具有奇异性的周期椭圆算子的绝对连续性

该文研究周期椭圆算子sun from(j,l=1) to d D_(jw)(x)a_(jl)D_l+V(x)在R~d(d≥3)中的谱性质,其中A=(a_(jl))是d×d阶的实常值正定矩阵,V(x)和w(x)是关于相同格点的周期标量函数,并且w(x)是正的.利用文中第一作者建立的d-环面上的一致Sobolev不等式,证明了该算子的谱是纯绝对连续的,如果V∈L_(loc)~(2pd/(d+2p))(R~d)且w∈A_(1+α)~(p,∞)(T~d)∩L~∞(T~d)(α0,p≥d),或者V∈L_(loc)~(2d/3)/(R~d),ω∈C~1(T~d),或者V∈L_(loc)~(d/2)(R~d),w∈L_(2,loc)~(d/2)(T~d).     

Recognition of finite simple groupsL 3(2 m ) ANDU 3(2 m ) by their element orders

It is proved that a finite group isomorphic to a simple non-Abelian group L₃(2^m) or U₃(2^m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S₄(2^m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1. Supported by RFFR grant No. 99-01-00550, by the National Natural Science Foundation of China (grant No. 19871066), and by the State Education Ministry of China (grant No. 98083). Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 567–585, September–October, 2000.     

Some examples of semigroup algebras of finite representation type

The semigroup algebras over a field K of the semigroups T_n of all permutations of a set of n elements are considered. It is proved: if n≤3 and (n!)^-1∈ K then the algebra KT_n has a finite representation type. Also the finiteness of the representation type of the semigroup algebra KS is established, where S is the sub-semigroup of T_n (n is arbitrary) such that S=J_n∪G where J_n={x∈T_n|rank x=1}, while G is a doubly transitive subgroup of the symmetric group S_n, the order of G being invertible in K. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 229–238, 1987.     

典型群的增广商群

设G 为有限域K 上的一般线性群(特殊线性群、酉群、辛群及正交群), 记整群环ZG 的n 次增广理想为△ⁿ(G). 本文着重研究有限域上的典型群的增广商群Q_n(G) = △ⁿ(G)/△ⁿ⁺¹(G), 并刻画了这些连续商群的结构.     

Minimal polynomials for compact sets of the complex plane

LetW _N(z)=a_Nz^N+... be a complex polynomial and letT _n be the classical Chebyshev polynomial. In this article it is shown that the polynomials (2a_N)^?n+1T_n(W_N), n ∈N, are minimal polynomials on all equipotential lines for {z∈C:|W _N(z)|≤1 Λ ImW _N(z)=0}     

On the graph for which there is a tree as the inverse interchange graph of a local graph

Let G be a finite, connected, undirected graph without loops and multiple edges. The note modifies slightly the concept of I^–1 (T_t), the inverse interchange graph of the local graph G(T_t) defined by a reference tree t G, and considers the properties of the graph G, when I^–1(T_t) is a tree.     

Threshold characterization of graphs with dilworth number two

A graph with nodes 1, …, n is a threshold signed graph if one can find two positive real numbers S, T and real numbers a₁, …, a_n associated with the vertices in such a way that i, j are linked iff either |a_i + a_j| ≥ S or |a_i - a_j| ≥ T. Such graphs generalize threshold graphs. It is shown that these graphs are precisely the graphs with Dilworth number at most two (the Dilworth number is the maximum number of pairwise incomparable vertices in the vicinal preorder). Some other properties of this subclass of perfect graphs are also presented. The graphs considered in this paper are finite simple graphs G = (V, E), where V is the vertex set of G and E a subset of pairs of G. For x V, N(x) denotes the neighbor set of x: N(x) = {y | {x, y} E}.     

Probabilistic Adaptive Width of a Multivariate Sobolev Space Equipped with a Gaussian Measure

In this paper, we determine the asymptotic values of the probabilistic adaptive widths of the space of multivariate functions with bounded mixed derivative (MW₂^r(T^d),μ) relative to the manifold (Y^N,ν) in the L_q(T^d)-norm, 1 < q ≤ 2, where μ and ν are two given Gaussian measures.     

The only convex body with extremal distance from the ball is the simplex

We can extend the Banach-Mazur distance to be a distance between non-symmetric sets by allowing affine transformations instead of linear transformations. It was proved in [J] that for every convex bodyK we haved(K, D)≤n. It is proved that ifK is a convex body in ℝ ⁿ such thatd(K, D)=n, thenK is a simplex. This article is an M.Sc. thesis written under the supervision of E. Gluskin and V.D. Milman at Tel Aviv University. Partially supported by a G.I.F. grant.     

Formule de Rice en dimension d

Summary Let {x(t): tR ^d} a stochastic process with parameter in R ^d, and u a fixed real number. Denote by C _u, A_u, B_u respectively the random sets {t: x(t)= u}, {t: x(t)}, {t: x(t)>u}. The paper contains two main results for processes with continuously differentiable paths plus some additional requirements: First, a formula for the expectation of Q _T(A_u) and Q _T(B_u), where for a given bounded open set T in R ^d, Q_T(B) denotes the perimeter of B relative to T and second, sufficient conditions on the process, so that it does not have local extrema on the barrier u. The second result can also be used to interpret the first in terms of C _u.     

On aschbacher’s localC(G; T) theorem

Aschbacher’s localC(G; T) theorem asserts that ifG is a finite group withF*(G)=O ₂(G), andTεSyl₂(G), thenG=C(G; T)K(G), whereC(G; T)=〈N _G (T ₀)|1≠T ₀ charT〉 andK(G) is the product of all near components ofG of typeL ₂(2 ⁿ ) orA ₂ ⁿ ₊₁. Near components are also known asχ-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher’s theorem in the case thatG is aK-group, i.e., in the case that every simple section ofG is isomorphic to one of the known simple groups. Our proof relies on a result of Meierfrankenfeld and Stroth [MS] on quadratic four-groups and on the Baumann-Glauberman-Niles theorem, for which Stellmacher [St2] has given an amalgam-theoretic proof. Apart from those results, our proof is essentially self-contained. For John Thompson Supported in part by NSF grant #DMS 89-03124, by DIMACS, an NSF Science and Technology Center, funded under contract STC-88-09648, and by NSA grant #MDA-904-91-H-0043. Prof. Gorenstein died on August 26, 1992.     

A note on the complexity and tractability of the heat equation

We wish to solve the heat equation u_t=Δu-qu in I^d×(0,T), where I is the unit interval and T is a maximum time value, subject to homogeneous Dirichlet boundary conditions and to initial conditions u(·,0)=f over I^d. We show that this problem is intractable if f belongs to standard Sobolev spaces, even if we have complete information about q. However, if f and q belong to a reproducing kernel Hilbert space with finite-order weights, we can show that the problem is tractable, and can actually be strongly tractable.     

奇异积分中的一些问题

设n≥2,Ω为R~n中单位球面S~(n-1)上的可积函数且Ω在S~(n-1)上的平均值为零,即∫_S~(n1)~Ω(x)dσ(x)=0.其中dσ为S~(n-1)上的体积元.定义奇异积分算子T_0,和相应的极大算子T~*,其中h∈L~∞(R~+).关于算子T和T~*已有许多研究([1]-[6]等).在1986年,Namazi利用Fourier变换的Hausdorff-Young不等式证明了     

On polynomial symbols for subdivision schemes

Given a dilation matrix A :ℤ^d→ℤ^d, and G a complete set of coset representatives of 2π(A ^−Tℤ^d/ℤ^d), we consider polynomial solutions M to the equation ∑ _g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order. Research partially supported by the Danish Technical Science Foundation, Grant No. 9701481, and by the Danish SNF-PDE network.     

Asymptotic behaviour of multiperiodic functions scaled by Pisot numbers

Let β > 1 be a Pisot number andg be a positive Hölder continuous function with period one and g(0) = 1. The multiperiodic functionG(ξ)=Π _n=0 ^∞ g(ξ/βⁿ) is studied and the asymptotic behaviour ofI _G(T) = ∫ ₀ ^T G(ξ)dξ investigated. We prove that the limit of logI(T)/ logT exists asT tends to infinity. We also provide a method to calculate this limit for the caseg(ξ) = cos² 2πξ, corresponding to the Fourier transform of the Bernoulli convolution associated to the golden number (or some of its generalizations).     

关于有限Abelian群的生成子集的基数

假若G =Z_m1 Z_m2… Z_mr 为 (m₁, m₂,…, m_r)型Abelian群, 其中Z_mi 为 m_i 阶的循环群且1≤i≤ r, m₁ |m₂|…| m_r, S 为G 的满足0∈ S=-S 的生成子集. 如果 |S|>|G|/ρ, 其中ρ≥l m_r /2l且m_r=e(G) 为群 G 的所有元素的阶的最小公倍数, 则ρS=G. 更进一步作者推广了Klopsch与lev ^[1]的一个结论,有:若 G=Z₂Z_m 为 (2, m) 型 Abelian 群(m ≥8), 则 t_m/2(G)=0.     

N-widths of Ω r ′ in L p

Let Ω _ϕ ^r ={f:f ^(r-1) abs. cont. on [0,1], ‖q_r(D)f‖_p≤1, f^(2K+σ) (0)=f^(2K+σ)=0, (k)=0,...,l-1}. where , and I is an identical operator. Denote Kolmogorov, linear, Geelfand and Bernstein n-widths of Ω _ϕ ^r in L^p byd _n (Ω _ϕ ^r ;L ^p ),δ _n (Ω _ϕ ^r ;L ^p ),d ⁿ (Ω _p ^r ;L ^p ) andb _n (Ω _p ^r ;L ^p ), respectively. In this paper, we find a method to get an exact estimation of these n-widths. Related optimal subspaces and an optimal linear operator are given. For another subset , similar results are also derrived.     

A counterexample in the cohomology of monoids

Let S be a commutative monoid, and 1et T be the maximal cancellative homomorphic image of S. The cohomology of S with coefficients in a left Z(S)-module D is given by Hⁿ(S,D)=Ext _Z(S) ⁿ (Z,D) as usual. [3] If from D we form the Z(T)-module D'=Hom_Z(S)(Z(T),D), then there is a natural homomorphism Hⁿ(T,D')→Hⁿ(S,D). If S is finite, then it follows from the results of [1] and [4] that this homomorphism is an isomorphism. The example presented below shows that this need no longer be an isomorphism if S is infinite. This research was supported in part by the National Science Foundation.