Recent zbMATH articles in MSC 44https://www.zbmath.org/atom/cc/442021-02-27T13:50:00+00:00WerkzeugSome properties of several functions involving polygamma functions and originating from the sectional curvature of the beta manifold.https://www.zbmath.org/1453.330032021-02-27T13:50:00+00:00"Qi, Feng"https://www.zbmath.org/authors/?q=ai:qi.fengThe sectional curvature \(K(x,y)\) of the Fisher metric \(ds^2=\psi'(x) dx^2+\psi'(y) dy^2-\psi'(x+y) (dx+dy)^2\) on the first quadrant \(M\) of the \(x, y\)-plane is
\[
K(x,y)=\frac{\psi''(x)\psi''(y)\psi''(x+y)A}{4\{(\psi'(x)+\psi'(y))\psi'(x+y)-\psi'(x)\psi'(y)\}^2},
\]
where \(A:=\psi'(x)/\psi''(x)+\psi'(y)/\psi''(y)-\psi'(x+y)/\psi''(x+y)\), with \(\psi(x)\) being the logarithmic derivative of \(\Gamma(x)\). It is shown that \(K(x,y)\) is analytic on \(M\). Various limits of \(K(x,y)\) are obtained such as \((x,y)\to(0,0)\), \((x,y)\to(\infty,\infty)\), \((x,y)\to (0,\infty)\) and \((x,y)\to(\infty,0)\).
The author states an open conjecture concerning the boundedness of \(K(x,y)\), namely that \(-1/2<K(x,y)<0\). The complete monotonicity on \((0,\infty)\) of two functions involving derivatives of \(\psi(x)\) is demonstrated. Finally, a sharp double inequality is obtained for the function
\[
(\psi'(x)+x\psi''(x))/((x\psi'(x)-1)^2)
\]
for \(x\in (0,\infty)\).
Reviewer: Richard B. Paris (Dundee)Integrals involving the aleph-function of several variables.https://www.zbmath.org/1453.330092021-02-27T13:50:00+00:00"Ayant, Frederic"https://www.zbmath.org/authors/?q=ai:ayant.frederic-ySummary: In this paper, we evaluate four integrals involving the product of elementary special functions and the multivariable Aleph-function. The integrals are quite general in nature and from them a large number of new results can be obtained simply by specializing the parameters of the multivariable Aleph-function.Fractional-order derivatives and integrals: introductory overview and recent developments.https://www.zbmath.org/1453.260082021-02-27T13:50:00+00:00"Srivastava, Hari Mohan"https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohanThis paper explores particular solutions to some classes of differential equations, which are generalizations of classic equations by including elements of fractional calculus. The title is somewhat misleading as it suggests a general overview of fractional calculus.
The first part of the paper, from Sections 3 to 6, resorts to Laplace or Sumudu transforms to find the solution; hence, it considers only linear time-invariant equations. The considered equation is called kinetic. The second part summarizes and exemplifies literature's solutions to a class of nonlinear fractional equations, using the Cauchy-Goursat fractional derivative. This part is more interesting, but it seems that no novel contribution is given.
There is no discussion of existence or uniqueness. The initial condition problem in fractional equations has been a topic of much discussion, due to several inconsistencies with physical considerations, a fact that is missing in this study. The equations in this paper are not motivated by modelling challenges and seem rather mathematical artifices.
Reviewer: Javier Gallegos (Santiago de Chile)A family of convolution-based generalized Stockwell transforms.https://www.zbmath.org/1453.420072021-02-27T13:50:00+00:00"Srivastava, H. M."https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohan"Shah, Firdous A."https://www.zbmath.org/authors/?q=ai:shah.firdous-ahmad"Tantary, Azhar Y."https://www.zbmath.org/authors/?q=ai:tantary.azhar-ySummary: The main purpose of this paper is to introduce a family of convolution-based generalized Stockwell transforms in the context of time-fractional-frequency analysis. The spirit of this article is completely different from two existing studies (see \textit{D. P. Xu} and \textit{K. Guo} [``Fractional \(S\)-transform-Part 1: Theory'', Appl. Geophys. 9, 73--79 (2012)] and \textit{S. K. Singh} [J. Pseudo-Differ. Oper. Appl. 4, No. 2, 251--265 (2013; Zbl 1272.42025)]) in the sense that our approach completely relies on the convolution structure associated with the fractional Fourier transform. We first study all of the fundamental properties of the generalized Stockwell transform, including a relationship between the fractional Wigner distribution and the proposed transform. In the sequel, we introduce both the semi-discrete and discrete counterparts of the proposed transform. We culminate our investigation by establishing some Heisenberg-type inequalities for the generalized Stockwell transform in the fractional Fourier domain.Semiclassical sampling and discretization of certain linear inverse problems.https://www.zbmath.org/1453.351952021-02-27T13:50:00+00:00"Stefanov, Plamen"https://www.zbmath.org/authors/?q=ai:stefanov.plamen-dBoundedness and compactness of localization operators for Weinstein-Wigner transform.https://www.zbmath.org/1453.430022021-02-27T13:50:00+00:00"Saoudi, Ahmed"https://www.zbmath.org/authors/?q=ai:saoudi.ahmed"Nefzi, Bochra"https://www.zbmath.org/authors/?q=ai:nefzi.bochraThe authors consider the notion of localization operators \({\mathcal{L}}_{\varphi, \psi}(\sigma)\) associated with
the Weinstein-Wigner transform
\[
{\mathcal{V}}(\varphi, \psi)(x, y)=\int_{{\mathbb{R}}_+^{d+1}}\varphi(\lambda)\Lambda_\alpha^d(y, \lambda){\tau^\alpha_x\psi(\lambda)}\,d\mu_\alpha(\lambda),
\]
where
\[
d\mu_\alpha(\lambda)={(2\pi)^{-d}2^{-2\alpha}\Gamma^{-2}(\alpha+1)}{\lambda_{d+1}^{2\alpha+1}}\,d\lambda,
\]
\[
\Lambda_\alpha^d(\lambda, x)=e^{-ix'\lambda'}j_\alpha(x_{d+1}\lambda_{d+1}),
\]
\(x=(x', x_{d+1})\), \(\lambda=(\lambda', \lambda_{d+1})\), \(j_\alpha\) is the normalized Bessel function of index \(\alpha\) and
\[
\tau^\alpha_x\varphi(y)=c_\alpha\int_0^\pi\varphi\left(x'+y'\sqrt{x_{d+1}^2+y_{d+1}^2+2x_{d+1}y_{d+1}\cos\theta}\right)\sin^{2\alpha}\theta\,d\theta
\]
is the translation operator associated with the Weinstein operator.
For two functions \(\varphi\), \(\psi\) that are measurable on \({\mathbb{R}}_+^{d+1}\) and a function \(\sigma\) that is measurable on \({\mathbb{R}}_+^{2d+2}\) the boundedness and compactness on \(L^p(\mathbb R_+^{d+1}, d\mu_\alpha)\), \(1\le p\le\infty\), of the localization operator
\[
{\mathcal{L}}_{\varphi, \psi}(\sigma)(f)(y)=\int_{{\mathbb{R}}_+^{2d+2}}\sigma(x, \lambda){\mathcal{V}}(f, \varphi)(x, \lambda)\Lambda_\alpha^d(\lambda, y)\overline{\tau^\alpha_y\psi(x)}\,d\mu_\alpha(x)d\mu_\alpha(\lambda),\ y\in {\mathbb{R}}_+^{d+1}
\]
are studied.
It is proved that the operators are in the trace class and a trace formula for them is given.
Reviewer: Vadim D. Kryakvin (Rostov-na-Donu)An analysis of a mathematical fractional model of hybrid viscous nanofluids and its application in heat and mass transfer.https://www.zbmath.org/1453.351442021-02-27T13:50:00+00:00"Ali, Rizwan"https://www.zbmath.org/authors/?q=ai:ali.rizwan"Asjad, Muhammad Imran"https://www.zbmath.org/authors/?q=ai:asjad.muhammad-imran"Akgül, Ali"https://www.zbmath.org/authors/?q=ai:akgul.aliOne of the well-know problems in hydrodynamics is modelling free convection of viscous fluid in rectangular channel with non-equal temperatures given on its walls. In the present paper, the authors generalize this problem and examine water- and engine-oil based nanofluids containing copper or aluminium particles.
As these media have more complicated properties than classical fluids, three governing laws are generalized using Caputo fractional-order derivatives in time: Newton's law for viscous stresses, Fourier's law of heat conduction and Fick's law for diffusion. To solve the resulting equations, the Laplace transform is used.
The authors examine dependencies of temperature, concentration and velocity on coordinate, time and orders of Caputo derivatives and illustrate them by numerous plots.
Reviewer: Aleksey Syromyasov (Saransk)Traveling waves of a diffusive SIR epidemic model with general nonlinear incidence and infinitely distributed latency but without demography.https://www.zbmath.org/1453.923022021-02-27T13:50:00+00:00"Hu, Haijun"https://www.zbmath.org/authors/?q=ai:hu.haijun"Zou, Xingfu"https://www.zbmath.org/authors/?q=ai:zou.xingfuSummary: In this paper, we are concerned with existence/non-existence of traveling waves of a diffusive SIR epidemic model with general incidence rate of the form of \(f (S) g (I)\) and infinitely distributed latency but without demography. We show that the existence of traveling waves only depends on the basic reproduction number of the corresponding spatial-homogeneous system of delay differential equations, which is determined by the recovery rate, the local properties of \(f\) and \(g\) and a minimal wave speed \(c^\ast\) that is affected by the distributed delay. The proof of existence of traveling waves is by employing Schauder's fixed point theorem, and the proof of nonexistence is completed with the aid of the bilateral Laplace transform.Complete Radon-Kipriyanov transform: some properties.https://www.zbmath.org/1453.440032021-02-27T13:50:00+00:00"Lyakhov, L. N."https://www.zbmath.org/authors/?q=ai:lyakhov.l-n"Lapshina, M. G."https://www.zbmath.org/authors/?q=ai:lapshina.marina-g"Roshchupkin, S. A."https://www.zbmath.org/authors/?q=ai:roshchupkin.s-aSummary: The even Radon-Kipriyanov transform \((K_\gamma\)-transform) is suitable for studying problems with the Bessel singular differential operator \(B_{\gamma_i} = \frac{\partial^2}{\partial x_i^2} + \frac{\gamma_i}{x_i}\frac{\partial }{\partial x_i}\), \(\gamma_i > 0\). In this work, the odd Radon-Kipriyanov transform and the complete Radon-Kipriyanov transform are introduced to study more general equations containing odd \(B\)-derivatives \(\frac{\partial }{\partial x_i} B_{\gamma_i}^k\), \(k = 0, 1, 2, \ldots \) (in particular, gradients of functions). Formulas of the \(K_\gamma \)-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B. M. Levitan and the odd Bessel transform introduced by I. A. Kipriyanov and V. V. Katrakhov, a relationship of the complete Radon-Kipriyanov transform with the Fourier transform and the mixed Fourier-Levitan-Kipriyanov-Katrakhov transform is deduced. An analogue of Helgason's support theorem and an analog of the Paley-Wiener theorem are given.On certain polynomials associated with the multivariable aleph-function.https://www.zbmath.org/1453.330082021-02-27T13:50:00+00:00"Ayant, Frederic"https://www.zbmath.org/authors/?q=ai:ayant.frederic-ySummary: In an attempt to unify various bilateral generating functions obtained earlier by
\textit{H. M. Srivastava} and \textit{R. Panda}, J. Reine Angew. Math. 283/284, 265--274 (1976; Zbl 0315.33003),
\textit{R. K. Raina}, Proc. Natl. Acad. Sci. India, Sect. A 46, 300--304 (1976; Zbl 0438.33004)
and \textit{H. M. Srivastava} and \textit{R. K. Raina}, Hokkaido Math. J. 10, 34--45 (1981; Zbl 0438.33005),
we study here a few new sets of polynomials associated with the multivariable Aleph-function and give certain theorems concerning the generating functions of these polynomials. In the sequel we also show that these theorems can be applied to yield several bilateral generating function for some polynomial sets.Poisson processes and a log-concave Bernstein theorem.https://www.zbmath.org/1453.260212021-02-27T13:50:00+00:00"Klartag, Bo'az"https://www.zbmath.org/authors/?q=ai:klartag.boaz"Lehec, Joseph"https://www.zbmath.org/authors/?q=ai:lehec.josephSummary: We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the Prékopa-Leindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.Convolution operators on weighted spaces of continuous functions and supremal convolution.https://www.zbmath.org/1453.440062021-02-27T13:50:00+00:00"Kleiner, T."https://www.zbmath.org/authors/?q=ai:kleiner.tillmann"Hilfer, R."https://www.zbmath.org/authors/?q=ai:hilfer.rudolfSummary: The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.A time multidomain spectral method for valuing affine stochastic volatility and jump diffusion models.https://www.zbmath.org/1453.653612021-02-27T13:50:00+00:00"Moutsinga, Claude Rodrigue Bambe"https://www.zbmath.org/authors/?q=ai:moutsinga.claude-rodrigue-bambe"Pindza, Edson"https://www.zbmath.org/authors/?q=ai:pindza.edson"Maré, Eben"https://www.zbmath.org/authors/?q=ai:mare.ebenA time-spectral domain decomposition method is developed that accommodates differential equations arising from financial models of affine type. The affine structure of the financial models is used to avoid solving the multi-dimensional partial integro-differential equation (PIDE) but rather to solve a system of Riccati equations. The method is based on the Tau-matrix approach using a differentiation matrix method on a time interval divided into disjoint domains. The Riccati equations are solved in the frequency domain using an operational matrix based on Chebyshev polynomials. In this way, the original problem is transformed into an iterative system of algebraic equations that is easier to solve. Three numerical examples are implemented and solutions are compared to numerical solutions from Chebfun [\textit{R. B. Platte} and \textit{L. N. Trefethen}, Math. Ind. 15, 69--87 (2010; Zbl 1220.65100)]. The numerical results show that the method maintains its spectral convergence even for large time-space intervals. The method can be applied to other affine models with jumps.
Reviewer: Bülent Karasözen (Ankara)On Lin's condition for products of random variables with singular joint distribution.https://www.zbmath.org/1453.600312021-02-27T13:50:00+00:00"Il'inskii, A."https://www.zbmath.org/authors/?q=ai:ilynsky.a-s|ilinskii.alexander|ilinskii.a-i"Ostrovska, S."https://www.zbmath.org/authors/?q=ai:ostrovska.sofiya|ostrovska.sofia-iSummary: Lin's condition is used to establish the moment determinacy/indeterminacy of absolutely continuous probability distributions. Recently, a number of papers related to Lin's condition for functions of random variables have appeared. In the present paper, this condition is studied for products of random variables with given densities in the case when their joint distribution is singular. It is proved, assuming that the densities of both random variables satisfy Lin's condition, that the density of their product may or may not satisfy this condition.Duality principle and new forms of the inverse Laplace transform for signal propagation analysis in inhomogeneous media with dispersion.https://www.zbmath.org/1453.440012021-02-27T13:50:00+00:00"Pavelyev, A. G."https://www.zbmath.org/authors/?q=ai:pavelev.a-g"Pavelyev, A. A."https://www.zbmath.org/authors/?q=ai:pavelev.a-aSummary: New equations for Laplace transform inversion are obtained. The equations satisfy the causality principle. The impulse response of a channel is determined in order to analyze dispersion distortions in inhomogeneous media. The impulse response excludes the possibility that the signal exceeds the speed of light in the medium. The transmission bandwidth, the angular spectrum, and the Doppler shift in the ionosphere are computed.Classes of power semicircle laws that are randomly weighted average distributions.https://www.zbmath.org/1453.623482021-02-27T13:50:00+00:00"Roozegar, Rasool"https://www.zbmath.org/authors/?q=ai:roozegar.rasool"Soltani, Ahmad Reza"https://www.zbmath.org/authors/?q=ai:soltani.ahmad-rezaSummary: We give an affirmative answer to the conjecture raised in our work [Stat. Probab. Lett. 82, No. 5, 1012--1020 (2012; Zbl 1241.62015)] that a certain class of power semicircle distributions, parameterized by \(n\), gives the distributions of the average of \(n\) independent and identically Arcsine random variables weighted by the cuts of \((0,1)\) by the order statistics of a uniform \((0, 1)\) sample of size \(n-1\), for each \(n\). Then we establish the central limit theorem for this class of distributions. We also use \textit{N. Demni}'s [Commun. Stoch. Anal. 3, No. 2, 197--210 (2009; Zbl 1331.60034)] results on the connection between the ordinary and generalized Cauchy or Stieltjes transforms, and introduce new classes of randomly weighted average distributions.One-dimensional and multi-dimensional integral transforms of Buschman-Erdélyi type with Legendre functions in kernels.https://www.zbmath.org/1453.440052021-02-27T13:50:00+00:00"Sitnik, Sergei M."https://www.zbmath.org/authors/?q=ai:sitnik.sergei-mihailovich"Skoromnik, Oksana V."https://www.zbmath.org/authors/?q=ai:skoromnik.oksana-vSummary: This paper consists of two parts. In the first part we give a brief survey of results on Buschman-Erdélyi operators, which are transmutations for the Bessel singular operator. Main properties and applications of Buschman-Erdélyi operators are outlined. In the second part of the paper we consider multi-dimensional integral transforms of Buschman-Erdélyi type with Legendre functions in kernels. Complete proofs are given in this part, main tools are based on Mellin transform properties and usage of Fox \(H\)-functions.
For the entire collection see [Zbl 1443.34001].Some inverse Laplace transforms that contain the Marcum \(Q\) function and an expanded property of the Marcum \(Q\) function.https://www.zbmath.org/1453.440022021-02-27T13:50:00+00:00"Veestraeten, Dirk"https://www.zbmath.org/authors/?q=ai:veestraeten.dirkIt appears that the current paper is an extension and generalization of [\textit{D.~Veestraeten}, Integral Transforms Spec. Funct. 26, No.~10, 859--871 (2015; Zbl 1331.33005)], where the inverse Laplace transform for the product of two parabolic cylinder functions is obtained using the Ornstein-Uhlenbeck process.
In Section~2, basic definitions of transition distributions and the transition density are presented.
Employing a Feller process, the Laplace transform of the transition density and of the transition distribution are given.
Using a new substitution in the transition distribution (2.5) and specifying the transition density (3.1), the transition distribution is derived in terms of the Marcum $Q$-function in Section~3.
In Section~4, the inverse Laplace transform is established from the transition density, which contains products of Kummer and Tricomi confluent hypergeometric functions (as Theorem~4.1) under certain initial conditions.
Using the transition distribution under different conditions six new inverse Laplace transforms are derived in Theorems 5.1--5.4, four of which contain the Marcum $Q$-function. Section~6 is divided into two subsections, one of which discusses the inverse Laplace transform (Theorem~6.1), which involves the relation between two Marcum $Q$-functions with reversed arguments, and in the other the Marcum $Q$-function is represented (Theorem~6.2) in terms of the generalized hypergeometric function involving integer and fractional order values.
Reviewer: Deshna Loonker (Jodhpur)An inversion formula for the horizontal conical Radon transform.https://www.zbmath.org/1453.440042021-02-27T13:50:00+00:00"Nguyen, Duy N."https://www.zbmath.org/authors/?q=ai:nguyen.duy-n"Nguyen, Linh V."https://www.zbmath.org/authors/?q=ai:nguyen.linh-viet|nguyen-viet-linh.Summary: In this paper, we consider the conical Radon transform on all one-sided circular cones in \(\mathbb{R}^3\) with horizontal central axis whose vertices are on a vertical line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical slice transform on a sphere and V-line transform on a plane.Some new integrals involving \(S\)-function and polynomials.https://www.zbmath.org/1453.330052021-02-27T13:50:00+00:00"Chand, Mehar"https://www.zbmath.org/authors/?q=ai:chand.meharSummary: In this paper, we aim to establish the new integrals involving \(S\)-function and Laguerre polynomials. On account of the most general nature of the functions involved herein, our main findings are capable of yielding a large number of new, interesting and useful integrals, expansion formula involving the \(S\)-function and the Laguerre polynomials as their special cases.