This is an announcement for the paper "Some results about the
Schroeder-Bernstein Property for separable Banach spaces" by Valentin
Ferenczi and Eloi Medina Galego.
Abstract: We construct a continuum of mutually non-isomorphic separable
Banach spaces which are complemented in each other. Consequently, the
Schroeder-Bernstein Index of any of these spaces is $2^{\aleph_0}$. Our
construction is based on a Banach space introduced by W. T. Gowers
and B. Maurey in 1997. We also use classical descriptive set theory
methods, as in some work of V. Ferenczi and C. Rosendal, to improve some
results of P. G. Casazza and of N. J. Kalton on the Schroeder-Bernstein
Property for spaces with an unconditional finite-dimensional Schauder
decomposition.
Archive classification: Functional Analysis
Mathematics Subject Classification: 46B03, 46B20
Remarks: 25 pages
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Submitted from: eloi(a)ime.usp.br
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http://front.math.ucdavis.edu/math.FA/0406479
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This is an announcement for the paper "Some equivalence relations which
are Borel reducible to isomorphism between separable Banach spaces"
by Valentin Ferenczi and Eloi Medina Galego.
Abstract: We improve the known results about the complexity of the
relation of isomorphism between separable Banach spaces up to Borel
reducibility, and we achieve this using the classical spaces $c_0$,
$\ell_p$ and $L_p$, $1 \leq p <2$. More precisely, we show that
the relation $E_{K_{\sigma}}$ is Borel reducible to isomorphism and
complemented biembeddability between subspaces of $c_0$ or $\ell_p,
1 \leq p <2$. We show that the relation $E_{K_{\sigma}} \otimes =^+$
is Borel reducible to isomorphism, complemented biembeddability, and
Lipschitz equivalence between subspaces of $L_p, 1 \leq p <2$.
Archive classification: Functional Analysis; Logic
Mathematics Subject Classification: 03E15; 46B03
Remarks: 22 pages; 2 figures
The source file(s), sjm16.tex: 74499 bytes, is(are) stored in gzipped
form as 0406477.gz with size 22kb. The corresponding postcript file has
gzipped size 86kb.
Submitted from: eloi(a)ime.usp.br
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http://front.math.ucdavis.edu/math.FA/0406477
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This is an announcement for the paper "The Busemann-Petty problem for
arbitrary measures" by Artem Zvavitch.
Abstract: The aim of this paper is to study properties of sections of
convex bodies with respect to different types of measures. We present
a formula connecting the Minkowski functional of a convex symmetric
body K with the measure of its sections. We apply this formula to study
properties of general measures most of which were known before only in
the case of the standard Lebesgue measure. We solve an analog of the
Busemann-Petty problem for the case of general measures. In addition,
we show that there are measures, for which the answer to the generalized
Busemann-Petty problem is affirmative in all dimensions. Finally,
we apply the latter fact to prove a number of different inequalities
concerning the volume of sections of convex symmetric bodies in $\R^n$
and solve a version of generalized Busemann-Petty problem for sections
by k-dimensional subspaces.
Archive classification: Metric Geometry; Functional Analysis
Mathematics Subject Classification: 52A15, 52A21, 52A38
The source file(s), GBP_Zvavitch.tex: 44254 bytes, is(are) stored in
gzipped form as 0406406.gz with size 12kb. The corresponding postcript
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Submitted from: zvavitch(a)math.kent.edu
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http://front.math.ucdavis.edu/math.MG/0406406
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http://arXiv.org/abs/math.MG/0406406
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This is an announcement for the paper "Lusin's Theorem and Bochner
integration" by Peter A. Loeb and Erik Talvila.
Abstract: It is shown that the approximating functions used to define
the Bochner integral can be formed using geometrically nice sets, such as
balls, from a differentiation basis. Moreover, every appropriate sum of
this form will be within a preassigned $\varepsilon$ of the integral, with
the sum for the local errors also less than $\varepsilon$. All of this
follows from the ubiquity of Lebesgue points, which is a consequence of
Lusin's theorem, for which a simple proof is included in the discussion.
Archive classification: Classical Analysis and ODEs; Functional Analysis
Mathematics Subject Classification: 28A20, 28B05; 26A39
Remarks: To appear in Scientiae Mathematicae Japonicae
The source file(s), bochnerbox.tex: 34366 bytes, is(are) stored in gzipped
form as 0406370.gz with size 11kb. The corresponding postcript file has
gzipped size 52kb.
Submitted from: etalvila(a)math.ualberta.ca
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.CA/0406370
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http://arXiv.org/abs/math.CA/0406370
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This is an announcement for the paper "Equilateral sets in
finite-dimensional normed spaces" by Konrad J. Swanepoel.
Abstract: This is an expository paper on the largest size of equilateral
sets in finite-dimensional normed spaces.
Archive classification: Metric Geometry; Functional Analysis
Mathematics Subject Classification: 52A21 (Primary) 46B20, 52C17
(Secondary)
Remarks: 30 pages
The source file(s), equilateral.tex: 94432 bytes, is(are) stored in
gzipped form as 0406264.gz with size 29kb. The corresponding postcript
file has gzipped size 128kb.
Submitted from: swanekj(a)unisa.ac.za
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http://front.math.ucdavis.edu/math.MG/0406264
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http://arXiv.org/abs/math.MG/0406264
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This is an announcement for the paper "Block combinatorics" by V. Farmaki
and S. Negrepontis.
Abstract: In this paper we extend the block combinatorics partition
theorems of Hindman and Milliken in the setting of the recursive system
of the block Schreier families (B^xi) consisting of families defined for
every countable ordinal xi. Results contain (a) a block partition Ramsey
theorem for every countable ordinal xi (Hindman's theorem corresponding
to xi=1, and Milliken's theorem to xi a finite ordinal), (b) a countable
ordinal form of the block Nash-Williams partition theorem, and (c)
a countable ordinal block partition theorem for sets closed in the
infinite block analogue of Ellentuck's topology.
Archive classification: Combinatorics; Functional Analysis
Mathematics Subject Classification: 05D10; 46B20
Remarks: 26 pages, AMS-LaTeX
The source file(s), fn04.tex: 83752 bytes, is(are) stored in gzipped
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gzipped size 98kb.
Submitted from: combs(a)mail.ma.utexas.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.CO/0406188
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