On the structure of the augmentation quotient group for some nonabelian 2-groups

Let G be a finite nonabelian group, ℤG its associated integral group ring, and Δ(G) its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups Q _n (G) = Δ ⁿ (G)/Δ ⁿ⁺¹(G) is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.     

On some new type generalized difference sequence spaces

In this paper we introduce a new type of difference operator Δ _m ⁿ for fixed m, n ∈ ℕ. We define the sequence spaces ℓ_∞(Δ _m ⁿ ), c(Δ _m ⁿ ) and c ₀(Δ _m ⁿ ) and study some topological properties of these spaces. We obtain some inclusion relations involving these sequence spaces. These notions generalize many earlier existing notions on difference sequence spaces.      

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex Δ^m(Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that Δ^m(Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part Δ₊^m(Φ) of Δ^m(Φ). An explicit homotopy equivalence is given between Δ₊^m(Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.     

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p~r,i.e.,a finite homocyclic abelian group.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(G) of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Q_n(G)=Δ~n(G)/Δ~(n 1)(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ℋ_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ℋ ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ℋ_q,1(B _n )- module D^λ remains irreducible on restriction to ℋ_q(D _n ).     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

On a Delay Reaction-Diffusion Difference Equation of Neutral Type

In this paper, we first consider a delay difference equation of neutral type of the form: Δ(y _n + py _n−k + q _n y _n−l = 0 for n∈ℤ⁺(0) (1*) and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241–248) to guarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n→∞. Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form: Δ₁(u _n,m + pu _n−k,m ) + q _n,m u _n−l,m = a ²Δ² ₂ u _{n +1, m−1} for (n,m) ∈ℤ⁺ (0) ×Ω, (2*) study various cases of p in the neutral term and obtain that if p≥−1 then every non-oscillatory solution of (2*) tends uniformly in m∈Ω to zero as n→∞; if p = −1 then every solution of (2*) oscillates and if p < −1 then every non-oscillatory solution of (2*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses. Received July 14, 1999, Revised August 10, 2000, Accepted September 30, 2000     

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999     

A uniqueness theorem and the myrberg phenomenon for a Zalcman domain

LetR=Δ₀\∪_nΔ_n be a Zalcman domain (or L-domain), where Δ₀ : 0<|z| <1, Δ_n : |z-c _n|≤r _n,c_n ↘0, Δ_n ⊂ Δ₀ and Δ_n ∩ Δ_m= φ(n≠m). 0217 0115 V 3 For an unlimited two-sheeted covering with the branch points {φ^-1(c _n)}, set . In the casec _n=2⁻ⁿ , it was proved that if a uniqueness theorem is valid forH ^∞ (R) atz=0, then the Myrberg phenomenon occurs. One might suspect that the converse also holds. In this paper, contrary to this intuition, we show that the converse of this previous result is not true. In addition, we generalize the previous result for more general sequences {c _n}. By this generalization we can even partly simplify the previous proof. To complete the present work the first and second (third, resp.) named authors were supported in part by Grant-in-Aid for Scientific Research, No. 10304010 (10640190, 11640187, resp.), Japanese Ministry of Education, Science and Culture.     

Oscillatory and asymptotic behaviour of fourth order nonlinear difference equations

This paper considers a class of fourth order nonlinear difference equations Δ²(r _n Δ²(y _n ) + Δ²(r _n ,f(n,n _n )=0,n ∈N(n ₀) wheref(n, y) may be classified as superlinear, sublinear, strongly superlinear and strongly sublinear. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. In strongly superlinear and strongly sublinear cases, sufficient conditions are given for all solutions to be oscillatory. Partially Supported by the National Science Foundation of China     

Cyclic elements of the shift operator in weighted anisotropic spaces of holomorphic functions in the polydisk

Let ϕ(r) = (ϕ₁(r₁), …, ϕ_n(r_n)) be a vector-valued function on R ₊ ⁿ . A necessary and sufficient condition is obtained under which any function f ∈, H^∞ (D ⁿ ), f(z) ≠ 0, z ∈, D ⁿ , is cyclic in the corresponding weighted space L^p(ϕ), where D ⁿ is the unit polydisk in C ⁿ. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 226–234.     

Comparison of germ expansion¶on inner forms of GL(n)

We prove that the germ expansion of a discrete series representation π^′ on GL _n (D) where D is a division algebra over k of index m and the germ expansion of the representation π of GL _mn (k) associated to π^′ by the Deligne–Kazhdan–Vigneras correspondence are closely related, and therefore certain coefficients in the germ expansion of a discrete series representation of GL _mn (k) can be interpreted (and therefore sometimes calculated) in terms of the dimension of a certain space of (degenerate) Whittaker models on GL _n (D). Received: 30 September 1999 / Revised version: 11 February 2000     

On some new generalized difference sequence spaces defined by a sequence of moduli

In this paper, we define the new generalized difference sequence spaces [V, λ, F, p, q]₀(Δ _v ^m ), [V, λ, F, p, q]₁(Δ _v ^m ) and [V, λ, F, p, q]_∞(Δ _v ^m ). We also study some inclusion relations between these spaces.     

Direct Constructions of Hyperplanes of Dual Polar Spaces Arising from Embeddings

Let e be one of the following full projective embeddings of a finite dual polar space Δ of rank n ≥ 2: (i) The Grassmann-embedding of the symplectic dual polar space Δ ≅ DW(2n – 1, q); (ii) the Grassmann-embedding of the Hermitian dual polar space Δ ≅ DH(2n – 1, q ²); (iii) the spin-embedding of the orthogonal dual polar space Δ ≅ DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Δ ≅DQ ⁻(2n + 1, q). Let H_e{\mathcal{H}_{e}} denote the set of all hyperplanes of Δ arising from the embedding e. We give a method for constructing the hyperplanes of H_e{\mathcal{H}_{e}} without implementing the embedding e and discuss (possible) applications of the given construction.     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

Balanced Convex Partitions of Measures in ℝ2

 Let n≥2 be an integer and let μ₁ and μ₂ be measures in ℝ² such that each μ _i is absolutely continuous with respect to the Lebesgue measure and μ₁(ℝ²)=μ₂(ℝ²)=n. Let u≠0 be a vector on the plane. We show that if μ₁(B)=μ₂(B)=n for some bounded domain B, then there exist positive integers n ₁,n ₂ with n ₁+n ₂=n and disjoint open half-planes D ₁,D ₂ such that , μ₁(D ₁)=μ₂(D ₁)=n ₁ and μ₁(D ₂)=μ₂(D ₂)=n ₂; or there exist positive integers n ₁,n ₂,n ₃ with n ₁+n ₂+n ₃=n and disjoint open convex domains D ₁,D ₂,D ₃ such that , μ₁(D ₁)=μ₂(D ₁)=n ₁, μ₁(D ₂)= μ₂(D ₂)=n ₂, μ₁(D ₃)=μ₂(D ₃)=n ₃ and such that the ray is parallel to u. We also show a similar result for partitioning of point sets on the plane. Received: November 24, 1999 Final version received: February 9, 2001     

On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone

In this paper, we study the initial-boundary value problem of the porous medium equation u _t  = Δu ^m  + V(x)u ^p in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|) ^σ . Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l ² + (n − 2)l = ω ₁. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u ₀ ≥ 0.