Summary . In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. Mizar language [1], [2], variables are strictly typed, their type conversion requires a complicated process. Our purpose is to formalize some of preserved transformations, contain complexity transformations into paper. Specifically, show such as algebraic structure, subsets, sequ...

Let k and b be integers with k > 1. A set A of integers is called (k, b)linear-free if x E A implies kx + b ¢ A. Such a set A is maximal in [I,n] = {1,2, ... ,n} if AU{t} is not (k,b)-linear-free for any t in [1,nJ\A. Let M = M(n, k, b) be the set of all maximal (k, b)-linear-free subsets of [1, n] and define f(n, k, b) = max{IAI : A E M} and g(n, k, b) = min{IAI : A EM}. In this paper a new me...

Abstract. Let A(n, d) denote the maximum number of codewords in a binary code of length n and minimum Hamming distance d. Upper and lower bounds on A(n, d) have been a subject for extensive research. In this paper we examine upper bounds on A(n, d) as a special case of bounds on the size of subsets in metric association scheme. We will first obtain general bounds on the size of such subsets, ap...

This paper is a survey of open problems and results involving extremal size of collections of subsets of a finite set subject to various restrictions, typically on intersections of members .

‎A topological space is called submaximal if each of its dense subsets is open and is called nodec if each of its nowhere dense ea subsets is closed‎. ‎Here‎, ‎we study a variety of spaces some of which have already been studied in $C(X)$‎. ‎Among them are‎, ‎most importantly‎, ‎quasi $P$-spaces and pointwise quasi $P$-spaces‎. ‎We obtain some new useful topological characterizations of quasi $...

In the present paper, we introduce the sequence space [{l_p}(E,Delta) = left{ x = (x_n)_{n = 1}^infty : sum_{n = 1}^infty left|  sum_{j in {E_n}} x_j - sum_{j in E_{n + 1}} x_jright| ^p < infty right},] where $E=(E_n)$ is a partition of finite subsets of the positive integers and $pge 1$. We investigate its topological properties and inclusion relations. Moreover, we consider the problem of fin...

Let $$\mathcal G$$ be a family of subsets an n-element set. The is called non-trivial 3-wise intersecting if the intersection any three in non-empty, but all empty. For real number $$p\in (0,1)$$ we define measure by sum $$p^{|G|}(1-p)^{n-|G|}$$ over $$G\in \mathcal . We determine maximum families. also discuss uniqueness and stability corresponding optimal structure. These results are obtained...