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 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.     

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.     

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 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.     

Disintegration and perfectness of measure spaces

For every finite measure space (X,A,P) we find a unique representation P=Q₁+Q₂+Q₃ such that Q₁ is compact, Q₂ is perfect and purely noncompact and Q₃ is purely nonperfect. We show that every Pachl-_O-disintegrable probability space is Ramachandran-_O-disintegrable and therefore perfect and under a certain condition we prove the equivalence between compactness and Ramachandran-_O-disintegrability.     

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.     

Semilinear hyperbolic systems and equations with singular initial data

We study the weak limits (t, .)of solutions to semilinear strictly hyperbolic systems and wave equations with initial datau (0, .) approximating a distribution, 0 < 1. We propose an optimal link between the singularity of and the growth of the nonlinear term in order that exists. In this way we extend some of the results in [3], [10], [13].     

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.     

On subseries convergent series and m-quasi-bases in topological linear spaces

We prove that the algebraic dimension of the linear span of sums of a subseries convergent series in a Hausdorff topological linear space is either finite or equals 2°. This result is applied to represent every infinite-dimensional metrizable linear spaee of cardinality 2° as the direct sum of a sequence of dense subspaces with a strange summability property. Moreover, we show that every infinite-dimensional separable metrizable linear space admits an m-quasi-basis which is not a quasi-basis.Some results of this paper were presented at the 9th Winter School on Abstract Analysis in Srni (Czechoslovakia), January 1981     

The number of linear orders on a field

An example is given of a field which can be ordered in exactly ways where is a given cardinal number.Translated from Matematicheskie Zametki, Vol. 6, No. 2, pp. 201–211, August, 1969.     

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.     

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)     

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.     

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.     

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.     

Singular parts of contractive analytic operator functions

One considers the class _G of holomorphic functions in a domain G, whose values are contractions in a separable Hilbert space. It is proved that if T(·) _G , T(z₀) is a weak contraction, its singular part T^s(z₀) is complete, and the increments T(z)–T(z₀) are not too large (for example, finite-dimensional), then the operator T^s(z₀) is complete for almost all zG. If, however, T(z₀) is, in addition, completely nonunitary and satisfies definite smoothness conditions, then in the nontrivial case the spectrum [z] of the contraction T^s(z) (zG) is a thin set: The proof of the mentioned results is based on the investigation of the formulas obtained in the paper, connecting the characteristic functions of the contractions T(z) for different values of zG.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 30–44, 1987.     

Some applications of illfoundedness

It is possible to completely characterize which countable models generated by 0^# exist inL. This in turn has applications in the study of analytic equivalence relations; for instance, ifE is ₁ ¹ and every invariant ₁ ¹ (0^#) set is ₁ ¹ , thenE has at most ₀ many equivalence classes.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

Abschnittspositive Matrizen und Positivitätsfaktoren in der Limitierungstheorie

Let A and B be normal matrices. In :={x=(x_k) ¦ x_k} we define the order relation _A by x_A0:<=> _k=0 ⁿ a_nkx_k0 (n ). Let T be a row-finite matrix. A is called T-section-positive, if _kt_mkx_ke^(k) _A0 (m ) for x_A0 (see [5]). We study the relation between T-sectional positivity and T-sectional boundedness. An (A,B)-summability factor sequence =(_k) is called positive, if (_kx_k)_B0 for each xc_A with x_A0. For B-section-positive matrices A we give a functional analytic characterization of positive (A,B)-summability factor sequences.

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Die Arbeit entstand während eines vom DAAD unterstützten Forschungsaufenthalts an der Fernuniversität-Gesamthochschule Hagen     

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).