Two- versus three-dimensional connectivity testing of first-order queries to semi-algebraic sets

This paper addresses the question whether one can determine the connectivity of a semi-algebraic set in three dimensions by testing the connectivity of a finite number of two-dimensional ``samples'' of the set, where these samples are defined by first-order queries. The question is answered negatively for two classes of first-order queries: cartesian-product-free, and positive one-pass.Comment: corrected minor confusion in Proof of Theorem

Moments of certain stochastic integrals occurring in mathematical physics

SynopsisWe give an expression for the n-th moment of certain Itô integrals. The integrands considered are nonanticipating functionals of the form s↦a(s, Xs), where a is a measurable time-dependent vector field in space satisfying mild regularity conditions, and Xs is standard translated Brownian motion. The expressions are similar to the Dyson-Phillips terms for magnetic Schrödinger semigroups.We use these expressions to establish properties of the solutions of certain Cauchy problems and we relate our results to the framework of generalised Dyson expansions as set up by Johnson and Lapidus.</jats:p

Winding number and the number of real zeros of a function

The theorem in this paper shows that the number of real simple zeros of a function of the form f ( x ) = q ( x ) + a x + b , x ∈ R f(x) = q(x) + ax + b,x \in \mathbb {R} , for not too wild q ( x ) q(x) can be obtained counting the winding number of a closed plane curve about the point ( a , b ) (a,b) .</p