Recent zbMATH articles in MSC 26Ahttps://zbmath.org/atom/cc/26A2022-09-13T20:28:31.338867ZUnknown authorWerkzeugBook 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].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)On 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)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.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)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 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.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)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.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.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 \(1/r + 1/p=2/q\).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)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 \(u(a) = u(b) = 0\).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.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.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.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)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.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)