We investigate the -randomness of unions and intersections of random sets under various notions of randomness corresponding to di erent probability measures. For example, the union of two relatively MartinL of random sets is not Martin-L of random but is random with respect to the Bernoulli measure 3 4 under which any number belongs to the set with probability 3 4 . Conversely, any 3 4 random...

In this paper the concept of prob'abilistic events is introduced and their closure properties under the operations on the set of all fuzzy events are studied. I t is shown that the mean of probabilistie events is a probabilistie event and the set of all probabilistie events is closed under the transposition and the operation of the convex-combination. A sufficient conditioll for the union and t...

In the basic paper defining the Gröbner Cover [16] for discussing parametric polynomial systems of equations, we 8 introduced algorithms that have been improved since then. We used our own algorithm buildtree for computing 9 the initial Comprehensive Gröbner System (CGS), needed for the Gröbner Cover, now substituted in the Singular [7] 10 library “grobcov.lib” by themore efficientKapur–Sun–Wan...

Let 2 ≤ d ≤ k be fixed and n be sufficiently large. Suppose that G is a collection of k-element subsets of an n-element set, and |G| > ( n−1 k−1 ) . Then G contains d sets with union of size at most 2k and empty intersection. This verifies a conjecture of the first author for large n.

This paper is concerned with the many deep and far reaching consequences of Ash’s positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite mono...

This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for nite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a nite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of nite monoids ha...

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture. The normal Moore space conjecture states that every normal Moore space is metrizable. This is known to be not provable from the usual axioms of set theory, since Silver [4] shows that Martin's ax...

Let Ω be a finite set and let S ⊆ P(Ω) be a set system on Ω. For x ∈ Ω, we denote by dS(x) the number of members of S containing x. A long-standing conjecture of Frankl states that if S is union-closed then there is some x ∈ Ω with dS(x) ≥ 12 |S|. We consider a related question. Define the weight of a family S to be w(S) := ∑ A∈S |A|. Suppose S is union-closed. How small can w(S) be? Reimer sho...

Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on Rn can be obtained from the polynomial ring R[x1, . . . , xn] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points α, β ∈ Sper R[x1, . . ...