An upper bound for the cardinality of ans-distance subset in real euclidean space,II

It is shown that ifX is ans-distance subset inR ^d , then |X|≦( _s ^d+s ). Supported in part by NSF grant MCS7903128 A01. Supported in part by NSF grant MCS.     

ADDITIVE PROPERTIES OF SETS OF REAL NUMBERS AND AN INFINITE GAME

Abstract

1. If X has strong measure zero aid if Y is contained in an F _σ, set of measure zero, then X + Y has measure zero (Proposition 9).

2. If X is a measure zero set with property s ₀ and Y is a Sierpinski set, then X + Y has property s ₀ (Theorem 12).

3. If X is a meager set with property s ₀ and Y is a Lusin set, then X + Y has property s ₀ (Theorem 17).

An infinite game is introduced, motivated by additive properties of certain classes of sets of real numbers.     

The singular locus of the secant varieties of a smooth projective curve

Let X be a smooth irreducible non-degenerated projective curve in some projective space P^N. Let r be a positive integer such that 2r + 1 < N and let S_r(X) be the r-th secant variety of X. It is a variety of dimension 2r + 1. In this paper we prove that the singular locus is the (r - 1)-th secant variety S_{r- 1}(X) if X does not have any (2r + 2)-secant 2r-space divisor. Received: 26 November 2002     

Turbulence Phenomena in Real Analysis

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and stands for the one-point compactification of X, while E^X is the equivalence relation which is defined on the Polish group C(X,R₊*) by where f, g are in C(X,R₊*), then E^X is induced by a turbulent Polish group action. Second we show that given any if we identify the n-dimensional unit sphere Sⁿ with the one-point compactification of Rⁿ via the stereographic projection, while E_n,r is the equivalence relation which is defined on the Polish group C^r(Rⁿ,R₊*) by where f, g are in C^r(Rⁿ,R₊*), then E_n,r is also induced by a turbulent Polish group action. Dedicated to my sister Alexandra and to her daughter Marianthi.     

An Upper Bound for the Cardinality of an <Emphasis Type="Italic">s</Emphasis>-Distance Set in Euclidean Space

In this paper we show that if X is an s-distance set in ^m and X is on p concentric spheres then Moreover if X is antipodal, then .     

Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities

Consider a multidimensional stochastic differential equation of the form X_t=x+ò₀^tb(X_s-) ds+ò₀^tf(X_s-) dZ_sX_{t}=x+\int_{0}^{t}b(X_{s-})\,ds+\int_{0}^{t}f(X_{s-})\,dZ_{s}, where (Z _s ) _s≥0 is a symmetric stable process. Under suitable assumptions on the coefficients, the unique strong solution of the above equation admits a density with respect to Lebesgue measure, and so does its Euler scheme. Using a parametrix approach, we derive an error expansion with respect to the time step for the difference of these densities.     

Tower techniques for cosimplicial resolutions

Let be a simplicial model category and J : a simplicial coaugmented functor. Given an object X, the assignment nJⁿ⁺¹X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J^sX = tot^s([n] Jⁿ⁺¹X). An object X is called J-injective if it is a retract of JX in Ho() via the natural map. We show that certain homotopy limits of J-injective objects are J_s-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}_s0{tot^SX}_{s 0}.The first author was supported in part by National Science Foundation grant DMS-0296117     

Some Maximal Function Fields and Additive Polynomials

We derive explicit equations for the maximal function fields F over 𝔽 _{q ²ⁿ} given by F = 𝔽 _{q ²ⁿ} (X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field 𝔽 _{q ²ⁿ} , and where A(Y) is q-additive and deg(f) = q ⁿ  + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over 𝔽 _{q ²ⁿ} (i.e., the extension H/F is Galois).     

On a problem of spencer

LetX ₁, ...,X _n be events in a probability space. Let ϱ_i be the probabilityX _i occurs. Let ϱ be the probability that none of theX _i occur. LetG be a graph on [n] so that for 1 ≦i≦n X _i is independent of ≈X _j ‖(i, j)∉G≈. Letf(d) be the sup of thosex such that if ϱ₁, ..., ϱ _n ≦x andG has maximum degree ≦d then ϱ>0. We showf(1)=1/2,f(d)=(d−1) ^d−1 d ^−d ford≧2. Hence df(d)=1/e. This answers a question posed by Spencer in [2]. We also find a sharp bound for ϱ in terms of the ϱ _i andG.     

Ann-dimensional search problem with restricted questions

The problem is the following: How many questions are necessary in the worst case to determine whether a pointX in then-dimensional Euclidean spaceR ⁿ belongs to then-dimensional unit cubeQ ⁿ, where we are allowed to ask which halfspaces of (n−1)-dimensional hyperplanes contain the pointX? It is known that ⌌3n/2⌍ questions are sufficient. We prove here thatcn questions are necessary, wherec≈1.2938 is the solution of the equationx log₂ x−(x−1) log₂ (x−1)=1.     

The multiplier ideal is a universal test ideal

Abstract

For a canonical threefold X, we know that h ⁰(X, 𝒪 _X (nK _X )) ≥ 1 for a sufficiently large n. When χ(𝒪 _X ) > 0, it is not easy to get such an integer n. Fletcher showed that h ⁰(X, 𝒪 _X (12K _X )) ≥ 1 and h ⁰(X, 𝒪 _X (24K _X )) ≥ 2 when χ(𝒪 _X ) = 1. He inquired about existence of a canonical threefold with given conditions which shows the result sharp. We show that such an example does not exist. Using a different technique, we prove h ⁰(X, 𝒪 _X (12K _X )) ≥ 2.     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Exponential trichotomy and p-admissibility for evolution families on the real line

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1,∞) and the pair (C_b(R,X),C_c(R,X)) is uniformly p-admissible for an evolution family ={U(t,s)}_t≥_s then is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C_b(R, X), C_c(R, X)) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C_b(R,X),L^p(R,X)) for an evolution family ={U(t,s)}_t≥_s is equivalent with its uniform exponential dichotomy.     

Versality of linearly normal non-special embeddings

LetX⊂P ⁿ be a smooth non-degenerate non special linearly normal projective curve. Here we classify all such embeddings ofX such that for every hyperplaneM ofP ⁿ the family of all hyperplane sections ofX is a versal deformation of the zerodimensional schemeX∩M.

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Sunto SiaX una curva liscia e proiettiva. Qui si classificano le immersioni non-speciali linearmente normali diX inP ⁿ tali che per ogni iperpianoM diP ⁿ la famiglia delle sezioni iperpiane diX induce una deformazione versale diX∩M.

     

Functions of elementary random variables and their application to system reliability

Summary LetX ₁,...,X _n be elementary random variables, i.e. random variables taking only finitely many values in . Then for an arbitray functionf(X ₁,...,X _n ) inX ₁,...,X _n a unique polynomial representation with the aid of Lagrange polynomials is given. This easily yields the moments as well as the distribution off(X ₁,...,X _n ) by terms of finitely many moments of the variablesX ₁,...,X _n . For n=1 a necessary and sufficient condition results thatm numbers are the firstm moments of a random variable takingm+1 different values. As an application of random functionsf(X ₁,...,X _n ) the reliability of technical systems with three states is treated.

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Zusammenfassung X ₁, ...,X _n seien elementare Zufallsvariable, d. h., Zufallsvariable, die nur endlich viele reelle Werte annehmen. Mit Hilfe von Lagrangepolynomen wird für eine beliebige Funktionf(X₁,...,X _n ) eine eindeutige polynomiale Darstellung angegeben. Daraus ergeben sich leicht die Momente wie auch die Verteilung von f(X₁,...,X _n ), ausgedrückt durch die Momente der VariablenX ₁,...,X _n . Fürn=1 erhält man eine notwendige und hinreichende Bedingung, daßm Zahlen die erstenm Momente einer Zufallsvariablen sind, diem+1 verschiedene Werte annimmt. Als Anwendung wird die Zuverlässigkeit eines technischen Systems mit drei Zuständen behandelt.

     

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.     

SOME LIMIT PROPERTIES OF LOCAL TIME FOR RANDOM WALK

Let X,X1,X2 be i. i. d. random variables with EX^2＋δ〈∞ （for some δ〉0）. Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ＊ （n）=supx∈zζ（x,n）,ζ（x,n） =#{0≤k≤n：[Sk]=x}. A strong approximation of ζ（n） by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ＊（n） are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ＊（n） is proved.     

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT

In this paper we calculate presentations for some natural monoids of transformations on a chain X _n  = {1 < 2 <?s < n}. First we consider 𝒪𝒟 _n [𝒫𝒪𝒟 _n ], the monoid of all full [partial] transformations on X _n that preserve or reverse the order. Two other monoids of partial transformations on X _n we look at are 𝒫𝒪𝒫 _n and 𝒫𝒪? _n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.     

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let （X, Xn; n ≥1） be a sequence of i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with covariance operator ∑. Set Sn = X1 ＋ X2 ＋ ... ＋ Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2（b＋1） ∞ ∑n=1 （logn）^b/n^3/2 E{｜｜Sn｜｜-σε√nlogn}=σ^-2（b＋1）/（2b＋3）（b＋1） B｜｜Y｜^2b＋3
holds if EX=0,and E｜｜X｜｜^2（log｜｜x｜｜）^3bv（b＋4）〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.     

Multipliers of classes of cauchy transforms

The analytic map g on the unit disk D is said to induce a multiplication operator L from the Banach space X to the Banach space Y if L(f)=f·g∈Y for all f∈X. For z ∈ D and α>0 the families of weighted Cauchy transforms F_α are defined by ?(z) = ∫_T K_x ^α (z)dμ(x) where μ(x) is complex Borel measures, x belongs to the unit circle T and the kernel K_x (z) = (1- xz)^?1. In this article we will explore the relationship between the compactness of the multiplication operator L acting on F ₁ and the complex Borel measures μ(x). We also give an estimate for the essential norm of L