Simultaneous Approximation in Positive Characteristic

 Let be the approximation exponent of a power series α (so that when α is algebraic of degree d, then by Dirichlet’s and Liouville’s Theorems). If the characteristic is positive, q is a power of the characteristic, and are related by a fractional linear transformation with polynomial coefficients, then by respective work of Voloch and of de Mathan, there are constants such that has no solution if , and infinitely many solutions if . We will formulate and prove generalizations to simultaneous approximation. (Received 7 December 1999; in revised form 31 March 2000)     

Student’s t-test for Gaussian scale mixtures

A Student-type test is constructed under a condition weaker than normal. We assume that the errors are scale mixtures of normal random variables and compute the critical values of the suggested s-test. Our s-test is optimal in the sense that if the level is at most α, then the s-test provides the minimum critical values. (The most important critical values are tabulated at the end of the paper.) For α ≤.05, the two-sided s-test is identical with Student’s classical t-test. In general, the s-test is a t-type test, but its degree of freedom should be reduced depending on α. The s-test is applicable for many heavy-tailed errors, including symmetric stable, Laplace, logistic, or exponential power. Our results explain when and why the P-value corresponding to the t-statistic is robust if the underlying distribution is a scale mixture of normal distributions. Bibliography: 24 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 5–19.     

On Shannon’s entropy power inequality

Summary We prove that the entropy power inequality follows from Blachman’s argument [1] if the densities have finite moments of order α, for some α>0, whenever Shannon’s variational approach can be applied if α>=2.

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Sommario Si dimostra che la disuguaglianza esponenziale dell’entropia può essere dedotta da alcuni precedenti risultati di Blachman [1] se le densità di probabilità hanno momenti finiti di ordine α, per qualche α>0. Si dimostra inoltre che l’argomento variazionale di Shannon può essere applicato se α≥2.

     

On the α-amenability of hypergroups

Let UC(K) denote the Banach space of all bounded uniformly continuous functions on a hypergroup K. The main results of this article concern the α-amenability of UC(K) and quotient and product of hypergroups. It is also shown that a Sturm-Liouville hypergroup with a positive index is α-amenable if and only if α = 1. Author’s address: Dietlinden Stra?e 16, 80802 München, Germany     

Major subsets ofα-recursively enumerable sets

Let s2cf(α), s2p(α) and ts2p(α) denote the Σ₂-confinality, Σ₂-projectum and the tame Σ₂-projectum of an admissible ordinalα. We show that if s2cf(α)<s2p(α), then noα-recursively enumerable set (α-r.e.) with complement of order type less than ts2p(α) can have a major subset. As a corollary, if s2cf(α)<s2p(α), then no hyperhypersimpleα-r.e. set can have a major subset.     

Maximal sets in α-recursion theory

Let α be an admissible ordinal, and leta ^* be the Σ₁-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α^* if α^*−M is undounded and for any α-r.e. setA, eitherA∩(α^*-M) or (⇌^*-A)∩(α^*-M) is unbounded in α^*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α^*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ ^L then it is shown that maximal sets do not exist. Research partially supported by NSF Grant GP-34088 X. Some of the results in this paper have been taken from the second author’s Ph. D. Thesis, written under the supervision of Gerald Sacks.     

On the Independence of Hartman Sequences

 Given and , we define by setting if and only if , where denotes the fractional part of α, i.e. α is considered as an element of the torus . If the topological boundary of A has Haar measure 0, then is called a Hartman sequence, which is a generalisation of Kronecker and Beatty sequences. In this article we answer a question of Winkler by showing explicitly for which sets , and vectors , we have . The main tool of the proof is Weyl’s theorem on uniform distribution.     

Nesterenko’s criterion when the small linear forms oscillate

In this paper we generalize Nesterenko’s criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to ζ(5), ζ(7), ζ(9), and ζ(11), using Zudilin’s proof that at least one of these numbers is irrational.     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .     

On the Independence of Hartman Sequences

 Given and , we define by setting if and only if , where denotes the fractional part of α, i.e. α is considered as an element of the torus . If the topological boundary of A has Haar measure 0, then is called a Hartman sequence, which is a generalisation of Kronecker and Beatty sequences. In this article we answer a question of Winkler by showing explicitly for which sets , and vectors , we have . The main tool of the proof is Weyl’s theorem on uniform distribution. Received 3 November 2000; in final form 24 April 2001     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .

(Received 14 March 2000; in revised form 16 November 2000)     

Completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight

We study the problem on the completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight. Theorems of general form are obtained. In particular, the necessary and sufficient conditions on α, β, q ₁, and q ₂ for which the known orthogonal systems are everywhere dense in asymmetric spaces L _(α,β);q ([0, 1]) are found. Theorem. Let α, β, q ₁, q ₂ ∈ [1,+∞]. The following orthogonal systems are dense in asymmetric spaces L _(α,β);q ([0, 1]) if and only if either max{α, β, q ₁, q ₂} < + ∞ or max {α, β} < +∞, q ₁ = q ₂ = +∞: trigonometric, algebraic, Haar’s system, Meyer’s system of wavelets, Shannon-Kotel’nikov wavelets, Stromberg and Lemarie-Battle wavelets, the Walsh system, and the Franklin system. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Consistency of change point estimators for symmetrical stable distribution with parameters shift

Assume that the characteristic indexαof stable distribution satisfies 1<α<2,and that the distribution is symmetrical about its mean.We consider the change point estimators for stable distribution withαor scale parameterβshift.For the one case that mean is a known constant,ifαorβchanges,then density function will change too.To this end,we suppose the kernel estimation for a change point.For the other case that mean is an unknown constant,we suppose to apply empirical characteristic function to estimate the change-point location.In the two cases,we consider the consistency and strong convergence rate of estimators.Furthermore,we consider the mean shift case.If mean changes,then corresponding characteristic function will change too.To this end,we also apply empirical characteristic function to estimate change point.We obtain the similar convergence rate.Finally,we consider its application on the detection of mean shift in financial market.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

Relations of core mass-stellar luminosity and core mass-stellar radius for AGB and FGB stars

The relations between core mass and stellar luminosity and radius for AGB and FGB stars are systematically investigated with the latest stellar models. A single unique relation between the core mass and giant’s luminosity is obtained. The core mass-giant’s radius relations are given for both Pop I and Pop II stars. The core mass-giant’s luminosity relation is almost independent of the stellar parameters, such as stellar metallicity, mixing length parameter α, and stellar winds. The core mass-giant’s radius relation is somewhat affected by the parameters. The giant’s radius at anM _c (core mass) for Pop II is smaller than that for Pop I (by a factor of ≈ 1.5). A larger mixing length parameter α also tends to reduce the giant’s radius. Project supported by the Climbing Program of the National Science Foundation of China.     

A Note on the Distribution of the Distance from a Lattice

Let ℒ be an n-dimensional lattice, and let x be a point chosen uniformly from a large ball in ℝ ⁿ . In this note we consider the distribution of the distance from x to ℒ, normalized by the largest possible such distance (i.e., the covering radius of ℒ). By definition, the support of this distribution is [0,1]. We show that there exists a universal constant α ₂ that provides a natural “threshold” for this distribution in the following sense. For any ε>0, there exists a δ>0 such that for any lattice, this distribution has mass at least δ on [α ₂−ε,1]; moreover, there exist lattices for which the distribution is tightly concentrated around α ₂ (and so the mass on [α ₂+ε,1] can be arbitrarily small). We also provide several bounds on α ₂ and its extension to other ℓ _p norms. We end with an application from the area of computational complexity. Namely, we show that α ₂ is exactly the approximation factor of a certain natural protocol for the Covering Radius Problem. I. Haviv’s research was supported by the Binational Science Foundation and by the Israel Science Foundation. V. Lyubashevsky’s research was supported by NSF ITR 0313241. O. Regev’s research was supported by an Alon Fellowship, by the Binational Science Foundation, by the Israel Science Foundation, and by the European Commission under the Integrated Project QAP funded by the IST directorate as Contract Number 015848.     

Schmidt’s game on fractals

We construct (α, β) and α-winning sets in the sense of Schmidt’s game, played on the support of certain measures (absolutely friendly) and show how to compute the Hausdorff dimension for some. In particular, we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of ℝ ^N satisfying the open set condition, (the Cantor’s ternary set, Koch’s curve and Sierpinski’s gasket to name a few known examples), then for any countable collection of non-singular affine transformations, Δ _i : ℝ ^N → ℝ ^N ,

A revised closed graph theorem for quasi-Suslin spaces

Some results about the continuity of special linear maps between F-spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological space X is said to have a (relatively countably) compact resolution if X admits a covering {A _α :α ∈ ℕ^ℕ} consisting of (relatively countably) compact sets such that A _α ⊆ A _β for α ⩽ β. Some applications and two open questions are provided.     

Beurling algebra analogues of the classical theorems of Wiener and Lévy on absolutely convergent fourier series

Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if φ is holomorphic on a neighbourhood of the range off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.     

Bernstein’s analyticity theorem for quantum differences

We consider real valued functions f defined on a subinterval I of the positive real axis and prove that if all of f’s quantum differences are nonnegative then f has a power series representation on I. Further, if the quantum differences have fixed sign on I then f is analytic on I.