Integral Representation Research Articles

Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations

Abstract This study investigates a number of integrals possessing products of different indices of the univariate Lommel function, s μ , ν ( a ) {s}_{\mu ,\nu }\left(a) , with various elementary and special functions. As a consequence, connections between the function and Chebyshev polynomials of the first kind are established.

Best approximation of Hartley-Bessel multiplier operators on weighted Sobolev spaces

The main goal of this paper is to introduce the Hartley-Bessel \(L^2_\alpha\)-multiplier operators and to give for them some new results as Plancherel’s, Calderon’s reproducing formulas and Heisenberg’s, Donoho-Stark’s uncertainty principles. Next, using the theory of reproducing kernels we give best approximation and an integral representation of the extremal functions related to these operators on weighted Sobolev spaces.

On the variational nature of the Anzellotti pairing

Abstract In this paper, we prove that the Anzellotti pairing can be regarded as a relaxed functional with respect to the weak⋆ convergence in the space BV \mathrm{BV} of functions of bounded variation. The crucial tool is a preliminary integral representation of this pairing by means of suitable cylindrical averages.

All-electron molecular tunnel ionization based on the weak-field asymptotic theory in the integral representation

All-electron molecular tunnel ionization based on the weak-field asymptotic theory in the integral representation

Space-Time Fractional Bessel Diffusion Equation

This study introduces a novel space-time fractional diffusion equation with the capacity to model a diverse spectrum of diffusion processes. The equation incorporates Caputo-type time derivatives with an arbitrary order and introduces a spatial-fractional operator known as the fractional Bessel operator. The fundamental solution of this equation is obtained utilizing harmonic analysis on the Kingman Bessel hypergroup. We present both Mellin-Barne integral and series representations for the fundamental solution and establish its positivity through the application of the Bernstein characterization of completely monotone functions and the positive definiteness property with respect to the Bessel operator.

Integral representation of balayage on locally compact spaces and its application

Integral representation of balayage on locally compact spaces and its application

New Modified Gamma and Beta Functions with an Application

This note presents a novel set of modified Gamma and Beta k functions. The authors introduce these new functions along with their first and second summation relations, functional relations, symmetry relations, Mellin transforms, and integral representations that incorporate algebraic powers and Mittag-Leffler functions, as well as those involving trigonometric terms, powers, and Mittag-Leffler functions. In contrast to other studies that generalize Gamma and Beta functions, this note highlights a statistical application, demonstrating the practical utility of the proposed functions across various fields.

The spectral ζ-function for quasi-regular Sturm–Liouville operators

In this work, we analyze the spectral ζ-function associated with the self-adjoint extensions, TA,B, of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension TA,B. The characteristic function is then employed to construct a contour integral representation for the spectral ζ-function of TA,B. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the ζ-function to a larger region of the complex plane. We also present a method for computing the value of the spectral ζ-function of TA,B at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral ζ-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s.

A study of two new integral representations for Gauss’s hypergeometric function with applications

A study of two new integral representations for Gauss’s hypergeometric function with applications

Existence of Solutions for a Class of Volterra Integral Equations on Time Scales

ABSTRACTIn this paper, we investigate a class of Volterra integral equations for existence of global classical solutions. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results, we propose a new approach based upon recent theoretical results. More precisely, we give a suitable integral representation of the solutions of the considered Volterra integral equation. Then, we construct two operators for which any fixed point of their sum is a solution of the considered Volterra integral equation.

Continuity of entropies via integral representations

Continuity of entropies via integral representations

Exact results on the number of gravitons radiated during binary inspiral

We derive an exact formula F(e) which provides a concrete estimate for the total number and angular momentum of gravitons emitted during the nonrelativistic inspiral of two black holes.We show that the function F(e) is a slowly growing monotonic function of the eccentricity 0 ≤ e ≤ 1and F(1) = 1.0128 ⋯.We confirm and extend the results obtained by Page for the function F(e). We also get an exact result for the ratioν (ei ) = 2ħN(Li , ei )/Li where the numerator 2ħN(Li , ei ) is the sum of the spin angular momentum magnitudes of the gravitons emitted and N(Li , ei ) is the total number of gravitons emitted in the gravitational waves during nonrelativistic inspiral from an initial eccentricity ei down to a final eccentricity e = 0 and the denominator Li is the magnitude of the initial orbital angular momentum.If the orbit starts off with unit eccentricity ei = 1,we get the value ν(1) = 1.002 268 666 2 ± 10-10 which confirms the Page's conjecture that the true value of ν(1) will lie between 1.001⋯ and 1.003⋯.We also show that the formula F(e) for gravitons emitted, originally expressed as an infinite series, can be representedby a single function through an integral representation.

Nonclassical dynamical behavior of solutions of partial differential-difference equations

<p>For partial differential-difference equations, a review of results regarding the relation between the type of the equation and dynamical properties of its solutions is provided. This includes the case of elliptic equations with timelike independent variables: Their solutions acquire dynamical properties (more exactly, behave as solutions of parabolic equations). The following approach to classify differential-difference equations into types, based on the property of differential-difference operators to be Fourier multipliers is applied in the following manner: An operator is treated to be elliptic if the real part of its symbol is positive, while the parabolic and hyperbolic types are defined correspondingly. It is shown that the proposed approach (being a natural extension of the classical notion of the ellipticity) is reasonable. On the other hand, fundamental novelties (compared with the classical theory of partial differential equations) occur as well. We provide conditions which guarantee the following results. For the half-space (half-plane) Dirichlet problem for elliptic equations, integral representations (of the Poisson type) of solutions are constructed, which are infinitely differentiable outside the boundary hyperplane (plane), and the asymptotic closeness of solutions (as the timelike independent variable unboundedly increases) takes place. For the Cauchy problem for parabolic equations, the same is valid, but we deal with the classical time instead of the timelike independent variable. For hyperbolic equations, multiparameter families of infinitely smooth global solutions are constructed. The said (sufficient) conditions restrict the sign of the real part of the symbol for the differential-difference operator with respect to spatial or spacelike independent variables. In a number of special cases, they might be weakened such that symbols with sign-changing real parts are admitted. The objective of the study is to observe the current stage of the classification issues for partial differential-difference equations in terms of the aforementioned approach: The ellipticity of a Fourier multiplier is defined by means of the sign (of there real part) of its symbol. Since both differential and translation operators are Fourier multipliers, methods of Fourier analysis are applicable in this study: We apply the Fourier transformation to the original partial differential-difference equation, solve the obtained ordinary differential equation, and apply the inverse Fourier transformation to the obtained solution. The main contribution obtained within this study is an efficient (workable) type concept for the fundamentally new extension of the class of partial differential equations, which is the class of partial differential-difference equations.</p>

Generating the true S-poly-Bargmann spaces by sliced creation operators and associated integral transforms

We deal with an operational realization of the so-called n-th true S-poly-Bargmann space. We show that it can be realized as the range of the iterated sliced creation operator, and that it is close to the spectral analysis of a sliced magnetic Laplacian for which we provide a geometrical realization à la Hodge. Some integral representations of the considered spaces are also investigated.

The Analysis of Pentagonal Fuzzy Numbers in a Neutrosophic Fuzzy Inventory Management Modelling with Minimal Insufficient Supply Required and Fuzzy Consumption

The fuzzy stock administration demonstrates displayed in this work employments neutrosophic set hypothesis, pentagonal fuzzy numbers, and the Graded mean Integration Representation (GMIR) strategy for defuzzification. Request rates, arrange amounts, utilization rates, holding costs, setup costs, and deficiency costs are all spoken to as fuzzy parameters within the demonstrate to account for the inborn instability and vacillation. To reduce by and large costs, the whole cost work is calculated, taking setup, holding, and shortage costs into consideration. In arrange to speak to the combined impacts of a few fetched components, the overall taken a toll work is rearranged and the ideal arrange amount is built up beneath fuzzy conditions utilizing pentagonal fuzzy parameters. The demonstrate is assessed beneath different degrees of instability through a case-based investigation, advertising an exhaustive system for making choices on stock administration in equivocal and dubious circumstances. The results appear how versatile and capable the show is for improving fetched advancement and stock control.

Non-Carathéodory analytic functions with respect to symmetric points

ABSTRACT The authors introduce new classes of analytic function with respect ( η , τ ) -symmetric points subordinate to a domain that is not Carathéodory. To use the existing infrastructure or framework, usually, the study of analytic function have been limited to a differential characterization subordinate to functions which are Carathéodory. Here, we try to obtain various interesting properties of functions which are not Carathéodory. Integral representation, interesting conditions for starlikeness and inclusion relations for functions in these classes are obtained.

A generalized Ramanujan master theorem and integral representations of meromorphic functions

A generalized Ramanujan master theorem and integral representations of meromorphic functions

A Fuzzy Inventory Model for Imperfect Process, Obsolescence and its Greenhouse Gas Emissions under Vendor Managed Inventory – Consignment Stock (VMI-CS) Policy

Objectives: To discuss a green inventory management system with greenhouse gas emissions, obsolescence, imperfect process, and Vendor Managed Inventory – Consignment Stock (VMI – CS) policy in a fuzzy nature by employing hexagonal fuzzy numbers to tackle uncertain situations. Methods: To find the optimum solution, we use two methods such as centroid method and the graded mean integration representation method for hexagonal fuzzy numbers. We use different sets of data for Marchi et al’s model 1 and for the proposed model with obsolescence to verify the results. Here two supply chain models are proposed, where the crisp model is first discussed with total cost and optimum order quantity. As a second part, the fuzzy model is formulated with fuzzy total cost and fuzzy optimal order quantity. Findings: The results of our discussion represent that the solution obtained from fuzzy by using two methods is slightly the same as the solution found from the crisp model. But since five parameters with ambiguous nature are solved under fuzzy by considering them in the form of a hexagonal fuzzy number, uncertainty is faced here. To show the applicability of the proposed model, numerical examples are given. Novelty: Implementation of obsolescence cost and its emissions in a green inventory system and the process of considering and solving parameters under fuzzy is a unique and important feature of our study. It helps sectors to know the effects of obsolete inventory, the need to include it, and also to rectify vagueness in their specified data. Hence this study is novel in the sense it discusses the above criteria under two fuzzy techniques. 2020 Mathematics Subject Classification: 90B05 Keywords: Hexagonal fuzzy numbers, Centroid method, Graded mean integration representation method, Obsolescence cost, Greenhouse gas emissions

On the solution of the interpretation tomography problem within the framework of the linear integral representation method and discrete gravity and magnetic potential theories

A new of principle approach to solving inverse problems of geophysics on the background of the linear integral representation method and discrete gravity and magnetic potential theories is proposed. This approach makes it possible to restore the continuously distributed masses with relatively high accuracy from the heterogeneous data in some network points.

Characterization of quasi-parabolic operators and their integral representation

Characterization of quasi-parabolic operators and their integral representation