Moments of the Mean of Dubins–Freedman Random Probability Distributions

Explicit formulas are given to recursively generate the moments of the mean M for Dubins–Freedman random distribution functions with arbitrary base measure . Using a standard inversion formula for moments of a distribution on the unit interval, the distribution of M is approximated for several natural choices of . The support of the mean is also considered. It is shown that the support of M is connected whenever is concentrated on the vertical bisector of the unit square S, but may have arbitrarily many gaps otherwise.     

Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching

In the present paper we first obtain the comparison principle for the nonlinear stochastic differential delay equations with Markovian switching. Later, using this comparison principle, we obtain some stability criteria, including stability in probability, asymptotic stability in probability, stability in thepth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Finally, an example is given to illustrate the effectiveness of our results.     

An Extension of Carlson's Inequality

An extension of Carlson's inequality is made by using the Euler-Maclaurin summation formula. The integral analogues of this inequality are also presented.     

Closed Geodesics in Homology Classes for Convex Co-Compact Hyperbolic Manifolds

Let be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space H ⁿ⁺¹. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = \H ⁿ⁺¹, under the assumption that H ¹(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes.     

Stochastic PDEs Driven by Nonlinear Noise and Backward Doubly SDEs

We study a new kind of backward doubly stochastic differential equations, where the nonlinear noise term is given by Itô–Kunita's stochastic integral. This allows us to give a probabilistic interpretation of classical and Sobolev's solutions of semilinear parabolic stochastic partial differential equations driven by a nonlinear space-time noise.     

Gaussian quadrature formulae on the unit circle

Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form

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For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

Minimal cubature formulae for a family of radial weight functions

For each fixed n a radial weight function on is given in closed form which admits minimal cubature formulae of both even and odd degree. The connection between the cubature formula and the polynomial interpolation is discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.