Volume 7, Issue 1

The most prominent examples of (operator-) selfdecomposable laws on vector spaces are (operator-) stable laws. In the past (operator-) semistability — a natural generalisation — had been intensively investigated, hence the description of the intersection of the classes of semistable and selfdecomposable laws turned out to be a challenging problem, which was finally solved by A. Łuczak's investigations [Probab. Theory Related Fields 90: 317–340, 1991]. For probabilities on groups, in particular on simply connected nilpotent Lie groups there exists meanwhile a satisfying theory of decomposability and semistability. Consequently it is possible to obtain a description of the intersection of these classes of measures — under additional commutativity assumptions — leading finally to partial extensions of the above-mentioned results for vector spaces to the group case.

The present paper is devoted to the study of the existence solution problem for a hemivariational inequality on vector-valued function space in the case when the nonlinear nonconvex part satisfies the unilateral growth condition. The critical point theory combined with the Galerkin approximation method have been used to establish the result.

Under reasonable assumptions on the data u , v and the function f , we show that the nonlinear periodic Goursat problem which cannot be posed in the general theory of distributions, may be studied and solved in a differential algebra of periodic new generalized functions on . This algebra contains, in a canonical way, the space of periodic distributions on as a linear subspace. When the data are Dirac measures, a particular study is done, showing the nature of the singularities involved in this case.

The almost global in time existence of regular solutions to equations of viscous compressible barotropic fluid motion for the initial value problem in the L p -framework is shown. To prove the result we apply the L p -estimate for the linearization of the equations which are obtained at the end of the paper.

A cardinal related to compositions of Sierpińnski–Zygmund functions from the left, c left (SZ), will be considered. We answer a question of K. Ciesielski and T. Natkaniec. In particular, we show that c l (SZ) = (2 𝔠 ) + if ℝ is not a union of less than 𝔠-many meager sets and 𝔠 is a limit cardinal. If 𝔠 = w 1 , then c l (SZ) is equal to the bounding number of 𝔠.

In this paper we introduce two topologies on the plane connected with the notions of density and I -density. Their definitions are based on the notion of a regular density point. We investigate connections between them and the density and I -density topologies on the plane and on the real line. We consider axioms of separation and functions continuous with respect to these topologies.

We investigate the topological entropy of a green interval map. Defining the complexity we estimate from above the topological entropy of a green interval map with a given complexity. The main result of the paper — stated in Theorem 0.2 — should be regarded as a completion of results of [Bobok, Fund. Math. 162: 1–36, 1999].

We consider the fourth order periodic problem with a functional parameter. Some sufficient conditions under which solutions of this problem continuously depend on parameters are given. Proofs of theorems are based on variational methods.

We give a complete characterization of sets of sums of absolutely convergent series with alternating signs. This result finishes works of various authors [Menon, Bull. Amer. Math. Soc. 54: 706–711, 1948 ], [Šalát, Math.-Phys. J. Slovak Acad. Sci. 9: 3–14, 1959]–[Šalát, Acta Univ. Comenian. Fac. Rerum Natur. 2: 71–74. 1957] who studied this problem. As we will see, the studied sets are exactly sets that can be obtained by generalizing the construction of the Cantor set.