Stochastic dynamics as a limit of Hamiltonian dynamics of hard spheres

We consider the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of a system of hard spheres. A new concept of averages over states of stochastic systems is introduced, in which the contribution of the hypersurfaces on which stochastic point particles interact is taken into account. We give a rigorous derivation of the infinitesimal operators of the semigroups of evolution operators.

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Стохастична динаміка як границя гамільтонової динаміки пружних куль

Визначена стохаатична динаміка, яка є границею Больцмана-Греда від гамільтонової динаміки системи пружних куль. Введено нову концепцію середніх від спостережуваних за станами стохастичних систем. В ньому враховуються вклади від гіперповерхонь, на яких взаємодіуть точкові стохастичні частки. Дано строге визначення інфінітезімальних операторов для півгрупи еволюційних операторів.

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This work was performed during the stay May–June 1997 of D. Ya. Petrina and during the stay September–December 1997 of K. D. Petrina as visiting professors at the Politecnico di Milano, supported by the Italian Consiglio Nazionale delle Ricerche. The authors would like to thank Prof. C. Cercignani for his comments and support.     

Decidability of first-order theories for groups and monoids of integral matrices

We deal with the decidability problem for first-order theories of a complete linear group GL(n,ℤ) of all integral matrices of order n ≥ 3. and of a respective complete linear monoid ML(n,ℤ). It is proved that theories ∀? ∧ GL(3,ℤ). ∃∀∧ GL(3,ℤ). ∀? ∧ ML(3,ℤ), and ∃? ∧ ML(3,ℤ) are critical. and that ∃∀ ∧ νGL(n,ℤ) and ∃∀ ∧ML(n,ℤ) are decidable for any n ≥ 3. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 480–504, July–August, 2000.     

Stability analysis of linear impulsive differential systems under structural perturbation

The stability and asymptotic stability of the solutions of large-scale linear impulsive systems under structural perturbations are investigated. Sufficient conditions for stability and instability are formulated in terms of the fixed signs of special matrices.

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Аналіз стійкості лінійних диференціальних імпу льсних систем із структурними зБуреннями

Досліджуються стійкість та асимптотична стійкість возв'язків великомасштабної лінійної імпульсної системи при структурних збуреннях. Достатні умови стійкості та нестійкості сформульовані на основі знаковизначеності спеціальних матриць.

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This work was done while the author was visiting the Department of Mathematics, University of Ioannina, in the framwork of the NATO Science Fellowships Programme through the Greek Ministry of National Economy.     

Automorphisms of Extensions of ℚ by a Direct Sum of Finitely Many Copies of ℚ

Let G be an extension of ℚ by a direct sum of r copies of ℚ. (1) If G is abelian, then G is a direct sum of r + 1 copies of ℚ and AutG ≅ GL(r + 1, Q); (2) If G is non-abelian, then G is a direct product of an extraspecial ℚ-group E and m copies of ℚ, where E/ζE is a linear space over Q with dimension 2n and m + 2n = r. Furthermore, let AutG′G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G′ of G, and AutG/ζG,ζGG be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center ζG of G. Then (i) The extension 1 → AutG′G → AutG → AutG′ → 1 is split; (ii) AutG′G/AutG/ζG,ζGG ≅ Sp(2n,Q) × (GL(m, Q) ⋉ ℚ(m)); (iii) AutG/ζG,ζGG/InnG ≅ ℚ(2nm).     

On the differential properties of continuous functions

We introduce and investigate some new differential properties of continuous functions by means of the geometrical properties of their derivatives.

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Про диференціальні властивості неперервних функцій

Введено та досліджено деякі нові диференціальні властивості неперервних функцій за допомогою геометричних властивостей їх похідних.

     

Description of sequences of zeros of one class of functions analytic in a half-plane

We describe sequences of zeros of functions ƒ ≠ 0 that are analytic in the right half-plane and satisfy the condition |ƒ(z)| ≤ 0(1) exp (σ| z |η(| z |)), 0 ≤ <+ ∞, Re z > 0, where η: [0; + ∞) → (- ∞; + ∞) is a function of bounded variation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1169–1176, September, 1998.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

Non-wandering sets of the powers of maps of a tree

Let T be a tree and let Ω ( f ) be the set of non-wandering points of a continuous map f: T→ T. We prove that for a continuous map f: T→ T of a tree T: ( i) if x∈ Ω( f) has an infinite orbit, then x∈ Ω( fⁿ) for each n∈ ℕ; (ii) if the topological entropy of f is zero, then Ω( f) = Ω( fⁿ) for each n∈ ℕ. Furthermore, for each k∈ ℕ we characterize those natural numbers n with the property that Ω(f^k) = Ω(f^kn) for each continuous map f of T.     

On a class of spaces of analytic functions

The spacesb (p, q, λ) (0<p<q⩽∞, 0<λ⩽∞) of functions, analytic in the circle |z|< 1, are introduced, and an unimprovable estimate is obtained for the Taylor coefficients of a functionf∃ b (p, q, λ). It is shown that B(p, q, λ) is the space of fractional derivatives f(α) of order α (−∞<α<1/p−1/q) of a function f of B(s, q, λ), where s=p/(1−αp). Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 141–150, February, 1977.     

On the local convergence of a family of Euler-Halley type iterations with a parameter

This paper aims to study the local convergence of a family of Euler-Halley type methods with a parameter α for solving nonlinear operator equations under the second-order generalized Lipschitz assumption. The radius r _α of the optimal convergence ball and the error estimation of the method corresponding to α are estimated for each α ∈ ( − ∞ , + ∞ ). For each α > 0, we get r _α  ≥ r _− α and the upper bound of the error estimation of the method with α > 0 is not larger than the one with α < 0. For each α ≤ 0, we get the precise value of r _α , which is closely linked to the dynamical property of the method applied to a real or a complex function, and the optimal error estimation, which decreases when α→0⁻. Results show that the method corresponding to α is better than the one corresponding to − α for each α > 0 and the Chebyshev-Euler method is the best among all methods in the family with α ∈ ( − ∞ , 0] from the view of both safe choice of the initial point and error estimation.     

Subsequences of normal sequences

In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume that_n→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θ _τ (0), whereθ _τ (i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.     

Euler sums of multiple hyperharmonic numbers

Lithuanian Mathematical Journal - For k ≔ (k1, …, kr) ∈ ℕr and n, m ∈ ℕ, we extend the definition of classical hyperharmonic...     

Maximizing a congruence with respect to its partition of idempotents

For a congruence σ on a semigroupS a congruence μ(σ) onS, containing σ, is defined such that the semigroupS/σ is fundamental if and only if σ=μ(σ). The congruence μ(σ) is shown to possess maximality properties and for idempotent-surjective semigroups, μ(σ) is the maximum congruence with respect to the partition of the idempotents determined by σ. Thus μ is the maximum idempotent-separating congruence on any idempotent-surjective semigroup. It is shown that μ(μ(σ))=μ(σ). If ρ is another congruence onS, possibly with the same partition of the idempotents as σ, then it is of interest to know when ρ⊆σ (or ρ⊆μ(σ)) implies μ(ρ)⊆μ(σ) or even μ(ρ)=μ(σ). These implications are not true in general but if σ⊆ρ⊆μ(σ) then μ(ρ)⊆μ(σ). IfS is an idempotent-surjective semigroup and ρ and σ have the same partition of the idempotents then μ(ρ)=μ(σ).     

Some remarks on estimation of parameters of uniform distribution

The best (in the sense of quadratic risk) unbiased estimators are constructed for the function f(x)=σ(2x/(n+1)−1)+μ from a sample of size n from the uniform distribution over [μ−σ, μ+σ] with unknown μ and σ. The best unbiased estimator for σ with μ being known is also presented. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 36–39, Perm, 1991.     

Complementary Sets of Finite Sets

Subsets 𝒜, 𝒮 of an additive group G are complementary if 𝒜 + 𝒮 = G. When 𝒜 is of finite cardinality ∣𝒜∣, and G is ℤ or ℝ, we give sufficient conditions for the existence of a complementary set 𝒮 with “density” not much larger than 1/∣𝒜∣. Supported in part by NSF DMS-0074531. Received February 14, 2002; in revised form July 18, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Covers in the lattice of varieties of ℓ-groups

We show that a variety ν in the lattice L of varieties of ℓ-groups has continuum many covers, and that the same is also true of an arbitrary o-approximable variety Χ with the property ν⊆Χ. It is proved that any o-approximable quasivariety Q of ℓ-groups, for which ν⊆Q, has the continuum of covers in the quasivariety lattice Λ. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 253–269, May–June, 1998.     

Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.     

Complementary Sets of Finite Sets

Subsets 𝒜, 𝒮 of an additive group G are complementary if 𝒜 + 𝒮 = G. When 𝒜 is of finite cardinality ∣𝒜∣, and G is ℤ or ℝ, we give sufficient conditions for the existence of a complementary set 𝒮 with “density” not much larger than 1/∣𝒜∣.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

Probabilities of high excursions of Gaussian fields

Let {ξ(t), t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E ξ(t) ≡ 0, D ξ(t) ≡ 1, and continuous trajectories defined on the m-dimensional interval T ì \mathbbR^m T \subset {\mathbb{R}^m} . The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{−v(t) < ξ(t) < u(t), t ∈ T}, when, for all t ∈ T, u(t), v(t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions of empirical processes and fields, Sov. Math., Dokl., 45(1):226–228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e^−Q  + Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R(t) = cov(ξ′(t), ξ′(t)).