Polynomial asymptotics near zero points of solutions of general elliptic equations

We extend a classical result of Lipman Bers concerning the local behavior of solutions to a wide class of elliptic equations, systems and inequalities with singular coefficients. The main theorem states that near any zero point of finite vanishing order, the solution is asymptotic to a homogeneous polynomial solution of the “osculating” equation, under mild hypotheses on coefficients. The analysis invloves homothety blow-up arguments with the aid of some elements of Lyapunov exponents in the theory of dynamical systems.     

Some optimal bounds in the central limit theorem using zero biasing

The convergence rate in the central limit theorem (CLT) is investigated in terms of a wide class of probability metrics. Namely, optimal estimates for the proximity between a probability distribution and its zero bias transformation are derived. These new inequalities allow one to establish optimal rates of convergence in the CLT for sums of independent random variables with finite moments of order s, s∈(2,3], in terms of ideal metrics introduced by V.M. Zolotarev.     

Image partition regularity near zero

Many of the classical results of Ramsey Theory are naturally stated in terms of image partition regularity of matrices. Many characterizations are known of image partition regularity over N and other subsemigroups of (R,+). We study several notions of image partition regularity near zero for both finite and infinite matrices, and establish relationships which must hold among these notions.     

Every three-point set is zero dimensional

This paper answers a question of Jan J. Dijkstra by giving a proof that all three-point sets are zero dimensional. It is known that all two-point sets are zero dimensional, and it is known that for all 3$">, there are -point sets which are not zero dimensional, so this paper answers the question for the last remaining case.

     

An Ikehara-type theorem for functions convergent to zero

We establish an analogue of the Ikehara theorem for positive non-increasing functions convergent to zero. In particular, this provides a complete proof of the results formulated in Diekmann & Kaper (1978) [5] and Carr & Chmaj (2004) [1], which are widely used nowadays to prove the uniqueness of traveling waves for various reaction–diffusion equations.     

A set intersection theorem and applications

Using Scarf's algorithm for “computing” a fixed point of a continuous mapping, the following is proved: LetM ₁ ? M _n be closed sets inR ⁿ which cover the standard simplexS, so thatM _i coversS _i , the face ofS opposite vertexi. We say a point inS iscompletely labeled if it belongs to everyM _i andk-almost-completely labeled if it belongs to all butM _k . Then there exists a closed setT ofk-almost-completely labeled points which connects vertexk with some completely labeled point. This result is used to prove Browder's theorem (a parametric fixed-point theorem) inR ⁿ . It is also used to generate “algorithms” for the nonlinear complementarity problem which are analogous to the Lemke—Howson algorithm and the Cottle—Dantzig algorithm, respectively, for the linear complementarity problem.     

Romanoff theorem in a sparse set

Let A be any subset of positive integers,and P the set of all positive primes.Two of our results are:(a) the number of positive integers which are less than x and can be represented as 2k + p(resp.p-2k) with k ∈ A and p ∈ P is more than 0.03A(log x/log 2)π(x) for all sufficiently large x;(b) the number of positive integers which are less than x and can be represented as 2q + p with p,q ∈ P is(1 + o(1))π(log x/log 2)π(x).Four related open problems and one conjecture are posed.     

Partition regularity of polynomial systems near zero

Semigroup Forum - Recently, S. Kanti Patra and Md. Moid Shaikh proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense...     

On Polynomial Sequences with Restricted Growth near Infinity

Let (P_n) be a sequence of polynomials which converges with ageometric rate on some arc in the complex plane to an analyticfunction. It is shown that if the sequence has restricted growthon a closed plane set E which is non-thin at , then the limitfunction has a maximal domain of existence, and (P_n) convergeswith a locally geometric rate on this domain. If is a sequence of partial sums of a power series,a similar growth restriction on E forces the power series tohave Ostrowski gaps. Moreover, the requirement of non-thinnessof E at is necessary for these conclusions.     

On the zero set of semi-invariants for quivers

Let be a prehomogeneous dimension vector for a finite quiver Q. We show that the set of common zeros of all semi-invariants of positive degree for the variety of representations of Q with dimension vector under the product of the general linear groups at all vertices is irreducible and a complete intersection for large natural numbers N.     

A non-commutative central limit theorem

This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 123)     

A topological central point theorem

In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.     

A Nikodym theorem for triangular set functions

Siberian Mathematical Journal -