Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

The number of powers of 2 in a representation of large even integers II

It is proved that for any integerk≥ 54 000, there isN _k >0 depending onk only such that every even integer ≥N _k is a sum of two odd prime numbers andk powers of 2. Project partially supported by RGC Research Grant (No.HKU 7122/97P) and Post-Doctoral Fellowship of the University of Hong Kong.     

Weight functions on groups and an amenability criterion for Beurling algebras

The paper is devoted to the study of weights on groups. A connection between weight functions and harmonic functions is established. A relationship between the weight theory on groups with the “Tychonoff property” and the theory of bounded cohomology is presented. It is proved that the Beurling algebraℓ1 (G, ω) constructed for the weightω is amenable if and only if the groupG is amenable and the weightω is equivalent to a multiplicative characterχ:G→ℝ₊. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 448–460, September, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00974 and by the INTAS Foundation under grant No. 94-3420.     

On generalized k-syzygy modules

In this paper, we first introduce the notion of generalized k-syzygy modules, and then give an equivalent characterization that the class of generalized k-syzygy modules coincides with that ofω-k-torsionfree modules. We further study the extension closure of the category consisting of generalized k-syzygy modules. Some known results are obtained as corollaries.     

The size of epimorphic extensions

A sharp bound is given for the size of epimorphic extensions in categories of models defined over elementary logic andL _κκ where κ is strongly compact. For fragments ofL _ω1ω an example is given of a category which has a countable model with epimorphic extensions whose cardinalities approach and include the first measurable cardinal. If no measurable cardinal exists then this category has a countable model with epimorphic extensions of unbounded cardinality. This work was supported in part by the National Research Council of Canada under grant numbers A8599, A5603 and A8190. Presented by J. D. Monk.     

There exists a maximal 3-C.E. enumeration degree

We construct an incomplete 3-c.e. enumeration degree which is maximal among then-c.e. enumeration degrees for everyn with 3≤n≤ω. Consequently then-c.e. enumeration degrees are not dense for any suchn. We show also that no lown-c.e. e-degree can be maximal among then-c.e. e-degrees, for 2≤n≤ω. The first two authors were partially supported by EPSRC Research Grant “Turing Definability” No. GR/M 91419 (UK), and the second author by NSF grant No. 69973048 and by NSF major grant No. 19931020 (P. R. China), and by an INDAM visiting professorship at the University of Siena. The fourth author was partially supported as a visiting scholar by the University of Siena. The first three authors were funded by the INTAS-RFBR joint projectComputability and Models, no. 972-139. The fourth authors would like to thank Marat Arslanov for useful discussions.     

Farrell cohomology of GL(n,Z)

The Tate-Farrell cohomology of GL(n,Z) with coefficients inZ/p is computed forp an odd prime andp−1 ≦n ≦ 2p−3. Its size depends on the Galois structure of the class group of the cyclotomic fieldQ(p√1) and is shown to be quite large in general. Research partially supported by NSF Grant No. DMS-8701758.     

A model in which there are Jech-Kunen trees but there are no Kurepa trees

By an ω₁-tree we mean a tree of power ω₁ and height ω₁. We call an ω₁-tree a Jech-Kunen tree if it hask-many branches for somek strictly between ω₁ and 2^ω1. In this paper we construct the models of CH plus 2^ω1 > ω₂, in which there are Jech-Kunen trees and there are no Kurepa trees. The research of the first author was partially supported by the Basic Research Fund, Israeli Academy of Science, Publ. No. 466.     

Banach ideals of operators with applications to the finite dimensional structure of Banach spaces

We present a few applications of the theory of Banach ideals of operators. In particular, we give operator characterizations of the ℒ _p spaces, compute the relative projection constant of isometric embeddings of Hilbert spaces inL _p -spaces, and show that Π₁ (E, F), the space of absolutely summing operators, is reflexive ifE andF are reflexive andE has the approximation property. Research supported by NSF-GP-34193 Research supported by NSF-Science Development Grant     

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

On the structure of stationary sets

We isolate several classes of stationary sets of [k]ωand investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.     

An ergodic theorem on Banach lattices

E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf _n=ϕ means that the infimum of |f _n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA _n=(1/n)(I+T+...+T ⁿ⁻¹). Without an additional assumption onE, the “truncated limit” TLA _nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE ₊ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE ₊, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β. Research of this author is partially supported by NSERC Grant A3974. Research of this author is partially supported by NSF Grant 8301619.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.     

Finite groups in which elements of the same order outside the center are conjugate

We prove that if G is a finite group in which the elements of the same order outside the center are conjugate,then either G is abelian or G(?)S_3.     

A fixed point theorem for a class of star-shaped sets inc 0

A subsetK ofc ₀ is coordinatewise star-shaped (c.s.s.) if there exists a center pointx ∈K such that fory ∈K andz ∈c ₀, ifz is coordinatewise betweenx andy thenz ∈K. We prove that a weakly compact c.s.s. subset ofc ₀ has the fixed point property for nonexpansive mappings and that a fixed point for such a mapping can be obtained in a constructive manner. Research of the first two authors was partially supported by NSF Grant MCS78-01344 and of the last author by MCS78-01501.     

On local frequencies related to farey fractions

LetP ₁ andP ₂ be two sets of prime numbers and let ω(m,P_i)=#{p: p/m, pεP_i} (i=1,2) be two related additive functions ofm. For an irreducible positive fractionm/n, defineh(m/n)=ω(m, P ₁)+ω(n, P₂). In this paper the local frequenciesv _x{h(m/n)=s}=#{m/n ∈ F_x:h(m/n)=s}/#F_x are considered, whereF _x denotes the classical Farey series. Using the mean-value theorem for multiplicative functions of rational argument, a local limit theorem forv _x{h(m/n)=s} is proved. Research supported by the Lithuanian State Science and Studies Foundation. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 113–131, January–March, 2000. Translated by V. Stakènas     

Not collapsing cardinals ≤<Emphasis Type="Italic">κ</Emphasis> in (<<Emphasis Type="Italic">κ</Emphasis>)-support iterations

We deal with the problem of preserving various versions of completeness in (<κ)-support iterations of forcing notions, generalizing the case “S-complete proper is preserved by CS iterations for a stationary costationaryS⊆ω ₁”. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinalκ and a stationary setS⊆κ with fat complement we can have uniformization for every (A _δ :δ ∈S′),A _δ ⊆δ = supA _δ , cf (δ) = otp(A _δ ) and a stationary non-reflecting setS′⊆S (see B.8.2). Research supported by The German-Israeli Foundation for Scientific Research & Development Grant No. G-294.081.06/93 and by The National Science Foundation Grant No. 144-EF67. Publication No. 587.     

More on countably compact,locally countable spaces

Following [5], aT ₃ spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+┐G(ω_ω+₁); (iii) the “Chang conjecture” (ω_ω+₁),→(ω ₁,ω) implies ┐S(κ) for allκ≧k≧ω_ω; (iv) ifP addsω ₁ dominating reals toV iteratively then, in , we haveG(λ^ω) for allλ. Research supported by Hungarian National Foundation for Scientific Research grant no. 1805.     

‘Fat’ triangulations,or solving certain nonconvex matrix optimization problems

In an earlier paper, the author proposed the problems of determining ‘optimal’ linear transformations of the triangulationsJ ₁ andK ₁, in the sense of minimizing their average directional density for a given mesh size. These tasks were also formulated as optimization problems where the variable is a matrix. Here we solve these problems, and another one which is analogously related to finding an ‘optimal’ linear transformation of the new triangulationJ′. We show thatJ ₁ andJ′ are themselves optimal, while the (α*β*) ofK ₁ developed by van der Laan and Talman is optimal. The latter theorem extends partial results of van der Laan and Talman and Eaves. The optimality of these linear transformations is quite robust: we may change the objective function to maximizing the volume of each simplex, or the constraints to limiting the sum of squares of edge lengths of each simplex, or both, without changing the optimal solutions. Research partially supported by a fellowship from the Alfred P. Sloan Foundation and by NSF Grant ENG82-15361     

Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures

The boundedness in Lebesgue spaces for commutators generated by multilinear singular integrals and RBMO(μ) functions of Tolsa with non-doubling measures is obtained, provided that‖μ‖=∞and multilinear singular integrals are bounded from L1(μ)×L1(μ)to L1/2,∞(μ).