Search Results (32)

This paper shows that certain ${}_{3}F_4$ hypergeometric functions can be expanded in sums of pair products of ${}_{1}F_2$ functions. In special cases, the ${}_{3}F_4$ hypergeometric functions reduce to ${}_{2}F_3$ functions. Further special cases allow one to reduce the ${}_{2}F_3$ functions to ${}_{1}F_2$ functions, and the sums to products of ${}_{0}F_1$ (Bessel) and ${}_{1}F_2$ functions. The class of hypergeometric functions with summation theorems are thereby expanded beyond those expressible as pair-products of ${}_{2}F_1$ functions, ${}_{3}F_2$ functions, and generalized Whittaker functions, into the realm of ${}_{p}F_q$ functions where $p<q$ for both the summand and terms in the series. Full article

It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are given. One observation is that a connected sum of two surface-links is a ribbon surface-link if and only if both the connected summands are ribbon surface-links. The other observation is a characterization of when a surface-link consisting of ribbon surface-knot components becomes a ribbon surface-link. Full article

In this paper, algorithms for different types of trigonometric power sums are developed and presented. Although interesting in their own right, these trigonometric power sums arise during the creation of an algorithm for the four types of twisted trigonometric power sums defined in the introduction. The primary aim in evaluating these sums is to obtain exact results in a rational form, as opposed to standard or direct evaluation, which often results in machine-dependent decimal values that can be affected by round-off errors. Moreover, since the variable, m, appearing in the denominators of the arguments of the trigonometric functions in these sums, can remain algebraic in the algorithms/codes, one can also obtain polynomial solutions in powers of m and the variable r that appears in the cosine factor accompanying the trigonometric power. The degrees of these polynomials are found to be dependent upon v, the value of the trigonometric power in the sum, which must always be specified. Full article

This paper shows that certain${ }_3{F}_4$ hypergeometric functions can be expanded in sums of pair products of${ }_2{F}_3$ functions, which reduce in special cases to${ }_2{F}_3$ functions expanded in sums of pair products of${ }_1{F}_2$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions,${ }_2{F}_1$ functions, and${ }_3{F}_2$ functions into the realm of${ }_p{F}_q$ functions where $p<q$ for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. Full article

The central limit theorem states that, in the limits of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to attain a stable distribution. The condition of independence, however, only holds in real systems as an approximation. To extend the theorem to more general situations, previous studies have derived a version of the central limit theorem that also holds for variables that are not independent. Here, we present numerical results that characterize how convergence is attained when the variables being summed are deterministically related to one another through the recurrent application of an ergodic mapping. In all the explored cases, the convergence to the limit distribution is slower than for random sampling. Yet, the speed at which convergence is attained varies substantially from system to system, and these variations imply differences in the way information about the deterministic nature of the dynamics is progressively lost as the number of summands increases. Some of the identified factors in shaping the convergence process are the strength of mixing induced by the mapping and the shape of the marginal distribution of each variable, most particularly, the presence of divergences or fat tails. Full article

The representation theory of a finite group, G, is an important area of research currently. This paper studied the modular representation of finite groups, which are direct products. There are three approaches to studying this representation: the ring approach, the character approach, and the module approach. Moreover, we learned some of the important conjectures in this representation, which link a representation of a finite group and its local subgroups, which are normalizer non-trivial p-subgroups. These conjectures are the McKay conjecture, Alperin’s weight conjecture, and the ordinary weight conjecture. The main aim of the proposed paper was to investigate these conjectures of direct products, the direct summands of which satisfy these conjectures for the associated tensor product of the p-block. We obtained the results by assuming the conjectures are true. Then, we used the properties of the direct products. Full article

Let m and n be fixed positive integers. Suppose that $\mathcal{A}$ is a von Neumann algebra with no central summands of type $I_1$, and $L_m:\mathcal{A}\to \mathcal{A}$ is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ${\varphi }_m:\mathcal{A}\to \mathcal{A}$ and an additive higher map ${\zeta }_m:\mathcal{A}\to Z(\mathcal{A})$, which annihilates every ${(n-1)}^{th}$ commutator $p_n({\mathfrak{S}}_1,{\mathfrak{S}}_2,\cdots ,{\mathfrak{S}}_n)$ with ${\mathfrak{S}}_1{\mathfrak{S}}_2=0$ such that $L_m(\mathfrak{S})={\varphi }_m(\mathfrak{S})+{\zeta }_m(\mathfrak{S})forall\mathfrak{S}\in \mathcal{A}.$ We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results. Full article

Let $\mathcal{A}$ be a unital ∗-algebra over the complex fields $\mathbb{C}$. For any $H_1,H_2\in \mathcal{A}$, a product ${[H_1,H_2]}_{•}=H_1{H}_2-H_2{H}_1^{*}$ is called the skew Lie product. In this article, it is shown that if a map $\xi$ : $\mathcal{A}\to \mathcal{A}$ (not necessarily linear) satisfies $\xi \left(P_n(H_1,H_2,\dots ,H_n)\right)={\sum }_{i=1}^n{P}_n(H_1,\dots ,$$H_{i-1},\xi \left(H_i\right),H_{i+1},\dots ,H_n)(n\ge 3)$ for all $H_1,H_2,\dots ,H_n\in \mathcal{A}$, then $\xi$ is additive. Moreover, if $\xi \left(i\frac{e}{2}\right)$ is self-adjoint, then $\xi$ is ∗-derivation. As applications, we apply our main result to some special classes of unital ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra, and von Neumann algebra with no central summands of type $I_1$. Full article

Great attention has been paid to the propagation of electromagnetic (EM) waves across the sea surface due to its important applications. Most of the previous research, however, focuses on the half-space model illustrating the deep sea environment. In this paper, EM field distribution in the extremely low frequency (ELF) near-region under horizontal electric dipole (HED) excitation in homogeneous half-space seawater is analyzed based on the general expression of the Sommerfeld integral using the quasistatic approximation method. The focus is on deriving complete and effective solutions in air and seawater regions under the cylindrical coordinates for the EM near-field, which is generated by an HED in a shallow sea. The resulting formulas can be given by a few summands in closed form as the well-known Fourier–Bessel integrals. The analytical approximate expression of ELF Sommerfeld EM field integral excited by the HED in the homogeneous half-space seawater is deduced under the condition that the propagation distance ρ satisfies kρ << 1. To this end, the EM field distribution in the range close to the HED antenna in seawater is simulated, the results have shown that the minimum attenuation value of the vertical electric component E_z is about 15 dB, and that of the radical magnetic components H_φ is about 30 dB, and these values are found to be of greatest potential for the near-field region propagation among the electric and magnetic components. Finally, the correctness of the proposed method is verified by comparison with Pan’s approximation method and Margetis’s exact expression approximation method, which demonstrated the correctness of the proposed method. Full article

A numerically efficient technique is presented for computing the backscattered fields from two spherical targets embedded in an underwater sediment. The bottom is assumed to be a homogeneous liquid attenuating half-space. The transmitter/receiver is located in a homogeneous water half-space. The distances between the transmitter/receiver and objects of interest are supposed to be large compared to the acoustic wavelengths in water and seabed. In simulations, the spherical scatterers of the same radius are assumed to be acoustically rigid. The interactions between two spheres are not taken into account because of the strong attenuation in the bottom. The scattering from one sphere in a wide frequency range is determined using the Hackman and Sammelmann’s general approach. The arising scattering coefficients of the sphere are evaluated using the steepest descent method. The obtained asymptotic expressions for the scattering coefficients essentially allowed to decrease a number of summands in the formula for the form-function of the backscattered acoustic field. Full article

A new class of probability distributions closely connected to generalized hyperbolic distributions is introduced. It is better adapted for studying the distributions of sums of a random number of random variables. The properties of these distributions are studied. It seems that this class may be useful for modeling asset returns. Full article

The tight focusing of an optical vortex with an integer topological charge (TC) and linear polarization was considered. We showed that the longitudinal components of the spin angular momentum (SAM) (it was equal to zero) and orbital angular momentum (OAM) (it was equal to the product of the beam power and the TC) vectors averaged over the beam cross-section were separately preserved during the beam propagation. This conservation led to the spin and orbital Hall effects. The spin Hall effect was expressed in the fact that the areas with different signs of the SAM longitudinal component were separated from each other. The orbital Hall effect was marked by the separation of the regions with different rotation directions of the transverse energy flow (clockwise and counterclockwise). There were only four such local regions near the optical axis for any TC. We showed that the total energy flux crossing the focus plane was less than the total beam power since part of the power propagated along the focus surface, while the other part crossed the focus plane in the opposite direction. We also showed that the longitudinal component of the angular momentum (AM) vector was not equal to the sum of the SAM and the OAM. Moreover, there was no summand SAM in the expression for the density of the AM. These quantities were independent of each other. The distributions of the AM and the SAM longitudinal components characterized the orbital and spin Hall effects at the focus, respectively. Full article

For each $t\in (-1,1)$, the exact value of the least upper bound $H\left(t\right)=sup\left\{\mathbb{E}\right|X{|}^3/\mathbb{E}\left|X-t{|}^3\right\}$ over all the non-degenerate distributions of the random variable X with a fixed normalized first-order moment $\mathbb{E}{X}_1/\sqrt{\mathbb{E}{X}_1^2}=t$, and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum $M:=sup{L}_1\left(X\right)/L_1(X-\mathbb{E}X)=\left(\sqrt{17+7\sqrt{7}}\right)/4$, where $L_1\left(X\right)=\mathbb{E}{\left|X\right|}^3/{\left(\mathbb{E}{X}^2\right)}^{3/2}$ is the non-central Lyapunov ratio, and hence proving S. Shorgin’s (2001) conjecture on the exact value of M. As a corollary, an analog of the Berry–Esseen inequality for the Poisson random sums of independent identically distributed random variables $X_1,X_2,\dots$ is proven in terms of the central Lyapunov ratio $L_1(X_1-\mathbb{E}{X}_1)$ with the constant $0.3031·H\left(t\right){(1-t^2)}^{3/2}\in [0.3031,0.4517)$, $t\in [0,1)$, which depends on the normalized first-moment $t:=\mathbb{E}{X}_1/\sqrt{\mathbb{E}{X}_1^2}$ of random summands and being arbitrarily close to $0.3031$ for small values of t, an almost $1.5$ size improvement from the previously known one. Full article

The convergence rate in the famous Rényi theorem is studied by means of the Stein method refinement. Namely, it is demonstrated that the new estimate of the convergence rate of the normalized geometric sums to exponential law involving the ideal probability metric of the second order is sharp. Some recent results concerning the convergence rates in Kolmogorov and Kantorovich metrics are extended as well. In contrast to many previous works, there are no assumptions that the summands of geometric sums are positive and have the same distribution. For the first time, an analogue of the Rényi theorem is established for the model of exchangeable random variables. Also within this model, a sharp estimate of convergence rate to a specified mixture of distributions is provided. The convergence rate of the appropriately normalized random sums of random summands to the generalized gamma distribution is estimated. Here, the number of summands follows the generalized negative binomial law. The sharp estimates of the proximity of random sums of random summands distributions to the limit law are established for independent summands and for the model of exchangeable ones. The inverse to the equilibrium transformation of the probability measures is introduced, and in this way a new approximation of the Pareto distributions by exponential laws is proposed. The integral probability metrics and the techniques of integration with respect to sign measures are essentially employed. Full article

In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the number of summands guaranteeing the prescribed value of the $(1-\alpha )$-quantile of the normalized Poisson sum for a given $\alpha \in (0,1)$. This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic $(1-\alpha )$-quantile of the sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution. Full article