Summation Of Infinite Series Research Articles

Performance analysis of orthogonal space‐time block codes over Nakagami‐qMIMO RFID backscattering channels

The authors employ the conditional moment generating function approach to analyse the performance of orthogonal space-time block codes over Nakagami- q (Hoyt) multiple-input multiple-output radio frequency identification backscattering channels. New exact and asymptotic symbol error rate expressions are derived for the case of two and four receiving antennas N = 2 , 4 . The exact expressions are in the form of a sum of infinite series while the asymptotic ones are in the closed form. The diversity order that the system can achieve is found to be L , where L is the number of tag antennas, and the performance of this system is found to be more sensitive to the channel condition (the q parameter) of the forward link than that of the backscattering link. The theoretical results (exact and asymptotic) are verified through comparison with simulation results.

Preliminary study on numerical solver of inhomogeneous burnup equations

The inhomogeneous burnup equation is often used for describing the time evolution of nuclides' depletion in nuclear systems which have a significant nuclide migration effect. However, lots of burnup calculations codes only deal with the homogeneous cases instead of the inhomogeneous ones, among them there are a few codes that can work only when the inhomogeneous term of the equation is constant. Based on the condition that the inhomogeneous term can be approximated by finite-order Taylor expansion, two methods are introduced to solve the inhomogeneous burnup equation whose inhomogeneous term is time dependent. For the first method, the transmutation trajectory analysis method is used to decompose the connections between nuclides into linear chains, for one chain the analytical solution is derived strictly by using the Laplace transform. For the second method, a solution of the inhomogeneous equation in the form of summation of infinite matrix series is first derived, and then the sum function of the series is found. Furthermore, the different-order nearly-best rational approximation function of the sum function is found by using Carathéodory-Fejér method. The error between the sum function and the rational function fluctuates in a certain range without exceeding a limit value, while the maximum error decreases exponentially with the order of rational function increasing. By adopting the nearly-best rational approximation, the summation of infinite matrix series converts into a finite expansion of matrix fraction, which is much easier to deal with. These two methods are implemented in the burnup calculation code JBURN and numerical tests are done through using two examples. The first example is a small-scale matrix example and the result shows that the results from the two methods agree well in at least 6 decimal precision together with the results from the reference solution. The second example is a large-scale problem based on real nuclides' reaction database, and the result shows that less than 1% among all nuclides have a deviation larger than 10% between two methods, while about 8% nuclides have a deviation larger than 0.01% and the remaining ones have a deviation smaller than 0.01%. These results validate the correctness and accuracy for each of the two methods. Finally, this paper provides a possible implementation process for solving inhomogeneous burnup equations which have other time-dependent forms of inhomogeneous term.

GLOBAL WELL-POSEDNESS OF THE PERIODIC CUBIC FOURTH ORDER NLS IN NEGATIVE SOBOLEV SPACES

We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces$H^{s}(\mathbb{T})$,$s>-\frac{1}{3}$, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in$H^{s}(\mathbb{T})$,$s>-\frac{9}{20}$, via the short-time Fourier restriction norm method. By following the argument in Guo–Oh for the cubic NLS, this also leads to nonexistence of solutions for the (nonrenormalized) 4NLS in negative Sobolev spaces. (ii) We then prove enhanced uniqueness in$H^{s}(\mathbb{T})$,$s>-\frac{1}{3}$, by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the$H^{s}$-energy functional, allowing us to introduce an infinite sequence of correction terms to the$H^{s}$-energy functional in the spirit of the$I$-method. In fact, the main novelty of this paper is this reduction of the$H^{s}$-energy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.

On interpolation functions for the number of k-ary Lyndon words associated with the Apostol–Euler numbers and their applications

The aim of this paper is to construct interpolation functions for the numbers of the k-ary Lyndon words which count n digit primitive necklace class representative on the set of the k-letter alphabet. By using the unified zeta-type function and the unification of the Apostol-type numbers which are defined by Ozden et al. (Comput Math Appl 60:2779–2787, 2010), we give an alternating series for the numbers of the k-ary Lyndon words, $$L_{k}\left( n\right) $$ in terms of the Apostol–Euler numbers and Frobenius–Euler numbers. We investigate various properties of these functions. Furthermore, applying higher order derivative operator to the interpolation functions for the Lyndon words, we derive ODEs including Stirling-type numbers, the Apostol–Euler numbers, the unified zeta-type functions and also combinatorial sums. By using recurrence relation of the Apostol–Euler numbers, we give computation algorithms for computing not only the Apostol–Euler numbers but also the interpolation functions of the numbers $$L_{k}\left( n\right) $$ . We also give some remarks, observations and computations for sums of infinite series including these interpolation functions.

Closed-Form Exact Solutions for Viscously Damped Free and Forced Vibrations of Longitudinal and Torsional Bars

This paper studies the viscously damped free and forced vibrations of longitudinal and torsional bars. The method is exact and yields closed form solution for the vibration displacement in contrast with the well-known eigenfunction superposition (ES) method, which requires expression of the distributed forcing functions and displacement response functions as infinite series sums of free vibration eigenfunctions. The viscously damped natural frequency equation and the critical viscous damping equation are exactly derived for the bars. Then the viscously damped free vibration frequencies and corresponding damped mode shapes are calculated and plotted, aside from the undamped free vibration and corresponding mode shapes typically computed and used in vibration problems. The longitudinal or torsional amplitude versus forcing frequency curves showing the forced response to distributed loadings are plotted for various viscous damping parameters. It is found that the viscous damping affects the natural frequencies and the corresponding mode shapes of longitudinal and torsional bars, especially for the fundamental frequency.

Solution of the Neumann Problem of Diffraction by a Strip Using the Wiener–Hopf Method: Short-Wave Asymptotic Solutions

Plane wave diffraction by a strip is considered by the Wiener–Hopf method and the simple short-wave asymptotic solutions are obtained. The Neumann boundary value problem is reduced to the solutions of the Fredholm integral equations of the second kind, and the exact solution in the form of the sum of infinite series has been found by the method of successive approximations. With the help of the saddle point method, the integral operators are calculated and the asymptotic solutions of system are found. The outcomes are compared with supra known asymptotic solutions taking into account tertiary diffraction. Quarternary diffraction was solved and the calculation methodology for the subsequent diffractions was shown.

Дослідження ефективності використання представлень функцій гіперкомплексного змінного

It is investigated the reduction of the amount of calculation when using representations for functions of hyper-complex variable such as exponent, trigonometric and hyperbolic ones as compared with their direct calculation utilizing the summation of infinite power series. Tabl.: 2. Refs: 14 titles.

Convergence and applications of some solutions of the confluent Heun equation

We study the convergence of a group of solutions in series of confluent hypergeometric functions for the confluent Heun equation. These solutions are expansions in two-sided infinite series (summation from minus to plus infinity) which are interpreted as a modified version of expansions proposed by Leaver (1986). We show that the two-sided solutions yield two nonequivalent groups of one-sided series solutions (summation from zero to plus infinity). In the second place, we find that one-sided solutions of one of these groups can be used to solve an equation which describes a time-dependent two-level system of Quantum Optics. For this problem, in addition to finite-series solutions, we obtain infinite-series wavefunctions which are convergent and bounded for any value of the time t, and vanish when t goes to infinity.

On Secrecy of a Multi-Antenna System with Eavesdropper in Close Proximity

This letter investigates secrecy performance of a wiretap channel where an eavesdropper is located in close proximity to legitimate receiver. In the system, both receivers equip multiple antennas and use maximal ratio combining (MRC) to harvest diversity gains. In order to analyze the impact of eavesdropping in close proximity, we derive closed-form expressions for the system performance in terms of both secrecy capacity and outage probability. Since the derived expressions involve infinite series sums, finite series approximations are then presented with guaranteed truncation error. Moreover, we also characterize the asymptotic behavior of secrecy outage. Some insightful observations are accordingly manifested with numerical verifications.

Acceleration of Spectral Domain Immitance Approach for Generalized Multilayered Shielded Microstrips Using the Levin’s Transformation

This letter reports on the acceleration of the infinite series summation in the multilayered shielded microstrip with multiple strips by using the spectral domain immitance approach (SDIA) and the Levin’s transformation. This technique does not need the asymptotic expansion for the Green’s functions and the basis functions by recasting the matrix elements into the form that is evaluated by the Levin’s transformation. It greatly saves the work for finding the asymptotic expansion coefficients. Using only the first leading term extraction, the overall acceleration performance is further improved. Convergence properties of SDIA by using this method match with or are even better than other acceleration techniques with high order leading term extraction.

A novel approach to accelerate spectral domain approach for shielded microstrip lines using the Levin transformations and summation-by-parts

A novel approach, which uses the Levin transformations or the hybrid of the Levin transformations and summation-by-parts, is presented for the acceleration of the slowly convergent series that asymptotically behave as 1/nk and sinusoidal functions divided by nk. This approach does not need the asymptotic expansion for the Green's functions and the Bessel functions, which saves the work for finding the asymptotic expansion coefficients. This approach has been applied to the acceleration of the infinite series summation in the shielded microstrip problem solved by the spectral domain approach (SDA) for obtaining accurate solutions of the propagation constant. Effective criteria of calculating the number of terms used in direct summation before applying the Levin transformations have been developed in this application. This approach can be easily extended to handle the multilayered shielded microstrip structure.

Application of Homotopy Analysis Method to Option Pricing Under Lévy Processes

Option pricing under the Levy process has been considered an important research direction in the field of financial engineering, where a closed-form expression for the standard European option is available due to the existence of analytically tractable characteristic function according to the Levy–Khinchin representation. However, this approach cannot be applied to exotic derivatives (such as barrier options) directly, although a large volume of exotic derivatives are actively traded in the current options market. An alternative approach is to solve the corresponding partial integro-differential equation (PIDE) numerically, which is, in fact, time-consuming and is not computationally tractable in general. In this paper, we apply the so-called homotopy analysis method (HAM) to solve the corresponding PIDE in a semi analytic form, being obtained from the following three steps: (1) Apply the Fourier transform to convert the PIDE to an ordinal differential equitation (ODE), and construct a differential system of ODEs. (2) Solve the system of ODEs, where each differential equation is shown to have an analytical solution. (3) Express the option price using the sum of infinite series, where each term may be expressed analytically and derived by applying Steps (1) and (2) recursively. To illustrate our technique more precisely, we take the variance gamma model as an example and provide the semi-analytic form. Numerical examples demonstrate a fast convergence of our proposed method to the prices of European and down-and-out call options with a few number of terms. Note that this method is easy to implement and can be applied to other types of options under general Levy processes.

Network deconvolution as a general method to distinguish direct dependencies in networks

Recognizing direct relationships between variables connected in a network is a pervasive problem in biological, social and information sciences as correlation-based networks contain numerous indirect relationships. Here we present a general method for inferring direct effects from an observed correlation matrix containing both direct and indirect effects. We formulate the problem as the inverse of network convolution, and introduce an algorithm that removes the combined effect of all indirect paths of arbitrary length in a closed-form solution by exploiting eigen-decomposition and infinite-series sums. We demonstrate the effectiveness of our approach in several network applications: distinguishing direct targets in gene expression regulatory networks; recognizing directly-interacting amino-acid residues for protein structure prediction from sequence alignments; and distinguishing strong collaborations in co-authorship social networks using connectivity information alone.

Efficient and Accurate Approximation of Infinite Series Summation Using Asymptotic Approximation and Super Convergent Series

We present an approach for very quick and accurate approximation of infinite series summation arising in electromagnetic problems. This approach is based on using asymptotic expansions of the arguments and the use of super convergent series to accelerate the convergence of each term. It has been validated by obtaining very accurate solution for propagation constant for shielded microstrip lines using spectral domain approach (SDA). In the spectral domain analysis of shielded microstrip lines, the elements of the Galerkin matrix are summations of infinite series of product of Bessel functions and Green's function. The infinite summation is accelerated by leading term extraction using asymptotic expansions for the Bessel function and the Green's function and the summation of the leading terms is carried out using the super convergent series.

On a treatment of coupled integral equations in nonlinear unilateral contact problem of plates

Application of finite Hankel integral transforms to the problem of a uniformly loaded unilaterally supported rectangular thin plates, the solution of partial differential equation can then be reduced to two coupled Fredholm integral equations of the second kind that obtained from the coupled dual-series equations of mixed boundary conditions, and also together with two constraint integral equations resulted from both of zero corner force condition and compatibility of edge displacements near each plate corner that loses contact. Owing to the summation of many different infinite series in the kernels and inhomogeneous parts of equations for the requirements of attainable highest accuracy and reducible computational time, the modified mathematical manipulations are improved and required asymptotic expansion for large arguments of the Bessel and Struve functions is treated to accelerate the convergence of series rapidly. The latter is to avoid the undesirable lengthy computer runs in numerical computations. Consequently, this paper is devoted to demonstrate the mathematical description and numerical algorithms in order to solve the system of coupled integral equations.

New Recursive Representations for the Favard Constants with Application to Multiple Singular Integrals and Summation of Series

There are obtained integral form and recurrence representations for some Fourier series and connected with them Favard constants. The method is based on preliminary integration of Fourier series which permits to establish general recursion formulas for Favard constants. This gives the opportunity for effective summation of infinite series and calculation of some classes of multiple singular integrals by the Favard constants.

EFFICIENT AND ACCURATE APPROXIMATION OF INFINITE SERIES SUMMATION USING ASYMPTOTIC APPROXIMATION AND FAST CONVERGENT SERIES

EFFICIENT AND ACCURATE APPROXIMATION OF INFINITE SERIES SUMMATION USING ASYMPTOTIC APPROXIMATION AND FAST CONVERGENT SERIES

A note on the Kluvánek integral

ABSTRACT For real valued functions, Igor Kluv´anek has introduced an integral, called Archimedean, which is equivalent to the Lebesgue integral. It is based on the summation of infinite series and it avoids measure. Modifying the definition by Kluv´anek, we introduce an integral which has some good and some bad properties and we compare it with the Lebesgue integral.

The Application of Probability Method on Mathematical Analysis

This paper summarizes the applications of probabilitymethod on mathematical analysis, including a class of generalized integra1 and acquiring the sum of infinite series £ so that contact up the mathematical analysis and probability and statistics knowledge ,explain the inner contacts among different branches of mathematics in some discussions. Key words : Probability method; Mathematical analysis; Generalized integral; The sum of infinite series

Euler’s Contributions to Mathematics Useful in Statistics

An enormously productive 18th century mathematician, Euler made a number of wide-ranging advances that remain useful in statistics. These include beta and gamma functions, summation of infinite series, calculus of variations, Euler’s phi-function, Latin and Graeco-Latin squares, and generating functions. All these are outlined and illustrated in this article.