Arithmetic Interpretability Types of Varieties and Some Additive Problems with Primes

We deal with varieties with one basic operation f(x₁,...,x_n) and one defining identity f(x₁,..., x_n) = f(x_π(1),...,x_π(n)), where π is a permutation whose cyclic set consists of distinct primes p₁,...,p_r, with the sum p₁+...+p_r = n. Their interpretability types, together with the greatest element 1 in a lattice ^int, are said to be arithmetic. It is proved that the arithmetic types constitute a distributive lattice _ar, which is dual to a lattice Sub _fΠ of finite subsets of the set Π of all primes. It is shown that for n ⩾ 2, the poset _ar( _n) of arithmetic types defined by permutations in _n, for n fixed, is a lattice iff n = 2, 3, 4, 6, 8, 9, 11. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 622–630, September–October, 2005.     

Sharp Jackson-Stechkin inequality inL2 for multidimensional spheres

In this paper we prove the Jackson-Stechkin inequalityE _n−1(f)<ω _n (f, 2τ _n ,λ),n≥1,m≥5,r≥1, f ∈L2( ),f ≢ const, which is sharp for eachn=2, 3, ...; hereE _n−1 (f) is the best approximation of a functionf by spherical polynomials of degree ≤n−1, ω _n (f, τ) is theτth modulus of continuity off based on the translations ,t ∈ ℝ,x ∈ , , is the measure of the unit Euclidean sphere , , andτ _n ,λ is the first positive zero of the Gegenbauer cosine polynomial (cost). Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 333–355, September, 1996. The present paper was discussed at Ural State University in a seminar headed by Professor Arestov. The author is grateful to Professor Arestov and Associate Professor Popov for useful conversations. This research was supported by the State Commission for Higher Education of the Russian Federation under grant No. 2-16-5-31 and by the Russian Foundation for Basic Research under grant No. 93-011-196.     

On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

Primitive systems of elements in the variety\mathfrak{A}\mathfrak{N}_2 and some locally finite varieties of groupsand some locally finite varieties of groups

Part of any basis of a relatively free group in the variety is called a primitive system of elements. We provide a criterion of being primitive for , where is a variety of Abelian groups satisfying x^m=1, and a variety generated by a finite group. Let be a variety of nilpotent groups of class ≤c. It is proved that, for the group , the property of being primitive for an element g is stronger than the condition of being unimodular on a vector composed of values of Fox derivatives in the ring . The group is not residually finite whenever a system of elements is primitive. Supported by RFFR grant No. 96-01-01948. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 687–699, November–December, 1998.     

The second main theorem for moving targets

The defect relation for holomorphic maps in respect to slowly moving target hyperplanes in is proved. The sharp defect boundn+1 is obtained. Communicated by Yoram Sagher     

On the orders of finite semisimple groups

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups. It is a theorem of Artin and Tits that a finite simple group is determined by its order, with the exception of the groups (A₃(2), A₂(4)) and(B _n (q), C _n (q)) forn ≥ 3,q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H( ) for a split semisimple algebraic groupH defined over , does not determine the groupH up to isomorphism, but it determines the field under some mild conditions. We then put a group structure on the pairs(H ₁,H ₂) of split semisimple groups defined over a fixed field such that the orders of the finite groups H₁( ) and H₂( ) are the same and the groupsH _i have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally show that the order coincidences for some of these generators can be understood by the inclusions of transitive actions of compact Lie groups.     

An example of pointwise non-convergence of iterated conditional expectation operators

Given ∈, we construct a sequence , … of Borel sub-sigma-algebras on the unit interval with the following property. Suppose the identity functionf(x)=x is transformed by successive conditioning on , then , then , Then the lim sup, with respect ton, will exceed (pointwise almost-everywhere) 1−∈ and its lim inf will be less than ∈. The sequence of functions also will fail to converge in the . This contrasts with the long-open conjecture that if all the come from a finite set of sigma-algebras, then the resulting sequence of functions must converge in . J. L. King was partially supported by NSF grant DMS-9112595.     

The Beckman-Quarles theorem for mappings from ℝ2 to $${\mathbb{F}}^2 $$ , where F is a subfield of a commutative field extending ℝ, where F is a subfield of a commutative field extending ℝ

Let F be a subfield of a commutative field extending ℝ. Let We say thatf : preserves distanced ≥ 0 if for eachx,y ∈ ℝ ∣x- y∣= d implies ϕ₂(f(x),f(y)) = d² . We prove that each unit-distance preserving mappingf : has a formI o (ρ,ρ), where is a field homomorphism and is an affine mapping with orthogonal linear part.     

Harmonic measure of curves in the disk

A powerful tool for studying the growth of analytic and harmonic functions is Hall's Lemma, which states that there is a constantC>0 so that the harmonic measure of a subsetE of the closed unit disk evaluated at 0 satisfies whereE _rad is the radial projection ofE onto . FitzGerald, Rodin and Warschawski proved that ifE is a continuum in whose radial projection has length at most π then (*) is true withC=1, and they asked how large the length, |E _rad|, can be in order for their result to be valid. We prove that (*) holds withC=1 for every continuum satisfying and θc cannot be replaced by a larger number. Fuchs asked for the largest constantC so that (*) holds for allE. We show that for every continuum , (*) holds withC=C _2π≅.977126698498665669…, whereC _2π is the harmonic measure of the two long sides of a 3∶1 rectangle evaluated at the center. There are Jordan curves for which equality holds in (*) withC=C _2π. The authors are supported in part by NSF grants DMS-9302823 and DMS-9401027, and while at MSRI by NSF grant DMS-9022140.     

Special finite-dimensional monocomposition algebras

Special monocomposition algebras introduced in [1] are studied. Using their properties, we prove that any nondegenerate monocomposition algebra ,dim ≥ 3, with unity contains no proper ideal of dimension ≤8. This implies that if 3≤dim ≤ 9, then is a central simple algebra. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 125–144, March–April, 1996.     

Autostability of hyperarithmetical models

Let be a Δ ₁ ¹ -constructivizable model. If its Scott rank is strictly less than ω ₁ ^CK , then it is proved autostable. But if , then there exists an ordinal α<ω ₁ ^CK for which is not autostable in any degree O^(γ+1) for all γ>α. We also consider some problems concerning Δ ₁ ¹ -autostability of Δ ₁ ¹ -constructivizable Boolean algebras. Supported through the FP “Integration” and the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 198–205, March–April, 2000.     

A note on certain fields of finite transcendence type

Let be a finitely generated extension field of ℚ, andα _i,β_j(1⩽i⩽m,1⩽j⩽n) be some complex numbers. Let (k=1,2,3) be fields obtained by adjoining to the numbers {α _i,β_j exp(α_iβ_j)}, {α_i, exp(α_iβ_j)}, and {exp(α_iβ_j)}, respectively. In the present note the relation between the transcendental degree of over and the transcendence type of over ℚ is given. This work was completed in Dpt. Math., Univ. of Southern Mississippi, Hattiesburg, USA.     

On a theorem of Seidel and Walsh

Given a sequence ( _n ) _n in with there are functions such that , is a dense subset of , and the set of functions with this property is residual in . We will show that in and some related Banach spaceX there are functionsf with is dense in , and we will give a sufficient condition when the set of such functions is residual inX.     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

Sui fibrati con struttura quaternionale generalizzata

Summary Quaternion generalized fiber bundles are studied, both isomorphic to global tensorial product ordinary quaternion fiber bundles right and left respectively) and quite general ones. A cohomology class is considered which represents the obstruction in order the fiber bundle be a tensorial product. Several properties and a splitting principle are proved for bundles . On this ground and founding on a convenient bundle B_E → X associated to jaz (that we call Bonan's bundle and for which ɛ( =ɛ(B_E)) relations are stated among Stiefel-Whitney classes of , B_E and the class ɛ.

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Entrata in Redazione il 14 agosto 1974.

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Lavoro eseguito con contributo del C.N.R., nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche e loro Applicazioni.     

Resampling method in nonparametric estimation related to renewal processes

Let the statistical data be a collection of n piecesx ₁, ...,x _n of simple renewal processes. The random times between renewals have an unknown cumulative hazard function (c.h.f.) H₀(·). Let τ(H(·)) be the value of some functional τ(·), given c.h.f. H(·). The value τ₀=τ(H₀(·)) is a parameter of interest. If is an estimator for τ₀, then its quality can be characterized by the distribution law of normed deviations . These laws L_n are unknown, but they can be consistently estimated, by using the resampling method, as n→∞. In the case considered, the resampling method can be justified as a special case of the central limit resampling theorem. Supported by the Bank of Sweden Tercentenary Foundation. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Pseudo algebraically closed fields over rings

We prove that for almost allσ ∈G ℚ the field has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V→ there exists a point a ∈ such thatϕ(a) ∈ ℤ^r. We then say that is PAC over ℤ. This is a stronger property then being PAC. Indeed we show that beside the fields other fields which are algebraic over ℤ and are known in the literature to be PAC are not PAC over ℤ.     

An optimal bound for high-quality conforming triangulations

This paper shows that, for any plane geometric graph withn vertices, there is a triangulation that conforms to , i.e., each edge of is the union of some edges of , where hasO(n²) vertices with each angle of its triangles measuring no more than 11/15π. Additionally, can be computed inO(n ² logn) time. This research was partially supported by the National University of Singapore under Grant RP940641.     

On Banach spaces with unconditional bases

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR _n:X→X such thatR _nR_m=R_min(_n,m) ifn≠m and lim _n→∞ R _n x=x for allx∈X. We prove that, ifR _n−R_n −1 factors uniformly through somel _p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL _Λ=closed span , where , has an unconditional basis. Examples include the Hardy space .     

Some inequalities concerning the characteristic frequencies of elastic continuum

Summary We consider the boundary value problem αz″(x)+m(x)y(x)=0, αy″(x)+p(x)z(x)=0, xε[0, 1], y(0)=y(1)=z(0)=0, where the functions m(x) and p(x) are assumed integrable and positive everywhere in [0, 1]. As the main result we obtain the inequalities for n=1, 2, ... where δ_n(m, p) stands for the product of the first n eigenvalues α_i(m, p) of the above system and where δ_n(m) abbreviates δ_n(m, m). Entrata in Redazione il 6 febbraio 1976.