Multiscale analysis and simulation of a reaction–diffusion problem with transmission conditions

This paper considers a transmission problem for partial differential equations obtained in the recent paper [M. Neuss-Radu, W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal. 39 (2007) 687–720] as the effective system modeling a reaction–diffusion process in two domains separated by a membrane. For the analysis of this problem an appropriate function space, which includes the coupling conditions for the concentrations on the interface, is introduced. The transmission conditions for the flux are included in the variational formulation with respect to this function space.The solution of the transmission problem is approximated by using the Galerkin method. Numerically the problem is reduced to a system of ordinary differential equations, where the coupling of the micro- and macro-variables leads to a special structure, distinguishing the variables relevant for the transmission. Results of numerical simulations are illustrating the effect of the microscopic process in the membrane on the macroscopic reaction–diffusion process in the bulk domains.     

On the separation of basic semialgebraic sets by polynomials

In this paper we show that two disjoint basic closed semialgebraic sets, defined over a real closed field R, can be separated by a polynomial, if one of them has dimension 2. Counterexamples are given in all higher dimensions.     

ON ENERGY CONSERVATION BY TRIGONOMETRIC INTEGRATORS IN THE LINEAR CASE WITH APPLICATION TO WAVE EQUATIONS

Trigonometric integrators for oscillatory linear Hamiltonian differential equations are considered. Under a condition of Hairer & Lubich on the filter functions in the method, a modified energy is derived that is exactly preserved by trigonometric integrators. This implies and extends a known result on all-time near-conservation of energy. The extension can be applied to linear wave equations.     

Intersection Local Times of Independent Brownian Motions as Generalized White Noise Functionals

A 'chaos expansion' of the intersection local time functional of two independent Brownian motions in R ^d is given. The expansion is in terms of normal products of white noise (corresponding to multiple Wiener integrals). As a consequence of the local structure of the normal products, the kernel functions in the expansion are explicitly given and exhibit clearly the dimension dependent singularities of the local time functional. Their L ^p -properties are discussed. An important tool for deriving the chaos expansion is a computation of the 'S-transform' of the corresponding regularized intersection local times and a control about their singular limit.     

On the Semicontinuity of Curvature Integrals

Let H_j(K, ·) be the j – th elementary symmetric function of the principal curvatures of a convex body K in Euclidean d – space. We show that the functionals ∫_bd f(H_j(K, x)) dℋ︁^d—1(x) depend upper semicontinuously on K, if f : [0, ∞) is concave, lim_t→0f(t) = 0, and lim_t→∞f(t)/t = 0. An analogous statement holds for integrals of elementary symmetric functions of the principal radii of curvature.     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative.     

Normal forms for stochastic differential equations

Summary.   We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx _t = ∑ _{j =0} ^m f _j (x _t )∘dW _t ^j and dx _t =∑ _{j =0} ^m g _j (x _t )∘dW _t ^j in ℝ ^d with smooth coefficients satisfying f _j (0)=g _j (0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,