Explicit Convergence Rates of the Embedded M/G/1 Queue

This paper investigates the explicit convergence rates to the stationary distribution π of the embedded M/G/1 queue; specifically, for suitable rate functions r（n） which may be polynomial with r（n） = n^l, l 〉 0 or geometric with r（n） = α^n, a 〉 1 and ＂moments＂ f ≥ 1, we find the conditions under which Σ∞n=0 r（n）｜｜P^n（i,·） - π（·）｜｜f ≤ M（i） for all i ∈ E. For the polynomial case, the explicit bounds on M（i） are given in terms of both ＂drift functions＂ and behavior of the first hitting time on the state O; and for the geometric case, the largest geometric convergence rate α＊ is obtained.     

Immersions of spheres and algebraically constructible functions

Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g _λ  =  g(· , λ) restricted to the sphere S ⁿ (r) is an immersion for every 0  <  r  <  r (λ), so that the intersection number I(g _λ|S ⁿ (r)) is defined. Then is an algebraically constructible function. I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7.     

Rates of convergence for the stability of large order statistics

Forr1 and eachnr, letM _nr be therth largest ofX ₁,X ₂, ...,X _n , where {X _n ,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of for all >0 and some –1, where {a _n } is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0.     

Normal growth of large groups

For a finitely generated group $\Gamma$, denote by the number of normal subgroups of index n. A. Lubotzky proved that for the free group F_r of rank r, is of type n^logn. We show that the same is true for a much larger class of groups. On the other hand we show that for almost all n, the inequality < holds true for every r-generated group $\Gamma$.Received: 30 October 2002     

A characterization of the periods of periodic points of 1-norm nonexpansive maps

In this paper the set of minimal periods of periodic points of 1-norm nonexpansive maps is studied. This set is denoted by R(n). The main goal is to present a characterization of R(n) by arithmetical and combinatorial constraints. More precisely, it is shown that , where denotes the set of periods of restricted admissible arrays on 2n symbols. The important point of this equality is that is determined by arithmetical and combinatorial constraints only, and that it can be computed in finite time. By using this equality the set R(n) is computed for . Furthermore it is shown that the largest element of R(n) satisfies:      

Problems similar to the additive divisor problem

For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants ,α>0 such that and Σ _p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:

Using Lagrangians of Hypergraphs to Find Non-Jumping Numbers (I)

Let r ≥ 2 be an integer. A real number is a jump for r if for any and any integer m, m ≥ r, any r-uniform graph with vertices and density at least contains a subgraph with m vertices and density at least α + c, where c = c(α) does not depend on or m. It follows from a result of Erdős, Stone, and Simonovits that every is a jump for r = 2. Erdőos asked whether the same is true for r ≥ 3. Frankl and R?dl gave a negative answer by showing an infinite sequence of non-jumping numbers for r ≥ 3. However, there are a lot of unknowns on determining whether a number is a jump for r ≥ 3. In this paper, we first find two infinite sequences of non-jumping numbers for r = 4, then we extend one of the results to every r ≥ 4. Our approach is still based on the approach developed by Frankl and R?dl. Received November 30, 2005     

Freeknot splines approximation of Sobolev-type classes of <Emphasis Type="Italic">s</Emphasis>-monotone functions

Let I be a finite interval, s ∈ ℕ₀, and r,ν,n ∈ ℕ. Given a set M, of functions defined on I, denote by M the subset of all functions y ∈ M such that the s-difference is nonnegative on I, ∀τ > 0. Further, denote by the Sobolev class of functions x on I with the seminorm . Also denote by Σ _ν,n , the manifold of all piecewise polynomials of order ν and with n – 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation and of the best s-monotonicity preserving approximation . Part of this work was done while the first author visited Tel Aviv University in May 2003 and in March 2004.     

Mixed Partitions of Projective Geometries

Starting from a linear collineation of PG(2n–1,q) suitably constructed from a Singer cycle of GL(n,q), we prove the existence of a partition of PG(2n–1,q) consisting of two (n–1)-subspaces and caps, all having size (qⁿ–1)/(q–1) or (qⁿ–1)/(q+1) according as n is odd or even respectively. Similar partitions of quadrics or hermitian varieties into two maximal totally isotropic subspaces and caps of equal size are also obtained. We finally consider the possibility of partitioning the Segre variety of PG(8,q) into caps of size q²+q+1 which are Veronese surfaces.     

Ample vector bundles with zero loci having a bielliptic curve section of low degree

Let be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of whose zero locus Z is a smooth subvariety of dimension n − r ≥ 3 of X. Let H be an ample line bundle on X such that its restriction H _Z to Z is very ample. Triplets are classified under the assumption that (Z,H _Z ) has a smooth bielliptic curve section of genus ≥ 3 with .      

<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

Far field velocity potential induced by a rapidly decaying vorticity distribution

We study the velocity field induced by a vorticity distribution decaying rapidly in the distancer from the origin. In the far field, the vector potential for the velocity field can be represented by a series A ⁽ⁿ⁾, withA ⁽ⁿ⁾ proportional tor ^–n–1, forn=1, 2, .... We show thatA ⁽ⁿ⁾ can be expressed as a linear combination ofM _n linearly independent vector functions. The numberM _n is equal to 3 forn=1 and 4n forn2 and the coefficient of a vector function is defined by a linear combination of nth moments of vorticity. We then show that only 2n+1 linear combinations of thoseM _n vector functions contribute to the far field velocity which is irrotational. The corresponding scalar potential ⁽ⁿ⁾ is then represented by a linear combination of 2n+1 spherical harmonics ofnth order whose coefficients are again linear combinations ofnth moments of vorticity.

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Zusammenfassung Die vorliegende Arbeit beschreibt das Geschwindigkeitsfeld fernab einer Wirbelverteilung, welche mit dem Abstandr vom Ursprung eines geeigneten Bezugssystems hinreichend schnell abklingt. Die Geschwindigkeit besitzt ein Vektorpotential, dessen Fernfeldverhalten einer Reihenentwicklung A ⁽ⁿ⁾, genügt. Dabei istA ⁽ⁿ⁾ proportional zur ^–n–1 fürn=1, 2, .... Wir entwickeln eine explizite Darstellung vonA ⁽ⁿ⁾ als Linearkombination vonM _n linear unabhängigen Vektorfunktionen. Die auftretenden Koeffizienten sind ihrerseits Kombinationenn-ter Momente der Wirbelverteilung. Die ZahlM ₁ ist gleich 3 und es istM _n=4n fürn2, während die Gesamtzahl dernten Momente beträgt. Weiterhin zeigen wir, da\ nur 2n+1 dieser Vektorfunktionen auch zum drehungsfreien Fernfeld der Geschwindigkeitn-ter Ordnung beitragen können und identifizieren die zugehörigen Kombinationen von Wirbelmomenten. Dieselben Kombinationen liefern dann auch die Koeffizienten in einer Entwicklung desskalaren Fernfeldpotentials nach Kugelfunktionen.

     

Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Let R be a commutative Noetherian ring, be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that for all i with i < n, then is finite for all i with i < n, and is finite for all i with i ≤ n, where for a subset T of Spec(R), we set . Also, among other things, we show that if n ≥ 0, R is semi-local and is finite for all i with i < n, then is finite for all i with i ≤ n. K. Khashyarmanesh was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 86130027).     

On Powers of Stieltjes Moment Sequences,I

For a Bernstein function f the sequence s_n=f(1)·...· f(n) is a Stieltjes moment sequence with the property that all powers s_n^c,c>0 are again Stieltjes moment sequences. We prove that is Stieltjes determinate for c≤ 2, but it can be indeterminate for c>2 as is shown by the moment sequence , corresponding to the Bernstein function f(s)=s. Nevertheless there always exists a unique product convolution semigroup such that ρ_c has moments . We apply the indeterminacy of for c>2 to prove that the distribution of the product of p independent identically distributed normal random variables is indeterminate if and only if p≥ 3     

Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For any finite family of simply connected orthogonal polygons in the plane and points x and y in , if every r (not necessarily distinct) members of contain a common staircase n-path from x to y, then contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths. Moreover, we establish the following dual result for unions of these sets: Let be any finite family of orthogonal polygons in the plane, with simply connected. If every three (not necessarily distinct) members of have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2.     

An optimal theorem for the spherical maximal operator on the Heisenberg group

Let be the Heisenberg group and μ _r be the normalized surface measure on the sphere of radiusr in ℂ ⁿ . Let . We prove an optimalL ^p-boundedness result for the spherical maximal functionMf, namely we prove thatM is bounded onL ^p(I ⁿ ) if and only ifp>2n/2n−1.     

The dimension of interior levels of the Boolean lattice

LetP(k,r;n) denote the containment order generated by thek-element andr-element subsets of ann-element set, and letd(k,r;n) be its dimension. Previous research in this area has focused on the casek=1.P(1,n–1;n) is the standard example of ann-dimensional poset, and Dushnik determined the value ofd(1,r;n) exactly, whenr2 . Spencer used the Erdös-Szekeres theorem to show thatd(1, 2;n) lg lgn, and he used the concept of scrambling families of sets to show thatd(1,r;n)=(lg lgn) for fixedr. Füredi, Hajnal, Rödl and Trotter proved thatd(1, 2;n)=lg lgn+(1/2+o(1))lg lg lgn. In this paper, we concentrate on the casek2. We show thatP(2,n–2;n) is (n–1)-irreducible, and we investigated(2,r;n) whenr2 , obtaining the exact value for almost allr.The research was supported in part by NSF grant DMS 9201467.The research was supported in part by the Universities in Russia program.     

An extension of almost sure central limit theorem for order statistics

Let {X _n ,n ≥ 1} be a sequence of i.i.d. random variables. Let M _n and m _n denote the first and the second largest maxima. Assume that there are normalizing sequences a _n  > 0, b _n and a nondegenerate limit distribution G, such that . Assume also that {d _k ,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have

A bound on the plurigenera of projective varieties

We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space P ^r , and classify the varieties attaining the bound, when n2, r2n+1, d>>r and i>>r. When n=2 and r=5, or n=3 and r=7, we give a complete classification, i.e. for any i1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n+1 and minimal degree in P ^r , with r>(n ² +3n)/(n–1), has not maximal i-th plurigenus, for i>>r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef. Mathematics Subject Classification (2000):Primary 14J99; Secondary 14N99