Definitions. Let X be an arbitrary set, J^ a Borel-field of some subsets B of X, and 9ii(jQ the family of all probability measures defined on J¿, i.e. the totality of all countably additive, non-negative set functions m(B), BE/¿, for which m(X) = 1. Henceforth the word "measure" denotes an element of M(jQ) and the expression "set of measures" a subset of Vïl(jQ. Two measures m and m' have been ...

The set equality problem is to decide whether two sets given by functions a and b such that A = {a(1), a(2), ..., a(n)} and B = {b(1), b(2), ..., b(n)} are equal or disjoint, under the promise that one of these is the case. Some other problems, like the Graph Isomorphism problem, is solvable by reduction to the set equality problem. This motivates to find a polynomial quantum lower bound for th...

⊇ Basic ⊇ Congruence ⊇ Division of sets of intervals, those sets closed under taking (a) similar intervals, (b) sub-intervals, (c) abutting intervals, (d) suprema, respectively. We let B(Λ) and D(Λ) be the family of all basic sets and division sets, respectively. Abstract and basic sets are not particularly important, but the do help with the organization. However, observe that the family B(Λ) ...

A permutation is an arrangement of a finite number of distinct elements of a linear order, for example, e, π, 0, √ 2 and 3412. Two permutations are order isomorphic if the have the same relative ordering. We say a permutation τ contains or involves the permutation β if deleting some of the entries of τ gives a permutation that is order isomorphic to β, and we write β ≤ τ . For example, 534162 (...

Let G be a graph. A set S ⊆ V (G) is hop dominating of if for every v ∈ (G)\S, there exists u such that dG(u, v) = 2. The minimum cardinality γh(G) the domination number G. Any γh-set which intersects transversal set. γbh(G) in In this paper, we initiate study domination. First, characterize graphs whose values are either n or − 1, and determine specific some graphs. Next, show positive integer...

Our main result is a construction of four families ${\cal C}_1,{\cal C}_2,{\cal B}_1,{\cal B}_2$ which are equipollent with the power set ${\Bbb R}$ and satisfy following properties. (i) The members proper subfields $K$ where algebraic over $K$. (ii) Each field in C}_1\cup{\cal C}_2$ contains {\it Cantor set}. (iii) B}_1\cup{\cal Bernstein (iv) All fields B}_1$ isomorphic. (v) If $K,L$ C}_2\cup...

Abstract. This investigation studies nonemptiness problems of plane square tiling. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges of p colors are arranged side by side such that adjacent tiles have the same colors. Given a set of Wang tiles B, the nonemptiness problem is to determine whether or not Σ(B) = ∅, where Σ(B) is the set of all global patterns on Z2 th...

A permutation is an arrangement of a finite number of distinct elements of a linear order, for example, e, π, 0, √ 2 and 3412. Two permutations are order isomorphic if the have the same relative ordering. We say a permutation τ contains or involves a permutation β if deleting some of the entries of π gives a permutation that is order isomorphic to β, and we write β ≤ τ . For example, 534162 (wh...

in this note we are going to survey several invariants of finite groups related either to their orders or to generating sets or to lattices of subgroups. some relations among these invariants will be exhibited. special attention will be paid to monotonicity of them.

For a finite graph G= (I’, E) Menger’s Theorem [6] states the following: if A, B c V then the minimal size of an A -B separating set of vertices (i.e., a set whose removal disconnects A from B), equals the maximal size of a set of disjoint A -B paths. This version of the theorem remains true, and quite easy to prove, also in the infinite case. To see this, take 9 to be a maximal (with respect t...