Projective surfaces withk-very ample line bundles of genus ≤3k+1

Let be the set of surfaces,S, polarized by a k-very ample line bundle,L, with genus≤3k+1. All the elements (S, L) of are listed. The classification of surfaces polarized by ak-very ample line bundle of degree ≤4k+4 is completed by proving that this class of surfaces is a subset of .     

The theory of multi-dimensional polynomial approximation

We consider the problem of polynomial approximation to a real valued functionf defined on a compact set . An approximation theorem is proven in terms of the newly defined modulus of approximation. It is shown to imply a multidimensional Jackson type theorem which is stronger than previously known results even for the interval [−1, 1]. A strong multidimensional Bernstein type inverse theorem is also proven. We allow quite general approximation quasi-norms including for 0<q≤∞. We have found that the space of polynomials ℙ on a compact setX induces a semimetric which encapsulates the local structure of ℙ. Any semimetric ρ equivalent to suffices for the rough theory presented here. Many examples of sets and their metrics are presented.     

Theta series of imaginary quadratic function fields

LetL be an imaginary quadratic extension of the rational function field . We prove transformation rules for the theta series corresponding to partial zeta functions of the extension .     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

Carleson measures and the heat equation

Let { }, where { } is the open unit disk on the complex plane { }. In G, we consider analytic solutions u(t, z) ({ }, { }) of the heat equation 2u_t=u_zz with initial data f(z)=u(0, z) belonging to the Fock space F, i.e., to the space of entire functions square summable with the weight e−|z|2.Conditions on a nonnegative measure μ on G are described under which for all f ∈ F we have { } Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 146–155. Translated by S. V. Kislyakov.     

Integral-valued rational functions on valued fields

Let K be a field and a non-trivial valuation ring of K withm as its maximal ideal. Denote by and the rings of polynomials f∈K[X] and rational functions f∈K(X) resp. such that . We prove that for one variable X we have if and only if the completion of (K, ) is locally compact or algebraically closed. In the second case—i.e. if K is dense in the algebraic closure of (K, )—we even get for any number of variables X=(X₁,...,X_n). This work contains parts of the second author's thesis [Ri] written under the supervision of the first author.     

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.     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let and be the valuation rings, and letG be the Galois group. We consider the powers of the maximal ideal of as modules over the group ring . We show that, ifG has orderp ^m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then and are isomorphic iffr≡r′ modp ^m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.     

Natural bounded concentrators

We give the first known direct construction for linear families of bounded concentrators. The construction is explicit and the results are simple natural bounded concentrators. Let be the field withq elements,g(x)∈F _q [x] of degree greater than or equal to 2, and . LetI _nputs=H/A,O _utputs=H/B, and draw an edge betweenaA andbB iffaA∩bB≠ϕ. We prove that for everyq≥5 this graph is an concentrator. Part of this research was done while the author was at the department of Computer Science, The University of British Columbia, Vancouver, B.C., Canada.     

Transvection free groups and invariants of polynomial tensor exterior algebras

Let :GGl(n, ) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra . It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of ), the invariants ofG acting on the tensor product of a polynomial and an exterior algebra are given by ,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in (G) are diagonalizable.     

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.     

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.     

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

A remark on H1 mappings

With , we here construct, for each positive integer N, a smooth function of degree zero so that there must be at least N singular points for any map that minimizes the energy in the family . The infimum of over U(g) is strictly smaller than the infimum of over the continuous functions in U(g). There are some generalizations to higher dimensions.Research partially supported by the National Science FoundationResearch supported by an Alfred P. Sloan Graduate Fellowship     

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.     

Estimate of fourier transforms with respect to the system of generalized eigenfunctions of the Schrödinger operator with Stummel-type potential

Suppose thatА is a nonnegative self-adjoint extension to { } of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {