This is an announcement for the paper "Parallelogram norm" by Madjid
Mirzavaziri and Mohammad Sal Moslehian.
Abstract: Replacing the triangle inequality by \|x+y\|^2\leq 2(\|x\|^2 +
\|y\|^2) in the definition of norm we obtain the notion of parallelogram
norm. We establish that every parallelogram norm is a norm in the
usual sense.
Archive classification: Functional Analysis
Mathematics Subject Classification: 46B20; 46C05
Remarks: 3 pages
The source file(s), Paral1.tex: 4582 bytes, is(are) stored in gzipped
form as 0503616.gz with size 2kb. The corresponding postcript file has
gzipped size 27kb.
Submitted from: msalm(a)math.um.ac.ir
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This is an announcement for the paper "Sampling from large matrices:
an approach through geometric functional analysis" by Mark Rudelson
and Roman Vershynin.
Abstract: We study random submatrices of a large matrix A. We show how to
approximately compute A from its random submatrix. This improves known
algorithms for computing low-rank approximations of large matrices. We
also estimate norms of random submatrices of A. This yields an improved
approximation algorithm for all MAX-2CSP problems (which includes MAX-CUT
and other graph problems). Our results are essentially dimension-free;
the picture is only controlled by the norms of the matrix and not by
its size or rank. We use methods of Probability in Banach spaces, in
particular the law of large numbers for random operators.
Archive classification: Functional Analysis; Numerical Analysis
Mathematics Subject Classification: 15A60, 68W20, 15A18
The source file(s), rv-random-submatrices.tex: 50699 bytes, is(are)
stored in gzipped form as 0503442.gz with size 16kb. The corresponding
postcript file has gzipped size 82kb.
Submitted from: vershynin(a)math.ucdavis.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.FA/0503442
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http://arXiv.org/abs/math.FA/0503442
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This is an announcement for the paper "Centroids and comparison of
volumes" by V.Yaskin and M.Yaskina.
Abstract: For $-1<p<1$ we introduce the concept of a polar $p$-centroid
body ${\Gamma^*_p K}$ of a star body $K$. We consider the question of
whether ${\Gamma^*_p K}\subset {\Gamma^*_p L}$ implies $\mathrm{vol}(L)\le
\mathrm{vol}(K).$ Our results extend the studies by Lutwak in the case
$p=1$ and Grinberg, Zhang in the case $p> 1$.
Archive classification: Functional Analysis
Mathematics Subject Classification: 52Axx
Remarks: 18 pages
The source file(s), centr.tex: 51970 bytes, is(are) stored in gzipped
form as 0503290.gz with size 13kb. The corresponding postcript file has
gzipped size 71kb.
Submitted from: yaskinv(a)math.missouri.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.FA/0503290
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http://arXiv.org/abs/math.FA/0503290
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This is an announcement for the paper "A solution to the lower dimensional
Busemann-Petty problem in the hyperbolic space" by V.Yaskin.
Abstract: The lower dimensional Busemann-Petty problem asks whether
origin symmetric convex bodies in $\mathbb{R}^n$ with smaller volume
of all $k$-dimensional sections necessarily have smaller volume. As
proved by Bourgain and Zhang, the answer to this question is negative
if $k>3$. The problem is still open for $k=2,3$. In this article we
formulate and completely solve the lower dimensional Busemann-Petty
problem in the hyperbolic space $\mathbb{H}^n$.
Archive classification: Functional Analysis; Metric Geometry
Mathematics Subject Classification: 52A55, 52A20, 46B20
Remarks: 12 pages, 2 figures
The source file(s), LDHBP.tex: 70816 bytes, pic04.eps: 9457 bytes,
pic06.eps: 9542 bytes, is(are) stored in gzipped form as 0503289.tar.gz
with size 25kb. The corresponding postcript file has gzipped size 59kb.
Submitted from: yaskinv(a)math.missouri.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.FA/0503289
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http://arXiv.org/abs/math.FA/0503289
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This is an announcement for the paper "On the intrinsic and the spatial
numerical range" by Miguel Martin, Javier Meri and Rafael Paya.
Abstract: For a bounded function $f$ from the unit sphere of a closed
subspace $X$ of a Banach space $Y$, we study when the closed convex
hull of its spatial numerical range $W(f)$ is equal to its intrinsic
numerical range $V(f)$. We show that for every infinite-dimensional
Banach space $X$ there is a superspace $Y$ and a bounded linear operator
$T:X\longrightarrow Y$ such that $\ecc W(T)\neq V(T)$. We also show
that, up to renormig, for every non-reflexive Banach space $Y$, one can
find a closed subspace $X$ and a bounded linear operator $T\in L(X,Y)$
such that $\ecc W(T)\neq V(T)$.
Finally, we introduce a sufficient condition for the closed convex
hull of the spatial numerical range to be equal to the intrinsic numerical
range, which we call the Bishop-Phelps-Bollobas property, and which is
weaker than the uniform smoothness and the finite-dimensionality. We
characterize strong subdifferentiability and uniform smoothness in terms
of this property.
Archive classification: Functional Analysis
Mathematics Subject Classification: 46B20; 47A12
Remarks: 12 pages
The source file(s), MartinMeriPaya.tex: 40725 bytes, is(are) stored in
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file has gzipped size 70kb.
Submitted from: mmartins(a)ugr.es
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