An alternative construction of Conway's ordered field No

An -universally extending ordered field of power is constructed for each regular power where 0 < On and . When is inaccessible, the structure is either a (set) model of J. H. Conway's ordered field No or an isomorphic copy of No depending on whether or not is a set or a proper class.Presented by Jan Mycielski.     

Models with second order properties. III. Omitting types forL(Q)

We generalize Keisler's omitting types theorem forL(Q) in the ₁-interpretation, to most cases in which Chang's two cardinal theorem applies. As an application we answer positively a question of Magidor and Malitz on the compactness of their logic in cardinalities higher than ₁.     

A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (_c)^± and (S _c)^± for 0 1. We investigate the relation between the spaces and show that they are topologically isomorphic when (^d) L2 (^d) (^d) is the underlying Gel'fand triple for (_c)^±. In this case we also give the explicit relation between the S-transform and -transform on (_c)^-1 and (S _c)^-1, respectively.     

An Investigation of the Fields of Bounded Formal Power Series by Means of Theory of Cuts

We consider a class K of real closed fields F, |F|=|G|=₁, where G is a group of Archimedean classes of F, and cofinality of each symmetric gap of F is ₁. We will show that this class is exactly a class of all bounded formal power series RG,₁, where G is a divisible Abelian group, card(G)=₁, under CH. A nonstandard real line ^*R, which is ₁-set belongs to this class; we will also consider a construction RG(L,P),₁ of fields from this class, where L is a totally ordered set, P is a totally ordered field, G(L,P) is a group of finite words. It will be describes symmetric gaps of such two fields in K, which are not ₁-set. Mathematics Subject Classifications (2000) 03E04, 12J15, 12J25.The work was supported by grant of Ministry of Education PD02-1.1-386.     

The structure of saturated free algebras

Requiring an algebraM to be both free (for the variety it generates) and ₁-saturated imposes very strong conditions onM. In the simplest examples (see below) there exist a finite number of relatively free algebrasA _o,...,A _n-1 whose theories are ₁-categorical such thatM is generated (as an algebra) by the UA _i In particular, this implies Th(M) has at most (+_o) models of cardinality . We will show a weaker structure theorem in the general case but deduce the same constraint on the spectrum ofT.Dedicated to Alfred Tarski on his 80th birthdayPartially supported by N.S.F. grant 77-01667.This research was partially supported by the United States Israel Binational Science Foundation grant 1110.Presented by R. McKenzie.     

Products of chains with monochromatic maximal chains and antichains

There is a product of two linear orders of size with the property that every subset or complement thereof contains a maximal chain. Furthermore, for regular , there is a product of two linear orders of size ₊₂ that when colored with fewer than colors always has a monochromatic maximal chain. As a corollary, for every uncountable strong limit cardinal , there is an ordered set of cardinality that must be colored with at least colors before no monochromatic maximal chains are present. Duals of these results show that at least as much is true for maximal antichains.Research supported in part by ONR Grant N00014-91-J-1150.     

The absence of bases from certain separable linear topological spaces

We give examples of separable linear topological spaces without Shauder-type bases. We prove that every linear set X of dimension ₀2₀ can be provided with a separable locally convex topology for which there is no Shauder-type basis.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 583–589, November, 1972.In conclusion the author wishes to express his gratitude to M. M. Dragilev for his attention to the paper.     

On the congruence problem in infinite dimensional sesquilinear spaces

If two subspaces V and V of a sesquilinear space E are congruent (i.e., there is an isometry : E E with (V)=V) then their corresponding quadratic lattices V(V, E) and V(V, E) are isomorphic. It is shown that the converse holds for important types of sesquilinear spaces E, provided that dim(E) ₃. However, the converse generally fails if dim(E) ₃.     

A finitely based theory of a non-trivial language,with exactly ℵ0 subcovers

LetL=f, g be the language with two unary operation symbols. I prove that the finitely based equational theory =[f₀=₀] ofL covers exactly ₀ others.Presented by S. Burris.Dedicated to George McNulty, my mentor in equational logic.     

Measures on product spaces and the existence of strong Baire liftings

Some notions are introduced for studying measures on product spaces, the main concept being that of property (*). In case when the topological factors are separable metric spaces, this property is equivalent to the completion regularity. We prove that (*) is preserved under arbitrary products of measure spaces. As a consequence, we deduce a series of related results in measure theory (some of which are known). In particular, the following extension of a result by Losert is obtained: Subject to CH, every product of ₂ many completion regular measures, each supported on any product of ₁ many compact metric spaces admits a strong Baire lifting.     

On the number of non conjugate subgroups

We shall prove that every group of cardinality ₁ has at least ₁ non conjugate subgroups, and we shall generalize this theorem to many more uncountable cardinalities. For example underGCH for every uncountable cardinal and every groupG of cardinality ,G has at least non conjugate subgroups.Presented by W. Taylor.I would like to thank Rami Grossberg for writing and rewriting this paper, and Wilfrid Hodges for removing many errors and suggesting improvements in presentation; many facts are proved only due to his explicit request.This research was supported by grant (No. 1110) from the United States-Israel Binational Science Foundation.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Finite cutsets and finite antichains

An ordered set (P,) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x} Fx meets every maximal chain of (P,). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardinality less than n. We specify n for m < w. Indeed, n-1=m= width P for m=1,2,n=5 if m=3 and n₁ if m 4. With the added hypothesis that every bounded chain has a supremum and infimum in P, it is shown that for 4m₀, n=₀. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain.This work was supported by the NSERC of Canada.     

Asymptotic behavior of the Jost function of a two-dimensional Schrödinger operator

We find the asymptotic behavior of the Jost function(Z,) of a two-dimensional Schrödinger operator for arbitrary and Z/|Z|S¹ as |Z| We discuss consequences of the asymptotic formulas for the inverse scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 96—103, 1988.     

Forcings with the countable chain condition and the covering number of the Marczewski ideal

We prove that the covering number of the Marczewski ideal is equal to ₁ in the extension with the iteration of Hechler forcing.     

Reflecting stationary sets and successors of singular cardinals

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a which is ⁺ⁿ -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the bad stationary set. It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for many PT(, ₁). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular which is a limit of supercompacts the bad stationary set concentrates on the right cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular {<⁺ : cf<} is good. It is proved in ZFC that if=cf>₁ then {<⁺ : cf<} is the union of sets on which there are squares.     

Über ein Problem von Mordell in der Geometrie der Zahlen

For a latticeL in ⁿ with determinantd(L), let (L) denote the supremum of the values 2^–2 V(P)/d(L), taken over theL-admissible parallelepidesP, symmetric with respect to the origin and with faces parallel to the coordinate-axes. In 1936, Mordell asked for the constants _n = min (L) over alln-dimensional lattices. In this paper we investigate isolated minima of (L) in all over alln-dimensional lattices. In this paper we (Satz 1) and some examples are given. In particular, forn<=4, the set of lattices with isolated turns out to be dense in the space of lattices. Conversely, the set of (algebraically generated) lattices with non-isolated is dense, at least in the case of a plane (Satz 2).     

Many simple cardinal invariants

Summary Forg in we definec(f,g) be the least number of uniform trees withg-splitting needed to cover a uniform tree withf-splitting. We show that we can simultaneously force ₁ many different values for different functions (f,g). In the language of [B1]: There may be ₁ many distinct uniformII ₁ ¹ characteristics.Supported by Israeli Academy of Sciences, Basic Research FundPublication 448. Supported partially by Israeli Academy of Sciences, Basic Research Fund and by the Edmund Landau Center for research in Mathematical Analysis (supported by the Minerva Foundation, Germany)     

Recurrence relations in the theory of Hecke operators

In this paper one computes the–2 power of the Frobenius element of the Hecke ring of the subgroup _n,1(q) of a modular group of genus n+1, which is the semidirect product of the Heisenberg group and the modular group _n(q) of genusn. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 65–73, 1983.     

Some coinductive graphs

Summary LetT be a universal theory of graphs such that Mod(T) is closed under disjoint unions. Let _T be a disjoint union _i such that each _i is a finite model ofT and every finite isomorphism type in Mod(T) is represented in{ _i i<3}. We investigate under what conditions onT, Th( _T ) is a coinductive theory, where a theory is called coinductive if it can be axiomatizated by -sentences. We also characterize coinductive graphs which have quantifier-free rank 1.