Recent zbMATH articles in MSC 26https://zbmath.org/atom/cc/262022-09-13T20:28:31.338867ZUnknown authorWerkzeugMathematics and statistics for sciencehttps://zbmath.org/1491.000062022-09-13T20:28:31.338867Z"Sneyd, James"https://zbmath.org/authors/?q=ai:sneyd.james"Fewster, Rachel M."https://zbmath.org/authors/?q=ai:fewster.rachel-m"McGillivray, Duncan"https://zbmath.org/authors/?q=ai:mcgillivray.duncanPublisher's description: Mathematics and statistics are the bedrock of modern science. No matter which branch of science you plan to work in, you simply cannot avoid quantitative approaches. And while you won't always need to know a great deal of theory, you will need to know how to apply mathematical and statistical methods in realistic scenarios. That is precisely what this book teaches. It covers the mathematical and statistical topics that are ubiquitous in early undergraduate courses, but does so in a way that is directly linked to science.
Beginning with the use of units and functions, this book covers key topics such as complex numbers, vectors and matrices, differentiation (both single and multivariable), integration, elementary differential equations, probability, random variables, inference and linear regression. Each topic is illustrated with widely-used scientific equations (such as the ideal gas law or the Nernst equation) and real scientific data, often taken directly from recent scientific papers. The emphasis throughout is on practical solutions, including the use of computational tools (such as Wolfram Alpha or R), not theoretical development. There is a large number of exercises, divided into mathematical drills and scientific applications, and full solutions to all the exercises are available to instructors.
Mathematics and Statistics for Science covers the core methods in mathematics and statistics necessary for a university degree in science, highlighting practical solutions and scientific applications. Its pragmatic approach is ideal for students who need to apply mathematics and statistics in a real scientific setting, whether in the physical sciences, life sciences or medicine.Book review of: S. M. Steward, How to integrate it. A practical guide to finding elementary integralshttps://zbmath.org/1491.000182022-09-13T20:28:31.338867Z"Pineda-Villavicencio, Guillermo"https://zbmath.org/authors/?q=ai:pineda-villavicencio.guillermoReview of [Zbl 1404.26004].Book review of: W. Urbina-Romero, Gaussian harmonic analysishttps://zbmath.org/1491.000222022-09-13T20:28:31.338867Z"Rindler, H."https://zbmath.org/authors/?q=ai:rindler.haraldReview of [Zbl 1421.42001].Book review of: V. Benci and M. Di Nasso, How to measure the infinite. Mathematics with infinite and infinitesimal numbershttps://zbmath.org/1491.000342022-09-13T20:28:31.338867Z"Wenmackers, Sylvia"https://zbmath.org/authors/?q=ai:wenmackers.sylviaReview of [Zbl 1429.26001].Means of fuzzy numbers in the fuzzy information evaluation problemhttps://zbmath.org/1491.030572022-09-13T20:28:31.338867Z"Khatskevich, V. L."https://zbmath.org/authors/?q=ai:khatskevich.vladimir-lSummary: Based on means of systems of fuzzy numbers, we introduce and study a class of averaging functionals for the implementation of the fuzzy information evaluation problem. It is shown that these functionals have a number of special properties: idempotency, monotonicity, continuity, etc., typical of scalar aggregating functions.On the smallest base in which a number has a unique expansionhttps://zbmath.org/1491.110102022-09-13T20:28:31.338867Z"Allaart, Pieter"https://zbmath.org/authors/?q=ai:allaart.pieter-c"Kong, Derong"https://zbmath.org/authors/?q=ai:kong.derongThis paper is concerned with the expansion of a real number \(x\in[0,1/(q-1)]\) in base \(q\in(1,2]\), called \(q\)-expansion. That is, we write
\[
x=\sum_{i\ge1}\frac{d_i}{q^i}, \quad\text{where } d_i\in\{0,1\}\quad\text{for all } i\geq 1.
\]
Such expansions are usually highly non-unique: for each \(q\in(1,2)\), almost all \(x\in[0,1/(q-1)]\) have continuum many \(q\)-expansions.
What is the infimum of the set of bases \(q\) such that \(x\) has a unique \(q\)-expansion? The central quantities in the paper under review are
\[
q_s(x):=\inf \bigl\{ q\in(1,2]: x\mbox{ has a unique \(q\)-expansion} \bigr\}
\]
and
\[
L(q):=\bigl\{x>0:q_s(x)=q\bigr\}.
\]
A number of results is proved for these quantities, including questions on cardinality, continuity, accumulation points, extreme values, monotonicity, the description and investigation of an algorithm for computing \(q_s(x)\), and a connection to de Vries-Komornik numbers. Moreover, the cases where the infimum in the definition of \(q_s(x)\) is in fact a minimum is investigated closely. Among other things, certain subsets of the the graph of \(q_s\) are studied, so-called \textit{Komornik-Loreti cascades}.
Reviewer: Lukas Spiegelhofer (Wien)2-variable Fubini-degenerate Apostol-type polynomialshttps://zbmath.org/1491.110312022-09-13T20:28:31.338867Z"Nahid, Tabinda"https://zbmath.org/authors/?q=ai:nahid.tabinda"Ryoo, Cheon Seoung"https://zbmath.org/authors/?q=ai:ryoo.cheon-seoungOn the maximum of cotangent sums related to the Riemann hypothesis in rational numbers in short intervalshttps://zbmath.org/1491.110772022-09-13T20:28:31.338867Z"Maier, Helmut"https://zbmath.org/authors/?q=ai:maier.helmut"Rassias, Michael Th."https://zbmath.org/authors/?q=ai:rassias.michael-thLet
\[
c_0\left(\frac{r}{b}\right)=-\sum_{m=1}^{b-1}\frac{m}{b}\cot\left(\frac{\pi mr}{b}\right).
\]
In this paper the authors obtain lower bounds for \(\max|c_0\left(\frac{r}{b}\right)|\) where (i) the numerator \(r\) is restricted to the sequence of prime numbers, and (ii) fractions \(\frac{r}{b}\) simultaneously varying the numerator \(r\) and the denominator \(b\).
Reviewer: Mehdi Hassani (Zanjan)Real analysis and foundationshttps://zbmath.org/1491.260012022-09-13T20:28:31.338867Z"Krantz, Steven G."https://zbmath.org/authors/?q=ai:krantz.steven-georgePublisher's description: Through four editions this popular textbook attracted a loyal readership and widespread use. Students find the book to be concise, accessible, and complete. Instructors find the book to be clear, authoritative, and dependable.
The primary goal of this new edition remains the same as in previous editions. It is to make real analysis relevant and accessible to a broad audience of students with diverse backgrounds while also maintaining the integrity of the course. This text aims to be the generational touchstone for the subject and the go-to text for developing young scientists.
This new edition continues the effort to make the book accessible to a broader audience. Many students who take a real analysis course do not have the ideal background. The new edition offers chapters on background material like set theory, logic, and methods of proof. The more advanced material in the book is made more apparent.
This new edition offers a new chapter on metric spaces and their applications. Metric spaces are important in many parts of the mathematical sciences, including data mining, web searching, and classification of images.
The author also revised the material on sequences and series adding examples and exercises that compare convergence tests and give additional tests.
The text includes rare topics such as wavelets and applications to differential equations. The level of difficulty moves slowly, becoming more sophisticated in later chapters. Students have commented on the progression as a favorite aspect of the textbook.
The author is perhaps the most prolific expositor of upper division mathematics. With over seventy books in print, thousands of students have been taught and learned from his books.
See the reviews of the 1st, 2nd and 3rd editions in [Zbl 0757.26002; Zbl 1056.26001; Zbl 1278.26001]. For the 4th edition see [Zbl 1348.26004].About the elements of analysis. Standard and non-standardhttps://zbmath.org/1491.260022022-09-13T20:28:31.338867ZPublisher's description: Nonstandard fordert Standard heraus. Das war immer so. So ist es auch in der Analysis.
Nonstandardanalysis fordert 100-jährige Routinen in den Elementen der Standardanalysis heraus. Wir entwickeln die Elemente -- standard neben nonstandard -- aus den Grundideen, zeigen ihre Anwendungen in vielen Beispielen und vergleichen beide Ansätze.
Wir stellen fest: Nonstandard erweitert und bereichert Standard. Zentral ist: Grenzprozesse, die Grundelemente der Standardanalysis, werden nonstandard zu Zahlen mit einer elementaren, anschaulich begleiteten Arithmetik. Sie bilden die Basis für die Bildung der Begriffe in Differential- und Integralrechnung. In einer Handreichung auf der Webseite nichtstandard.de stellen wir erprobte Unterrichtsgänge vor.
Nonstandard ist mathematisch längst Standard, aber noch selten in der Lehre und im Unterricht anzutreffen. Dass sich hier etwas bewegt, dafür ist dieses Lehrbuch geschrieben -- für die Praxis aus der Praxis. Es berichtet aus dem mathematischen, methodischen und historischen Hintergrund und wendet sich an Lehrende, Studierende und alle, die alte Routinen durchschauen und neue Elemente der Analysis entdecken wollen.
The articles of this volume will not be indexed individually.Comparison of some families of real functions in algebraic termshttps://zbmath.org/1491.260032022-09-13T20:28:31.338867Z"Filipczak, Małgorzata"https://zbmath.org/authors/?q=ai:filipczak.malgorzata"Ivanova, Gertruda"https://zbmath.org/authors/?q=ai:ivanova.gertrudaSummary: We compare families of functions related to the Darboux property (functions having the \(\mathcal A\)-Darboux property) with family of strong Świątkowski functions using the notions of strong \(\mathfrak c\)-algebrability. We also compare families of functions associated with density topologies.A note to the Sierpiński first class of functionshttps://zbmath.org/1491.260042022-09-13T20:28:31.338867Z"Menkyna, Robert"https://zbmath.org/authors/?q=ai:menkyna.robertSummary: The purpose of this paper is to establish some theorems concerning approximation and representation of a function of the Sierpiński first class by Darboux function of the Sierpiński first class.A coupled Caputo-Hadamard fractional differential system with multipoint boundary conditionshttps://zbmath.org/1491.260052022-09-13T20:28:31.338867Z"Aibout, Samir"https://zbmath.org/authors/?q=ai:aibout.samir"Abbas, Saïd"https://zbmath.org/authors/?q=ai:abbas.said"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Bohner, Martin"https://zbmath.org/authors/?q=ai:bohner.martin-jSummary: This paper deals with existence of solutions for a coupled system of Caputo-Hadamard fractional differential equations with multipoint boundary conditions in Banach spaces. Some applications are made using some fixed point theorems on Banach spaces. An illustrative example is presented in the last section.On a discrete composition of the fractional integral and Caputo derivativehttps://zbmath.org/1491.260062022-09-13T20:28:31.338867Z"Płociniczak, Łukasz"https://zbmath.org/authors/?q=ai:plociniczak.lukaszSummary: We prove a discrete analogue for the composition of the fractional integral and Caputo derivative. This result is relevant in numerical analysis of fractional PDEs when one discretizes the Caputo derivative with the so-called L1 scheme. The proof is based on asymptotic evaluation of the discrete sums with the use of the Euler-Maclaurin summation formula.On two times differentiable preinvex and prequasiinvex functionshttps://zbmath.org/1491.260072022-09-13T20:28:31.338867Z"Işcan, Imdat"https://zbmath.org/authors/?q=ai:iscan.imdat"Kadakal, Mahir"https://zbmath.org/authors/?q=ai:kadakal.mahir"Kadakal, Huriye"https://zbmath.org/authors/?q=ai:kadakal.huriyeSummary: The main goal of this paper is to establish a new identity for functions defined on an open invex subset of real numbers. By using this identity, the Hölder integral inequality and power mean integral inequality, we introduce some new type integral inequalities for functions whose powers of second derivatives in absolute values are preinvex and prequasiinvex.Note on different kinds of Schur convexities of the Heinz type meanhttps://zbmath.org/1491.260082022-09-13T20:28:31.338867Z"Sridevi, K."https://zbmath.org/authors/?q=ai:sridevi.k"Nagaraja, K. M."https://zbmath.org/authors/?q=ai:nagaraja.k-m"Reddy, P. Siva Kota"https://zbmath.org/authors/?q=ai:reddy.p-siva-kotaSummary: The Schur convexity of functions relating to special means is a very significant research subject and has attracted the interest of many mathematicians. In this note, a new family of one parameterized Heinz type mean is introduced and we discuss the different kinds of Schur convexity and concavity of Heinz type mean.Row-summable matrices with application to generalization of Schröder's and Abel's functional equationshttps://zbmath.org/1491.260092022-09-13T20:28:31.338867Z"Eshkaftaki, Ali Bayati"https://zbmath.org/authors/?q=ai:eshkaftaki.ali-bayatiIn this paper, the functional equation \[ \sum_n \alpha_n(x) f(u_n(x))=g(x) \] is considered. The unknown function is \(f:Y\to \mathbb{C},\) while \(u_n:X\to Y,\) \(\alpha_n:X\to \mathbb{C},\) \(n\in \mathbb{N},\) and \(g:X\to \mathbb{C}\) are given; \(X,Y\) are nonempty sets. In the main theorem, it is shown that, under suitable conditions on the functions involved, there exists a unique bounded solution to the functional equation. Essential tools in the proof are row-summable matrices and their related bounded linear operators.
Reviewer: Rita Pini (Milano)On the linear structures induced by the four order isomorphisms acting on \(\mathrm{Cvx}_0(\mathbb{R}^n)\)https://zbmath.org/1491.260102022-09-13T20:28:31.338867Z"Florentin, Dan I."https://zbmath.org/authors/?q=ai:florentin.dan-itzhak"Segal, Alexander"https://zbmath.org/authors/?q=ai:segal.alexander.2Motivated by the fact that the volume functional \(\phi \mapsto \int e^{-\phi}\) satisfies some concavity/convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on \(Cvx_0(\mathbb{R}^n)\), the authors propose a fourth linear structure in the same setting as the pullback of the standard linear structure under the \(\mathcal{J}\) transform, and show that no concavity or convexity inequalities can be shown with respect to it, a quasi-convexity inequality being actually violated only by up to a factor of 2. The order relations satisfied by the four different interpolations are provided, too.
Reviewer: Sorin-Mihai Grad (Paris)Affine invariant maps for log-concave functionshttps://zbmath.org/1491.260112022-09-13T20:28:31.338867Z"Li, Ben"https://zbmath.org/authors/?q=ai:li.ben"Schütt, Carsten"https://zbmath.org/authors/?q=ai:schutt.carsten"Werner, Elisabeth M."https://zbmath.org/authors/?q=ai:werner.elisabeth-mAffine invariant points and maps for sets, which were introduced by B. Grünbaum in the 1960s to study the symmetry structure of convex sets, are central in affine differential geometry and convex geometry and they and their associated inequalities have far reaching consequences for many other areas of mathematics. The authors of the present paper extend these notions to functions. While the corresponding definitions can be given for any function, the authors first concentrate on log-concave functions. The definitions of affine contravariant points and affine covariant mappings for log-concave functions are given and some of their basic properties are established. For instance, the role of symmetry in the setting of convex bodies is now taken by the notion of evenness in the functional setting. Some classical examples of affine contravariant points for functions are presented. It is shown that the centroid and the Santaló point of a log-concave function are examples of affine contravariant points and that the newly developed notions of floating function, John function, and Löwner function are examples of affine covariant mappings. This leads naturally to new affine contravariant points.
Reviewer: Alexey Alimov (Moskva)Annular bounds for the zeros of a polynomial from companion matriceshttps://zbmath.org/1491.260122022-09-13T20:28:31.338867Z"Bhunia, Pintu"https://zbmath.org/authors/?q=ai:bhunia.pintu"Paul, Kallol"https://zbmath.org/authors/?q=ai:paul.kallolThe authors find several upper bounds for the moduli of the zeros of a monic univariate complex polynomial in terms of the coefficients and the degree, and hence determine annuli in the complex plane that contain all the zeros. The proofs are based on matrix inequalities for the spectral and numerical radius and the spectral norm which are applied to the Frobenius companion matrix of the polynomial.
Reviewer: Armin Rainer (Wien)A note on smooth transcendental approximation to \(|x|\)https://zbmath.org/1491.260132022-09-13T20:28:31.338867Z"Bagul, Yogesh J."https://zbmath.org/authors/?q=ai:bagul.yogesh-j"Khairnar, Bhavna K."https://zbmath.org/authors/?q=ai:khairnar.bhavna-kSummary: In this review paper, we present a pellucid proof of how \(x\tanh(x/\mu)\) approximates \(|x|\) and is better than \(\sqrt{x^2+ \mu}\) when we are concerned with accuracy.Unifications of continuous and discrete fractional inequalities of the Hermite-Hadamard-Jensen-Mercer type via majorizationhttps://zbmath.org/1491.260142022-09-13T20:28:31.338867Z"Faisal, Shah"https://zbmath.org/authors/?q=ai:faisal.shah"Khan, Muhammad Adil"https://zbmath.org/authors/?q=ai:khan.muhammad-adil"Khan, Tahir Ullah"https://zbmath.org/authors/?q=ai:khan.tahir-ullah"Saeed, Tareq"https://zbmath.org/authors/?q=ai:saeed.tareq"Mohammad Mahdi Sayed, Zaid Mohammmad"https://zbmath.org/authors/?q=ai:mohammad-mahdi-sayed.zaid-mohammmadAn innovative idea of bringing the continuous and discrete inequalities into a unified form is presented. In particular, new unified forms of Hermite-Hadamard-Jensen-Mercer type inequalities are derived and proved using the concept of majorization techniques in the context of Caputo fractional operators. These results are achieved by embedding majorization theory with the existing notion of continuous and discrete inequalities. In addition, some new identities for differentiable functions are derived. Using these identities and by considering the convexity of \(\mid \phi^{(n+1)}\mid\) and \(\phi^{(n+1)}\mid^q\)~ (q > 1), bounds for the absolute difference of the right- and left-sides of the main results established in this paper are provided.
Reviewer: James Adedayo Oguntuase (Abeokuta)Bessel-type operators and a refinement of Hardy's inequalityhttps://zbmath.org/1491.260152022-09-13T20:28:31.338867Z"Gesztesy, Fritz"https://zbmath.org/authors/?q=ai:gesztesy.fritz"Pang, Michael M. H."https://zbmath.org/authors/?q=ai:pang.michael-m-h"Stanfill, Jonathan"https://zbmath.org/authors/?q=ai:stanfill.jonathanSummary: The principal aim of this paper is to employ Bessel-type operators in proving the inequality
\[
\begin{aligned} \int_0^\pi dx |f'(x)|^2 \geq \frac{1}{4} \int_0^\pi dx \frac{|f(x)|^2}{\sin^2 (x)}+\frac{1}{4} \int_0^\pi dx |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{aligned}
\]
where both constants \(1/4\) appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if \(f \equiv 0\). This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schrödinger operator associated with the differential expression
\[
\begin{aligned} \tau_s=-\frac{d^2}{dx^2}+\frac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), x \in (0,\pi). \end{aligned}
\]
The new inequality represents a refinement of Hardy's classical inequality
\[
\begin{aligned} \int_0^\pi dx |f'(x)|^2 \geq \frac{1}{4}\int_0^\pi dx \frac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{aligned}
\]
and it also improves upon one of its well-known extensions in the form
\[
\begin{aligned} \int_0^\pi dx |f'(x)|^2 \geq \frac{1}{4}\int_0^\pi dx \frac{|f(x)|{}^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{aligned}
\]
where \(d_{(0,\pi)}(x)\) represents the distance from \(x \in (0,\pi)\) to the boundary \(\{0,\pi\}\) of \((0,\pi)\).
For the entire collection see [Zbl 1479.47003].Certain inequalities for fractional \((p,q)\)-calculushttps://zbmath.org/1491.260162022-09-13T20:28:31.338867Z"Jain, Pankaj"https://zbmath.org/authors/?q=ai:jain.pankaj"Manglik, Rohit"https://zbmath.org/authors/?q=ai:manglik.rohit(no abstract)Some Caputo \(k\)-fractional derivatives of Ostrowski type concerning \((n+1)\)-differentiable generalized relative semi-\((r; m, p, q, h_1, h_2)\)-preinvex mappingshttps://zbmath.org/1491.260172022-09-13T20:28:31.338867Z"Kashuri, Artion"https://zbmath.org/authors/?q=ai:kashuri.artion"Liko, Rozana"https://zbmath.org/authors/?q=ai:liko.rozanaSummary: In this article, we first presented some integral inequalities for Gauss-Jacobi type quadrature formula involving generalized relative semi-\((r; m,p, q, h_1, h_2)\)-preinvex mappings. And then, a new identity concerning \((n+1)\)-differentiable mappings defined on \(m\)-invex set via Caputo \(k\)-fractional derivatives is derived. By using the notion of generalized relative semi-\((r; m, p, q,h_1, h_2)\)-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Ostrowski type inequalities via Caputo \(k\)-fractional derivatives are established. It is pointed out that some new special cases can be deduced from main results of the article.New bounds for generalized Taylor expansionshttps://zbmath.org/1491.260182022-09-13T20:28:31.338867Z"Barić, J."https://zbmath.org/authors/?q=ai:baric.josipa"Kvesić, Ljiljanka"https://zbmath.org/authors/?q=ai:kvesic.ljiljanka"Pečarić, Josip"https://zbmath.org/authors/?q=ai:pecaric.josip-e"Penava, M. Ribičić"https://zbmath.org/authors/?q=ai:penava.m-ribicicSummary: We give inequalities for higher order convex functions involving harmonic sequence of polynomials. As a consequence, we obtain bounds for generalized Taylor expansions.Refinements of some classical inequalities on time scaleshttps://zbmath.org/1491.260192022-09-13T20:28:31.338867Z"Bibi, Rabia"https://zbmath.org/authors/?q=ai:bibi.rabiaSummary: We obtain refinement of the Hölder's inequality and its related inequalities including integral Minkowski's inequality and some Hardy type inequalities on time scales.Approximation of integral operators for absolutely continuous functions whose derivatives are essentially boundedhttps://zbmath.org/1491.260202022-09-13T20:28:31.338867Z"Dragomir, S. S."https://zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: Approximations via Ostrowski type inequalities for the integral transform with the kernel \(K(\cdot,\cdot)\) of absolutely continuous functions \(g:[a,b]\to\mathbb{R}\) whose derivative \(g':[a,b]\to\mathbb{R}\) belongs to \(L_\infty[a,b]\) are obtained. Applications for particular integral transforms such as the Finite Mellin transform, the Finite Sine and Cosine transforms are also given.New integral inequalities pertaining convex functions and their applicationshttps://zbmath.org/1491.260212022-09-13T20:28:31.338867Z"Kashuri, Artion"https://zbmath.org/authors/?q=ai:kashuri.artionSummary: In this paper, first we prove a new generalized midpoint identity. By applying this identity some interesting midpoint type integral inequalities via \(s\)-convex functions are given. Some special cases obtained from our main results are discussed in details. Finally, some applications on the Bessel functions, special means of distinct positive real numbers and error estimation about midpoint quadrature formula are presented to support our theoretical results.On generalized fractional integral inequalities for functions of bounded variation with two variableshttps://zbmath.org/1491.260222022-09-13T20:28:31.338867Z"Kashuri, Artion"https://zbmath.org/authors/?q=ai:kashuri.artion"Budak, Hüseyin"https://zbmath.org/authors/?q=ai:budak.huseyin"Liko, Rozana"https://zbmath.org/authors/?q=ai:liko.rozana"Ali, Muhammad Aamir"https://zbmath.org/authors/?q=ai:ali.muhammad-aamir"Özçelik, Kubilay"https://zbmath.org/authors/?q=ai:ozcelik.kubilaySummary: In this paper, the authors establish some identities for generalized fractional integrals. Utilizing these identities, some Ostrowski and midpoint type inequalities for generalized fractional integrals for functions of bounded variation with two variables are obtained. Moreover, some new inequalities involving \(k\)-Riemann-Liouville fractional integrals are presented as special cases of our main results.A sharp form of the discrete Hardy inequality and the Keller-Pinchover-Pogorzelski inequalityhttps://zbmath.org/1491.260232022-09-13T20:28:31.338867Z"Krejčiřík, David"https://zbmath.org/authors/?q=ai:krejcirik.david"Štampach, František"https://zbmath.org/authors/?q=ai:stampach.frantisekRecently, an improvement of the classical discrete Hardy inequality was established in [\textit{M. Keller}, Am. Math. Mon. 125, No. 4, 347--350 (2018; Zbl 1392.26036)]. The present note provides a straightforward proof of this result. Moreover, optimality is also shown by reformulating the inequality as an identity.
Reviewer: Adam Besenyei (Budapest)Some Simpson's Riemann-Liouville fractional integral inequalities with applications to special functionshttps://zbmath.org/1491.260242022-09-13T20:28:31.338867Z"Nasir, Jamshed"https://zbmath.org/authors/?q=ai:nasir.jamshed"Qaisar, Shahid"https://zbmath.org/authors/?q=ai:qaisar.shahid"Butt, Saad Ihsan"https://zbmath.org/authors/?q=ai:butt.saad-ihsan"Khan, Khuram Ali"https://zbmath.org/authors/?q=ai:khan.khuram-ali"Mabela, Rostin Matendo"https://zbmath.org/authors/?q=ai:mabela.rostin-matendoSummary: Based on the Riemann-Liouville fractional integral, a new form of generalized Simpson-type inequalities in terms of the first derivative is discussed. Here, some more inequalities for convexity as well as concavity are established. We expect that present outcomes are the generalization of already obtained results. Applications to beta, \(q\)-digamma, and Bessel functions are also provided.On new Hermite-Hadamard-Fejér type inequalities for harmonically quasi convex functionshttps://zbmath.org/1491.260252022-09-13T20:28:31.338867Z"Turhan, Sercan"https://zbmath.org/authors/?q=ai:turhan.sercan"İşcan, İmdat"https://zbmath.org/authors/?q=ai:iscan.imdatSummary: In this paper, we give the theorems and results for the trapezoidal and midpoint type inequality of new Hermite-Hadamard-Fejér for harmonically-quasi convex functions via fractional integrals.Generalized fractional integral inequalities for MT-non-convex and \(pq\)-convex functionshttps://zbmath.org/1491.260262022-09-13T20:28:31.338867Z"Wang, Wei"https://zbmath.org/authors/?q=au:Wang, Wei"Ul Haq, Absar"https://zbmath.org/authors/?q=ai:ul-haq.absar"Saleem, Muhammad Shoaib"https://zbmath.org/authors/?q=ai:saleem.muhammad-shoaib"Zahoor, Muhammad Sajid"https://zbmath.org/authors/?q=ai:zahoor.muhammad-sajidThe authors study the concepts of \(MT\)-non-convex and \(pq\)-convex functions. They derive and prove some new generalized fractional integral inequalities for \(MT\)-non-convex fuctions and \(pq\)-convex functions. The results obtained generalize the recent results of \textit{S. Salas} et al. [``On some generalized fractional integral inequalities for \(p\)-convex functions'', Fract. Fractional 3, No. 2, Paper No. 29, 9 p. (2019; \url{doi:10.3390/fractalfract3020029})] and similar results of this type in the literature.
Reviewer: James Adedayo Oguntuase (Abeokuta)Choquet integral Jensen's inequalities for set-valued and fuzzy set-valued functionshttps://zbmath.org/1491.260272022-09-13T20:28:31.338867Z"Zhang, Deli"https://zbmath.org/authors/?q=ai:zhang.deli"Guo, Caimei"https://zbmath.org/authors/?q=ai:guo.caimei"Chen, Degang"https://zbmath.org/authors/?q=ai:chen.degang"Wang, Guijun"https://zbmath.org/authors/?q=ai:wang.guijunSummary: This article attempts to establish Choquet integral Jensen's inequality for set-valued and fuzzy set-valued functions. As a basis, the existing real-valued and set-valued Choquet integrals for set-valued functions are generalized, such that the range of the integrand is extended from \(P_0(R^+)\) to \(P_0(R)\), the upper and lower Choquet integrals are defined, and the fuzzy set-valued Choquet integral is introduced. Then Jensen's inequalities for these Choquet integrals are proved. These include reverse Jensen's inequality for nonnegative real-valued functions, real-valued Choquet integral Jensen's inequalities for set-valued functions, and two families of set-valued and fuzzy set-valued Choquet integral Jensen's inequalities. One is that the related convex function is set-valued or fuzzy set-valued, and the integrand is real-valued, the other is that the related convex function is real-valued, and the integrand is set-valued or fuzzy set-valued. The obtained results generalize earlier works [\textit{T. M. Costa}, Fuzzy Sets Syst. 327, 31--47 (2017; Zbl 1382.28017); \textit{D. Zhang} et al., ibid. 404, 178--204 (2021; Zbl 1464.26025)].Quasi-Cauchy quotients and meanshttps://zbmath.org/1491.260282022-09-13T20:28:31.338867Z"Matkowski, Janusz"https://zbmath.org/authors/?q=ai:matkowski.januszThis paper contains the study of quasi-Cauchy quotients of different types. If \(I\subseteq \mathbb{R}\) is an interval (closed under addition or multiplications, respectively) and \(k\in\mathbb{N}\) with \(k\geq 2\), \(f:I\rightarrow(0,\infty)\), \(F(x)=\frac{f(T(x,\ldots,x))}{L(f(x),\ldots,f(x))}\) then \(M_f(x_1,\ldots,x_k)=F^{-1}\left(\frac{f(T(x_1,\ldots,x_k))}{L(f(x_1),\ldots,f(x_k))}\right)\) is a quasi-Cauchy quotient. If \(T\) and \(L\) are addition, then \(M_f\) is of additive type, if \(T\) and \(L\) are multiplication, \(M_f\) is of multiplicative type, if \(T\) is addition \(L\) is multiplication then \(M_f\) is of exponential type, and if \(T\) is multiplication and \(L\) is addition, then \(M_f\) is of logarithmic type.
The author studies these quasi-Cauchy quotients for various type of functions, showing when they are premeans on means and poses the question when \(M_f=M_g\). For additive type as the author points out, continuous additive functions \(f\) have no meaning but the cases where \(f\) is of exponential (new class of means), logarithmic (single premean) and multiplicative type (no premeans) are studied. Similar results are obtained for the other types of quasi-Cauchy quotients.
Reviewer: Thomas Riedel (Louisville)Sharp power mean bounds for the lemniscate type meanshttps://zbmath.org/1491.260292022-09-13T20:28:31.338867Z"Zhao, Tie-Hong"https://zbmath.org/authors/?q=ai:zhao.tiehong"Shen, Zhong-Hua"https://zbmath.org/authors/?q=ai:shen.zhonghua"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yumingSummary: In this paper, we present sharp power mean bounds for the so-called lemniscate type means, which were introduced by \textit{E. Neuman} [Math. Pannonica 18, No. 1, 77--94 (2007; Zbl 1164.33007)]. The obtained results measure what the distance is between the lemniscate-type means and power means. As applications, several new bounds for the arc lemniscate functions are established.Structural derivatives on time scaleshttps://zbmath.org/1491.260302022-09-13T20:28:31.338867Z"Bayour, Benaoumeur"https://zbmath.org/authors/?q=ai:bayour.benaoumeur"Torres, Delfim F. M."https://zbmath.org/authors/?q=ai:torres.delfim-f-mSummary: We introduce the notion of structural derivative on time scales. The new operator of differentiation unifies the concepts of fractal and fractional order derivative and is motivated by lack of classical differentiability of some self-similar functions. Some properties of the new operator are proved and illustrated with examples.Analysis of mixed Weyl-Marchaud fractional derivative and box dimensionshttps://zbmath.org/1491.280032022-09-13T20:28:31.338867Z"Chandra, Subhash"https://zbmath.org/authors/?q=ai:chandra.subhash-ajay"Abbas, Syed"https://zbmath.org/authors/?q=ai:abbas.syed-wasim|abbas.syed-mohsin|abbas.syed-afsar|abbas.syed-alam|abbas.syed-hussnain|abbas.syed-saiden|abbas.syed-zaheer|abbas.syed|abbas.syed-muzahir|abbas.syed-fFractal dimensions of Katugampola fractional integral of continuous functions satisfying Hölder conditionhttps://zbmath.org/1491.280052022-09-13T20:28:31.338867Z"Yao, Kui"https://zbmath.org/authors/?q=ai:yao.kui"Wang, Zekun"https://zbmath.org/authors/?q=ai:wang.zekun"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xia"Peng, Wenliang"https://zbmath.org/authors/?q=ai:peng.wenliang"Yao, Jia"https://zbmath.org/authors/?q=ai:yao.jiaFractal convolution on the rectanglehttps://zbmath.org/1491.280112022-09-13T20:28:31.338867Z"Pasupathi, R."https://zbmath.org/authors/?q=ai:pasupathi.r"Navascués, M. A."https://zbmath.org/authors/?q=ai:navascues.maria-antonia"Chand, A. K. B."https://zbmath.org/authors/?q=ai:chand.arya-kumar-bedabrataThe authors prime objective with this paper is the investigation of fractal bases and frames for the Lebesgue space \(L^2(I\times J)\) where \(I\) and \(J\) are compact intervals of \(\mathbb{R}\). In particular, the (left and right) fractal convolution operator is used to derive results such as the existence of Bessel sequences, Riesz bases and frames consisting of products of self-referential functions.
Reviewer: Peter Massopust (München)On vector valued multipliers for the class of strongly \(\mathcal {HK}\)-integrable functionshttps://zbmath.org/1491.280122022-09-13T20:28:31.338867Z"Singh, Surinder Pal"https://zbmath.org/authors/?q=ai:kainth.surinder-pal-singh"Bhatnagar, Savita"https://zbmath.org/authors/?q=ai:bhatnagar.savitaSummary: We investigate the space of vector valued multipliers of strongly Henstock-Kurzweil integrable functions. We prove that if \(X\) is a commutative Banach algebra with identity \(e\) such that \(\Vert e \Vert = 1\) and \(g\colon [a,b]\longrightarrow X\) is of strongly bounded variation, then the multiplication operator defined by \(M_g(f):=fg\) maps \(\mathcal{SHK}\) to \(\mathcal{HK}\). We also prove a partial converse, when \(X\) is a Gel'fand space.Choquet integration by Simpson's rule with application in Hellinger distancehttps://zbmath.org/1491.280132022-09-13T20:28:31.338867Z"Agahi, Hamzeh"https://zbmath.org/authors/?q=ai:agahi.hamzeh"Behroozifar, Mahmoud"https://zbmath.org/authors/?q=ai:behroozifar.mahmoudSummary: In non-additive measure theory, there are a few studies on the numerical Choquet integral in continuous case on real line. Recently, based on the statistics software R, \textit{V. Torra} and \textit{Y. Narukawa} [``Numerical integration for the Choquet integral'', Inf. Fusion 31, 137--145 (2016; \url{doi:10.1016/j.inffus.2016.02.007})] considered the problem of computing a numerical Choquet integral and some algorithms for Hellinger distance between two monotone measures. In this paper, the composite Simpson rule for numerical Choquet integration is proposed. Then some algorithms in \textit{Mathematica} software are given. As well as, CPU time of the Examples in terms of seconds is reported which shows, in computational view, the presented method has a high speed.Fractional calculus operators of the product of generalized modified Bessel function of the second typehttps://zbmath.org/1491.330042022-09-13T20:28:31.338867Z"Agarwal, Ritu"https://zbmath.org/authors/?q=ai:agarwal.ritu"Kumar, Naveen"https://zbmath.org/authors/?q=ai:kumar.naveen"Parmar, Rakesh Kumar"https://zbmath.org/authors/?q=ai:parmar.rakesh-kumar"Purohit, Sunil Dutt"https://zbmath.org/authors/?q=ai:purohit.sunil-duttSummary: In this present paper, we consider four integrals and differentials containing the Gauss' hypergeometric \({}_2 F_1 (x)\) function in the kernels, which extend the classical Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators. Formulas (images) for compositions of such generalized fractional integrals and differential constructions with the \(n\)-times product of the generalized modified Bessel function of the second type are established. The results are obtained in terms of the generalized Lauricella function or Srivastava-Daoust hypergeometric function. Equivalent assertions for the Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential are also deduced.Fractional differentiations and integrations of quadruple hypergeometric serieshttps://zbmath.org/1491.330062022-09-13T20:28:31.338867Z"Bin-Saad, Maged G."https://zbmath.org/authors/?q=ai:bin-saad.maged-gumaan"Nisar, Kottakkaran S."https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran"Younis, Jihad A."https://zbmath.org/authors/?q=ai:younis.jihad-aSummary: The hypergeometric series of four variables are introduced and studied by Bin-Saad and Younis recently. In this line, we derive several fractional derivative formulas, integral representations and operational formulas for new quadruple hypergeometric series.Functional inequalities and monotonicity results for modified Lommel functions of the first kindhttps://zbmath.org/1491.330072022-09-13T20:28:31.338867Z"Gaunt, Robert E."https://zbmath.org/authors/?q=ai:gaunt.robert-edwardSummary: We establish some monotonicity results and functional inequalities for modified Lommel functions of the first kind. In particular, we obtain new Turán type inequalities and bounds for ratios of modified Lommel functions of the first kind, as well as the function itself. These results complement and in some cases improve on existing results, and also generalize a number of the results from the literature on monotonicity patterns and functional inequalities for the modified Struve function of the first kind.On first and second order linear Stieltjes differential equationshttps://zbmath.org/1491.340032022-09-13T20:28:31.338867Z"Fernández, Francisco J."https://zbmath.org/authors/?q=ai:fernandez.francisco-javier"Marquéz Albés, Ignacio"https://zbmath.org/authors/?q=ai:marquez-albes.ignacio"Tojo, F. Adrián F."https://zbmath.org/authors/?q=ai:tojo.f-adrian-fSummary: This work deals with the obtaining of solutions of first and second order Stieltjes differential equations. We define the notion of Stieltjes derivative on the whole domain of the functions involved, provide a notion of \(n\)-times continuously Stieltjes-differentiable functions and prove existence and uniqueness results of Stieltjes differential equations in the space of such functions. We also present the Green's functions associated to the different problems and an application to the Stieltjes harmonic oscillator.Green's function and an inequality of Lyapunov-type for conformable boundary value problemhttps://zbmath.org/1491.340092022-09-13T20:28:31.338867Z"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-i"Basua, Debananda"https://zbmath.org/authors/?q=ai:basua.debananda"Jonnalagadda, Jagan Mohan"https://zbmath.org/authors/?q=ai:jonnalagadda.jaganmohanSummary: In this article, we consider a conformable boundary value problem associated with Robin type boundary conditions and present a Lyapunov-type inequality for the same. Further, we attain a lower bound on the smallest eigenvalue for the corresponding conformable eigenvalue problem using the established result, semi maximum norm and Cauchy-Schwartz inequality.Solution sets for fractional differential inclusionshttps://zbmath.org/1491.340102022-09-13T20:28:31.338867Z"Beddani, Moustafa"https://zbmath.org/authors/?q=ai:beddani.moustafa"Hedia, Benaouda"https://zbmath.org/authors/?q=ai:hedia.benaoudaSummary: The aim of this paper is to study an initial value problem for a fractional differential inclusions using the Riemann-Liouville fractional derivative. We apply appropriate fixed point theorems for multivalued maps to obtain the existence results for the given problems covering convex as well as non-convex cases for multivalued maps. We also obtain some topological properties of the solution sets.Existence of solutions to a Kirchhoff \(\psi\)-Hilfer fractional \(p\)-Laplacian equationshttps://zbmath.org/1491.340122022-09-13T20:28:31.338867Z"Ezati, Roozbeh"https://zbmath.org/authors/?q=ai:ezati.roozbeh"Nyamoradi, Nemat"https://zbmath.org/authors/?q=ai:nyamoradi.nematSummary: In this paper, using the genus properties in critical point theory, we study the existence and multiplicity of solutions to the following Kirchhoff \(\psi\)-Hilfer fractional \(p\)-Laplacian:
\[
\begin{cases}
\left(a + b \int_0^T \left|{}^H D_{0^+}^{\alpha, \beta; \psi} \xi (x)\right|^p d x\right){}^H D_T^{\alpha, \beta; \psi} \left(\left|{}^H D_{0^+}^{\alpha, \beta;\psi} \xi (x)\right|^{p - 2}{}^HD_{0^+}^{\alpha, \beta; \psi} \xi (x)\right) \\
\quad \quad - \lambda |\xi (x) |^{p - 2} \xi (x) = g (x, \xi (x)), \\
I_{0^+}^{\beta (\beta - 1); \psi} \xi (0) = I_T^{\beta (\beta - 1); \psi} \xi (T),
\end{cases}
\] where \(^H D_{0^+}^{\alpha, \beta; \psi} \xi(x)\) and \(^H D_T^{\alpha, \beta; \psi}\) are \(\psi\)-Hilfer fractional derivatives left-sided and right-sided of order \(1/p < \alpha < 1, a, b > 0\) are constants, \(0 \leq \beta \leq 1\) and \(I_{0^+}^{\beta (\beta -1); \psi}(.)\) and \(I_T^{\beta (\beta -1); \psi}(.)\) are \(\psi\)-Riemann-Liouville fractional integrals left-sided and right-sided, and \(g : [0.T] \times \mathbb{R} \to \mathbb{R}\) is a continuous function.Analytical approach to a class of Bagley-Torvik equationshttps://zbmath.org/1491.340162022-09-13T20:28:31.338867Z"Mahmudov, Nazim I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisoglu"Huseynov, Ismail T."https://zbmath.org/authors/?q=ai:huseynov.ismail-t"Aliev, Nihan A."https://zbmath.org/authors/?q=ai:aliev.nihan-a"Aliev, Fikret A."https://zbmath.org/authors/?q=ai:aliev.fikret-akhmedaliogluSummary: Multi-term fractional differential equations have been studied because of their applications in modelling, and solved using miscellaneous mathematical methods. We present explicit analytical solutions for several families of generalized multidimensional Bagley-Torvik equations with permutable matrices and two various fractional orders which are satisfying \(\alpha \in(1,2]\), \(\beta \in (0,1]\) and \(\alpha \in(1,2]\), \(\beta \in (1,2]\), both homogeneous and inhomogeneous cases. The results are obtained by means of Mittag-Leffler type matrix functions with double infinite series. In addition, we acquire general solutions of the Bagley-Torvik scalar equations with \(\frac{1}{2}\)-order and \(\frac{3}{2}\)-order derivatives. At the end, we present different examples to verify the efficiency to our main results.Lyapunov functions for fractional-order nonlinear systems with Atangana-Baleanu derivative of Riemann-Liouville typehttps://zbmath.org/1491.340172022-09-13T20:28:31.338867Z"Martínez-Fuentes, Oscar"https://zbmath.org/authors/?q=ai:martinez-fuentes.oscar"Fernández-Anaya, Guillermo"https://zbmath.org/authors/?q=ai:fernandez-anaya.guillermo"Muñoz-Vázquez, Aldo Jonathan"https://zbmath.org/authors/?q=ai:munoz-vazquez.aldo-jonathanSummary: Stability analysis plays an essential role in control systems design. This analysis can be done using different techniques that show the equilibrium points are stable (or unstable). This paper focuses on fractional systems of order \(0 < \alpha < 1\) modeled by the Atangana-Baleanu derivative of Riemann-Liouville type (ABR), which allows consistent modeling of a large class of physical systems with complex dynamics. The main contribution of the paper consists of some novel inequalities for the Atangana-Baleanu derivative of the Riemann-Liouville type. Furthermore, the proposed study allows considering both quadratic and convex Lyapunov functions to analyze stability in ABR systems by applying the Direct Lyapunov Method.A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operatorshttps://zbmath.org/1491.340182022-09-13T20:28:31.338867Z"Mert, Raziye"https://zbmath.org/authors/?q=ai:mert.raziye"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Peterson, Allan"https://zbmath.org/authors/?q=ai:peterson.allan-cSummary: In this work, we use integration by parts formulas derived for fractional operators with Mittag-Leffler kernels to formulate and investigate fractional Sturm-Liouville Problems (FSLPs). We analyze the self-adjointness, eigenvalue and eigenfunction properties of the associated Fractional Sturm-Liouville Operators (FSLOs). The discrete analogue of the obtained results is formulated and analyzed by following nabla analysis.A new type of Sturm-Liouville equation in the non-Newtonian calculushttps://zbmath.org/1491.340432022-09-13T20:28:31.338867Z"Goktas, Sertac"https://zbmath.org/authors/?q=ai:goktas.sertacSummary: In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.On generation of family of resolving operators for a distributed order equation analytic in sectorhttps://zbmath.org/1491.340702022-09-13T20:28:31.338867Z"Fedorov, V. E."https://zbmath.org/authors/?q=ai:fedorov.v-e.1|fedorov.v-eSummary: The questions of the existence and uniqueness of solution to the Cauchy problem for an equation in a Banach space of distributed order at most one are investigated. Necessary and sufficient conditions for the existence of a resolving family of operators of this equation analytic in the sector are obtained. An explicit form of these operators is found. Two versions of the theorem on unique solvability of the Cauchy problem for the corresponding inhomogeneous equation are obtained: with condition of increased smoothness in spatial variables (the condition of continuity in the norm of the graph of the generator of the resolving family) and with condition of increased smoothness in the time variable (the Hölder condition). Abstract results are obtained using theory of the Laplace transform and generalization some results from the theory of analytic operator semigroups and its extensions to the case of integral equations and fractional differential equations. The conditions for unique solvability of an equation in a Banach space are used to study a class of initial boundary value problems for equations with polynomials in an elliptic differential operator with respect to spatial variables.On the initial value problem of impulsive differential equation involving Caputo-Katugampola fractional derivative of order \(q\in (1, 2)\)https://zbmath.org/1491.340882022-09-13T20:28:31.338867Z"Zhang, Xian-Min"https://zbmath.org/authors/?q=ai:zhang.xianminSummary: This paper mainly focuses on the non-uniqueness of solution to the initial value problem (IVP) of impulsive fractional differential equations (IFrDE) with Caputo-Katugampola derivative (of order \(q\in (1, 2)\)). The system of impulsive higher order fractional differential equations may involve two different kinds of impulses, and the obtained result shows that its equivalent integral equations include two arbitrary constants, which means that its solution is non-unique. Next, two numerical examples are used to show the non-uniqueness of solution for the IVP of IFrDE.Introduction to Halanay lemma, via weakly Picard operator theoryhttps://zbmath.org/1491.340892022-09-13T20:28:31.338867Z"Petruşel, A."https://zbmath.org/authors/?q=ai:petrusel.adrian"Rus, I. A."https://zbmath.org/authors/?q=ai:rus.ioan-aIn his monograph \textit{A. Halanay} [Differential equations: Stability, oscillations, time lags. Elsevier, Amsterdam (1966; Zbl 0144.08701)] introduced a Lemma ``presenting an interest in itself'' [loc. cit., 378--380]:
Let \(\alpha>\beta>0\), \(h>0\), \(t_0\in R\). If \(x\in C([t_0-h,\infty[)\) and \(x|_{[t_0,\infty[}\in C^1[t_0,\infty[\) and if \[ \displaystyle{x'(t)\leq -\alpha x(t) + \beta\max_{\theta\in[t-h,t]}x(\theta)\ ,\ x\in[t_0,\infty[} \] then there exist \(k>0\), \(\gamma>0\) such that \[ \displaystyle{x(t)\leq ke^{-\gamma(t-h)}\ ,\ \forall t\in[t_0-h,\infty[} \]
The present paper examines this lemma in the context of the functional differential equations with maximum of the form
\[ \displaystyle{x' = f(t,x(t), \max_I (x)(t))\ t\geq t_0}, \]
where \(\max_I(x)(t)=\max_{\theta\in I(t)} x(\theta)\), the operator \(\max_I\) being defined on a space of continuous functions. These equations are considered in the framework of fixed point theory. In this way there is considered the abstract Gronwall lemma, the Cauchy problem for the Halanay equation \[ \displaystyle{x'(t) = -\alpha x(t) + \beta\max_{\theta\in[t-h,t]} x(\theta)}. \]
The concluding remark embeds Halanay Lemma in the framework of the Chaplygin and Gronwall type results.
For the entire collection see [Zbl 1471.41001].
Reviewer: Vladimir Răsvan (Craiova)Rigidity and trace properties of divergence-measure vector fieldshttps://zbmath.org/1491.351222022-09-13T20:28:31.338867Z"Leonardi, Gian Paolo"https://zbmath.org/authors/?q=ai:leonardi.gian-paolo"Saracco, Giorgio"https://zbmath.org/authors/?q=ai:saracco.giorgioSummary: We consider a \(\varphi\)-rigidity property for divergence-free vector fields in the Euclidean \(n\)-space, where \(\varphi(t)\) is a non-negative convex function vanishing only at \(t=0\). We show that this property is always satisfied in dimension \(n=2\), while in higher dimension it requires some further restriction on \(\varphi\). In particular, we exhibit counterexamples to \textit{quadratic rigidity} (i.e. when \(\varphi(t)=ct^2 )\) in dimension \(n\geq 4\). The validity of the quadratic rigidity, which we prove in dimension \(n=2\), implies the existence of the trace of a divergence-measure vector field \(\xi\) on an \(\mathcal{H}^1 \)-rectifiable set \(S\), as soon as its weak normal trace \([\xi\cdot\nu_S]\) is maximal on \(S\). As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.An integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensionshttps://zbmath.org/1491.352632022-09-13T20:28:31.338867Z"Watschinger, Raphael"https://zbmath.org/authors/?q=ai:watschinger.raphael"Of, Günther"https://zbmath.org/authors/?q=ai:of.guntherSummary: While an integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions is available in the literature, a proof of this formula seems to be missing. Moreover, the available formula contains an integral term including the time derivative of the fundamental solution of the heat equation, whose interpretation is difficult at second glance. To fill these gaps, we provide a rigorous proof of a general version of the integration by parts formula and an alternative representation of the mentioned integral term, which is valid for a certain class of functions including the typical tensor-product discretization spaces.A second-order evolution equation and logarithmic operatorshttps://zbmath.org/1491.352862022-09-13T20:28:31.338867Z"Bezerra, F. D. M."https://zbmath.org/authors/?q=ai:bezerra.flank-david-moraisSummary: In this paper we introduce a matrix representation of the logarithmic wave operator and we study a second-order semilinear evolution equation governed by this operator. We present a result of local well-posedness for this problem and properties of the logarithmic wave operator in terms of the logarithmic negative Dirichlet Laplacian operator.Inviscid limit of the inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in \(\mathbb{R}^3\)https://zbmath.org/1491.353222022-09-13T20:28:31.338867Z"Wang, Dixi"https://zbmath.org/authors/?q=ai:wang.dixi"Yu, Cheng"https://zbmath.org/authors/?q=ai:yu.cheng"Zhao, Xinhua"https://zbmath.org/authors/?q=ai:zhao.xinhuaSummary: In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in \(\mathbb{R}^3\). In particular, this limit is a weak solution of the corresponding Euler equations. We first deduce the Kolmogorov-type hypothesis in \(\mathbb{R}^3\), which yields the uniform bounds of \(\alpha^{th}\)-order fractional derivatives of \(\sqrt{\rho^\mu} \mathbf{u}^\mu\) in \(L^2_x\) for some \(\alpha > 0\), independent of the viscosity. The uniform bounds can provide strong convergence of \(\sqrt{\rho^\mu} \mathbf{u}^\mu\) in \(L^2\) space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations.Pointwise convergence of the fractional Schrödinger equation in \(\mathbb{R}^2\)https://zbmath.org/1491.353722022-09-13T20:28:31.338867Z"Cho, Chu-Hee"https://zbmath.org/authors/?q=ai:cho.chu-hee"Ko, Hyerim"https://zbmath.org/authors/?q=ai:ko.hyerimSummary: We investigate the pointwise convergence of the solution to the fractional Schrödinger equation in \(\mathbb{R}^2\). By establishing \(H^s(\mathbb{R}^2) - L^3(\mathbb{R}^2)\) estimates for the associated maximal operator provided that \(s > 1/3\), we improve the previous result obtained by \textit{C. Miao} et al. [Stud. Math. 230, No. 2, 121--165 (2015; Zbl 1343.42027)]. Our estimates extend the refined Strichartz estimates obtained by \textit{X. Du} et al. [Ann. Math. (2) 186, No. 2, 607--640 (2017; Zbl 1378.42011)] to a general class of elliptic functions.Local existence and nonexistence for fractional in time reaction-diffusion equations and systems with rapidly growing nonlinear termshttps://zbmath.org/1491.354382022-09-13T20:28:31.338867Z"Suzuki, Masamitsu"https://zbmath.org/authors/?q=ai:suzuki.masamitsuSummary: We study the fractional in time reaction-diffusion equation
\[
\begin{cases}
\partial_t^\alpha u = \Delta u + f (u) & \text{in } \mathbb{R}^N \times (0, T), \\
u (x, 0) = u_0 (x) & \text{in } \mathbb{R}^N,
\end{cases}
\] where \(0 < \alpha < 1\), \(N \geq 1\), \(T > 0\) and \(u_0 \geq 0\). The fractional derivative \(\partial_t^\alpha\) is meant in a generalized Caputo sense. We mainly consider the case where \(f\) has an exponential or a superexponential growth, and \(u_0\) has a singularity. We obtain integrability conditions on \(u_0\) which explicitly determine local in time existence/nonexistence of a nonnegative mild solution. Moreover, our analysis can be applied to time fractional systems.Regularization of the backward stochastic heat conduction problemhttps://zbmath.org/1491.354432022-09-13T20:28:31.338867Z"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Lesnic, Daniel"https://zbmath.org/authors/?q=ai:lesnic.daniel"Thach, Tran Ngoc"https://zbmath.org/authors/?q=ai:thach.tran-ngoc"Ngoc, Tran Bao"https://zbmath.org/authors/?q=ai:ngoc.tran-baoSummary: In this paper, we study the backward problem for the stochastic parabolic heat equation driven by a Wiener process. We show that the problem is ill-posed by violating the continuous dependence on the input data. In order to restore stability, we apply a filter regularization method which is completely new in the stochastic setting. Convergence rates are established under different a priori assumptions on the sought solution.On a stochastic nonclassical diffusion equation with standard and fractional Brownian motionhttps://zbmath.org/1491.354692022-09-13T20:28:31.338867Z"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Ngoc, Tran Bao"https://zbmath.org/authors/?q=ai:ngoc.tran-bao"Thach, Tran Ngoc"https://zbmath.org/authors/?q=ai:thach.tran-ngoc"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan.Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systemshttps://zbmath.org/1491.370302022-09-13T20:28:31.338867Z"Goufo, Emile Franc Doungmo"https://zbmath.org/authors/?q=ai:doungmo-goufo.emile-francSummary: Chaotic dynamical attractors are themselves very captivating in Science and Engineering, but systems with multi-dimensional and saturated chaotic attractors with many scrolls are even more fascinating for their multi-directional features. In this paper, the dynamics of a Caputo three-dimensional saturated system is successfully investigated by means of numerical techniques. The continuity property for the saturated function series involved in the model preludes suitable analytical conditions for existence and stability of the solution to the model. The Haar wavelet numerical method is applied to the saturated system and its convergence is shown thanks to error analysis. Therefore, the performance of numerical approximations clearly reveals that the Caputo model and its general initial conditions display some chaotic features with many directions. Such a chaos shows attractors with many scrolls and many directions. Then, the saturated Caputo system is indeed chaotic in the standard integer case (Caputo derivative order \(\alpha = 1)\) and this chaos remains in the fractional case \((\alpha = 0.9)\). Moreover the dynamics of the system change depending on the parameter \(\alpha\), leading to an important observation that the saturated system is likely to be regulated or controlled via such a parameter.Analysis of a Caputo HIV and Malaria co-infection epidemic modelhttps://zbmath.org/1491.370782022-09-13T20:28:31.338867Z"Ahmed, Idris"https://zbmath.org/authors/?q=ai:ahmed.idris"Yusuf, Abdullahi"https://zbmath.org/authors/?q=ai:yusuf.abdullahi-a"Sani, Musbahu Aminu"https://zbmath.org/authors/?q=ai:sani.musbahu-aminu"Jarad, Fahd"https://zbmath.org/authors/?q=ai:jarad.fahd"Kumam, Wiyada"https://zbmath.org/authors/?q=ai:kumam.wiyada"Thounthong, Phatiphat"https://zbmath.org/authors/?q=ai:thounthong.phatiphatSummary: In this paper, we investigate a fractional-order compartmental HIV and Malaria co-infection epidemic model using the Caputo derivative. The existence and uniqueness of the solution to the proposed fractional-order model were investigated using fixed point theorem techniques. To demonstrate that the proposed fractional-order model is both mathematically and epidemiologically well-posed, we compute the model's positivity and boundedness, which is an important feature in epidemiology. Finally, we analyze the dynamic behavior of each of the state variables using a recent and powerful computational technique known as the fractional Euler method.On dynamic behavior of a discrete fractional-order nonlinear prey-predator modelhttps://zbmath.org/1491.370792022-09-13T20:28:31.338867Z"Aldurayhim, A."https://zbmath.org/authors/?q=ai:aldurayhim.abdullah"Elsadany, A. A."https://zbmath.org/authors/?q=ai:elsadany.abdelalim-a"Elsonbaty, A."https://zbmath.org/authors/?q=ai:elsonbaty.amr-rA method for minimizing the control calculation time of fractional systemshttps://zbmath.org/1491.370822022-09-13T20:28:31.338867Z"Hcheichi, Khaled"https://zbmath.org/authors/?q=ai:hcheichi.khaled"Bouani, Faouzi"https://zbmath.org/authors/?q=ai:bouani.faouziSummary: This article deals with the non-commensurate fractional systems represented by a state-space model. It presents the advantages and disadvantages of using this type of model. It focuses on the disadvantages of fractional systems, especially the increase of computing time due to the accumulation of history. A method is proposed to minimize the calculation time, it consists in limiting the use of history of the state variables to a reduced number and taking into account the uncertainty of model in the predictive control. This method will be compared to the classical method in term of performance and calculation time.Existence and uniqueness of solutions for a class of discrete-time fractional equations of order \(2<\alpha \leq 3\)https://zbmath.org/1491.390022022-09-13T20:28:31.338867Z"Leal, Claudio"https://zbmath.org/authors/?q=ai:leal.claudio"Murillo-Arcila, Marina"https://zbmath.org/authors/?q=ai:murillo-arcila.marinaSummary: In this paper we consider a class of linear difference equations of fractional order \(2<\alpha \leq 3\) in the sense of Riemann-Liouville. The explicit solution for this model is provided in terms of a fractional resolvent sequence which allows to write the solution to this equation as a variation of constant formula. We also characterize the existence and uniqueness of solutions in \(\ell_p (\mathbb{N}_0, X)\) spaces with \(X\) being a \textit{UMD}-space in terms of the \(R\)-boundedness of the operator symbol of the model. Moreover, we are able to relax this condition in the case of Hilbert spaces. Finally, we illustrate our results with an example that involves the generator of a contraction semigroup.Stability analysis of fixed point of fractional-order coupled map latticeshttps://zbmath.org/1491.390062022-09-13T20:28:31.338867Z"Bhalekar, Sachin"https://zbmath.org/authors/?q=ai:bhalekar.sachin"Gade, Prashant M."https://zbmath.org/authors/?q=ai:gade.prashant-mSummary: We study the stability of synchronized fixed-point state for linear fractional-order coupled map lattice (CML). We observe that the eigenvalues of the connectivity matrix determine the stability of this system as in integer-order CML. These eigenvalues can be determined exactly in certain cases. We find exact bounds in a one-dimensional lattice with translationally invariant coupling using the theory of circulant matrices. This can be extended to any finite dimension. Similar analysis can be carried out for the synchronized fixed point of nonlinear coupled fractional maps where eigenvalues of the Jacobian matrix play the same role. The analysis is generic and demonstrates that the eigenvalues of connectivity matrix play a pivotal role in the stability analysis of synchronized fixed point even in coupled fractional maps.Stability of systems of fractional-order difference equations and applications to a Rulkov-type neuronal modelhttps://zbmath.org/1491.390072022-09-13T20:28:31.338867Z"Brandibur, Oana"https://zbmath.org/authors/?q=ai:brandibur.oana"Kaslik, Eva"https://zbmath.org/authors/?q=ai:kaslik.eva"Mozyrska, Dorota"https://zbmath.org/authors/?q=ai:mozyrska.dorota"Wyrwas, Małgorzata"https://zbmath.org/authors/?q=ai:wyrwas.malgorzataSummary: Necessary and sufficient conditions for the asymptotic stability and instability of two-dimensional linear autonomous incommensurate systems of fractional-order Caputo difference equations are presented. Moreover, the occurrence of discrete Flip and Hopf bifurcations is also discussed, choosing the fractional orders as bifurcation parameters. The theoretical results are then applied to the investigation of the stability and instability properties of a fractional-order version of the Rulkov neuronal model. Numerical simulations are further presented to illustrate the theoretical findings, revealing complex bursting behavior in the fractional-order Rulkov model.
For the entire collection see [Zbl 1470.74004].Chaotic dynamics of a novel 2D discrete fractional order Ushiki maphttps://zbmath.org/1491.390082022-09-13T20:28:31.338867Z"Higazy, M."https://zbmath.org/authors/?q=ai:higazy.m-sh"Selvam, George Maria"https://zbmath.org/authors/?q=ai:selvam.george-maria"Janagaraj, R."https://zbmath.org/authors/?q=ai:janagaraj.rajendranHyperchaotic dynamics of a new fractional discrete-time systemhttps://zbmath.org/1491.390092022-09-13T20:28:31.338867Z"Khennaoui, Amina-Aicha"https://zbmath.org/authors/?q=ai:khennaoui.amina-aicha"Ouannas, Adel"https://zbmath.org/authors/?q=ai:ouannas.adel"Momani, Shaher"https://zbmath.org/authors/?q=ai:momani.shaher-m"Dibi, Zohir"https://zbmath.org/authors/?q=ai:dibi.zohir"Grassi, Giuseppe"https://zbmath.org/authors/?q=ai:grassi.giuseppe"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-i"Pham, Viet-Thanh"https://zbmath.org/authors/?q=ai:pham.viet-thanhImage encryption based on two-dimensional fractional quadric polynomial maphttps://zbmath.org/1491.390122022-09-13T20:28:31.338867Z"Liu, Ze-Yu"https://zbmath.org/authors/?q=ai:liu.zeyu"Xia, Tie-Cheng"https://zbmath.org/authors/?q=ai:xia.tie-cheng"Feng, Hua-Rong"https://zbmath.org/authors/?q=ai:feng.huarong"Ma, Chang-You"https://zbmath.org/authors/?q=ai:ma.changyouReduction of the Kolmogorov inequality for a non negative part of the second derivative on the real line to the inequality for convex functions on an intervalhttps://zbmath.org/1491.390162022-09-13T20:28:31.338867Z"Payuchenko, Nikita Slavich"https://zbmath.org/authors/?q=ai:payuchenko.nikita-slavichSummary: In this paper we delve into connection between sharp constants in the inequalities
\begin{align*}
\| y' \|_{L_q(\mathbb{R})}&\leq K_+ \sqrt{\| y\|_{L_r(\mathbb{R})}\| y''_+\|_{L_p(\mathbb{R})}}\\
\| u' \|_{L_q(0,1)} &\leq \overline{K}_+ \sqrt{\| u\|_{L_r(0,1)}\| u''_+\|_{L_p(0,1)}}
\end{align*}
where the second one is considered for convex functions \(u(x)\) \(x\in[0,1]\) with an absolutely continuous derivative that vanishes at the point \(x=0\). We prove that \(K_+ =\overline{K}\) under conditions \(1\leq q,r,p <\infty\) and \(1/r + 1/p=2/q\).Ulam-Hyers stability results of \(\lambda\)-quadratic functional equation with three variables in non-Archimedean Banach space and non-Archimedean random normed spacehttps://zbmath.org/1491.390172022-09-13T20:28:31.338867Z"An, Ly Van"https://zbmath.org/authors/?q=ai:an.ly-van"Tamilvanan, Kandhasamy"https://zbmath.org/authors/?q=ai:tamilvanan.kandhasamy"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.r|udhayakumar.radhagayathri-k"Kabeto, Masho Jima"https://zbmath.org/authors/?q=ai:kabeto.masho-jima"Ngoc, Ly Van"https://zbmath.org/authors/?q=ai:ngoc.ly-vanSummary: In this paper, we introduce the \(\lambda \)-quadratic functional equation with three variables and obtain its general solution. The main aim of this work is to examine the Ulam-Hyers stability of this functional equation in non-Archimedean Banach space by using direct and fixed point techniques and examine the stability results in non-Archimedean random normed space.Variations on the strongly lacunary quasi Cauchy sequenceshttps://zbmath.org/1491.400022022-09-13T20:28:31.338867Z"Kaplan, Huseyin"https://zbmath.org/authors/?q=ai:kaplan.huseyinSummary: In this paper,we introduce concepts of a strongly lacunary \(p\)-quasi-Cauchy sequence and strongly lacunary \(p\)-ward continuity. We prove that a subset of \(\mathbb{R}\) is bounded if and only if it is strongly lacunary \(p\)-ward compact. It is obtained that any strongly lacunary \(p\)-ward continuous function on a subset \(A\) of \(\mathbb{R}\) is continuous in the ordinary sense. We also prove that the uniform limit of strongly lacunary \(p\)-ward continuous functions on a subset \(A\) of \(\mathbb{R}\) is strongly lacunary \(p\)-ward continuous.\( \mu \)-statistical convergence and the space of functions \(\mu \)-stat continuous on the segmenthttps://zbmath.org/1491.400032022-09-13T20:28:31.338867Z"Sadigova, S. R."https://zbmath.org/authors/?q=ai:sadigova.sabina-rahibThe author introduces $\mu$-statistical density of a point and the concept of $\mu$-statistical fundamentality at a point and also states that this method is equivalent to the concept of $\mu$-stat convergence. On the other hand, the concept of $\mu$-stat continuity is defined. Some properties of the space of all $\mu$-stat continuous functions are examined.
Reviewer: Emre Taş (Kırşehir)A numerical comparative study of generalized Bernstein-Kantorovich operatorshttps://zbmath.org/1491.410032022-09-13T20:28:31.338867Z"Kadak, Uğur"https://zbmath.org/authors/?q=ai:kadak.ugur"Özger, Faruk"https://zbmath.org/authors/?q=ai:ozger.farukA generalization of the Bernstein-Kantorovich operators involving multiple shape parameters is introduced, establishing Voronovskaja and Grüss-Voronovskaja type theorems, the statistical convergence and the statistical rate of convergence obtained by means of a regular summability matrix. The new operators are compared with the classical Bernstein, Bernstein-Kantorovich, \(\lambda\)-Bernstein, \(\lambda\)-Bernstein-Kantorovich, \(\alpha\)-Bernstein and \(\alpha\)-Bernstein-Kantorovich operators, respectively.
Reviewer: Zoltán Finta (Cluj-Napoca)A Bernstein inequality for differential and integral operators on Orlicz spaceshttps://zbmath.org/1491.410062022-09-13T20:28:31.338867Z"Ha Huy Bang"https://zbmath.org/authors/?q=ai:ha-huy-bang."Vu Nhat Huy"https://zbmath.org/authors/?q=ai:vu-nhat-huy."Nguyen Ngoc Huy"https://zbmath.org/authors/?q=ai:nguyen-ngoc-huy.Summary: In this paper, we obtain a Bernstein inequality for polynomial differential operators and polynomial integral operators on Orlicz spaces. Let \(\Phi: [0,+\infty)\to [0,+\infty]\) be an arbitrary Young function, \(K\) be an arbitrary compact set in \(\mathbb{R}\) and \(P(x)\) be a polynomial. Then there exists a constant \(C\) independent of \(\Phi\) such that
\[
\|P^m(D)f\|_{(\Phi)}\le Cm \sup\limits_{x\in K} |P^m(x)| \|f\|_{(\Phi)}
\]
for all \(m\in\mathbb{N}\) and all \(f\in \mathcal{L}_{\phi,K}\), where \(\mathcal{L}_{\Phi,K}=\{f\in L^\Phi(\mathbb{R}):\operatorname{supp} \hat{f}\subset K\}\), \(\hat{f}\) is the Fourier transform of \(f\) and \(\|\cdot\|_{(\Phi)}\) is the Luxemburg norm.
The corresponding result for polynomial integral operators and an application are also given.Weighted inequalities for multilinear fractional operators in Dunkl settinghttps://zbmath.org/1491.420092022-09-13T20:28:31.338867Z"Mukherjee, Suman"https://zbmath.org/authors/?q=ai:mukherjee.suman"Parui, Sanjay"https://zbmath.org/authors/?q=ai:parui.sanjaySummary: In this paper we define multilinear fractional integral operator and multilinear fractional maximal function in Dunkl setting and present a weighted theory for these operators. Sufficient conditions for two-weight inequalities are found using ``power-bump'' conditions and for one-weight inequalities Muckenhoupt-type conditions are obtained as sufficient condition.Bi-parameter trilinear Fourier multipliers and pseudo-differential operators with flag symbolshttps://zbmath.org/1491.420122022-09-13T20:28:31.338867Z"Lu, Guozhen"https://zbmath.org/authors/?q=ai:lu.guozhen"Pipher, Jill"https://zbmath.org/authors/?q=ai:pipher.jill-c"Zhang, Lu"https://zbmath.org/authors/?q=ai:zhang.luSummary: The main purpose of this paper is to study \(L^r\) Hölder type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for \(f, g, h\in{\mathcal{S}}(\mathbb{R}^2)\), the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by
\[
T_{m_1,m_2}(f,g,h)(x):=\int_{\mathbb{R}^6}m_1(\xi ,\eta ,\zeta)m_2(\eta ,\zeta){\hat{f}}(\xi) {\hat{g}}(\eta){\hat{h}}(\zeta)e^{2\pi i(\xi +\eta +\zeta)\cdot x}d\xi d\eta d\zeta,
\]
when \(m_1\), \(m_2\) are two bi-parameter symbols. We study Hölder type estimates: \(L^{p_1}\times L^{p_2}\times L^{p_3} \rightarrow L^r\) for \(1<p_1\), \(p_2\), \(p_3< \infty\) with \(1/p_1+1/p_2+1/p_3=1/r\), and \(0<r<\infty\). We will show that our problem can be reduced to establish the \(L^r\) estimate for the special multiplier \(m_1(\xi_1, \eta_1, \zeta_1) m_2(\eta_2, \zeta_2)\) (see Theorem 1.7). We also study these \(L^r\) estimates for the corresponding bi-parameter trilinear pseudo-differential operators defined by
\[
T_{ab}(f,g,h)(x):=\int_{\mathbb{R}^6}a(x,\xi ,\eta ,\zeta)b(x,\eta ,\zeta){\hat{f}}(\xi){\hat{g}}(\eta){\hat{h}}(\zeta)e^{2\pi i x(\xi +\eta +\zeta)}d\xi d\eta d\zeta,
\]
where the smooth symbols \(a\), \(b\) satisfy certain bi-parameter Hörmander conditions. We will also show that the \(L^r\) estimate holds for \(T_{ab}\) as long as the \(L^r\) estimate for the flag multiplier operator holds when the multiplier has the special form \(m_1(\xi_1, \eta_1, \zeta_1) m_2(\eta_2, \zeta_2)\) (see Theorem 1.10). Using our reduction of the flag multiplier, we also provide an alternative proof of some of the mixed norm estimates recently established by \textit{C. Muscalu} and \textit{Y. Zhai} [``Five-linear singular integrals of Brascamp-Lieb type'', Preprint, \url{arXiv:2001.09064}] when the functions \(g\) and \(h\) are of tensor product forms (Theorem 1.8). Moreover, our method also allows us to establish the weighted mixed norm estimates (Theorem 1.9). The bi-parameter and trilinear flag Fourier multipliers considered in this paper do not satisfy the conditions of the classical bi-parameter trilinear Fourier multipliers considered by \textit{C. Muscalu} et al. [Acta Math. 193, No. 2, 269--296 (2004; Zbl 1087.42016)] and \textit{C. Muscalu} et al. [Rev. Mat. Iberoam. 22, No. 3, 963--976 (2006; Zbl 1114.42005)]. They may also be viewed as the bi-parameter trilinear variants of estimates obtained for the one-parameter flag paraproducts by \textit{C. Muscalu} [ibid. 23, No. 2, 705--742 (2007; Zbl 1213.42071)].Compactness, or lack thereof, for the harmonic double layerhttps://zbmath.org/1491.420222022-09-13T20:28:31.338867Z"Mitrea, Dorina"https://zbmath.org/authors/?q=ai:mitrea.dorina"Mitrea, Irina"https://zbmath.org/authors/?q=ai:mitrea.irina"Mitrea, Marius"https://zbmath.org/authors/?q=ai:mitrea.mariusSummary: Ahlfors regular domains which are infinitesimally flat, in a scale-invariant fashion, constitute the most general geometric context in which the harmonic double layer potential as well as other similar singular integral operators are compact in the framework of Lebesgue spaces. We review recent progress in this direction and, working with a drop-shaped domain, we give a direct, self-contained proof of the fact that the harmonic double layer potential fails to be compact on Lebesgue spaces in the presence of a single corner (or conical) singularity.
For the entire collection see [Zbl 1479.47003].Characterizations of weak reverse Hölder inequalities on metric measure spaceshttps://zbmath.org/1491.420362022-09-13T20:28:31.338867Z"Kinnunen, Juha"https://zbmath.org/authors/?q=ai:kinnunen.juha"Kurki, Emma-Karoliina"https://zbmath.org/authors/?q=ai:kurki.emma-karoliina"Mudarra, Carlos"https://zbmath.org/authors/?q=ai:mudarra.carlosSummary: We present ten different characterizations of functions satisfying a weak reverse Hölder inequality on an open subset of a metric measure space with a doubling measure. Among others, we describe these functions as a class of weak \(A_\infty\) weights, which is a generalization of Muckenhoupt weights that allows for nondoubling weights. Although our main results are modeled after conditions that hold true for Muckenhoupt weights, we also discuss two conditions for Muckenhoupt \(A_\infty\) weights that fail to hold for weak \(A_\infty\) weights.\(p\)-adic weak central Morrey spaces on differential formshttps://zbmath.org/1491.420382022-09-13T20:28:31.338867Z"Wang, Jianwei"https://zbmath.org/authors/?q=ai:wang.jianwei"Wang, Linlin"https://zbmath.org/authors/?q=ai:wang.linlin"Xing, Yuming"https://zbmath.org/authors/?q=ai:xing.yumingSummary: In this article, the theory of differential forms on \(\mathbb{R}^n\) was extended to the filed \(\mathbb{Q}^n_p\) of \(p\)-adic numbers. The imbedding inequalities for differential forms were derived on \(\mathbb{Q}^n_p\). Then, we show the definitions of \(p\)-adic weak central Morrey spaces and \(p\)-adic \(\lambda\)-central BMO
spaces on differential forms. The boundedness of Hardy operator and its adjoint operator were given in the new space. Finally, we give the characterization of the two operators in \(p\)-adic \(\lambda\)-central BMO spaces by using imbedding inequalities on differential forms.Generalized fractional integral operators on Campanato spaces and their bi-predualshttps://zbmath.org/1491.420392022-09-13T20:28:31.338867Z"Yamaguchi, Satoshi"https://zbmath.org/authors/?q=ai:yamaguchi.satoshi"Nakai, Eiichi"https://zbmath.org/authors/?q=ai:nakai.eiichiSummary: In this paper we prove the boundedness of the generalized fractional integral operator \(I_\rho\) on generalized Campanato spaces with variable growth condition, which is a generalization and improvement of previous results, and then, we establish the boundedness of \(I_\rho\) on their bi-preduals. We also prove the boundedness of \(I_\rho\) on their preduals by the duality.On the coefficients of multiple series with respect to Vilenkin systemhttps://zbmath.org/1491.420432022-09-13T20:28:31.338867Z"Skvortsov, Valentin A."https://zbmath.org/authors/?q=ai:skvortsov.valentin-a"Tulone, Francesco"https://zbmath.org/authors/?q=ai:tulone.francescoSummary: We give a sufficient condition for coefficients of double series \(\sum\sum_{n,m}a_{n,m}\chi_{n,m}\) with respect to Vilenkin system to be convergent to zero when \(n+m\rightarrow \infty\). This result can be applied to the problem of recovering coefficients of a Vilenkin series from its sum.Lower bound of sectional curvature of Fisher-Rao manifold of beta distributions and complete monotonicity of functions involving polygamma functionshttps://zbmath.org/1491.440022022-09-13T20:28:31.338867Z"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.fengSummary: In the paper, by virtue of convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the Fisher-Rao manifold of beta distributions.On an extension of the Mikusiński type operational calculus for the Caputo fractional derivativehttps://zbmath.org/1491.440052022-09-13T20:28:31.338867Z"Al-Kandari, M."https://zbmath.org/authors/?q=ai:alkandari.maryam"Hanna, L. A.-M."https://zbmath.org/authors/?q=ai:hanna.latif-a-m"Luchko, Yu."https://zbmath.org/authors/?q=ai:luchko.yuri|luchko.yurii-fSummary: In this paper, a two-parameter extension of the operational calculus of Mikusiński's type for the Caputo fractional derivative is presented. The first parameter is connected with the rings of functions that are used as a basis for construction of the convolution quotients fields. The convolutions by themselves are characterized by another parameter. The obtained two-parameter operational calculi are compared each to other and some homomorphisms between the fields of convolution quotients are established.Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditionshttps://zbmath.org/1491.450012022-09-13T20:28:31.338867Z"Li, Yulong"https://zbmath.org/authors/?q=ai:li.yulongSummary: The generalized (or coupled) Abel equations exist in many BVPs of fractional-order differential equations and play a key role during the process of converting weak solutions to the true solutions. Motivated by the analysis of double-sided fractional diffusion-advection-reaction equations, this article develops the mapping properties of generalized Abel operators \({\alpha_a}D_x^{- s} + {\beta_x}D_b^{- s}\) in fractional Sobolev spaces, where \(0 < \alpha\), \(\beta\), \(\alpha + \beta = 1\), \(0 < s < 1\) and \(_aD_x^{- s}\), \(_xD_b^{- s}\) are fractional Riemann-Liouville integrals. It is mainly concerned with the regularity property of \(({{\alpha_a}D_x^{- s} + {\beta_x}D_b^{- s}})u = f\) by taking into account homogeneous boundary conditions. Namely, we investigate the regularity behavior of \(u(x)\) while letting \(f(x)\) become smoother and imposing homogeneous boundary restrictions \(u(a) = u(b) = 0\).On the existence and uniqueness of solution to Volterra equation on a time scalehttps://zbmath.org/1491.450032022-09-13T20:28:31.338867Z"Kluczyński, Bartłomiej"https://zbmath.org/authors/?q=ai:kluczynski.bartlomiejSummary: Using a global inversion theorem we investigate properties of the following operator
\[
V (x)(\cdot) := x^\Delta(\cdot) + \int^\cdot_0\nu(\cdot,\tau, x(\tau))\Delta\tau,\qquad x(0) = 0,
\]
in a time scale setting. Under some assumptions on the nonlinear term \(\nu\) we then show that there exists exactly one solution \(x_y\in W^{1,p}_{\Delta,0}\left([0, 1]_{\mathbb{T}},\mathbb{R}^N\right)\) to the associated integral equation
\[
\begin{cases}
x^\Delta(t) +\int^t_0\nu(t,\tau, x(\tau))\Delta\tau = y(t) \text{ for }\Delta\text{-a.e. } t \in [0,1]_{\mathbb{T}},\\
x(0) = 0,
\end{cases}
\]
which is considered on a suitable Sobolev space.Qualitative analyses of fractional integrodifferential equations with a variable order under the Mittag-Leffler power lawhttps://zbmath.org/1491.450112022-09-13T20:28:31.338867Z"Jeelani, Mdi Begum"https://zbmath.org/authors/?q=ai:jeelani.mdi-begum"Alnahdi, Abeer S."https://zbmath.org/authors/?q=ai:alnahdi.abeer-s"Almalahi, Mohammed A."https://zbmath.org/authors/?q=ai:almalahi.mohammed-a"Abdo, Mohammed S."https://zbmath.org/authors/?q=ai:abdo.mohammed-salem"Wahash, Hanan A."https://zbmath.org/authors/?q=ai:wahash.hanan-abdulrahman"Alharthi, Nadiyah Hussain"https://zbmath.org/authors/?q=ai:alharthi.nadiyah-hussainSummary: This research paper intends to study some qualitative analyses for a nonlinear fractional integrodifferential equation with a variable order in the frame of a Mittag-Leffler power law. At first, we convert the considered problem of variable order into an equivalent standard problem of constant order using generalized intervals and piecewise constant functions. Next, we prove the existence and uniqueness of analytic results by application of Krasnoselskii's and Banach's fixed point theorems. Besides, the guarantee of the existence of solutions is shown by different types of Ulam-Hyer's stability. Then, we investigate sufficient conditions of positive solutions for the proposed problem. In the end, we discuss an example to illustrate the applicability of our obtained results.Fundamental results about the fractional integro-differential equation described with Caputo derivativehttps://zbmath.org/1491.450122022-09-13T20:28:31.338867Z"Sene, Ndolane"https://zbmath.org/authors/?q=ai:sene.ndolaneSummary: In this paper, we study the existence and uniqueness of the mild solution of the fractional integro-differential with the nonlocal initial condition described by the Caputo fractional operator. Note that here the order of the Caputo derivative satisfies the condition that \(\alpha\in\left( 1 , 2\right)\). The existence of \(\alpha \)-resolvent operator in Banach space and fixed point theorem has been utilized in the proof of the existence of the mild solution. We have established in this paper the Hyers-Ulam stability of the mild solution of the considered fractional integro-differential equation. An illustrative example has been provided to support the main findings of the paper.Solvability of implicit fractional order integral equation in \(\ell_p\) (\(1 \leq p < \infty\)) space via generalized Darbo's fixed point theoremhttps://zbmath.org/1491.450182022-09-13T20:28:31.338867Z"Haque, Inzamamul"https://zbmath.org/authors/?q=ai:haque.inzamamul"Ali, Javid"https://zbmath.org/authors/?q=ai:ali.javid"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad-ayman|mursaleen.mohammadSummary: We present a generalization of Darbo's fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations in \(\ell_p\) (\(1 \leq p < \infty\)) space. The fundamental tool in the presentation of our proofs is the measure of noncompactness (mnc) approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations.On the factorable spaces of absolutely \(p\)-summable, null, convergent, and bounded sequenceshttps://zbmath.org/1491.460072022-09-13T20:28:31.338867Z"Başar, Feyzi"https://zbmath.org/authors/?q=ai:basar.feyzi"Roopaei, Hadi"https://zbmath.org/authors/?q=ai:roopaei.hadiSummary: Let \(F\) denote the factorable matrix and \(X\in\{ \ell_p,c_0,c, \ell_\infty\}\). In this study, we introduce the domains \(X(F)\) of the factorable matrix in the spaces \(X\). Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces \(X(F)\). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes \((\ell_p(F),\ell_\infty)\), \((\ell_p(F),f)\) and \((X,Y(F))\) of matrix transformations, where \(Y\) denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix \(F\) and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix \(F\). Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.Functions of bounded variation on complete and connected one-dimensional metric spaceshttps://zbmath.org/1491.460272022-09-13T20:28:31.338867Z"Lahti, Panu"https://zbmath.org/authors/?q=ai:lahti.panu"Zhou, Xiaodan"https://zbmath.org/authors/?q=ai:zhou.xiaodan|zhou.xiaodan.1Summary: In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by \textit{M.~Miranda} [J. Math. Pures Appl. (9) 82, No.~8, 975--1004 (2003; Zbl 1109.46030)]. Furthermore, we study the necessity of conditions on the underlying space in Federer's characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.Around Jensen's inequality for strongly convex functionshttps://zbmath.org/1491.470162022-09-13T20:28:31.338867Z"Moradi, Hamid Reza"https://zbmath.org/authors/?q=ai:moradi.hamid-reza"Omidvar, Mohsen Erfanian"https://zbmath.org/authors/?q=ai:omidvar.mohsen-erfanian"Khan, Muhammad Adil"https://zbmath.org/authors/?q=ai:khan.muhammad-adil"Nikodem, Kazimierz"https://zbmath.org/authors/?q=ai:nikodem.kazimierzSummary: In this paper, we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen-Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen's operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp(A) \subset (1,\infty)\), then
\[
\langle Ax,x \rangle^r\leq \langle A^rx,x \rangle -\frac{r^2-r}{2}\left(\langle A^2x,x\rangle -\langle Ax,x \rangle ^2 \right),\quad r\geq 2
\]
and if \(Sp(A) \subset (0,1)\), then
\[
\langle A^rx,x\rangle \leq\langle Ax,x \rangle^r+\frac{r-r^2}{2}\left(\langle Ax,x \rangle^2-\langle A^2x,x\rangle \right),\quad 0<r<1
\]
for each positive operator \(A\) and \(x\in \mathcal {H}\) with \(\| x\| =1\).Operator functions and the operator harmonic meanhttps://zbmath.org/1491.470172022-09-13T20:28:31.338867Z"Uchiyama, Mitsuru"https://zbmath.org/authors/?q=ai:uchiyama.mitsuruSummary: The objective of this paper is to investigate operator functions by making use of the operator harmonic mean `!'. For \(0<A\leqq B\), we construct a unique pair \(X, Y\) such that \(0<X\leqq Y\), \(A=X\operatorname{!}Y\), \(B=\frac{X+Y}{2} \). We next give a condition for operators \(A, B, C\geqq 0\) in order that \(C \leqq A\operatorname{!}B\) and show that \(g\ne 0\) is strongly operator convex on \(J\) if and only if \(g(t)>0\) and \(g (\frac{A+B}{2}) \leqq g(A)\operatorname{!}g(B)\) for \(A, B\) with spectra in \(J\). This inequality particularly holds for an operator decreasing function on the right half line. We also show that \(f(t)\) defined on \((0, b)\) with \(0<b\leqq \infty\) is operator monotone if and only if \(f(0+)<\infty\), \(f (A\operatorname{!}B)\leqq \frac{1}{2}(f(A) + f(B))\). In particular, if \(f>0\), then \(f\) is operator monotone if and only if \(f (A\operatorname{!}B) \leqq f(A)\operatorname{!}f(B)\). We lastly prove that if a strongly operator convex function \(g(t)>0\) on a finite interval \((a, b)\) is operator decreasing, then \(g\) has an extension \(\tilde{g}\) to \((a, \infty )\) that is positive and operator decreasing.Generalized fractional integral operators and their commutators with functions in generalized Campanato spaces on Orlicz spaceshttps://zbmath.org/1491.470382022-09-13T20:28:31.338867Z"Shi, Minglei"https://zbmath.org/authors/?q=ai:shi.minglei"Arai, Ryutaro"https://zbmath.org/authors/?q=ai:arai.ryutaro"Nakai, Eiichi"https://zbmath.org/authors/?q=ai:nakai.eiichiSummary: We investigate the commutators \([b,I_{\rho}]\) of generalized fractional integral operators \(I_{\rho}\) with functions \(b\) in generalized Campanato spaces and give a necessary and sufficient condition for the boundedness of the commutators on Orlicz spaces. To do this, we define Orlicz spaces with generalized Young functions and prove the boundedness of generalized fractional maximal operators on the Orlicz spaces.Existence of local fractional integral equation via a measure of non-compactness with monotone property on Banach spaceshttps://zbmath.org/1491.470472022-09-13T20:28:31.338867Z"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Ibrahim, Rabha W."https://zbmath.org/authors/?q=ai:ibrahim.rabha-waell"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-p"Can, N. H."https://zbmath.org/authors/?q=ai:can.ngo-huy|can.nguyen-huuSummary: In this paper, we discuss fixed point theorems for a new \(\chi\)-set contraction condition in partially ordered Banach spaces, whose positive cone \(\mathbb{K}\) is normal, and then proceed to prove some coupled fixed point theorems in partially ordered Banach spaces. We relax the conditions of a proper domain of an underlying operator for partially ordered Banach spaces. Furthermore, we discuss an application to the existence of a local fractional integral equation.On generalized Newton's aerodynamic problemhttps://zbmath.org/1491.490032022-09-13T20:28:31.338867Z"Plakhov, A."https://zbmath.org/authors/?q=ai:plakhov.a-yuSummary: We consider the generalized Newton's least resistance problem for convex bodies: minimize the functional \(\iint_\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx\, dy\) in the class of concave functions \(u\colon \Omega \to [0,M]\), where the domain \(\Omega \subset \mathbb{R}^2\) is convex and bounded and \(M > 0\). It has been known (see [\textit{G. Buttazzo} et al., Math. Nachr. 173, 71--89 (1995; Zbl 0835.49001)]) that if \(u\) solves the problem, then \(|\nabla u(x,y)| \ge 1\) at all regular points \((x,y)\) such that \(u(x,y) < M\). We prove that if the upper level set \(L = \{ (x,y)\colon u(x,y) = M \}\) has nonempty interior, then for almost all points of its boundary \((\bar{x}, \bar{y}) \in \partial L\) one has \(\lim_{\substack{(x,y)\to (\bar{x},\bar{y})\\u(x,y)<M}}|\nabla u(x,y)| = 1\). As a by-product, we obtain a result concerning local properties of convex surfaces near ridge points.Fractional variational problems on conformable calculushttps://zbmath.org/1491.490162022-09-13T20:28:31.338867Z"Öğrekçi, Süleyman"https://zbmath.org/authors/?q=ai:ogrekci.suleyman"Asliyüce, Serkan"https://zbmath.org/authors/?q=ai:asliyuce.serkanThe authors discuss the optimality conditions of the variational problems including conformable fractional derivatives. They obtain the optimality conditions for Öxed end-point variational problems, and for variable end-point variational problems. Then they investigate the isoperimetric problem, and variational problems with holonomic constraints. Finally, they give a sufficient condition for optimality results of variational problems.
Reviewer: Suvra Kanti Chakraborty (Kolkata)Strict convergence with equibounded area and minimal completely vertical liftingshttps://zbmath.org/1491.490342022-09-13T20:28:31.338867Z"Mucci, Domenico"https://zbmath.org/authors/?q=ai:mucci.domenicoSummary: Minimal lifting measures of vector-valued functions of bounded variation were introduced by Jerrard-Jung. They satisfy strong continuity properties with respect to the strict convergence in \(BV\). Moreover, they can be described in terms of the action of the optimal Cartesian currents enclosing the graph of \(u\). We deal with a good notion of completely vertical lifting for maps with values into the two dimensional Euclidean space. We then prove lack of uniqueness in the high codimension case. Relationship with the relaxed area functional in the strict convergence is also discussed.Applications of fractional calculus in equiaffine geometry: plane curves with fractional orderhttps://zbmath.org/1491.530132022-09-13T20:28:31.338867Z"Aydin, Muhittin Evren"https://zbmath.org/authors/?q=ai:aydin.muhittin-evren"Mihai, Adela"https://zbmath.org/authors/?q=ai:mihai.adela"Yokus, Asif"https://zbmath.org/authors/?q=ai:yokus.asifIn the present paper, new geometry differential notions are introduced in the field of fractional calculus. The present paper particularly addresses differential geometry problems by utilizing fractional derivative operators. The novel notions of equiaffine arclength and curvature are introduced. After the introduction of the equiaffine notion, the so-called equiaffine Frenet frame of a nondegenerate smooth curve in two-dimensional spaces has been provided. A comparative study between the geometrics notions introduced in this paper and their standard associated notions is provided. The importance of these new notions is that these notions are defined using a fractional operator and will permit to use the theory of fractional operators. For the illustration of the equiaffine notion, the paper provides an example illustrating the equiaffine curvature function described by a fractional-order derivative. Several illustrative examples are provided to illustrate the main findings of the present paper.
Reviewer: Ndolane Sene (Dakar)On some density topology with respect to an extension of Lebesgue measurehttps://zbmath.org/1491.540052022-09-13T20:28:31.338867Z"Flak, Katarzyna"https://zbmath.org/authors/?q=ai:flak.katarzyna"Hejduk, Jacek"https://zbmath.org/authors/?q=ai:hejduk.jacek"Tomczyk, Sylwia"https://zbmath.org/authors/?q=ai:tomczyk.sylwiaSummary: This paper presents a density type topology with respect to an extension of Lebesgue measure involving sequence of intervals tending to zero. Some properties of such topologies are investigated.On some generalizations of porosity and porouscontinuityhttps://zbmath.org/1491.540182022-09-13T20:28:31.338867Z"Kowalczyk, Stanisław"https://zbmath.org/authors/?q=ai:kowalczyk.stanislaw"Turowska, Małgorzata"https://zbmath.org/authors/?q=ai:turowska.malgorzataSummary: In the present paper, we introduce notions of v-porosity and v-porouscontinuity. We investigate some properties of v-porouscontinuity and its connections with porouscontinuity introduced by \textit{J. Borsík} and \textit{J. Holos} [Math. Slovaca 64, No. 3, 741--750 (2014; Zbl 1340.54028)]. Moreover, we show that v-porouscontinuous functions may not belong to Baire class one.Mean value theorems for the noncausal stochastic integralhttps://zbmath.org/1491.600772022-09-13T20:28:31.338867Z"Ogawa, Shigeyoshi"https://zbmath.org/authors/?q=ai:ogawa.shigeyoshiSummary: We are concerned with the validity of the mean value theorem for the noncausal stochastic integral \(\int_s^t f(X_r)d_\ast W_r\) with respect to Brownian motion \(W_t(\omega)\), where \(X_t\) is an Itô process. We establish first a mean value theorem for the noncausal stochastic integral \(\int_s^t f(X_r)dX_r\) and based on the result we show the corresponding formulae for the noncausal integral \(\int_s^t f(X_r)d_\ast W_r\) or for Itô integral \(\int_s^t f(X_r)d_0W_r\) as well. We also study the problem for such a genuin noncausal case where the process \(X_t\) is noncausal, that is, not adapted to the natural filtration associated to Brownian motion. The discussions are developed in the framework of the noncausal calculus. Hence some materials and basic facts in the theory of noncausal stochastic calculus are briefly reviewed as preliminary.Handbook of fractional calculus for engineering and sciencehttps://zbmath.org/1491.650012022-09-13T20:28:31.338867ZPublisher's description: Fractional calculus is used to model many real-life situations from science and engineering. The book includes different topics associated with such equations and their relevance and significance in various scientific areas of study and research. In this book readers will find several important and useful methods and techniques for solving various types of fractional-order models in science and engineering. The book should be useful for graduate students, PhD students, researchers and educators interested in mathematical modelling, physical sciences, engineering sciences, applied mathematical sciences, applied sciences, and so on.
This Handbook:
\begin{itemize}
\item Provides reliable methods for solving fractional-order models in science and engineering.
\item Contains efficient numerical methods and algorithms for engineering-related equations.
\item Contains comparison of various methods for accuracy and validity.
\item Demonstrates the applicability of fractional calculus in science and engineering.
\item Examines qualitative as well as quantitative properties of solutions of various types of science- and engineering-related equations.
\end{itemize}
Readers will find this book to be useful and valuable in increasing and updating their knowledge in this field and will be it will be helpful for engineers, mathematicians, scientist and researchers working on various real-life problems.
The articles of this volume will not be indexed individually.Fractional derivative based nonlinear diffusion model for image denoisinghttps://zbmath.org/1491.650202022-09-13T20:28:31.338867Z"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh.3|kumar.santosh.4|kumar.santosh.2|kumar.santosh.1|kumar.santosh"Alam, Khursheed"https://zbmath.org/authors/?q=ai:alam.khursheed"Chauhan, Alka"https://zbmath.org/authors/?q=ai:chauhan.alkaSummary: In this article, a new conformable fractional anisotropic diffusion model for image denoising is presented, which contains the spatial derivative along with the time-fractional derivative. This model is a generalization of the diffusion model [\textit{M. Welk} et al., Lect. Notes Comput. Sci. 3459, 585--597 (2005; Zbl 1119.68511)] with forward-backward diffusivities. The proposed model is very efficient for noise removal of the noisy images in comparison to the classical anisotropic diffusion model. The numerical experiments are performed using an explicit scheme for different-different values of fractional order derivative \(\alpha \). The experimental results are obtained in terms of peak signal to noise ratio (PSNR) as a metric.Fast and improved scaled HSS preconditioner for steady-state space-fractional diffusion equationshttps://zbmath.org/1491.650252022-09-13T20:28:31.338867Z"Chen, Fang"https://zbmath.org/authors/?q=ai:chen.fang"Li, Tian-Yi"https://zbmath.org/authors/?q=ai:li.tianyiSummary: For the discrete linear system resulted from the considered steady-state space-fractional diffusion equations, we propose an improved scaled HSS (ISHSS) iteration method and discuss its convergence theory. Then, we construct a fast ISHSS (FISHSS) preconditioner to accelerate the convergence rates of the Krylov subspace iteration methods. We discuss the spectral properties of the FISHSS preconditioning matrix. Numerical experiments show the good performance of the FISHSS preconditioner.A novel discrete fractional Grönwall-type inequality and its application in pointwise-in-time error estimateshttps://zbmath.org/1491.651122022-09-13T20:28:31.338867Z"Li, Dongfang"https://zbmath.org/authors/?q=ai:li.dongfang"She, Mianfu"https://zbmath.org/authors/?q=ai:she.mianfu"Sun, Hai-wei"https://zbmath.org/authors/?q=ai:sun.haiwei"Yan, Xiaoqiang"https://zbmath.org/authors/?q=ai:yan.xiaoqiangSummary: We present a family of fully-discrete schemes for numerically solving nonlinear sub-diffusion equations, taking the weak regularity of the exact solutions into account. Using a novel discrete fractional Grönwall inequality, we obtain pointwise-in-time error estimates of the time-stepping methods. It is proved that as \(t\rightarrow 0\), the convergence orders can be \(\sigma_k\), where \(\sigma_k\) is the regularity parameter. The initial convergence results are sharp. As \(t\) is far away from 0, the schemes give a better convergence results. Numerical experiments are given to confirm the theoretical results.The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applicationshttps://zbmath.org/1491.651132022-09-13T20:28:31.338867Z"Pourbabaee, Marzieh"https://zbmath.org/authors/?q=ai:pourbabaee.marzieh"Saadatmandi, Abbas"https://zbmath.org/authors/?q=ai:saadatmandi.abbasSummary: In this paper, the properties of Chebyshev polynomials and the Gauss-Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given.A reduced basis method for fractional diffusion operators. IIhttps://zbmath.org/1491.651312022-09-13T20:28:31.338867Z"Danczul, Tobias"https://zbmath.org/authors/?q=ai:danczul.tobias"Schöberl, Joachim"https://zbmath.org/authors/?q=ai:schoberl.joachimSummary: We present a novel numerical scheme to approximate the solution map \(s \mapsto u(s) := \mathcal{L}^{-s} f\) to fractional PDEs involving elliptic operators. Reinterpreting \(\mathcal{L}^{-s}\) as an interpolation operator allows us to write \(u(s)\) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation \(L\) of the operator whose inverse is projected to the \(s\)-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously.
A second algorithm is presented to avoid inversion of \(L\). Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.
For Part I, see [\textit{T. Danczul} and \textit{J. Schöberl}, Numer. Math. 151, No. 2, 369--404 (2022; Zbl 07536676)].Minimal gauge invariant couplings at order \(\ell_p^6\) in M-theoryhttps://zbmath.org/1491.810262022-09-13T20:28:31.338867Z"Garousi, Mohammad R."https://zbmath.org/authors/?q=ai:garousi.mohammad-rSummary: Removing the field redefinitions, the Bianchi identities and the total derivative freedoms from the general form of the gauge invariant couplings at order \(\ell_p^6\) for the bosonic fields of M-theory, we have found that the minimum number of independent couplings in the structures with even number of the three-form, is 1062. We find that there are schemes in which there is no coupling involving \(R\), \(R_{\mu\nu}\), \(\nabla_\mu F^{\mu\alpha\beta\gamma}\). In these schemes, there are sub-schemes in which, except one coupling which has the second derivative of \(F^{(4)}\), the couplings can have no term with more than two derivatives. We find some of the parameters by dimensionally reducing the couplings on a circle and comparing them with the known couplings of the one-loop effective action of type IIA superstring theory. In particular, we find the coupling which has term with more than two derivatives is zero.On quasiconvex functions which are convexifiable or nothttps://zbmath.org/1491.910762022-09-13T20:28:31.338867Z"Crouzeix, Jean-Pierre"https://zbmath.org/authors/?q=ai:crouzeix.jean-pierreSummary: A quasiconvex function \(f\) being given, does there exist an increasing and continuous function \(k\) which makes \(k\circ f\) convex? How to build such a \(k\)? Some words on least convex (concave) functions. The ratio of two positive numbers is neither locally convexifiable nor locally concavifiable. Finally, some considerations on the approximation of a preorder from a finite number of observations and on the revealed preference problem are discussed.Outliers in control engineering. Fractional calculus perspective. Based on the 20th world congress of the International Federation of Automatic Control (IFAC), Toulouse, France, July 9--14, 2017https://zbmath.org/1491.930032022-09-13T20:28:31.338867ZPublisher's description: Outliers play an important, though underestimated, role in control engineering. Traditionally they are unseen and neglected. In opposition, industrial practice gives frequent examples of their existence and their mostly negative impacts on the control quality. The origin of outliers is never fully known. Some of them are generated externally to the process (exogenous), like for instance erroneous observations, data corrupted by control systems or the effect of human intervention. Such outliers appear occasionally with some unknow probability shifting real value often to some strange and nonsense value. They are frequently called deviants, anomalies or contaminants. In most cases we are interested in their detection and removal.
However, there exists the second kind of outliers. Quite often strange looking data observations are not artificial data occurrences. They may be just representatives of the underlying generation mechanism being inseparable internal part of the process (endogenous outliers). In such a case they are not wrong and should be treated with cautiousness, as they may include important information about the dynamic nature of the process. As such they cannot be neglected nor simply removed. The Outlier should be detected, labelled and suitably treated. These activities cannot be performed without proper analytical tools and modeling approaches. There are dozens of methods proposed by scientists, starting from Gaussian-based statistical scoring up to data mining artificial intelligence tools. The research presented in this book presents novel approach incorporating non-Gaussian statistical tools and fractional calculus approach revealing new data analytics applied to this important and challenging task.
The proposed book includes a collection of contributions addressing different yet cohesive subjects, like dynamic modelling, classical control, advanced control, fractional calculus, statistical analytics focused on an ultimate goal: robust and outlier-proof analysis. All studied problems show that outliers play an important role and classical methods, in which outlier are not taken into account, do not give good results. Applications from different engineering areas are considered such as semiconductor process control and monitoring, MIMO peltier temperature control and health monitoring, networked control systems, and etc.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Domański, Paweł D.; Chen, Yangquan; Ławryńczuk, Maciej}, Outliers in control engineering -- they exist, like it or not, 1-24 [Zbl 07543834]
\textit{Domek, Stefan}, On the possibilities of using fractional-order differential calculus in linear and nonlinear model predictive control, 27-46 [Zbl 07543835]
\textit{Duncan, Tyrone E.; Pasik-Duncan, Bozenna}, Stochastic control systems with long-range dependent noise, 47-59 [Zbl 07543836]
\textit{Huang, Yulong; Zhu, Fengchi; Zhang, Yonggang; Zhao, Yuxin; Shi, Peng; Chambers, Jonathon}, Outlier-robust Kalman filtering framework based on statistical similarity measure, 61-98 [Zbl 07543837]
\textit{Oświęcimka, Paweł; Minati, Ludovico}, Multifractal characteristics of singular signals, 99-112 [Zbl 07543838]
\textit{Domański, Paweł D.; Ławryńczuk, Maciej}, Study on robustness of nonlinear model predictive control performance assessment, 115- [Zbl 07543839]
\textit{Guc, Furkan; Chen, Yangquan}, Backlash quantification in control systems using noises with outliers: a benchmark study, 149-156 [Zbl 07543841]
\textit{Liu, Kai; Chen, Yangquan; Domański, Paweł D.}, A novel method for control performance assessment with fractional-order signal processing, 167-186 [Zbl 07543843]
\textit{Viola, Jairo; Rodriguez, Carlos; Hollenbeck, Derek; Chen, Yangquan}, A radio frequency impedance matching control benchmark and optimal fractional-order stochastic extremum seeking method, 237-257 [Zbl 07543846]A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertaintyhttps://zbmath.org/1491.930232022-09-13T20:28:31.338867Z"Ullah, Nasim"https://zbmath.org/authors/?q=ai:ullah.nasim"Al-Ahmadi, Ahmad Aziz"https://zbmath.org/authors/?q=ai:al-ahmadi.ahmad-azizA second order nonlinear control system with measurement noise and uncertainty is considered. A fractional order sliding manifold is introduced and a control law is proposed, which can operate as a classical sliding mode control, or integral, or fractional order integral sliding mode control. The convergence of the state errors to zero in finite time is proved. An application is discussed in detail.
Reviewer: Tullio Zolezzi (Genova)System identification of MISO fractional systems: parameter and differentiation order estimationhttps://zbmath.org/1491.930282022-09-13T20:28:31.338867Z"Victor, Stéphane"https://zbmath.org/authors/?q=ai:victor.stephane"Mayoufi, Abir"https://zbmath.org/authors/?q=ai:mayoufi.abir"Malti, Rachid"https://zbmath.org/authors/?q=ai:malti.rachid"Chetoui, Manel"https://zbmath.org/authors/?q=ai:chetoui.manel"Aoun, Mohamed"https://zbmath.org/authors/?q=ai:aoun.mohamedSummary: This paper deals with continuous-time system identification of multiple-input single-output (MISO) fractional differentiation models. When differentiation orders are assumed to be known, coefficients are estimated using the simplified refined instrumental variable method for continuous-time fractional models extended to the MISO case. For unknown differentiation orders, a two-stage optimization algorithm is proposed with the developed instrumental variable for coefficient estimation and a gradient-based algorithm for differentiation order estimation. A new definition of structured-commensurability (or S-commensurability) is introduced to better cope with differentiation order estimation. Three variants of the algorithm are then proposed: (i) first, all differentiation orders are set as integer multiples of a global S-commensurate order, (ii) then, the differentiation orders are set as integer multiples of a local S-commensurate orders (one S-commensurate order for each subsystem), (iii) finally, all differentiation orders are estimated by releasing the S-commensurability constraint. The first variant has the smallest number of parameters and is used as a good initial hit for the second variant which in turn is used as a good initial hit for the third variant. Such a progressive increase of the number of parameters allows better performance of the optimization algorithm evaluated by Monte Carlo simulation analysis.A digraph approach to the state-space model realization of MIMO non-commensurate fractional order systemshttps://zbmath.org/1491.930632022-09-13T20:28:31.338867Z"Zhao, Dongdong"https://zbmath.org/authors/?q=ai:zhao.dongdong"Hu, Yang"https://zbmath.org/authors/?q=ai:hu.yang"Sun, Weiguo"https://zbmath.org/authors/?q=ai:sun.weiguo"Zhou, Xingwen"https://zbmath.org/authors/?q=ai:zhou.xingwen"Xu, Li"https://zbmath.org/authors/?q=ai:xu.li"Yan, Shi"https://zbmath.org/authors/?q=ai:yan.shiSummary: This paper proposes a novel approach that can generate a state-space model with low inner dimension for an MIMO non-commensurate fractional order (NCFO) system. Specifically, the notion of an admissible digraph is firstly introduced associated with a fractional order transfer (function column) vector. Then, new state-space model realization conditions and corresponding procedures based on this admissible digraph are proposed for the state-space model realization of an NCFO polynomial transfer matrix. Finally, a new necessary and sufficient state-space model realization condition is proposed for the rational transfer matrix of an MIMO NCFO system, and it is shown, based on a matrix fractional description (MFD) of the given rational transfer matrix, a state-space model realization can be obtained by firstly converting it to the polynomial case and then utilizing the digraph approach for polynomial case. Symbolic and numerical examples are provided to demonstrate the main ideas and effectiveness of the proposed digraph approach.Origami at the intersection of algebra, geometry and calculushttps://zbmath.org/1491.970132022-09-13T20:28:31.338867Z"Wares, Arsalan"https://zbmath.org/authors/?q=ai:wares.arsalan"Valori, Giovanna"https://zbmath.org/authors/?q=ai:valori.giovanna(no abstract)Differential and integral proportional calculus: how to find a primitive for \(f(x)=1/\sqrt{2\pi}e^{-(1/2)x^2}\)https://zbmath.org/1491.970222022-09-13T20:28:31.338867Z"Campillay-Llanos, William"https://zbmath.org/authors/?q=ai:campillay-llanos.william"Guevara, Felipe"https://zbmath.org/authors/?q=ai:guevara.felipe"Pinto, Manuel"https://zbmath.org/authors/?q=ai:pinto.manuel"Torres, Ricardo"https://zbmath.org/authors/?q=ai:torres.ricardo-pratoSummary: We present a type of arithmetic called Proportional Arithmetic. The main properties and objects that emerge with this way of operating quantities are exposed. Finally, the antiderivative and the indefinite integral are defined in order to calculate the primitive of \(f(x)=1/\sqrt{2\pi}e^{-(1/2)x^2}\) in the Proportional context.Some variants of the integral mean value theoremhttps://zbmath.org/1491.970232022-09-13T20:28:31.338867Z"Lozada-Cruz, German"https://zbmath.org/authors/?q=ai:lozada-cruz.german(no abstract)Arc length of function graphs via Taylor's formulahttps://zbmath.org/1491.970242022-09-13T20:28:31.338867Z"Nystedt, Patrik"https://zbmath.org/authors/?q=ai:nystedt.patrikSummary: We use Taylor's formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. As a means for generating interesting examples of exact arc length calculations in calculus courses, we recall two large classes of functions \(f\) with the property that \(\sqrt{1+(f^\prime)^2}\) has a primitive, including classical examples by Neile, van Heuraet and Fermat, as well as more recent ones induced by Pythagorean triples of functions. We also discuss potential benefits for our proposed definition of arc length in introductory calculus courses.Integrating rational functions of sine and cosine using the rules of Biochehttps://zbmath.org/1491.970272022-09-13T20:28:31.338867Z"Stewart, Seán M."https://zbmath.org/authors/?q=ai:stewart.sean-mark(no abstract)Curve parametrization and triple integrationhttps://zbmath.org/1491.970282022-09-13T20:28:31.338867Z"Rabiei, Nima"https://zbmath.org/authors/?q=ai:rabiei.nima"Saleeby, Elias G."https://zbmath.org/authors/?q=ai:saleeby.elias-george(no abstract)