New estimates of double trigonometric sums with exponential functions

We establish a new bound for the exponential sum

Variance and exponential estimates via coupling

We give a proof of Devroye and exponential inequalities based on a coupling. We mostly deal with the case of continuous random variables with dynamical systems in mind (and their rather special mixing properties) although the approach is more general.     

Global error estimates for exponential splitting

Global errors of principal exponential splitting formulae, i.e.,first-order splitting, Strang's splitting and parallel splitting,are investigated. It is found that the commutativity of theunderlying matrices in the exponents plays an important rolein the analysis. It is shown that the global error estimatesbehave as polynomials in the splitting steps when the stepsare small and decay exponen tially when the splitting stepsare chosen to be large. They are much more reliable than thosederived from the traditional local truncation error analysiswhen relatively large splitting steps are adopted. This corresponds,for example, to solving partial differential equations usingrelatively large Courant numbers. In this case, the global erroranalysis provides sharp bounds in contrast to the local errorestimates, which give inaccurate or even deceptive results.Numerical examples are given to demonstrate our results.     

STRUCTURES OF SYSTEMS WITH EXPONENTIAL LIFE DISTRIBUTIONS

Abstract. Structures of monotone systems and cold standby systems with     

Completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight

We study the problem on the completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight. Theorems of general form are obtained. In particular, the necessary and sufficient conditions on α, β, q ₁, and q ₂ for which the known orthogonal systems are everywhere dense in asymmetric spaces L _(α,β);q ([0, 1]) are found. Theorem. Let α, β, q ₁, q ₂ ∈ [1,+∞]. The following orthogonal systems are dense in asymmetric spaces L _(α,β);q ([0, 1]) if and only if either max{α, β, q ₁, q ₂} < + ∞ or max {α, β} < +∞, q ₁ = q ₂ = +∞: trigonometric, algebraic, Haar’s system, Meyer’s system of wavelets, Shannon-Kotel’nikov wavelets, Stromberg and Lemarie-Battle wavelets, the Walsh system, and the Franklin system. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Rate estimates of gradient blowup for a heat equation with exponential nonlinearity

We consider a one-dimensional semilinear parabolic equation , for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish estimates of blowup rate upper and lower bounds. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.     

Completeness of perturbed systems of trigonometric functions

The paper addresses the completeness of perturbed systems of sines and cosines with non-integer indices and, moreover, complex, in the space of Lebesgue-integrable functions. The criteria for the variation from the integer value have been found such that sine and cosine systems are complete.     

Conversion estimates for gas-solid reactions: Improved estimates for irreversible systems

Solid conversion has been of much interest to research workers and the Cumulative Gas Concentration Method (CGCM) has been developed to provide bounds for the reactant solid in irreversible systems having power-law type kinetics. This CGCM is used in conjunction with the Pseudo-Steady-State (PSS) assumption and these bounds appear to be good under certain conditions and for long times.

In this work, the PSS condition is not assumed when deriving the bounds on the cumulative gas concentration and the effect of the inclusion of the transient part is analysed. A technique, based on an iterative scheme incorporating the CGCM, is developed to provide improved estimates for the reactant solid.     

Tail estimates for exponential functionals and applications to SDEs

In this paper, based on techniques of Malliavin calculus, we obtain an explicit bound for tail probabilities of a general class of exponential functionals. We apply the obtained results to derive asymptotic behaviors for the tail of the exponential functional of stochastic differential equations.     

Asymptotic estimates of Gerber-Shiu functions in the renewal risk model with exponential claims

This paper continues to study the asymptotic behavior of Gerber-Shiu expected discounted penalty functions in the renewal risk model as the initial capital becomes large. Under the assumption that the claim-size distribution is exponential, we establish an explicit asymptotic formula. Some straightforward consequences of this formula match existing results in the field.     

Polynomial first integrals of Hamiltonian systems with exponential interaction

Moscow Institute of Radio Engineering, Electronics, and Automation. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 88–89, July–September, 1991.     

Trimmed estimates in simultaneous estimation of parameters in exponential families

Let X₁,…, X_p be p (≥ 3) independent random variables, where each X_i has a distribution belonging to the one-parameter exponential family of distributions. The problem is to estimate the unknown parameters simultaneously in the presence of extreme observations. C. Stein (Ann. Statist.9 (1981), 1135–1151) proposed a method of estimating the mean vector of a multinormal distribution, based on order statistics corresponding to the |X_i|'s, which permitted improvement over the usual maximum likelihood estimator, for long-tailed empirical distribution functions. In this paper, the ideas of Stein are extended to the general discrete and absolutely continuous exponential families of distributions. Adaptive versions of the estimators are also discussed.