Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

Asymptotics of degrees of someS n-sub regular representations

The numbers % MathType!End!2!1!, λ ⊢n appear in the enumeration of various objects, as well as coefficients inS _nrepresentations associated with products of higher commutators. We study their asymptotics asn→∞ and show that if (λ₁, λ₂, …)≈(α ₁,α ₂, …)n, if (λ′₁, λ′₂, …)≈(β ₁,β ₂, …)n and ifγ=1− Σ _k⩽1(α _k⩽1+β _k⩽1), then % MathType!End!2!1!. Work partially supported by N.S.F. Grant No. DMS 94-01197.     

Asymptotic formulas for the hyperġeometric function2 F 1 of matrix argument,useful in multivariate analysis

Summary LetS _i have the Wishart distributionW _p(∑_i,n_i) fori=1,2. An asymptotic expansion of the distribution of for large n=n₁+n₂ is derived, when∑ ₁∑ ₂ ⁻¹ =I+n^−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function₂ F ₁, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S ₂(S ₁+S ₂)⁻¹| under fixed∑ ₁∑ ₂ ⁻¹ , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for₁ F ₁ were investigated by Sugiura [7].     

Completely monotone multisequences, symmetric probabilities and a normal limit theorem

Let G _n,k be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc _n(β₁,β₂,…, β _k ), β₁,β₂,…, β_k = 0,1,2,…, β₁+β₂ + … +β _k ≤n,c _n(0,0,…, 0) = 1 and whenever β₀ ≤n - (β₁ + β₂ + … + β _k ) where Δc _n(β₁,β₂,…, β _k ) =c _n(β₁ + 1, β₂,…, β _k )+c _n(β₁,β₂+1,…, β _k )+…+c _n (β₁,β₂,…, β _k +1) -c _n(β₁,β₂,…, β _k ). Further, let Π _n,k be the set of all symmetric probabilities on {0,1,2,…,k} ⁿ . We establish a one-to-one correspondence between the sets G _n,k and Π _n,k and use it to formulate and answer interesting questions about both. Assigning to G _n,k the uniform probability measure, we show that, asn→∞, any fixed section {it{c_n}(β₁,β₂,…, β _k ), 1 ≤ Σβ _i ≤m}, properly centered and normalized, is asymptotically multivariate normal. That is, converges weakly to MVN[0, Σ _m ]; the centering constantsc ₀(β₁, β₂,…, β _k ) and the asymptotic covariances depend on the moments of the Dirichlet (1, 1,…, 1; 1) distribution on the standard simplex inR ^k.     

Maximal Dimension of Invariant Subspaces to Systems of Nonlinear Evolution Equations

In this paper,the dimension of invariant subspaces admitted by nonlinear systems is estimated under certain conditions.It is shown that if the two-component nonlinear vector differential operator F=(F 1,F 2) with orders {k 1,k 2 } (k 1 ≥ k 2) preserves the invariant subspace W 1 n 1 × W 2 n 2 (n 1 ≥ n 2),then n 1 n 2 ≤ k 2,n 1 ≤ 2(k 1 + k 2) + 1,where W q n q is the space generated by solutions of a linear ordinary differential equation of order n q (q=1,2).Several examples including the (1+1)-dimensional diffusion system and Ito 's type,Drinfel'd-Sokolov-Wilson's type and Whitham-Broer-Kaup's type equations are presented to illustrate the result.Furthermore,the estimate of dimension for m-component nonlinear systems is also given.     

Inverse problem for zeros of certain koebe-related functions

If w₁,…,w _N is a finite sequence of nonzero points in the unit disk, then there are distinct points λ₁,…, λ_N on the unit circle and positive numbers Μ₁,…,Μ _N such that is the zero sequence of the function 1 — . The points λ₁,…, λ_N and numbers Μ₁,…,Μ_N are unique (except for reorderings).     

Exact asymptotic behaviour of the codimensions of some P.I. algebras

Letc _n (A) denote the codimensions of a P.I. algebraA, and assumec _n (A) has a polynomial growth: . Then, necessarily,q∈ℚ [D3]. If 1∈A, we show that , wheree=2.71…. In the non-unitary case, for any 0<q∈ℚ, we constructA, with a suitablek, such that . In memory of S. A. Amitsur, our teacher and friend Partially supported by Grant MM404/94 of Ministry of Education and Science, Bulgaria and by a Bulgarian-American Grant of NSF. Partially supported by NSF grant DMS-9101488.     

On the asymptotic convergence of sequences of analytic functions

We consider sequences {f _n } of analytic self mappings of a domain and the associated sequence {Θ _n } of inner compositions given by . The case of interest in this paper concerns sequences {f _n } that converge assymptotically to a function f, in the sense that for any sequence of integers {n _k } with n ₁ < n ₂ < ... one has that locally uniformly in Ω. Most of the discussion concerns the case where the asymptotic limit f is the identity function in Ω. Received: 16 December 2006     

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.     

On statistical properties of finite continued fractions

Statistical properties of continued fractions for numbers a/b, where a and b lie in the sector a, b ≥ 1, a² + b² ≤ R², are studied. The main result is an asymptotic formula with two meaning terms for the quantity

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

Estimates for spectral projection operators of the sub-laplacian on the heisenberg group

In this paper, we use Laguerre calculus to find theLP spectrum (λ, Μ) of the pair (L, iT). Here md T = ∂/∂t with a basis for the left-invariant vector fields on the Heisenberg group. We find kernels for the spectral projection operators on the ray λ > 0 in the Heisenberg brush and show that they are Calderón-Zygmund-Mikhlin operators. Estimates for these operators in L _k ^p (H_n), H^P(H_n), and S _k ^pv (H_n) spaces can therefore be deduced.     

Uniformly Accurate Quantile Bounds for Sums of Arbitrary Independent Random Variables

Fix any n≥1. Let X ₁,…,X _n be independent random variables such that S _n =X ₁+⋅⋅⋅+X _n , and let S^*_n=sup_{1 ￡ k ￡ n}S_kS^{*}_{n}=\sup_{1\le k\le n}S_{k} . We construct upper and lower bounds for s _y and s_y^*s_{y}^{*} , the upper \frac1y\frac{1}{y} th quantiles of S _n and S^*_nS^{*}_{n} , respectively. Our approximations rely on a computable quantity Q _y and an explicit universal constant γ _y , the latter depending only on y, for which we prove that

On the rates of the other law of the logarithm

Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn).     

Reverse inequalities

There are reverse inequalities for square functions of differences arising in ergodic theory and differentiation of functions. For example, it is shown that if A_n is the usual average in ergodic theory, and (n_k∶k=1,2,3,...) is an increasing lacunary sequence with no non-trivial common divisor, then one has for any p, 1<p<∞, there is a constant C_p such that for all f∃ L_p(X),

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional in the class of functionsf(z)=z+a ₂z²+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk ₁,…,k_m contains an even number. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997. Translated by S. S. Anisov     

Graphic sequences with a realization containing intersecting cliques

Let r≥ 1, k≥ 2 and Fm1 ,...,mki;r denote the most general definition of a friendship graph, that is, the graph of Kr+m1 , . . . , Kr+mk meeting in a common r set, where Kr+mi is the complete graph on r + mi vertices. Clearly, | Fm1 ,...,mki;r | = m1+ ··· + mk + r. Let σ(Fm1 ,...,mki;r , n) be the smallest even integer such that every n-term graphic sequence π = (d1, d2, . . . , dn) with term sum σ(π) = d1 + d2 + ··· + dn ≥σ(Fm1 ,...,mki;r,n) has a realization G containing Fm1 ,...,mki;r as a subgraph. In this paper, we determine σ(Fm1 ,...,mki;r,n) for n sufficiently large.