Some local properties of ω-stable groups

In this note we study some local properties of-stable groups of finite Morley rank.     

2 ω-finite automata and sets of obstructions of their languages

Nondeterministic finite Rabin-Scott’s automata without initial and final states (2ω-FA) are considered. In this paper, they are used to define so called sets of obstructions, used also in various algebraic systems, and to consider similar problems for the formal languages theory. Thus, we define sets of obstructions of languages (or, rather, 2ω-languages) of such automata. We obtain that each 2ω-language defined by 2 ω-FA has the set of obstruction being a regular language. And, vice versa, for each regular languageL (containing no proper subword of its another word), there exists a 2ω-FA havingL as the set of obstructions.     

On Ω-closed sets and Ωs-closed sets in topological spaces

Summary New classes of sets called -closed sets and s-closed sets are introduced and studied. Also, we introduce and study -continuous functions and s-continuous functions and prove pasting lemma for these functions. Moreover, we introduce classes of topological spaces -T_1/2 and -T_s.     

The topological type of the α-sections of convex sets

An α=(α₁,…,αk)(0?αi?1) section of a family {K₁,…,Kk} of convex bodies in Rd is a transversal halfspace H⁺ for which Vold(Ki∩H⁺)=αi⋅Vold(Ki) for every 1?i?k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sd−k.     

Some properties of Γ-limits of integral functionals

Summary We prove some integral representation theorems for the Γ limit in L¹ of sequences of the form dx in coercivity and bounded growth hypothesis which are optimal as it is checked with examples. These results are utilized to describe the L¹ lower semicontinuous envelope of a given functional. We consider also the stability of Γ limits with respect to obstacle type perturbations and prove the homogenization formulas in conditions more generals of those already considered by several authors. Entrata in Redazione il 10 gennaio 1978.     

Chaotic and topological properties of β-transformations

In this paper, we prove that the β-transformations are chaotic in the sense of both Li-Yorke and Devaney. The topological and metric properties of the sets of points with dense or non-dense orbits are investigated. We also prove the result that the set of points with non-dense orbits under the β-transformation is of full Hausdorff dimension for any β>1.     

Some properties of Van Koch’s curves

We investigate the properties of an integral operator T with a Cauchy kernel. The operator acts from L ^∞(Γ, μ), where Γ is a Van Koch curve, to the space of functions ? → ?. We prove that the range of T is nontrivial and lies in the space AC(Γ) of functions continuous in ?, vanishing at ∞, and analytic outside Γ. We also show that T is injective and compact while satisfying some special functional equation. These results may be regarded as a natural continuation of our research on the problem of AC-removability of quasiconformal curves whose solution was announced in [1] for the first time and supplemented later with some other properties of Van Koch’s curves [2, 3]. In this paper the problem is discussed in a more general setting and, in particular, all important details lacking in [1] are given. Some open problems are formulated.     

Some properties of the Schröder numbers

In the paper, the authors present some properties, including convexity, complete monotonicity, product inequalities, and determinantal inequalities, of the large Schröder numbers and find three relations between the Schröder numbers and central Delannoy numbers. Moreover, the authors sketch generalizing the Schröder numbers and central Delannoy numbers and their generating functions.     

Some properties of the family Γ of modular Lie superalgebras

In this paper, we continue to investigate some properties of the family Γ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras Γ are intrinsic. Thereby, we classify these Lie superalgebras in the sense of isomorphism. Finally, we study the associative forms and Killing forms of these Lie superalgebras and determine which superalgebras in the family are restrictable.     

Some properties of the psi function and evaluations of γ

In this paper, the authors show some monotonicity and concavity of the classical psi function, by which several known results are improved and some new asymptotically sharp estimates are obtained for this function. In addition, applying the new results to the psi function,the authors improve the well-known lower and upper bounds for the approximate evaluation of Euler's constant γ.     

Twofold fuzzy sets and rough sets—Some issues in knowledge representation

This paper deals with the representation of sets where the membership of some elements may be ill-known rather than just a matter of degree as in a fuzzy set. The notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete. A twofold fuzzy set is made of a nested pair of fuzzy sets: the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong. Twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by Pawlak. Set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined. Twofold fuzzy relations are also introduced. Finally, various applications of twofold fuzzy sets in knowledge representation are briefly discussed.     

Some results on the ℒ-fuzzy topological isomorphism

In this paper, we define ℒ-fuzzy boundedness for linear operators and we prove that every finite dimensional ℒ-fuzzy normed space is complete.