For a ring R, denote by SpecΓ(κ,R) the κ-spectrum of the Γ-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that SpecΓ(א1 , R) is full for a suitable von Neumann regular algebra R, but the techniques do not extend to cardinals κ > א1. By a direct construction, we prove that for any field F and any regular uncountable cardinal κ there is a...

We consider a bipartite distance-regular graph Γ with vertex set X, diameter D ≥ 4, and valency k ≥ 3. For 0 ≤ i ≤ D, let Γ i (x) denote the set of vertices in X that are distance i from vertex x. We assume there exist scalars r, s, t ∈, not all zero, such that r|Γ 1 (x) ∩ Γ 1 (y) ∩ Γ 2 (z)| + s|Γ 2 (x) ∩ Γ 2 (y) ∩ Γ 1 (z)| + t = 0 let Γ 2 2 denote the graph with vertex set˜X = {y ∈ X | ∂(x, y)...

By using combinatorial properties of stationary sets, we give a simple proof of some generalization of Silver’s Theorem i.e. if κ is an uncountable regular cardinal such that אκ is a singular strong limit cardinal, then the following hold. (1). If {α < κ : א<κ α ≤ אα·2} is stationary, then 2אκ ≤ אκ·2. (2). If {α < κ : א<κ α ≤ אα+γ}, where 0 < γ < κ, is stationary, then 2אκ ≤ אκ+γ.

We present a general framework for Matrix theory compactified on a quotient space R/Γ, with Γ a discrete group of Euclidean motions in R. The general solution to the quotient conditions gives a gauge theory on a noncommutative space. We characterize the resulting noncommutative gauge theory in terms of the twisted group algebra of Γ associated with a projective regular representation. Present A...

In these lectures we plan to present a survey of certain aspects of harmonic analysis on a Heisenberg nilmanifold Γ\Hn. Using Weil-Brezin-Zak transform we obtain an explicit decomposition of L(Γ\H) into irreducible subspaces invariant under the right regular representation of the Heisenberg group. We then study the Segal-Bargmann transform associated to the Laplacian on a nilmanifold and charac...

Let Γ be a distance-regular graph with diameterd ≥ 3 and heighth = 2, whereh = max{i : pd,i 6= 0}. Suppose that for every α in Γ and every β in Γd(α), the induced subgraph on Γd(α)∩ Γ2(β) is isomorphic to a complete multipartite graph Kt×2 with t ≥ 2. Then d = 4 and Γ is isomorphic to the Johnson graph J(10, 4).

For a graph Γ with adjacency matrix A, we consider a switching operation that takes Γ into a graph Γ′ with adjacency matrix A′, defined by A′ = Q > AQ, where Q is a regular orthogonal matrix of level 2 (that is, Q > Q = I, Q1 = 1, 2Q is integral, and Q is not a permutation matrix). If such an operation exists, and Γ is nonisomorphic with Γ′, then we say that Γ′ is semi-isomorphic with Γ. Semiis...

We call a Cayley digraph Γ = Cay(G,S) normal for G if R(G), the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on the groups of order pq and also on non-abelian finite groups G such that every proper subgroup of G is abelian.

A shell of neutron detectors was designed, constructed, and employed in γ−ray spectroscopy with Gammasphere. It consists of up to 35 tapered regular hexagons that replace the same number of forward Ge-detector modules in Gammasphere. The shell was designed for high detection efficiency and very good neutron−γ discrimination. The simultaneous use of time of flight, and two methods of pulse shape...

Let γ(G) and γ◦(G) denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then γ(G) ≤ n d (log d+ 1) In this paper, the main result is that if G is any n-vertex d-regular graph of girth at least five then γ◦(G) ≤ n d (...