Linear functional differential equations of neutral type asymptotically autonomous

Summary It is studied the relationship between the solutions of the linear functional differential equations(1) (d/dx) D(x_t)=L(x_t) and its perturbed equation(2) [(d/dx) D(x_t)−G(t, x_t)]= =L(x_t)+F(t, x_t) and is proved, under certain hypotheses which will be precised bellow that, if μ is a simple characteristic root of(1), then there exist a σ > 0 and a non zero vector a such that system(2) has a solution satisfying where δ(t)=αd{F(t, ϕ_μ)+μG(t, ϕ_μ)+F(t, X₀G(t, ϕ_μ))}, ϕ_μ(θ)=c·exp (μθ), −r⩾θ⩾0 and α, d, X₀ are given constants. Entrata in Redazione il 5 gennaio 1972.     

Further results on simultaneous confidence intervals for the normal distribution

Summary Based on a random sample from the normal cumulative distribution function ϕ(x; μ, σ) with unknown parameters μ and σ, one-sided confidence contours for ϕ(x; μ, σ), −∞<x<∞, and simultaneous confidence intervals for ϕ(y; μ, σ)−ϕ(x; μ, σ), −∞<x<y<∞, are constructed using the method outlined in [3]. Small sample and asymptotic distributions of the relevant statistics are provided so that the construction could be completely carried out in any practical situation.     

On one bifurcation in relaxation systems

We establish conditions under which, in three-dimensional relaxation systems of the form {fx066-01}, where 0 < ε << 1, |μ| << 1, and ƒ, g ∈ C ^∞, the so-called “blue-sky catastrophe” is observed, i.e., there appears a stable relaxation cycle whose period and length tend to infinity as μ tends to a certain critical value μ_*(ε), μ_*(0) 0 = 0. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 63–72, January, 2008.     

A Maximal Function Approach to Christoffel Functions and Nevai��s Operators

Let μ be a compactly supported positive measure on the real line, with associated Christoffel functions λ _n (d μ,⋅). Let g be a measurable function that is bounded above and below on supp[μ] by positive constants. We show that λ _n (g  d μ,⋅)/λ _n (d μ,⋅)→g in measure in {x:μ′(x)>0} and consequently in all L _p norms, p<∞. The novelty is that there are no local or global restrictions on μ. The main idea is a new maximal function estimate for the “tail” in Nevai’s operators.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

Differential subordinations concerning starlike functions

Denote byS ^* (⌕), (0≤⌕<1), the family consisting of functionsf(z)=z+a ₂z²+...+a_nzⁿ+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎<1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)^μ−1<h(z) implies f∈S^*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.     

On the theorem of M Golomb

Let X ₁, …, X _n be compact spaces and X = X ₁ × … × X _n . Consider the approximation of a function ƒ ∈ C(X) by sums g ₁(x ₁)+…+g _n (x _n ), where g _i ∈ C(X _i ), i = 1, …, n. In [8], Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of X, called ‘projection cycles’. However, his proof had a gap, which was pointed out by Marshall and O’Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb’s formula holds in a stronger form.     

Dominated estimates of convex combinations of commuting isometries

The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofL _p(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofL _p (0, 1) induced by point transformations of the form τx=x ^k,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1. Research supported in part by NSF Grant No. GP-7475. A portion of the contents of this paper is based on the author’s doctoral dissertation written under the direction of Professor R. V. Chacon of the University of Minnesota.     

Instability of Set Recurrence and Green’s Function on Groups with the Liouville Property

Let μ and ν be probability measures on a group Γ and let G _μ and G _ν denote Green’s function with respect to μ and ν. The group Γ is said to admit instability of Green’s function if there are symmetric, finitely supported measures μ and ν and a sequence {x _n } such that G _μ (e, x _n )/G _ν (e,x _n ) →0, and Γ admits instability of recurrence if there is a set S that is recurrent with respect to ν but transient with respect to μ. We give a number of examples of groups that have the Liouville property but have both types of instabilities. Previously known groups with these instabilities did not have the Liouville property.     

Un teorema di Radon-Nikodym per funzioni d’insieme subadditive

Riassunto In questo articolo si danno delle condizioni necessarie e sufficienti affinchè per una fissata coppia di funzioni d’insieme ν, μ crescenti esista una funzionef tale che ν=∫fdμ. Si ottiene cosi una proposizione comprendente il teorema di R-N. classico e dei teoremi di R-N., presentati da altri autori, riguardanti le funzioni d’insieme finitamente additive e le funzioni d’insieme subadditive e continue per successioni crescenti.

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Résumé Soient ν, μ:A→[0,+∞) deux fonctions d’ensemble croissantes sur une σ-algèbre d’ensemblesA⊂T(X), telles que pour chaqueA∈A avec ν(A)=μ(A)=0 on a l’égalité μ(A)=μ(A∪S) ∀S∈A (c’est le cas des fonctions sousadditives!). Dans cet article on démontre qu’il existe une fonctionf A-measurable telle que ν=∫fdμ si et seulement si pour chaquer∈(0, + ∞) il y a un ensembleA _r∈A qui vérifie les trois conditions suivantes: (1) ,B∈A avecB⊂A; (2) A (3) limν(A _r)=0. On déduit ainsi une proposition qui a été donnée parI. Forana: ?Si ν, μ sont simplement additives, il existe une fonctionf telle que ν=∫fdμ si et seulement si ν≪μ et la fonction d’ensemble additive a une decomposition de Hahn pour chaquer∈(0, + ∞), c’est-á-dire il y aA _r∈A tel que ?.

     

Functional equations on abelian groups with involution

Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ _I ² =₁ g _l (x)h _l (y),x, y∈G, where the functionsf,g ₁,h ₁ to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar functional equations for a general involution σ.     

A Spectral Gap Property for Random Walks Under Unitary Representations

Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $\pi ,\mathcal{H}Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation p,H\pi ,\mathcal{H} of G, we study spectral properties of the operator π(μ) acting on H\mathcal{H} Assume that μ is adapted and that the trivial representation 1 _G is not weakly contained in the tensor product p?[`(p)]\pi\otimes \overline\pi We show that π(μ) has a spectral gap, that is, for the spectral radius r_spec(p(m))r_{\rm spec}(\pi(\mu)) of π(μ), we have r_spec(p(m)) < 1.r_{\rm spec}(\pi(\mu))< 1. This provides a common generalization of several previously known results. Another consequence is that, if G has Kazhdan’s Property (T), then r_spec(p(m)) < 1r_{\rm spec}(\pi(\mu))< 1 for every unitary representation π of G without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.     

On the applicability conditions of the strong law of large numbers for sequences of independent random variables

We investigate relationship between Kolmogorov–s condition and Petrov–s condition in theorems on the strong law of large numbers for a sequence of independent random variables X ₁, X ₂, … with finite variances. The convergence (S _n − ES _n )/n → 0 holds a.s. (here, S _n = Σ _k=1 ⁿ X _k ), provided that Σ _n=1^∞ DX _n /n ² < ∞ (Kolmogorov’s condition) or DS _n = O(n ²/ψ(n)) for some positive non-decreasing function ψ(n) such that Σ1/(nψ(n)) < ∞ (Petrov’s condition). Kolmogorov’s condition is shown to follow from Petrov’s condition. Besides, under some additional restrictions, Petrov’s condition, in turn, follows from Kolmogorov’s condition.     

Edgeworth expansions for compound Poisson processes and the bootstrap

One-term Edgeworth Expansions for the studentized version of compound Poisson processes are developed. For a suitably defined bootstrap in this context, the so called one-term Edgeworth correction by bootstrap is also established. The results are applicable for constructing second-order correct confidence intervals (which make correction for skewness) for the parameter “mean reward per unit time”. Research work of Gutti Jogesh Babu was supported in part by NSF grants DMS-9626189 and DMS-0101360.     

Bernstein�CWalsh Type Theorems for Real Analytic Functions in Several Variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝ ^N . In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure F=g[`(h)]F=g\overline{h}, where g and h are holomorphic in a neighborhood of a compact set K⊂ℂ <sup>N</sup> . To this end the maximal convergence number ρ(S <sub>c</sub> ,f) for continuous functions f defined on a compact set S <sub>c</sub> ⊂ℂ <sup>N</sup> is connected to a maximal convergence number ρ(S <sub>r</sub> ,F) for continuous functions F defined on a compact set S <sub>r</sub> ⊂ℝ <sup>N</sup> . We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions F=g[`(h)]F=g\overline{h} if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.     

Uniqueness of Embedding into a Gaussian Semigroup and a Poisson Semigroup with Determinate Jump Law on a Simply Connected Nilpotent Lie Group

Let {μ _t ⁽ⁱ⁾} _t≥0 (i=1,2) be continuous convolution semigroups (c.c.s.) on a simply connected nilpotent Lie group G. Suppose that μ ₁⁽¹⁾=μ ₁⁽²⁾. Assume furthermore that one of the following two conditions holds:

On inferring the probability of misclassification by the linear discriminant function

Summary Let the two alternative populationsP ₁ andP ₂ from which the individual with measurements χ may have come beN(μ(1), Σ) andN(μ(2), Σ). Then the classification rule with minimum risk is to assign the individual toP ₁ orP ₂ according as (μ(2)-μ(1))′Σ^-1 x≶(1/2)(μ(2)-μ(1))′Σ^-1(μ(1)+μ(2))+c wherec is a constant depending on the prior probabilities ofP ₁ andP ₂ and the costs of the two kinds of misclassification. The probability of misclassifying an individual fromP ₂ by this rule is π₂₁=Φ(-δ/2+cδ^-1), where Φ(.) is the distribution function of anN(0, 1) and . (Since we are free to choose which population we shall callP ₂, it is not necessary to consider separately the probability of misclassifying an individual fromP ₁.) LetP ₂₁ denote the probability of misclassification of an individual fromP ₂ by the rule derived from the one mentioned by fixing μ(1), μ(2) and Σ at estimates andV and letP ₂₁ ^* be the probability of misclassification of an individual fromP ₂ when the classification rule is the one with minimum risk among those based on . The fiducial distributions of π₂₁,P ₂₁ andP ₂₁ ^* are determined. Point estimates and confidence intervals for π₂₁,P ₂₁ andP ₂₁ ^* are derived. Only easily available tables are needed to make fiducial inferences. An incidental result of some interest elsewhere as well is the distribution of a linear combination of a chi and an independent normal variable.     

Upper bounds for growth in the Ryll-Nardzewski function of anω-categorical,ω-stable theory

We establish a sharp upper bound for growth in the sequences _k (T):=the number ofk-types consistent withT, forT ω-categorical andω-stable. This paper forms part of the author’s doctoral dissertation written at the University of Maryland under the direction of David W. Kueker.     

Integral inequalities with weights for maxiamal operators

Let μ be a measure on the upper half-space R ₊ ⁿ⁺¹ , and v a weight onR ⁿ, we give a characterization for the pair (v, μ) such that ∥M(fv)∥_{L Θ (μ}) ⩽ c ∥f∥_{L Θ (μ}), where Φ is an N-function satisfying Δ₂ condition andMf(x,t), is the maximal function onR ₊ ⁿ⁺¹ , which was introduced by Ruiz, F. and Torrea, J.. Supported by NSFC.     

Convex Solid Subsets of <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">X</Emphasis>, μ)

If A⊂ L₀(X, μ) is a convex solid subset of L₀(X, μ), then there exist disjoint X₀ and X₁ with X = X₀∪ X₁ such that A_{| X_0} is dense in L₀(X₀, μ) and A_{|X_1} is bounded in measure in L₀(X₁, μ).