In this article we prove a conjecture of A. Lubotzky: let G = G0(K), where K is a local field of characteristic p ≥ 5, G0 is a simply connected, absolutely almost simple K-group of K-rank at least 2. We give the rate of growth of ρx(G) := |{Γ ⊆ G|Γ a lattice in G, vol(G/Γ) ≤ x}/ ∼ |, where Γ1 ∼ Γ2 if and only if there is an abstract automorphism θ of G such that Γ2 = θ(Γ1). We also study the ra...

We determine the number of maximal intersecting families on a 9-set and find 423295099074735261880. We determine the number of independent sets of the Kneser graph K(9, 4) and find 366996244568643864340. Finally, we determine the number of intersecting families on an 8-set and find 14704022144627161780744368338695925293142507520.

We prove uniform L estimates (Theorem 1.1) for a family of paraproducts and corresponding maximal operators.

The dimension of a poset (X, P), denoted dim (A", P), is the minimum number of linear extensions of P whose intersection is P. It follows from Dilworth's decomposition theorem that dim (X, P)& width (X, P). Hiraguchi showed that dim(X, P)s \X\/Z In this paper, A denotes an antichain of (A", P) and E the set of maximal elements. We then prove that dim {X, P) s |X A\; dim(X, P) < 1 + width (X E);...

Assuming that every set is constructible, we find a Π1 maximal cofinitary group of permutations of N which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a Π1 maximal cofinitary group in L.

In 2007, Olsson and Stanton gave an explicit form for the largest (a, b)-core partition, for any relatively prime positive integers a and b, and asked whether there exists an (a, b)-core that contains all other (a, b)-cores as subpartitions; this question was answered in the affirmative first by Vandehey and later by Fayers independently. In this paper we investigate a generalization of this qu...

Let S be a non-empty subset of a group G. We say S is product-free if S ∩ SS = ∅, and S is locally maximal if whenever T is product-free and S ⊆ T , then S = T . Finally S fills G if G∗ ⊆ S t SS (where G∗ is the set of all non-identity elements of G), and G is a filled group if every locally maximal product-free set in G fills G. Street and Whitehead [8] investigated filled groups and gave a cl...

We describe the general class of N -extended D = (2 + 1) Galilean supersymmetries obtained, respectively, from the N -extended D = 3 Poincaré superalgebras with maximal sets of central charges. We confirm the consistency of supersymmetry with the presence of the ‘exotic’ second central charge θ. We show further how to introduce a N = 2 Galilean superfield equation describing nonrelativistic spi...

In this paper it is described a method to compute the distance between sets, that implies the formation of distance functions different from Hausdorff metric. Two functions with metric properties, which describe quantitatively distances between sets, are formed. First function can be used for sets arbitrary situated from each other. Second distance is more suited for sets clustered by rank link...

Blbzsik, Z., M. Hujter, A. Pluhir and Z. Tuza, Graphs with no induced C4 and 2K,, Discrete Mathematics 115 (1993) 51-55. We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class B of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members...