Power Series Type Research Articles

Exact receptance function of a tapered AFG beam with nonlinearly varying ratios of beam properties carrying concentrated masses

This paper presents a method for establishing the exact receptance function of a tapered axially functionally graded (AFG) beam with nonlinear ratios of properties using the Adomian method. In current papers, the Adomian method was applied for linearly tapered beams where the geometric series was used conveniently. However, for nonuniform AFG beams with nonlinearly varying ratios of properties, the geometric series cannot be used, thus the other type of power series needs to be established and applied. In this paper, the derivation of the power series applied for obtaining the exact receptance function of a nonuniform AFG beam with nonlinearly varying ratios of properties is presented. Numerical simulation results of the receptance function of a tapered AFG beam with nonlinearly varying ratios of beam properties carrying concentrated masses are conducted and provided. The influences of the concentrated masses and the varying ratios of properties of beam on the receptance matrix are also investigated and presented.

Certain properties of a class of analytic functions involving the Mathieu type power series

<abstract><p>In this paper, we studied some geometric properties of a class of analytic functions related to the generalized Mathieu type power series. Furthermore, we have identified interesting consequences and some examples accompanied by graphical representations to illustrate the results achieved.</p></abstract>

Geometric Properties of Some Generalized Mathieu Power Series inside the Unit Disk

We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejér and Ozaki, we provide sufficient conditions for these functions to be close-to-convex or starlike inside the unit disk, and thus univalent. One of our proofs is assisted by symbolic computation.

Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series

A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

Linear recurrent relations, power series distributions, and generalized allocation scheme

Abstract We consider distributions of the power series type determined by the generating functions of sequences satisfying linear recurrence relations with nonnegative coefficients. These functions are represented by power series with positive radius of convergence. An integral limit theorem is proved on the convergence of such distributions to the exponential distribution. For the generalized allocation scheme generated by these linear relations a local normal theorem for the total number of components is proved. As a consequence of more general results of the author, a limit theorem is stated containing sufficient conditions under which the distributions of the number of components having a given volume converge to the Poisson distribution.

A fractional q-integral operator associated with a certain class of q-Bessel functions and q-generating series

This paper deals with Al-Salam fractional q-integral operator and its application to certain q-analogues of Bessel functions and power series. Al-Salam fractional q-integral operator has been applied to various types of q-Bessel functions and some power series of special type. It has been obtained for basic q-generating series, q-exponential and q-trigonometric functions as well. Various results and corollaries are provided as an application to this theory.

Fractional power series and the method of dominant balances

This paper derives an explicit formula for a type of fractional power series, known as a Puiseux series, arising in a wide class of applied problems in the physical sciences and engineering. Detailed consideration is given to the gaps which occur in these series (lacunae); they are shown to be determined by a number-theoretic argument involving the greatest common divisor of a set of exponents appearing in the Newton polytope of the problem, and by two number-theoretic objects, called here Sylvester sets, which are complements of Frobenius sets. A key tool is Faà di Bruno’s formula for high derivatives, as implemented by Bell polynomials. Full account is taken of repeated roots, of arbitrary multiplicity, in the leading-order polynomial which determines a fractional-power expansion, namely the facet polynomial. For high multiplicity, the fractional powers are shown to have large denominators and contain irregularly spaced gaps. The orientation and methods of the paper are those of applications, but in a concluding section we draw attention to a more abstract approach, which is beyond the scope of the paper.

Generalized Cauchy–Kovalevskaya extension and plane wave decompositions in superspace

The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the biaxial Dirac operator $$\partial _{\mathbf{x}} +\partial _{\mathbf{y}}$$ . In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of the generalized CK-extension into plane waves integrated over the supersphere. Moreover, these results are applied to obtain a decomposition of the Cauchy kernel in superspace into monogenic plane waves, which shall be useful for inverting the super Radon transform.

Limit theorems for some classes of power series type distributions

AbstractSeveral classes of distributions of power series type with finite and infinite radii of convergence are considered. For such distributions local limit theorems are obtained as the parameter of distribution tends to the right end of the interval of convergence. For the case when the convergence radius equals to 1, we prove an integral limit theorem on the convergence of distributions of random variables (1 − x)ξxas x → 1− to the gamma-distribution (ξx is a random variable with corresponding distribution of the power series type). The proofs are based on the steepest descent method.

A nuclear matter calculation with the tensor-optimized Fermi sphere method with central interaction

Abstract The tensor-optimized Fermi sphere (TOFS) theory is applied first for the study of the property of nuclear matter using the Argonne V4$^\prime$$NN$ potential. In the TOFS theory, the correlated nuclear matter wave function is taken to be a power-series type of the correlation function $F$, where $F$ can induce central, spin–isospin, tensor, etc. correlations. This expression has been ensured by a linked cluster expansion theorem established in the TOFS theory. We take into account the contributions from all the many-body terms arising from the product of the nuclear matter Hamiltonian $\mathcal{H}$ and $F$. The correlation function is optimally determined in the variation of the total energy of nuclear matter. It is found that the density dependence of the energy per particle in nuclear matter is reasonably reproduced up to the nuclear matter density $\rho \simeq 0.20$ fm$^{-3}$ in the present numerical calculation, in comparison with other methods such as the Brueckner–Hartree–Fock approach.

Autonomous power generation facilities, hybrid structures comprising renewable energy sources

Introduction: the implementation of land development programmes in the polar regions, Siberia andFar East turns more complex due to power shortages in these areas. High electric energy rates and the need to maintain special purpose buildings encourage engineers to develop autonomous energy supply systems.Methods: the co-authors have analyzed several types of autonomous power supply facilities and series of autonomous power supply units designated for the power supply to the production machinery installed as part of various economic and special-purpose facilities, located at a long distance from industrial sources of electric energy.Findings and discussion: the co-authors present an algorithm for the selection of the configuration of an autonomous power supply system. The co-authors employ the findings of their analysis to substantiate the choice of a solar power station as a supplementary source of energy used as part of hybrid power generation systems.Conclusion: the algorithm developed by the co-authors may be used to design an efficient autonomous power supply system comprising hybrid units.

Fractional transform methods for coupled system of time fractional derivatives of non-homogeneous Burgers’ equations arise in diffusive effects

The coupled system of time fractional derivatives of non-homogeneous Burgers’ equations is solved both analytically and numerically. The approximate analytical solutions of power series type are obtained by the fractional homotopy analysis transform method, which is verified numerically by the coupled fractional reduced differential transform method. Particularly, the present analytical solutions are compared with the results available in the literature for several special cases, and very excellent agreement was found. An error analysis is done to compute the average squared residual errors of those analytical approximations for other cases. The present analysis shows that the proposed technique has very excellent efficiency to give solutions with high precision. On the other hand, it is found that the optimal convergence control parameter derived here not only controls the convergence of the present solutions but also leads to identifying multiple solutions.

Tensor optimized Fermi sphere method for nuclear matter—Power series correlated wave function and a cluster expansion

A new formalism, called “tensor optimized Fermi sphere (TOFS) method”, is developed to treat the nuclear matter using a bare interaction among nucleons. In this method, the correlated nuclear matter wave function is taken to be a power series type, ΨN=[∑n=0N(1∕n!)Fn]Φ0 and an exponential type, Ψex=exp(F)Φ0, with the uncorrelated Fermi-gas wave function Φ0, where the correlation operator F can induce central, spin–isospin, tensor, etc. correlations, and Ψex corresponds to a limiting case of ΨN (N→∞). In the TOFS formalism based on Hermitian form, it is shown that the energy per particle in nuclear matter with Ψex can be expressed in terms of a linked-cluster expansion. On the basis of these results, we present the formula of the energy per particle in nuclear matter with ΨN. We call the Nth-order TOFS calculation for evaluating the energy with ΨN, where the correlation functions are optimally determined in the variation of the energy. The TOFS theory is applied for the study of symmetric nuclear matter using a central NN potential with short-range repulsion. The calculated results are fairly consistent to those of other theories such as the Brueckner–Hartree–Fock approach etc.

Differential Inverse Power Series Rings with Quasi-Armendariz-like Condition

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, [Formula: see text]-quasi-Armendariz), is introduced and studied. It is shown that the [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime [Formula: see text]-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

Limit theorems for some classes of power series type distributions

Рассмотрены некоторые классы распределений типа степенного ряда с конечным и бесконечным радиусами сходимости. Для этих распределений получены локальные предельные теоремы в ситуации, когда параметр распределения стремится к правой граничной точке интервала сходимости. Для случая, когда радиус сходимости равен 1, получена интегральная предельная теорема о сходимости распределений случайных величин $(1-x)\xi_x$ при $x\to 1-$ ($\xi_x$ - случайная величина, имеющая соответствующее распределение типа степенного ряда) к гамма-распределению. В доказательствах используется метод перевала.

On nilpotent elements of ore extensions

Let [Formula: see text] be an associative ring with unity, [Formula: see text] be an endomorphism of [Formula: see text] and [Formula: see text] an [Formula: see text]-derivation of [Formula: see text]. We introduce the notion of [Formula: see text]-nilpotent p.p.-rings, and prove that the [Formula: see text]-nilpotent p.p.-condition extends to various ring extensions. Among other results, we show that, when [Formula: see text] is a nil-[Formula: see text]-compatible and [Formula: see text]-primal ring with [Formula: see text] nilpotent, then [Formula: see text]; and when [Formula: see text] is a nil Armendriz ring of skew power series type with [Formula: see text] nilpotent, then [Formula: see text] where [Formula: see text] is the set of nilpotent elements of [Formula: see text]. These results extend existing results to a more general setting.

Power Series Rings Over Prüfer v-multiplication Domains. II

AbstractLet D be an integral domain, X1(D) be the set of height-one prime ideals of D, {Xβ} and {Xα} be two disjoint nonempty sets of indeterminates over D, D[{Xβ}] be the polynomial ring over D, and D[{Xβ}][[{Xα}]]1 be the first type power series ring over D[{Xβ}]. Assume that D is a Prüfer v-multiplication domain (PvMD) in which each proper integral t-ideal has only finitely many minimal prime ideals (e.g., t-SFT PvMDs, valuation domains, rings of Krull type). Among other things, we show that if X1(D) = Ø or DP is a DVR for all P ∊ X1(D), then D[{Xβ}][[{Xα}]]1D−{0} is a Krull domain. We also prove that if D is a t-SFT PvMD, then the complete integral closure of D is a Krull domain and ht(M[{Xβ}][[{Xα}]]1) = 1 for every height-one maximal t-ideal M of D.

Nilpotent Elements in Skew Polynomial Rings

Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We also prove that, if is nil--compatible and nil-Armendariz ring of power series type with nilpotent, then . We show that, if is a nil-Armendariz ring of power series type, with nilpotent and nil--compatible ring, then As a consequence, several known results are unified and extended to the more general setting. Also examples are provided to illustrate our results.

POWER SERIES RINGS OVER PRÜFER v-MULTIPLICATION DOMAINS

Let D be an integral domain, {} be a nonempty set of indeterminates over D, and be the first type power series ring over D. We show that if D is a t-SFT v-multiplication domain, then is a Krull domain, and is a v-multiplication domain if and only if D is a Krull domain.

Log-convexity and log-concavity for series in gamma ratios and applications

Polynomial sequence {Pm}m≥0 is q-logarithmically concave if Pm2−Pm+1Pm−1 is a polynomial with nonnegative coefficients for any m≥1. We introduce an analogue of this notion for formal power series whose coefficients are nonnegative continuous functions of a parameter. Four types of such power series are considered where the parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of q-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions. In the last section of the paper we present new inequalities for the Kummer function, the ratio of the Gauss functions and the generalized hypergeometric function obtained as direct applications of the general theorems.