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.     

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.     

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.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

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     

Sufficient conditions on the existence of trapped modes in problems of the linear theory of surface waves

An asymptotic model is found for the Neumann problem for the second-order differential equation with piecewise constant coefficients in a composite domain Ω∪ω, which are small, of order ε, in the subdomain ω. Namely, a domain Ω(ε) with a singular perturbed boundary is constructed, the solution for which provides a two-term asymptotic, that is, of increased accuracy O(ε²| log ε|^3/2), approximation to the restriction to Ω of the solution of the original problem. As opposed to other singularly perturbed problems, in the case of contrasting stiffness, the modeling requires the construction of a contour ∂Ω(ε) with ledges, i.e., with boundary fragments of curvature O(ε⁻¹). Bibliography: 33 titles.     

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.     

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.     

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.     

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.     

Stability of lurie-type non-linear equations

Summary Conditions are given for the indirect control system x′=a(x)+bμ, μ′=φ(σ), σ=c^Tx−ϱμ, to be absolutely stable. These conditions reduce to LaSalle and Lefschetz's in the linear case: a(x)=Ax. The conditions obtained for the stability of the direct control system x′=a(x)+bφ(σ), σ=c^Tx, reduce also to Lurie's condition in the linear case. The case of the direct control system x′=a(x, t)+bφ(σ), σ=c^Tx is also investigated. Entrata in Redazione il 18 febbraio 1976.     

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.     

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.     

A posteriori analysis of the finite element discretization of some parabolic equations

We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes.

RÉSUMÉ. Nous considérons la discrétisation d'équations paraboliques, soit linéaires soit semi-linéaires, par un schéma d'Euler implicite en temps et par éléments finis en espace. L'idée de cet article est de construire des indicateurs d'erreur liés à l'approximation en temps et en espace et de prouver leur équivalence avec l'erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d'éléments finis adaptés à la solution.

     

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.     

L P-Integrability of derivatives of Riemann mappings on Ahlfors-David regular curves

In 1981, Hayman and Wu proved that for any simply connected domain Ω and any Riemann mappingF: Ω →D,F′ ∈ L¹ (L ∩ Ω), whereL is any line in the complex plane. Several years later, Fernández, Heinonen and Martio showed that there is anε > 0 such thatF′ ∈ L^1+∈(L ∩ Ω). The question arises as to which curves other than lines satisfy such a statement. A curve Γ is said to be Ahlfors-David regular if there is a constantA such that for any B(x, r) (the disk of radiusr centered atx), l(Γ ∩ B(x, r))≤ Ar. The major result of the paper is the following theorem: Let Γ be an Ahlfors-David regular curve with constantA. Then there exists an∈ > 0, depending only onA, such thatF′ ∈ L^1+∈(Γ ∩ Ω). This result is the synthesis of the extension of Fernández, Heinonen and Martio, and the result of Bishop and Jones showing thatF′ ∈ L¹(Γ ∩ Ω). The proof of the results uses a stopping-time argument which seeks out places in the curve where small pieces may be added in order to control the portions of the curve where |F′ | is large. This is accomplished with an estimate on the vanishing of the harmonic measure of the curve in such places. The paper also includes simpler arguments for the special cases where Γ = ∂Ω and Γ ⊂Ω.     

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).     

On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain

We study the spectrum of the boundary-value problem for the Laplace operator in a thin domain Ω(ε) obtained by small perturbation of the cylinder Ω(ε)=ω×(-ε/2.ε/2) ⊂ ℝ³in a neighborhood of the lateral surface. The Dirichlet condition is imposed on the bases of the cylinder, and the Dirichlet condition or the Neumann condition is imposed on the remaining part of ∂Ω(ε). We construct and justify asymptotic formulas (as ε→+0) for eigenvalues and eigenfunctions. In view of a special form of the lateral surface, there are eigenfunctions of boundary-layer type that exponentially decrease far from the lateral surface. For the mixed boundary-value problem such a localization is possible in neighborhoods of local maxima of the curvature of the contour ∂ω. This property of eigenfunctions is a characteristic feature of the first points of the spectrum (in particular, the first eigenvalue) and, under the passage from Ω(h)₍₎ to Ω(h), the spectrum itself has perturbation O(h⁻²). Bibliography: 29 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 105–149.     

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).