Let P be a compact n-dimensional convex polyhedron in R n containing the origin in its interior and let e H(t) = Z 1 0 Z vP e 2it ddv, t2 R n ;where vP is the characteristic function of the dilated polyhedron vP. Let H N (t) = X m2Z n e H 1 N+1 (t+m), t 2 T n , where e H " (t) = " ?n e H(t="). We prove that (e H " f)(t) ! f(t) a.e., as " ! 0, for any f 2 L 1 (R n), and that (H N f)(t) ! f(t) a....

In characteristic function games, an agent can potentially join many different coalitions, and so must choose which coalition to join. To compare each potential coalition, the agents need to calculate a value for each coalition. As the number of coalitions grows exponentially with the number of agents, the burden of requiring every agent to compute all of the values is high. This can be reduced...

The usual proof that the product of a collection of probability measures exists uses Fubini’s theorem. This is unsatisfying because one ought not need to use Fubini’s theorem to prove things having only to do with σ-algebras and measures. In this note I work through the proof given by Saeki of the existence of the product of a collection of probability measures. We speak only about the Lebesgue...

Let G be a closed subgroup of S∞ and X be a Polish G-space with a countable basisA of clopen sets. Each x ∈ X defines a characteristic function τx on A by τx(A) = 1 ⇔ x ∈ A. We consider computable complexity of τx and some related questions.

There are two cases where it is well known that Schriidinger operators have non-degenerate eigenvalues: The lowest eigenvalue in general dimension and all one-dimensional eigenvalues. One can ask about making this quantitative, i.e., obtain explicit lower bounds on the distance to the nearest eigenvalues. Obviously, one cannot hope to do this without any restrictions on V since, for example, if...

Consider a function f : f0;1g n ! f0;1g. The sensitivity of a point v 2 f0;1g n is jfv 0 : f (v 0) 6 = f (v); dist(v; v 0) = 1gj, i.e. the number of neighbors of the point in the discrete cube on which the value of f diiers. The average sensitivity of f is the average of the sensitivity of all points in f0;1g n. (This can also be interpreted as the sum of the innuences of the n variables on f ,...

We study C0-operators relative to a multiply connected domain using a substitute of the characteristic function. This method allows us to prove certain relations between the Jordan model of an operator and that of its restriction to an invariant subspace.

We propose stochastic N -component synchronization models (x1(t), . . . , xN (t)), xj ∈ Rd, t ∈ R+, whose dynamics is described by Lévy processes and synchronizing jumps. We prove that symmetric models reach synchronization in a stochastic sense: differences between components d (N) kj (t) = xk(t)−xj(t) have limits in distribution as t → ∞. We give conditions of existence of natural (intrinsic)...

Let D be a bounded domain in Rn such that ∂D is a union of a finite number of smooth hypersurfaces in general position. The asymptotics of the Fourier transform of a piece-wise smooth function f(x) with discontinuities of the first kind on ∂D, in particular of the characteristic function of D, is found in almost all directions. The results are based on our recent work on the singularities of th...

We consider the convolution model where i.i.d. random variables Xi having unknown density f are observed with additive i.i.d. noise, independent of the X’s. We assume that the density f belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically. We consider the ...