Measures of affinity of a sequence for a number

We define strong and weak affinities of a number a for a sequence (x_k ) denoted by L (a,(x_k )) and U (a, (x_k )) respectively. We show U (a,(x_k )) > 0 if and only if the number a is a statistical limit point of the sequence (x_k ). We consider the distribution of sequences with positive weak and strong measures of affinity within the space l ^∞ of bounded sequences. The main result is that the set of bounded sequences with U (a,(x_k )) > 0, that is, the set of sequences with statistical limit points, is a dense subset in l ^∞ of the first category. We also show the set of sequences with positive strong affinities is a nowhere dense subset of l ^∞.     

Central limit theorem for perturbed empirical distribution functions evaluated at a random point

Let be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n from a (smooth) distribution function F. Sufficient conditions are given for the central limit theorem to hold for the target statistic where {U_n} is a sequence of U-statistics.     

Convergence of self-normalized generalizedU-statistics

Conditions are given for almost certain and distribution convergence of self-normalized generalizedU-statistics composed of random variables without particular probabilistic structure. The set of almost certain limit points of some classicalU-statistics is obtained. A variant of theU-statistic involving squares of some of the random variables is also treated. Applications include Martingale differences, stationary sequences, and the classical i.i.d. case where a Marcinkiewicz-Zygmund-type strong law is obtained.     

Asymptotic Behavior of the Perturbed Empirical Distribution-Functions Evaluated at a Random Point for Absolutely Regular Sequences

Let _n be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n from smooth distribution function F. A central limit theorem is established for the target statistic _n(U_n) where the underlying random variable form an absolutely regular stationary process and where {U_n} is a sequence of U-statistics. The result obtained generalizes Puri and Ralescu (1986, J. Multivariate Anal.19, 273-279) under the iid set-up.     

Rates of Convergence and Law of the Iterated Logarithm for U-Processes

We study in this paper some limit theorems for U-processes. We calculate rates of convergence in the central limit theorem of nondegenerate U-processes under metric entropy with bracketing condition. In application, we improve upon the law of the iterated logarithm of Arcones. All calculations use the Ossiander chaining procedure.     

On approximation of affine Baire-one functions

It is known (G. Choquet, G. Mokobodzki) that a Baire-one affine function on a compact convex set satisfies the barycentric formula and can be expressed as a pointwise limit of a sequence of continuous affine functions. Moreover, the space of Baire-one affine functions is uniformly closed. The aim of this paper is to discuss to what extent analogous properties are true in the context of general function spaces. In particular, we investigate the function spaceH(U), consisting of the functions continuous on the closure of a bounded open setU⊂ℝ ^m and harmonic onU, which has been extensively studied in potential theory. We demonstrate that the barycentric formula does not hold for the spaceB ₁ ^b (H(U)) of bounded functions which are pointwise limits of functions from the spaceH(U) and thatB ₁ ^b (H(U)) is not uniformly closed. On the other hand, every Baire-oneH(U)-affine function (in particular a solution of the generalized Dirichlet problem for continuous boundary data) is a pointwise limit of a bounded sequence of functions belonging toH(U). It turns out that such a situation always occurs for simplicial spaces whereas it is not the case for general function spaces. The paper provides several characterizations of those Baire-one functions which can be approximated pointwise by bounded sequences of elements of a given function space. Research supported in part by grants GA ČR No. 201/00/0767 from the Grant Agency of the Czech Republic, GA UK 165/99 from the Grant Agency of Charles University, and in part by grant number MSM 113200007 from the Czech Ministry of Education.     

Nonuniqueness of solutions of Riemann problems

We investigate a general mechanism, utilizing nonclassical shock waves, for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws. This nonuniqueness occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., two statesU ₁ andU ₂ such that a traveling wave leads fromU ₁ toU ₂ and another leads fromU ₂ toU ₁. We prove that a 2-cycle gives rise to an open region of Riemann data for which there exist multiple solutions of the Riemann problem, and we determine all solutions within a certain class. We also present results from numerical experiments that illustrate how these solutions arise in the time-asymptotic limit of solutions of the conservation laws, as augmented by viscosity terms.     

A class of asymptotically normal degenerate quasi <Emphasis Type="Italic">U</Emphasis>-statistics

Some quasi U-statistics, unlike other variants of U-statistics, arising in distance based tests for homogeneity of groups, have first-order stationary kernels of degree 2, and yet they enjoy asymptotic normality under suitable hypotheses of invariance. Central limit theorems for a more general class of quasi U-statistics with possibly higher order stationarity (and degree) are formulated with the aid of appropriate martingale (array) characterizations as well as permutational invariance structures.     

Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

Let U_m be an m×m Haar unitary matrix and U_[m,n] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U_[m,n] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U_[m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.     

The Bar-Radical of a Class of Almost Alternative Baric Algebras

In this article we prove that, if (U, ω) is a finite dimensional baric algebra of (γ, δ) type over a field F of characteristic ≠ 2,3,5 such that γ² ? δ² + δ = 1 and δ ≠ 0,1, then rad(U) = R(U) ∩ (bar(U))², where R(U) is the nilradical (maximal nil ideal) of U.     

Holomorphic vector bundles on a neighborhood of non-Stein compact subsets of Stein spaces

LetK be a compact subset of a complex spaceX. Here we give conditions onX andK assuring the existence of a fundamental systemU of open neighborhoods, ofK such that for everyU∈U there is a holomorphic vector bundleE onU which is not holomorphically trivial.

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Sunto SiaX uno spazio complesso eK∩X un compatto. In questo lavoro diamo condizioni suX eK che garantiscono l'esistenza di un sistema fondamentale di intorni apertiU diK inX tali che per ogniU∈U esiste un fibrato olomorfo non-triviale suU.

     

Some observations on the substructure lattice of a Δ1 ultrapower

Given a (nontrivial) Δ₁ ultrapower ?/??, let ??_U denote the set of all Π₂‐correct substructures of ?/??; i.e., ??_U is the collection of all those subsets of |?/??| that are closed under computable (in the sense of ?/??) functions. Defining in the obvious way the lattice ??(?/??)) with domain ??_U, we obtain some preliminary results about lattice embeddings into – or realization as – an ??(?/??). The basis for these results, as far as we take the matter, consists of (1) the well‐known class of (non‐trivial) minimal ?/??'s, which function as atoms, and (2) the class of minimalfree ?/??'s, to whose nonemptiness a substantial section of the paper is devoted. It is shown that an infinite, convergent monotone sequence together with its limit point is embeddable in an ??(?/??), and that the initial segment lattices {0, 1,…, n } are not just embeddable in (as is trivial), but in fact realizable as, lattices ??(?/??). Finally, the diamond is (easily) embeddable; and if it is not realizable, then either the 1 ‐ 3 ‐ 1 lattice or the pentagon is at least embeddable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Random sparse unary predicates

Random unary predicates U on [n] holding with probability p are examined with respect to sentences A in a first-order language containing U and “less than.” When p = p(n) satisfies n^k+1 ? 1 ? np^k it is shown that Pr[A] approaches a limit dependent only on k and A. In a similar circular model the limit is shown to be zero or one. © 1994 John Wiley & Sons, Inc.     

A Functional Law of the Iterated Logarithm for U-Statistic Type Processes

We prove a functional law of the iterated logarithm for U-statistics type processes. The result is used to determine the almost sure set of limit points for change-point estimators.     

Extension of holomorphic vector bundles across a boundary point

Let U ? C ⁿ , n ≥ 3, be a domain and P ∈ ?U such that U is 2-concave at P. Here we prove the existence of a holomorphic vector bundle on U which does not extend across P, but it extends across every Q ∈ ?U with Q ≠ P. We also prove a similar result taking a Stein space X instead of C ⁿ .     

The law of the iterated logarithm and Marcinkiewicz law of large numbers forB-valuedU-statistics

Suppose thatB is a separable Banach space and (S,l,P) a probability space.H is a measurable symmetric kernel function fromS ^m intoB. In this paper we shall further study some limit theorems forB-valuedU-statisticsU _m ⁿ H based onP andH. Special attention is paid upon the Marcinkiewicz type law of large numbers and the law of the iterated logarithm. Our results can be regarded as extensions of corresponding results for sums of independentB-valued random variables toU-statistics.Research supported by National Natural Science Foundation of China and Zhejiang Province.     

Low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces

We consider the Dirichlet problem for the reduced wave equation ΔU_x + x²U_x = 0 in a two-dimensional exterior domain with boundary C, where C consists of a finite number of smooth closed curves C₁,…,C_m. The question of interest is the behavior of U_x as ? → 0. We show that U converges to the solution of the corresponding exterior Dirichlet problem of potential theory if the boundary data converge to a limit uniformly on C. This generalizes a well-known result of R. C. MacCamy for the case m = 1.     

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form w(z) = exp(−|z|² + U^μ(z)), where U^μ(z) is the logarithmic potential of a compactly supported positive measure μ. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials U^μ(z). It is also shown that the 2 × 2 matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights w(z) supported on a compact domain. Submitted: July 25, 2008. Accepted: October 1, 2008. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche sur la nature et les technologies du Québec (FQRNT).     

Rings of quotients of f-rings by gabriel filters of ideals

The article examines the role of Gabriel filters of ideals in the ontext of semiprime f-rings. It is shown that for every 2-convex semiprime f-ring Aand every multiplicative filter B of dense ideals the ring of quotients of A by B, namely the direct limit of the Hom _A (I, A) over all I∈ B, is an l-subring of QA, the maximum ring of quotients. Relative to the category of all commutative rings with identity, it is shown that for every 2-convex semiprime f-ring A qA, the classical ring of quotients, is the largest flat epimorphic extension of A. If Ais also a Prüfer ring then it follows that every extension of Ain qA is of the form S ^-1A for a suitable multiplicative subset S. The paper also examines when a Utumi ring of quotients of a semiprime f-ring is obtained from a Gabriel filter. For a ring of continuous functions C(X), with Xcompact, this is so for each C(U) and C ^*(U), when Uis dense open, but not for an arbitrary direct limit of C(U),taken over a filter base of dense open sets. In conclusion, it is shown that, for a complemented semiprime f-ring A, the ideals of Awhich are torsion radicals with respect to some hereditary torsion theory are precisely the intersections of minimal prime ideals of A.