A statistic w on Sn is a weighted-inversion (w-i) statistic if there exist weights w i;j such that w() = P i j ]w i;j for each 2 Sn. Two well-known examples are the major index and inversion count statistics. These two statistics share the same distribution over Sn, and many bijections Sn ! Sn have been described to prove this. These bijections thus have the property that they map a certa...

Given a divisor A of a function field, there is a unique divisor of minimum degree that defines the same vector space of rational functions as A and there is a unique divisor of maximum degree that defines the same vector space of rational differentials as A. These divisors are called the floor and the ceiling of A. A method is given for finding both the floor and the ceiling of a divisor. The ...

Claim 1. In any equilibrium, we have  1  0. Proof. First, we show that in any equilibrium we must have ∗  0. Suppose to the contrary that we had ∗ = 0. Then by posting a price 0 ∈ ( ̄) in period 1, an inactive firm would be sure to sell, making the deviation profitable, a contradiction to equilibrium. Hence we must have ∗  0 . Next, suppose to the contrary to the statement of Claim 1 ...

We consider the Fröbenius-Perron semigroup of linear operators associated to a semidynamical system defined in a topological space X endowed with a finite or a σ -finite regular measure. We prove that if there exists a faithful invariant measure for the semidynamical system, then the Fröbenius-Perron semigroup of linear operators is C0-continuous in the space Lμ(X). We also give a geometrical c...

It is proved that for any centrally symmetric convex polygonal domain P and for any natural number r, there exists a constant k = k(P, r) such that any k-fold covering of the plane with translates of P can be split into r simple coverings.

We prove the following theorems: 1. There exists an ω-covering with the property s0. 2. Under cov(N ) = 2 there exists X such that ∀B∈Bor[B ∩ X is not an ω-covering or X \B is not an ω-covering]. 3. Also we characterize the property of being an ω-covering.

In the paper we show that all combinatorial triangle-free configurations for v ≤ 18 are geometrically realizable. We also show that there is a unique smallest astral (183) triangle-free configuration and its Levi graph is the generalized Petersen graph G(18, 5). In addition, we present geometric realizations of the unique flag transitive triangle-free configuration (203) and the unique point tr...

In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no modelcomplete theory of II1 factors.

An (r, w; d) cover-free family (CFF ) is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. The minimum number of elements for which there exists an (r, w; d) − CFF with t blocks is denoted by N((r, w; d), t). In this paper, we show that the value of N((r, w; d), t) is equal...

In combinatorial games, few results are known about the overall structure of multi-player games. We prove that given any finite set S of multi-player games, the set of games all of whose immediate options belong to S forms a completely distributive lattice with respect to every partial order relation ≤C, where C is an arbitrary coalition of players.