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The transformations of supersymmetric quantum mechanics are discussed within a formalism that employs a six-parameter function, from which the superpotential and the supersymmetric partner potentials V − (x) and V + (x) are constructed in a general form. By specific choice of the parameters, V − (x) and V + (x) are matched with the general form of PI class potentials and their rationally extended versions. The choice of the parameters also determines which of the four possible SUSY transformations T i , i = 1 , ... 4 is in effect. After this general discussion, the formulae are specified to the three members of this potential class, the Scarf I, Scarf II and generalized Pöschl–Teller potentials. Due to the different domains of definition and their consequences on the boundary conditions, the results turn out to be rather diverse for the three potentials, while the mathematical formalism and the network of the potentials interconnected by the SUSYQM transformations still remains common to a large extent. The general framework allows a unified and consistent interpretation of earlier isolated findings. It also helps to connect the results to further potential classes and to place them into a more general context within the zoo of exactly solvable potentials. [ABSTRACT FROM AUTHOR]

This note introduces a new class of preinvexity called (h 1 , h 2) -Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in order to verify all the developed results. In addition, we discussed some applications related to the trapezoidal formula, probability density functions, special functions and special means. Lastly, we discussed the importance of order relations and left two open problems for future research. As an additional benefit, we believe that the present work can provide a strong catalyst for enhancing similar existing literature. [ABSTRACT FROM AUTHOR]

The fractional derivatives have been widely applied in many fields and has attracted widespread attention. This paper extracts a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKe) with the local fractional derivative (LFD) for the first time. Two special functions, namely, the LT δ (Ξ δ) and LC δ (Ξ δ) functions that are derived on the basis of the Mittag-Leffler function (MLF) defined on the Cantor set (CS), are employed to construct the auxiliary trial function to look into the exact solutions (ESs). Aided by Yang's non-differentiable (ND) transformation, six groups of the ND ESs are found. The ND ESs on the CS for δ = ln 2 / ln 3 are depicted graphically. Additionally, as a comparison, the ESs of the classic (3+1)-dimensional MZKe for δ = 1 are also illustrated. The outcomes reveal that the derived method is powerful and effective, and can be used to deal with the other local fractional PDEs. [ABSTRACT FROM AUTHOR]

We consider the stochastic non-autonomous Kuramoto-Sivashinsky equation with multiplicative white noise and colored coefficients. Due to the anti-dissipative term in the equation, we restrict the state space on the Lebesgue space of odd functions and then obtain a pullback random attractor $ \mathcal{A} $ in the odd Lebesgue space, where a special bridge function plays an crucial role in a priori estimate. Moreover, we prove the continuity in expectation for the set-valued mapping $ (t ,s) \to \mathcal{A}(t,\theta_s\omega) $ with respect to the Hausdorff metric and establish the residual dense continuity in the pathwise sense. In the application, we verify four conditions including the joint continuity of the non-autonomous random dynamical system, the union closedness of a tempered universe, local compactness and boundedness in expectation for the random attractor. [ABSTRACT FROM AUTHOR]

The primary goal of current study is to formulate a novel mathematical model employing the Prabhakar fractional operator, aimed at examining the dynamic behaviour of Jeffrey fluid flow and heat transfer phenomena under ramped wall temperature conditions. Our investigation entails a meticulous investigation of magnetohydrodynamic (MHD) natural convective flow within Jeffrey fluids, where we derive precise analytical solutions utilizing the Prabhakar fractional derivative, characterized by a non-singular type kernel. Furthermore, our study integrates fundamental principles such as Fick’s and Fourier’s laws into the model, leveraging a multi-parameter Mittag-Leffler kernel associated with the fractional operator. We delve into the intricacies of fluid flow near an infinitely vertical plate, considering characteristics such as velocity u0. To address the complexities of the problem, we express it in context the of partial differential equations along side generalized boundary conditions, employing a set of appropriate variables to transform these equations into a dimensionless form. Utilizing Laplace transform, we analyse the equations of fractional system, presenting outcomes both in the form of series and through specialized functions. We systematically explore the influence of key parameters α, Pr, β, Gm, Sc, γ, Gr on fluid flow dynamics, unveiling significant insights. Our comparative analysis reveals the superior performance of the Prabhakar-like non-integer approach over existing operators, substantiated by graphical representations of the results. Additionally, we extend our investigation to various limiting cases, including Newtonian fluids and second-grade, in both fractionalized and classical forms, thus highlighting the versatility and applicability of our proposed model within fluid dynamics research.

The aim of this paper is to derive formulae for the generating functions of the Bernstein type polynomials. We give a PDE equation for this generating function. By using this equation, we give recurrence relations for the Bernstein polynomials. Using generating functions, we also derive some identities including a symmetry property for the Bernstein type polynomials. We give some relations among the Bernstein type polynomials, Bernoulli numbers, Stirling numbers, Dahee numbers, the Legendre polynomials, and the coefficients of the classical superoscillatory function associated with the weak measurements. We introduce some integral formulae for these polynomials. By using these integral formulae, we derive some new combinatorial sums involving the Bernoulli numbers and the combinatorial numbers. Moreover, we define Bezier type curves in terms of these polynomials.

In this paper, we introduce the concept of generalized Fourier series, generated by the p-trigonometric functions, namely cosp and sinp, recently introduced related to the generalized complex numbers systems. The aim of this study is to represent a periodic signal as a sum of p-sine and p-cosine functions. In order to achieve this, we first present the integrals of the product of the same or different family of p-trigonometric functions over the full period of these functions to understand the orthogonality properties. Next, we use these integrals to derive the coefficients of the generalized p-Fourier series along with a few examples. The generalized Fourier series can be used to expand an arbitrary forcing function in the solution of a non-homogeneous linear ordinary differential equation (ODE) with constant coefficients.

Various microswimmers move along circles rather than straight lines due to their swimming mechanisms, body shapes, or hydrodynamic effects. In this paper, we adopt the concepts of stochastic thermodynamics to analyze circle swimmers confined to a two-dimensional plane and study the trade-off relations between various physical quantities, such as precision, energy cost, and rotational speed. Based on these findings, we predict principles and strategies for designing microswimmers of special optimized functions under limited energy resource conditions, which will bring new experimental inspiration for designing smart motors. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of a scalar bosonic field within the backdrop of Eddington-inspired Born–Infeld gravity with a global monopole configuration. Additionally, we consider the presence of both scalar and vector potentials. To introduce the scalar potential, we perform a transformation on the mass term, replacing M 2 with (M + S) 2 . For the vector potential, we apply a minimal coupling scheme, transforming p μ → D μ ≡ (p μ − e A μ) within the relativistic Klein–Gordon equation. We opt for a harmonic oscillator scenario where the scalar and vector potentials are equal, i.e. S = V ∝ x 2 . Through this setup, we analytically solve the wave equation using the confluent Heun equation form. Subsequently, we delve into the quantum system in the absence of any potential, determining the permissible values for bound-state relativistic energy levels and the scalar field's wave function. Our analysis reveals that the energy levels and wave function are influenced by various parameters present in the eigenvalue expression. [ABSTRACT FROM AUTHOR]

In this paper we introduce generalized phase retrievable frames or simply g-phase retrievable frames in real or complex n-dimensional Hilbert space H n , which include ordinary phase retrievable frames in H n . Specifically, a g-phase retrievable frame is a λ-phase retrievable frame, where λ is an special function, which is called phase coefficient function. Using the λ-phase retrievable frames, every vector in H n can be reconstructed up to a constant phase coefficient factor from the action of λ on the its frame coefficients, which can be bounded for bounded phase coefficient functions. We obtain some equivalent conditions to g-phase retrievable frames and we study on stability of special g-phase retrievable frames under small perturbation of their vectors and under small perturbation of their phase coefficient functions. [ABSTRACT FROM AUTHOR]

In the survey Kiryakova: "A Guide to Special Functions in Fractional Calculus" (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions p Ψ q and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all "Classical Special Functions" of the class of the Meijer's G- and p F q -functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another "fractionalized" variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an "Extended Class of SF of FC". This includes the I-functions of Rathie and, in particular, the H ¯ -functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and H ¯ -functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of Γ -functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these "generalized integrations" can be extended as kinds of generalized operators of fractional integration, and are also compositions of "Le Roy type" Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and H ¯ -functions still needs clarification of some details, we consider this work as a "Discussion Survey" and also provide a list of open problems. [ABSTRACT FROM AUTHOR]

In this work, we investigate the quantum motions of nonrelativistic particles interacting with a potential in the presence of the Aharonov–Bohm (AB) flux field within a topological defect geometry, for example space–time with a distortion of a vertical line into a vertical spiral. We begin by deriving the radial Schrödinger wave equation, incorporating an anharmonic oscillator potential, which is a superposition of a harmonic oscillator and an inverse square potential, along with a constant term. The eigenvalue solution is obtained through the confluent Heun equation focusing on the ground state energy level and the radial wave function for the radial mode n = 1 as an example and analyze the results. Subsequently, we use these results to molecular potential models, considering pseudoharmonic and shifted pseudoharmonic potentials. The derived eigenvalue solutions provide insights into the behavior of particles within these potentials. Expanding our exploration, we study the quantum system featuring only an inverse square potential in the presence of the quantum flux field in the same geometry background. Employing the same procedure, we determine the ground state energy level and the radial wave function. Notably, our findings reveal that the eigenvalue solutions are significantly influenced by the topological defect characterized by the parameter β , and the quantum flux field Φ AB . This influence manifests as a shift in the energy spectrum, drawing parallels to the gravitational analog of the AB effect. [ABSTRACT FROM AUTHOR]

Fuzzy graphs as labeled graphs (fuzzy vertex labeling and fuzzy edge labeling) have many applications in real life such as complex networks, coding theory, medical sciences, communication networks, and management sciences. Also, triangular norms as a special class of functions, have many applications in fuzzy set theory, probability and statistics, and other areas. This paper considers the notations of an inverse fuzzy graph and triangular norms to introduce the new type of graphs as valued-inverse Dombi fuzzy graphs. The valued-inverse Dombi fuzzy graphs are a generalization of inverse fuzzy graphs and are dual to Dombi fuzzy graphs. For any given greater than or equal to one real number, we construct a type of Dombi inverse fuzzy graph and investigate some conditions such that the product and union of Dombi inverse fuzzy graphs be a Dombi inverse fuzzy graph. [ABSTRACT FROM AUTHOR]

The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the nonlinear parameters. The matrix pencil method, which reformulates the problem statement into a generalized eigenvalue problem for the nonlinear parameters and a structured linear system for the linear parameters, is generally considered as the more stable method to solve the problem computationally. In Section 2 the matrix pencil associated with the classical complex exponential fitting or sparse interpolation problem is summarized and the concepts of dilation and translation are introduced to obtain matrix pencils at different scales. Exponential analysis was earlier generalized to the use of several polynomial basis functions and some operator eigenfunctions. However, in most generalizations a computational scheme in terms of an eigenvalue problem is lacking. In the subsequent Sections 3–6 the matrix pencil formulation, including the dilation and translation paradigm, is generalized to more functions. Each of these periodic, polynomial or special function classes needs a tailored approach, where optimal use is made of the properties of the parameterized elementary or special function used in the sparse interpolation problem under consideration. With each generalization a structured linear matrix pencil is associated, immediately leading to a computational scheme for the nonlinear and linear parameters, respectively from a generalized eigenvalue problem and one or more structured linear systems. Finally, in Section 7 we illustrate the new methods. [ABSTRACT FROM AUTHOR]

Viazovska's solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska's work is no exception. In this article, we'll examine how it has led to new interpolation theorems in Fourier analysis, specifically a theorem of Radchenko and Viazovska. [ABSTRACT FROM AUTHOR]

New (linear and nonlinear) integral-functional equations, the solutions of which are a truncated spectral characteristic of a fourth order coherence function of a laser beam propagating in a turbulent medium are derived. The linear integral-functional includes an auxiliary function W on which the solution of the given equation does not depend (it is invariant with respect to the choice of W). For the first time formally strict and fairly simple representations are obtained for the fourth-order coherence function and its truncated spectral characteristics. In searching for representations of these functions, in particular, we have used the invariance of the solutions of the second order differential equation in partial derivatives for the four-point coherence function and a new linear integral-functional equation with respect to the choice of the function W. It is shown that by a special choice of this function and the parameters determining the choice of the position of the "observation" points and the transverse cross sections of the laser beam, it is possible to obtain different exact, asymptotic, and semianalytic representations for the truncated spectral characteristics, integral characteristics of the fourth-order coherence function and the function itself. Here semianalytic representations are taken to be ones that contain information on the unknown quantities partially and in inexplicit form, but which simultaneously make it possible to find them in analytic (or numerical) form by direct use of such (e.g. iterative) constructive procedures. In particular, a semianalytic representation is the recurrence formula obtained with which an analytic expression and numerical values for the above functions can be found effectively for a laser beam propagating in a turbulent medium. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of integral geometry in three-dimensional space over a family of cones with a weight function of a special form. The theorem of uniqueness of the solution is proved, and several examples are considered in order to obtain more precise solutions to the problem of integral geometry. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of possibly non-convex and non-smooth optimization problems arising in many contemporary applications such as machine learning, variable selection and image processing. To solve this class of problems, we propose a proximal gradient method with extrapolation and line search (PGels). This method is developed based on a special potential function and successfully incorporates both extrapolation and non-monotone line search, which are two simple and efficient acceleration techniques for the proximal gradient method. Thanks to the non-monotone line search, this method allows more flexibility in choosing the extrapolation parameters and updates them adaptively at each iteration if a certain criterion is not satisfied. Moreover, with proper choices of parameters, our PGels reduces to many existing algorithms. We also show that, under some mild conditions, our line search criterion is well defined and any cluster point of the sequence generated by the PGels is a stationary point of our problem. In addition, by making assumptions on the Kurdyka–Łojasiewicz exponent of the objective in our problem, we further analyze the local convergence rate of two special cases of the PGels, including the widely used non-monotone proximal gradient method as one case. Finally, we conduct some preliminary numerical experiments for solving the ℓ 1 regularized logistic regression problem and the ℓ 1 - 2 regularized least squares problem. The obtained numerical results show the promising performance of the PGels and validate the potential advantage of combining two acceleration techniques. [ABSTRACT FROM AUTHOR]

This paper presents research results in the field of industrial die forging, mostly related to the use of advanced measuring techniques and tools, numerical simulations, and other IT tools and methods for a geometrical analysis of the forged items as well as detection of forging flaws and their prevention, and optimization of the hot-forging processes. The results of the conducted investigations were divided into three main areas. The first area refers to the application of, e.g., optical scanners and programs related to their operation, data analysis, including the construction of virtual gauges, measurements of selected geometrical features of both the manufactured forgings and their physical and virtual models, as well as an analysis of the durability of the forging tools based on the proprietary reverse scanning method. The second area presents the results of measurements and analyses performed with the use of finite element modeling and by means of some special functions in the calculation packages, such as contact, flow lines, trap, or fold, for the detection of forging defects and an analysis of the force parameters. In turn, the third area presents a combination of different methods of measurement and analysis, both FEM and scanning, as well as other IT methods (physical modeling, image analysis, etc.) for the analysis of the geometry and defects of the forgings. The presented results point to the great potential of these types of tools and techniques in forging industry applications as they significantly shorten the time and increase the accuracy of the measurement, as well as providing a lot of valuable information, physical variables, and technological parameters that are difficult or impossible to determine either analytically or through experimental means. The use and development of these techniques and methods are fully justified, both in the aspect of science and the increased effectiveness and efficiency of production. [ABSTRACT FROM AUTHOR]

In this work, the geometric nature of solutions to two second-order differential equations, z y ′ ′ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ′ ′ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) , is studied. Here, a (z) , b (z) , and d (z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F (z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1 + z . We construct several examples by the judicious choice of a (z) , b (z) , and d (z) . The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion. [ABSTRACT FROM AUTHOR]

In this work the matrix exponential function is solved analytically for the special orthogonal groups S O (n) up to n = 9 . The number of occurring k-th matrix powers gets limited to 0 ≤ k ≤ n − 1 by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of S O (3) , a quadratic equation needs to be solved for n = 4 , 5 , a cubic equation for n = 6 , 7 , and a quartic equation for n = 8 , 9 . As an interesting subgroup of S O (7) , the exceptional Lie group G 2 of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ = 1 . The traces of the S O (n) -matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities. [ABSTRACT FROM AUTHOR]

Recently, the realm related to Euler's Beta function has played a significant role in the development of special function theory. In this study, a new extension of the special function known as Euler's Beta function with respect to the Mittag-Leffler-Kummer function is introduced. Another formula of this new Beta function in terms of the Fox-Wright function is also presented. Numerous analytical properties of this new special function, such as several functional and summation relations, Mellin transforms, integral representations, and several derivative formulas, are studied. Furthermore, significant statistical implementations of Euler's Beta distribution are also discussed. [ABSTRACT FROM AUTHOR]

Animal phenotypic traits are utilised in a variety of studies. Often the traits are measured from images. The processing of a large number of images can be challenging; nevertheless, image analytical applications, based on neural networks, can be an effective tool in automatic trait collection.Our aim was to develop a stand‐alone application to effectively segment an arthropod from an image and to recognise individual body parts: namely, head, thorax (or prosoma), abdomen and four pairs of appendages. It is based on convolutional neural network with U‐Net architecture trained on more than a thousand images showing dorsal views of arthropods (mainly of wingless insects and spiders). The segmentation model gave very good results, with the automatically generated segmentation masks usually requiring only slight manual adjustments.The application, named MAPHIS, can further (1) organise and preprocess the images; (2) adjust segmentation masks using a simple graphical editor; and (3) calculate various size, shape, colouration and pattern measures for each body part organised in a hierarchical manner. In addition, a special plug‐in function can align body profiles of selected individuals to match a median profile and enable comparison among groups. The usability of the application is shown in three practical examples.The application can be used in a variety of fields where measures of phenotypic diversity are required, such as taxonomy, ecology and evolution (e.g. mimetic similarity). Currently, the application is limited to arthropods, but it can be easily extended to other animal taxa. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive the expansion formulae for the incomplete I-function. Furthermore, their special cases are illustrated in terms of various types of special functions (incomplete I-function, incomplete H-function, and incomplete H-function) that are common in nature and very useful for further analysis. [ABSTRACT FROM AUTHOR]

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How to coordinate the tourism industry with the ecological environment has become an important issue. However, there is a lack of attention to special ecological function areas, such as the Beijing-Hangzhou Grand Canal Basin. This study evaluated the degree of coupling coordination and distinguished obstacle factors between two systems, the tourism industry and the ecological environment in the Beijing-Hangzhou Grand Canal Basin from 2008 to 2017. The development level index of the tourism industry showed "sustained growth-slight decline-rapid growth." Jiangsu showed the highest annual growth rate (17.26%) for the development level index of the ecological environment. The comprehensive evaluation index changed in stages, showing a trend of large overall growth and small partial decline. The coupling coordination degree improved rapidly on the whole, developing from close to incoordination in 2008 to good coordination and quality coordination in 2017. In most cases, the provinces demonstrated the ecological lag type during this period. The greatest increase in obstacle degree was in the ecological environment pressure in the control layer. Generally, the main obstacles hindering the coupled and coordinated development of these two systems were the volume of household garbage clearance, wastewater emissions, A-level scenic spots, total consumption of domestic tourists and the number of inbound tourists. This study clearly reveals the evolution, degree and types of coupling coordination in the Beijing-Hangzhou Grand Canal Basin of the two systems, which contributes to healthy basin development. [ABSTRACT FROM AUTHOR]

In this paper we deduced Turán-type inequalities for nthderivative of k-gamma function, k-gauss hypergeometric functions, kconfluent hypergeometric functions, and k-Appell series F1,k by using different generalizations of Cauchy-Bunyakovsky-Schwarz inequalities with parameter k > 0. Some special cases are also derived. [ABSTRACT FROM AUTHOR]

A lower semicontinuity result and a relaxation formula for free discontinuity functionals with non-standard growth in the bulk energy are provided. Our analysis is based on a non-trivial adaptation of the blow-up (Ambrosio in Nonlinear Anal 23:405–425, 1994) and of the global method for relaxation (Bouchitté in Arch Ration Mech Anal 165:187–242, 2002) to the setting of generalized special function of bounded variation with Orlicz growth. Key tools developed in this paper are an integral representation result and a Poincaré inequality under non-standard growth. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three types of generalized special functions: beta, Gauss hypergeometric, and confiuent hypergeometric, all involving two generalized M-series at their kernels. We then give several properties of these functions, such as integral representations, functional relations, summation relations, derivative formulas, transformation formulas, and double Laplace transforms. Furthermore, we obtain solutions of fractional partial differential equations involving these new generalized special functions and then we present graphs of the approximate behavior of the solutions. Also, we introduce a new generalized beta distribution and incomplete beta function. Finally, we establish relationships between the new generalized special functions and other generalized special functions found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, graphs of entire functions are considered as parabolic manifolds in the sense of the existence of a special exhaustion function. On the graph it is defined polynomials, as well as extremal functions of the Siciak and Green type. On the analytic graphs a significant equality between the extremal functions is proved. [ABSTRACT FROM AUTHOR]

The schemata for applications of the integral transforms of mathematical physics to recurrence relations, differential, integral, integro-differential equations and in the theory of special functions has been developed. The article aims to introduce and present operational representations for a new class of extended Hermite-Apostol type Frobenius-Genocchi polynomials via integral transforms. The recurrence relations and some identities involving these polynomials are established. The article concludes by establishing a determinant form and quasi-monomial properties for the Hermite-Apostol type Frobenius-Genocchi polynomials and for their extended forms. [ABSTRACT FROM AUTHOR]

In this note we find optimal one-sided majorants of exponential type for the signum function subject to certain monotonicity conditions. As an application, we use these special functions to obtain a simple Fourier analysis proof of the (non-sharp) weighted Hilbert-Montgomery-Vaughan inequality. [ABSTRACT FROM AUTHOR]

In this paper, our principal objective is to investigate the impact of disclination and throat radius of a three-dimensional traversable wormhole on quantum oscillator fields. Specifically, we focus on Perry–Mann-type wormhole with disclination while also considering the influence of rainbow gravity's. We derive the radial equation of the relativistic Klein–Gordon oscillator within this wormhole background under the effects of gravity's rainbow and the analytical eigenvalue solution is obtained using the confluent Heun function. In fact, we show that the behavior of the oscillator fields is significantly influenced not only by the presence of disclination and the throat radius but also by the parameter of rainbow gravity's. We choose various such rainbow functions to present and analyze the eigenvalue solutions of the quantum oscillator fields. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional Petrov-Galerkin finite element interface (3D PGFEI) method with non-body-fitted grids is proposed for solving the inhomogeneous anisotropic time-harmonic Maxwell's equations in irregular regions. We derive the weak variational form of the anisotropic Maxwell interface problem. The irregular region is discretized by uniform tetrahedral grids while the media interface is not aligned with the grids. The level-set functions are applied to capture the discontinuity on the anisotropic media interface. Meanwhile, we construct the special basis functions according to the jump conditions across the inhomogeneous interface, where the permittivity and permeability are discontinuous three-dimensional tensors. The resulting linear system is a banded sparse matrix and demands fewer memory resources than dense matrix. Numerical experiments illustrate the efficiency and accuracy of the proposed 3D PGFEI method and the solutions can achieve second-order accuracy in terms of the relative L ∞ (Ω) and relative L 2 (Ω) error norms. [ABSTRACT FROM AUTHOR]

This paper designs a unified control framework to achieve the fixed-time and preassigned-time synchronization of fuzzy inertial neural networks (FINNs) with time-varying delays. First of all, by using the generalized variable transformation, the FINNs model defined by a second-order equation is reduced to two first-order equations. Then, the influence of the fuzzy operations and time delays is offset by devising a proper scaling term. Moreover, by constructing appropriate Lyapunov functions and applying the mean inequality technique, several sufficient conditions are derived to ensure the fixed-time and preassigned-time synchronization for the discussed FINNs. As a result, several more accurate estimations for the settling time are obtained by means of some special functions. Finally, numerical examples are given to show the correctness of the results. [ABSTRACT FROM AUTHOR]

In this research work, the relativistic quantum dynamics of oscillator field in wormhole background with a topological defect produced by a cosmic string is investigated. We consider an example of Morris–Thorne-type wormhole including a osmic string and derived the wave equation of the relativistic quantum oscillator. Through the Heun equation, we solve analytically the radial equation and obtain the ground state energy level E 1 , m and the radial wave function ψ 1 , m as particular cases. In fact, it is shown that the eigenvalue solution of the oscillator field is influenced by the topological defect of cosmic string and shifted the result. Furthermore, the wormhole throat radius also modifies the energy levels and the wave function of the relativistic quantum oscillator. [ABSTRACT FROM AUTHOR]

Methods of differential games are used to find the optimal solution in a multi-agent system with a network structure. A class of differential games on cycle-free graphs is described. A characteristic function of a special form is used, which takes into account the network structure of the game. The average tree solution is considered as a cooperative optimality principle. An illustrative example is given. [ABSTRACT FROM AUTHOR]

Methods for calculating the special function of wave catastrophe B4 and its first derivatives, which describe edge focusing of the caustic-tip type with an edge, are studied. The caustic and ray structures of this singularity are constructed. The characteristic amplitude–phase sections of the special function and its derivatives are calculated. The method of rotation of the integration contour and the method of ordinary differential equations are considered and compared both in terms of accuracy and speed. It is shown that the method of ordinary differential equations works stably in the region of negative values of the first two parameters of the special function, and the method of rotation of the integration contour exhibits stability in the region of positive values. Thus, the approaches are complementary. However, the calculation time for the method of ordinary differential equations is an order of magnitude less than the time for the method of rotation of the integration contour. [ABSTRACT FROM AUTHOR]

Simple and accurate closed-form approximations to the probability density function of the sum of independent identically distributed random variables are derived. We propose a unified approach capable of providing the pdf of the sum when the summands involve the generalized Nakagami-m, generalized Gamma, Rayleigh and Weibull random variables. Unlike others work, the approximations do not use any infinite series or require any complicated special functions. Numerical results show that the new approximations have satisfactory accuracies. As an application, the outage probability and the M-QAM average SER performance of pre-detection EGC diversity are presented. [ABSTRACT FROM AUTHOR]

Harmonic numbers and generalized harmonic numbers have been studied in connection with combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. We introduce Lerch-harmonic numbers which generalize the harmonic numbers and the generalized harmonic numbers. The aim of this note is to derive some identities expressing certain finite sums as the infinite sums involving the Lerch-harmonic numbers. Then, by taking limits of such identities we obtain the corresponding infinite sums of the finite sums as the infinite sums involving the Lerch transcendents. [ABSTRACT FROM AUTHOR]

The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases. [ABSTRACT FROM AUTHOR]