Advances on the theory of generalized Bessel functions and applications to multiphoton processes

The generalized Bessel functions (GBF) are presented within the context of a more comprehensive formalism. We also discuss the partial differential equations defining GBF and their modified versions of integer order (MGBF). Numerical results are given for the first-kind MGBF as well as for the GBF with the imaginary parameter, whose importance in multiphoton processes is extensively discussed.     

On the computation of spherical Bessel functions of complex arguments

In this short paper, problems with existing algorithms for computing spherical Bessel functions of complex arguments are reported. As a remedy, a revised algorithm based on the recurrence is proposed. The upper and lower limits for the starting order that can be used in the backward recurrence to reach a desired accuracy are given. The proposed algorithm is stable and is capable of computing a wide range of arguments approaching machine accuracy.     

Structural modification of systems using discretization and generalized sampled-data hold functions

This paper investigates the decentralized control of LTI continuous-time plants using generalized sampled-data hold functions (GSHF). GSHFs can be used to modify the structure of the digraph of the resultant discrete plant, by removing certain interconnections in the discrete-time equivalent model to form a hierarchical system model of the plant. This is a new application of discretization and has, as its motivation, the design of decentralized controllers using centralized methods.     

Some infinite integrals involving products of exponential and bessel functions

This paper is concerned with the evaluation of some infinite integrals involving products of exponential and Bessel functions. These integrals are transformed, through some identities, into the expressions containing modified Bessel functions. In this way, the difficulties associated with the computations of infinite integrals with oscillating integrands are eliminated.     

On the numerical evaluation of an oscillating infinite series III

An oscillating infinite series involving product of Bessel function J _o(x) and an oscillating infinite series involving trigonometric function sin(x) were evaluated and computed numerically in [1] and [2] respectively. In this paper, an oscillating infinite series involving product of exponential, Bessel and trigonometric functions is evaluated. The series is transformed first into the sum of two infinite integrals by using contour integration and then the infinite integral with oscillating integrand is transformed through some identities into a finite integral containing modified Bessel function K ₁(x). Finally, theset two integrals are evaluated numerically without any computational difficulties at all.     

两类多输出逻辑函数的关系

首次给出了多输出广义部分Bent函数的定义并论证了其的存在,得到了多输出广义部分Bent函数的等价判别条件,给出了多输出p值广义部分Bent函数与多输出p值广义Bent函数的关系,并讨论了这两者的广义一阶Chrestenson谱的关系,为多输出p值广义部分Bent函数的构造提供了一种方法。     

Symbolic integration of a product of two spherical Bessel functions with an additional exponential and polynomial factor

We present a Mathematica package that performs the symbolic calculation of integrals of the form

(1)     

Gain/phase margin improvement using static generalized sampled-data hold functions

This paper considers the robust control of a finite-dimensional linear time-invariant (FDLTI) continuous-time plant by a static generalized sampled-data hold function (GSHF) controller. It is shown that it is possible to design a static GSHF controller to provide a gain margin as large as desired, and a phase margin of up to 90°. The design is straight-forward, and we illustrate it with an example.     

lsjk—a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions

Generalized log-sine functions appear in higher order ?-expansion of different Feynman diagrams. We present an algorithm for the numerical evaluation of these functions for real arguments. This algorithm is implemented as a C++ library with arbitrary-precision arithmetics for integer 0?k?9 and j?2. Some new relations and representations of the generalized log-sine functions are given.

Program summary

Title of program:lsjkCatalogue number:ADVSProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVSProgram obtained from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing terms: GNU General Public LicenseComputers:allOperating systems:POSIXProgramming language:C++Memory required to execute:Depending on the complexity of the problem, at least 32 MB RAM recommendedNo. of lines in distributed program, including testing data, etc.:41 975No. of bytes in distributed program, including testing data, etc.:309 156Distribution format:tar.gzOther programs called:The CLN library for arbitrary-precision arithmetics is required at version 1.1.5 or greaterExternal files needed:noneNature of the physical problem:Numerical evaluation of the generalized log-sine functions for real argument in the region 0<θ<π. These functions appear in Feynman integralsMethod of solution:Series representation for the real argument in the region 0<θ<πRestriction on the complexity of the problem:Limited up to , and j is an arbitrary integer number. Thus, all function up to the weight 12 in the region 0<θ<π can be evaluated. The algorithm can be extended up to higher values of k(k>9) without modificationTypical running time:Depending on the complexity of problem. See text below.     

Discrete-time loop transfer recovery via generalized sampled-data hold functions based compensator

Loop transfer recovery (LTR) techniques are known to enhance the input or output robustness properties of linear quadratic gaussian (LQG) designs. One restriction of the existing discrete-time LQG/LTR methods is that they can obtain arbitrarily good recovery only for minimum-phase plants. A number of researchers have attempted to devise new techniques to cope with non-minimum-phase plants and have achieved some degrees of success.^6-9 Nevertheless, their methods only work for a restricted class of non-minimum-phase systems. Here, we explore the zero placement capability of generalized sampled-data hold functions (GSHF) developed in Reference 14 and show that using the arbitrary zero placement capability of GSHF, the discretized plant can always be made minimum-phase. As a consequence, we are able to achieve discrete-time perfect recovery using a GSHF-based compensator irrespective of whether the underlying continuous-time plant is minimum-phase or not.     

Error bound for the generalized complementarity problem with analytic functions

In this paper, we consider the global error bound for the generalized complementarity problem (GCP) with analytic functions. Based on the new technique, we establish computable global error bound under milder conditions, which refines the previously known results.     

The positive real lemma and construction of all realizations of generalized positive rational functions

We here extend the well known positive real lemma (also known as the Kalman-Yakubovich-Popov lemma) to a complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, this approach enables us to characterize systems which can be brought, through a static output feedback, to be generalized positive.     

A numerical method for deadbeat control of generalized state-space systems

In this paper we give a new numerical method for constructing a rank m correction BF to an n × n matrix A, such that the generalized eigenvalues of λE−(A+BF) are all at λ = 0. In the control literature, this problem is known as ‘deadbeat control’ of a generalized state-space system Ex_i+1 = Ax_i + Bu_i, whereby the matrix F is the ‘feedback matrix’ to be constructed.     

The novel generating algorithm and properties of hybrid-P-ary generalized bridge functions

1 Introduction Copy theory is a kind of signal generation technology, built on the research of Walsh function. The main content of the copy theory includes generating non-sine orthogonal functions, utilizing shift and symmetric copy methods and control in…     

A note on generalized hold functions

In this note, we study the generalized hold functions for use of digital control of continuous-time systems. First we define and solve the problem of simultaneous stabilization and minimization. It is shown that every plant with a strictly proper transfer function is simultaneously stabilizable and minimizable, provided that the sampling interval is small enough. We also consider robust designs of generalized hold functions. A condition is given for the stability of a system using generalized hold functions in the presence of uncertainty.     

A new approach to the epsilon expansion of generalized hypergeometric functions

Assuming that the parameters of a generalized hypergeometric function depend linearly on a small variable ε$\epsilon$, the successive derivatives of the function with respect to that small variable are evaluated at ε=0$\epsilon =0$ to obtain the coefficients of the ε$\epsilon$-expansion of the function. The procedure, which is quite naive, benefits from simple explicit expressions of the derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols with respect to their argument. The algorithm may be used algebraically, irrespective of the values of the parameters. It reproduces the exact results obtained by other authors in cases of especially simple parameters. Implemented numerically, the procedure improves considerably, for higher orders in ε$\epsilon$, the numerical expansions given by other methods.     

单片机系统数字滤波的BESSEL算法

为克服随机干扰引入的误差，硬件上可采用滤波技术，软件上可采用软件算法实现数字滤波，但每种单片机软件算法都有自己的优劣和相应适用范围。本文根据误差理论中的Bessel公式和滑动平均数字滤波法的思想，提出一种数字滤波Bessel算法，具有高可靠性，并且给出采用MCS-51指令编写的汇编子程序。     

Jordan type inequalities involving the Bessel and modified Bessel functions

In this work, two general forms of Jordan’s inequalities for the Bessel and modified Bessel functions are established, and proved by using l’Hospital’s rule for monotonicity and some properties of the spherical Bessel and the modified spherical Bessel functions of the first kind. The applications of the results above give two new infinite series for the Bessel and modified Bessel functions.