Recent zbMATH articles in MSC 44https://zbmath.org/atom/cc/442024-09-27T17:47:02.548271ZWerkzeugPositivstellensätze for semiringshttps://zbmath.org/1541.130262024-09-27T17:47:02.548271Z"Schmüdgen, Konrad"https://zbmath.org/authors/?q=ai:schmudgen.konrad"Schötz, Matthias"https://zbmath.org/authors/?q=ai:schotz.matthiasSince the seminal work [\textit{K. Schmüdgen}, Math. Ann. 289, No. 2, 203--206 (1991; Zbl 0744.44008)], there has been an increasing interest in understanding representations of strictly positive polynomial functions on compact, (basic) semialgebraic sets. Remarkably, such representations have no denominator, and they are also interesting for their applications in optimization.
Classically, this problem has been investigated using quadratic modules and preorderings, which are convex cones of polynomials containing the sums of squares cone. In this article, the existence of representations in a semiring \(\mathcal{S}\) for general commutative unital \(\mathbb{R}\)-algebras \(\mathcal{A}\) is investigated. Semirings, also known as preprimes e.g. in [\textit{M. Marshall}, Positive polynomials and sums of squares. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1169.13001)], are convex cones \(\mathcal{S}\subset \mathcal{A}\) which contain the identity and are closed under multiplication (and therefore do not necessarily contain the sums of squares).
The first contribution of the manuscript is a unified approach to the Positivstellensätze in the case of semirings and quadratic modules. Such a unified and general approach leads to new representations without denominators, under a compactness assumption. For instance, the classical Positivstellensätz of K. Schmüdgen is refined, allowing only sums of squares multipliers of a special form. Other applications are presented with the additional convexity assumption, or when the semialgebraic set is known to be included in compact polyhedra.
In the second part of the manuscript, the authors investigate the non-compact cases. To do this, a new \textit{filtered} Positivstellensätz is presented. Intuitively, the filtration allows to treat some of the points at infinity, and the final result is a representation theorem with controlled denominators. As a final special application, a Positivstellensätz for \textit{cylinders} over semialgebraic sets is proven. This is a vast generalization of \textit{V. Powers} [J. Pure Appl. Algebra 188, No. 1--3, 217--226 (2004; Zbl 1035.14022)].
Reviewer: Lorenzo Baldi (Leipzig)Abstract algebraic construction in fractional calculus: parametrised families with semigroup propertieshttps://zbmath.org/1541.260222024-09-27T17:47:02.548271Z"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arranSummary: What structure can be placed on the burgeoning field of fractional calculus with assorted kernel functions? This question has been addressed by the introduction of various general kernels, none of which has both a fractional order parameter and a clear inversion relation. Here, we use ideas from abstract algebra to construct families of fractional integral and derivative operators, parametrised by a real or complex variable playing the role of the order. These have the typical behaviour expected of fractional calculus operators, such as semigroup and inversion relations, which allow fractional differential equations to be solved using operational calculus in this general setting, including all types of fractional calculus with semigroup properties as special cases.New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine functionhttps://zbmath.org/1541.260232024-09-27T17:47:02.548271Z"Geng, Lu-Lu"https://zbmath.org/authors/?q=ai:geng.lu-lu"Yang, Xiao-Jun"https://zbmath.org/authors/?q=ai:yang.xiao-jun"Alsolami, Abdulrahman Ali"https://zbmath.org/authors/?q=ai:alsolami.abdulrahman-ali(no abstract)Some properties of \(k\)-Riemann-Liouville fractional integral operatorhttps://zbmath.org/1541.260272024-09-27T17:47:02.548271Z"Prajapat, Radhe Shyam"https://zbmath.org/authors/?q=ai:prajapat.radhe-shyam"Bapna, Indu Bala"https://zbmath.org/authors/?q=ai:bapna.indu-balaSummary: In this paper we will introduce some properties of \(k\)-Riemann Liouville fractional integral operator involving convolution property. The fractional derivative of \(k\)-Riemann Liouville fractional integral operator of integral transforms will be obtained. Applications of this operator will be introduced. All results of nature will be discussed as special cases.A new fractional derivative operator and its application to diffusion equationhttps://zbmath.org/1541.260292024-09-27T17:47:02.548271Z"Sharma, Ruchi"https://zbmath.org/authors/?q=ai:sharma.ruchi"Goswami, Pranay"https://zbmath.org/authors/?q=ai:goswami.pranay"Dubey, Ravi Shanker"https://zbmath.org/authors/?q=ai:dubey.ravi-shanker"Belgacem, Fethi Bin Muhammad"https://zbmath.org/authors/?q=ai:belgacem.fethi-bin-muhammad(no abstract)A new weighted fractional operator with respect to another function via a new modified generalized Mittag-Leffler lawhttps://zbmath.org/1541.260302024-09-27T17:47:02.548271Z"Thabet, Sabri T. M."https://zbmath.org/authors/?q=ai:thabet.sabri-t-m"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Kedim, Imed"https://zbmath.org/authors/?q=ai:kedim.imed"Ayari, M. Iadh"https://zbmath.org/authors/?q=ai:ayari.mohamed-iadhSummary: In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann-Liouville involving a new modified version of a generalized Mittag-Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann-Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.Regularized limit, analytic continuation and finite-part integrationhttps://zbmath.org/1541.300012024-09-27T17:47:02.548271Z"Galapon, Eric A."https://zbmath.org/authors/?q=ai:galapon.eric-aSummary: Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [\textit{E. A. Galapon}, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 473, No. 2197, Article ID 20160567, 18 p. (2017; Zbl 1404.26013)]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin.A family of Horn-Bernstein functionshttps://zbmath.org/1541.330012024-09-27T17:47:02.548271Z"Berg, Christian"https://zbmath.org/authors/?q=ai:berg.christian"Pedersen, Henrik L."https://zbmath.org/authors/?q=ai:pedersen.henrik-laurbergSummary: A family of recently investigated Bernstein functions is revisited and those functions for which the derivatives are logarithmically completely monotonic are identified. This leads to the definition of a class of Bernstein functions, which we propose to call Horn-Bernstein functions because of the results of \textit{R. A. Horn} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 8, 219--230 (1967; Zbl 0314.60017)].Complete monotonicity for a ratio of finitely many gamma functionshttps://zbmath.org/1541.330032024-09-27T17:47:02.548271Z"Chen, Hai-Sheng"https://zbmath.org/authors/?q=ai:chen.haisheng"Zhu, Ye-Cheng"https://zbmath.org/authors/?q=ai:zhu.yecheng"Wang, Jia-Hui"https://zbmath.org/authors/?q=ai:wang.jiahuiSummary: In this paper, we solve the question completely raised by \textit{F. Qi} and \textit{D. Lim}
in paper [Adv. Difference Equ. 2020, Paper No. 193, 9 p. (2020; Zbl 1482.33001)] published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value.Srivastava-Luo-Raina \(M\)-transform involving the incomplete \(I\)-functionshttps://zbmath.org/1541.330052024-09-27T17:47:02.548271Z"Bhatter, Sanjay"https://zbmath.org/authors/?q=ai:bhatter.sanjay"Shyamsunder, Nishant"https://zbmath.org/authors/?q=ai:shyamsunder.nishant"Purohit, Sunil Dutt"https://zbmath.org/authors/?q=ai:purohit.sunil-duttThe authors compute the Srivastava-Luo-Raina \(M\)-transform for the incomplete \(I\)-function and derive the Srivastava-Luo-Raina \(M\)-transform for the product of the incomplete \(I\)-function and the Srivastava polynomial. Since the incomplete \(I\)-function and the Srivastava polynomials generalize many classical special functions, the results obtained are important. When \(p=0\), the results reduce to well-known formulas for the classical Stieltjes, Laplace, natural, and Sumudu transforms for the incomplete \(I\)-function and some classical orthogonal polynomials.
Reviewer: Pan Lian (Tianjin)The sequential conformable Langevin-type differential equations and their applications to the RLC electric circuit problemshttps://zbmath.org/1541.340092024-09-27T17:47:02.548271Z"Aydin, M."https://zbmath.org/authors/?q=ai:aydin.mustafa"Mahmudov, N. I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisogluSummary: In this paper, the sequential conformable Langevin-type differential equation is studied. A representation of a solution consisting of the newly defined conformable bivariate Mittag-Leffler function to its nonhomogeneous and linear version is obtained via the conformable Laplace transforms' technique. Also, existence and uniqueness of a global solution to its nonlinear version are obtained. The existence and uniqueness of solutions are shown with respect to the weighted norm defined in compliance with (conformable) exponential function. The concept of the Ulam-Hyers stability of solutions is debated based on the fixed-point approach. The LRC electrical circuits are presented as an application to the described system. Simulated and numerical instances are offered to instantiate our abstract findings.Differential equations with fractional derivatives with fixed memory lengthhttps://zbmath.org/1541.340152024-09-27T17:47:02.548271Z"Ledesma, César T."https://zbmath.org/authors/?q=ai:torres-ledesma.cesar-e"Rodríguez, Jesús A."https://zbmath.org/authors/?q=ai:rodriguez.jesus-a.1"da C. Sousa, J. Vanterler"https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.joseSummary: In the present paper, we have as main purpose to contribute significantly with deep analysis of the properties of the fractional operators with fixed memory length, in particular, involving the Laplace transform of the Riemann-Liouville fractional integral and derivative with fixed memory length. In this sense, we tackle a problem of the existence and uniqueness of a fractional differential equation with fixed memory length, by means of two fundamental lemmas in the discussion of the integral equation and the Banach fixed point.Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas methodhttps://zbmath.org/1541.350422024-09-27T17:47:02.548271Z"Chatziafratis, Andreas"https://zbmath.org/authors/?q=ai:chatziafratis.andreas"Grafakos, Loukas"https://zbmath.org/authors/?q=ai:grafakos.loukas"Kamvissis, Spyridon"https://zbmath.org/authors/?q=ai:kamvissis.spyridonSummary: We study the inhomogeneous Airy partial differential equation (also called Stokes or linearized Korteweg-de Vries equation with a negative sign) on the half-line with generic initial and boundary data in a classical smooth setting, via the formula provided by the Fokas unified transform method for linear evolution equations. We first present a suitable decomposition of that formula in the complex plane in order to appropriately interpret various terms appearing in it, thus securing convergence in a strict sense. Writing the solution in an Ehrenpreis-Palamodov form, our analysis allows for rigorous \textit{a posteriori} verification of the full initial-boundary-value problem and a thorough investigation of the behavior of the solution near the boundaries of the spatiotemporal domain. We prove that the integrals in this representation converge uniformly to prescribed values and the solution admits a smooth extension up to the boundary only under certain data compatibility conditions (with implications for well-posedness, control theory and efficient numerical computations). Importantly, based on this analysis, we perform an effective asymptotic study of far-field dynamics. This yields new explicit asymptotic formulae which characterize the properties of the solution in terms of (in)compatibilities of the data at the `corner' of the quadrant. In particular, the asymptotic behavior of the solution is sensitive to perturbations of the data at the origin. In all cases, even assuming the initial data to belong to the Schwartz class, the solution loses this property at soon as time becomes positive. Hereby, we report on the discovery of a novel type of a long-range instability phenomenon for linear dispersive differential equations. Our ideas are extendable to other Airy-like and more general problems for dispersive evolution equations.Bent-half space model problem for Lamé equation with surface tensionhttps://zbmath.org/1541.353852024-09-27T17:47:02.548271Z"Maryani, Sri"https://zbmath.org/authors/?q=ai:maryani.sri"Wardayani, Ari"https://zbmath.org/authors/?q=ai:wardayani.ari"Renny"https://zbmath.org/authors/?q=ai:renny.Summary: The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the \(\mathcal{R}\)-boundedness of the solution operator families of the Lamé equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of \(N\)-dimensional Euclidean space \((N \geq 2)\). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [the first author et al., Math. Stat. 10, No. 3, 498--514 (2022; \url{doi:10.13189/ms.2022.100305})]. They investigated the \(\mathcal{R}\)-boundedness of the solution operator families in the half-space case for the model problem of the Lamé equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that \(\mathcal{R}\)-boundedness implies the maximal \(L_p\)-\(L_q\) regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli systemhttps://zbmath.org/1541.354332024-09-27T17:47:02.548271Z"Maltseva, Diana S."https://zbmath.org/authors/?q=ai:maltseva.diana-s"Popovych, Roman O."https://zbmath.org/authors/?q=ai:popovych.roman-oSummary: We carry out extended symmetry analysis of the (1+2)-dimensional Boiti-Leon-Pempinelli system, which corrects, enhances and generalizes many results existing in the literature. The point-symmetry pseudogroup of this system is computed using an original megaideal-based version of the algebraic method. A number of meticulously selected differential constraints allow us to construct families of exact solutions of this system, which are significantly larger than all known ones. After classifying one- and two-dimensional subalgebras of the entire (infinite-dimensional) maximal Lie invariance algebra of this system, we study only its essential Lie reductions, which give solutions beyond the above solution families. Among reductions of the Boiti-Leon-Pempinelli system using differential constraints or Lie symmetries, we identify a number of famous partial and ordinary differential equations. We also show how all the constructed solution families can significantly be extended by Laplace and Darboux transformations.Quantitative uncertainty principles associated with the \(k\)-Hankel wavelet transform on \(\mathbb{R}^d\)https://zbmath.org/1541.420372024-09-27T17:47:02.548271Z"Mejjaoli, Hatem"https://zbmath.org/authors/?q=ai:mejjaoli.hatem"Shah, Firdous A."https://zbmath.org/authors/?q=ai:shah.firdous-ahmadSummary: The \(k\)-Hankel wavelet transform ( \(k\)-HWT) is a novel addition to the class of wavelet transforms, which has gained a respectable status in the realm of time-frequency signal analysis within a short span of time. Knowing the fact that the study of uncertainty principles is both theoretically interesting and practically useful, we formulate several qualitative uncertainty principles for the \(k\)-Hankel wavelet transform. First, we formulate the Heisenberg's uncertainty principle governing the simultaneous localization of a signal and the corresponding \(k\)-HWT via three approaches: \({L}^2\)-type, \({L}^p\)-type, and \(k\)-entropy based. Second, we derive some weighted uncertainty inequalities such as the Pitt's and Beckner's uncertainty inequalities [\textit{W. Beckner}, Proc. Am. Math. Soc. 123, No. 6, 1897--1905 (1995; Zbl 0842.58077)] for the \(k\)-HWT. We culminate our study by formulating several concentration-based uncertainty principles, including the Amrein-Berthier-Benedicks's and local inequalities for the \(k\)-Hankel wavelet transform.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Moment functions of higher rank on some types of hypergroupshttps://zbmath.org/1541.430102024-09-27T17:47:02.548271Z"Fechner, Żywilla"https://zbmath.org/authors/?q=ai:fechner.zywilla"Gselmann, Eszter"https://zbmath.org/authors/?q=ai:gselmann.eszter"Székelyhidi, László"https://zbmath.org/authors/?q=ai:szekelyhidi.laszloSummary: We consider moment functions of higher order. In our earlier paper [\textit{Ż.~Fechner} et al., Result. Math. 76, No.~4, Paper No.~171, 16~p. (2021; Zbl 1510.44009)], we have already investigated the moment functions of higher order on groups. The main purpose of this work is to prove characterization theorems for moment functions on the multivariate polynomial hypergroups and on the Sturm-Liouville hypergroups. In the first case, the moment generating functions of higher rank are partial derivatives (taken at zero) of the composition of generating polynomials of the hypergroup and functions whose coordinates are given by the formal power series. On Sturm-Liouville hypergroups the moment functions of higher rank are restrictions of even smooth functions that also satisfy certain boundary value problems. The second characterization of moment functions of higher rank on Sturm-Liouville hypergroups is given by means of an exponential family. In this case, the moment functions of higher rank are partial derivatives of an appropriately modified exponential family again taken at zero.Theory and applications on a new generalized Laplace-type integral transformhttps://zbmath.org/1541.440012024-09-27T17:47:02.548271Z"Albayrak, Durmuş"https://zbmath.org/authors/?q=ai:albayrak.durmusSummary: In this paper, several theorems and relations are examined by using a generalized Laplace-type integral transform. A generalization of the harmonic oscillator in a non-resisting and resisting medium problems, initial-boundary problems, and integral equations is solved via this integral transform. Furthermore, the well-known series entitled as Basel problem is obtained in a similar way. Moreover, we compare numerically classical and newly introduced by us integral transform.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Generalized derivatives and Laplace transform in \((k, \psi)\)-Hilfer formhttps://zbmath.org/1541.440022024-09-27T17:47:02.548271Z"Başcı, Yasemin"https://zbmath.org/authors/?q=ai:basci.yasemin"Mısır, Adil"https://zbmath.org/authors/?q=ai:misir.adil"Öğrekçi, Süleyman"https://zbmath.org/authors/?q=ai:ogrekci.suleymanSummary: In this work, we discuss the most generalized derivatives and integrals and their features in \(\left(k,\psi \right)\)-Hilfer form. Furthermore, we define the new generalized Laplace transform to the generalized derivatives and integrals in \(\left(k,\psi \right)\)-Hilfer form. Also, we have obtained the new generalized Laplace transforms of some expressions. These statements obtained cover many previous studies. Finally, we have given an example that will both use some of the results obtained and emphasize the importance of parameters such as \(k , \rho , \psi\) of the \(\left(k,\psi \right)\)-generalized Laplace transform.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Interval Laplace transform and its application in production inventoryhttps://zbmath.org/1541.440032024-09-27T17:47:02.548271Z"Das, Subhajit"https://zbmath.org/authors/?q=ai:das.subhajit"Rahman, Md Sadikur"https://zbmath.org/authors/?q=ai:rahman.md-sadikur"Shaikh, Ali Akbar"https://zbmath.org/authors/?q=ai:shaikh.ali-akbar"Bhunia, Asoke Kumar"https://zbmath.org/authors/?q=ai:bhunia.asoke-kumar"Konstantaras, Ioannis"https://zbmath.org/authors/?q=ai:konstantaras.ioannisSummary: This work demonstrates the Laplace and inverse transforms of interval valued functions with exaggerating its necessary properties under interval flexibility. After proposing the formal definition interval Laplace transform (i.e., Laplace transform of interval valued functions), some of its important properties are derived. Thereafter, the sufficient condition for the existence of interval Laplace transforms is established. Then, some important results regarding switching points of interval Laplace transform are discussed and illustrated with some numerical examples. Finally, the definition of inverse transform of interval valued functions is proposed, and as an application, a production inventory model in interval uncertainty is studied using all of the proposed theoretical results.
{{\copyright} 2022 John Wiley \& Sons Ltd.}On Laplace transforms with respect to functions and their applications to fractional differential equationshttps://zbmath.org/1541.440042024-09-27T17:47:02.548271Z"Fahad, Hafiz Muhammad"https://zbmath.org/authors/?q=ai:fahad.hafiz-muhammad"ur Rehman, Mujeeb"https://zbmath.org/authors/?q=ai:ur-rehman.mujeeb"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arranSummary: An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called \(\Psi\)-fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}A novel approach to evaluating improper integralshttps://zbmath.org/1541.440052024-09-27T17:47:02.548271Z"Gordon, Russell A."https://zbmath.org/authors/?q=ai:gordon.russell-a"Stewart, Séan M."https://zbmath.org/authors/?q=ai:stewart.sean-markSummary: We explain and apply a recently developed method for evaluating improper integrals of the form \(\int^\infty_0 f(t) dt\) using Laplace transforms. A number of examples are provided to illustrate the method, along with some results that streamline the computations. We show how the method can be used to readily determine values for entire classes of certain integrals which, using other more familiar methods, are difficult to find. We also indicate how the method can determine the values of integrals for which other methods fail.On a system of \(q\)-modified Laplace transform and its applicationshttps://zbmath.org/1541.440062024-09-27T17:47:02.548271Z"Kilicman, Adem"https://zbmath.org/authors/?q=ai:kilicman.adem"Sinha, Arvind Kumar"https://zbmath.org/authors/?q=ai:sinha.arvind-kumar"Panda, Srikumar"https://zbmath.org/authors/?q=ai:panda.srikumarSummary: We introduce $q$-modified Laplace transform and establish theoretical results. We also give some applications of $q$-modified Laplace transform for solving homogeneous and non-homogeneous Mboctara partial differential equations with initial and boundary values problems to show its effectiveness and performance of the proposed transform.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}Qualitative uncertainty principles for the windowed Opdam-Cherednik transform on weighted modulation spaceshttps://zbmath.org/1541.440072024-09-27T17:47:02.548271Z"Mondal, Shyam Swarup"https://zbmath.org/authors/?q=ai:mondal.shyam-swarup"Poria, Anirudha"https://zbmath.org/authors/?q=ai:poria.anirudhaSummary: The aim of this paper is to establish a few qualitative uncertainty principles for the windowed Opdam-Cherednik transform on weighted modulation spaces associated with this transform. In particular, we obtain Cowling-Price's, Hardy's and Morgan's uncertainty principles for this transform on weighted modulation spaces. The proofs of the results are based on versions of the Phragmén-Lindelöf type result for several complex variables on weighted modulation spaces and the properties of the Gaussian kernel associated with the Jacobi-Cherednik operator.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Fareeha transform: a new generalized Laplace transformhttps://zbmath.org/1541.440082024-09-27T17:47:02.548271Z"Sami Khan, Fareeha"https://zbmath.org/authors/?q=ai:khan.fareeha-sami"Khalid, M."https://zbmath.org/authors/?q=ai:khalid.muhammad-hamza|khalid.mohd-aman|khalid.mudassar|khalid.maira|khalid.mohsin|khalid.marzuki|khalid.muhammad-zeeshan|khalid.memoona|khalid.madiha|khalid.mohd-masood|khalid.maryam|khalid.madeeha|khalid.mohammad|khalid.muhammad-saif-ullah|khalid.mahmood|khalid.muhammad-usmanSummary: In this paper, a new integral-type transformation is being introduced, which is the generalization of Laplace and Fourier transform. This integral transform is named as ``Fareeha transform''. It is successfully applied on ordinary differential equation to show its efficacy and simplicity. Also, the possible applications of Fareeha transform in control theory, electric circuits, and data compression have been discussed in detail.
{{\copyright} 2023 John Wiley \& Sons Ltd.}Non-classical Tauberian and abelian type criteria for the moment problemhttps://zbmath.org/1541.440092024-09-27T17:47:02.548271Z"Patie, P."https://zbmath.org/authors/?q=ai:patie.pierre"Vaidyanathan, A."https://zbmath.org/authors/?q=ai:vaidyanathan.adityaSummary: The aim of this paper is to provide some new criteria for the determinacy problem of the Stieltjes moment problem. We first give a Tauberian type criterion for moment indeterminacy that is expressed purely in terms of the asymptotic behavior of the moment sequence (and its extension to imaginary lines). Under an additional assumption this provides a converse to the classical Carleman's criterion, thus yielding an equivalent condition for moment determinacy. We also provide a criterion for moment determinacy that only involves the large asymptotic behavior of the distribution (or of the density if it exists), which can be thought of as an abelian counterpart to the previous Tauberian type result. This latter criterion generalizes Hardy's condition for determinacy, and under some further assumptions yields a converse to the Pedersen's refinement of the celebrated Krein's theorem. The proofs utilize non-classical Tauberian results for moment sequences that are analogues to the ones developed in [\textit{P.~D. Feigin} and \textit{E.~Yashchin}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 35--48 (1983; Zbl 0506.60008)] and [\textit{A.~A. Balkema} et al., J. Lond. Math. Soc., II. Ser. 51, No.~2, 383--400 (1995; Zbl 0821.60025)] for the bi-lateral Laplace transforms in the context of asymptotically parabolic functions. We illustrate our results by studying the time-dependent moment problem for the law of log-Lévy processes viewed as a generalization of the log-normal distribution. Along the way, we derive the large asymptotic behavior of the density of spectrally-negative Lévy processes having a Gaussian component, which may be of independent interest.
{{\copyright} 2022 Wiley-VCH GmbH.}\(q\)-Abel integral equation with \(n\) termshttps://zbmath.org/1541.450032024-09-27T17:47:02.548271Z"Younus, Awais"https://zbmath.org/authors/?q=ai:younus.awaisSummary: In this paper solution of Abel's integral equation with \(n\) terms is discussed by using Laplace transform. Furthermore we derive the results about the solution of \(q\)-Abel integral equation as well as \(q\)-Abel integral equations with \(n\)-terms using different techniques.On Buschman-Erdelyi and Mehler-Fock transforms related to the group \(SO_0(3,1)\)https://zbmath.org/1541.810092024-09-27T17:47:02.548271Z"Shilin, Il'ya Anatol'evich"https://zbmath.org/authors/?q=ai:shilin.ilya-anatolevichSummary: By using a functional defined on a pair of the assorted represention spaces of the connected subgroup of the proper Lorentz group, a formula for the Buschman-Erdelyi transform of the Legendre function (up to a factor) is derived. Also a formula for the Mehler-Fock transform of the Legendre function of an inverse argument is obtained. Moreover, a generalization of one known formula for the Mehler-Fock transform is derived.A contribution to the mathematical theory of diffraction: a note on double Fourier integralshttps://zbmath.org/1541.810372024-09-27T17:47:02.548271Z"Assier, R. C."https://zbmath.org/authors/?q=ai:assier.raphael-c"Shanin, A. V."https://zbmath.org/authors/?q=ai:shanin.andrey-v"Korolkov, A. I."https://zbmath.org/authors/?q=ai:korolkov.andrey-iSummary: We consider a large class of physical fields \(u\) written as double inverse Fourier transforms of some functions \(F\) of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of \(u\). In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve our aim, we only need to study a finite number of real points in the Fourier space: the contributing points. This result is called the locality principle. We provide an extensive set of results allowing one to decide whether a point is contributing or not. Moreover, to each contributing point, we associate an explicit closed-form far-field asymptotic component of \(u\). We conclude the article by validating this theory against full numerical computations for two specific examples.Generalised graph Laplacians and canonical Feynman integrals with kinematicshttps://zbmath.org/1541.810642024-09-27T17:47:02.548271Z"Brown, Francis"https://zbmath.org/authors/?q=ai:brown.francis-c-sSummary: To any graph with external half-edges and internal masses, we associate canonical integrals which depend non-trivially on particle masses and momenta, and are always finite. They are generalised Feynman integrals which satisfy graphical relations obtained from contracting edges in graphs, and a coproduct involving both ultra-violet and infra-red subgraphs. Their integrands are defined by evaluating bi-invariant forms, which represent stable classes in the cohomology of the general linear group, on a generalised graph Laplacian matrix which depends on the external kinematics of a graph.