Successive derivatives of Whittaker functions with respect to the first parameter

Algorithms are given for the computation of the nth derivatives of the Whittaker functions M_κ,μ(z) and W_κ,μ(z) with respect to the parameter κ. The algorithms are based on a convergent expansion, due to Buchholz, of M_κ,μ(z) in series of Bessel functions. Properties of the Buchholz polynomials and algorithms for evaluating the n-derivative of the reciprocal Gamma function are discussed in two appendices.     

fgh, a code for the calculation of Coulomb radial wave functions from series expansions

The code fgh is an up-dated version of a code coulfg (see [Seaton, Comput. Phys. Comm. 25 (1982) 87]), used for the calculation of the Coulomb functions f and g, analytic in the energy, for attractive potentials. The new code works for attractive and repulsive potentials and also gives the functions h which have simple asymptotic forms. There is an option to use either the variables (?,r) customary in atomic physics, or (for positive energies) (η,ρ) customary in nuclear physics. When (η,ρ) are used, the code also gives the functions F_?(η,ρ) and G_?(η,ρ).Use of series solutions can lead to loss of accuracy due to cancellation effects. fgh provides an indication of the number of significant figures lost due to cancellations.     

Multidimensional nuclear spectra compression using fast adaptive Fourier-based transforms

In the paper a class of fast adaptive Fourier-based transforms were used for spectroscopic data compression. These transforms are based on adaptive modification of the Cooley-Tukey's signal flow graph. The adaptive versions of the cosine, cosine-Haar and cosine-Walsh transform of various degrees were taken as a base for the experiments. The transform kernels are modified according to reference vectors representing a given class of processed data. The results obtained using these transforms for γ-γ ray coincidence spectra compression are presented and compared with the results obtained by use of classical transforms. Both classical and adaptive transforms can be used for off-line as well as for on-line compression.     

Integral minimization of the second derivative module of the boundary control of string vibrations with the fixed end

A problem of boundary control of string vibrations with the fixed end with a bounded control resource is studied. The integral of the second derivative module of the boundary control is minimized. Estimates of upper and lower bounds are found for the extremum. In the vibration suppression problem, at the time period T formulas of boundary control that is close to the optimal control with large values of the time period T are given.     

Design of dynamic suboptimal low-order controllers on basis of the H∞-criterion

A design problem of dynamic reduced-dimension controllers according to the boundedness condition of the H_∞-norm of a transfer matrix of a closed system is studied. The problem of controller deflation is related to the solution to degenerate filtering problems (no noise of measuring) and to degenerate control problems (no control in a manipulated output). It was demonstrated that these problems can be solved on basis of the modified 2-Riccati approach, in which the “filtrating” (for a singular filtrating problem) or the “control” (for a singular control problem) Lurie-Riccati equations have a low order. An example of an optimal reduced-dimension controller design is given that illustrates the obtained results.     

Processor sharing: A survey of the mathematical theory

During last few decades the Egalitarian Processor Sharing (EPS) has gained a prominent role in applied probability, especially, in queueing theory and its computer applications. While the EPS paradigm emerged in 1967 as an idealization of round-robin (RR) scheduling algorithm in time-sharing computer systems, it has recently capture renewed interest as a powerful concept for modeling WEB servers. This paper summarizes the most important results concerning the exact solutions for the M/GI/1 queue with egalitarian processor sharing. The material is drawn, mainly, from recent authors’ papers which are supplemented, in small degree, by other related results. Many of the further results are established under the direct influence of our earlier papers. Our main purpose is to give a survey of state-of-the-art with regard to main achievements of the contemporary theory of the M/GI/1 queueing system with processor sharing. The focus is on the methods and techniques of exact and asymptotic analysis of this queueing system. In contrast to the standard surveys, the abridged proofs (or their ideas) of some key theorems and corollaries are included in the paper. We outline recent developments in exact analysis of the M/GI/1-EPS queue with further emphasis on time-dependent (transient) probability distributions of the main characteristics. In particular, the present paper includes the results on the joint time-dependent distribution of the sojourn time of a job arriving at time t with the service demand (length) u, and of the number of jobs at time t- in the M/GI/1 queue with egalitarian processor sharing, which obtained in form of multiple transforms. We also show how the non-stationary solutions can be used to obtain known and new results which allow to predict the behaviour of the EPS queue and to yield additional insights into its new unexpected properties. We also discuss a number of limit theorems arising under the study of the processor sharing queues.     

Hidden solitons in the Zabusky–Kruskal experiment: Analysis using the periodic,inverse scattering transform

Recent numerical work on the Zabusky–Kruskal experiment has revealed, amongst other things, the existence of hidden solitons in the wave profile. Here, using Osborne’s nonlinear Fourier analysis, which is based on the periodic, inverse scattering transform, the hidden soliton hypothesis is corroborated, and the exact number of solitons, their amplitudes and their reference level is computed. Other “less nonlinear” oscillation modes, which are not solitons, are also found to have nontrivial energy contributions over certain ranges of the dispersion parameter. In addition, the reference level is found to be a non-monotone function of the dispersion parameter. Finally, in the case of large dispersion, we show that the one-term nonlinear Fourier series yields a very accurate approximate solution in terms of Jacobian elliptic functions.     

Omnidirectional edge detection

In this paper we propose a new method for extending 1-D step edge detection filters to two dimensions via complex-valued filtering. Complex-valued filtering allows us to obtain edge magnitude and direction simultaneously. Our method can be viewed either as an extension of n-directional complex filtering of Paplinski to infinite directions or as a variant of Canny’s gradient-based approach. In the second view, the real part of our filter computes the gradient in the x direction and the imaginary part computes the gradient in the y direction. Paplinski claimed that n-directional filtering is an improvement over the gradient-based method, which computes gradient only in two directions. We show that our omnidirectional and Canny’s gradient-based extensions of the 1-D DoG coincide. In contrast to Paplinski’s claim, this coincidence shows that both approaches suffer from being confined to the subspace of two 2-D filters, even though n-directional filtering hides these filters in a single complex-valued filter. Aside from these theoretical results, the omnidirectional method has practical advantages over both n-directional and gradient-based approaches. Our experiments on synthetic and real-world images show the superiority of omnidirectional and gradient-based methods over n-directional approach. In comparison with the gradient-based method, the advantage of omnidirectional method lies mostly in freeing the user from specifying the smoothing window and its parameter. Since the omnidirectional and Canny’s gradient-based extensions of the 1-D DoG coincide, we have based our experiments on extending the 1-D Demigny filter. This filter has been proposed by Demigny as the optimal edge detection filter in sampled images.     

一种基于数据流环境感知的共享过滤算法

为了有效过滤数据流中的有害信息,通常在数据流上注册大量查询,同时构建过滤器来计算这些查询.在多媒体流环境中,查询和过滤器常常是一种“多对多”的连接,也就是说,对于单个过滤器的计算可能会同时给出多个查询的结果.在这种情况下,如何排序所有的过滤器来获得最小的过滤代价变得非常重要.对于过滤器的排序一般依赖于3个指标：过滤器本身的执行代价c、过滤器连接的查询数目p以及过滤器对于随机样本判断为真的概率s.目前基于贪心的排序算法虽然在一定程度上给出了近似最优的结果,但是还存在以下两个问题：1) 指标s只是简单依据经验值设定,不能随着流的变化而自适应变化;2)将3个指标融合成一个代价函数进行排序,而没有深入分析各个指标之间的关系.考虑到以上方法存在的不足,提出一个层次排序算法(adaptive hierarchal ordering, AHO)来高效地过滤多媒体数据流.该算法首先依据过滤器的指标c和p进行分类,然后再在每个类别上按照s进行二次排序.在真实多媒体流环境中的过滤结果证明: AHO可以在不降低准确度的情况下,自适应调整过滤器顺序,其性能优于已有的贪心排序算法.     

TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95

We present TaylUR, a Fortran 95 module to automatically compute the numerical values of a complex-valued function's derivatives with respect to several variables up to an arbitrary order in each variable, but excluding mixed derivatives. Arithmetic operators and Fortran intrinsics are overloaded to act correctly on objects of a defined type taylor, which encodes a function along with its first few derivatives with respect to the user-defined independent variables. Derivatives of products and composite functions are computed using Leibniz's rule and Faà di Bruno's formula. TaylUR makes heavy use of operator overloading and other Fortran 95 features such as elemental functions.

Program summary

Program title: TaylURCatalogue identifier:ADXR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXR_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:noneProgramming language:Fortran 95Computer:Any computer with a conforming Fortran 95 compilerOperating system:Any system with a conforming Fortran 95 compilerNo. of lines in distributed program, including test data, etc.:6286No. of bytes in distributed program, including test data, etc:14 994Distribution format:tar.gzNature of problem:Problems that require potentially high orders of derivatives with respect to some variables, such as e.g. expansions of Feynman diagrams in particle masses in perturbative Quantum Field Theory, and which cannot be treated using existing Fortran modules for automatic differentiation [C.W. Straka, ADF95: Tool for automatic differentiation of a FORTRAN code designed for large numbers of independent variables, Comput. Phys. Comm. 168 (2005) 123-139, arXiv:cs.MS/0503014; S. Stamatiadis, R. Prosmiti, S.C. Farantos, auto_deriv: Tool for automatic differentiation of a FORTRAN code, Comput. Phys. Comm. 127 (2000) 343-355].Solution method:Arithmetic operators and Fortran intrinsics are overloaded to act correctly on objects of a defined type taylor, which encodes a function along with its first few derivatives with respect to the user-defined independent variables. Derivatives of products and composite functions are computed using Leibniz's rule and Faà di Bruno's formula.Restrictions:Memory and CPU time constraints may restrict the number of variables and Taylor expansion order that can be achieved. Loss of numerical accuracy due to cancellation may become an issue at very high orders.Unusual features:No mixed higher-order derivatives are computed. The complex conjugation operation assumes all independent variables to be real.Running time:The running time of TaylUR operations depends linearly on the number of variables. Its dependence on the Taylor expansion order varies from linear (for linear operations) through quadratic (for multiplication) to exponential (for elementary function calls).     

Design of consensus and adaptive consensus filters for distributed parameter systems

This work establishes an abstract framework that considers the distributed filtering of spatially varying processes using a sensor network. It is assumed that the sensor network consists of groups of sensors, each of which provides a number of state measurements from sensing devices that are not necessarily identical and which only transmit their information to their own sensor group. A modification to the local spatially distributed filters provides the non-adaptive case of spatially distributed consensus filters which penalize the disagreement amongst themselves in a dynamic manner. A subsequent modification to this scheme incorporates the adaptation of the consensus gains in the disagreement terms of all local filters. Both the well-posedness of these two consensus spatially distributed filters and the convergence of the associated observation errors to zero in appropriate norms are presented. Their performance is demonstrated on three different examples of a diffusion partial differential equation with point measurements.     

Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations

A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh. Examples illustrate the key steps of the algorithm. Through discussion and example, parallels are drawn to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed.

Program summary

Title of program: DDESpecialSolutions.mCatalogue identifier:ADUJProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUJProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: Created using a PC, but can be run on UNIX and Apple machinesOperating systems under which the program has been tested: Windows 2000 and Windows XPProgramming language used: Mathematica, version 3.0 or higherMemory required to execute with typical data: 9 MBNumber of processors used: 1Has the code been vectorised or parallelized?: NoNumber of lines in distributed program, including test data, etc.: 3221Number of bytes in distributed program, including test data, etc.: 23 745Nature of physical problem: The program computes exact solutions to differential-difference equations in terms of the tanh function. Such solutions describe particle vibrations in lattices, currents in electrical networks, pulses in biological chains, etc.Method of solution: After the differential-difference equation is put in a traveling frame of reference, the coefficients of a candidate polynomial solution in tanh are solved for. The resulting traveling wave solutions are tested by substitution into the original differential-difference equation.Restrictions on the complexity of the program: The system of differential-difference equations must be polynomial. Solutions are polynomial in tanh.Typical running time: The average run time of 16 cases (including the Toda, Volterra, and Ablowitz-Ladik lattices) is 0.228 seconds with a standard deviation of 0.165 seconds on a 2.4 GHz Pentium 4 with 512 MB RAM running Mathematica 4.1. The running time may vary considerably, depending on the complexity of the problem.     

GeM software package for computation of symmetries and conservation laws of differential equations

We present a recently developed Maple-based “GeM” software package for automated symmetry and conservation law analysis of systems of partial and ordinary differential equations (DE). The package contains a collection of powerful easy-to-use routines for mathematicians and applied researchers. A standard program that employs “GeM” routines for symmetry, adjoint symmetry or conservation law analysis of any given DE system occupies several lines of Maple code, and produces output in the canonical form. Classification of symmetries and conservation laws with respect to constitutive functions and parameters present in the given DE system is implemented. The “GeM” package is being successfully used in ongoing research. Run examples include classical and new results.

Program summary

Title of program: GeMCatalogue identifier: ADYK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYK_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputers: PC-compatible running Maple on MS Windows or Linux; SUN systems running Maple for Unix on OS SolarisOperating systems under which the program has been tested: Windows 2000, Windows XP, Linux, SolarisProgramming language used: Maple 9.5Memory required to execute with typical data: below 100 MegabytesNo. of lines in distributed program, including test data, etc.: 4939No. of bytes in distributed program, including test data, etc.: 166 906Distribution format: tar.gzNature of physical problem: Any physical model containing linear or nonlinear partial or ordinary differential equations.Method of solution: Symbolic computation of Lie, higher and approximate symmetries by Lie's algorithm. Symbolic computation of conservation laws and adjoint symmetries by using multipliers and Euler operator properties. High performance is achieved by using an efficient representation of the system under consideration and resulting symmetry/conservation law determining equations: all dependent variables and derivatives are represented as symbols rather than functions or expressions.Restrictions on the complexity of the problem: The GeM module routines are normally able to handle ODE/PDE systems of high orders (up to order seven and possibly higher), depending on the nature of the problem. Classification of symmetries/conservation laws with respect to one or more arbitrary constitutive functions of one or two arguments is normally accomplished successfully.Typical running time: 1-20 seconds for problems that do not involve classification; 5-1000 seconds for problems that involve classification, depending on complexity.     

Direct data-driven filter design for uncertain LTI systems with bounded noise

This paper investigates the filter design problem for linear time-invariant dynamic systems when no mathematical model is available, but a set of initial experiments can be performed where also the variable to be estimated is measured. Instead of using the initial experimental data to identify a model on the basis of which a filter is designed, these data are used to directly design a filter. Assuming norm-bounded disturbances and noises, a Set Membership formulation is followed. For classes of filters with exponentially decaying impulse response, approximating sets are determined that guarantee to contain all the solutions to the optimal filtering problem, where the aim is the minimization of the induced norm from disturbances to the estimation error. A method is proposed for designing almost-optimal linear filters with finite impulse response, whose worst-case filtering error is at most twice the lowest achievable one. In the H_∞ SISO case, an efficient technique is presented, that allows the evaluation of bounds on the guaranteed worst-case filtering error of the designed filter. Numerical examples illustrate the effectiveness of the proposed solution.     

Determining Liouvillian first integrals for dynamical systems in the plane

Here we present/implement a semi-algorithm to find Liouvillian first integrals of dynamical systems in the plane. The algorithm is based on a Darboux-type procedure to find the integrating factor for the system. Since the particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE), the Lsolver package presents a set of software routines in Maple for dealing with rational first order ODEs. The package present commands permitting research investigations of some algebraic properties of the system that is being studied.

Program summary

Program title:LsolverCatalogue number:ADZF_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZF_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:1513No. of bytes in distributed program, including test data, etc.:17 367Distribution format:tar.gzProgramming language:MapleComputer:Any computer running MapleOperating system:Windows ME, Windows XPClassification:4.3Nature of problem: Solution of rational first order ordinary differential equations.Solution method:The method of solution is based on a Darboux/PS type approach.Restrictions:If the integrating factor for the ODE under consideration presents Darboux Polynomials of high degree (>3) in the dependent and independent variables, the package may spend an impractical amount of time to obtain the solution.Unusual features:Our implementation not only searches for Liouvillian conserved quantities, but can also be used as a research tool that allows the user to follow all the steps of the procedure (for example, we can calculate the algebraic invariants curves and associated co-factors, the integrating factor etc). In addition, since our package is based on recent theoretical developments [J. Avellar, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Integrating first-order differential equations with Liouvillian solutions via quadratures: a semi-algorithmic method, J. Comput. Appl. Math. 182 (2005) 327-332], it can successfully solve a class of rational first order ODEs that were not solved by some of the best-known ODE solvers available.Running time:This depends strongly on the ODE, but usually under 4 seconds.     

HypExp 2, Expanding hypergeometric functions about half-integer parameters

In this article, we describe a new algorithm for the expansion of hypergeometric functions about half-integer parameters. The implementation of this algorithm for certain classes of hypergeometric functions in the already existing Mathematica package HypExp is described. Examples of applications in Feynman diagrams with up to four loops are given.

New version program summary

Program title:HypExp 2Catalogue identifier:ADXF_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXF_v2_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:106 401No. of bytes in distributed program, including test data, etc.:2 668 729Distribution format:tar.gzProgramming language:MathematicaComputer:Computers running MathematicaOperating system:Linux, Windows, MacRAM:Depending on the complexity of the problemSupplementary material:Library files which contain the expansion of certain hypergeometric functions around their parameters are availableClassification:4.7, 5Does the new version supersede the previous version?:YesNature of problem:Expansion of hypergeometric functions about parameters that are integer and/or half-integer valued.Solution method:New algorithm implemented in Mathematica.Reasons for new version:Expansion about half-integer parameters.Summary of revisions:Ability to expand about half-integer valued parameters added.Restrictions:The classes of hypergeometric functions with half-integer parameters that can be expanded are listed below.Additional comments:The package uses the package HPL included in the distribution.Running time:Depending on the expansion.     

A new framework for robust speech recognition in complex channel environments

Channel distortion is one of the major factors which degrade the performances of automatic speech recognition (ASR) systems. Current compensation methods are generally based on the assumption that the channel distortion is a constant or slowly varying bias in an utterance or globally. However, this assumption is not sustained in a more complex circumstance, when the speech records being recognized are from many different unknown channels and have parts of the spectrum completely removed (e.g. band-limited speech). On the one hand, different channels may cause different distortions; on the other, the distortion caused by a given channel varies over the speech frames when parts of the speech spectrum are removed completely. As a result, the performance of the current methods is limited in complex environments. To solve this problem, we propose a unified framework in which the channel distortion is first divided into two subproblems, namely, spectrum missing and magnitude changing. Next, the two types of distortions are compensated with different techniques in two steps. In the first step, the speech bandwidth is detected for each utterance and the acoustic models are synthesized with clean models to compensate for spectrum missing. In the second step, the constant term of the distortion is estimated via the expectation-maximization (EM) algorithm and subtracted from the means of the synthesized model to further compensate for magnitude changing. Several databases are chosen to evaluate the proposed framework. The speech in these databases is recorded in different channels, including various microphones and band-limited channels. Moreover, to simulate more types of spectrum missing, various low-pass and band-pass filters are used to process the speech from the chosen databases. Although these databases and their filtered versions make the channel conditions more challenging for recognition, experimental results show that the proposed framework can substantially improve the performance of ASR systems in complex channel environments.     

On ruin probability minimization under excess reinsurance

The problem of ruin probability minimization in the Cramer-Lundberg risk model under excess reinsurance is studied. Together with traditional maximization of the Lundberg characteristic coefficient R is considered the problem of direct calculation of insurer’s ruin probability ? _r (x) as an initial-capital function x under the prescribed level of net-retention r. To solve this problem, we propose the excess variant of the Cramer integral equation which is an equivalent to the Hamilton-Jacobi-Bellman equation. The continuation method is used for solving this equation; by means of it is found the analytical solution to the Markov risk model. We demonstrated on a series of standard examples that with any admissible value of x the ruin probability ? _x (r): = ? _r (x) is usually a unimodal function r. A comparison of the analytic representation of ruin probability ? _r(x) with its asymptotic approximation with x → ∞ was conducted.