Volume 9, Issue 1

I investigate an asymmetric product construction for σ -ideals of σ -algebras of sets, corresponding to one-sided Fubini and Kuratowski-Ulam theorems, examining the cellularities of these products.

In the paper we formulate an axiom , which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to w 1 , while 𝔠 = w 2 : the independence number 𝔦, the reaping number 𝔯, the almost disjoint number 𝔞, and the ultrafilter base number 𝔲. We will also show that implies the existence of crowded and selective ultrafilters as well as nonselective P -points. In addition we prove that under every selective ultrafilter is w 1 -generated. The paper finishes with the proof that holds in the iterated perfect set model.

The conditions of solvability and description of all solutions of the truncated Stieltjes moment problem are obtained using as the starting point earlier results on the Hamburger truncated moment problem. An algebraic algorithm for the explicit solution of both problems is proposed.

In the paper the existence and continuous dependence of a kind of minimax solution to the dual Hamilton-Jacobi equations is proved. The main difficulties which appear here are a special type of the boundary conditions and the transversality conditions which that solution must satisfy. That type of problems come from optimal control and game theory.

We consider the problem u tt + δu t + εa Δ u + ϕ ( ∫ Ω |∇ u | 2 dx )Δ u ≥ f ( x , t ), posed in Ω × (0, +∞). Here is a an open smooth bounded domain and ϕ is like ϕ ( s ) = bs γ , γ > 0, a > 0 and ε = ±1. We prove, in certain conditions on f and ϕ that there is absence of global solutions. The method of proof relies on a simple analysis of the ordinary inequality of the type w ″ + δw ′ ≥ αw + βw p . It is also shown that a global positive solution, when it exists, must decay at least exponentially.

We examine functions of two variables whose all vertical sections are equiderivatives. In particular we show that a bounded function whose horizontal sections are strongly measurable and vertical sections are equiderivatives, is strongly measurable. The theorems we prove are generalizations of the results of Z. Grande [Real Anal. Exchange 21: 637–647, 1995–96].

In this paper, we give two new maximal element theorems in l.c. metric spaces, and as their applications, a new coincidence theorem and two new existence theorems of solutions for generalized quasivariational inequalities are obtained.

We provide a generalization of a well known Krasnosel'skii theorem on continuity of the Nemytskii operator for functions taking values in separable Banach spaces. We follow the results obtained in [Evans, Partial Differential Equations, Graduate Studies in Mathematics, Amer. Math. Soc., 1998] for the finite dimensional case.