A Family of Generalized Legendre-Based Apostol-Type Polynomials

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.

Molecular Dynamics Study of Diffusion Coefficient for Low-Temperature Dusty Plasmas in the Presence of External Electric Fields

The effects of external electric field (E) on the diffusion coefficient of dust particles in low-temperature dusty plasmas (LT-DPs) have been computed through nonequilibrium molecular dynamics (NEMD) simulations. The new simulation result was obtained by employing the integral formula of velocity autocorrelation functions (VACF) using the Green-Kubo relation. The normalized self-diffusion coefficient (D*) is investigated for different combinations of plasma coupling (Γ) and Debye screening (κ) parameters. The simulation outcome shows that the decreasing position of D* shifts toward Γ and also increased with the increase of κ. The D* linearly decreased with Γ and increased when applied external E increases. It is observed that the increasing trend of D* depends on the E strength. These investigations show that the present algorithm provides precise data with fast convergence and effects of κ, Γ, E. It is shown that the current NEMD techniques with applied external E can be employed to understand the microscopic mechanism of dusty plasmas.

Two-Sided Random Matching Markets: Ex-Ante Equivalence of the Deferred Acceptance Procedures

Stable matching in a community consisting of N men and N women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal work by Gale and Shapley. When the input preference profile is generated from a distribution, we study the output distribution of two stable matching procedures: women-proposing-deferred-acceptance and men-proposing-deferred-acceptance. We show that the two procedures are ex-ante equivalent—that is, under certain conditions on the input distribution, their output distributions are identical. In terms of technical contributions, we generalize (to the non-uniform case) an integral formula, due to Knuth and Pittel, which gives the probability that a fixed matching is stable. Using an inclusion-exclusion principle on the set of rotations, we give a new formula that gives the probability that a fixed matching is the women/men-optimal stable matching.

Self-Inductance Calculation of the Archimedean Spiral Coil

In this paper, a new method to calculate the self-inductance of the Archimedean spiral coil is presented. The proposed method is derived by solving Neumann’s integral formula, and the numerical tool is used to calculate the inductance value. The calculation results are verified with several conventional formulas derived from the Wheeler formula or its modified form and 3D finite element analyses. The comparison with simulation results shows that the conventional formula has an error of above 40% compared to the proposed method, which has below 7% when the wire diameter is reduced. To further check the validity, different sizes of the spiral coil are fabricated by changing the geometrical parameters such as the number of turns, turn spacing, inner radius, outer radius, and wire diameter. Litz wire is chosen for making the spiral coil, and bobbins are made using a 3D printer. Finally, the calculation results are compared with the experimental result. The error between them is less than 2%. The comparison with the conventional formulas, simulation, and measurement results shows the accuracy of the proposed method. This method can be used to calculate the self-inductance of wireless power coils, inductors and antenna design.

The generalized stokes integral theorem for a covariant pseudotensor field

Ориентируемые континуумы играют важную роль в микрополярной теории упругости, все реализации которой возможны только в рамках псевдотензорного формализма и представления об ориентируемом многообразии. Особенно это касается теории микрополярных гемитропных упругих сред. В настоящей работе рассматриваются различные формулировки интегральной теоремы Стокса для асимметричного ковариантного пседотензорного поля, заданного веса. Тем самым достигается распространение известной интегральной формулы Стокса на случай псевдотензоров. Последнее обстоятельство позволяет использовать, указанное обобщение для микрополярных континуумов. Исследование существенно опирается на класс специальных координатных систем. Oriented continua play an important role in the micropolar theory of elasticity, all realizations of which are possible only within the framework of the pseudotensor formalism and the orientable manifold concept. This especially concerns the theory of micropolar hemitropic elastic media. In this paper, we consider various formulations of the Stokes integral theorem for an asymmetric covariant pseudotensor field of a given weight. This extends the well-known Stokes integral formula to the case of pseudotensors. The latter circumstance makes it possible to use the manifistated generalization for micropolar continua. The study relies heavily on the class of special coordinate systems.

Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

AdS one-loop partition functions from bulk and edge characters

We show that the one-loop partition function of any higher spin field in (d + 1)-dimensional Anti-de Sitter spacetime can be expressed as an integral transform of an SO(2, d) bulk character and an SO(2, d − 2) edge character. We apply this character integral formula to various higher-spin Vasiliev gravities and find miraculous (almost) cancellations between bulk and edge characters that lead to agreement with the predictions of HS/CFT holography. We also discuss the relation between the character integral representation and the Rindler-AdS thermal partition function.

A further generalization of the Catalan numbers and its explicit formula and integral representation

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy’s integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.