An index theorem for gauge-invariant families: The case of solvable groups

We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family of Lie groups (these families are called ``gauge-invariant families' in what follows). If the fibers of are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah–Singer type formula that incorporates also topological information on the bundle . The algebras of invariant pseudodifferential operators that we study, and , are generalizations of ``parameter dependent' algebras of pseudodifferential operators (with parameter in R q), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex. This revised version was published online in June 2006 with corrections to the Cover Date.     

Green Integrals on Manifolds with Cracks

We prove the existence of a limit in H ^m (D)of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchyproblem in D with data on S, for an elliptic operator A of order m 1, whenever these solutions exist.This representation involves the sum of a series whose terms are iterationsof the double layer potential. A similar regularisation is constructed also for a mixed problem in D.     

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.     

On Conservative Matrix Methods for Double Sequence Spaces

We consider two types of summability methods of double sequences defined by 4-dimensional matrices. Our concern is with methods preserving the regular convergence (in Hardy"s sense) and the -convergence of double sequences defined as follows: every column is convergent and the sequence of column limits converges as well. Of primary interest is whether the following well-known facts of summability concerning the summation of bounded sequences by conservative matrices can be extended to 4-dimensional matrices: (1) if the space of all convergent sequences is closed in a summability domain then the matrix does not sum bounded divergent sequences, (2) if a conservative matrix sums a divergent sequence then it sums an unbounded sequence and (3) two regular matrices are consistent on bounded sequences. We emphasize that the negative answer to (2), in the case of regular convergence, shows that the main result in [8] fails in general.     

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.