Finiteness properties of chevalley groups overFq[t]

LetG be a simple Chevalley group of rankn and Γ=G( ). Then the finiteness length of Γ shall be determined by studying the action of Γ on the Bruhat-Tits buildingX ofG . This is always possible provided that certain subcomplexes of the links of simplices inX are spherical. As a consequence, one obtains that Γ is of typeF _n−1 but not of typeFP _n ifG is of typeA _n, B_n, C_n orD _n andq≥2²ⁿ⁻¹.     

The sequence of codimensions of PI-algebras

Bounds and asymptotic formulas are given for the size of the irreducible representations of the symmetric groups. These are applied to obtain information on the identities and codimension sequencec _n(R) of a PI-algebraR, of a PI-algebraR of characteristic zero, e.g., the “ultimate” width of the hook in which the diagrams of the cocharacters ofR lies is <=(lim c _n (R)^1/n ) ² , and lim c_n(R)^1/n≦ 2(lim c_n(R)^1/n)² for rings with no right (or left) total annihilators.     

部分和乘积的几乎处处中心极限定理

设X_n, n≥1是独立同分布正的随机变量序列, E(X₁)=u >0, Var(X₁)=σ², E|X₁|³<∞, 记S_n==∑^N_k=1X_k, 变异系数γ=σ/u.g是满足一定条件的无界可测函数, 证明了 lim_N→∞1/logN∑^N_n=11/n g((∏ⁿ_k=1S_k/n!uⁿ )^1/γ√n )=∫^∞₀g(x)dF(x),a.s., 其中 F(•) 是随机变量e^√2ξ 的分布函数, ξ 是服从标准正态分布的随机变量.     

负相依随机变量的自正则化中心极限定理与部分和的方差估计

§ 1　 IntroductionWe firstintroduce some concepts.Random variables X and Y are called negative dependent ( ND) if for any pair ofmonotonically non-decresing functions f and g,Cov{ f( X) ,g( Y) }≤ 0 .Clearly itis equivalenttoP( X≤ x,Y≤ y)≤ P( X≤ x) P( Y≤ y)for all x,y∈R.A random sequence{ Xi,i≥ 1 } is said to be negative quadrant dependent( NQD) if any pairof variables Xi,Xj( i≠j) are ND.A sequence of random variables{ Xi,i≥ 1 } is said to be linear negative quadrand depend…     

A Lower Bound for the Norm of the Minimal Residual Polynomial

Let S be a compact infinite set in the complex plane with 0∉S, and let R _n be the minimal residual polynomial on S, i.e., the minimal polynomial of degree at most n on S with respect to the supremum norm provided that R _n (0)=1. For the norm L _n (S) of the minimal residual polynomial, the limit k(S):=lim_n?￥ⁿ?{L_n(S)}\kappa(S):=\lim_{n\to\infty}\sqrt[n]{L_{n}(S)} exists. In addition to the well-known and widely referenced inequality L _n (S)≥κ(S) ⁿ , we derive the sharper inequality L _n (S)≥2κ(S) ⁿ /(1+κ(S)²ⁿ ) in the case that S is the union of a finite number of real intervals. As a consequence, we obtain a slight refinement of the Bernstein–Walsh lemma.     

Rigidity of codimension two indefinite spheres

We describe all isometric immersions f:S _n ^s S _s ⁿ +2/S _s ⁿ n–s4, whenever the set of totally geodesic points does not disconnect S _s ⁿ , where S _n ^s denotes the complete n-dimensional indefinite Riemannian space form of constant positive curvature 1 and signature s.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

QUINTESSENTIAL PRIMES AND IDEAL TOPOLOGIES OVER A MODULE

Let I be an ideal of a Noetherian ring R, N a finitely generated R-module and let S be a multiplicatively closed subset of R. We define the Nth (S)-symbolic power of I w.r.t. N as S(I ⁿ N) = ∪ _s∈S (I ⁿ N: _N s). The purpose of this paper is to show that the topologies defined by {Iⁿ N} _n≥0 and {S(Iⁿ N)} _n≥0 are equivalent (resp. linearly equivalent) if and only if S is disjoint from the quintessential (resp. essential) primes of I w.r.t. N.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.     

Estimation of the variance for strongly mixing sequences

Abstract. Let {Xn,n≥1} be a stationary strongly mixing random sequence satisfying EX1=u,     

M?bius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z_K^{(\mathbb D, \mathbb S)}\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z_K\mathcal{Z}_{K} and the real moment-angle complex \mathbbRZ_K\mathbb{R}\mathcal {Z}_{K} as special examples. We show that H^*(Z_K^{(\mathbb D, \mathbb S)};k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor  ^k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for Z_K\mathcal{Z}_{K} (resp. \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T ^m -action on Z_K\mathcal{Z}_{K} (resp. (ℤ₂) ^m -action on \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}).     

A local limit theorem for random walks conditioned to stay positive

We consider a real random walk S_n=X₁+...+X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n/a_n⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S₁>0,...,S_n>0) and let S_n⁺ denote the random variable S_n conditioned on : it is known that S_n⁺/a_n ↠ ϕ⁺(x) dx, where ϕ⁺(x):=x exp (−x²/2)1_(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X₁ has an absolutely continuous law: in this case the uniform convergence of the density of S_n⁺/a_n towards ϕ⁺(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.     

Ein Null-Zwei-Gesetz auf lokalkompakten abelschen Halbgruppen

Let λ, μ be regular probability measures on a locally compact abelian semigroup S, λ * μ the convolution of λ and μ, λⁿ the n^th iterated convolution of λ, δ_x the point measure of x?S. We study the totalvariation of λⁿ–δ_x * λⁿ for n → ∞. We shall see that for a certain class of semigroups the limit of this sequence is either 0 or 2.     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.     

Classification ofS n -normal semigroups

IfS _n andC _n denote, respectively, the symmetric group and inverse semigroup onn symbols, thenS _n⊂C_n and a semigroupT⊂C_n isS _n -normal ifα ⁻¹ Tα ⊂Tfor every α∈S _n . TheS _n -normal semigroups are classified.     

Moments of the maximum of normed partial sums of ρ^--mixing random variables

Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 ｜Sn/n^1/r｜^p（0 〈 r 〈 2,p 〉 0） are given,which are the same as that in the independent case.     

A class of Hamiltonian and edge symmetric Cayley graphs on symmetric groups

Abstract. Let Sn be the symmetric group     

Strong Approximation Theorems for Sums of Random Variables when Extreme Terms are Excluded

Let {X _n ;n≥1} be a sequence of i.i.d. random variables and let X ^(r) _n = X _j if |X _j | is the r-th maximum of |X ₁|, ..., |X _n |. Let S _n = X ₁+⋯+X _n and ^(r) S _n = S _n −(X ⁽¹⁾ _n +⋯+X ^(r) _n ). Sufficient and necessary conditions for ^(r) S _n approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates of the strong law of large numbers for ^(r) S _n are studied. Received March 22, 1999, Revised November 6, 2000, Accepted March 16, 2001     

Homeomorphism classification of complex projective complete intersections of dimensions 5, 6 and 7

In this paper we prove that two complete intersections X_n(d){X_n(\underline{d})} and X_n(d￠){X_n(\underline{d}^\prime)} are homeomorphic if and only if they have the same total degree, Pontrjagin classes and Euler characteristics, provided n = 5, 6, 7. This extends earlier result of Fang and Klaus (Manuscr Math 90:139–147, 1996).