Volume 1, Issue 1

L. Fejes Tóth [Acta Math. Acad. Sci. Hungar, 13: 379–382, 1962] introduced the notion of fixing system for a compact, convex body M ⊂ R n . Such a system F ⊂ bd M stabilizes M with respect to translations. In particular, every minimal fixing system F is primitive , i.e., no proper subset of F is a fixing system. In [Boltyanski and Martini, Combinatorial geometry of belt bodies] lower and upper bounds for cardinalities of mimimal fixing systems are indicated. Here we give an improved lower bound and show by examples, now both the bounds are exact. Finally, we formulate a Fejes Tóth Problem .

In this paper, a connection between Bellman's spheres for piecewise continuous and measurable controls is considered. The investigation is conducted for linear controlled objects of the second order with a rather complicated two–dimensional control region. A collection of examples is included to illustrate the obtained results.

The Hahn–Banach extension theorem is generalized to the case of continuous linear operators mapping a subspace Y of a normed space X into a normed space V . In contrast with known results of this kind, we do not equip V with a partial ordering neither impose any restrictions on V . The extension property is fully characterized by the sign of the one sided Gateaux derivative of the norm ∥ · ∥ x . Other characterizations, involving e.g. Birkhoff's orthogonality, are also provided.

We find exponents of independent marginals of operator Lévy's measures, and show that those measures which are convolutions of one-dimensional factors are multivariate Lévy's with the factors being Lévy's too. A characterization of exponents of such measures is also given.

The local existence of classical solutions for a characteristic initial boundary value problem for the equations of ideal compressible polytropic fluids is proved. The problem is replaced by a system of well–posed problems and then the method of successive approximations is used.

The goal of this note is to construct — via the notion of Young measure–minimizing sequences for problems of calculus of variations that do not admit minimizers.

Studying homoclinic solutions of equations is one of the steps to go deeper in the understanding of dynamics. As it is known to the authors there are no papers studying homoclinic solutions of hyperbolic systems. In the paper we present a new variational method general enough to treat the problem of the existence of homoclinic solutions for the following semi-linear wave equation: x tt ( t, y ) – x yy ( t, y ) + g ( t, y, x ( t, y )) = 0 for 0 < y < Y , t ∈ R , x ( t , 0) = 0, x ( t , Y ) = 0 for t ∈ R . Our approach covers both sublinear and super linear cases.

It was shown by Butnariu and Flåm [J. Nnmer. Funct. Anal. Optim. 15: 601–636, 1995] that, under some conditions, sequences generated by the expected projection method (EPM) in Hilbert spaces approximate almost common points of measurable families of closed convex subsets provided that such points exist. In this work we study the behavior of the EPM in the more general situation when the involved sets may or may not have almost common points and we give necessary and sufficient conditions for weak and strong convergence. Also, we show how the EPM can be applied to finding solutions of linear operator equations and to solving convex optimization problems.