We discuss the extension to the multi-dimensional case of the Wick-Itô integral with respect to fractional Brownian motion, introduced by [6] in the 1-dimensional case. We prove a multidimensional Itô type isometry for such integrals, which is used in the proof of the multi-dimensional Itô formula. The results are applied to study the problem of minimal variance hedging in a market driven by fractional Brownian motions.

Existence results are established for a second order boundary value problem on the half line motivated from the model of a slender dry patch in a liquid film draining under gravity down an inclined plane.

Starting from a nonempty set X and a commutative semigroup G acting on X we construct a new space B(X,G) whose algebraic character is similar to a quotient field. The construction of the quotient field from an integral domain is a special case of our construction. Other interpretations of the construction include the space of Schwartz distributions of finite order, tempered distributions, Radon measures, and Boehmians.

In this paper we describe the construction of B(X,G), discuss some general properties of B(X,G), and present some applications of the construction.

Orthogonal polynomials associated with H_q- semiclassical linear form will be studied as a generalization of the H_q-classical linear forms. The concept of class and a criterion for determining it will be given. The q-difference equation that the corresponding formal Stieltjes series satisfies is obtained. Also, the structure relation as well as the second order linear q-difference equation are obtained. Some examples of H_q-semiclassical of class 1 were highlighted.

We study the asymptotic expansion, as λ → 0⁺, of integrals of the form J_H,Χ(λ) =∫exp(H(χ)/λ). Χ(χ)dχ, where H and Χare smooth from R^p to R, H has a unique (degenerate) maximum at 0, Χ has compact support a neighborhood of 0.

If p = 2 or if the Newton Diagram of H contains only one facet, we give an algorithm to compute explicitely the complete asymptotic expansion of J_H,Χ(λ). In the general case, we show how to write J_H,Χ(λ) as a linear combination of simpler integrals, involving only the fundamental part of H. We give an equivalent of the first term of the expansion of J_H,Χ(λ), and specify the exact form of this first term under a simple additional condition.

We study the positivity of the second shape derivative around an equilibrium for a functional defined on exterior domains in the plane and which involves the perimeter of the domains and their Dirichlet energy under volume constraint. We prove that small analytic perturbations of circles may be stable or not, depending on the positivity of a simple and explicit two-variable quadratic form. The approach is general and involves a numerical criterion of independent interest for the positivity of a quadratic form on a given hyperplane.

Two sharp inequalities are derived. The first of them is a sharp inequality which gives an error bound for a Gauss-Legendre quadrature rule. The second is a sharp inequality which gives an error bound for a Radau quadrature rule. These inequalities enlarge the applicability of the corresponding quadrature rules with respect to the obtained error bounds. Applications in numerical integration are also given.