Summation Of Infinite Series Research Articles

Sums of infinite series involving the Riemann zeta function

Sums of infinite series involving the Riemann zeta function

Some infinite series summations involving linear recurrence relations of order 2 and 3

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of $\sum_{n=0}^{\infty}{\frac{a_{mn}}{10^{n+1}}, \ m=1,2,3}$, including their convergence criteria, and applied to many standard sequences, as particular cases of a generic $\left\{a_n\right\}$. The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

A simple yet efficient solution for TE scattering from a right trapezoidal groove

AbstractA simple yet efficient solution to the problem of transverse electric (TE) scattering by a two‐dimensional right trapezoidal groove placed in a perfect electric conductor (PEC) surface is presented. Using superposition of plane waves and applying the Neumann boundary condition at the groove walls, the transverse magnetic field is expressed as the sums of infinite series of waveguide modes. Considering the equivalent magnetic current on the aperture, a Fredlholm's integral equation with logarithmic singular kernel is extracted and then discretised by a collocation method based on a piecewise constant approximant. The extracted linear system of equations can be easily solved by the matrix methods for the unknown coefficients. Numerous examples are presented to verify the results and show the ability of the proposed method. Finally, this method is employed to examine the effect of the groove parameters and incidence angle on the scattering signatures.

Sums of infinite series involving the Riemann zeta function II

Sums of infinite series involving the Riemann zeta function II

Planck length in classical and quantum Hamiltonian formulations of general relativity

The physical meaning of the Planck length (ell _{P}) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the Oleft( ell _{P}^{0}right) contribution of the series is ultimately permitted, namely the unique one which is independent of ell _{P}. Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through Oleft( ell _{P}^{2}right) . Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.

Artificial Intelligence-Aided Computation for Green’s Function of the Spherical Layered Earth

Currently, there are considerable challenges associated with the theoretical derivation and numerical calculation of Green’s function for spherical layered earths. In order to study the coupling effect between wide-area interconnected systems and the earth, this earth model is widely used. Artificial intelligence methods have been applied in order to overcome these challenges. First, Green’s function in the form of infinite Legendre series is derived using intelligent symbolic operations, and the weight function of Legendre series is recursively computed by means of a recursive algorithm. It is on this basis that the intelligent complex image method has been developed for solving spherical layered Green’s functions, which transforms the summation of infinite series of Green’s functions into the superposition of complex image potentials. Additionally, the upper error limit of the algorithm has been determined, and examples have been provided to demonstrate the accuracy of the algorithm. A solution based on multiple precision algorithms has been proposed to resolve the numerical singularity problem and the extremely slow convergence of the spherical layered Green’s function at the earth’s scale. It has been demonstrated that the intelligent complex image method has advantages in terms of computing speed and accuracy. A spherical layered Green’s function at the earth scale is presented in this paper as an effective and intelligent solution.

Partial sums of analytic functions defined by binomial distribution and negative binomial distribution

ABSTRACT The study of statistical distributions in a complex variable is one of the most vibrant areas of research. The complex analogue of many distributions has been studied. This article introduces the complex analogue of Binomial distribution and Negative Binomial distribution and studies certain geometrical properties of these analogues. It comprises the study of analytic functions defined by using the complex versions of Binomial distribution and Negative Binomial distribution functions. It includes the problems of finding the lower bounds of real parts of certain ratios of partial sums to the infinite series sums for these analytic functions. Such lower bounds are determined for the said analytic functions, for their first derivatives and for the Alexander transformation of these analytic functions.

On modified Anderson-Darling test statistics with asymptotic properties

This study proposes two modified Anderson-Darling test statistics that emphasize the upper or lower tails of distribution. Under the finite sample sizes, various models are used to estimate the upper tail percentage points of the proposed test statistic. The limiting distribution of the proposed test statistics is derived with a sum of infinite series. The polynomial approximation, the method of Durbin and Knott and the normal and non normal saddlepoint approximations are used to approximate the limiting distribution of the suggested test statistics. Simulations are used to investigate the accuracy of various approximations. The asymptotic power of the proposed statistics is obtained to compare with various statistics. The proposed test statistics are illustrated by the analysis of real data.

Calculation of indoor air temperature using dimensionless parameters for integrated climate control systems

Introduction. It is noted that the improvement of the technology for calculating the thermal regime of premises in the conditions of automation of climate systems is still relevant. The aim of the work is to find a universal dependence of the air temperature in the building premises on time in conditions of a jump in heat access or heat loss for the integral law of regulating the compensatory heat flow from microclimate systems. In the form of a scientific hypothesis, we consider the statement about the power nature of such a dependence with the presence of a maximum and an asymptotic tendency to zero.
 Materials and methods. The study involves the use of basic equations that relate the most significant components of heat flows in rooms serviced by microclimate systems equipped with integrated controllers under conditions of abrupt changes in thermal disturbances. The method of dimension analysis is used to identify dimensionless complexes that are essential for obtaining engineering dependencies, as well as software calculation of sums of infinite series with a given accuracy, numerical solution of nonlinear equations, and the method of power series economization.
 Results. An analytical expression is found for changes in room temperature under integrated control of climate equipment under conditions of a jump in heat availability, which has the form of an infinite series in degrees of a dimensionless parameter that characterizes the properties of the room and the automation system. A simplified expression for the deviation of air temperature is obtained and a formula for the required control time is derived, as well as its estimation is given on the example of one residential building in the climatic conditions of Moscow.
 Conclusions. It is shown that the dependence of the air temperature in a room serviced by microclimate systems with integrated controllers on time is represented in a universal dimensionless form, suitable for any objects regardless of their specific characteristics. The previously discovered relationships for the moment of maximum deviation and the value of the dynamic control error depending on the air exchange of the room, the transmission coefficient of the controller and the room’s own heat stability are confirmed and refined.

Approximating the Sum of Infinite Series of Non Negative Terms with reference to Integral Test

This paper describes a method of obtaining approximate sum of infinite series of positive terms by using integrals under its historical background. It has shown the application of improper integrals to determine whether the given innate series is convergent or divergent. Here, the limits of the integrals and the series usually extend to infinity though they may be slowly convergent. We have also established a relation to approximate the sum of infinite series of positive terms with a suitable example.

The solution of vibroacoustic linear systems as a finite sum of transmission paths

Linear systems are frequently encountered in low, mid and high vibroacoustics modelling of mechanical built-up structures. It has recently been proved that the solution to those systems can be always factorized as an infinite (weighted) Neumann series summation, which accounts for signal transmission through paths connecting system elements. The key to path expansion relies on the concept of direct transmissibility. In this work, we explore some additional theoretical aspects of transmissibility-based transmission path analysis (TPA), which is known to constitute a valuable tool to remedy noise and vibration problems. In particular, we show that it is also possible to expand the solution of a matrix linear system as a finite summation of transmission paths. Furthermore, our goal is to provide mathematical and physical insight into such path factorization. As regards the former, we exploit the relationship between graph theory and matrix algebra to interpret the terms appearing in the series expansion as combinations of open and closed paths in a graph. In what concerns the second, two benchmark examples are addressed that benefit from the graph theory outcomes. The first one consists of a mass-damping-stiffness system representative of vibroacoustic modelling at low frequencies. A relation is established between the relative weights of the paths, the global system resonances and the resonances of complementary systems, which contain elements not belonging to the paths. The second example involves a statistical energy analysis (SEA) model made of connected plates. The meaning of the relative weights of open paths in the finite expansion for energy transmission between SEA subsystems is analyzed and compared to the results of infinite SEA path factorization.

Reliability properties of [formula omitted]-out-of-[formula omitted] systems with one cold standby unit

In this paper, we study reliability properties of a k-out-of-n system with a single cold standby unit. We mainly focus on the case when the system operates in discrete time. In order to describe its aging behavior we consider three different mean residual life functions. By using some properties of order statistics we present several monotonicity results associated with these reliability characteristics. Since the calculation of the described quantities requires finding sums of infinite series, we provide a procedure to approximate them with an error not greater than a desired value. As an illustration we consider three special cases when the component lifetimes have geometric, negative binomial and discrete Weibull distributions.

ON APPROXIMATION OF SUM OF CONVERGENT SERIES

The paper has given a clear opinion on the progress of environmental protection and sustainability in the Nigerian context. The environmental regulations scenario in the country is marred by malpractices and corruption more stringent policy enforcement will help in the achievement of environmental protection. This paper deals with a specialized method of approximating the sum of an infinite series containing positive terms which are monotonically decreasing. The analysis has been done by taking some references done by the great mathematician Leonhard Euler with some special examples. Consequently, we have established a relation to estimate the sum of convergent infinite series

A Mode-Matching Solution for TE-Backscattering from an Arbitrary 2D Rectangular Groove in a PEC

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.

On supercongruences for truncated sums of squares of basic hypergeometric series

Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.

Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach

In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous methods to derive known and new properties of the Mittag-Leffler functions. Moreover, it is shown that sums of infinite series of the Mittag-Leffler functions can be expressed as convolution integrals, while the derivatives of the Mittag-Leffler functions with respect to their parameters are expressible as double convolution integrals. The derivatives can also be obtained from integral representations of the Mittag-Leffler functions. On the other hand, direct differentiation of the Mittag-Leffler functions with respect to parameters produces an infinite power series, whose coefficients are quotients of the digamma and gamma functions. Closed forms of these series can be derived when the parameters are set to be integers.

Transmission Line Representation of the Capacitive Via-Plate Interaction Toward a Capacitor-Free Via Model

Conventional physics-based via circuit model takes into account the capacitive coupling between via barrel and power/ground plane by a via-plate capacitor, whose calculation relies on a complex expression involving an infinite summation of Hankel functions. In this paper, a capacitor-free via model is proposed and realized by way of a novel transmission line representation of the via-plate capacitance. In particular, the via-plate capacitor is replaced by an open-circuited transmission line (OC-TL) with equal input admittance. The resultant capacitor-free model obviates the need of calculating the sum of infinite series. Instead, it only requires identifying the length of the OC-TL by simple closed-form expressions. Hence, the proposed efficient capacitor-free via model is able to facilitate the modeling of realistic multilayer printed circuit boards (PCBs) and packages. Both numerical and measurement results validate the proposed OC-TL-based via model.

Laplace Adomian and Laplace Modified Adomian Decomposition Methods for Solving Nonlinear Integro-Fractional Differential Equations of the Volterra-Hammerstein Type

In this work, we will combine the Laplace transform method with the Adomian decomposition method and modified Adomian decomposition method for semi-analytic treatments of the nonlinear integro-fractional differential equations of the Volterra-Hammerstein type with difference kernel and such a problem which the kernel has a first order simple degenerate kind which the higher-multi fractional derivative is described in the Caputo sense. In these methods, the solution of a functional equation is considered as the sum of infinite series of components after applying the inverse of Laplace transformation usually converging to the solution, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. Finally, examples are prepared to illustrate these considerations.

Analytical scheme for the solution of multi-higher Fractional Order Linear Integro-Differential Equations of Fredholm type with Variable Coefficients using Adomian and Modified Adomian Decomposition Methods

In this paper, Adomian decomposition method (ADM) and modified Adomian decomposition method (MADM) has been introduced to find the exact or approximate solution to a wide class of multi-higher fractional order linear integro-differential problems of Fredholm type with variable coefficients in which the fractional derivative is described in the Caputo sense. In this methods the solution of a functional equation is considered as the sum of infinite series of components usually converging to the solution based on the appearance of the noise terms, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. Finally, Numerical experiment prepared that modified Adomian decomposition solutions converges faster than Adomian decomposition solution and by several examples illustrate these considerations.

Classification of Irrational Numbers

An irrational number expressed in decimal form / infinite continued fractions or sum of infinite series of rational numbers, can only evaluate its approximate magnitude, but cannot represent it. There are different kinds of irrational numbers. One of the kinds is generated by extraction of ischolar_mains of the positive rational numbers. The deliberation shows that all irrational numbers of this kind are not of similar category, which has been defined as the ‘order of irrational numbers’.