This is an announcement for the paper "On maps which preserve
equality of distance in F-spaces" by Dongni Tan.
Abstract: In order to generalize the results of Mazur-Ulam and Vogt,
we shall prove that any map T which preserves equality of distance
with T(0)=0 between two F-spaces without surjective condition is
linear. Then , as a special case linear isometries are characterized
through a simple property of their range.
Archive classification: math.FA math.MG
Mathematics Subject Classification: 46A16
Remarks: 11 pages, 385 figures
The source file(s), DongniTan.tex: 17852 bytes, is(are) stored in
gzipped form as 0709.3620.gz with size 6kb. The corresponding
postcript file has gzipped size 66kb.
Submitted from: 0110127(a)mail.nankai.edu.cn
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This is an announcement for the paper "Note on distortion and
Bourgain $\ell_1$ index" by Anna Maria Pelczar.
Abstract: The relation between different notions measuring proximity
to $\ell_1$ and distortability of a Banach space is studied. The
main result states that a Banach space, whose all subspaces have
Bourgain $\ell_1$ index greater than $\omega^\alpha$, $\alpha<\omega_1$,
contains either an arbitrary distortable subspace or an
$\ell_1^\alpha$-asymptotic subspace.
Archive classification: math.FA
Mathematics Subject Classification: 46B20 (primary), 46B03 (secondary)
Remarks: 10 pages
The source file(s), distortion_bourgain.tex: 36771 bytes, is(are)
stored in gzipped form as 0709.2272.gz with size 11kb. The corresponding
postcript file has gzipped size 92kb.
Submitted from: anna.pelczar(a)im.uj.edu.pl
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http://front.math.ucdavis.edu/0709.2272
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This is an announcement for the paper "Some classes of rational
functions and related Banach spaces" by R. M. Dudley, Sergiy Sidenko,
Zuoqin Wang, and Fangyun Yang.
Abstract: For positive integers d, r, and M, we consider the class
of rational functions on real d-dimensional space whose denominators
are products of at most r functions of the form 1+Q(x) where each
Q is a quadratic form with eigenvalues bounded above by M and below
by 1/M. Each numerator is a monic monomial of the same degree as
the corresponding denominator. Then we form the Banach space of
countable linear combinations of such rational functions with
absolutely summable coefficients, normed by the infimum of sums of
absolute values of the coefficients. We show that for rational
functions whose denominators are rth powers of a specific 1+Q, or
differences of two such rational functions with the same numerator,
the norm is achieved by and only by the obvious combination of one
or two functions respectively. We also find bounds for coefficients
in partial-fraction decompositions of some specific rational
functions, which in some cases are quite sharp.
Archive classification: math.FA
Mathematics Subject Classification: 46B99 (primary), 46B22 (secondary)
Remarks: LaTex, 18 pages, no figures
The source file(s), bspsrtlfncts.tex: 74856 bytes, is(are) stored
in gzipped form as 0709.2449.gz with size 25kb. The corresponding
postcript file has gzipped size 93kb.
Submitted from: rmd(a)math.mit.edu
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This is an announcement for the paper "Conical square functions in
UMD Banach spaces" by Tuomas Hytonen, Jan van Neerven, and Pierre
Portal.
Abstract: We study conical square function estimates for Banach-valued
functions, and introduce a vector-valued analogue of the
Coifman-Meyer-Stein tent spaces. Following recent work of
Auscher-McIntosh-Russ, the tent spaces in turn are used to construct
a scale of vector-valued Hardy spaces associated with a given
bisectorial operator \(A\) with certain off-diagonal bounds, such
that \(A\) always has a bounded \(H^{\infty}\)-functional calculus
on these spaces. This provides a new way of proving functional
calculus of \(A\) on the Bochner spaces \(L^p(\R^n;X)\) by checking
appropriate conical square function estimates, and also a conical
analogue of Bourgain's extension of the Littlewood-Paley theory to
the UMD-valued context. Even when \(X=\C\), our approach gives
refined \(p\)-dependent versions of known results.
Archive classification: math.FA math.SP
Mathematics Subject Classification: Primary: 46B09; Secondary:
42B25, 42B35, 46B09, 46E40, 47A60, 47F05
Remarks: 28 pages; submitted for publication
The source file(s), tent/newsymbol.sty: 440 bytes tent/tent.bbl:
5616 bytes tent/tent.tex: 91867 bytes, is(are) stored in gzipped
form as 0709.1350.tar.gz with size 29kb. The corresponding postcript
file has gzipped size 167kb.
Submitted from: J.M.A.M.vanNeerven(a)tudelft.nl
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This is an announcement for the paper "The extremal volume ellipsoids
of convex bodies, their symmetry properties, and their determination
in some special cases" by Osman Gueler and Filiz Guertuna.
Abstract: A convex body K has associated with it a unique circumscribed
ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid
IE(K) with maximum volume. We first give a unified, modern exposition
of the basic theory of these extremal ellipsoids using the semi-infinite
programming approach pioneered by Fritz John in his seminal 1948
paper. We then investigate the automorphism groups of convex bodies
and their extremal ellipsoids. We show that if the automorphism
group of a convex body K is large enough, then it is possible to
determine the extremal ellipsoids CE(K) and IE(K) exactly, using
either semi-infinite programming or nonlinear programming. As
examples, we compute the extremal ellipsoids when the convex body
K is the part of a given ellipsoid between two parallel hyperplanes,
and when K is a truncated second order cone or an ellipsoidal
cylinder.
Archive classification: math.OC math.FA
Mathematics Subject Classification: 90C34; 46B20; 90C30; 90C46;
65K10
Remarks: 36 pages
The source file(s), Ellipsoid35.bbl: 8177 bytes
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This is an announcement for the paper "Sharp constants related to
the triangle inequality in Lorentz spaces" by Sorina Barza, Viktor
Kolyada, and Javier Soria.
Abstract: We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range
$1<p<s\le \infty$, for which the standard functional $$
||f||_{p,s}=\left(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t}\right)^{1/s}
$$ is only a quasi-norm. We find the optimal constant in the
triangle inequality
for this quasi-norm, which leads us to consider the following
decomposition norm:
$$ ||f||_{(p,s)}=\inf\bigg\{\sum_{k}||f_k||_{p,s}\bigg\}, $$ where
the infimum is taken over all finite representations $f=\sum_{k}f_k.
$ We also prove that the decomposition norm and the dual norm $$
||f||_{p,s}'= \sup\left\{ \int_R fg\,d\mu: ||g||_{p',s'}=1\right\}
$$ agree for all values $p,s>1$.
Archive classification: math.FA math.CA
Mathematics Subject Classification: 46E30, 46B25
Remarks: 24 pages
The source file(s), Norms-Constants.tex: 47398 bytes
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http://front.math.ucdavis.edu/0709.0647
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This is an announcement for the paper "On the linear independence
of spikes and sines" by Joel A. Tropp.
Abstract: The purpose of this work is to survey what is known about
the linear independence of spikes and sines. The paper provides new
results for the case where the locations of the spikes and the
frequencies of the sines are chosen at random. This problem is
equivalent to studying the spectral norm of a random submatrix drawn
from the discrete Fourier transform matrix. The proof involves
methods from geometric functional analysis.
Archive classification: math.FA math.MG
Mathematics Subject Classification: 46B07, 47A11, 15A52
Remarks: 4 figures
The source file(s), art/old/square-unnorm.eps: 11263 bytes, etc.,
is(are) stored in gzipped form as 0709.0517.tar.gz with size 344kb.
The corresponding postcript file has gzipped size 173kb.
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This is an announcement for the paper "Counterexamples for the
convexity of certain matricial inequalities" by Marius Junge and
Quanhua Xu.
Abstract: In \cite{CL} Carlen and Lieb considered Minkowski type
inequalities in the context of operators on a Hilbert space. More
precisely, they considered the homogenous expression
\[ f_{pq}(x_1,...,x_n) \lel \big(tr\big((\sum_{k=1}^n
x_k^q)^{p/q}\big)\big)^{1/p} \pl \] defined for positive matrices.
The concavity for $q=1$ and $p<1$ yields
strong subadditivity for quantum entropy. We discuss the convexity
of $f_{pq}$ and show that, contrary to the commutative case, there
exists a $q_0>1$ such that $f_{1q}$ is not convex for all $1<q<q_0$.
This is achieved by constructing a family of interesting channels
on $2\times 2$ matrices.
Archive classification: math.FA math-ph math.MP
Mathematics Subject Classification: 46L25 15A48
The source file(s), cedriv.tex: 58533 bytes, is(are) stored in
gzipped form as 0709.0433.gz with size 18kb. The corresponding
postcript file has gzipped size 129kb.
Submitted from: junge(a)math.uiuc.edu
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