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Two-level quantum systems are fundamental physical models that continue to attract growing interest due to their crucial role as a building block of quantum technologies. The exact analytical solution of the dynamics of these systems is central to control theory and its applications, such as that to quantum computing. In this study, we reconsider the two-state charge transfer problem by extending and using a methodology developed to study (pseudo)spin systems in quantum electrodynamics contexts. This approach allows us to build a time evolution operator for the charge transfer system and to show new opportunities for the coherent control of the system dynamics, with a particular emphasis on the critical dynamic region around the transition state coordinate, where the avoided crossing of the energy levels occurs. We identify and propose possible experimental implementations of a class of rotations of the charge donor (or acceptor) that endow the electronic coupling matrix element with a time-dependent phase that can be employed to realize controllable coherent dynamics of the system across the avoided level crossing. The analogy of these rotations to reference frame rotations in generalized semiclassical Rabi models is discussed. We also show that the physical rotations in the charge-transfer systems can be performed so as to implement quantum gates relevant to quantum computing. From an exquisitely physical–mathematical viewpoint, our approach brings to light situations in which the time-dependent state of the system can be obtained without resorting to the special functions appearing in the Landau–Zener approach. [ABSTRACT FROM AUTHOR]

A strategy that combines the global‐local version of the generalized finite element method (GFEM gl$$ {}^{\mathrm{gl}} $$) with a mixed‐dimensional coupling iterative method is proposed to simulate two‐dimensional crack propagation in structures globally represented by Timoshenko‐frame models. The region of interest called the local problem, where the crack propagates, is represented by a 2D elasticity model, where a fine mesh of plane stress/strain elements and special enrichment functions are used to describe this phenomenon accurately. A model of Timoshenko‐frame elements simulates the overall behavior of the structure. A coarse mesh of plane stress/strain elements provides a bridge between these two representation scales. The mixed‐dimensional coupling method imposes displacement compatibility and stress equilibrium at the interface between the two different element types by an iterative procedure based on the principle of virtual work. After establishing the constraint equations for the interface, the 2D elasticity model is related to the small‐scale model by the global‐local enrichment strategy of GFEM gl$$ {}^{\mathrm{gl}} $$. In such a strategy, the numerical solutions of the local problem subjected to boundary conditions derived from the global‐scale problem enrich the approximation of this same global problem in an iterative procedure. Each step of the crack propagation requires a new sequence of this iterative global‐local procedure. On the other hand, the constraint equation for the interface is defined only once. The crack representation in a confined region by the global‐local strategy avoids a remeshing that would require new constraint equations. Two numerical problems illustrate the proposed strategy and assess the influence of the analysis parameters. [ABSTRACT FROM AUTHOR]

Mock theta functions were first introduced by Ramanujan in his last paper to Hardy. Moreover, some other mock theta functions were presented in his lost notebook. It is well known that all the classical mock theta functions can be expressed by the universal mock theta functions g 2 (x , q) and g 3 (x , q). In this paper, we establish some generalized mock theta functions and express them in terms of Appell–Lerch sums. Meanwhile, we show that some of Ramanujan's two-parameter mock theta functions are the special cases of these functions. [ABSTRACT FROM AUTHOR]

We introduce new extension of the extended Pochhammer symbol and gamma function by using the extended Mittag-Leffler function. We also present extension of the generalized hypergeometric function as well as some of their special cases by using this extended Pochhammer symbol. Further, we define the extension of the τ-Gauss hypergeometric function. Integral and derivative formulas involving the Mellin transform and fractional calculus techniques associated with this extended τ-Gauss hypergeometric function are also given. Also, new extended τ-Gauss hypergeometric function also provides a few more interesting and well-known results. This enriches the theory of special functions. The obtained results are believed to be newly presented. [ABSTRACT FROM AUTHOR]

In this article, we propose the fractional weak adversarial networks (f-WANs) for the stationary fractional advection dispersion equations based on their weak formulas. This enables us to handle less regular solutions for the fractional equations. To handle the non-local property of the fractional derivatives, convolutional layers and special loss functions are introduced in this neural network. Numerical experiments for both smooth and less regular solutions show the validity of f-WANs. [ABSTRACT FROM AUTHOR]

We present efficient algorithms for multi-precision and multi-accuracy calculation of error functions and the Dawson integral, all with complex arguments. These algorithms achieve exceptional accuracies, ranging from 26 significant digits (SD) up to 30 SD depending on the function. In addition to the Faddeyeva or Faddeeva function, or the scaled complementary error function, , the list of functions considered include the error function, erf z ; the complementary error function, erfc z ; the imaginary error function, erfi (z) ; and the Dawson integral Daw (z) . These algorithms are integrated into a modern Fortran module confirming the claimed accuracies and superior efficiency compared to other competitive codes in the literature. Additionally, we highlight an observation regarding the built-in " Erfc _ Scaled (x) " function included in a widely used Fortran compiler, which exhibits a significant decline in accuracy when employed in quadruple-precision arithmetic. [ABSTRACT FROM AUTHOR]

In this work, we investigate the behavior of non-relativistic quantum particles immersed in a cosmic string space-time background. Our study involves the examination of these particles as they interact with a range of influences, including potential, magnetic, and quantum flux fields. We employ analytical methods to solve the associated wave equation, leading to the derivation of eigenvalue solutions for this quantum system. Subsequently, we leverage these eigenvalue solutions to scrutinize several potential models. For each model, we present and engage in a thorough discussion of the corresponding eigenvalue solutions. In an extension of our investigation, we explore the thermodynamic and magnetic properties of the quantum system when it is exposed to non-zero temperature conditions, denoted by T ≠ 0. Our analysis encompasses the calculation of essential parameters such as the partition function for the system and other pertinent functions. Following these calculations, we meticulously examine and interpret the outcomes, shedding light on the system's behavior and characteristics in the presence of temperature variations. Furthermore, we calculate entropic information to investigate the location of particles in the system. [ABSTRACT FROM AUTHOR]

In this note, we provide an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the Kohler-Jobin resolution of a conjecture of Pólya-Szegö: when the Gaussian torsional rigidity of a domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a "modified" torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure. We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle. [ABSTRACT FROM AUTHOR]

The concept of special functions plays an important role in mathematical analysis and physics as well. In this paper, we study some different types of the special Gamma function defined on the two-fold fuzzy complex field, where we combine the classical Gamma function with the two-fold fuzzy algebra defined on complex numbers. On the other hand, many elementary properties of this new special function will be determined in terms of theorems and proofs. [ABSTRACT FROM AUTHOR]

This paper focuses on integrals and Borel transforms concerning λ$$ \lambda $$‐Bessel and λ$$ \lambda $$‐Tricomi functions, employing an umbral perspective as a central approach. Differintegral techniques, presently utilized in calculus, offer a rich reserve of tools that can be utilized across a broad spectrum of problems encompassing the realm of special functions and beyond. Employing integral transforms akin to the Borel type and the related formalism proves to be a potent approach, facilitating a connection between umbral and operational methodologies. By integrating these perspectives, we establish a novel and effective method for deriving integrals of hybrid special functions and achieving the summation of their corresponding generating functions. [ABSTRACT FROM AUTHOR]

This study focuses on investigating the relativistic quantum dynamics of a fermionic oscillator field in the background of a minimal surface, specifically a catenoid bridge. Our primary objective is to analyze the bound-state solutions of this system by solving the non-perturbative radial wave equation. To achieve this, we consider two distinct scenarios. First, we explore the case where the particle is located inside the catenoid bridge and search for solutions to the wave equation. Subsequently, we extend our analysis to scenarios where the particle is positioned at a significant distance from the origin, comparable to the size of the catenoid bridge. In this regime, we obtain solutions to the wave equation to understand the behavior of the fermionic oscillator field. By examining these solutions, we investigate the influence of both the oscillator coupling and the spacetime background on the system. [ABSTRACT FROM AUTHOR]

Biunivalent holomorphic functions form an interesting class in geometric function theory and are connected with special functions and solutions of complex differential equations. This class has been investigated by many authors, mainly to find the coefficient estimates. The assumption of biunivalence is rigid; this rigidity means that, for example, only the initial Taylor coefficients have been estimated. The aim of this paper is to develop a variational technique for biunivalent functions, which provides a power tool for solving the general extremal problems on the classes of such functions. It involves quasiconformal analysis. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Due to the similar characteristics of chaotic systems and cryptography, public key encryption algorithms based on chaotic systems are worth in-depth research and have high value for the future. Chebyshev polynomials have good properties and are often used in the design of public key algorithms. This paper improves the Bose Multi-Chaotic Public Key Cryptographic Algorithm (BMPKC) by applying Chebyshev polynomials. The proposed algorithm (CMPKC- k i ) introduces the selective coefficient k i based on the properties of Chebyshev polynomials, allowing the special functions that need to be negotiated in the original system to be freely and randomly chosen as Chebyshev polynomials, and can also be expanded to m levels. The improved cryptographic algorithm also utilizes chaotic hash functions and logistic mapping to generate pseudo-random sequences and overcomes shortcomings of the Bose algorithm by iteratively iterating the selected Chebyshev polynomials based on the number of 0s or 1s in the pseudo-random sequence, thus providing better security. Analysis and software testing results indicate that this algorithm has strong robustness against brute force attacks, achieving a higher attack time for breaking the private key compared to the CEPKC, BMPKC, and CMPKC algorithms. Compared to the CMPKC algorithm, our proposal algorithm achieves better performance in the encryption and decryption phases. Furthermore, we combine this Multi-Chaotic System Key Exchange Protocol with the Advanced Encryption Standard (AES) algorithm, while providing a demonstration, offering more possibilities for practical applications of this system. [ABSTRACT FROM AUTHOR]

The paper studies the regularities of wave propagation in an inhomogeneous elastic layer with gradient properties varying along the layer thickness. The laws of change in elastic parameters are established at which the waves propagating in the direction of parameter change are harmonic. The waves propagating in the layer with a linear law of change in the elastic parameters are noт-harmonic. Analytical solutions obtained for a monochromatic wave defined on the boundary of a plane layer with a rigidly fixed second boundary are represented by special functions. In the case of gradient variation of the layer density, the solution is determined by a superposition of the Airy functions, in the case of equal relative variations of the density and elastic moduli, they are expressed through the modified Bessel functions, and in the case of arbitrary linear laws of variation in the elastic parameters, they are expressed through the confluent hypergeometric functions. Based on the analytical solution obtained for a layer with inhomogeneous density, the conditions for extreme values of displacements and strains are determined, and their distributions over the layer thickness are calculated and analyzed. [ABSTRACT FROM AUTHOR]

In this work, novel Ostrowski-type inequalities for dissimilar function classes and generalized fractional integrals (FITs) are presented. We provide a useful identity for differentiable functions under FITs, which results in special expressions for functions whose derivatives have convex absolute values. A new condition for bounded variation functions is examined, as well as expansions to bounded and Lipschitzian derivatives. Our comprehension is improved by comparison with current findings, and recommendations for future study areas are given. [ABSTRACT FROM AUTHOR]

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Polyelectrolytes are charged polymers comprising macromolecules in which substantial portions of the constituent units contain cationic (e.g., pyridinium, ammonium) or anionic (e.g., sulfonate, carboxylate) groups, which possess special functions from the features of counterions, such as dissociation to charged species, mechanical stability, phase behavior, etc. Therefore, functional polyelectrolytes have been widely applied in many fields. In this perspective, we present some progresses in the studies of poly(polyoxometalate)s, denoted as poly(POM)s, as a kind of new charged polymers/polyelectrolytes, by covalent bonding between the inorganic polyoxometalate (POM) clusters and the organic polymer chains. According to the distinct positions of POMs in polymer chain and functions of poly(POM)s, they are divided into the following four categories: crosslinked poly(POM); side-chain poly(POM); backbone poly(POM), including poly(POM)-conjugated polymer hybrid and block poly(POM)-polymer; and POM-based covalent organic framework (PCOF). This perspective introduces the synthesis methods of poly(POM) polyelectrolytes and their macromolecular and aggregate structural characteristics, while also focusing on their properties and functions. Their application areas include catalysis, thermal resistance, optical functions, fuel cells and batteries, etc. [ABSTRACT FROM AUTHOR]

We propose new two-sided estimates with sharp constants for the number e. Minorants and majorants in the estimates are expressed as continued fractions. To justify the estimate, we essentially use the integral representation of a special analytic function obtained earlier by the authors. We discuss the validity of similar inequalities and relationship with known results. [ABSTRACT FROM AUTHOR]

This paper is devoted to reduce the conservativeness of distributionally robust optimization with moments information. Since the optimal solution of distributionally robust optimization is required to be feasible for all uncertain distributions in a given ambiguity distribution set and so the conservativeness of the optimal solution is inevitable. To address this issue, we introduce the globalized distributionally robust counterpart (GDRC) which allows constraint violations controlled by functional distance of the true distribution to the inner distribution set. We obtain the deterministic equivalent forms for several GDRCs under the moment-based framework. To be specific, we show the deterministic equivalent systems of inequalities for GDRCs under second order moment information with a separable convex distance function and a special jointly convex function, respectively. We also obtain the deterministic equivalent inequality for GDRC under first order moment and support information. The computationally tractable examples are presented for these GDRCs. Numerical tests of a portfolio optimization problem are given to show the effectiveness of our method and the results demonstrate that the globalized distributionally robust solution is non-conservative and flexible compared to the distributionally robust solution. [ABSTRACT FROM AUTHOR]

We study the BMO-type functional \kappa _{\varepsilon }(f,\mathbb {R}^n), which can be used to characterize bounded variation functions f\in \mathrm {BV}(\mathbb {R}^n). The \Gamma-limit of this functional, taken with respect to L^1_{\mathrm {loc}}-convergence, is known to be \tfrac 14 |Df|(\mathbb {R}^n). We show that the \Gamma-limit with respect to L^{\infty }_{\mathrm {loc}}-convergence is \[ \tfrac 14 |D^a f|(\mathbb {R}^n)+\tfrac 14 |D^c f|(\mathbb {R}^n)+\tfrac 12 |D^j f|(\mathbb {R}^n), \] which agrees with the "pointwise" limit in the case of special functions of bounded varation. [ABSTRACT FROM AUTHOR]

Newly discovered, incomplete forms of special functions are increasing the interest of both pure and applied mathematicians. The main purpose of this work is to derive four theorems on partial derivatives with incomplete Aleph functions of two variables and generalize them up to r-variables. In addition to these theorems, we also established some novel formulae on the partial derivatives that play a key role in deriving the main results in terms of finite sum. Further, we generalize the result and obtain the finite sum for the incomplete Aleph functions with r-variables. Here, we also established some particular cases that are in most general character and including the results given earlier by Buschman and Deshpande and may prove significant in numerous interesting situations appearing in the literature on mathematical analysis, applied mathematics and mathematical physics. [ABSTRACT FROM AUTHOR]

Let \mathcal L denote the twisted Laplacian in \mathbb {C}^d. We study almost everywhere convergence of the Bochner–Riesz mean S^\delta _{t}(\mathcal L) f of f\in L^p(\mathbb C^d) as t\to \infty, which is an expansion of f in the special Hermite functions. For 2\le p\le \infty, we obtain the sharp range of the summability indices \delta for which the convergence of S^\delta _{t}(\mathcal L) f holds for all f\in L^p(\mathbb {C}^d). [ABSTRACT FROM AUTHOR]

Conformal predictions make it possible to define reliable and robust learning algorithms. But they are essentially a method for evaluating whether an algorithm is good enough to be used in practice. To define a reliable learning framework for classification from the very beginning of its design, the concept of scalable classifier was introduced to generalize the concept of classical classifier by linking it to statistical order theory and probabilistic learning theory. In this paper, we analyze the similarities between scalable classifiers and conformal predictions by introducing a new definition of a score function and defining a special set of input variables, the conformal safety set, which can identify patterns in the input space that satisfy the error coverage guarantee, i.e., that the probability of observing the wrong (possibly unsafe) label for points belonging to this set is bounded by a predefined ε error level. We demonstrate the practical implications of this framework through an application in cybersecurity for identifying DNS tunneling attacks. Our work contributes to the development of probabilistically robust and reliable machine learning models. [ABSTRACT FROM AUTHOR]

In this study, we explore quantum dynamics of non-relativistic particles within the context of a Morris–Thorne-type wormhole space-time featuring a cosmic string. Our investigation focuses on the eigenvalue solution of Schrödinger equation by utilizing the confluent Heun equation. We demonstrate that the cosmic string and the wormhole throat radius leads to significant modifications in the energy levels and wave functions of non-relativistic quantum particles. As a consequence, the degeneracy of the energy spectrum is disrupted. Subsequently, we extend our analysis to the harmonic oscillator problem, where non-relativistic particles interact harmonically within the same wormhole space-time background with a cosmic string. Employing a similar approach, we obtain the eigenvalue solution for the harmonic oscillator and analyze this solution to show the influence of the cosmic string parameter and the radius of the wormhole throat. In both scenarios, we obtain exact analytical solutions for the wave equation and specifically, we present the ground-state energy level E 1 , m and the corresponding wave function ψ 1 , m as particular cases. [ABSTRACT FROM AUTHOR]

First we prove that the main theorem for absolutely monotonic functions on (0, ∞) from the book Mitrinović D.S., Pečarić J.E., Fink A.M. ”Classical And New Inequalities In Analysis”, Kluwer Academic Publishers, 1993, is not valid without additional restrictions. After that connected inequalities for special functions of hypergeometric type is studied. [ABSTRACT FROM AUTHOR]

We examine the impact of China's announcements of upcoming government environmental audits (GEAs) on firms' corporate social responsibility (CSR). We hypothesize that firms, especially heavily polluting firms, engage in more CSR to proactively burnish their public image to weather forthcoming adverse outcomes from a GEA. Our baseline findings confirm that firms respond to the government's GEA announcement by engaging in more CSR. Additional analysis suggests that reputation concerns and environmental fines mediate the impact of GEA announcements on CSR. Cross‐sectional analyses show that the effect of GEA announcements on CSR is more salient for firms: (1) in a business environment in which GEAs are more effective, (2) that CSR engagement is critically important, or (3) that CSR can serve some special function. Overall, our findings imply that it is generally a good government policy to implement GEAs. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a generalization of (η1,η2)-convex functions using the local fractional calculus developed by Yang [Advanced Local Fractional Calculus and its Applications (World Science Publisher, New York, 2012)], namely generalized (η1,η2)-convex functions. Among other results, we obtain some Fejér-type inequalities for this class of functions. As applications, we present some inequalities with generalized probability density functions and generalized special means. [ABSTRACT FROM AUTHOR]

In this effort, we present a new definition of the Steiner symmetrization by using special analytic functions in a complex domain (the open unit disk) with respect to the origin. This definition will be used to optimize the class of univalent analytic functions. Our method is based on the concept of differential subordination and the Caratheodory theory. Examples are illustrated in the sequel involving the modified Libera-Livingston-Bernardi integral operator over the open unit disk. The result gives that this integral satisfies the definition of bounded turning function (univalent analytic function). [ABSTRACT FROM AUTHOR]

We examine hypergeometric functions in the finite field, p-adic and classical settings. In each setting, we prove a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. We provide multiple applications of these results, including new reduction and summation formulas for finite field hypergeometric functions, along with classical analogues; evaluations of special values of these functions which apply in both the finite field and p-adic settings; and new relations to Fourier coefficients of modular forms. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials. [ABSTRACT FROM AUTHOR]

This study delves into special class of generalized p-trigonometric functions, examining their connection to the established counterparts like p-cosine and p-sine functions. We here explore the new class of functions called the p-versine, p-coversine, p-haversine, and p-hacovercosine, by providing comprehensive definitions and properties. Grounded in the characteristics of p-cosine and p-sine functions, the newly proposed functions offer unique mathematical insights. Our work contributes towards a thorough understanding of these new special functions, showcasing their potential applications in diverse scientific domains, from mathematical analysis to physics and engineering. This paper contributes as a valuable resource for future applied mathematics researchers, engaging with these new mathematical functions, enhancing the ability to model complex patterns from diverse real-world applications. [ABSTRACT FROM AUTHOR]

Source identities are fundamental identities between multivariable special functions. We give a geometric derivation of rational and trigonometric source identities. We also give a systematic derivation and extension of various determinant representations for source functions which appeared in previous literature as well as introducing the elliptic version of the determinants, and obtain identities between determinants. We also show several symmetrization formulas for the rational version. [ABSTRACT FROM AUTHOR]

Bound state energy eigensolutions of nine different diatomic molecules, specifically, H2$$ {\mathrm{H}}_2 $$, TiH, N2$$ {\mathrm{N}}_2 $$, CO, NO, ScH, CrH, HCl and LiH, placed in a point‐like global monopole, have been calculated by solving the time‐independent Schrödinger equation (SE). The molecules are confined by the Aharonov–Bohm (AB) flux field and the interaction potential among the charged particles is governed by the combined screened modified Kratzer potential (SMKP) plus Hulthén potential model. Three other special cases of the interaction potential have also been discussed here. Asymptotic iteration method has been used for the mathematical calculations. It is difficult to solve the SE exactly for any non‐zero values of the azimuthal quantum number l$$ l $$ due to the presence of the centrifugal barrier term. The well‐known Pekeris approximation technique for the terms 1r2$$ \frac{1}{r^2} $$ and 1r$$ \frac{1}{r} $$ has taken into consideration for the purpose of performing numerical computations connected to an arbitrary l$$ l $$ state. The outcomes of this study are in reasonable agreement with the results of previous research on SMKP, Kratzer‐Fues potential, and modified Kratzer potential in the Minkowski space. It is observed that the energy spectra of the diatomic molecules suffer considerable change due to the effects of the background AB‐flux field and topological defect parameter. [ABSTRACT FROM AUTHOR]

In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size p 0 tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski's inequality, trapezoidal inequality, midpoint inequality, Simpson's inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of γ and parameter ξ. [ABSTRACT FROM AUTHOR]

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer's G-function and Fox's H-functions and their utilization in mathematical physics are the prime focus of this paper. [ABSTRACT FROM AUTHOR]

In a recent paper, we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical $ \bar {\partial } $ ∂ ¯ -problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander–Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by $ \eta =(\eta _n)_{n\geq ~0} $ η = (η n) n ≥ 0 leading to a special entire function $ \mathsf {E}(z) $ E (z) that is used to express the kernel function of the Hörmander–Fock space. We present also an example of a special function belonging to the class Mittag-Leffler (ML) introduced recently by Alpay et al. and apply a Bochner–Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the bouncing universe in a modified gravity model f (Q , T). The bouncing universe theory posits that the universe goes through periodic expansions and contractions, with a "bounce" occurring at the end of each contraction that leads to a new expansion. This theory suggests that our universe didn't emerge on its own out of nothing, but is instead the latest in a series of universes that have been expanding and contracting for eternity. It has gained renewed interest in recent years as a potential alternative to the Big Bang theory, which suggests that our universe began with a massive expansion from a singularity. According to the above explanations, first, we review of f (Q , T) gravity model, then the bouncing universe scenario according to the f (Q , T) gravity model in some special functions is analyzed in detail. We find that in which condition of parameters space, bouncing occurs. [ABSTRACT FROM AUTHOR]

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass O μ (Δ 1 , Δ 2 , Δ 3 , Δ 4) of univalent functions defined in the unit disk Λ = { τ ∈ C : τ < 1 } . More specifically, we investigate some sufficient conditions for Rabotnov functions, Poisson functions, Bessel functions and Wright functions to be in this subclass. Some corollaries of our main results are given. The novelty and the advantage of this research could inspire researchers of further studies to find new sufficient conditions to be in the subclass O μ (Δ 1 , Δ 2 , Δ 3 , Δ 4) not only for the aforementioned special functions but for different types of special functions, especially for hypergeometric functions, Dini functions, Sturve functions and others. [ABSTRACT FROM AUTHOR]

The Kullback–Leibler divergence is a measure of the divergence between two probability distributions, often used in statistics and information theory. However, exact expressions for it are not known for multivariate or matrix-variate distributions apart from a few cases. In this paper, exact expressions for the Kullback–Leibler divergence are derived for over twenty multivariate and matrix-variate distributions. The expressions involve various special functions. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive analysis and solution methodology for simultaneous triple series equations involving generalized Bateman-k functions. The study extends the classical Bateman-k function, a notable special function in mathematical physics, to a more generalized form, enabling the exploration of its applications in solving complex simultaneous series equations. We derive explicit solutions by employing advanced mathematical techniques, including the method of integral transforms and series expansion. The results not only contribute to the theoretical understanding of Bateman-k functions but also offer practical computational tools for problems in mathematical physics, engineering, and applied mathematics where such functions naturally arise. The paper also discusses the convergence conditions and uniqueness of the solutions, providing a robust framework for future research in this area. In this paper an exact solution is obtained for the simultaneous triple series equations involving generalized Bateman k-Functions by multiplying factor method. [ABSTRACT FROM AUTHOR]

Entropic Value-at-Risk (EVaR) measure is a convenient coherent risk measure. Due to certain difficulties in finding its analytical representation, it was previously calculated explicitly only for the normal distribution. We succeeded to overcome these difficulties and to calculate Entropic Value-at-Risk (EVaR) measure for Poisson, compound Poisson, Gamma, Laplace, exponential, chi-squared, inverse Gaussian distribution and normal inverse Gaussian distribution with the help of Lambert function that is a special function, generally speaking, with two branches. [ABSTRACT FROM AUTHOR]

AbstractIn this article, a new approach to study the fractionalized second grade fluid flow is described by the different fractional derivative operators near an exponentially accelerated vertical plate together with exponentially variable velocity, energy and mass diffusion through a porous media is critically examined. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimentional form. For the sake of better rheology of second grade fluid, developed a fractional model by applying the new definition of Constant Proportional-Caputo hybrid derivative (CPC), Atangana Baleanu in Caputo sense (ABC) and Caputo Fabrizio (CF) fractional derivative operators that describe the generalized memory effects. For seeking exact solutions in terms of Mittag-Leffler and G-functions for velocity, temperature and concentration equations, Laplace integral transformation technique is applied. For physical significance of various system parameters on fluid velocity, concentration and temperature distributions are demonstrated through various graphs by using graphical software. Furthermore, for being validated the acquired solutions, accomplished a comparative analysis with some published work. It is also analyzed that for exponential heating and non-uniform velocity conditions, the CPC fractional operator is the finest fractional model to describe the memory effect of velocity, energy and concentration profile. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, in the literature, it is observed that to derived analytical results from fractional fluid models developed by the various fractional operators, is difficult and this article contributing to answer the open problem of obtaining analytical solutions the fractionalized fluid models.

We consider one variant of stable assignment problems in a bipartite graph endowed with nonnegative capacities on the edges and quotas on the vertices. It can be viewed as a generalization of the stable allocation problem introduced by Baïou and Balinsky, which arises when strong linear orders of preferences on the vertices in the latter are replaced by weak ones. At the same time, our stability problem can be stated in the framework of a theory by Alkan and Gale on stable schedule matchings generated by choice functions of a wide scope. In our case, the choice functions are of a special, so-called mixed , type. The main content of this paper is devoted to a study of rotations in our mixed model, functions on the edges determining "elementary" transformations between close stable assignments. These look more sophisticated compared with rotations in the stable allocation problem (which are generated by simple cycles). We efficiently construct a poset of rotations and show that the stable assignments are in bijection with the so-called closed functions for this poset; this gives rise to a "compact" affine representation for the lattice of stable assignments and leads to an efficient method to find a stable assignment of minimum cost. [ABSTRACT FROM AUTHOR]

The article describes the institutional mechanism for ensuring the right to development at the international universal level. The functions and mandate of the Office of the United Nations High Commissioner for Human Rights, the Working Group on the Right to Development, the High-level Task Force on the Implementation of the Right to Development, the Special Rapporteur on the Right to Development, the Expert Mechanism on the Right to Development, etc. are described. It is noted that the right to development is the only human right specifically mentioned in resolution 48/141 of the UN General Assembly, by which it established the position of High Commissioner and the Office of the High Commissioner for Human Rights. It is highlighted the functions of the Office of the High Commissioner for Human Rights in ensuring the right to development. It is described the mandate of the Working Group on the Right to Development, which includes: monitoring and analysis of progress made in the promotion and implementation of the right to development at the national and international levels, making recommendations and further analysis of obstacles to its full implementation; review of reports and any other information submitted by States, UN agencies, other relevant international organisations and nongovernmental organisations on the relationship between their activities and the right to development, etc. It is noted that the High-level Task Force on the Implementation of the Right to Development was established within the framework of the Working Group on the Right to Development to assist it in fulfilling its mandate. It is stated that the Special Rapporteur on the Right to Development contributes to the realisation of the right to development by developing practical guidance to implement the right and to integrate it within development policies and programmes at local, national, regional and international levels. The functions and mandate of the Special Rapporteur are highlighted. It is described the activities of the Expert Mechanism on the Right to Development, which provides the Human Rights Council with thematic expertise on the right to development in searching for, identifying and sharing best practices with Member States and promotes the implementation of the right to development worldwide. [ABSTRACT FROM AUTHOR]

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form. [ABSTRACT FROM AUTHOR]

inspecting the attributes of derivatives and integrals of fractional orders known as fractional derivatives and integrals. In this article, a far-out complex integral transform known as the Rohit transform (RT) is put into use for working out general homogeneous and non-homogeneous differential equations of fractional or non-integral orders of particular forms entailing the Caputo fractional derivative operator and the Riemann-Liouville fractional derivative operator. The Rohit transform (RT) of the Mittag-Leffler function, the Caputo fractional derivative operator. and the Riemann-Liouville fractional derivative operator are obtained, and then the solutions of fractional systems characterized by fractional homogeneous and fractional non-homogeneous differential equations entailing the Caputo fractional derivative operator and the Riemann-Liouville fractional derivative operator are obtained by utilizing the Rohit transform (RT). This article showcases the ability and efficacy of the Rohit transform (RT) to straighten out fractional systems characterized by fractional homogeneous and fractional non-homogeneous differential equations. While other methods, present in the literature such as the homotopy-perturbation method, Adomian decomposition method, fractional variational iteration method, Lyapunov direct method, and generalized Mittag Leffler stability, may also be capable of solving the examples presented in the paper, the Rohit transform introduced innovative concepts or methodologies that offer new insights or perspectives on the problems examined in the paper, distinguishing itself from existing transforms and potentially opening up new research directions. The simplicity and ease of implementation of the Rohit transform make it a preferable choice for practical applications, requiring fewer computational steps, less complex algorithms, and simpler parameter tuning.

Analytical solutions of some contact problems are infinite functional series according to the system of basic functions. When constructing such solutions, it becomes necessary to represent the kernels of integral equations describing the process of interaction in the form of two-dimensional series on a given basis. Often the kernels have a rather complex appearance, therefore, the process of finding the decomposition coefficients is a rather complex and labor-intensive process, on which the accuracy and speed of obtaining final results depend. The paper proposes a calculation method that allows calculating the coefficients of decomposition of the kernels of plane contact problems according to a special orthonormal basis that takes into account the features of contacting bodies. Other approximate formulas are also derived for the special case when coating characteristics are constant. Based on the received presentation, conclusions and recommendations are formulated.