Some remarks on the analyticity of minimizers of free discontinuity problems

In this paper we give a partial answer to a conjecture of De Giorgi, namely we prove that in dimension two the regular part of the discontinuity set of a local minimizer of the homogeneous Mumford–Shah functional is analytic with the exception of at most a countable number of isolated points.     

Some remarks on the free interpolation by bounded and slowly increasing analytic functions

One constructs a new linear bounded operator which solves the problem of free interpolation in the Hardy space H. This operator is analogous to the well-known interpolation operator of P. W. Jones. One of the fundamental distinctions of the operator constructed in this paper consists in the fact that it acts in the smallest (in a certain sense) linear closed subspace of the space H which, in turn, gives the possibility to obtain new results on free interpolation for several subclasses of the spaces H. In the second part of the paper one proves a theorem which, in a significant number of cases, allows us to reduce the solution of the problem on multiple interpolation (when the multiplicities are bounded in their totality) to the problem of free interpolation with simple nodes. One gives several examples of the application of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 35–46, 1983.     

Some remarks on the ring

Some properties like factoriality, seminormality and being a Krull domain, … are studied on power series rings , and over a commutative ring A. If \(\mathbb{X}\) is an uncountable set, there is an other sub-ring of that stands strictly between and , we denote it by . In this paper, we study properties mentioned before on the ring .     

Some more on the finite automata

Some new problems of the theory of the finite automata are considered. Applying the finite automata in various tasks of the formal languages theory is observed. Special proof of Kleene’s theorem is obtained. This proof is used for the defining the star-height of the finite automaton. The properties of the last object are considered. The star-height of the finite automaton is used for reformulating the star-height problem of regular expressions for finite automata. The method of the reduction of the star-height problem to the task of making special finite automaton is obtained. This reformulating can help to solve the star-height problem by new way.     

Some remarks on the global structure of proper Lie groupoids in low codimensions

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation, i.e. one whose kernel consists only of ineffective arrows, on a smooth vector bundle. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension, by some bundle of compact Lie groups, of some action groupoid G?X with G compact.     

Some remarks on the Clebsch's system

The almost Hamilton-Poisson realization, the stability problem, the existence of periodic solutions and the numerical integration via the Lie-Trotter integrator for the Clebsch system are discussed and some of their properties are pointed out.     

Some remarks on the Fefferman-Stein inequality

We investigate the Fefferman-Stein inequality related to a function f and the sharp maximal function f ^# on a Banach function space X. It is proved that this inequality is equivalent to a certain boundedness property of the Hardy-Littlewood maximal operator M. The latter property is shown to be self-improving. We apply our results in several directions. First, we show the existence of nontrivial spaces X for which the lower operator norm of M is equal to 1. Second, in the case when X is the weighted Lebesgue space L ^p (w), we obtain a new approach to the results of Sawyer and Yabuta concerning the C _p condition.