Motivated by Rosenthal’s famous $$l^1$$ -dichotomy in Banach spaces, Haydon’s theorem, and additionally recent works on tame dynamical systems, we introduce the class of locally convex spaces. This is a natural analogue Rosenthal spaces (for which any bounded sequence contains weak Cauchy subsequence). Our approach based bornology subsets turn closely related to eventual fragmentability. leads,...

We study splitting chains in $$\mathcal{P}(\omega)$$ , that is, families of subsets ω which are linearly ordered by ⊆* and splitting. prove their existence is independent axioms ZFC. show they can be used to construct certain peculiar Banach spaces: twisted sums C(ω*) = l∞/c0 c0(c). Also, we consider a topological setting, where give rise the so called tunnels.

We analyze a class of L∞ vector fields, called divergence-measure fields. We establish the Gauss-Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of L∞ fields. Then we apply this theory to analyze L∞ entropy solutions of initial-boundary-value problems for hyperbolic conservation laws and to study the ways in which the solutions assume t...

These numbers will be regarded as distinct although two or more may be equal in value. Let «,■ (t=l, 2, • • • , k) be any fixed set of k positive integers whose sum is n, let C= Yli-i w»' and let R = n\/C. We shall consider partitions P* = {Bf, B2*, ■ ■ ■ , Bk*\ of the set B into disjoint subsets Bf, each B* containing n,elements of B (* = 1, 2, • • • , k). By a partition here is meant an order...

We consider the Schrödinger operator on a finite interval with an $L^1$-potential. prove that potential can be uniquely recovered from one spectrum and subsets of another point masses spectral measure (or norming constants) corresponding to first spectrum. also solve this Borg–Marchenko-type problem under some conditions two spectra, when missing part second known have different index sets.

We consider the algorithmic problem of generating each subset of [n] := {1, 2, . . . , n} whose size is in some interval [k, l], 0 ≤ k ≤ l ≤ n, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k = 0 and l = n this is the classical problem of generating all 2 subsets of [n] by element additions/removals, and for k = l this is the...

Linear index coding can be formulated as an interference alignment problem, in which precoding vectors of the minimum possible length are to be assigned to the messages in such a way that the precoding vector of a demand (at some receiver) is independent of the space of the interference (non sideinformation) precoding vectors. An index code has rate 1 l if the assigned vectors are of length l. ...

Much of the existing work on the broadcast channel focuses only on the sending of private messages. In this work we examine the scenario where the sender also wishes to transmit common messages to subsets of receivers. For an L-user broadcast channel there are 2L−1 subsets of receivers and correspondingly 2L−1 independent messages. The set of achievable rates for this channel is a 2 − 1-dimensi...

Let n ≥ l ≥ 2 and q ≥ 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose l-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hy...

Let n ≥ l ≥ 2 and q ≥ 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose l-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hy...