Exact multiplicity and ordering properties of positive solutions of a p-Laplacian Dirichlet problem and their applications

We study the exact multiplicity and ordering properties of positive solutions of the p-Laplacian Dirichlet problem

     

Generalized Padé-Type Approximation and Integral Representations

In several complex variables, the multivariate Padé-type approximation theory is based on the polynomial interpolation of the multidimensional Cauchy kernel and leads to complicated computations. In this paper, we replace the multidimensional Cauchy kernel by the Bergman kernel function K (z,x) into an open bounded subset of C ⁿ and, by using interpolating generalized polynomials for K (z,x), we define generalized Padé-type approximants to any f in the space OL ²() of all analytic functions on which are of class L ². The characteristic property of such an approximant is that its Fourier series representation with respect to an orthonormal basis for OL ²() matches the Fourier series expansion of f as far as possible. After studying the error formula and the convergence problem, we show that the generalized Padé-type approximants have integral representations which give rise to the consideration of an integral operator – the so-called generalized Padé-type operator – which maps every f OL ²() to a generalized Padé-type approximant to f. By the continuity of this operator, we obtain some convergence results about series of analytic functions of class L ². Our study concludes with the extension of these ideas into every functional Hilbert space H and also with the definition and properties of the generalized Padé-type approximants to a linear operator of H into itself. As an application we prove a Painlevé-type theorem in C ⁿ and we give two examples making use of generalized Padé-type approximants.     

On the nonexistence of finite time bubble trees in symmetric harmonic map heat flows from the disk to the 2-sphere

Let and be the unit disk and the unit sphere, and let be a radially symmetric harmonic map heat flow, whose singularities coincide with downward energy jumps. Then its finite time singularities are simple in the sense that precisely one harmonic sphere separates at a time.     

Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

A remark on intersection of convex sets

Let X be a real Banach space. Let be a family of closed, convex subsets of X. We show that either the intersection ?_γ∈Γ(G_γ)^? of the ?-neighborhood of the sets G_γ is bounded for each ?>0, or it is unbounded for each ?>0. From this we derive a fixed point theorem for suitable maps that move some points along a bounded direction in Hilbert spaces.     

New Component of the Moduli Space M(2;0,3) of Stable Vector Bundles on the Double Space P 3 of Index Two

We study the moduli scheme M(2;0,n) of rank-2 stable vector bundles with Chern classes c ₁=0, c ₂=n, on the Fano threefold X – the double space P ³ of index two. New component of this scheme is produced via the Serre construction using certain families of curves on X. In particular, we show that the Abel–Jacobi map :HJ(X) of any irreducible component H of the Hilbert scheme of X containing smooth elliptic quintics on X into the intermediate Jacobian J(X) of X factors by Stein through the quasi-finite (probably birational) map g:M of (an open part of) a component M of the scheme M(2;0,3) to a translate of the theta-divisor of J(X).     

Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry

We consider a one-dimensional lattice of expanding antisymmetric maps [–1, 1][–1, 1] with nearest neighbor diffusive coupling. For such systems it is known that if the coupling parameter is small there is unique stationary (in time) state, which is chaotic in space-time. A disputed question is whether such systems can exhibit Ising-type phase transitions as grows beyond some critical value _c. We present results from computer experiments which give definite indication that such a transition takes place: the mean square magnetization appears to diverge as approaches some critical value, with a critical exponent around 0.9. We also study other properties of the coupled map system.     

General Blowings-up of ℙ2 and Injectivity of Gauss Maps

We consider the blowing-up Y _k of the projective plane along k general points P ₁,...,P _k . Let _k : Y _k ² be the projection map and E _i the exceptional divisor corresponding to P _i for 1ik. For m2 and km(m+3)/2–4 let _k be the invertible sheaf _k ^*( ²(m)) _{Y k} (–E ₁–···–E _k ) on Y _k , and let _k: Y _k ^N be the morphism corresponding to _k . As _k is a local embedding, the Gauss map _k corresponding to _k is defined on Y _k by _k (x)=(d _{x k} )(T _x (Y _k )) for all xY _k . We prove that this Gauss map _k is injective.     

Necessary and sufficient conditions for the lipschitzian invertibility of nonlinear difference operators in the spacesl p (ℤ, ℝ) with 1≤p≤∞with 1≤p≤∞

Necessary and sufficient conditions for the Lipschitzian invertibility of the difference map

Light paths with an odd number of vertices in polyhedral maps

Let P _k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2-manifold with Euler characteristic contains a path P _k such that each vertex of this path has, in G, degree . Moreover, this bound is attained for k = 1 or k 2, k even. In this paper we prove that for each odd , this bound is the best possible on infinitely many compact 2-manifolds, but on infinitely many other compact 2-manifolds the upper bound can be lowered to .