A representation of solution of stochastic differential equations

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.     

The relation between invariant integrals of the linear isotropic theory of elasticity and integrals defined by the duality principle

Invariant integrals of the linear isotropic theory of elasticity, determined by a certain specified elastic field, are considered, and also invariant integrals generated by the interaction of the specified field with an arbitrary secondary field. For all types of invariant integral, generated by the interaction of the specified elastic field and an arbitrary secondary elastic field, transformations of the secondary fields are found for which the invariant integrals considered turn out to be equal to the RG-integrals, determined by the duality principle, of the specified elastic field and the transformed secondary elastic field. The invariant J-, L- and M-integrals themselves are also expressed in terms of the RG-integrals of the specified elastic field and its corresponding transformation.     

On stochastic integration by series of Wiener integrals

Stochastic integrals of random functions with respect to a white-noise random measure are defined in terms of random series of usual Wiener integrals. Conditions for the existence of such integrals are obtained in terms of the nuclearity of certain operators onL ² -spaces. The relation with the Fisk-Stratonovich symmetric integral is also discussed.This research was supported by AFOSR Contract No. F49620 82 C 0009.     

Eigenanalysis on a bivariate covariance kernel

Certain constructions of copulas can be interpreted as an eigendecomposition of a kernel. We study some properties of the eigenfunctions and their integrals of a covariance kernel related to a bivariate distribution. The covariance between functions of random variables in terms of the cumulative distribution function is used. Some bounds for the trace of the kernel and some inequalities for a continuous random variable concerning a function and its derivative are obtained. We also obtain relations to diagonal expansions and canonical correlation analysis and, as a by-product, series of constants for some particular distributions.     

Chaos decomposition of multiple fractional integrals and applications

Chaos decomposition of multiple integrals with respect to fractional Brownian motion (with H > 1/2) is given. Conversely the chaos components are expressed in terms of the multiple fractional integrals. Tensor product integrals are introduced and series expansions in those are considered. Strong laws for fractional Brownian motion are proved as an application of multiple fractional integrals. Received: 22 September 1998 / Revised version: 20 April 1999     

On the summation of Schlömilch's series

ABSTRACT

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.     

Closed form expressions for some series over certain trigonometric integrals

The series (3) and (4), where T(x) denotes trigonometric integrals (2), are represented as series in terms of Riemann zeta and related functions using the sums of the series (5) and (6), whose terms involve one trigonometric function. These series can be brought in closed form in some cases, where closed form means that the series are represented by finite sums of certain integrals. By specifying the function φ(y) appearing in trigonometric integrals (2) we obtain new series for some special types of functions as well as known results.     

Revisiting the box counting algorithm for the correlation dimension analysis of hyperchaotic time series

We undertake the correlation dimension analysis of hyperchaotic time series using the box counting algorithm. We show that the conventional box counting scheme is inadequate for the accurate computation of correlation dimension (D₂) of a hyperchaotic attractor and propose a modified scheme which is automated and gives better convergence of D₂ with respect to the number of data points. The scheme is first tested using the time series from standard chaotic systems, pure noise and data added with noise. It is then applied on the time series from three standard hyperchaotic systems for computing D₂. Our analysis clearly reveals that a second scaling region appears at lower values of box size as the system makes a transition into the hyperchaotic phase. This, in turn, suggests that correlation dimension analysis can also give information regarding chaos-hyperchaos transition.     

Computing elliptic integrals by duplication

Summary Logarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the convergence is improved by including a fixed number of terms of Taylor's series, the error ultimately decreases by a factor of 4096 in each cycle of iteration. Except for Cauchy principal values there is no separation of cases according to the values of the variables, and no serious cancellations occur if the variables are real and nonnegative. Only rational operations and square roots are required. An appendix contains a recurrence relation and two new representations (in terms of elementary symmetric functions and power sums) forR-polynomials, as well as an upper bound for the error made in truncating the Taylor series of anR-function.     

Two non-holonomic integrable problems tracing back to Chaplygin

The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange’s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.     

Simulation of stochastic integrals with respect to Lévy processes of type G

We study the simulation of stochastic processes defined as stochastic integrals with respect to type G Lévy processes for the case where it is not possible to simulate the type G process exactly. The type G Lévy process as well as the stochastic integral can on compact intervals be represented as an infinite series. In a practical simulation we must truncate this representation. We examine the approximation of the remaining terms with a simpler process to get an approximation of the stochastic integral. We also show that a stochastic time change representation can be used to obtain an approximation of stochastic integrals with respect to type G Lévy processes provided that the integrator and the integrand are independent.     

The Lebesgue summability of trigonometric integrals

Motivated by the notion of Lebesgue summability of trigonometric series, we define the Lebesgue summability of trigonometric integrals in terms of the symmetric differentiability of the sum of the formally integrated trigonometric integral in question. We extend two theorems of Zygmund from trigonometric series to integrals, and one of them even in a more general form.     

Product formulas for solutions of initial value partial differential equations,I

Families of operators which approximate semi-groups or evolution systems generated by partial differential operators are constructed. Product formulas are used to recover these semi-groups or evolution systems through product integrals. Conditions on generators are provided under which its semi-group or evolution system can be approximated in this way by families of specific types of operators.     

Mathematical and physical modelling of seismic faults (Poster Presentation)

Modelling of seismic faults by using spring block systems on a frictional surface, requires to solve coupled differential equation systems with a great number of equations, in order to avoid this problem, the spring-block model is mapped into a continuous, non-conservative cellular automaton. Every time the automaton is calculated a synthetic earthquake is obtained. We obtained catalogues of synthetic earthquake magnitude, which we can represent as a time series. Such series exhibit power law behavior so much for the magnitudes (Gutenberg-Richter law) as for the duration times. The analysis of these time series with methods of non-linear dynamics provides interesting results; for example, the time series show multifractality for large values of the conservation level. We present the correlation analysis of three kinds of time series: recurrence times, jumps or distance between consecutive earthquakes and magnitude for different thresholds. We obtained long correlations in recurrence times as it has been observed in real seismicity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A pathway from Bayesian statistical analysis to superstatistics

Superstatistics and Tsallis statistics in statistical mechanics are given an interpretation in terms of Bayesian statistical analysis. Subsequently superstatistics is extended by replacing each component of the conditional and marginal densities by Mathai’s pathway model and further both components are replaced by Mathai’s pathway models. This produces a wide class of mathematically and statistically interesting functions for prospective applications in statistical physics. It is pointed out that the final integral is a particular case of a general class of integrals introduced by the authors earlier. Those integrals are also connected to Krätzel integrals in applied analysis, inverse Gaussian densities in stochastic processes, reaction rate integrals in the theory of nuclear astrophysics and Tsallis statistics in nonextensive statistical mechanics. The final results are obtained in terms of Fox’s H-function. Matrix variate analogue of one significant specific case is also pointed out.     

The Eisenstein series for <Emphasis Type="Italic">GL</Emphasis>(3,<Emphasis Type="Bold">Z</Emphasis>) induced from cusp forms

We study the Eisenstein series for GL(3,Z) induced from cusp forms. We give the expression of the Fourier-Whittaker coefficients of the Eisenstein series in terms of the Jacquet integrals. Moreover, by evaluating the Jacquet integrals, we give the Mellin-Barnes type integral expressions of those at the minimal K-type.     

Isomorphism and Hamilton representation of some nonholonomic systems

We consider some questions connected with the Hamiltonian form of the two problems of nonholonomic mechanics: the Chaplygin ball problem and the Veselova problem. For these problems we find representations in the form of the generalized Chaplygin systems that can be integrated by the reducing multiplier method. We give a concrete algebraic form of the Poisson brackets which, together with an appropriate change of time, enable us to write down the equations of motion of the problems under study. Some generalization of these problems are considered and new ways of implementation of nonholonomic constraints are proposed. We list a series of nonholonomic systems possessing an invariant measure and sufficiently many first integrals for which the question about the Hamiltonian form remains open even after change of time. We prove a theorem on isomorphism of the dynamics of the Chaplygin ball and the motion of a body in a fluid in the Clebsch case.     

Almost Sure Convergence of the Numerical Discretization of Stochastic Jump Diffusions

Based on the shuffle product expansion of exponential Lie series in terms of a Philip Hall basis for the stochastic differential equations of jump-diffusion type, we can establish Stratonovich–Taylor–Hall (STH) schemes. However, the STHr scheme converges only at order r in the mean-square sense. In order to have the almost sure Stratonovich–Taylor–Hall (ASTH) schemes, we have to include all the terms related to multiple Poissonian integrals as the moments of multiple Poissonian integrals always have lower orders of magnitudes as compared with those of multiple Brownian integrals.