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).     

The semiregular polytopes

A convexd-polytype inE ^d is called semiregular, if its facets are regular and its vertices equivalent. A list of semiregular polytopes ford≥4 is known since 1900. Recently it has been proved by П. В. Макаров [сб. Вопр. дискр. геом., Мат. исслед. ИМ АН Молд. ССР вып. 103, C. 139–150, Кищинев 1988], that this list is complete ford=4. We present here a simple proof for that this list is complete in any dimension.     

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∣.     

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).     

Elastic waveguides: history and the state of the art. II

In this paper, the normal modes of an elastic rectangular waveguide are analyzed. We retrace the key aspects of the almost 150-year history of this problem. Using the superposition method, we have obtained an analytical solution of the problem for four types of symmetry of the wave field. In addition, we have established important differences of the dispersion characteristics of normal modes in a rectangle from the Rayleigh–Lamb modes for an infinite plate and the Pochhammer–Chree modes for a cylinder. We give also an estimate of a series of approximate theories for a rectangular waveguide.

The numerical interpretation of the results of analysis is however necessary, and it is a degree of perfection which it would be very important to give to every application of analysis to the natural sciences. So long as it is not obtained, the solutions may be said to remain incomplete and useless, and the truth which it is proposed to discover is no less hidden in the formulas of analysis than it was in the physical problem itself.

                             J. Fourier [28, Sec. 13]

     

Zeros in Tables of Characters for the Groups <Emphasis Type="Italic">S</Emphasis><Subscript>n</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>. II

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for S_n. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χ^α(g_β) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χ^α(g_γ) ≠ 0, where χ^α is an irreducible character of S_n, indexed by α, and g_β is an element in the conjugacy class of S_n, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χ^α(g_h′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χ^α(g_f′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in A_n. Supported by RFBR grant No. 04-01-00463. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 643–663, November–December, 2005.     

The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

For any (noncommutative) skew field T, the lattice of subgroups of the special linear group Λ=SL(n,T) that contain the subgroup Δ=SD(n,T) of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup H, Δ≤H≤Λ, there exists a uniquely determined unital net σ such that Λ(σ)≤H≤N(σ), where Λ(σ) is the net subgroup associated with the net σ and N(σ) is its normalizer in Λ. Bibliography: 11 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 91–103. Translated by Bui Xuan Hai.     

一类无限矩阵拓扑代数

Let λ and μ be sequence spaces and have both the signed-weak gliding hump property, (λ,μ) the algebra of the infinite matrix operators which transform λ into μ. In this paper, it is proved that if λ and μ are β-spaces and λ^β and ,μ^β have also the signed-weak gliding hump property, then for any polar topology τ, ((λ,μ),τ) is always sequentially complete locally convex topological algebra.     

The Change-Base Issue for Ω-Categories

Let G ： Ω→Ω＇ be a closed unital map between commutative, unital quantales. G induces a functor G^- from the category of Ω-categories to that of Ω＇-categories. This paper is concerned with some basic properties of G^-. The main results are： （1） when Ω, Ω＇ are integral, G ： Ω→Ω＇ and F ： Ω＇→Ω are closed unital maps, F is a left adjoint of G^- if and only if F is a left adjoint of G; （2） G^- is an equivalence of categories if and only if G is an isomorphism in the category of commutative unital quantales and closed unital maps; and （3） a sufficient condition is obtained for G^- to preserve completeness in the sense that GA is a complete Ω＇-category whenever A is a complete Ω-category.     

On groups factorized by finitely many subgroups

We prove that every group factorizable into a product of finitely many pairwise permutable central-by-finite minimax subgroups is a soluble-by-finite group.

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Про групи, факторизовані скінченним числом шдгруп

Розвивається спектральна теорія та теорія розсіяння для одпого класу самоспряжених матричних диференціальних операторів змішаного порядку.

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This work was done while the author was visiting the Ukrainian Academy of Sciences in Kiev. He is grateful to the Institute of Mathematics for its warm hospitality.     

On the embedding problem with non-Abelian kernel of orderp 4. IV

In the present paper, the embedding problem is considered for number fields with p-groups whose kernel is either of two groups with two generators α and β and with the following relations: (1) α^ρ=1, α^ρ=1, [α,β,β]=1, [α,β,α,α]=1, or (2) α^ρ=[α, β α], β^ρ=1, [α,β,β]=1. It is shown that for the solvability of the original embedding problem it is necessary and sufficient to have the solvability of the associated Abelian and local problems for all completions of the base fields. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 120–126. Translated by V. V. Ishkhanov.     

Riemann surfaces in fibered polynomial hulls

Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and letA(Δ) be the algebra of complex functions continuous on Δ and analytic in int Δ. LetK be a compact set in C² such that Π(K)=Γ, and letK _λ≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ,K _λ is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))²-S(λ) quadratic in w withR,S∈A(Δ) such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ intK _λ, whereS has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component ofK _λ. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull ofK. Furthermore, we show that БК/K is the disjoint union of such disks.     

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.     

T 0-группа и ее место в теории групп

Охарактеризовано класT ₀-груп, тісно пов'язаний із вільними бернсайдівськими групами непарного періоду не менш ніж 665. Наведено приклади, що грунтуються на відомих конструкціях С. І. Адяна та О. Ю. Ольшанцького. Крім цього, вказано місце скінченної групи у класі всіх груп.

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A class ofT ₀-groups is characterized which is closely associated with free Burnside groups with odd period not less than 665. Examples based on the well-known Adyan and Ol'shanskii constructions are given. In addition, the place of a finite group in the class of all groups is indicated.

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Выполнена при поддержке Красноярского краевого фонда науки (грант № 7F0008) и Российского фонда фундаментальных исследований (грант № 96-01-00400).     

Self-injective rings and linear (weak) inverses of linear finite automata over rings

LetR be a finite commutative ring with identity and τ be a nonnegative integer. In studying linear finite automata, one of the basic problems is how to characterize the class of rings which have the property that every (weakly) invertible linear finite automaton ℳ with delay τ over R has a linear finite automaton ℳ′ over R which is a (weak) inverse with delay τ of ℳ. The rings and linear finite automata are studied by means of modules and it is proved that *-rings are equivalent to self-injective rings, and the unsolved problem (for τ=0) is solved. Moreover, a further problem of how to characterize the class of rings which have the property that every invertible with delay τ linear finite automaton ℳ overR has a linear finite automaton ℳ′ over R which is an inverse with delay τ′ for some τ′⩾τ is studied and solved. Project supported by the National Natural Science Foundation of China(Grant No. 69773015).     

Erdős measures,sofic measures,and Markov chains

We consider the random variable ζ = ξ₁ρ+ξ₂ρ²+…, where ξ₁, ξ₂, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξ_i = 0) = p₀, P(ξ_i = 1) = p₁, 0 < p₀ < 1. Let β = 1/ρ be the golden number. The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η₁ρ + η₂ρ² + … where the random variables η_k are {0, 1}-valued and η_kη_k+1 = 0. The infinite random word η = η₁η₂ … η_n … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous. We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure. We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ₁, ξ₂, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.     

Each regular number structure is biregular

Roughly speaking we show that for certain number structures ℋ, ℬ with ℬ ⊆ ℋ, if ℬ is bounded above in ℋ then ℬ is bounded below in ℋ.     

There are infinitely many limit points of the fractional parts of powers

Suppose that α > 1 is an algebraic number and ξ > 0 is a real number. We prove that the sequence of fractional partsξα ⁿ , n = 1, 2, 3, …, has infinitely many limit points except when α is a PV-number and ξ ∈ ℚ(α). For ξ = 1 and α being a rational non-integer number, this result was proved by Vijayaraghavan.