Recent zbMATH articles in MSC 26https://www.zbmath.org/atom/cc/262021-03-30T15:24:00+00:00WerkzeugNew integral inequalities through the \(\varphi\)-preinvexity.https://www.zbmath.org/1455.260182021-03-30T15:24:00+00:00"Meftah, Badreddine"https://www.zbmath.org/authors/?q=ai:meftah.badreddineSummary: In this note, we give some estimates of the generalized quadrature formula of Gauss-Jacobi type for \(\varphi\)-preinvex functions.On paranormed type fuzzy \(I\)-convergent double multiplier sequences.https://www.zbmath.org/1455.460092021-03-30T15:24:00+00:00"Sen, M."https://www.zbmath.org/authors/?q=ai:sen.mausumi|sen.matilal|sen.manimay|sen.murat|de-la-sen.manuel|sen.mihir|sen.monalisa|sen.moitri|sen.mira|sen.mridul-kanti"Roy, S."https://www.zbmath.org/authors/?q=ai:roy.soma|roy.sambudha|roy.santanu|roy.shongkour|roy.serge|roy.subhadeep|roy.swalpa-kumar|roy.stephen-c|roy.sandip|roy.sunny|roy.s-n|roy.subrata-shyam|roy.shubho-r|roy.siuli|roy.subhrajit|roy.sourin|roy.sanjit-kumar|roy.shobhan|roy.sylvain|roy.sasanka|roy.souktik|roy.sukesh|roy.samit|roy.shovonlal|roy.sukomal-chandra|roy.subhamoy-singha|roy.smarajit|roy.swati|roy.sanghamitra|roy.swapnoneel|roy.sudeepa|roy.supriyo|roy.sonali|roy.soumen-kumar|roy.sudipto|roy.sitikantha|roy.subhankar|roy.saptarshi|roy.shyamal|roy.shaibal|roy.samudra|roy.scott|roy.sisir|roy.shaswati|roy.siddhant|roy.sumit|roy.soumyaroop|roy.subhajit|roy.shibendu-shekhar|roy.suddhasatwa|roy.sarah-m|roy.sanjukta|roy.sutanu|roy.sawpna|roy.souvik.1|roy.suchismita|roy.subir|roy.sabyasachi|roy.samriddho|roy.sanjay-k|roy.spandan|roy.sujoy-sinha|roy.sankar-kumar|roy.sugata-sen|roy.samir|roy.shraddha|roy.supratik|roy.sarbani|roy.suman|roy.sunanda|roy.sanat-kumar|roy.susmita|roy.soumya|roy.shibdas|roy.surupa|roy.sovik|roy.shuvo|roy.sayan-basu|roy.sourov|roy.sudeshna|roy.sambuddha|roy.subash-chandra|roy.somnath|roy.senjuti-basu|roy.swapna|roy.subhradeep|roy.subhas-chandra|roy.subhro|roy.sankhadip|roy.sandipan|roy.s-c-dutta|roy.shasanka-mohan|roy.sebastien|roy.sourav|roy.shibaji|roy.subhransu|roy.shouvik|roy.saikat|roy.soumendu|roy.sanjiban-sekhar|roy.shourya|roy.saswati|roy.swarupa|roy.suvam|roy.subhadip|roy.sthitadhi|roy.satyajit|roy.sajalSummary: In this article, we introduce the classes of fuzzy real valued double sequences \(_2c^{I(F)}(\Lambda,p)\) and \(_2c_0^{I(F)}(\Lambda,p)\), where \(\Lambda = (\lambda_{nk})\) is a multiplier sequence of non-zero real numbers and \(p=(p_{nk})\) is a double sequence of bounded strictly positive numbers. We study different topological properties of these classes of sequences. Also we characterize the multiplier problem and obtain some inclusion relation involving these classes of sequences.Sharp rational bounds for the gamma function.https://www.zbmath.org/1455.330022021-03-30T15:24:00+00:00"Shen, Jian-Mei"https://www.zbmath.org/authors/?q=ai:shen.jian-mei"Yang, Zhen-Hang"https://www.zbmath.org/authors/?q=ai:yang.zhenhang"Qian, Wei-Mao"https://www.zbmath.org/authors/?q=ai:qian.weimao"Zhang, Wen"https://www.zbmath.org/authors/?q=ai:zhang.wen.3"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yumingThe main result of the paper offers a best upper bound of the type $(x^2+p)/(x+p)$ of the gamma function at $x+1$, where $x$ is in the interval $(0,1)$.
Reviewer's remark: The authors are not aware, that this result has been proved in 2014 by \textit{P. A. Kupán} and \textit{R. Szász} [Integral Transforms Spec. Funct. 25, No. 7, 562--570 (2014; Zbl 1301.26013)].
Reviewer: József Sándor (Cluj-Napoca)Formulate a relationship between saddle points on surfaces and inflection points on curves.https://www.zbmath.org/1455.260092021-03-30T15:24:00+00:00"Mohd, I."https://www.zbmath.org/authors/?q=ai:mohd.ismail-bin"Dasril, Y."https://www.zbmath.org/authors/?q=ai:dasril.yosza-binSummary: It is known that there is a much closed relationship between saddle and inflection points. It was shown in one of the research papers that a connection between the saddle points of functions of two variables with the inflection points of functions of one variable and the researcher claimed that he has not found any references to this result in the literature. However, the author himself worried by asking whether there always exists such a one variable function that is differentiable at the saddle point or not. In this paper, it will be proposed two results for relationship between the saddle and inflection points through the quadratic functions of two variables and two linear and non-linear functions of one variable. These results will be supported with several numerical examples.Optimal control of the linear wave equation by time-depending BV-controls: a semi-smooth Newton approach.https://www.zbmath.org/1455.490022021-03-30T15:24:00+00:00"Engel, Sebastian"https://www.zbmath.org/authors/?q=ai:engel.sebastian"Kunisch, Karl"https://www.zbmath.org/authors/?q=ai:kunisch.karlSummary: An optimal control problem for the linear wave equation with control cost chosen as the BV semi-norm in time is analyzed. This formulation enhances piecewise constant optimal controls and penalizes the number of jumps. Existence of optimal solutions and necessary optimality conditions are derived. With numerical realisation in mind, the regularization by \(H^1\) functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the \(H^1\)-regularized problems the semi-smooth Newton algorithm can be used to solve the first order optimality conditions with super-linear convergence rate. Examples are constructed which show that the distributional derivative of an optimal control can be a mix of absolutely continuous measures with respect to the Lebesgue measure, a countable linear combination of Dirac measures, and Cantor measures. Numerical results illustrate and support the analytical results.A complete monotonicity property of the multiple gamma function.https://www.zbmath.org/1455.330012021-03-30T15:24:00+00:00"Das, Sourav"https://www.zbmath.org/authors/?q=ai:das.souravThis paper is concerned with the complete monotonicity property of
some complicated, implicitly defined functions.
Rather than mention these functions, I note that since logarithms and
exponentials occur on many places, it would be advisable to define
a branch of the logarithm. The multiple gamma function, as well as the
gamma function and the (multiple) \(q\)-gamma function, which occurs in
another paper of the author [C. R., Math., Acad. Sci. Paris 358, No. 3, 327--332 (2020; Zbl 1450.33004)], are by definition complex functions.
Graphs of the functions are given in the end.
Reviewers remarks:
\begin{itemize}\item[1.]
The complex logarithm is denoted by log, not ln. Both notations are used in the paper.
\item[2.]
There is a misprint in the definition on line 5 on page 919. It should say:
Similarly, one can define multiple \(\Phi\) function \(\Phi_n=\frac{G_n'}{G_n}\).
\item[3.]
The application of Leibnitz' rule in the proof of theorem 5 is not clear,
since it is not possible to distinguish the factors \(u\) and \(v\) in the proof.
\end{itemize}
Reviewer: Thomas Ernst (Uppsala)Solving two initial-boundary value problems including fractional partial differential equations by spectral and contour integral methods.https://www.zbmath.org/1455.352872021-03-30T15:24:00+00:00"Jahanshahi, M."https://www.zbmath.org/authors/?q=ai:jahanshahi.morteza-afshar|jahanshahi.mohamad|jahanshahi.mohsen|jahanshahi.mohammad-h|jahanshahi.marjan"Aliyev, N."https://www.zbmath.org/authors/?q=ai:aliev.n-ya|aliev.n-l|aliev.nihan-a|aliyev.najaf|aliev.n-m|aliyev.nicat|aliev.nizomiddin-makhmasaitovich|aliev.n-t"Jahanshahi, F."https://www.zbmath.org/authors/?q=ai:jahanshahi.fSummary: Initial-boundary value problems including fractional partial differential equations are the mathematical models of physical problems and natural phenomena. In this paper, at first we consider a fractional partial differential equation which has no mixed term derivative with respect to spatial and time variable. We first consider the spectral problem, then its eigenvalues and eigenfunctions are calculated. After that the eigenvalues and eigenfunctions of the adjoint problem are calculated. By using these eigenfunctions and Mittag-Leffler functions the approximate solution is constructed. In second section, we consider differential equation which has a mixed term derivative. In this case, by using Laplace transformation, the analytic solution and approximate solution are calculated as integral expression over suitable closed contours by contour integral method. At the end, some examples are presented for several cases of different distributions of eigenvalues in complex plane.Weighted Riesz bounded variation spaces and the Nemytskii operator.https://www.zbmath.org/1455.260052021-03-30T15:24:00+00:00"Cruz-Uribe, D."https://www.zbmath.org/authors/?q=ai:cruz-uribe.david-v"Guzman, O. M."https://www.zbmath.org/authors/?q=ai:guzman.oscar-mauricio"Rafeiro, H."https://www.zbmath.org/authors/?q=ai:rafeiro.humbertoSummary: We define a weighted version of the Riesz bounded variation space. We show that a generalization of the Riesz theorem relating these spaces to the Sobolev space \(W^{1,p}(I)\) holds for weighted Riesz bounded variation spaces when the weight belongs to the Muckenhoupt class. As an application, for weights belonging to the Muckenhoupt class, we characterize the globally Lipschitz Nemytskii operators acting in the weighted Riesz bounded variation spaces.An efficient algorithm for numerical inversion of system of generalized Abel integral equations.https://www.zbmath.org/1455.652322021-03-30T15:24:00+00:00"Dixit, Sandeep"https://www.zbmath.org/authors/?q=ai:dixit.sandeepSummary: In this article a direct method is introduced, which is based on orthonormal Bernstein polynomials, to present an efficient and stable algorithm for numerical inversion of the system of singular integral equations of Abel type. The appropriateness of earlier numerical inversion methods was restricted to the one portion of singular integral equations of Abel type. The proposed method is absolutely accurate, and numerical illustrations are given to show the convergence and utilization of the suggested method and comparisons are made with some other existing numerical solution.Nemytskii operator on \((\phi,2,\alpha )\)-bounded variation space in the sense of Riesz.https://www.zbmath.org/1455.260042021-03-30T15:24:00+00:00"Castillo, René E."https://www.zbmath.org/authors/?q=ai:castillo.rene-erlin"Rojas, Edixon M."https://www.zbmath.org/authors/?q=ai:rojas.edixon-m"Trousselot, Eduard"https://www.zbmath.org/authors/?q=ai:trousselot.eduardThis paper deals with a generalization of functions of bounded variation; namely functions with bounded \((\phi,2,\alpha)\)-variation in the sense of Riesz.
Let \(\phi: [0,\infty)\to [0,\infty)\) be a function such that
\begin{itemize}
\item[a)] \(\phi(x)=0\) if and only if \(x=0\),
\item[b)] \(\lim_{x\to+\infty}\phi(x)=+\infty\),
\end{itemize}
known as a Young function. Consider now a real function \(f\) defined on \([a,b]\) and let \(\alpha\) be any strictly increasing continuous function defined on \([a,b]\). Let \(\Pi\) be a block partition of the interval \([a,b]\), that is,
\begin{align*}
\Pi: a&=x_{1,1}<x_{1,2}\le x_{1,3}<x_{1,4}=x_{2,1}<x_{2,2}\le x_{2,3}<x_{2,4}\\
&=x_{3,1}<\dotsb<x_{n-1,4}=x_{n,1}<x_{n,2}\le x_{n,3}<x_{n,4}=b.
\end{align*}
Writing
\[
f_{\alpha}[p,q]=\frac{f(q)-f(p)}{\alpha(q)-\alpha(p)}
\]
one defines:
\[
\sigma^{R}_{(\phi,2,\alpha)}(f,\Pi)=\sum^{n}_{j=1}\phi\left(
\frac{|f_{\alpha}[x_{j,4},x_{j,3}]-f_{\alpha}[x_{j,2},x_{j,1}]|}{|\alpha(x_{j,4})-\alpha(x_{j,1})|}\right)\cdot |\alpha(x_{j,4})-\alpha(x_{j,1})|.
\]
When \(V^{R}_{(\phi,2,\alpha)}(f)=\sup_{\Pi}\sigma^{R}_{(\phi,2,\alpha)}(f,\Pi)<+\infty\), one says that \(f\) is of \((\phi,2,\alpha)\)-variation in the sense of Riesz.
Introducing the \(\alpha\)-derivative
\[
f'_{\alpha}(x_{0})=\lim_{x\to x_{0}}\frac{f(x)-f(x_{0})}{\alpha(x)-\alpha(x_{0})}
\]
one proves the following result:
\begin{itemize}\item
If \(\phi\) satisfies the conditions \(\lim_{x\to +\infty}\frac{\phi(x)}{x}=+\infty\) and \(f\in V^{R}_{(\phi,2,\alpha)}[a,b]\) (the space of functions with finite \(V^{R}_{(\phi,2,\alpha)}\) variation), then \(f'_{\alpha}(x_{0})\) exists at each point~\(x_{0}\in [a,b]\) and \(f'_{\alpha}\in V^{R}_{(\phi,2,\alpha)}[a,b]\). Moreover
\[
\int^{b}_{a}\phi (|f''_{\alpha}(t)|)\,d\alpha(t)\le V_{(\phi,2,\alpha)}(f).
\]
\end{itemize}
If \(\phi\) is a convex function, the set of functions \(f: [a,b]\to \mathbb{R}\) such that there exists \(\lambda>0\) with \(V^{R}_{(\phi,2,\alpha)}(\lambda f)<+\infty\) is denoted by \(RV_{(\phi,2,\alpha)}([a,b])\). It turns out that \(RV_{(\phi,2,\alpha)}([a,b])\) is a Banach algebra:
\begin{itemize}\item
If \(f,g,\in RV_{(\phi,2,\alpha)}([a,b])\), then \(f\cdot g\in RV_{(\phi,2,\alpha)}([a,b])\).
\end{itemize}
The superposition operator, or Nemytskii operator, defined by \(F(u(s)) = f (s, u(s))\), is the simplest among the nonlinear operators. In the space \(RV_{(\phi,2,\alpha)}([a,b])\), the Nemytskii operator is globally Lipschitz as it shows the following result:
\begin{itemize}\item
Let \(\phi\) be a convex \(\phi\)-function which satisfies \(\lim_{x\to +\infty}\frac{\phi(x)}{x}=+\infty\). Let \(f: [a,b]\times \mathbb{R}\to \mathbb{R}\). The Nemyskii operator associated to \(f\) defined by
\begin{align*}
F: RV_{(\phi,2,\alpha)}([a,b])&\longrightarrow \mathbb{R}\\
u &\longmapsto F(u)
\end{align*}
with \(F(u)=f(t,u(t))\), \(t\in [a,b]\) acts on \(RV_{(\phi,2,\alpha)}([a,b])\) and is globally Lipschitz, that is there exists \(K>0\) such that
\[
\|F(u_{1})-F(u_{2})\|^{R}_{(\phi,2,\alpha)}\le K\|u_{1}-u_{2}\|^{R}_{(\phi,2,\alpha)},\quad t\in [a,b]
\]
if and only if there exists \(g,h\in RV_{(\phi,2,\alpha)}([a,b])\) such that
\[
f(t,y)=g(t)y+h(t),\quad t\in [a,b],\,y\in \mathbb{R}.
\]
\end{itemize}
Reviewer: Julià Cufí (Bellaterra)Differentiability of continuous functions in terms of Haar-smallness.https://www.zbmath.org/1455.260022021-03-30T15:24:00+00:00"Kwela, Adam"https://www.zbmath.org/authors/?q=ai:kwela.adam"Wołoszyn, Wojciech Aleksander"https://www.zbmath.org/authors/?q=ai:woloszyn.wojciech-aleksanderThe authors give the following description of the present investigations:
``We prove that the set of somewhere differentiable functions (i.e., functions which are differentiable at some point) is not Haar-countable.
We consider functions differentiable on a set of positive Lebesgue's measure and functions differentiable almost everywhere with respect to Lebesgue's measure. Furthermore, we study multidimensional case, i.e., differentiability of continuous functions defined on \([0, 1]^k\). Finally, we pose an open question concerning Takagi's function.''
A brief survey is devoted to Takagi's function, to the notion of Haar-null sets, and to somewhere differentiable functions. The special attention is given to the notions of Haar-countable, Haar-finite, and Haar-\(n\) sets. Also, the notion of Haar-meager sets is briefly considered.
One can note that the special attention also is given to sets of nowhere differentiable functions on \([0, 1]^k\) (i.e., nowhere differentiable functions on \([0, 1]^k\) are ``functions defined on \([0, 1]^k\) which do not have a finite directional derivative at any point along any vector'').
The main results are proven with explanations, the main notions are given with brief surveys. Some open questions related with topics of this research, are noted.
Reviewer: Symon Serbenyuk (Kyïv)Approximation of maps into spheres by piecewise-regular maps of class \(C^k\).https://www.zbmath.org/1455.141082021-03-30T15:24:00+00:00"Bilski, Marcin"https://www.zbmath.org/authors/?q=ai:bilski.marcinSummary: The aim of this paper is to prove that every continuous map from a compact subset of a real algebraic variety into a sphere can be approximated by piecewise-regular maps of class \(\mathcal{C}^k,\) where \(k\) is an arbitrary nonnegative integer.Inequalities for the derivative of polynomials with restricted zeros.https://www.zbmath.org/1455.260152021-03-30T15:24:00+00:00"Rather, N. A."https://www.zbmath.org/authors/?q=ai:rather.nisar-ahemad|rather.nisar-ahmed|rather.nisar-ahmad"Dar, Ishfaq"https://www.zbmath.org/authors/?q=ai:dar.ishfaq"Iqbal, A."https://www.zbmath.org/authors/?q=ai:iqbal.atif|iqbal.adam-s|iqbal.adnan|iqbal.amer|iqbal.ayesha|iqbal.anam|iqbal.azhar.1|iqbal.ather|iqbal.asif|iqbal.akhlad|iqbal.azhar|iqbal.arshad|iqbal.afshanSummary: For a polynomial \(\mathit{P(z)=\sum_{\nu =0}^na_{\nu}z^{\nu}}\) of degree \(\mathit{n}\) having all its zeros in \(\mathit{|z|\leq k,k \geq 1} \), it was shown by \textit{N. A. Rather} and \textit{I. Dar} [``Some applications of the boundary Schwarz lemma for polynomials with restricted zeros'', Appl. Math. E-Notes 20, 422--431 (2020)] that
\[
\max_{|z|=1} |P^{\prime}(z)|\geq \frac{1}{1+k^n}\bigg(n+\frac{k^n|a_n|-|a_0|}{k^n|a_n|+|a_0|}\bigg)\max_{|z|=1}|P(z)|.
\]
In this paper, we shall obtain some sharp estimates, which not only refine the above inequality but also generalize some well known Turán-type inequalities.Finite quasi-barycentric means in an abstract setting.https://www.zbmath.org/1455.260212021-03-30T15:24:00+00:00"Girotto, Bruno"https://www.zbmath.org/authors/?q=ai:girotto.bruno"Holzer, Silvano"https://www.zbmath.org/authors/?q=ai:holzer.silvanoBy taking inspiration from Archimedes' theory of centre of gravity, the authors introduce for non-null masses with finite support an arbitrary non empty set \(A\), the notion of quasi-barycentric mean, which is a map from the set of all such masses into \(A\) which satisfies the properties of consistency, shift sensitivity and associativity. In the case of vector spaces, explicit computation by the barycentre of finite distribution of masses is provided.
Reviewer: József Sándor (Cluj-Napoca)Adjoint fuzzy partition and generalized sampling theorem.https://www.zbmath.org/1455.940872021-03-30T15:24:00+00:00"Perfilieva, Irina"https://www.zbmath.org/authors/?q=ai:perfilieva.irina-g"Holčapek, Michal"https://www.zbmath.org/authors/?q=ai:holcapek.michal"Kreinovich, Vladik"https://www.zbmath.org/authors/?q=ai:kreinovich.vladik-yaSummary: A new notion of adjoint fuzzy partition is introduced and the reconstruction of a function from its F-transform components is analyzed. An analogy with the Nyquist-Shannon-Kotelnikov sampling theorem is discussed.
For the entire collection see [Zbl 1385.68004].Vector Lyapunov-like functions for multi-order fractional systems with multiple time-varying delays.https://www.zbmath.org/1455.931742021-03-30T15:24:00+00:00"Gallegos, Javier A."https://www.zbmath.org/authors/?q=ai:gallegos.javier-a"Aguila-Camacho, Norelys"https://www.zbmath.org/authors/?q=ai:aguila-camacho.norelys"Duarte-Mermoud, Manuel"https://www.zbmath.org/authors/?q=ai:duarte-mermoud.manuel-armandoThe authors propose a general method for establishing the asymptotic stability of systems with fractional derivatives, which is based on the use of vector Lyapunov functions. In Section 2, the authors study the stability of multi-order multiple time-varying delays positive fractional differential systems, which are later used in Section 3. In Section 3, they present the main result, which establishes that if a system has a vector Lyapunov-like function, the asymptotic stability can be asserted. In final Section 4, the authors provide illustrative examples of multi-order nonlinear systems having vector Lyapunov-like functions.
Reviewer: Anatoly Martynyuk (Kyïv)First extremal point comparison for a fractional boundary value problem with a fractional boundary condition.https://www.zbmath.org/1455.340062021-03-30T15:24:00+00:00"Henderson, Johnny"https://www.zbmath.org/authors/?q=ai:henderson.johnny"Neugebauer, Jeffrey T."https://www.zbmath.org/authors/?q=ai:neugebauer.jeffrey-tThis work is on a fractional differential equation with Riemann-Liouville fractional derivative. The order of derivative is taken between $n-1$ to $n$, where \(n\) is any natural number greater than or equal to two. The authors compare the first extremal point of the considered differential equations. The mathematical tools used are the concept of Green's function and some inequalities.
Reviewer: Syed Abbas (Mandi)A new design method for observer-based control of nonlinear fractional-order systems with time-variable delay.https://www.zbmath.org/1455.930602021-03-30T15:24:00+00:00"Phat, Vu"https://www.zbmath.org/authors/?q=ai:vu-ngoc-phat."Niamsup, Piyapong"https://www.zbmath.org/authors/?q=ai:niamsup.piyapong"Thuan, Mai V."https://www.zbmath.org/authors/?q=ai:thuan.mai-viet|thuan.mai-vetSummary: In this paper, an LMI-based design is proposed for observer control problem of nonlinear fractional-order systems subject to time-variable delay, where the delay function is non-differentiable, but continuous and bounded. Our novel technique is based on a new lemma concerning Caputo derivative estimation of quadratic functions. In this proposed approach, delay-dependent sufficient conditions in terms of linear matrix inequalities are obtained for the design state feedback controller and observer gains. A simulation-based example is given to illustrate the effectiveness of the theoretical result.The sharp affine \(L_2\) Sobolev trace inequality and affine energy in the fractional Sobolev spaces.https://www.zbmath.org/1455.460372021-03-30T15:24:00+00:00"Nguyen, Van Hoang"https://www.zbmath.org/authors/?q=ai:nguyen-van-hoang.1Summary: We establish an affine invariant version of the sharp \(L_2\) Sobolev trace inequality which contains a recent result of \textit{P.~L.~De Nápoli} et al. [Math. Ann. 370, No.~1--2, 287--308 (2018; Zbl 1390.46036)] as special case. We also give a notion of the so-called affine fractional energy for functions in the fractional homogeneous Sobolev space \(\dot{H}^s( \mathbb{R}^n)\) for \(s\in(0,1)\) and \(n > 2s\). We investigate some properties of this affine fractional energy such as a representation formula and the affine fractional Pólya-Szegő principle. Finally, we obtain a sharp affine fractional \(L_2\) Sobolev inequality in the fractional homogeneous Sobolev space \(\dot{H}^s(\mathbb{R}^n)\) for \(s\in(0,1)\) and \(n > 2s\).An integro quadratic spline-based scheme for solving nonlinear fractional stochastic differential equations with constant time delay.https://www.zbmath.org/1455.650172021-03-30T15:24:00+00:00"Moghaddam, B. P."https://www.zbmath.org/authors/?q=ai:moghaddam.behrouz-parsa"Mostaghim, Z. S."https://www.zbmath.org/authors/?q=ai:mostaghim.zeinab-salamat"Pantelous, Athanasios A."https://www.zbmath.org/authors/?q=ai:pantelous.athanasios-a"Tenreiro Machado, J. A."https://www.zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: This paper proposes an accurate and computationally efficient technique for the approximate solution of a rich class of fractional stochastic differential equations with constant delay driven by Brownian motion. In this regard, a piecewise integro quadratic spline interpolation approach is adopted for approximating the fractional-order integral. The performance of the computational scheme is evaluated by statistical indicators of the exact solutions. Moreover, the computational convergence is also analysed. Three families of models with stochastic excitations illustrate the accuracy of the new approach as compared with the M-scheme.On a preorder relation for Schur-convex functions and a majorization inequality for their gradients and divergences.https://www.zbmath.org/1455.260102021-03-30T15:24:00+00:00"Niezgoda, Marek"https://www.zbmath.org/authors/?q=ai:niezgoda.marekAfter proposing a preordering for Schur-convex functions on \(\mathbb{R}^n\), a majorization statement involving gradients and divergences of two Gâteaux differentiable Schur-convex functions whose difference is Schur-convex as well is provided. Various implications of this result are provided, in particular for \(c\)-strongly convex functions (for some positive real \(c\)).
Reviewer: Sorin-Mihai Grad (Wien)Equivalent definitions of Caputo derivatives and applications to subdiffusion equations.https://www.zbmath.org/1455.352882021-03-30T15:24:00+00:00"Krasnoschok, Mykola"https://www.zbmath.org/authors/?q=ai:krasnoschok.mykola-valeriiovych"Pata, Vittorino"https://www.zbmath.org/authors/?q=ai:pata.vittorino"Siryk, Sergii V."https://www.zbmath.org/authors/?q=ai:siryk.sergii-v"Vasylyeva, Nataliya"https://www.zbmath.org/authors/?q=ai:vasylyeva.nataliya-vSummary: An equivalent definition of the fractional Caputo derivative \(D^\nu_t g\), for \(\nu \in (0, 1)\), is found, within suitable assumptions on \(g\). Some applications to the fractional calculus and to the theory of fractional partial differential equations are then discussed. In particular, this alternative definition is used to prove the maximum principle for the classical solutions to the linear subdiffusion equation subject to nonhomogeneous boundary conditions. This approach also allows to construct numerical solutions to the initial-boundary value problem for the subdiffusion equation with memory.Fractional \(q\)-deformed chaotic maps: a weight function approach.https://www.zbmath.org/1455.390022021-03-30T15:24:00+00:00"Wu, Guo-Cheng"https://www.zbmath.org/authors/?q=ai:wu.guocheng"Niyazi Çankaya, Mehmet"https://www.zbmath.org/authors/?q=ai:cankaya.mehmet-niyazi"Banerjee, Santo"https://www.zbmath.org/authors/?q=ai:banerjee.santoSummary: The fractional derivative holds long-time memory effects or non-locality. It successfully depicts the dynamical systems with long-range interactions. However, it becomes challenging to investigate chaos in the deformed fractional discrete-time systems. This study turns to fractional quantum calculus on the time scale and reports chaos in fractional \(q\)-deformed maps. The discrete memory kernels are used, and a weight function approach is proposed for fractional modeling. Rich \(q\)-deformed dynamics are demonstrated, which shows the methodology's efficiency.
{\copyright 2020 American Institute of Physics}A sharp lower bound for the complete elliptic integrals of the first kind.https://www.zbmath.org/1455.330142021-03-30T15:24:00+00:00"Yang, Zhen-Hang"https://www.zbmath.org/authors/?q=ai:yang.zhenhang"Tian, Jing-Feng"https://www.zbmath.org/authors/?q=ai:tian.jingfeng"Zhu, Ya-Ru"https://www.zbmath.org/authors/?q=ai:zhu.yaruThe complete elliptic integral of the first kind is defined as
\[\mathcal{K}(r)=\int_{0}^{\pi / 2} \frac{1}{\sqrt{1-r^{2} \sin ^{2} t}} d t.\]
It is shown in the paper that
\[\frac{2}{\pi} \mathcal{K}(r)>\left[1-\lambda+\lambda\left(\frac{\operatorname{arth} r}{r}\right)^{q}\right]^{1 / q}\]
holds for \(r\in(0,1)\), and the best constants are
\[\lambda=\frac34,\quad\mbox{and}\quad q=\frac1{10}.\]
Here \(\operatorname{arth} r\) the inverse hyperbolic tangent function.
Reviewer: István Mező (Nanjing)Towards a non-conformable fractional calculus of \(n\)-variables.https://www.zbmath.org/1455.260082021-03-30T15:24:00+00:00"Martínez, Francisco"https://www.zbmath.org/authors/?q=ai:martinez.francisco-j|martinez.francisco-manuel-bernal"Valdés Nápoles, Juan E."https://www.zbmath.org/authors/?q=ai:valdes-napoles.juan-eUsing a rather complicated approach, the authors define a differential operator that they claim to be of fractional order but that, in fact, can easily be seen to be just a multiple of the classical first derivative. Specifically, in the one-dimensional case, their operator is nothing but \(N^\alpha_x f(a) = \exp(a^{-\alpha}) f'(a)\), and the extension to the case of multivariate functions follows the usual strategy for partial derivatives. Some of the properties of this operator are derived, but in view of its simple relation to the classical differential operator, all of this is very straightforward.
Reviewer: Kai Diethelm (Schweinfurt)Existence of compositional square-roots of functions.https://www.zbmath.org/1455.260062021-03-30T15:24:00+00:00"Kannan, V."https://www.zbmath.org/authors/?q=ai:kannan.vThe authors obtains 9 classes of functions $f$ that admit square roots $g$, i.e., $g\circ g=f$; some new, some already known.
For the entire collection see [Zbl 1446.65004].
Reviewer: George Stoica (Saint John)Asymptotic stability of fractional neutral stochastic systems with variable delays.https://www.zbmath.org/1455.931582021-03-30T15:24:00+00:00"Lu, Ziqiang"https://www.zbmath.org/authors/?q=ai:lu.ziqiang"Zhu, Yuanguo"https://www.zbmath.org/authors/?q=ai:zhu.yuanguo"Xu, Qinqin"https://www.zbmath.org/authors/?q=ai:xu.qinqinSummary: This paper mainly concerns with the stability of nonlinear Caputo fractional neutral stochastic differential system with variable delay. A sufficient condition is derived to ensure that the trivial solution is mean-square asymptotically stable by employing the Banach's contraction principle, in which the boundedness of delays is not required. An illustrative example is provided to show the efficiency of our results.Asymptotic expansion and bounds for complete elliptic integrals.https://www.zbmath.org/1455.330132021-03-30T15:24:00+00:00"Wang, Miao-Kun"https://www.zbmath.org/authors/?q=ai:wang.miaokun"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yuming"Li, Yong-Min"https://www.zbmath.org/authors/?q=ai:li.yongmin"Zhang, Wen"https://www.zbmath.org/authors/?q=ai:zhang.wen.3This paper is concerned with an asymptotic expansion for the complete elliptic integrals of the first kind as \(k\to 1\).
Furthermore, sharp symmetrical bounds for the arithmetic-geometric
mean and a new symmetric mean \(E(a, b)\) are found.
Observe that unusual notations
for the complete elliptic integrals of the first and second kind are used, and
the modulus \(k\) is denoted by \(r\).
The proofs use L'hospital rule and several tedious computations involving the
area tangens hyperbolicus function, which sometimes are not shown.
Reviewer: Thomas Ernst (Uppsala)Finite-time inter-layer projective synchronization of Caputo fractional-order two-layer networks by sliding mode control.https://www.zbmath.org/1455.931792021-03-30T15:24:00+00:00"Wu, Xifen"https://www.zbmath.org/authors/?q=ai:wu.xifen"Bao, Haibo"https://www.zbmath.org/authors/?q=ai:bao.haibo"Cao, Jinde"https://www.zbmath.org/authors/?q=ai:cao.jindeSummary: Finite-time inter-layer projective synchronization (FIPS) of Caputo fractional-order two-layer networks (FTN) based on sliding mode control (SMC) technique is investigated in this article. Firstly, in order to realize the FIPS of FTN, a fractional-order integral sliding mode surface (SMS) is established. Then, through the theory of SMC, we design a sliding mode controller (SMCr) to ensure the appearance of sliding mode motion. According to the fractional Lyapunov direct method, the trajectories of the system are driven to the proposed SMS, and some novel sufficient conditions for FIPS of FTN are derived. Furthermore, as two special cases of FIPS, finite-time inter-layer synchronization and finite-time inter-layer anti-synchronization for the FTN are studied. Finally, this paper takes the fractional-order chaotic Lü's system and the fractional-order chaotic Chen's system as the isolated node of the first layer system and the second layer system, respectively. And the numerical simulations are given to demonstrate the feasibility and validity of the proposed theoretical results.Sliding mode active disturbance rejection control for uncertain nonlinear fractional-order systems.https://www.zbmath.org/1455.930182021-03-30T15:24:00+00:00"Djeghali, Nadia"https://www.zbmath.org/authors/?q=ai:djeghali.nadia"Bettayeb, Maamar"https://www.zbmath.org/authors/?q=ai:bettayeb.maamar"Djennoune, Said"https://www.zbmath.org/authors/?q=ai:djennoune.saidSummary: It is recognized today that active disturbance rejection control (ADRC) is an effective control strategy in the presence of uncertainties and disturbances and especially in the absence of a model. Its advantages and its power have been demonstrated practically on numerous engineering applications. This control technique has been widely developed in the case of integer-order systems. On the other hand, fractional-order systems are gaining more and more interest due to their use in modeling of many physical phenomena. Some works have been dedicated to the design of the ADRC for linear and nonlinear fractional-order systems. However, the sliding mode technique has not yet been used for the design of ADRC of fractional-order systems. The aim of this paper is to propose a sliding mode active disturbance rejection controller (SMADRC) for nonlinear fractional-order systems with uncertainties and external disturbances for stabilization and tracking purposes. First, a step by step sliding mode extended state observer (SMESO) for both state variables and total disturbance (uncertainties and external disturbances) estimation is proposed. The finite time convergence of the proposed extended state observer is established. Then, a sliding mode controller using the estimated states and total disturbance is presented to realize stabilization and desired references tracking with compensation of the total disturbance. The closed loop stability is analyzed. Simulation results of the proposed SMADRC applied to the control of fractional-order chaotic systems are compared to those of the conventional sliding mode control scheme.Application of the fractional differential transform method to the first kind Abel integral equation.https://www.zbmath.org/1455.450012021-03-30T15:24:00+00:00"Mondal, Subhabrata"https://www.zbmath.org/authors/?q=ai:mondal.subhabrata"Mandal, B. N."https://www.zbmath.org/authors/?q=ai:mandal.baidya-nath|mandal.birendra-nathSummary: The fractional differential transform method is employed here for solving first kind Abel integral equation. Abel integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics and applied sciences. An analytic technique for solving Abel integral equation of first kind by the proposed method is introduced here. Also illustrative examples with exact solutions are considered to show the validity and applicability of the proposed method. Numerical results reveal that the proposed method works well and has good accuracy. The method introduces a promising tool for solving many linear and nonlinear fractional integral equation.
For the entire collection see [Zbl 1411.65006].On generalizations of Hadamard inequalities for fractional integrals.https://www.zbmath.org/1455.260172021-03-30T15:24:00+00:00"Farid, Gh."https://www.zbmath.org/authors/?q=ai:farid.ghulam"Ur, Rehman A. U."https://www.zbmath.org/authors/?q=ai:ur.rehman-a-u"Zahra, M."https://www.zbmath.org/authors/?q=ai:zahra.moquddsaSummary: Fejér Hadamard inequality is generalization of Hadamard inequality. In this paper we prove certain Fejér Hadamard inequalities for \(k\)-fractional integrals. We deduce Fejér Hadamard-type inequalities for Riemann-Liouville fractional integrals. Also as a special case Hadamard inequalities for \(k\)-fractional as well as fractional integrals are given.An illustrative example of complete connected space with a continuous function having an extremum at every point.https://www.zbmath.org/1455.540142021-03-30T15:24:00+00:00"Storozhuk, K. V."https://www.zbmath.org/authors/?q=ai:storozhuk.konstantin-valerevichSummary: We construct a complete metric space ``realized'' in \(\mathbb{R}^3\) at which the height function has a local (nonstrict) extremum at every point. The question of the existence of such a space was posed in [\textit{E. Behrends} et al., Real Anal. Exch. 33, No. 2, 467--470 (2008; Zbl 1170.26002)]. Such examples were constructed (see, for example, [\textit{A. Fedeli} and \textit{A. Le Donne}, Topology Appl. 156, No. 13, 2196--2199 (2009; Zbl 1173.54006) and \textit{T. Banakh} et al., ``On locally extremal functions on connected spaces'', Preprint, \url{arXiv:0811.1771}]) but seem to be less transparent since they are ``realized'' in \(\mathbb{R}^n, n \geq 4\), and are hard to draw.To the question of fractional differentiation. II.https://www.zbmath.org/1455.260032021-03-30T15:24:00+00:00"Gladkov, Sergeĭ Oktyabrinovich"https://www.zbmath.org/authors/?q=ai:gladkov.sergei-oktyabrinovich"Bogdanova, Sof'ya Borisovna"https://www.zbmath.org/authors/?q=ai:bogdanova.sofya-borisovnaSummary: In the paper the investigation continues with the help of definition Fourier fractional differentiation setting in the previous paper [the authors, Vestn. Samar. Univ., Estestvennonauchn. Ser. 24, No. 3, 7--13 (2018; Zbl 07141721)]. There were given explicit expressions of a fairly wide class of periodic functions and for functions represented in the form of wavelet decompositions. It was shown that for the class of exponential functions all derivatives with non-integer exponent are equal to zero. The found derivatives have a direct relationship to practical problems and let them use to solve a large class of problems associated with the study of phenomena such as thermal conduction, transmissions, electrical and magnetic susceptibility for a wide range of materials with fractal dimensions.Ky-Fan and Sion theorems for the lexicographic order and applications to vectorial games and min-max control problems.https://www.zbmath.org/1455.901512021-03-30T15:24:00+00:00"Caroff, Nathalie"https://www.zbmath.org/authors/?q=ai:caroff.nathalie"Serea, Oana Silvia"https://www.zbmath.org/authors/?q=ai:serea.oana-silviaThe aim of this paper is to show some classical results of nonlinear analysis and optimization in a vectorial setting, through the use of the lexicographic order. First, the infimum and the supremum of a set under the lexicographic order is introduced. Based on this, one defines the notions of vector level lower-semicontinuity and vector lower-semicontinuity. Then, a vectorial
version of the Ky Fan inequality is shown, with two corollaries: a vectorial version of the min-max theorem for convex-concave functions, and a result on the existence of a Nash equilibrium.
The first main result is an extension of Sion's theorem to this vectorial setting. Two more applications concern the min-max control problem: one shows an inequality between the upper and lower value functions, and the control-against control value functions; also, one shows the existence of saddle points for this problem.
Reviewer: Nicolas Hadjisavvas (Ermoupoli)Approximating sums of consecutive integral roots.https://www.zbmath.org/1455.110432021-03-30T15:24:00+00:00"Ruankong, Pongpol"https://www.zbmath.org/authors/?q=ai:ruankong.pongpol"Kuhapatanakul, Kantaphon"https://www.zbmath.org/authors/?q=ai:kuhapatanakul.kantaphonSummary: We present an alternative proof of \textit{P. W. Saltzman} and \textit{P. Yuan}'s result on the sums of consecutive integral roots [Am. Math. Mon. 115, No. 3, 254--261 (2008; Zbl 1239.11028)]. We use the AM-GM-HM inequality to prove the main result. Moreover, the lower bound for which the result holds is greatly improved.On max-min representations of ordered median functions.https://www.zbmath.org/1455.260112021-03-30T15:24:00+00:00"Grzybowski, J."https://www.zbmath.org/authors/?q=ai:grzybowski.jerzy"Kalcsics, J."https://www.zbmath.org/authors/?q=ai:kalcsics.jorg"Nickel, S."https://www.zbmath.org/authors/?q=ai:nickel.stefan"Pallaschke, D."https://www.zbmath.org/authors/?q=ai:pallaschke.diethard"Urbański, R."https://www.zbmath.org/authors/?q=ai:urbanski.ryszardSummary: An ordered median function is a continuous piecewise linear function. It is well known, that in finite dimensional spaces every continuous piecewise linear function admits a max-min representation in terms of its linear functions. We give an explicit representation of an ordered median function in max-min form using a purely combinatorial approach.A characterization of polynomials whose high powers have non-negative coefficients.https://www.zbmath.org/1455.260122021-03-30T15:24:00+00:00"Michelen, Marcus"https://www.zbmath.org/authors/?q=ai:michelen.marcus"Sahasrabudhe, Julian"https://www.zbmath.org/authors/?q=ai:sahasrabudhe.julianSummary: Let \(f\in \mathbb{R}[z]\) be a polynomial with real coefficients. We say that \(f\) is \textit{eventually non-negative} if \(f^m\) has non-negative coefficients for all sufficiently large \(m\in \mathbb{N}\). In this short paper, we give a classification of all eventually non-negative polynomials. This generalizes a theorem of De Angelis, and proves a conjecture of Bergweiler, Eremenko and Sokal.A new approach for solving nonlinear Volterra integro-differential equations with Mittag-Leffler kernel.https://www.zbmath.org/1455.651062021-03-30T15:24:00+00:00"Ganji, Roghayeh Moallem"https://www.zbmath.org/authors/?q=ai:ganji.roghayeh-moallem"Jafari, Hossein"https://www.zbmath.org/authors/?q=ai:jafari.hosseinSummary: In this work, we consider a general class of nonlinear Volterra integro-differential equations with Atangana-Baleanu derivative. We use the operational matrices based on the shifted Legendre polynomials to obtain numerical solution of the considered equations. By approximating the unknown function and its derivative in terms of the shifted Legendre polynomials and substituting these approximations into the original equation and using the collocation points, the original equation is reduced to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, some examples are included to show the accuracy and validity of the proposed method.The distributional hyper-Jacobian determinants in fractional Sobolev spaces.https://www.zbmath.org/1455.460472021-03-30T15:24:00+00:00"Wu, Chuanxi"https://www.zbmath.org/authors/?q=ai:wu.chuanxi"Tu, Qiang"https://www.zbmath.org/authors/?q=ai:tu.qiang"Qiu, Xueting"https://www.zbmath.org/authors/?q=ai:qiu.xuetingSummary: In this paper we give a positive answer to the question about hyper-Jacobian determinants and associated minors raised by \textit{E. Baer} and \textit{D. Jerison} [J. Funct. Anal. 269, No. 5, 1482--1514 (2015; Zbl 1335.46027)]. Inspired by recent works of \textit{H. Brezis} and \textit{H.-M. Nguyen} [Invent. Math. 185, No. 1, 17--54 (2011; Zbl 1230.46029)] and Baer-Jerison [loc.\,cit.]\ on Jacobian and Hessian determinants, we establish the weak continuity and fundamental representation for the distributional \(m\)th-Jacobian minors of degree \(r\) in the fractional Sobolev space \(W^{m - \frac{m}{r}, r}(\Omega, \mathbb{R}^N)\). Moreover, the result is optimal in the framework of fractional Sobolev spaces, i.e., all \(m\)th-Jacobian minors of degree \(r\) are well defined in \(W^{s, p}(\Omega, \mathbb{R}^N)\) if and only if \(W^{s , p}(\Omega, \mathbb{R}^N) \subseteq W^{m - \frac{m}{r} , m}(\Omega, \mathbb{R}^N)\).Continuity with respect to fractional order of the time fractional diffusion-wave equation.https://www.zbmath.org/1455.352952021-03-30T15:24:00+00:00"Tuan, Nguyen Huy"https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan."O'Regan, Donal"https://www.zbmath.org/authors/?q=ai:oregan.donal"Ngoc, Tran Bao"https://www.zbmath.org/authors/?q=ai:ngoc.tran-baoSummary: This paper studies a time-fractional diffusion-wave equation with a linear source function. First, some stability results on parameters of the Mittag-Leffler functions are established. Then, we focus on studying the continuity of the solution of both the initial problem and the inverse initial value problems corresponding to the fractional-order in our main results. One of the difficulties encounteblack comes from estimating all constants independently of the fractional orders. Finally, we present some numerical results to confirm the effectiveness of our methods.On approximate controllability of impulsive fractional semilinear systems with deviated argument in Hilbert spaces.https://www.zbmath.org/1455.930112021-03-30T15:24:00+00:00"Aimene, D."https://www.zbmath.org/authors/?q=ai:aimene.d"Laoubi, K."https://www.zbmath.org/authors/?q=ai:laoubi.karima"Seba, D."https://www.zbmath.org/authors/?q=ai:seba.djamilaSummary: In this paper we apply a fixed-point theorem to study the existence and uniqueness of a mild solution and the approximate controllability of a fractional order impulsive differential equation with deviated argument in Hilbert spaces. An example is provided to show the effectiveness of the theory.Sobolev, Hardy, Gagliardo-Nirenberg, and Caffarelli-Kohn-Nirenberg-type inequalities for some fractional derivatives.https://www.zbmath.org/1455.260132021-03-30T15:24:00+00:00"Kassymov, Aidyn"https://www.zbmath.org/authors/?q=ai:kassymov.aidyn-adilovich"Ruzhansky, Michael"https://www.zbmath.org/authors/?q=ai:ruzhansky.michael-v"Tokmagambetov, Niyaz"https://www.zbmath.org/authors/?q=ai:tokmagambetov.niyaz-esenzholovich"Torebek, Berikbol T."https://www.zbmath.org/authors/?q=ai:torebek.berikbol-tillabayulyThe authors prove various inequalities for fractional-order diferential operators, e.g. Sobolev, Hardy, Gagliardo-Nirenberg, and Cafarelli-Kohn-Nirenberg-type, for the Caputo, Riemann-Liouville, and Hadamard derivatives. Applications are given to the uncertainty principle, embeddings of spaces and apriori estimates for diffusion problems.
Reviewer: George Stoica (Saint John)On generalization of mean value.https://www.zbmath.org/1455.400022021-03-30T15:24:00+00:00"Pocherevin, Roman Vladimirovich"https://www.zbmath.org/authors/?q=ai:pocherevin.roman-vladimirovichSummary: In paper we discuss the solution of mean value general form problem in case of all variables symmetry absence. In 1930 A. N. Kolmogorov proved the formula for general form of mean value. He formulated four axioms: continuity and monotony on each variable, symmetry on each variable, mean value of equal variables is equal to these variables, any substitution of any group of variables with their mean value does not change the mean value. In Kolmogorov's theorem all arguments are equitable, this means that the mean value is symmetric on each variable. V. N. Chubarikov set the task of generalization to this result in case of all variables symmetry absence. We divide all the variables on groups and the mean value is a symmetric function for variables in each group separately. For example, if we have only one group the mean value will be Kolmogorov's mean value, so we have a generalization of Kolmogorov's theorem. In paper we show the general form of mean value in our case and we note the connection with uniform distribution modulo 1.Real analysis: foundations.https://www.zbmath.org/1455.260012021-03-30T15:24:00+00:00"Ovchinnikov, Sergei"https://www.zbmath.org/authors/?q=ai:ovchinnikov.sergei-g|ovchinnikov.sergei-vPublisher's description: This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis.
Comprising six chapters, the book opens with a rigorous presentation of the theories of rational and real numbers in the framework of ordered fields. This is followed by an accessible exploration of standard topics of elementary real analysis, including continuous functions, differentiation, integration, and infinite series. Readers will find this text conveniently self-contained, with three appendices included after the main text, covering an overview of natural numbers and integers, Dedekind's construction of real numbers, historical notes, and selected topics in algebra.\textit{Real Analysis: Foundations} is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one-semester course on elementary real analysis, as well as independent study.Complex fractional moments for the characterization of the probabilistic response of non-linear systems subjected to white noises.https://www.zbmath.org/1455.352582021-03-30T15:24:00+00:00"Di Paola, Mario"https://www.zbmath.org/authors/?q=ai:di-paola.mario"Pirrotta, Antonina"https://www.zbmath.org/authors/?q=ai:pirrotta.antonina"Alotta, Gioacchino"https://www.zbmath.org/authors/?q=ai:alotta.gioacchino"Di Matteo, Alberto"https://www.zbmath.org/authors/?q=ai:di-matteo.alberto"Pinnola, Francesco Paolo"https://www.zbmath.org/authors/?q=ai:pinnola.francesco-paoloThe core of the chapter consists in a demonstration of an employment of Complex Fractional Moments (CFM) in an analysis of the PDF of a system response under an excitation of a white noise of several types. Although the chapter has a character of an overview, the rich evaluation of literature resources indicates that the set of papers published by authors of this chapter makes more or less a closed package of information about a new non-conventional approach. Nevertheless, the chapter itself is worthy to be recommended to those, who are interested in FPE analysis of various types. The chapter is transparently written and provides valuable information about the problem and adequate literature.
In the first two sections, a general form of the FPE solution using decomposition with respect to stochastic moments of the non-integer order type is explained clarifying a relation with the Mellin integral transform in continuous or discrete versions. Important details concerning mathematical limitations and pitfalls are discussed before individual applications are presented. Three examples of FPE originating from a nonlinear diffusion dynamic system (Langevin type equation) with different type of a white noise excitation are demonstrated: (i) conventional FPE with additive Gaussian white noise; (ii) FPE with \(\alpha\)-stable white noise excitation; (iii) Generalized FPE (Kolmogorov-Feller) with Poissonian white noise. In the case (i) advantages of the CFM solution procedure are demonstrated in comparison with integer order moments. Problems of non-guaranteed convergence of integer order moments are avoided, hierarchy problems are solved as well, compromises with various types of closure disappeared, etc. Furthermore, convergence seems to be faster and more convenient type. Whereas in the (i) the CFM application offer more convenient and elegant solution, the cas (ii), Lévy \(\alpha\)-stable white noise excitation, explicitly requires to be analyzed by means of the CFM. It follows immediately from the FPE itself, where fractional derivative of the α-order is present. In such a case an application of the CFM is doubtlessly inherent. Similarly the case (iii) is much more related with fractional order moments than those with integer order. Although a procedure with classical moments is also possible, numerical evaluation is very clumsy and convergence problematic. Analytical results are verified by means of stochastic simulation. Comparison of both is excellent. In general, the chapter should be considered as an excellent pioneering work, which gives not only an effective alternative, but also new possibilities in searching of a weak solution of the FPE. The chapter doubtlessly attracts many readers working in area of random vibration of nonlinear dynamic systems as well as people involved in basic theory of random processes.
For the entire collection see [Zbl 1430.74006].
Reviewer: Jiri Náprstek (Praha)The sharp Sobolev type inequalities in the Lorentz-Sobolev spaces in the hyperbolic spaces.https://www.zbmath.org/1455.460412021-03-30T15:24:00+00:00"Nguyen, Van Hoang"https://www.zbmath.org/authors/?q=ai:nguyen-van-hoang.1Summary: Let \(W^1 L^{p, q}(\mathbb{H}^n)\), \(1 \leq q, p < \infty\), denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces \(\mathbb{H}^n\). Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in \(W^1 L^{p, q}( \mathbb{H}^n)\) with \(1 \leq q \leq p\) which generalizes the result in
[\textit{Q.~A.~Ngô} and \textit{V.~A.~Nguyen}, Acta Math. Vietnam. 44, No.~3, 781--795 (2019; Zbl 1426.26038)]
to the setting of Lorentz-Sobolev spaces. Second, we prove several sharp Poincaré-Sobolev type inequalities in \(W^1 L^{p, q}(\mathbb{H}^n)\) with \(1 \leq q \leq p < n\) which generalize the results in
[\textit{V.~H.~Nguyen}, J. Math. Anal. Appl. 462, No.~2, 1570--1584 (2018; Zbl 1396.46031)]
to the setting of Lorentz-Sobolev spaces. Finally, we provide the improved Moser-Trudinger type inequalities in \(W^1 L^{n, q}( \mathbb{H}^n)\) in the critical case \(p = n\) with \(1 \leq q \leq n\) which generalize the results in
[\textit{V.~H.~Nguyen}, Nonlinear Anal., Theory Methods Appl., Ser.~A, Theory Methods 168, 67--80 (2018; Zbl 1381.26021)]
and improve the results in
[\textit{Q.-H. Yang} and \textit{Y.~Li}, J. Math. Anal. Appl. 472, No.~1, 1236--1252 (2019; Zbl 1423.46054)].
In the proof of the main results, we shall prove a Pólya-Szegő type principle in \(W^1 L^{p, q}(\mathbb{H}^n)\) with \(1 \leq q \leq p\) which maybe is of independent interest.A new approach for solving linear fractional integro-differential equations and multi variable order fractional differential equations.https://www.zbmath.org/1455.651042021-03-30T15:24:00+00:00"Ghomanjani, F."https://www.zbmath.org/authors/?q=ai:ghomanjani.fatemeSummary: In the sequel, the numerical solution of linear fractional integrodifferential equations (LFIDEs) and multi variable order fractional differential equations (MVOFDEs) are found by Bezier curve method (BCM) and operational matrix. Some numerical examples are stated and utilized to evaluate the good and accurate results.\(q\)-fractional Dirac type systems.https://www.zbmath.org/1455.390012021-03-30T15:24:00+00:00"Allahverdiev, Bilender P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, Hüseyin"https://www.zbmath.org/authors/?q=ai:tuna.huseyinSummary: This paper is devoted to study a regular \(q\)-fractional Dirac type system. We investigate the properties of the eigenvalues and the eigenfunctions of this system. By using a fixed point theorem we give a sufficient condition on eigenvalues for the existence and uniqueness of the associated eigenfunctions.Bounds on moments of weighted sums of finite Riesz products.https://www.zbmath.org/1455.420032021-03-30T15:24:00+00:00"Bonami, Aline"https://www.zbmath.org/authors/?q=ai:bonami.aline"Latała, Rafał"https://www.zbmath.org/authors/?q=ai:latala.rafal"Nayar, Piotr"https://www.zbmath.org/authors/?q=ai:nayar.piotr"Tkocz, Tomasz"https://www.zbmath.org/authors/?q=ai:tkocz.tomaszSummary: Let \(n_j\) be a lacunary sequence of integers, such that \(n_{j+1}/n_j\geq r\). We are interested in linear combinations of the sequence of finite Riesz products \(\prod_{j=1}^N(1+\cos (n_j t))\). We prove that, whenever the Riesz products are normalized in \(L^p\) norm \((p\geq 1)\) and when \(r\) is large enough, the \(L^p\) norm of such a linear combination is equivalent to the \(\ell^p\) norm of the sequence of coefficients. In other words, one can describe many ways of embedding \(\ell^p\) into \(L^p\) based on Fourier coefficients. This generalizes to vector valued \(L^p\) spaces.Some weighted trapezoidal type inequalities via \(h\)-preinvexity.https://www.zbmath.org/1455.260142021-03-30T15:24:00+00:00"Meftah, B."https://www.zbmath.org/authors/?q=ai:meftah.boudjelal|meftah.badreddine"Mekalfa, K."https://www.zbmath.org/authors/?q=ai:mekalfa.kSummary: In this paper, a new identity is given, some weighted trapezoidal type inequalities via \(h\)-preinvexity are established, and several known results are derived.Combinatorial extensions of Popoviciu's inequality via Abel-Gontscharoff polynomial with applications in information theory.https://www.zbmath.org/1455.260072021-03-30T15:24:00+00:00"Butt, Saad Ihsan"https://www.zbmath.org/authors/?q=ai:butt.saad-ihsan"Rasheed, Tahir"https://www.zbmath.org/authors/?q=ai:rasheed.tahir"Pečarić, Đilda"https://www.zbmath.org/authors/?q=ai:pecaric.dilda"Pečarić, Josip"https://www.zbmath.org/authors/?q=ai:pecaric.josip-eSummary: We establish new refinements and improvements of Popoviciu's inequality for \(n\)-convex functions using Abel-Gontscharoff interpolating polynomial along with the aid of new Green functions. We construct new inequalities for \(n\)-convex functions and compute new upper bounds for Ostrowski and Grüss type inequalities. As an application of our work in information theory, we give new estimations for Shannon, Relative and Zipf-Mandelbrot entropies using generalized Popoviciu's inequality.Hermite-Hadamard inequality for semiconvex functions of rate \((k_1,k_2)\) on the coordinates and optimal mass transportation.https://www.zbmath.org/1455.260162021-03-30T15:24:00+00:00"Chen, Ping"https://www.zbmath.org/authors/?q=ai:chen.ping"Cheung, Wing-Sum"https://www.zbmath.org/authors/?q=ai:cheung.wingsum|cheung.wing-sumSummary: We give a new Hermite-Hadamard inequality for a function \(f:[a,b] \times [c,d] \subset \mathbb{R}^2 \to \mathbb{R}\) which is semiconvex of rate \((k_1, k_2)\) on the coordinates. This generalizes some existing results on Hermite-Hadamard inequalities of S. S. Dragomir. In addition, we explain the Hermite-Hadamard inequality from the point of view of optimal mass transportation with cost function \(c (x,y):=f(y-x) + \frac{k_1}{2} | x_1-y_1 |^2 + \frac{k_2}{2}|x_2-y_2|^2\), where \(f(\cdot):[a,b] \times [c,d] \to [0, \infty)\) is semiconvex of rate \((k_1,k_2)\) on the coordinates and \(x=(x_1,x_2),y=(y_1,y_2) \in [a,b] \times [c,d]\).Poisson image denoising based on fractional-order total variation.https://www.zbmath.org/1455.940132021-03-30T15:24:00+00:00"Chowdhury, Mujibur Rahman"https://www.zbmath.org/authors/?q=ai:chowdhury.mujibur-rahman"Zhang, Jun"https://www.zbmath.org/authors/?q=ai:zhang.jun.6|zhang.jun.9|zhang.jun.10|zhang.jun.3|zhang.jun.2|zhang.jun|zhang.jun.1|zhang.jun.5|zhang.jun.7"Qin, Jing"https://www.zbmath.org/authors/?q=ai:qin.jing"Lou, Yifei"https://www.zbmath.org/authors/?q=ai:lou.yifeiSummary: Poisson noise is an important type of electronic noise that is present in a variety of photon-limited imaging systems. Different from the Gaussian noise, Poisson noise depends on the image intensity, which makes image restoration very challenging. Moreover, complex geometry of images desires a regularization that is capable of preserving piecewise smoothness. In this paper, we propose a Poisson denoising model based on the fractional-order total variation (FOTV). The existence and uniqueness of a solution to the model are established. To solve the problem efficiently, we propose three numerical algorithms based on the Chambolle-Pock primal-dual method, a forward-backward splitting scheme, and the alternating direction method of multipliers (ADMM), each with guaranteed convergence. Various experimental results are provided to demonstrate the effectiveness and efficiency of our proposed methods over the state-of-the-art in Poisson denoising.Solving exterior boundary value problems for the Laplace equation.https://www.zbmath.org/1455.652252021-03-30T15:24:00+00:00"Galanin, M. P."https://www.zbmath.org/authors/?q=ai:galanin.m-p"Sorokin, D. L."https://www.zbmath.org/authors/?q=ai:sorokin.d-lThe present paper develops an algorithm for solving exterior boundary value problems for the Laplace equation in two- and three-dimensional cases. An iterative algorithm is constructed for solving integro-differential equations. The paper is outlined as follows. The beginning is an Introduction. In Section 1, solving exterior boundary value problems for the Laplace equation in the case of a known Green function is given. The iterative solution algorithm is discussed in Section 2. In greater detail, the exterior problem for the two-dimensional domain is considered in the case where it is impossible to write Green's function in closed form. In Section 3, the exterior boundary value problem using the iterative process for a complex-shaped three-dimensional domain is solved. Examples of solving evolution problems are given in Section 4. Finally, some conclusions are fixed at the end of the paper.
Reviewer: Temur A. Jangveladze (Tbilisi)Positive solution of nonlinear fractional differential equations with Caputo-like counterpart hyper-Bessel operators.https://www.zbmath.org/1455.340122021-03-30T15:24:00+00:00"Zhang, Kangqun"https://www.zbmath.org/authors/?q=ai:zhang.kangqunIn this paper, the author obtain some Grownwall-type integral inequalities involving Mittag-Leffler functions. Using these inequalities and fixed point theorems the author establish the existence and uniqueness of positive solution of initial value problem to nonlinear fractional differential equation with Caputo-like counterpart hyper-Bessel operators.
Reviewer: Krishnan Balachandran (Coimbatore)Refinements of Young's integral inequality via fundamental inequalities and mean value theorems for derivatives.https://www.zbmath.org/1455.260202021-03-30T15:24:00+00:00"Qi, Feng"https://www.zbmath.org/authors/?q=ai:qi.feng"Li, Wen-Hui"https://www.zbmath.org/authors/?q=ai:li.wenhui"Wu, Guo-Sheng"https://www.zbmath.org/authors/?q=ai:wu.guo-sheng"Guo, Bai-Ni"https://www.zbmath.org/authors/?q=ai:guo.bai-niFrom the book's preface: This chapter reviews several refinements of Young's integral inequality via several mean value theorems, such as Lagrange's and Taylor's mean value theorems of Lagrange's and Cauchy's type remainders, and via several fundamental inequalities, such as Čebyšev's integral inequality, Hermite-Hadamard's type integral inequalities, Hölder's integral inequality, and Jensen's discrete and integral inequalities, in terms of higher-order derivatives and their norms. It also surveys several applications of several refinements of Young's integral inequality and further refines Young's integral inequality via Pólya's type integral inequalities.
For the entire collection see [Zbl 1453.00001].Characterizations of rough fractional-type integral operators on variable exponent vanishing Morrey-type spaces.https://www.zbmath.org/1455.420092021-03-30T15:24:00+00:00"Gürbüz, Ferit"https://www.zbmath.org/authors/?q=ai:gurbuz.ferit"Ding, Shenghu"https://www.zbmath.org/authors/?q=ai:ding.shenghu"Huili, Han"https://www.zbmath.org/authors/?q=ai:huili.han"Long, Pinhong"https://www.zbmath.org/authors/?q=ai:long.pinhongSummary: This chapter mainly focuses on some operators and commutators on the variable exponent-generalized Morrey-type space. It aims to characterize the boundedness for the maximal operator, fractional integral operator, and fractional maximal operator with rough kernel as well as the corresponding commutators on the variable exponent vanishing generalized Morrey spaces. Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces cannot handle. But classical Morrey spaces are not totally enough to describe the boundedness properties. The fractional-type operators and their weighted boundedness theory play important roles in harmonic analysis and other fields, and the multilinear operators arise in numerous situations involving product-like operations. The chapter shows that various classical operators -- such as rough maximal, potential, and singular integral operators -- and their commutators are bounded in different proper closed subspaces of variable exponent-generalized Morrey spaces.
For the entire collection see [Zbl 1453.00001].On some important inequalities.https://www.zbmath.org/1455.260192021-03-30T15:24:00+00:00"Pavić, Zlatko"https://www.zbmath.org/authors/?q=ai:pavic.zlatkoFrom the book's preface: This chapter aims to present some useful mathematical inequalities such as the Jensen, Hermite-Hadamard, Rogers-Hölder, and Minkowski inequalities. The aforementioned four inequalities have been presented in discrete and integral forms, and their generalizations have also been discussed. The Jensen and Hermite-Hadamard inequalities have been considered in more detail. An expansion of the initial integral form of Jensen's inequality is promoted for convex functions of several variables. The Rogers-Hölder and Minkowski inequalities have been derived from the integral form of Jensen's inequality for convex functions of several variables. The Minkowski inequality is realized independently of the Rogers-Hölder inequality.
For the entire collection see [Zbl 1453.00001].Class of integrals involving generalized hypergeometric function.https://www.zbmath.org/1455.330052021-03-30T15:24:00+00:00"Suthar, D. L."https://www.zbmath.org/authors/?q=ai:suthar.daya-l"Hailay, Teklay"https://www.zbmath.org/authors/?q=ai:hailay.teklay"Amsalu, Hafte"https://www.zbmath.org/authors/?q=ai:amsalu.hafte"Singh, Jagdev"https://www.zbmath.org/authors/?q=ai:singh.jagdevIn this paper, the authors evaluate some definite integrals involving generalized hypergeometric functions, a product of algebraic functions, Jacobi functions, Legendre functions and a general class of polynomials.
Reviewer: Osman Yürekli (Ithaca)Approximation by piecewise-regular maps.https://www.zbmath.org/1455.141092021-03-30T15:24:00+00:00"Bilski, Marcin"https://www.zbmath.org/authors/?q=ai:bilski.marcin"Kucharz, Wojciech"https://www.zbmath.org/authors/?q=ai:kucharz.wojciechSummary: A real algebraic variety \(W\) of dimension \(m\) is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of \(\mathbb{R}^m\). Let \(l\) be any nonnegative integer. We prove that every map of class \(\mathcal{C}^l\) from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the \(\mathcal{C}^l\) topology by piecewise-regular maps of class \(\mathcal{C}^k\), where \(k\) is an arbitrary integer satisfying \(k \geq l\). Next we derive consequences regarding algebraization of topological vector bundles.Uncertain viscoelastic models with fractional order: a new spectral tau method to study the numerical simulations of the solution.https://www.zbmath.org/1455.760142021-03-30T15:24:00+00:00"Ahmadian, A."https://www.zbmath.org/authors/?q=ai:ahmadian.ali"Ismail, F."https://www.zbmath.org/authors/?q=ai:ismail.fudziah-bt|ismail.fudiah"Salahshour, S."https://www.zbmath.org/authors/?q=ai:salahshour.soheil"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-i"Ghaemi, F."https://www.zbmath.org/authors/?q=ai:ghaemi.ferialSummary: The analysis of the behaviors of physical phenomena is important to discover significant features of the character and the structure of mathematical models. Frequently the unknown parameters involve in the models are assumed to be unvarying over time. In reality, some of them are uncertain and implicitly depend on several factors. In this study, to consider such uncertainty in variables of the models, they are characterized based on the fuzzy notion. We propose here a new model based on fractional calculus to deal with the Kelvin-Voigt (KV) equation and non-Newtonian fluid behavior model with fuzzy parameters. A new and accurate numerical algorithm using a spectral tau technique based on the generalized fractional Legendre polynomials (GFLPs) is developed to solve those problems under uncertainty. Numerical simulations are carried out and the analysis of the results highlights the significant features of the new technique in comparison with the previous findings. A detailed error analysis is also carried out and discussed.