The character of an irreducible admissible representation of a p-adic reductive group is known to be a constant function in some neighborhood of any regular semisimple element γ in the group. Under certain mild restrictions on γ, we give an explicit description of a neighborhood of γ on which the character is constant.

Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 4. Assume that the intersection numbers of Γ satisfy ai = 0 for 0 ≤ i ≤ D − 1 and aD 6= 0. We show that Γ is a polygon, a folded cube, or an Odd graph.

A Cayley graph Γ = Cay(G,S) is said to be normal for a finite group G, if the right regular representation R(G) is normal in the full automorphism group Aut(Γ) of Γ. In this paper we investigate the normality of Cayley graphs of groups of order a product of two distinct primes, by determining all nonnormal Cayley graphs of these groups.

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m for every positive integer m. If m > 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m, 2m +m, m +m, m +m). Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defined by ...

J.H. Koolen and J. Park proved a lower bound for the intersection number c2 of a distance-regular graph Γ. Moreover, they showed that a graph Γ, for which equality is attained in this bound, is a Terwilliger graph. We prove that Γ is the icosahedron, the Doro graph or the Conway–Smith graph if equality is attained and c2 > 2.

Let Γ denote a Q-polynomial distance-regular graph with diameter at least three and standard module V . We introduce two direct sum decompositions of V . We call these the displacement decomposition for Γ and the split decomposition for Γ. We describe how these decompositions are related.

Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 4. Assume that the intersection numbers of Γ satisfy ai = 0 for 0 ≤ i ≤ D − 1 and aD 6= 0. We show that Γ is a polygon, a folded cube, or an Odd graph.

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m4 for every positive integerm. Ifm > 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m4, 2m4 + m2,m4 + m2,m4 + m2). Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defi...

A Schwarz function on an open domain Ω is a holomorphic satisfying S(ζ)=ζ ¯ Γ, which part of the boundary Ω. Sakai in 1991 gave complete characterization admitting function. In fact, if simply connected and Γ=∂Ω∩D(ζ 0 ,r), then Γ has to be regular real analytic (with possible cusps). Sakai’s result natural applications 1) quadrature domains, 2) free problem for Δu=1 equation. our scenarios can ...

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...