functions and establish several new \((p,q)\)-quantum error estimations for various integral inequalities via \((\alpha, m)\)-convexity. We also compare our results with the previously known results and provide two examples to show the superiority of our obtained results.Hadamard type inequalities via fractional calculus in the space of exp-convex functions and applicationshttps://www.zbmath.org/1475.260242022-01-14T13:23:02.489162Z"Ma, Li"https://www.zbmath.org/authors/?q=ai:ma.li|ma.li.1"Yang, Guangzhengao"https://www.zbmath.org/authors/?q=ai:yang.guangzhengaoConvex functions play very important and significant roles in mathematical analysis and in particular in the theory of inequalities. Fractional integral inequalities are important in studying the existence, uniqueness, stability, and other properties of fractional differential equations. In recent years, many authors have studied and extended many classical inequalities to fractional calculus. Several applications of these inequalities are provided in the literature. Convex functions play very important and significant roles in mathematical analysis and in particular in the theory of inequalities. Several generalizations of convex functions and inequalities associated with these new classes of convex functions have been provided in the literature.
In this paper, the authors study some properties of the class of exp-convex functions. Some fractional integral inequalities of Hermite-Hadamard type associated with exp-convex functions are also established.
The results in this paper are quite interesting and significant in my opinion.
Reviewer: Seth Kermausuor (Montgomery)More on operator Bellman inequalityhttps://www.zbmath.org/1475.260252022-01-14T13:23:02.489162Z"Mirzapour, F."https://www.zbmath.org/authors/?q=ai:mirzapour.farzollah"Morassaei, A."https://www.zbmath.org/authors/?q=ai:morassaei.ali"Moslehian, M. S."https://www.zbmath.org/authors/?q=ai:moslehian.mohammad-salSummary: We present a Bellman inequality involving operator means for operators acting on a Hilbert space. We also give some Bellman inequalities concerning sesquilinear forms. Finally, we refine the Jensen's operator inequality and use it for obtaining a refinement of the Bellman operator inequality.Quantum integral inequalities with respect to Raina's function via coordinated generalized \(\Psi\)-convex functions with applicationshttps://www.zbmath.org/1475.260262022-01-14T13:23:02.489162Z"Rashid, Saima"https://www.zbmath.org/authors/?q=ai:rashid.saima"Butt, Saad Ihsan"https://www.zbmath.org/authors/?q=ai:butt.saad-ihsan"Kanwal, Shazia"https://www.zbmath.org/authors/?q=ai:kanwal.shazia"Ahmad, Hijaz"https://www.zbmath.org/authors/?q=ai:ahmad.hijaz"Wang, Miao-Kun"https://www.zbmath.org/authors/?q=ai:wang.miaokunThe Hermite-Hadamard inequality in its original form provides an estimate of the mean value of an integral of a convex function. The inequality has been generalized in numerous ways, and this paper gives us an additional generalization. It combines the extension of a classical inequality in the case of the two-variable inequality with extensions within the framework of quantum integrals. The authors present some applications of their results concerning Raina's function, the hypergeometric function, and the Mittag-Leffler function.
Reviewer: Hrvoje Šikić (Zagreb)Discrete fractional Bihari inequality and uniqueness theorem of solutions of nabla fractional difference equations with non-Lipschitz nonlinearitieshttps://www.zbmath.org/1475.260272022-01-14T13:23:02.489162Z"Wang, Mei"https://www.zbmath.org/authors/?q=ai:wang.mei"Jia, Baoguo"https://www.zbmath.org/authors/?q=ai:jia.baoguo"Chen, Churong"https://www.zbmath.org/authors/?q=ai:chen.churong"Zhu, Xiaojuan"https://www.zbmath.org/authors/?q=ai:zhu.xiaojuan"Du, Feifei"https://www.zbmath.org/authors/?q=ai:du.feifeiSummary: In this paper, the discrete fractional Bihari inequality is developed. Based on this inequality, uniqueness theorem of solutions of fractional difference equations with non-Lipschitz nonlinearities is derived. In addition, the effectiveness of proposed results is illustrated by a nonlinear numerical example.On a new generalization of some Hilbert-type inequalitieshttps://www.zbmath.org/1475.260282022-01-14T13:23:02.489162Z"You, Minghui"https://www.zbmath.org/authors/?q=ai:you.minghui"Song, Wei"https://www.zbmath.org/authors/?q=ai:song.wei"Wang, Xiaoyu"https://www.zbmath.org/authors/?q=ai:wang.xiaoyuSummary: In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.On the intuitionistic fuzzy representations of rough real functionshttps://www.zbmath.org/1475.260292022-01-14T13:23:02.489162Z"Csajbók, Zoltán Ernő"https://www.zbmath.org/authors/?q=ai:csajbok.zoltan-ernoSummary: Studying rough calculus was initiated by Z. Pawlak in his many papers. He defined the concept of rough real functions and investigated their different properties. In this paper, first, two different representations of rough real functions will be presented. Then, a possible bridge between rough real functions, fuzzy sets, and intuitionistic fuzzy sets will be proposed. Some elementary properties of these connections will be investigated. In consequence of the proposed connections, intuitionistic fuzzy calculus may be applied to the rough calculus.
For the entire collection see [Zbl 1470.68020].A double inequality for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic meanhttps://www.zbmath.org/1475.260302022-01-14T13:23:02.489162Z"Jiang, Wei-Dong"https://www.zbmath.org/authors/?q=ai:jiang.weidong"Qi, Feng"https://www.zbmath.org/authors/?q=ai:qi.fengSummary: We find the greatest value and the least value μ such that the double inequality
\begin{align*}
C(\lambda a+(1-\lambda)b,\, \lambda b+(1-\lambda)a) & <\alpha A(a,b)+(1-\alpha)T(a,b)\\
& <C(\mu a+(1-\mu)b,\, \mu b+(1-\mu)a)
\end{align*}
holds for all \(\alpha\in (0,1)\) and \(a,b>0\) with \(a\neq b\), where \(C(a,b)\), \(A(a,b)\), and \(T(a,b)\) denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers \(a\) and \(b\).Comparison of arithmetic, geometric, and harmonic meanshttps://www.zbmath.org/1475.260312022-01-14T13:23:02.489162Z"Rozovsky, L. V."https://www.zbmath.org/authors/?q=ai:rozovskii.leonid-viktorovichSummary: The main purpose of the paper is to strengthen the results of P. R. Mercer (2003) concerning the comparison of arithmetic, geometric, and harmonic weighted means.Sharp bounds for the Toader mean in terms of arithmetic and geometric meanshttps://www.zbmath.org/1475.260322022-01-14T13:23:02.489162Z"Yang, Zhen-Hang"https://www.zbmath.org/authors/?q=ai:yang.zhenhang"Tian, Jing-Feng"https://www.zbmath.org/authors/?q=ai:tian.jingfengThe authors offer sharp lower and upper bounds for a mean of two arguments considered by \textit{G. Toader} [J. Math. Anal. Appl. 218, No. 2, 358--368 (1998; Zbl 0892.26015)], in terms of the arithmetic and geometric mean. In the proofs, the well-known connection between the Toader mean and the complete elliptic integral of second type is used. A basic method of proof is the recurrence method combined with the classical result of \textit{M. Biernacki} and \textit{J. Krzyz} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 9, 135--147 (1957; Zbl 0078.26402)] on the monotonicity of the ratio of power series. Another tool is a monotonicity criteria due to \textit{Z.-H. Yang} et al. [J. Math. Anal. Appl. 428, No. 1, 587--604 (2015; Zbl 1321.26019)].
Reviewer: József Sándor (Cluj-Napoca)Approximations for the ratio of the gamma functions via the digamma function and its applicationshttps://www.zbmath.org/1475.330022022-01-14T13:23:02.489162Z"Han, Min"https://www.zbmath.org/authors/?q=ai:han.min"Zhang, Hongliang"https://www.zbmath.org/authors/?q=ai:zhang.hongliang"You, Xu"https://www.zbmath.org/authors/?q=ai:you.xu"Sun, Zhaoxu"https://www.zbmath.org/authors/?q=ai:sun.zhaoxuSummary: In this paper, the authors establish some asymptotic formulas and two-sided inequalities for the ratio of the gamma function in terms of the digamma function. Considering the application of the gamma function to central Bernoulli coefficients, the authors give some better approximations and two-sided inequalities of central Bernoulli coefficients. Finally, for demonstrating the superiority of our results, some numerical computations are given.An analog of Titchmarsh's theorem for the q-Dunkl transform in the space \(L_{q,\alpha }^2({\mathbb{R}}_q)\)https://www.zbmath.org/1475.330082022-01-14T13:23:02.489162Z"Daher, Radouan"https://www.zbmath.org/authors/?q=ai:daher.radouan"Tyr, Othman"https://www.zbmath.org/authors/?q=ai:tyr.othmanThe authors use the \(q\)-harmonic analysis associated with the \(q\)-Dunkl operator and some properties of \(q\)-Bessel function to prove a \(q\)-analog of Titchmarsh's theorem for the functions satisfying the \(q\)-Dunkl Lipschitz condition.
Reviewer: Zhi-Guo Liu (Shanghai)Stability region of fractional differential systems with Prabhakar derivativehttps://www.zbmath.org/1475.340022022-01-14T13:23:02.489162Z"Alidousti, Javad"https://www.zbmath.org/authors/?q=ai:alidousti.javadSummary: This paper analyzes the stability of fractional differential equations with Prabhakar derivative, which is a generalization of the fractional differential equation with Caputo and Riemann-Liouville derivative. As a result, the sufficient condition for asymptotic stability has been obtained by studying the eigenvalues of system related matrix and the position of these eigenvalues in the complex plane. To show the application of the results a two-dimensional predator-prey model has been studied with Prabhakar derivative and the dynamic behavior has been extracted. Then, a numerical method along with its error has been presented for solving differential equations with Prabhakar derivative. The predator-prey model simulation has been conducted by using this numerical method.Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernelhttps://www.zbmath.org/1475.340052022-01-14T13:23:02.489162Z"Kachhia, Krunal B."https://www.zbmath.org/authors/?q=ai:kachhia.krunal-b"Atangana, Abdon"https://www.zbmath.org/authors/?q=ai:atangana.abdonSummary: The concept of differential operator with variable order has attracted attention of many scholars due to their abilities to capture more complexities like anomalous diffusion. While these differential operators are useful in real life, they can only be handled numerically. In this work, we used a newly introduced variable order differential operators that can be used analytically and numerically, has connection with all integral transform to model some interesting mathematical models arising in electromagnetic wave in plasma and dielectric. The differential operators used are non-singular and have the crossover properties therefore the models studied can explain the propagation of the wave in two different layers which cannot be achieved with those differential variable order operators with singular kernels. Using the Laplace transform and its connection with the new differential operator, we derive the exact solution of the models under investigation.Unique weak solutions of the \(d\)-dimensional micropolar equation with fractional dissipationhttps://www.zbmath.org/1475.352572022-01-14T13:23:02.489162Z"Ben Said, Oussama"https://www.zbmath.org/authors/?q=ai:ben-said.oussama"Wu, Jiahong"https://www.zbmath.org/authors/?q=ai:wu.jiahongSummary: This article examines the existence and uniqueness of weak solutions to the \(d\)-dimensional micropolar equations \((d=2\) or \(d=3)\) with general fractional dissipation \((- \Delta)^{\alpha} u\) and \((- \Delta)^{\beta} w\). The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when \(\alpha \geq \frac{ 1}{ 2}\) and \(\beta \geq \frac{ 1}{ 2} \), any initial data \((u_0,w_0)\) in the critical Besov space \(u_0 \in B_{2 , 1}^{1 + \frac{ d}{ 2} - 2 \alpha}( \mathbb{R}^d)\) and \(w_0 \in B_{2 , 1}^{1 + \frac{ d}{ 2} - 2 \beta}( \mathbb{R}^d)\) yields a unique weak solution. For \(\alpha \geq 1\) and \(\beta =0\), any initial data \(u_0 \in B_{2 , 1}^{1 + \frac{ d}{ 2} - 2 \alpha}( \mathbb{R}^d)\) and \(w_0 \in B_{2 , 1}^{\frac{ d}{ 2}}( \mathbb{R}^d)\) also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, \( \alpha =\beta =1\), have a unique weak solution for \(( u_0, w_0) \in B_{2 , 1}^0\). The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivativehttps://www.zbmath.org/1475.352972022-01-14T13:23:02.489162Z"Zhou, Yong"https://www.zbmath.org/authors/?q=ai:zhou.yong.1"Wang, Jing Na"https://www.zbmath.org/authors/?q=ai:wang.jing-naSummary: The Rayleigh-Stokes problem has gained much attention with the further study of non-Newtonain fluids. In this paper, we are interested in discussing the existence of solutions for nonlinear Rayleigh-Stokes problem for a generalized second grade fluid with Riemann-Liouville fractional derivative. We firstly show that the solution operator of the problem is compact and continuous in the uniform operator topology. Furtherly, we give an existence result of mild solutions for the nonlinear problem.Direct methods for pseudo-relativistic Schrödinger operatorshttps://www.zbmath.org/1475.353142022-01-14T13:23:02.489162Z"Dai, Wei"https://www.zbmath.org/authors/?q=ai:dai.wei.4"Qin, Guolin"https://www.zbmath.org/authors/?q=ai:qin.guolin"Wu, Dan"https://www.zbmath.org/authors/?q=ai:wu.danSummary: In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schrödinger operators \((- \Delta +m^2)^s\) with \(s \in (0,1)\) and mass \(m>0\). As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry and uniqueness results for solutions to various equations involving the operators \((- \Delta +m^2)^s\) in bounded or unbounded domains with certain geometrical structures (e.g., coercive epigraph and epigraph), including pseudo-relativistic Schrödinger equations, 3D boson star equations and the equations with De Giorgi-type nonlinearities. When \(m=0\) and \(s=1\), equations with De Giorgi-type nonlinearities are related to De Giorgi conjecture connected with minimal surfaces and the scalar Ginzburg-Landau functional associated to harmonic map.Stability of periodic waves for the fractional KdV and NLS equationshttps://www.zbmath.org/1475.353202022-01-14T13:23:02.489162Z"Hakkaev, Sevdzhan"https://www.zbmath.org/authors/?q=ai:hakkaev.sevdzhan-a"Stefanov, Atanas G."https://www.zbmath.org/authors/?q=ai:stefanov.atanas-gSummary: We consider the focussing fractional periodic Korteweg-deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with \(L^2\) sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each \(\lambda > 0\), there is a travelling wave solution to fKdV and fNLS \(\phi:\|\phi\|_{L^2[-T,T]}^2=\lambda\), which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in \(H^{\alpha/2}[-T,T]\) and a natural technical condition. This is done rigorously, without any \textit{a priori} assumptions on the smoothness of the waves or the Lagrange multipliers.An implicit semi-linear discretization of a bi-fractional Klein-Gordon-Zakharov system which conserves the total energyhttps://www.zbmath.org/1475.353212022-01-14T13:23:02.489162Z"Martínez, Romeo"https://www.zbmath.org/authors/?q=ai:martinez.romeo"Macías-Díaz, Jorge E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardo"Sheng, Qin"https://www.zbmath.org/authors/?q=ai:sheng.qinSummary: In this work, we propose an implicit finite-difference scheme to approximate the solutions of a generalization of the well-known Klein-Gordon-Zakharov system. More precisely, the system considered in this work is an extension to the spatially fractional case of the classical Klein-Gordon-Zakharov model, considering two different orders of differentiation and fractional derivatives of the Riesz type. The numerical model proposed in this work considers fractional-order centered differences to approximate the spatial fractional derivatives. The energy associated to this discrete system is a non-negative invariant, in agreement with the properties of the continuous fractional model. We establish rigorously the existence of solutions using fixed-point arguments and complex matrix properties. To that end, we use the fact that the two difference equations of the discretization are decoupled, which means that the computational implementation is easier than for other numerical models available in the literature. We prove that the method has square consistency in both time and space. In addition, we prove rigorously the stability and the quadratic convergence of the numerical model. As a corollary of stability, we are able to prove the uniqueness of numerical solutions. Finally, we provide some illustrative simulations with a computer implementation of our scheme.A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation with critical Sobolev exponenthttps://www.zbmath.org/1475.353322022-01-14T13:23:02.489162Z"Mederski, Jarosław"https://www.zbmath.org/authors/?q=ai:mederski.jaroslaw"Szulkin, Andrzej"https://www.zbmath.org/authors/?q=ai:szulkin.andrzejSummary: Let \(\Omega \subset \mathbb{R}^3\) be a Lipschitz domain and let \(S_\text{curl}(\Omega )\) be the largest constant such that
\[ \int_{\mathbb{R}^3}|\nabla \times u|^2\, \text{d}x\ge S_{\text{curl}}(\Omega ) \inf_{\substack{w\in W_0^6(\text{curl};\mathbb{R}^3)\\ \nabla \times w=0}} \Big (\int_{\mathbb{R}^3}|u+w|^6\,\text{d}x\Big )^{\frac{1}{3}} \]
for any \(u\) in \(W_0^6(\text{curl};\Omega )\subset W_0^6(\text{curl};\mathbb{R}^3)\), where \(W_0^6(\text{curl};\Omega )\) is the closure of \(\mathcal{C}_0^{\infty }(\Omega ,\mathbb{R}^3)\) in \(\{u\in L^6(\Omega ,\mathbb{R}^3): \nabla \times u\in L^2(\Omega ,\mathbb{R}^3)\}\) with respect to the norm \((|u|_6^2+|\nabla \times u|_2^2)^{1/2}\). We show that \(S_{\text{curl}}(\Omega )\) is strictly larger than the classical Sobolev constant \(S\) in \(\mathbb{R}^3\). Moreover, \(S_{\text{curl}}(\Omega )\) is independent of \(\Omega\) and is attained by a ground state solution to the curl-curl problem
\[ \nabla \times (\nabla \times u) = |u|^4u \]
if \(\Omega =\mathbb{R}^3\). With the aid of these results we also investigate ground states of the Brezis-Nirenberg-type problem for the curl-curl operator in a bounded domain \(\Omega\)
\[ \nabla \times (\nabla \times u) +\lambda u = |u|^4u\quad \hbox{in }\Omega , \]
with the so-called metallic boundary condition \(\nu \times u=0\) on \(\partial \Omega \), where \(\nu\) is the exterior normal to \(\partial \Omega \).Semi-analytical analysis of Allen-Cahn model with a new fractional derivativehttps://www.zbmath.org/1475.353342022-01-14T13:23:02.489162Z"Deniz, Sinan"https://www.zbmath.org/authors/?q=ai:deniz.sinanSummary: The Allen-Cahn model equation is extended to the fractional form by using Atangana-Baleanu derivative. The modified nonlinear equation is analyzed via optimal perturbation iteration technique and Laplace transform. Some new analytical approximate solutions are derived for different cases of order \(\alpha \). Absolute residual errors of different order of approximations are presented to check the effectiveness and power of the proposed method and new derivative.Some exact solutions of a variable coefficients fractional biological population modelhttps://www.zbmath.org/1475.353452022-01-14T13:23:02.489162Z"Abdel Kader, Abass H."https://www.zbmath.org/authors/?q=ai:abdel-kader.abass-h"Abdel Latif, Mohamed S."https://www.zbmath.org/authors/?q=ai:abdel-latif.mohamed-soror"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: In this paper, we investigate the exact solutions of a nonlinear variable coefficients time fractional biological population model using the invariant subspace method. The subspaces with dimensions one, two, and three are derived for certain cases of the variable coefficients. The exact solutions of the nonlinear time fractional biological population model are obtained in some cases.Pattern formation in superdiffusion predator-prey-like problems with integer- and noninteger-order derivativeshttps://www.zbmath.org/1475.353602022-01-14T13:23:02.489162Z"Owolabi, Kolade M."https://www.zbmath.org/authors/?q=ai:owolabi.kolade-matthew"Karaagac, Berat"https://www.zbmath.org/authors/?q=ai:karaagac.berat"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: This paper focuses on the modeling and application of fractional derivative to model the interactions between two different species in which the one named predator depends on the other called prey solely for survival. The interaction between predator and prey has been one of the most intriguing and interesting subjects in applied mathematical biology and ecology. In the models, the classical reaction-diffusion equations subject to the Neumann boundary conditions are formulated on a finite but large domain \(x \in [0, L]\) by replacing the second-order spatial derivatives with the fractional Laplacian operator of order \(1 < \alpha \leq 2\), which is classified as superdiffusion process. We examine the resulting coupled reaction-diffusion models for linear stability analysis and derive conditions under which the spatial patterns is evolved. In a view to understand our theoretical findings, the species spatial interactions is described in one and two dimensions. Through numerical experiments, we observe that a number of patterns can arise, including Turing spots, spiral-like structures, and seemingly complex spatiotemporal distributions.Analytic normalized solutions of 2D fractional Saint-Venant equations of a complex variablehttps://www.zbmath.org/1475.353812022-01-14T13:23:02.489162Z"Alarifi, Najla M."https://www.zbmath.org/authors/?q=ai:alarifi.najla-m"Ibrahim, Rabha W."https://www.zbmath.org/authors/?q=ai:ibrahim.rabha-waellSummary: Saint-Venant equations describe the flow below a pressure surface in a fluid. We aim to generalize this class of equations using fractional calculus of a complex variable. We deal with a fractional integral operator type Prabhakar operator in the open unit disk. We formulate the extended operator in a linear convolution operator with a normalized function to study some important geometric behaviors. A class of integral inequalities is investigated involving special functions. The upper bound of the suggested operator is computed by using the Fox-Wright function, for a class of convex functions and univalent functions. Moreover, as an application, we determine the upper bound of the generalized fractional 2-dimensional Saint-Venant equations (2D-SVE) of diffusive wave including the difference of bed slope.Boundedness and decay of solutions for some fractional magnetic Schrödinger equations in \(\mathbb{R}^N\)https://www.zbmath.org/1475.353832022-01-14T13:23:02.489162Z"Ambrosio, Vincenzo"https://www.zbmath.org/authors/?q=ai:ambrosio.vincenzoSummary: We prove that nontrivial weak solutions of a class of fractional magnetic Schrödinger equations in \(\mathbb{R}^N\) are bounded and vanish at infinity.Dispersive estimates for time and space fractional Schrödinger equationshttps://www.zbmath.org/1475.353982022-01-14T13:23:02.489162Z"Su, Xiaoyan"https://www.zbmath.org/authors/?q=ai:su.xiaoyan"Zhao, Shiliang"https://www.zbmath.org/authors/?q=ai:zhao.shiliang"Li, Miao"https://www.zbmath.org/authors/?q=ai:li.miaoSummary: In this paper, we consider the Cauchy problem for the fractional Schrödinger equation \(iD^\alpha_t u - (- (\Delta)^\frac{\beta}{2} u = 0 \) with
\(0 < \alpha < 1\), \(\beta > 0\). We establish the dispersive estimates by a carefully study of the Mittag-Leffler functions and give some applications as well. In particular, we prove that the decay rates are sharp.On a class of semilinear nonclassical fractional wave equations with logarithmic nonlinearityhttps://www.zbmath.org/1475.354002022-01-14T13:23:02.489162Z"Van, Au Vo"https://www.zbmath.org/authors/?q=ai:van.au-vo"Thi, Kim Van Ho"https://www.zbmath.org/authors/?q=ai:thi.kim-van-ho"Nguyen, Anh Tuan"https://www.zbmath.org/authors/?q=ai:nguyen.anh-tuanSummary: In this paper, we consider the initial boundary value problem for time-fractional subdiffusive equations with Caputo derivative. Our problem has many applications in population dynamics. The source function is given in the logarithmic form. We examine the existence, uniqueness of local solutions, and their ability to continue to a maximal interval of existence. The main tool and analysis here are of applying some Sobolev embedding and some fixed point theorems.Nehari manifold and bifurcation for a \(\psi \)-Hilfer fractional \(p\)-Laplacianhttps://www.zbmath.org/1475.354012022-01-14T13:23:02.489162Z"Vanterler da C. Sousa, J."https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.joseConsider the problem
\[
\left\{ \begin{array} [c]{ll} {}^H \mathbb{D}_T^{\alpha,\beta;\psi}\left(\left|{}^H\mathbb{D}_{0+}^{\alpha,\beta;\psi}\xi(x)\right|^{p-2}{}^H\mathbb{D}_{0+}^{\alpha,\beta;\psi}\xi(x)\right)=\lambda |\xi(x)|^{p-2}\xi(x)+\mathbf{b}(x)|\xi(x)|^{q-1}\xi(x), \\
\mathbf{I}_{0+}^{\beta(\beta-1);\psi}\xi(0)=\mathbf{I}_{T}^{\beta(\beta-1);\psi}\xi(T), \end{array} \right.
\]
where \({}^H\mathbb{D}_{0+}^{\alpha,\beta;\psi}(\cdot),{}^H \mathbb{D}_T^{\alpha,\beta;\psi}(\cdot)\) are \(\psi\)-Hilfer fractional derivatives left sided and right sided of order \(1/p<\alpha<1\), type \(0\le \beta\le 1\), \(1<q<p-1\), \(\mathbf{b}\in L^\infty(\Omega)\), \(\Omega=[0,T]\), \(\mathbf{I}_{0+}^{\beta(\beta-1);\psi}(\cdot)=\mathbf{I}_{T}^{\beta(\beta-1);\psi}(\cdot)\) are \(\psi\)-Riemann-Liouville integrals left sided and right sided and \(\lambda\) is positive parameter. By using the Nehari manifold and standard techniques the author shows that the first eigenvalue of the \(\psi\)-fractional \(p\)-Laplacian operator corresponds to a bifurcation from infinity.
The idea is old and reminiscent of, for example, \textit{P. H. Rabinowitz}, J. Differ. Equations 14, 462--475 (1973; Zbl 0272.35017). With respect to variational arguments, the following works should also be mentioned:
\textit{P. Drábek} and \textit{S. I. Pohozaev}, Proc. R. Soc. Edinb., Sect. A, Math. 127, No. 4, 703--726 (1997; Zbl 0880.35045);
\textit{J. Chabrowski} and \textit{D. G. Costa}, Commun. Partial Differ. Equations 33, No. 8, 1368--1394 (2008; Zbl 1180.35089).
Reviewer: Kaye Silva (Goiânia)Logarithmic transformation between (variable-order) Caputo and Caputo-Hadamard fractional problems and applicationshttps://www.zbmath.org/1475.354042022-01-14T13:23:02.489162Z"Zheng, Xiangcheng"https://www.zbmath.org/authors/?q=ai:zheng.xiangchengSummary: We present a logarithmic transformation reducing the (variable-order) Caputo-Hadamard fractional problems to their Caputo analogues. Then the analysis of the former may be directly obtained from the existing results for the latter by the inverse transformation. By employing this transformation method, we connect these two kinds of problems and obtain new analysis results for variable-order Caputo-Hadamard fractional operators and models.Numerical solution of an inverse random source problem for the time fractional diffusion equation via PhaseLifthttps://www.zbmath.org/1475.354322022-01-14T13:23:02.489162Z"Gong, Yuxuan"https://www.zbmath.org/authors/?q=ai:gong.yuxuan"Li, Peijun"https://www.zbmath.org/authors/?q=ai:li.peijun.1"Wang, Xu"https://www.zbmath.org/authors/?q=ai:wang.xu.4|wang.xu.5|wang.xu|wang.xu.1|wang.xu.3|wang.xu.2"Xu, Xiang"https://www.zbmath.org/authors/?q=ai:xu.xiangIn this paper, the following initial boundary value problem for the one-dimensional stochastic time fractional diffusion equation is considered:
\begin{align*}
& \partial_t^\alpha u(x,t)- \partial_{xx} u(x,t) = F(t)\dot{W}_x, \quad (x,t) \in (0,1)\times \mathbb{R}_+, \\
& u(x,0) = 0, \quad x \in [0,1], \\
& \partial_x u(0,t) = 0, \quad u(1,t) = 0, \quad t \in \mathbb{R}_+,
\end{align*}
where \( \partial_t^\alpha \) denotes the Caputo fractional derivative of order \( 0 < \alpha < 1 \) with respect to the variable \( t \), and \( F \) is a deterministic function satisfying \( F(0) = 0 \). In addition, \( W_x \) is the spatial Brownian motion satisfying \( \mathbb{E}[W_xW_y] = \min\{x, y\} \) for \( x,y \in (0,1) \), and \( \dot{W}_x \) denotes the formal derivative of \( W_x \) known as the white noise. The authors deduce the Green's function for the following equivalent problem in frequency domain:
\begin{align*}
& \partial_{xx} U(x,\omega) -(\text{i}\omega)^\alpha U(x,\omega) = -\hat{F}(\omega)\dot{W}_x, \quad x \in (0,1), \ \omega \in \mathbb{R}, \\
& \partial_x U(0,\omega) = 0, \quad U(1,\omega) = 0, \quad \omega \in \mathbb{R},
\end{align*}
where \( \hat{F} \) denotes the Fourier transform of the zero extention of \( F \) in \( (-\infty, 0) \). This provides the necessary tools for showing well-posedness of the direct problem. Subsequently, the inverse problem is considered which is to reconstruct the diffusion coefficient \( F \) of the random source from the measured data \( u(0,t) \) for \( t > 0 \). It is shown that the modulus \( \vert \hat{F}(\omega) \vert \) is uniquely and unstable determined by the data \( \mathbb{V}[U(0,\omega)] \). The phase retrieval for the inverse problem is also discussed. The paper concludes with some numerical illustrations that make use of a finite difference method to discretize the problem, and in addition a regularized convex optimization scheme is used as a stabilizer.
Reviewer: Robert Plato (Siegen)Hamiltonian formulation of systems described using fractional singular Lagrangianhttps://www.zbmath.org/1475.370602022-01-14T13:23:02.489162Z"Song, Chuanjing"https://www.zbmath.org/authors/?q=ai:song.chuanjing"Agrawal, Om Prakash"https://www.zbmath.org/authors/?q=ai:agrawal.om-prakashSummary: Fractional singular systems defined using mixed integer and Caputo fractional derivative are analyzed. Using these derivatives, fractional primary constraints, fractional constrained Hamilton equations and the corresponding Poisson brackets are established. Several examples are presented to demonstrate applications of the formulations.KAM, \(\alpha\)-Gevrey regularity and the \(\alpha\)-Bruno-Rüssmann conditionhttps://www.zbmath.org/1475.370672022-01-14T13:23:02.489162Z"Bounemoura, Abed"https://www.zbmath.org/authors/?q=ai:bounemoura.abed-bounemoura"Féjoz, Jacques"https://www.zbmath.org/authors/?q=ai:fejoz.jacquesSummary: We prove a new invariant torus theorem, for \(\alpha\)-Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the \(\alpha\)-Bruno-Rüssmann condition, and which reduces to the classical Bruno-Rüssmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this work are new functional estimates in the Gevrey class.Metric Fourier approximation of set-valued functions of bounded variationhttps://www.zbmath.org/1475.420082022-01-14T13:23:02.489162Z"Berdysheva, Elena E."https://www.zbmath.org/authors/?q=ai:berdysheva.elena-e"Dyn, Nira"https://www.zbmath.org/authors/?q=ai:dyn.nira"Farkhi, Elza"https://www.zbmath.org/authors/?q=ai:farkhi.elza-m"Mokhov, Alona"https://www.zbmath.org/authors/?q=ai:mokhov.alonaThe main topic of the paper is an adaptation of the trigonometric Fourier series to set-valued functions of bounded variation with compact images. The authors also try to gain error bounds under minimal regularity requirements on the approximated multifunctions. This analysis exploits properties of maps of bounded variation from [\textit{V. V. Chistyakov}, J. Dyn. Control Syst. 3, No. 2, 261--289 (1997; Zbl 0940.26009)]. The authors define a metric analogue of the partial sums of the Fourier series of a~multifunction via convolutions with the Dirichlet kernel. To define these convolutions the authors introduce a new weighted metric integral. The study of the error bounds of the approximations leads to the introduction of a~new one-sided local moduli of continuity and a~one-sided local quasi-moduli of continuity. The main result of the paper is an analogy of the classical Dirichlet-Jordan Theorem for real functions. In particular, if a multifunction \(F\) of bounded variation is continuous at a point \(x\), then its metric Fourier approximants at~\(x\) converge to \(F(x)\). This convergence is uniform on bounded closed intervals in which \(F\) is continuous. At a point of discontinuity the limit set is determined by the values of the metric selections of \(F\).
Reviewer: Miroslav Repický (Košice)Modulus of continuity and modulus of smoothness related to the deformed Hankel transformhttps://www.zbmath.org/1475.420132022-01-14T13:23:02.489162Z"Negzaoui, Selma"https://www.zbmath.org/authors/?q=ai:negzaoui.selma"Oukili, Sara"https://www.zbmath.org/authors/?q=ai:oukili.saraSummary: In this paper, we consider the deformed Hankel transform \(\mathscr{F}_{\kappa}\), which is a deformation of the Hankel transform by a parameter \(\kappa>\frac{1}{4}\). We introduce, via modulus of continuity, a function subspace of \(L^p(d\mu_{\kappa})\) that we call deformed Hankel Dini-Lipschitz space. In the case \(p=2\), we provide equivalence theorem: we get a characterization of those spaces by means of asymptotic estimate growth of the norm of their \(\mathscr{F}_{\kappa}\) transform for \(0<\gamma<1\) and \(\alpha\geq 0\). As a consequence we have the analogous of generalized Titchmarsh theorem in \(L^2(d\mu_{\kappa})\). Moreover, we introduce the modulus of smoothness related to \(\mathscr{F}_\kappa\) for which we study some properties on the Sobolev type space.Inequalities for derivatives with the Fourier transformhttps://www.zbmath.org/1475.420152022-01-14T13:23:02.489162Z"Osipenko, K. Yu."https://www.zbmath.org/authors/?q=ai:osipenko.konstantin-yuThe Landau-Kolmogorov type inequalities are generalized in this paper in two directions. First, the norm of the function on the right is replaced by that of the Fourier transform. Secondly, multidimensional extensions are obtained. The main goal of such inequalities, finding sharp constants, is achieved in many important cases.
Reviewer: Elijah Liflyand (Ramat-Gan)Explicit counterexamples to the weak Muckenhoupt-Wheeden conjecturehttps://www.zbmath.org/1475.420212022-01-14T13:23:02.489162Z"Osękowski, Adam"https://www.zbmath.org/authors/?q=ai:osekowski.adamSummary: We present an explicit construction of examples showing that the estimate \[\Vert T^\epsilon \Vert_{L^1(w)\rightarrow L^{1,\infty }(w)}\lesssim [w]_{A_1}\log (1+[w]_{A_1})\] for Haar multipliers is sharp in terms of the characteristic \([w]_{A_1}\).Sharp inequalities for maximal operators on finite graphs. IIhttps://www.zbmath.org/1475.420332022-01-14T13:23:02.489162Z"González-Riquelme, Cristian"https://www.zbmath.org/authors/?q=ai:gonzalez-riquelme.cristian"Madrid, José"https://www.zbmath.org/authors/?q=ai:madrid.jose-a-jimenezSummary: Let \(M_G\) be the centered Hardy-Littlewood maximal operator on a finite graph \(G\). We find \(\lim_{p \to \infty} \| M_G \|_p^p\) when \(G\) is the start graph \(( S_n)\) and the complete graph \(( K_n)\), and we fully describe \(\| M_{S_n} \|_p\) and the corresponding extremizers for \(p \in(1, 2)\). We prove that \(\lim_{p \to \infty} \| M_{S_n} \|_p^p = \frac{ 1 + \sqrt{ n}}{ 2}\) when \(n \geq 25\). Also, we compute the best constant \(\mathcal{C}_{S_n , 2}\) such that for every \(f : V \to \mathbb{R}\) we have \(\operatorname{Var}_2 M_{S_n} f \leq \mathcal{C}_{S_n , 2} \operatorname{Var}_2 f\). We prove that \(\mathcal{C}_{S_n , 2} = \frac{ ( n^2 - n - 1 )^{1 / 2}}{ n}\) for all \(n \geq 3\) and characterize the extremizers. Moreover, when \(M\) is the Hardy-Littlewood maximal operator on \(\mathbb{Z} \), we compute the best constant \(\mathcal{C}_p\) such that \(\operatorname{Var}_p M f \leq \mathcal{C}_p \| f \|_p\) for \(p \in(\frac{ 1}{ 2}, 1)\) and we describe the extremizers.
For Part I, see [the authors, J. Geom. Anal. 31, No. 10, 9708--9744 (2021; Zbl 07389654)].Sparse Brudnyi and John-Nirenberg spaceshttps://www.zbmath.org/1475.420362022-01-14T13:23:02.489162Z"Domínguez, Óscar"https://www.zbmath.org/authors/?q=ai:dominguez.oscar"Milman, Mario"https://www.zbmath.org/authors/?q=ai:milman.marioSummary: A generalization of the theory of \textit{Yu. A. Brudnyĭ} [Tr. Mosk. Mat. O.-va 24, 69--132 (1971; Zbl 0254.46018)], and \textit{A. Brudnyi} and \textit{Y. Brudnyi} [J. Approx. Theory 251, Article ID 105346, 70 p. (2020; Zbl 1436.26012); Diss. Math. 548, 1--52 (2020; Zbl 1445.26012)], is presented. Our construction connects Brudnyi's theory, which relies on local polynomial approximation, with new results on sparse domination. In particular, we find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by \textit{N. Kruglyak} and \textit{E. A. Kuznetsov} [Ark. Mat. 44, No. 2, 309--326 (2006; Zbl 1156.42308)]. Our spaces shed light on the structure of the John-Nirenberg spaces. We show that \(SJN_p\) (sparse John-Nirenberg space) coincides with \(L^p\), \(1<p<\infty\). This characterization yields the John-Nirenberg inequality by extrapolation and is useful in the theory of commutators.Haar and Shannon wavelet expansions with explicit coefficients of the Takagi functionhttps://www.zbmath.org/1475.420492022-01-14T13:23:02.489162Z"Fukuda, Naohiro"https://www.zbmath.org/authors/?q=ai:fukuda.naohiro"Kinoshita, Tamotu"https://www.zbmath.org/authors/?q=ai:kinoshita.tamotu"Suzuki, Toshio"https://www.zbmath.org/authors/?q=ai:suzuki.toshioThe blancmange curve or the Takagi curve is the attractor of a system of two iterated functions. The Takagi (or blancmange) function is continuous, 1-periodic and nowhere differentiable. The paper presents the explicit computations of the wavelet coefficients of the Takagi function using the Haar wavelet and the Shannon wavelet.
Reviewer: Françoise Bastin (Liège)Chandrasekhar quadratic and cubic integral equations via Volterra-Stieltjes quadratic integral equationhttps://www.zbmath.org/1475.450082022-01-14T13:23:02.489162Z"El-Sayed, Ahmed M. A."https://www.zbmath.org/authors/?q=ai:el-sayed.ahmed-mohamed-ahmed"Omar, Yasmin M. Y."https://www.zbmath.org/authors/?q=ai:omar.yasmin-m-yIn this paper, the following quadratic integral equation of Volterra-Stieltjes type is studied: \[ x(t)=a(t)+\int\limits_{0}^{\varphi(t)}f_1(t,s,x(s))d_sg_1(t,s)\int\limits_{0}^{1}f_2(t,s,x(s))d_sg_2(t,s), \quad t\in[0,1]. \] The authors discuss the existence of solutions of this equation in the class of continuous functions and the continuous dependence of its unique solution on the functions \(g_1\), \(g_2\) and \(\varphi\).
The obtained results are then applied to the fractional-order quadratic integral equation \[ x(t)=a(t)+I^{\alpha}f_1(t,s,x)\cdot\int\limits_{0}^{1}\frac{t}{t+s}f_2(t,s,x(s))ds \] and to the Chandrasekhar quadratic and cubic integral equations \[ x(t)=a(t)+x^p(t)\cdot\int\limits_{0}^{1}\frac{t}{t+s}a(s)x(s)ds, \quad p=1,2. \]
Reviewer: Alexander N. Tynda (Penza)Non-instantaneous impulsive fractional integro-differential equations with proportional fractional derivatives with respect to another functionhttps://www.zbmath.org/1475.450122022-01-14T13:23:02.489162Z"Abbas, Mohamed I."https://www.zbmath.org/authors/?q=ai:abbas.mohamed-iSummary: This paper concerns the existence and uniqueness of solutions of non-instantaneous impulsive fractional integro-differential equations with proportional fractional derivatives with respect to another function. By the aid of the nonlinear alternative of Leray-Schauder type and the Banach contraction mapping principle, the main results are demonstrated. Two examples are inserted to illustrate the applicability of the theoretical results.Solvability of control problem for fractional nonlinear differential inclusions with nonlocal conditionshttps://www.zbmath.org/1475.450132022-01-14T13:23:02.489162Z"Chadha, Alka"https://www.zbmath.org/authors/?q=ai:chadha.alka"Sakthivel, Rathinasamy"https://www.zbmath.org/authors/?q=ai:sakthivel.rathinasamy"Bora, Swaroop Nandan"https://www.zbmath.org/authors/?q=ai:bora.swaroop-nandanThis paper is concerned with the approximate controllability of a nonlocal problem involving Caputo-type integro-differential inclusions of order \(q\in (1, 2)\) in a Hilbert space. The main results are obtained under a series of assumptions and by using the ideas of sectorial operator, multivalued analysis, and measure of noncompactness. The paper does not contain any illustrative example.
Reviewer: Bashir Ahmad (Jeddah)Boundary integral equation formulation for fractional order theory of thermo-viscoelasticityhttps://www.zbmath.org/1475.450152022-01-14T13:23:02.489162Z"Elhagary, M. A."https://www.zbmath.org/authors/?q=ai:elhagary.mohammed-ahmedSummary: This work presents a formulation of the boundary integral equation method for fractional order theory of thermo-viscoelasticity. Fundamental solutions of corresponding differential equations are obtained in the Laplace transform domain. A reciprocity theorem is established. The implementation of the boundary element method is discussed for the solution of the above boundary equations Special attention is given to the representation of primary fields, namely temperature and displacements. The initial, mixed boundary value problem is considered as an example illustrating the BIE formulation.
For the entire collection see [Zbl 1473.45002].The stability of the fractional Volterra integro-differential equation by means of \(\Psi \)-Hilfer operator revisitedhttps://www.zbmath.org/1475.450192022-01-14T13:23:02.489162Z"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-i"Saadati, Reza"https://www.zbmath.org/authors/?q=ai:saadati.reza"Sousa, José"https://www.zbmath.org/authors/?q=ai:sousa.joseSummary: In this note, we have as main purpose to investigate the Ulam-Hyers stability of a fractional Volterra integral equation through the Banach fixed point theorem and present an example on Ulam-Hyers stability using operator theory \(\alpha \)-resolvent in order to elucidate the investigated result. Our results modify the Theorem 4 of \textit{J. V. Da C. Sousa} et al. [Math. Methods Appl. Sci. 42, No. 9, 3033--3043 (2019; Zbl 1428.45010)] and present a corrected proof with a modified approximation.Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equationhttps://www.zbmath.org/1475.450202022-01-14T13:23:02.489162Z"Vanterler da C. Sousa, J."https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"de Oliveira, E. Capelas"https://www.zbmath.org/authors/?q=ai:de-oliveira.edmundo-capelasSummary: Using the \(\psi\)-Hilfer fractional derivative, we present a study of the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the fractional Volterra integro-differential equation by means of fixed-point method.Hypo-\(q\)-norms on a Cartesian product of algebras of operators on Banach spaceshttps://www.zbmath.org/1475.460162022-01-14T13:23:02.489162Z"Dragomir, Silvestru Sever"https://www.zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: In this paper we consider the hypo-\(q\)-operator norm and hypo-\(q\)-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the \(q\)-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.Weighted fractional and Hardy type operators in Orlicz-Morrey spaceshttps://www.zbmath.org/1475.460272022-01-14T13:23:02.489162Z"Burtseva, Evgeniya"https://www.zbmath.org/authors/?q=ai:burtseva.evgeniyaLet \(\Phi:[0,\infty)\to[0,\infty)\) be an increasing, continuous and convex function with \(\Phi(0)=0\) and let \(\varphi:[0,\infty)\to[0,\infty)\) be a measurable function satisfying \(\varphi(r)=0\) iff \(r=0\), \(\lim_{r\to 0}\varphi(r)=\varphi(0)=0\) and \(\varphi(r)\ge Cr^n\) for some constant \(C>0\) and all \(r\in(0,1]\). The generalized Orlicz-Morrey space \(M^{\Phi,\varphi}(\mathbb{R}^n)\) consists of all functions \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^n)\) such that \[ \|f\|_{M^{\Phi,\varphi}} := \sup_{x\in\mathbb{R}^n,r>0} \inf\left\{\lambda>0:\frac{1}{\varphi(r)}\int_{B(x,r)} \Phi\left(\frac{|f(y)|}{\lambda}\right)dy\le 1\right\}<\infty, \] where \(B(x,r)\) denotes the open ball in \(\mathbb{R}^n\) centered at \(x\in\mathbb{R}^n\) of radius \(r>0\). Sufficient conditions for the boundedness of the following operators between distinct generalized Orlicz-Morrey spaces are obtained: the Riesz fractional integral operator defined by \(I_\alpha f(x)=\int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n-\alpha}}\,dy\), where \(0<\alpha<n\); its weighted analogue \(w(|\cdot|) I_\alpha\frac{1}{w}\), where \(w:(0,\infty)\to(0,\infty)\) is a suitable weight; and the weighted Hardy operators defined by \(H_w^\alpha f(x)=|x|^{\alpha-n}w(|x|)\int_{|y|\le|x|} \frac{f(y)}{w(|y|)}\,dy\) and \(\mathcal{H}_w^\alpha f(x)=|x|^\alpha w(|x|)\int_{|y|>|x|}\frac{f(y)}{|y|^nw(|y|)}\,dy\).
Reviewer: Alexei Yu. Karlovich (Lisboa)Inequalities for \(L^p\)-norms that sharpen the triangle inequality and complement Hanner's inequalityhttps://www.zbmath.org/1475.460282022-01-14T13:23:02.489162Z"Carlen, Eric A."https://www.zbmath.org/authors/?q=ai:carlen.eric-anders"Frank, Rupert L."https://www.zbmath.org/authors/?q=ai:frank.rupert-l"Ivanisvili, Paata"https://www.zbmath.org/authors/?q=ai:ivanisvili.paata"Lieb, Elliott H."https://www.zbmath.org/authors/?q=ai:lieb.elliott-hSummary: In [J. Oper. Theory 62, No. 1, 151--158 (2009; Zbl 1199.47095)], \textit{A. Carbery} raised a question about an improvement on the naïve norm inequality \(\Vert f+g\Vert_p^p\le 2^{p-1}(\Vert f\Vert_p^p+\Vert g\Vert_p^p)\) for two functions \(f\) and \(g\) in \(L^p\) of any measure space. When \(f=g\) this is an equality, but when the supports of \(f\) and \(g\) are disjoint the factor \(2^{p-1}\) is not needed. Carbery's question concerns a proposed interpolation between the two situations for \(p>2\) with the interpolation parameter measuring the overlap being \(\Vert fg\Vert_{p/2}\). Carbery proved that his proposed inequality holds in a special case. Here, we prove the inequality for all functions and, in fact, we prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for \textit{all} real \(p\ne 0\).Limit of fractional power Sobolev inequalitieshttps://www.zbmath.org/1475.460342022-01-14T13:23:02.489162Z"Chang, Sun-Yung Alice"https://www.zbmath.org/authors/?q=ai:chang.sun-yung-alice"Wang, Fang"https://www.zbmath.org/authors/?q=ai:wang.fangSummary: We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by \textit{T. P. Branson} [Trans. Am. Math. Soc. 347, No. 10, 3671--3742 (1995; Zbl 0848.58047)].Power-aggregation of pseudometrics and the McShane-Whitney extension theorem for Lipschitz \(p\)-concave mapshttps://www.zbmath.org/1475.460732022-01-14T13:23:02.489162Z"Rodríguez-López, J."https://www.zbmath.org/authors/?q=ai:rodriguez-lopez.jesus"Sánchez-Pérez, E. A."https://www.zbmath.org/authors/?q=ai:sanchez-perez.enrique-alfonsoSummary: Given a countable set of families \(\{\mathcal{D}_k:k\in\mathbb{N}\}\) of pseudometrics over the same set \(D\), we study the power-aggregations of this class, that are defined as convex combinations of integral averages of powers of the elements of \(\bigcup_k \mathcal{D}_k\). We prove that a Lipschitz function \(f\) is dominated by such a power-aggregation if and only if a certain property of super-additivity involving the powers of the elements of \(\bigcup_k\mathcal{D}_k\) is fulfilled by \(f\). In particular, we show that a pseudo-metric is equivalent to a power-aggregation of other pseudometrics if this kind of domination holds. When the super-additivity property involves a \(p\)-power domination, we say that the elements of \(\mathcal{D}_k\) are \(p\)-concave. As an application of our results, we prove under this requirement a new extension result of McShane-Whitney type for Lipschitz \(p\)-concave real valued maps.Numerical radius inequalities and its applications in estimation of zeros of polynomialshttps://www.zbmath.org/1475.470052022-01-14T13:23:02.489162Z"Bhunia, Pintu"https://www.zbmath.org/authors/?q=ai:bhunia.pintu"Bag, Santanu"https://www.zbmath.org/authors/?q=ai:bag.santanu"Paul, Kallol"https://www.zbmath.org/authors/?q=ai:paul.kallolThe authors give an upper bound and a lower bound for the numerical radius of a bounded linear operator $T$ acting on a Hilbert space, which improves the existing bounds given in [\textit{A. Abu-Omar} and \textit{F. Kittaneh}, Ann. Funct. Anal. 5, No. 1, 56--62 (2014; Zbl 1298.47011)] and [\textit{F. Kittaneh}, Stud. Math. 168, No. 1, 73--80 (2005; Zbl 1072.47004)], respectively. Moreover, they present an upper bound of the numerical radius in terms of $\Vert\Re(e^{i\theta}T)\Vert$ and a lower bound of the numerical radius in terms of the spectral values of the real part $\Re(T)$ and the imaginary part $\Im(T)$ of $T$. In addition, they estimate the spectral radius of sum of product of $n$ pairs of operators. As an application, they estimate the zeros of a polynomial.
Reviewer: Mohammad Sal Moslehian (Mashhad)Semigroups of Hadamard multipliers on the space of real analytic functionshttps://www.zbmath.org/1475.470242022-01-14T13:23:02.489162Z"Golińska, Anna"https://www.zbmath.org/authors/?q=ai:golinska.annaSummary: An operator \(M\) acting on the space of real analytic functions \(\mathscr{A}(\mathbb {R})\) is called a multiplier if every monomial is its eigenvector. In this paper we state some results concerning strongly continuous semigroups generated by Hadamard multipliers. In particular we show when an Euler differential operator of finite order is a generator and when it is not.Natural boundary conditions for a class of generalized fractional variational problemhttps://www.zbmath.org/1475.490092022-01-14T13:23:02.489162Z"Singha, N."https://www.zbmath.org/authors/?q=ai:singha.neelam"Nahak, C."https://www.zbmath.org/authors/?q=ai:nahak.chandalSummary: In this paper, we analyze a problem of the generalized fractional calculus of variations with free boundary values. Specifically, we focus on the generalized fractional variational problems with (i) finite subsidiary conditions, (ii) simple variable end point, (iii) end points lying on the given curves. For all of the mentioned problems, we derive the necessary optimality conditions. The formulation of these conditions involves Riemann-Liouville and Caputo's fractional order derivatives. With free terminal points in a simple variable end point problem, we obtain the natural boundary conditions that provide the additional information on the boundaries of the concerned problem. Furthermore, we derive the transversality conditions for the fractional variational problem with end point lying on the given curves. Both the natural boundary conditions and transversality conditions provide the missing boundary conditions for the variational problem with variable terminal points.Natural boundary conditions for a class of generalized fractional variational problemhttps://www.zbmath.org/1475.490102022-01-14T13:23:02.489162Z"Singha, N."https://www.zbmath.org/authors/?q=ai:singha.neelam"Nahak, C."https://www.zbmath.org/authors/?q=ai:nahak.chandalSummary: In this paper, we analyze a problem of the generalized fractional calculus of variations with free boundary values. Specifically, we focus on the generalized fractional variational problems with (i) finite subsidiary conditions, (ii) simple variable end point, (iii) end points lying on the given curves. For all of the mentioned problems, we derive the necessary optimality conditions. The formulation of these conditions involves Riemann-Liouville and Caputo's fractional order derivatives. With free terminal points in a simple variable end point problem, we obtain the natural boundary conditions that provide the additional information on the boundaries of the concerned problem. Furthermore, we derive the transversality conditions for the fractional variational problem with end point lying on the given curves. Both the natural boundary conditions and transversality conditions provide the missing boundary conditions for the variational problem with variable terminal points.On the approximation of monotone variational inequalities in \(L^P\) spaces with probability measurehttps://www.zbmath.org/1475.490162022-01-14T13:23:02.489162Z"Passacantando, Mauro"https://www.zbmath.org/authors/?q=ai:passacantando.mauro"Raciti, Fabio"https://www.zbmath.org/authors/?q=ai:raciti.fabioThis paper is concerned with an \(L^p\) approach for random variational inequalities. The authors propose an approximation procedure for a class of monotone variational inequalities in probabilistic Lebesgue spaces. The results are applied to the random traffic network equilibrium problem with polynomial cost functions.
For the entire collection see [Zbl 1470.49002].
Reviewer: Zijia Peng (Nanning)Compactness and lower semicontinuity in \(GSBD\)https://www.zbmath.org/1475.490182022-01-14T13:23:02.489162Z"Chambolle, Antonin"https://www.zbmath.org/authors/?q=ai:chambolle.antonin"Crismale, Vito"https://www.zbmath.org/authors/?q=ai:crismale.vitoThe authors prove a compactness and semicontinuity result in GSBD for sequences with bounded Griffith energy. This generalises classical results in (G)SBV by \textit{L. Ambrosio} [Boll. Unione Mat. Ital., VII. Ser., B 3, No. 4, 857--881 (1989; Zbl 0767.49001); Arch. Ration. Mech. Anal. 111, No. 4, 291--322 (1990; Zbl 0711.49064); Calc. Var. Partial Differ. Equ. 3, No. 1, 127--137 (1995; Zbl 0837.49011)] and SBD by \textit{G. Bellettini} et al. [Math. Z. 228, No. 2, 337--351 (1998; Zbl 0914.46007)]. As a result, the static problem by \textit{G. A. Francfort} and \textit{J. J. Marigo}'s variational approach to crack growth [J. Mech. Phys. Solids 46, No. 8, 1319--1342 (1998; Zbl 0966.74060)] admits (weak) solutions.
Reviewer: Georgios Psaradakis (Mannheim)Differential geometry of curves in Euclidean 3-space with fractional orderhttps://www.zbmath.org/1475.530112022-01-14T13:23:02.489162Z"Aydin, Muhittin Evren"https://www.zbmath.org/authors/?q=ai:aydin.muhittin-evren"Bektaş, Mehmet"https://www.zbmath.org/authors/?q=ai:bektas.mehmet"Öğremiş, Alper"https://www.zbmath.org/authors/?q=ai:ogremis.alper"Yokuş, Asıf"https://www.zbmath.org/authors/?q=ai:yokus.asifThe paper introduces new results in differential geometry of curves, including the fractional-order derivative. A modified Caputo fractional derivative is used in the investigations. Considering Euclidean 3-space, the arc-length, the curvature, and the tension are presented. The Frenet-Serret formulas are provided. The methods of determining a curve in 2 and 3-dimensional space using the previously cited invariants (arc length, curvature, and tension) are proposed and explained. Illustrative examples and graphical representations are offered to support the main findings of the paper.
Reviewer: Ndolane Sene (Dakar)On the distribution of the numbers of solutions of random inclusionshttps://www.zbmath.org/1475.600342022-01-14T13:23:02.489162Z"Kopytsev, V. A."https://www.zbmath.org/authors/?q=ai:kopytsev.v-a"Mikhaĭlov, V. G."https://www.zbmath.org/authors/?q=ai:mikhajlov.v-gSummary: For given sets \(D\) and \(B\) of vectors in linear spaces \(V^n\) and \(V^T\) over the field \(K=GF(q)\) we consider the number of solutions \(\xi(D,F,B)\) of the system of inclusions \(x\in D\), \(A_1x+A_2 f(x)\in B\), where \(A_1\) and \(A_2\) are random \(T\times n\) and \(T\times m\) matrices over \(K\) with independent elements and \(f\colon V^n\to V^m\) is a given mapping. Sufficient conditions for the convergence of distributions of \(\xi(D,F,B)\) to the Poisson or compound Poisson distributions are found. Results are applied to the number of solutions of a system of random polynomial equations.Stable polynomials and sums of dependent Bernoulli random variables: application to Hoeffding inequalitieshttps://www.zbmath.org/1475.600402022-01-14T13:23:02.489162Z"Ennafti, Hazar"https://www.zbmath.org/authors/?q=ai:ennafti.hazar"Louhichi, Sana"https://www.zbmath.org/authors/?q=ai:louhichi.sanaSummary: We give, in this paper, a characterization of the independent representation in law for a sum of dependent Bernoulli random variables. This characterization is related to the stability property of the probability-generating function of this sum, which is a polynomial with positive coefficients. As an application, we give a Hoeffding inequality for a sum of dependent Bernoulli random variables when its probability-generating function has all its roots with negative real parts. Some sufficient conditions on the law of the sum of dependent Bernoulli random variables guaranteeing the negativity of the real parts of the roots are discussed. This paper generalizes some results in [\textit{T. M. Liggett}, Stochastic Processes Appl. 119, No. 1, 1--15 (2009; Zbl 1172.60031)].Concentration inequalities for polynomials in \(\alpha\)-sub-exponential random variableshttps://www.zbmath.org/1475.600452022-01-14T13:23:02.489162Z"Götze, Friedrich"https://www.zbmath.org/authors/?q=ai:gotze.friedrich-w"Sambale, Holger"https://www.zbmath.org/authors/?q=ai:sambale.holger"Sinulis, Arthur"https://www.zbmath.org/authors/?q=ai:sinulis.arthurLet $X_1,\dots,X_n$ be independent random variables and $f:\mathbb R^n\to \mathbb R$ be a measurable function. One of the main questions of probability theory has been finding a good function $h:[0,\infty]\to[0,1]$ such that
\[
P(|f(X_1,\dots,X_n)|-Ef(X_1,\dots,X_n)|\geq t)\leq h(t),
\]
and the following concentration inequality, regarding the tail decay of the normal distribution is perhaps the most well-known one: If $X_1,\dots,X_n$ are independent $N(0,1)$ random variables, and $f(X_1,\dots,X_n)=n^{-1/2}\sum_{i=1}^n X_i$, then $f(X_1,\dots,X_n)$ is distributed as $N(0,1)$ and
\[
P(|f(X_1,\dots,X_n)|-Ef(X_1,\dots,X_n)|\geq t)\leq 2e^{-t^2/2}.
\]
The present authors derive concentration inequalities for polynomials $f(X_1,\dots,X_n)$ in independent random variables $X_1,\dots,X_n$ with an $\alpha$-sub-exponential tail decay. A particularly interesting case is given by quadratic forms $f(X_1,\dots,X_n)=\langle X, AX \rangle$ for which they prove Hanson-Wright-type inequalities [\textit{D. L. Hanson} and \textit{F. T. Wright}, Ann. Math. Stat. 42, 1079--1083 (1971; Zbl 0216.22203)] with explicit dependence on various norms of the matrix $A$. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in $\alpha$-sub-exponential random variables, such as quadratic Poisson chaos. They provide various applications of these inequalities. Among them are generalizations of some results proven by \textit{M. Rudelson} and \textit{R. Vershynin} [Electron. Commun. Probab. 18, Paper No. 82, 9 p. (2013; Zbl 1329.60056)] from sub-normal to $\alpha$-sub-exponential random variables, i.e., concentration of the Euclidean norm of the linear image of a random vector and concentration inequalities for the distance between a random vector and a fixed subspace.
Reviewer: Andreas N. Philippou (Patras)Spectral gap of Boltzmann measures on unit circlehttps://www.zbmath.org/1475.600462022-01-14T13:23:02.489162Z"Ma, Yutao"https://www.zbmath.org/authors/?q=ai:ma.yutao"Zhang, Zhengliang"https://www.zbmath.org/authors/?q=ai:zhang.zhengliangLet \(\mu_h\) be a Boltzmann measure on the unit circle in \(\mathbb{R}^2\) with parameter \(h\in\mathbb{R}\), and \(\lambda_1(\mu_h)\) be its spectral gap. The authors prove an explicit two-sided bound on this spectral gap:
\[
1\vee\frac{|h|}{7}\leq\lambda_1(\mu_h)\leq\sqrt{3}|h|+\frac{2h^2+3}{h^2+3}\,.
\]
This bound is sharp at \(h=0\) and gives a rate of convergence of \(\lambda_1(\mu_h)\) to infinity as \(h\to\infty\).
Reviewer: Fraser Daly (Edinburgh)Submartingale property of set-valued stochastic integration associated with Poisson process and related integral equations on Banach spaceshttps://www.zbmath.org/1475.600992022-01-14T13:23:02.489162Z"Zhang, Jinping"https://www.zbmath.org/authors/?q=ai:zhang.jinping"Mitoma, Itaru"https://www.zbmath.org/authors/?q=ai:mitoma.itaru"Okazaki, Yoshiaki"https://www.zbmath.org/authors/?q=ai:okazaki.yoshiakiSummary: In an M-type 2 Banach space, firstly we explore some properties of the set-valued stochastic integral associated with the stationary Poisson point process. By using the Hahn decomposition theorem and bounded linear functional, we obtain the main result: the integral of a set-valued stochastic process with respect to the compensated Poisson measure is a set-valued submartingale but not a martingale unless the integrand degenerates into a single-valued process. Secondly we study the strong solution to the set-valued stochastic integral equation, which includes a set-valued drift, a single-valued diffusion driven by a Brownian motion and the set-valued jump driven by a Poisson process.An efficient energy-preserving method for the two-dimensional fractional Schrödinger equationhttps://www.zbmath.org/1475.650682022-01-14T13:23:02.489162Z"Fu, Yayun"https://www.zbmath.org/authors/?q=ai:fu.yayun"Xu, Zhuangzhi"https://www.zbmath.org/authors/?q=ai:xu.zhuangzhi"Cai, Wenjun"https://www.zbmath.org/authors/?q=ai:cai.wenjun"Wang, Yushun"https://www.zbmath.org/authors/?q=ai:wang.yushunSummary: In this paper, we study the Hamiltonian structure and develop a novel energy-preserving scheme for the two-dimensional fractional nonlinear Schrödinger equation. First, we present the variational derivative of the functional with fractional Laplacian to derive the Hamiltonian formula of the equation and obtain an equivalent system by defining a scalar variable. An energy-preserving scheme is then presented by applying exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space. The proposed scheme is a linear system and can be solved efficiently. Numerical experiments are displayed to verify the conservation, efficiency, and good performance at a relatively large time step in long time computations.A fast multi grid algorithm for 2D diffeomorphic image registration modelhttps://www.zbmath.org/1475.650702022-01-14T13:23:02.489162Z"Han, Huan"https://www.zbmath.org/authors/?q=ai:han.huan"Wang, Andong"https://www.zbmath.org/authors/?q=ai:wang.andongSummary: A 2D diffeomorphic image registration model is proposed to eliminate mesh folding by the first author and \textit{Z. Wang} [SIAM J. Imaging Sci. 13, No. 3, 1240--1271 (2020; Zbl 1451.65126)]. To solve the 2D diffeomorphic model, a diffeomorphic fractional-order image registration algorithm (DFIRA for short) is proposed in Han and Wang (2020). DFIRA achieves a satisfactory image registration result but it costs too much CPU time. To accelerate DFIRA, we propose a fast multi grid algorithm for 2D diffeomorphic image registration model in this paper. This algorithm achieves a satisfactory image registration result by using only one-tenth CPU time of DFIRA. At the same time, no mesh folding occurs in proposed algorithm. Furthermore, convergence analysis of the proposed algorithm is also presented. Moreover, numerical tests are also performed to show that the proposed algorithm is competitive compared with some other algorithms.On a nonlinear energy-conserving scalar auxiliary variable (SAV) model for Riesz space-fractional hyperbolic equationshttps://www.zbmath.org/1475.650712022-01-14T13:23:02.489162Z"Hendy, Ahmed S."https://www.zbmath.org/authors/?q=ai:hendy.ahmed-s"Macías-Díaz, J. E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardoSummary: In this work, we consider a fractional extension of the classical nonlinear wave equation, subjected to initial conditions and homogeneous Dirichlet boundary data. We consider space-fractional derivatives of the Riesz type in a bounded real interval. It is known that the problem has an associated energy which is preserved through time. The mathematical model is presented equivalently using the scalar auxiliary variable (SAV) technique, and the expression of the energy is obtained using the new scalar variable. The new differential system is discretized then following the SAV approach. The proposed scheme is a nonlinear implicit method which has an associated discrete energy, and we prove that the discrete model is also conservative. The present work is the first report in which the SAV method is used to design nonlinear conservative numerical method to solve a Hamiltonian space-fractional wave equations.Banded preconditioners for Riesz space fractional diffusion equationshttps://www.zbmath.org/1475.650792022-01-14T13:23:02.489162Z"She, Zi-Hang"https://www.zbmath.org/authors/?q=ai:she.zihang"Lao, Cheng-Xue"https://www.zbmath.org/authors/?q=ai:lao.cheng-xue"Yang, Hong"https://www.zbmath.org/authors/?q=ai:yang.hong"Lin, Fu-Rong"https://www.zbmath.org/authors/?q=ai:lin.furong|lin.fu-rongIn this paper, numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations are considered. Crank-Nicolson technique is applied to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. They transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. They prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners.
In my opinion, this work is interesting and makes some progress in fast algorithm for solving fractional partial differential equation.
Reviewer: Qifeng Zhang (Hangzhou)Splitting schemes for non-stationary problems with a rational approximation for fractional powers of the operatorhttps://www.zbmath.org/1475.650822022-01-14T13:23:02.489162Z"Vabishchevich, Petr N."https://www.zbmath.org/authors/?q=ai:vabishchevich.petr-nSummary: In this paper we study the numerical approximation of the solution of a Cauchy problem for a first-order-in-time differential equation involving a fractional power of a self-adjoint positive operator. One popular approach for the approximation of fractional powers of such operators is based on rational approximations. The purpose of this work is to construct special approximations in time so that the solution at a new time level is produced by solving a set of standard problems involving the self-adjoint positive operator rather than its fractional power. Stable splitting schemes with weight parameters are proposed for the additive representation of the rational approximation of the fractional power of the operator. Finally, numerical results for a two-dimensional non-stationary problem with a fractional power of the Laplace operator are also presented.High-order conservative schemes for the space fractional nonlinear Schrödinger equationhttps://www.zbmath.org/1475.650842022-01-14T13:23:02.489162Z"Wang, Junjie"https://www.zbmath.org/authors/?q=ai:wang.junjieSummary: In the paper, the high-order conservative schemes are presented for space fractional nonlinear Schrödinger equation. First, we give two class high-order difference schemes for fractional Risze derivative by compact difference method and extrapolating method, and show the convergence analysis of the two methods. Then, we apply high-order conservative difference schemes in space direction, and Crank-Nicolson, linearly implicit and relaxation schemes in time direction to solve fractional nonlinear Schrödinger equation. Moreover, we show that the arising schemes are uniquely solvable and approximate solutions converge to the exact solution at the rate \(O(\tau^2+h^4)\), and preserve the mass and energy conservation laws. Finally, we given numerical experiments to show the efficiency of the conservative finite difference schemes.A diagonalization-based parareal algorithm for dissipative and wave propagation problemshttps://www.zbmath.org/1475.650982022-01-14T13:23:02.489162Z"Gander, Martin J."https://www.zbmath.org/authors/?q=ai:gander.martin-j"Wu, Shu-Lin"https://www.zbmath.org/authors/?q=ai:wu.shulinThis paper proposes a new parareal algorithm for solving initial value problems of the form \(\partial_t u+f(t,u)=0\), where \(f:(0,T)\times \mathbb{R}^m\to \mathbb{R}^m\), \(m\ge 1\). The algorithm allows the use of a coarse propagator that discretizes the underlying problem on the same mesh as the fine propagator. The coarse propagator is approximated with a head-tail coupled condition such that it can be parallelized using diagonalization in time. It is shown that with an optimal choice of the parameter in the head-tail condition, the new parareal algorithm converges rapidly for both linear and nonlinear problems under certain conditions. Numerical experiments for solving PDEs with linear and nonlinear fractional Laplacian and the nonlinear wave equation are included.
Reviewer: Zhiming Chen (Beijing)A meshless computational approach for solving two-dimensional inverse time-fractional diffusion problem with non-local boundary conditionhttps://www.zbmath.org/1475.651042022-01-14T13:23:02.489162Z"Ghehsareh, Hadi Roohani"https://www.zbmath.org/authors/?q=ai:ghehsareh.hadi-roohani"Zabetzadeh, Sayyed Mahmood"https://www.zbmath.org/authors/?q=ai:zabetzadeh.sayyed-mahmoodSummary: This paper is devoted to investigating a two-dimensional inverse anomalous diffusion problem. The missing solely time-dependent Dirichlet boundary condition is recovered by imposing an additional integral measurement over the domain. An efficient computational technique based on a combination of a time integration scheme and local meshless Petrov-Galerkin method is implemented to solve the governing inverse problem. Firstly, an implicit time integration scheme is used to discretize the model in the temporal direction. To fully discretize the model, the primary spatial domain is represented by a set of distributed nodes and data-dependent basis functions are constructed by using the radial point interpolation method. Then, the local meshless Petrov-Galerkin method is used to discretize the problem in the spatial direction. Numerical examples are presented to verify the accuracy and efficiency of the proposed technique. The stability of the method is examined when the input data are contaminated with noise.A fractional-order quasi-reversibility method to a backward problem for the time fractional diffusion equationhttps://www.zbmath.org/1475.651112022-01-14T13:23:02.489162Z"Shi, Wanxia"https://www.zbmath.org/authors/?q=ai:shi.wanxia"Xiong, Xiangtuan"https://www.zbmath.org/authors/?q=ai:xiong.xiangtuan"Xue, Xuemin"https://www.zbmath.org/authors/?q=ai:xue.xueminSummary: In this paper, we consider the regularization of the backward problem of diffusion process with time-fractional derivative. Since the equation under consideration involves the time-fractional derivative, we introduce a new perturbation which is related to the time-fractional derivative into the original equation. This leads to a fractional-order quasi-reversibility method. In theory, we give the regularity of the regularized solution and the corresponding convergence rate is also proved under the appropriate regularization parameter choice rule. In numerics, some numerical experiments are presented to illustrate the effectiveness of our method and some numerical comparison with the existing quasi-reversibility method is conducted. Both theoretical and numerical results show the advantage of the new method.Identifying an unknown source term in a time-space fractional parabolic equationhttps://www.zbmath.org/1475.651122022-01-14T13:23:02.489162Z"Van Thang, Nguyen"https://www.zbmath.org/authors/?q=ai:thang.nguyen-van"Van Duc, Nguyen"https://www.zbmath.org/authors/?q=ai:duc.nguyen-van"Minh, Luong Duy Nhat"https://www.zbmath.org/authors/?q=ai:minh.luong-duy-nhat"Thành, Nguyen Trung"https://www.zbmath.org/authors/?q=ai:thanh.nguyen-trungSummary: An inverse problem of identifying an unknown space-dependent source term in a time-space fractional parabolic equation is considered in this paper. Under reasonable boundedness assumptions about the source function, a Hölder-type stability estimate of optimal order is proved. To regularize the inverse source problem, a mollification regularization method is applied. Error estimates of the regularized solution are proved for both \textit{a priori} and \textit{a posteriori} rules for choosing the mollification parameter. A direct numerical method for solving the regularized problem is proposed and numerical examples are presented to illustrate its effectiveness.Optimal regularity and error estimates of a spectral Galerkin method for fractional advection-diffusion-reaction equationshttps://www.zbmath.org/1475.651212022-01-14T13:23:02.489162Z"Hao, Zhaopeng"https://www.zbmath.org/authors/?q=ai:hao.zhaopeng"Zhang, Zhongqiang"https://www.zbmath.org/authors/?q=ai:zhang.zhongqiangNonlocal models of advection-diffusion-reaction equations with fractional Laplacian are studied, which can be considered as a simplified model for the fractional Navier-Stokes equations. The convergence rate of a spectral Galerkin method is analyzed for the one-dimensional case. Sharp regularity estimates of the solution are proven in weighted Sobolev spaces. Hereby a factorization of the solution is employed. A matrix free iterative solver is likewise proposed which has almost optimal complexity. Different numerical experiments considering smooth right-hand sides, weakly singular ones, or a boundary singularity case, illustrate the theoretical results for different parameter settings.
Reviewer: Kai Schneider (Marseille)Solving a non-linear fractional convection-diffusion equation using local discontinuous Galerkin methodhttps://www.zbmath.org/1475.651302022-01-14T13:23:02.489162Z"Safdari, Hamid"https://www.zbmath.org/authors/?q=ai:safdari.hamid"Rajabzadeh, Majid"https://www.zbmath.org/authors/?q=ai:rajabzadeh.majid"Khalighi, Moein"https://www.zbmath.org/authors/?q=ai:khalighi.moeinSummary: We propose a local discontinuous Galerkin method for solving a nonlinear convection-diffusion equation consisting of a fractional diffusion described by a fractional Laplacian operator of order \(0<p<2\), a nonlinear diffusion, and a nonlinear convection term. The algorithm is developed by the local discontinuous Galerkin method using Spline interpolations to achieve higher accuracy. In this method, we convert the main problem to a first-order system and approximate the outcome by the Galerkin method. In this study, in contrast to the direct Galerkin method using Legender polynomials, we demonstrate that the proposed method can be suitable for the general fractional convection-diffusion problem, remarkably improve stability and provide convergence order \(O(h^{k+1})\), when \(k\) indicates the degree of polynomials. Numerical results have illustrated the accuracy of this scheme and compare it for different conditions.A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equationshttps://www.zbmath.org/1475.651362022-01-14T13:23:02.489162Z"Yin, Baoli"https://www.zbmath.org/authors/?q=ai:yin.baoli"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.11"Li, Hong"https://www.zbmath.org/authors/?q=ai:li.hong"Zeng, Fanhai"https://www.zbmath.org/authors/?q=ai:zeng.fanhaiSummary: A family of novel time-stepping methods for the fractional calculus operators is presented with a shifted parameter. The truncation error with second-order accuracy is proved under the framework of the shifted convolution quadrature. To improve the efficiency, two aspects are considered, that i) a fast algorithm is developed to reduce the computation complexity from \(O(N_t^2)\) to \(O( N_t\log N_t)\) and the memory requirement from \(O(N_t)\) to \(O(\log N_t)\), where \(N_t\) denotes the number of successive time steps, and ii) correction terms are added to deal with the initial singularity of the solution. The stability analysis and error estimates are provided in detail where in temporal direction the novel time-stepping methods are applied and the spatial variable is discretized by the finite element method. Numerical results for \(d\)-dimensional examples \((d=1,2,3)\) confirm our theoretical conclusions and the efficiency of the fast algorithm.An efficient second order stabilized scheme for the two dimensional time fractional Allen-Cahn equationhttps://www.zbmath.org/1475.651452022-01-14T13:23:02.489162Z"Jia, Junqing"https://www.zbmath.org/authors/?q=ai:jia.junqing"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.1|zhang.hui.8|zhang.hui.3|zhang.hui|zhang.hui.7|zhang.hui.11|zhang.hui.2|zhang.hui.10|zhang.hui.4|zhang.hui.5|zhang.hui.6|zhang.hui.9"Xu, Huanying"https://www.zbmath.org/authors/?q=ai:xu.huanying"Jiang, Xiaoyun"https://www.zbmath.org/authors/?q=ai:jiang.xiaoyunSummary: In this paper, we give a stabilized second order scheme for the time fractional Allen-Cahn equation. The scheme uses the fractional backward difference formula (FBDF) for the time fractional derivative and the Legendre spectral method for the space approximation. The nonlinear terms are treated implicitly with a second order stabilized term. Based on the fractional Grönwall inequality, we strictly prove that the proposed scheme converges to second order accuracy in time and spectral accuracy in space. To save computation time and storage, a fast evaluation is developed. Finally, we give some numerical examples to show the configurations of phase field evolution and verify the effectiveness of the proposed methods.Flat morphological operators from non-increasing set operators. I: General theoryhttps://www.zbmath.org/1475.684262022-01-14T13:23:02.489162Z"Ronse, Christian"https://www.zbmath.org/authors/?q=ai:ronse.christianSummary: Flat morphology is a general method for obtaining increasing operators on grey-level or multivalued images from increasing operators on binary images (or sets). It relies on threshold stacking and superposition; equivalently, Boolean max and min operations are replaced by lattice-theoretical sup and inf operations. In this paper we consider the construction a flat operator on grey-level or colour images from an operator on binary images that is not increasing. Here grey-level and colour images are functions from a space to an interval in \(\mathbb{R}^m\) or \(\mathbb{Z}^m\) \((m \geq 1)\). Two approaches are proposed. First, we can replace threshold superposition by threshold summation. Next, we can decompose the non-increasing operator on binary images into a linear combination of increasing operators, then apply this linear combination to their flat extensions. Both methods require the operator to have bounded variation, and then both give the same result, which conforms to intuition. Our approach is very general, it can be applied to linear combinations of flat operators, or to linear convolution filters. Our work is based on a mathematical theory of summation of real-valued functions of one variable ranging in a poset. In a second paper, we will study some particular properties of non-increasing flat operators.Advanced analysis of local fractional calculus applied to the Rice theory in fractal fracture mechanicshttps://www.zbmath.org/1475.740082022-01-14T13:23:02.489162Z"Yang, Xiao-Jun"https://www.zbmath.org/authors/?q=ai:yang.xiao-jun"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-i"Srivastava, H. M."https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohanSummary: In this chapter, the recent results for the analysis of local fractional calculus are considered for the first time. The local fractional derivative (LFD) and the local fractional integral (LFI) in the fractional (real and complex) sets, the series and transforms involving the Mittag-Leffler function defined on Cantor sets are introduced and reviewed. The uniqueness of the solutions of the local fractional differential and integral equations and the local fractional inequalities are considered in detail. The local fractional vector calculus is applied to describe the Rice theory in fractal fracture mechanics.
For the entire collection see [Zbl 1471.93009].The slip flow of generalized Maxwell fluids with time-distributed characteristics in a rotating microchannelhttps://www.zbmath.org/1475.760062022-01-14T13:23:02.489162Z"Feng, Chenqing"https://www.zbmath.org/authors/?q=ai:feng.chenqing"Si, Xinhui"https://www.zbmath.org/authors/?q=ai:si.xinhui"Cao, Limei"https://www.zbmath.org/authors/?q=ai:cao.limei"Zhu, Beibei"https://www.zbmath.org/authors/?q=ai:zhu.beibeiSummary: A time-distributed order fractional continuity model is proposed to simulate the rotating electro-osmotic slip flow of generalized Maxwell fluids in an alternating electric field. The model fully exhibits the wider memory characteristics of viscoelastic fluid. Complex momentum equation is discretized by L1 and L2 algorithms and verified by constructing analytical solutions method. Due to the stronger memory of generalized Maxwell fluids, reverse flow is restrained, and the time for oscillation reaching the steady state is postponed. Moreover, with the increase of Debye-Hückle coefficient the velocity will increase rapidly.Numerical study for the unsteady space fractional magnetohydrodynamic free convective flow and heat transfer with Hall effectshttps://www.zbmath.org/1475.760852022-01-14T13:23:02.489162Z"Chi, Xiaoqing"https://www.zbmath.org/authors/?q=ai:chi.xiaoqing"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui|zhang.hui.4|zhang.hui.6|zhang.hui.2|zhang.hui.10|zhang.hui.3|zhang.hui.5|zhang.hui.7|zhang.hui.9|zhang.hui.8|zhang.hui.1|zhang.hui.11Summary: In this paper, a numerical research for the problem of unsteady space fractional magnetohydrodynamic (MHD) free convective flow and heat transfer with Hall effects is investigated. We firstly establish a space fractional MHD flow and heat transfer model, which is coupled by the incompressible space fractional Navier-Stokes equations and the heat conduction equation. Then we develop a finite difference spectral decomposition method based on the pressure correction algorithm to solve this nonlinear coupled model. The validity of the proposed numerical scheme is verified and some flow field, temperature field are obtained. We also detailedly analyze the effects of relevant parameters on the space fractional MHD flow and heat transfer.Karush-Kuhn-Tucker type optimality condition for quasiconvex programming in terms of Greenberg-Pierskalla subdifferentialhttps://www.zbmath.org/1475.901312022-01-14T13:23:02.489162Z"Suzuki, Satoshi"https://www.zbmath.org/authors/?q=ai:suzuki.satoshi.3|suzuki.satoshi.2|suzuki.satoshi.1This work deals with the finite-dimensional constrained optimization problem of minimizing an objective function \(f\) on the set \(K=\{x\in \mathbb{R}^{n}:\,g_{i}(x)\leq 0,\,\forall i\in I\}\) where all functions \(f\) and \(g_{i}\), \(i\in I,\) are upper semicontinuous and essentially quasiconvex (that is, they have convex sublevel sets and every local minimum is global). The paper presents necessary and sufficient conditions for the optimality in terms of the Greenberg-Pierskalla subdifferential, which is a subdifferential that had previously used in quasiconvex analysis.
Reviewer: Aris Daniilidis (Vienna)Application of Caputo-Fabrizio derivative to a cancer model with unknown parametershttps://www.zbmath.org/1475.920452022-01-14T13:23:02.489162Z"El-Dessoky, M. M."https://www.zbmath.org/authors/?q=ai:el-dessoky.mohamed-m"Khan, Muhammad Altaf"https://www.zbmath.org/authors/?q=ai:khan.muhammad-altafSummary: The present work explore the dynamics of the cancer model with fractional derivative. The model is formulated in fractional type of Caputo-Fabrizio derivative. We analyze the chaotic behavior of the proposed model with the suggested parameters. Stability results for the fixed points are shown. A numerical scheme is implemented to obtain the graphical results in the sense of Caputo-Fabrizio derivative with various values of the fractional order parameter. Further, we show the graphical results in order to study that the model behave the periodic and quasi periodic limit cycles as well as chaotic behavior for the given set of parameters.Stability analysis of a fractional-order cancer model with chaotic dynamicshttps://www.zbmath.org/1475.920512022-01-14T13:23:02.489162Z"Naik, Parvaiz Ahmad"https://www.zbmath.org/authors/?q=ai:naik.parvaiz-ahmad"Zu, Jian"https://www.zbmath.org/authors/?q=ai:zu.jian"Naik, Mehraj-ud-din"https://www.zbmath.org/authors/?q=ai:naik.mehraj-ud-dinA fractional-order model with time delay for tuberculosis with endogenous reactivation and exogenous reinfectionshttps://www.zbmath.org/1475.921512022-01-14T13:23:02.489162Z"Chinnathambi, Rajivganthi"https://www.zbmath.org/authors/?q=ai:chinnathambi.rajivganthi"Rihan, Fathalla A."https://www.zbmath.org/authors/?q=ai:rihan.fathalla-a"Alsakaji, Hebatallah J."https://www.zbmath.org/authors/?q=ai:alsakaji.hebatallah-jSummary: In this paper, we propose a fractional-order delay differential model for tuberculosis (TB) transmission with the effects of endogenous reactivation and exogenous reinfections. We investigate the qualitative behaviors of the model throughout the local stability of the steady states and bifurcation analysis. A discrete time delay is introduced in the model to justify the time taken for diagnosis of the disease. Existence and positivity of the solutions are investigated. Some interesting sufficient conditions that ensure the local asymptotic stability of infection-free and endemic steady states are studied. The fractional-order TB model undergoes Hopf bifurcation with respect to time delay and disease transmission rate. The presence of fractional order and time delay in the model improves the model behaviors and develops the stability results. A numerical example is provided to support our theoretical results.Analyzing a novel coronavirus model (COVID-19) in the sense of Caputo-Fabrizio fractional operatorhttps://www.zbmath.org/1475.921542022-01-14T13:23:02.489162Z"Dokuyucu, Mustafa Ali"https://www.zbmath.org/authors/?q=ai:dokuyucu.mustafa-ali"Çelik, Ercan"https://www.zbmath.org/authors/?q=ai:celik.ercanSummary: In this paper, a new model has been proposed to analyse the infection due to the coronavirus (COVID-19). The model emphasizes the importance of environmental reservoir in spreading the infection and infecting others. It also keeps control measures regarding infection at the highest level by using non-constant transmission rates in the model. The analysis of the coronavirus model has been done via Caputo-Fabrizio fractional derivative operator. The existence of solutions of the model has been examined by using a fixed-point approach and the uniqueness of the solution has also been obtained. Further, the stability analysis of the model has been performed in the sense of Hyers-Ulam stability. Finally, the numerical solution has been obtained by using the Adam-Basford numerical approach, and also simulations for different fractional derivative values have been carried out. As a result, the mathematical modeling of the new type of coronavirus (COVID-19) has been applied to fractional-order derivatives and integral operators and its simulations with the real data have been shown.Chaotic behaviors of the prevalence of an infectious disease in a prey and predator system using fractional derivativeshttps://www.zbmath.org/1475.921632022-01-14T13:23:02.489162Z"Ghanbari, Behzad"https://www.zbmath.org/authors/?q=ai:ghanbari.behzadSummary: The risk of spread infectious diseases in the environment is always one of the main threats to the life of living organisms. This point can be assumed as a clear proof of the importance of studying such problems from various aspects such as computational mathematical models. In this contribution, we examine a mathematical model to investigate the prevalence of an infectious disease in a prey and predator system, including three subpopulations through a fractional system of nonlinear equations. The model characterizes a possible interaction between predator and prey where infectious disease outbreaks in a community. Further, the prey population is divided into two: susceptible and the infected population. This model involves the Caputo-Fabrizio derivative. One of the basic features of this type of derivative is the use of a non-singular (exponential) kernel, which increases its ability to describe phenomena compared to other existing operators. To the best of the author's knowledge, the use of fractional derivative operators for this model has not yet been investigated. Therefore, the presented results can be considered as new and interesting results for this model. In some of the acquired simulations, the chaotic behaviors are clearly detectable.Controllability of fractional non-instantaneous impulsive differential inclusions without compactnesshttps://www.zbmath.org/1475.930182022-01-14T13:23:02.489162Z"Wang, JinRong"https://www.zbmath.org/authors/?q=ai:wang.jinrong"Ibrahim, A. G."https://www.zbmath.org/authors/?q=ai:ibrahim.ahmed-gamal"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michal"Zhou, Yong"https://www.zbmath.org/authors/?q=ai:zhou.yongSummary: In this paper, we study the controllability for a system governed by fractional non-instantaneous nonlinear impulsive differential inclusions in Banach spaces. We adopt a new approach to derive the controllability results under weak conditions by establishing a new version weakly convergent criteria in the piecewise continuous functions spaces. In particular, we emphasize that we do not assume any regularity conditions on the multivalued non-linearity expressed in terms of measures of non-compactness. Moreover, unlike the previous literatures, we also do not restrict that the invertibility of the linear controllability operator satisfies a condition expressed in terms of measures of non-compactness. It allows us to apply the weakly topology theory for weakly sequentially closed graph operator and to obtain the controllability results for both upper weakly sequentially closed and relatively weakly compact types of non-linearity.Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaceshttps://www.zbmath.org/1475.930192022-01-14T13:23:02.489162Z"Zhou, Yong"https://www.zbmath.org/authors/?q=ai:zhou.yong|zhou.yong.2|zhou.yong.4|zhou.yong.3|zhou.yong.1"Suganya, S."https://www.zbmath.org/authors/?q=ai:suganya.selvaraj"Arjunan, M. Mallika"https://www.zbmath.org/authors/?q=ai:arjunan.mani-mallika|mallika-arjunan.m"Ahmad, B."https://www.zbmath.org/authors/?q=ai:ahmad.babar|ahmad.bakhtiar|ahmad.bashir.2|ahmad.bashar-i|ahmad.bilal|ahmad.bashir.1|ahmad.bashir.3|ahmad-khuda-bakhsh.bashir|ahmad.bilas|ahmad.bashair|ahmad.bashir.4Summary: In this paper, the problem of approximate controllability for non-linear impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces is investigated. We study the approximate controllability for non-linear impulsive integro-differential systems under the assumption that the corresponding linear control system is approximately controllable. By utilizing the methods of fractional calculus, semigroup theory, fixed-point theorem coupled with solution operator, sufficient conditions are formulated and proved. Finally, an example is provided to illustrate the proposed theory.Conditions and a computation method of the constrained regulation problem for a class of fractional-order nonlinear continuous-time systemshttps://www.zbmath.org/1475.930572022-01-14T13:23:02.489162Z"Si, Xindong"https://www.zbmath.org/authors/?q=ai:si.xindong"Yang, Hongli"https://www.zbmath.org/authors/?q=ai:yang.hongli"Ivanov, Ivan G."https://www.zbmath.org/authors/?q=ai:ivanov.ivan-ganchevThe authors study stability conditions and methods of controller synthesis to ensure the stability performance of the resulting system under some special state-and-input constraints. The authors call this problem the constraint regulation problem (CRP). A control law is a solution to the CRP for the system if and only if the closed-loop system is asymptotically stable at the origin, while the corresponding trajectory does not violate the constraints.
In the paper CRP for fractional-order nonlinear continuous-time systems is investigated. The system is assumed to be affine in control. The linear part of the system is employed for designing a linear state feedback controller that provides asymptotic stability of the closed-loop system.
First, three theorems with conditions guaranteeing positive invariance of some polyhedral sets for the systems are established, using the comparison principle and positively invariant set theory.
Then, the authors propose a controller model for the CRP and the corresponding algorithm, by using the obtained invariance conditions. The presented model is formulated as a linear programming problem, which can be easily implemented from a computational point of view. At last, numerical examples illustrate the proposed method.
Reviewer: Nataliya O. Sedova (Ulyanovsk)Robust stability analysis of a general class of interval delayed fractional order plants by a general form of fractional order controllershttps://www.zbmath.org/1475.930862022-01-14T13:23:02.489162Z"Ghorbani, Majid"https://www.zbmath.org/authors/?q=ai:ghorbani.majid"Tavakoli-Kakhki, Mahsan"https://www.zbmath.org/authors/?q=ai:tavakoli-kakhki.mahsanSummary: In this paper, the robust stability of interval fractional order plants with one time delay controlled by fractional order controllers is investigated in a general form. For robust stability analysis of the closed loop system by the zero exclusion principle, the distance between the origin and the value set of the characteristic function needs to be checked. It is known that the outer vertices of this value set may change at some switching frequencies and the repetitive calculation of these vertices at switching frequencies leads to additional calculations. In this study initially, new necessary and sufficient conditions are proposed to check the robust stability of a delayed fractional order closed loop system. Then, a novel robust stability testing function is presented based on some vertices, which are fixed for all positive frequencies. Therefore, no additional calculation is needed to obtain the outer vertices of the characteristic function value set for any pair of the switching frequencies. Also, a finite frequency range is presented to reduce the computational cost noticeably. Eventually, three numerical examples are given to verify the efficiency of the results of this paper.The geometric mean value theoremhttps://www.zbmath.org/1475.970252022-01-14T13:23:02.489162Z"de Camargo, André Pierro"https://www.zbmath.org/authors/?q=ai:de-camargo.andre-pierroSummary: In a previous article published in the American Mathematical Monthly, \textit{T. W. Tucker} [Am. Math. Mon. 104, No. 3, 231--240 (1997; Zbl 0871.26004)] made severe criticism on the Mean Value Theorem and, unfortunately, the majority of calculus textbooks also do not help to improve its reputation. The standard argument for proving it seems to be applying Rolle's theorem to a function like
\[
h(x) : = f (x) - f (a) - \frac{f(b) - f (a)}{b - a} ( x - a ).
\]
Although short and effective, such reasoning is not intuitive. Perhaps for this reason, Tucker classified the Mean Value Theorem as \textit{a technical existence theorem used to prove intuitively obvious statements.} Moreover, he argued that \textit{there is nothing obvious about the Mean Value Theorem without the continuity of the derivative}. Under so unfair discrimination, we felt the need to come to the defense of this beautiful theorem in order to clear up these misunderstandings.Iterated exponential inequalities: a classroom-tailored approachhttps://www.zbmath.org/1475.970272022-01-14T13:23:02.489162Z"Ghergu, Marius"https://www.zbmath.org/authors/?q=ai:ghergu.mariusSummary: We explore the connection between the notion of critical point for a function of one variable and various inequalities for iterated exponentials defined on the positive semiline of real numbers.Some variants of Cauchy's mean value theoremhttps://www.zbmath.org/1475.970282022-01-14T13:23:02.489162Z"Lozada-Cruz, German"https://www.zbmath.org/authors/?q=ai:lozada-cruz.germanSummary: In this note, some variants of Cauchy's mean value theorem are proved. The main tools to prove these results are some elementary auxiliary functions.Why does trigonometric substitution work?https://www.zbmath.org/1475.970302022-01-14T13:23:02.489162Z"Cunningham, Daniel W."https://www.zbmath.org/authors/?q=ai:cunningham.daniel-wSummary: Modern calculus textbooks carefully illustrate how to perform integration by trigonometric substitution. Unfortunately, most of these books do not adequately justify this powerful technique of integration. In this article, we present an accessible proof that establishes the validity of integration by trigonometric substitution. The proof offers calculus instructors a simple argument that can be used to show their students that trigonometric substitution is a valid technique of integration.A shortcut for solving the ubiquitous integral \(\int x^n\mathrm{e}^{\alpha x} \mathrm{d}x\) using only derivativeshttps://www.zbmath.org/1475.970322022-01-14T13:23:02.489162Z"Lima, F. M. S."https://www.zbmath.org/authors/?q=ai:lima.fabio-m-sSummary: In this note, I present an `easy-to-be-remembered' shortcut for promptly solving the ubiquitous integral \(\int x^n \mathrm{e}^{\alpha x} \mathrm{d}x\) for any integer \(n>0\) using only the successive derivatives of \(x^n\). Some interesting applications are indicated. The shortcut is so simple that it could well be included in calculus textbooks and classes directed to first year undergraduates.Poisson's fundamental theorem of calculus via Taylor's formulahttps://www.zbmath.org/1475.970332022-01-14T13:23:02.489162Z"Nystedt, P."https://www.zbmath.org/authors/?q=ai:nystedt.patrikSummary: We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left endpoints which are equally spaced. We discuss potential benefits for such an approach in basic calculus courses.On the smoothness condition in Euler's theorem on homogeneous functionshttps://www.zbmath.org/1475.970352022-01-14T13:23:02.489162Z"Dobbs, David E."https://www.zbmath.org/authors/?q=ai:dobbs.david-earlSummary: For a function \( f : \mathbb{R}^n \setminus \{ ( 0 , \cdots , 0 ) \} \to \mathbb{R}\) with continuous first partial derivatives, a theorem of Euler characterizes when \(f\) is a homogeneous function. This note determines whether the conclusion of Euler's theorem holds if the smoothness of \(f\) is not assumed. An example is given to show that if \(n \geq 2\), a homogeneous function (of any degree) need not be differentiable (and so the conclusion of Euler's theorem would fail for such a function). By way of contrast, it is shown that if \(n = 1\), a homogeneous function (of any degree) must be differentiable (and so Euler's theorem does not need to assume the smoothness of \(f\) if \(n = 1)\). Additional characterizations of homogeneous functions, remarks and examples illustrate the theory, emphasizing differences in behaviour between the contexts \(n \geq 2\) and \(n = 1\). This note could be used as enrichment material in calculus courses and possibly some science courses.