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Let Γ be a finitely generated group. Given a finite set of generators S of Γ, the word length lS(γ) for an element γ ∈ Γ is defined to be the smallest positive integer for which there exist s1, · · · , sn ∈ S ∪ S −1 such that γ = s1 · · · sn. For each n ∈ N, denote by BS(n) the set of elements in Γ whose word length with respect to S is at most n. It(More)
Let k be a local field of characteristic not 2, and let G be the group of k-rational points of a connected reductive linear algebraic group defined over k with a simple derived group of k-rank at least 2. We construct new uniform pointwise bounds for the matrix coefficients of all infinite-dimensional irreducible unitary representations of G. These bounds(More)
1.1. Motivation and overview. Let G denote the identity component of the special orthogonal group SO(n, 1), n ≥ 2, and V a finite-dimensional real vector space on which G acts linearly from the right. A discrete subgroup of a locally compact group with finite covolume is called a lattice. For v ∈ V and a subgroup H of G, let Hv = {h ∈ H : vh = v} denote the(More)
Let f be an integral homogeneous polynomial of degree d, and let Vm = {X : f(X) = m} be the level set for each m ∈ N. For a compact subset Ω in V1(R), set Nm(f, Ω) = #Vm(Z) ∩ mΩ. We define the notion of Hardy-Littlewood system for the sequence {Vm}, according as the asymptotic of Nm(f, Ω) as m → ∞ coincides with the one predicted by Hardy-Littlewood circle(More)
A white rot fungus was isolated from rotted leaves and identified as Pestalotiopsis sp. KCTC 8637P. It produced a flocculant biopolymer. A flocculant was partially purified from the culture broth by series of precipitations with 95% ethanol and named as Pestan. The components of Pestan were consisted of glucose : glucosamine : glucuronic acid : rhamnose(More)
When whole-cell extracts are analyzed, proton nuclear magnetic resonance (1H NMR) spectroscopy provides biochemical profiles that contain overlapping signals of the majority of the compounds. To determine whether cyanobacteria could be taxonomically discriminated on the basis of metabolic fingerprinting, we subjected whole-cell extracts of the cyanobacteria(More)
We show that for a given bounded Apollonian circle packing P, there exists a constant c > 0 such that the number of circles of curvature at most T is asymptotic to c ·T as T →∞. Here α ≈ 1.30 is the residual dimension of the packing. For P integral, let πP(T ) denote the number of circles with curvature bounded by T and prime. Similarly let πP 2 (T ) be the(More)
Let M be a complete Riemannian locally symmetric space of nonpositive curvature and of finite volume. We show that there are only finitely many compact maximal flats in M of volume bounded by a given number. As a corollary in the case M = SLn(Z)\ SLn(R)/ SOn, we give a different proof of a theorem of Remak that for any n ∈ N, there are only finitely many(More)