We prove a Harnack inequality for eigenfunctions of certain homogeneous graphs and subgraphs which we call strongly convex. This inequality can be used to derive a lower bound for the (nontrivial) Neumann eigenvalues by l/(8kD) where k is the maximum degree and D denotes the diameter of the graph.

A complete two-dimensional local field K of mixed characteristic with finite second residue field is considered. It is shown that the rank of the quotient U(1)K 2 K/TK , where TK is the closure of the torsion subgroup, is equal to the degree of the constant subfield of K over Qp. Also, a basis of this quotient is constructed in the case where there exists a standard field L containing K such th...

In this paper, we define the almost uniform convergence and the almost everywhere convergence for cone-valued functions with respect to an operator valued measure. We prove the Egoroff theorem for Pvalued functions and operator valued measure θ : R → L(P, Q), where R is a σ-ring of subsets of X≠ ∅, (P, V) is a quasi-full locally convex cone and (Q, W) is a locally ...

The purpose of this article is two-fold. In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {Dn : n ∈ N}, a non-decreasing family of non empty, closed, convex and bounded sets such that Dn + Dn+2 ⊆ 2Dn+1 for every n ∈ N. The second part ...

Let X be a metrizable compact convex subset of a locally convex space. Using Choquet's Theorem, wc determine the structure of the support point set of X when X has countably many extreme points. We also characterize the support points of certain families of analytic functions.

If X is a convex surface in a Euclidean space, then the squared (intrinsic) distance function dist(x, y) is d.c. (DC, delta-convex) on X×X in the only natural extrinsic sense. For the proof we use semiconcavity (in an intrinsic sense) of dist(x, y) on X × X if X is an Alexandrov space with nonnegative curvature. Applications concerning r-boundaries (distance spheres) and the ambiguous locus (ex...

In 1956 Shiffman [14] proved that every minimally immersed annulus in 3 bounded by convex curves in parallel planes is embedded. He proved this theorem by showing that the minimal annulus was foliated by convex curves in parallel planes. We are able to prove a related embeddedness theorem for extremal convex planar curves. Recall that a subset of 3 is extremal if it is contained on the boundary...

We define a class of L-convex-concave subsets of RP n , where L is a projective sub-space of dimension l in RP n. These are sets whose sections by any (l+1)-dimensional space L ′ containing L are convex and concavely depend on L ′. We introduce an L-duality for these sets, and prove that the L-dual to an L-convex-concave set is an L *-convex-concave subset of (RP n) *. We discuss a version of A...

Let S be closed, m-convex subset of R d S locally a full ddimensional, with Q the corresponding set of Inc points of S If q is an essential inc point of order k then for some neighborhood U of q Q u is expressible as a union of k or fewer (d2)-dimenslonal manifolds, each containing q For S compact, if to every q E Q there corresponds a k > 0 such that q is an essential inc point of order k then...