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.     

Greedy Algorithm Computing Minkowski Reduced Lattice Bases with Quadratic Bit Complexity of Input Vectors

The authors present an algorithm which is a modification of the Nguyen-Stehle greedy reduction algorithm due to Nguyen and Stehle in 2009. This algorithm can be used to compute the Minkowski reduced lattice bases for arbitrary rank lattices with quadratic bit complexity on the size of the input vectors. The total bit complexity of the algorithm is $O(n^2 \cdot (4n!)^n \cdot (\tfrac{{n!}} {{2^n }})^{\tfrac{n} {2}} \cdot (\tfrac{4} {3})^{\tfrac{{n(n - 1)}} {4}} \cdot (\tfrac{3} {2})^{\tfrac{{n^2 (n - 1)}} {2}} \cdot \log ^2 A) $O(n^2 \cdot (4n!)^n \cdot (\tfrac{{n!}} {{2^n }})^{\tfrac{n} {2}} \cdot (\tfrac{4} {3})^{\tfrac{{n(n - 1)}} {4}} \cdot (\tfrac{3} {2})^{\tfrac{{n^2 (n - 1)}} {2}} \cdot \log ^2 A) , where n is the rank of the lattice and A is maximal norm of the input base vectors. This is an O(log² A) algorithm which can be used to compute Minkowski reduced bases for the fixed rank lattices. A time complexity n! · 3 ⁿ (log A) ^O(1) algorithm which can be used to compute the successive minima with the help of the dual Hermite-Korkin-Zolotarev base was given by Blomer in 2000 and improved to the time complexity n! · (log A) ^O(1) by Micciancio in 2008. The algorithm in this paper is more suitable for computing the Minkowski reduced bases of low rank lattices with very large base vector sizes.     

Local limit theorem for the first crossing time of a fixed level by a random walk

Let X,X(1),X(2),... be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + ... + X(n), n = 1, 2,..., and define the Markov stopping time η _y = inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,.... In the case $ \mathbb{E} $ \mathbb{E} |X|³ < ∞, the following relation was obtained in [8]: $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) as n → ∞, where the constant R and the bounded sequence ν _n were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0 under the condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ only; In [1], an explicit form of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) was found under the same condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that this corrected version was formulated in [8] as a conjecture.     

Packing 4-Cycles in Eulerian and Bipartite Eulerian Tournaments with an Application to Distances in Interchange Graphs

We prove that every Eulerian orientation of K_m,n contains arc-disjoint directed 4-cycles, improving earlier lower bounds. Combined with a probabilistic argument, this result is used to prove that every regular tournament with n vertices contains arc-disjoint directed 4-cycles. The result is also used to provide an upper bound for the distance between two antipodal vertices in interchange graphs.Received February 6, 2004     

Higher derivations of ore extensions

Let R be a prime ring and δ a derivation of R. Divided powers $ D_n ^{\underline{\underline {def.}} } \tfrac{1} {{n!}}\tfrac{{d^n }} {{dx^n }} $ D_n ^{\underline{\underline {def.}} } \tfrac{1} {{n!}}\tfrac{{d^n }} {{dx^n }} of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that $ S^{\underline{\underline {def.}} } Q[x;\delta ] $ S^{\underline{\underline {def.}} } Q[x;\delta ] is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form

On the Fon-Der-Flaass interpretation of extremal examples for Turán’s (3, 4)-problem

Fon-Der-Flaass (1988) presented a general construction that converts an arbitrary [(C)\vec]₄\vec C_4 -free oriented graph Γ into a Turán (3, 4)-graph. He observed that all Turán-Brown-Kostochka examples result from his construction, and proved the lower bound $\tfrac{4} {9} $\tfrac{4} {9} (1 − o(1)) on the edge density of any Turán (3, 4)-graph obtainable in this way. In this paper we establish the optimal bound $\tfrac{3} {7} $\tfrac{3} {7} (1 − o(1)) on the edge density of any Turán (3, 4)-graph resulting from the Fon-Der-Flaass construction under any of the following assumptions on the undirected graph G underlying the oriented graph Γ: (i) G is complete multipartite; (ii) the edge density of G is not less than $\tfrac{2} {3} - \varepsilon $\tfrac{2} {3} - \varepsilon for some absolute constant ε > 0. We are also able to improve Fon-Der-Flaass’s bound to $\tfrac{7} {{16}} $\tfrac{7} {{16}} (1 − o(1)) without any extra assumptions on Γ.     

Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f ₁(z), f ₂(z), …, f _n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ ⁿ and

Boundedness of solutions for a class of oscillating potentials without the twist condition

In this paper, we prove that the second order differential equation d^2x/dt^2＋x^2n_1f（x）＋p（t）=0with p（t ＋ 1） = p（t）, f（x ＋ T） = f（x） smooth and f（x） 〉 0, possesses Lagrangian stability despite of the fact that the monotone twist condition is violated.     

Estimates of the capacity of orthogonal arrays of large strength

D.G. Fon-Der-Flaass showed that Boolean correlation-immune n-variable functions of order m are resilient for $ m \geqslant \frac{{2n - 2}} {3} $ m \geqslant \frac{{2n - 2}} {3} . In this paper this theorem is generalized to orthogonal arrays. It is shown that orthogonal arrays of strength m not less than $ \frac{{2n - 2}} {3} $ \frac{{2n - 2}} {3} , where n is a number of factors having size at least 2 ⁿ⁻¹ and all arrays of size 2 ⁿ⁻¹, are simple.     

On rigidity of Clifford torus in a unit sphere

We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S n+1 satisfying Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n + δ(n), then S ≡ n, i.e., M is one of the Clifford torus S~k ((k/n)~1/2 ) ×S~...     

A better semi-online algorithm for Q3/s1 = s2≤ s3/Cmin with the known largest size

This paper investigates the semi-online machine covering problem on three special uniform machines with the known largest size. Denote by sj the speed of each machine, j = 1, 2, 3. Assume 0 < s1 = s2 = r ≤ t = s3, and let s = t/r be the speed ratio. An algorithm with competitive ratio max{2, (3s+6)/(s+6) } is presented. We also show the lower bound is at least max{2, (3s)/(s+6)}. For s ≤ 6, the algorithm is an optimal algorithm with the competitive ratio 2. Besides, its overall competitive ratio is 3 which matches the overall lower bound. The algorithm and the lower bound in this paper improve the results of Luo and Sun.     

Gauges for the cookie-cutter sets

Let E be a cookie-cutter set with dimH E =s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 〈 liminft→0 g（t）/ts 〈 ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 〈 lim supt→0 g（t）/ts 〈 ∞.     

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

The impact of poles on the convergence of best rational approximants

The convergence behavior of best uniform rational approximants r _n,m ^* with numerator degree n and denominator degree m on a compact set E is investigated for functions f continuous on E and ray sequences above the diagonal of the Walsh table, i. e. sequences {n,m(n)} _n=1^∞ with

Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

Let {X _i } _i=1^∞ be a standardized stationary Gaussian sequence with covariance function r(n) = EX ₁ X _n+1, S _n = Σ _i=1 ⁿ X _i , and $\bar X_n = \tfrac{{S_n }} {n} $\bar X_n = \tfrac{{S_n }} {n} . And let N _n be the point process formed by the exceedances of random level $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n by X ₁,X ₂,…, X _n . Under some mild conditions, N _n and S _n are asymptotically independent, and N _n converges weakly to a Poisson process on (0,1].     

Normality and quasinormality of zero-free meromorphic functions

Let k, K ∈ N and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F , f(k)-1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most ν = K k+1 , where ν is equal to the largest integer not exceeding K/k+1 . In particular, if K = k, then F is normal. The results are sharp.     

Precise Asymptotics in the Baum-Katz and Davis Laws of Large Numbers of ρ-mixing Sequences

Let {X,Xn;n ≥ 1} be a strictly stationary sequence of ρ-mixing random variables with mean zeros and finite variances. Set Sn =∑k=1^n Xk, Mn=maxk≤n｜Sk｜,n≥1.Suppose limn→∞ESn^2/n=：σ^2〉0 and ∑n^∞=1 ρ^2/d（2^n）〈∞,where d=2 if 1≤r〈2 and d〉r if r≥2.We prove that if E｜X｜^r 〈∞,for 1≤p〈2 and r〉p,then limε→0ε^2（r-p）/2-p ∑∞n=1 n^r/p-2 P{Mn≥εn^1/p}=2p/r-p ∑∞k=1（-1）^k/（2k＋1）^2（r-p）/（2-p）E｜Z｜^2（r-p）/2-p,where Z has a normal distribution with mean 0 and variance σ^2.     

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.     

On the rate of convergence of the expected spectral distribution function of a Wigner matrix to the semi-circular law

Let X:= (X _jk ) denote a Hermitian random matrix with entries X _jk which are independent for all 1 ≤ j ≤ k. We study the rate of convergence of the expected spectral distribution function of the matrix X to the semi-circular law under the conditions E X _jk = 0, E X _jk ² = 1, and E|X _jk |^2+η ≤ M _η < ∞, 0 < η ≤ 2. The bounds of order $ O(n^{ - \frac{\eta } {{2 + \eta }}} ) $ O(n^{ - \frac{\eta } {{2 + \eta }}} ) for 1 ≤ η ≤ 2, and those of order $ O(n^{ - \frac{{2\eta }} {{(2 + \eta )(3 - \eta )}}} ) $ O(n^{ - \frac{{2\eta }} {{(2 + \eta )(3 - \eta )}}} ) for 0 < η ≤ 1, are obtained.     

On sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers on sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers

New results about some sums s _n (k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Y, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s _n (k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.