Criteria for Convergence of the Number of Near Maxima for Long Tails

Let M_n denote the maximum of a random sample of size n and K_n(a) be the number of near maxima, i.e. the number of sample observations in the fixed-width window (M_n – a, M_n]. There is a known integral criterion for almost sure convergence (to unity) of K_n(a), and we establish a similar criterion for complete convergence. We obtain simple but quite general sufficient conditions on the survivor function for satisfying the integral criteria. Further insight is obtained by seeking the rate at which P(K_n(a > 1)) tends to zero.AMS 2000 Subject Classification. 62G30, 60F15     

Matrix Summability and Positive Linear Operators

In the present paper we prove a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space C_ρ1 into a weighted space B_ρ2 with the use of a matrix summability method which includes both convergence and almost convergence. We also study the rates of convergence of these operators.     

Boundedness of Marcinkiewicz integral on the weighted Herz-type Hardy spaces

In this paper, we discuss the boundedness of Marcinkiewicz integral μΩ with homogeneous kernel on the weighted Herz-type Hardy spaces, and prove that μΩ is bounded from H K αq ,p ( w1 ; w2 ) into Kαq ,p (w 1; w2).     

The Hausdorff measure of generalized Sierpinski carpets

Let 0<a<1/2,T_a,2(x)=ax and T_0,2(x)=ax+1-a, and let K_a be a unique nomepty compact subset of R satisfying K_a=T_a,1(K_a)∪T_a,2(K_a). If 1/4<a<1/2, then the Hausdorff measure of K_a×K_a is strictly less than 2 ^{- log_a 2}2^{ - log_a 2} . If0<a≤1/4, then the Hausdorff measure of K_a×K_a is equal to 2 ^{- log_a 2}2^{ - log_a 2} .     

Equilibrium states for factor maps between subshifts

Let π:X→Y be a factor map, where (X,σX) and (Y,σY) are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a=(a₁,a₂)∈R² with a₁>0 and a₂?0. Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure μ on X that maximizes . In particular, taking f≡0 we see that there is a unique invariant measure μ on X that maximizes the weighted entropy a₁hμ(σX)+a₂hμ°π−1(σY), which answers an open question raised by Gatzouras and Peres (1996) in [15]. An extension is given to high dimensional cases. As an application, we show that for each compact invariant set K on the k-torus under a diagonal endomorphism, if the symbolic coding of K satisfies weak specification, then there is a unique invariant measure μ supported on K so that dimHμ=dimHK.     

Rates of -statistical convergence of positive linear operators

In this paper we study the rates of A-statistical convergence of sequences of positive linear operators mapping the weighted space C_ρ1 into the weighted space B_ρ2.     

Subnormality of Aluthge Transforms of Weighted Shifts

In this note we study the k-hyponormality and the subnormality of Aluthge transforms of weighted shifts. It is shown that Aluthge transforms of weighted shifts need not preserve the k-hyponormality. Moreover, we show that if W _α is a subnormal weighted shift with 2-atomic Berger measure then its Aluthge transform [(W)\tilde]_a{\widetilde{W}_\alpha} is subnormal if and only if at least one of two atoms is zero.     

On the strong convergence of weighted sums

Let {X _n ,n≥1} be a sequence of independent and identically distributed random variables and {a _ni ,1≤i≤n,n≥1} an array of constants. Some strong convergence results for the weighted sums ∑ _i=1 ⁿ a _ni X _i are obtained.     

Convolution operator and maximal function for the Dunkl transform

For a family of weight functionsh _K invariant under a finite reflection group onR ^d, analysis related to the Dunkl transform is carried out for the weightedL ^p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence. ST wishes to thank YX for the warm hospitality during his stay in Eugene. The work of YX was supported in part by the National Science Foundation under Grant DMS-0201669.     

External rays to periodic points

We prove that for every polynomial-like holomorphic mapP, ifaεK (filled-in Julia set) and the componentK _aofK containinga is either a point ora is accessible along a continuous curve from the complement ofK andK _ais eventually periodic, thena is accessible along an external ray. Ifa is a repelling or parabolic periodic point, then the set of arguments of the external rays converging toa is a nonempty closed “rotation set”, finite (ifK _ais not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in ℂ. In the Appendix we discuss constructions via cutting and glueing, fromP to its external map with a “hedgehog”, and backward. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis, sponsored by the Minerva Foundation (Germany). Supported by the Polish KBN Grants 210469101 “Iteracje i Fraktale” and 210909101 “Uklady Dynamiczne”.     

Halphen pencils on weighted Fano threefold hypersurfaces

On a general quasismooth well-formed weighted hypersurface of degree Σ _i=1⁴ a _i in ℙ(1, a ₁, a ₂, a ₃, a ₄), we classify all pencils whose general members are surfaces of Kodaira dimension zero.      

On the Cheney and Sharma Operator

In this paper, we consider the double-indexed Bernstein power series operatorP_{a, t}introduced by Cheney and Sharma and related to the classical Laguerre polynomials. We obtain sharp estimates of the first two moments which allow us to prove that, when acting on a continuous functionf, the convergence ofP_{a, t}ftofis uniform on the whole interval [0, 1]. Moreover, we show that the rates of convergence depend on the way in whicht/agoes to 0. We also show thatP_{a, t}preserves monotonicity and global smoothness. Finally, we consider the monotonicity properties ofP_{a, t}with respect to both parameters. To achieve the mentioned results, we use a probabilistic approach based on the representation of the operator in terms of a suitable multi-indexed stochastic process. The path properties and the martingale-type properties of this process are key points to give short proofs.     

Edge-disjoint Open Trails in Complete Bipartite Multigraphs

Let ^r K _a,b be a complete bipartite multigraph. We show a necessary and sufficient condition for a multigraph ^r K _a,b to be arbitrarily decomposable into open trails. We extend the results obtained by Balister for complete graphs (Balister in Probab Comput 10:463–499, 2001). Moreover we show that a multigraph ^r K _a,b with even r and a ≥ 3 or b ≥ 3 is arbitrarily decomposable into open and closed trails.     

Upper-Lower Class Tests for Weighted i.i.d. Sequences and Martingales

We study the upper-lower class behavior of weighted sums ∑ _k=1 ⁿ a _k X _k , where X _k are i.i.d. random variables with mean 0 and variance 1. In contrast to Feller’s classical results in the case of bounded X _j , we show that the refined LIL behavior of such sums depends not on the growth properties of (a _n ) but on its arithmetical distribution, permitting pathological behavior even for bounded (a _n ). We prove analogous results for weighted sums of stationary martingale difference sequences. These are new even in the unweighted case and complement the sharp results of Einmahl and Mason obtained in the bounded case. Finally, we prove a general upper-lower class test for unbounded martingales, improving several earlier results in the literature.     

Relative -difference sets in p-subgroups of

In this note, we study relative (p^a,p^b,p^a,p^a−b)-relative difference sets in certain p-subgroups of SL(n,K), K=F_q, where q is a prime power.     

A criterion for weak convergence on Berkovich projective space

We give a criterion for the weak convergence of unit Borel measures on the N-dimensional Berkovich projective space P^N_K{{\bf P}^{N}_K} over a complete non-archimedean field K. As an application, we give a sufficient condition for a certain type of equidistribution on P^N_K{{\bf P}^{N}_K} in terms of a weak Zariski-density property on the scheme-theoretic projective space \mathbb P^N_[(K)\tilde]{{\mathbb P}^N_{\tilde{K{}_{\vphantom{0}}}}} over the residue field [(K)\tilde]{\tilde{K}} . As a second application, in the case of residue characteristic zero we give an ergodic-theoretic equidistribution result for the powers of a point a in the N-dimensional unit torus \mathbb T^N_K{{\mathbb T}^N_K} over K. This is a non-archimedean analogue of a well-known result of Weyl over \mathbb C{\mathbb C} , and its proof makes essential use of a theorem of Mordell-Lang type for \mathbb G_m^N{{\mathbb G}_m^N} due to Laurent.     

Accelerating convergence of limit periodic continued fractionsK(a n /1)

Summary It is shown that the convergence of limit periodic continued fractionsK(a _n /1) with lima _n =a can be substantially accelerated by replacing the sequence of approximations {S _n (0)} by the sequence {S _n (x ₁)}, where . Specific estimates of the improvement are derived.     

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field \Bbb K \Bbb K of characteristic zero, we associate a set (\mathbbK) (\mathbb{K}) and a group G₀(\Bbb K) {\cal G}_0(\Bbb K) . Elements of (\mathbbK) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of G₀(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over \Bbb K \Bbb K . We construct a bijection between (\mathbbK)×G₀(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over \Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of (\mathbbK) (\mathbb{K}) , we associate a functor \frak a?\operatornameSh^v(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras; \operatornameSh^v(\frak a) \operatorname{Sh}^\varpi(\frak a) contains U\frak a U\frak a .? 2) When \frak a \frak a and \frak b \frak b are Lie algebras, and r_{\frak a\frak b} ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element ?^v (r_{\frak a\frak b}) {\cal R}^{\varpi} (r_{\frak a\frak b}) of \operatornameSh^v(\frak a)?\operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular, ?^v(r_{\frak a\frak b}) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from \operatornameSh^v(\frak a)^* \operatorname{Sh}^\varpi(\frak a)^* to \operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When \frak a = \frak b \frak a = \frak b and r_\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series r^v(r_\frak a) \rho^\varpi(r_\frak a) such that ?^v(r^v(r_\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of r^v(r_\frak a) \rho^\varpi(r_\frak a) in terms of r_\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a Lie bialgebra \frak g \frak g as the image of the morphism defined by ?^v(r^v(r)) {\cal R}^\varpi(\rho^\varpi(r)) , where r ? \mathfrakg ?\mathfrakg^* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P>     

On q-Parametric Szász-Mirakjan Operators

In the present paper, we introduce q-parametric Szász-Mirakjan operators. We study convergence properties of these operators S _n,q (f). We obtain inequalities for the weighted approximation error of q-Szász-Mirakjan operators. Such inequalities are valid for functions of polynomial growth and are expressed in terms of weighted moduli of continuity. We also discuss Voronovskaja-type formula for q-Szász-Mirakjan operators.     

The Picard iteration and its application

The convergence behavior of the Picard iteration X_k+1=AX_k+B and the weighted case Y_k=X_k/b_k is investigated. It is shown that the convergence of both these iterations is related to the so-called effective spectrum of A with respect to some matrix. As an application of our convergence results we discuss the convergence behavior of a sequence of scaled triangular matrices {D^NT^N }.