This is an announcement for the paper "Inequalities for mixed $p$-affine
surface area" by Elisabeth Werner and Deping Ye.
Abstract: We prove new Alexandrov-Fenchel type inequalities and new
affine isoperimetric inequalities for mixed $p$-affine surface areas. We
introduce a new class of bodies, the illumination surface bodies, and
establish some of their properties. We show, for instance, that they are
not necessarily convex. We give geometric interpretations of $L_p$ affine
surface areas, mixed $p$-affine surface areas and other functionals via
these bodies. The surprising new element is that not necessarily convex
bodies provide the tool for these interpretations.
Archive classification: math.MG math.FA
Mathematics Subject Classification: 52A20, 53A15
Remarks: 39 pages
The source file(s), MixedLp.tex: 97032 bytes, is(are) stored in gzipped
form as 0812.4550.gz with size 26kb. The corresponding postcript file
has gzipped size 162kb.
Submitted from: elisabeth.werner(a)case.edu
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http://front.math.ucdavis.edu/0812.4550
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This is an announcement for the paper "Operator algebras with unique
preduals" by Kenneth R. Davidson and Alex Wright.
Abstract: We show that every free semigroup algebras has a (strongly)
unique Banach space predual. We also provide a new simpler proof that a
weak*-closed unital operator operator algebra containing a weak* dense
subalgebra of compact operators has a unique Banach space predual.
Archive classification: math.OA math.FA
Mathematics Subject Classification: 47L50; 46B04; 47L35
Remarks: 13 pages
The source file(s), DavidsonWright3a.tex: 44578 bytes, is(are) stored in
gzipped form as 0812.3159.gz with size 14kb. The corresponding postcript
file has gzipped size 96kb.
Submitted from: krdavids(a)uwaterloo.ca
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/0812.3159
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http://arXiv.org/abs/0812.3159
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This is an announcement for the paper "Spectral norm of products of
random and deterministic matrices" by Roman Vershynin.
Abstract: We study the spectral norm of matrices M that can be factored
as M=BA, where A is a random matrix with independent mean zero entries,
and B is a fixed matrix. Under the (4+epsilon)-th moment assumption on the
entries of A, we show that the spectral norm of such an m by n matrix M is
bounded by \sqrt{m} + \sqrt{n}, which is sharp. In other words, in regard
to the spectral norm, products of random and deterministic matrices behave
similarly to random matrices with independent entries. This result along
with the previous work of M. Rudelson and the author implies that the
smallest singular value of a random m times n matrix with i.i.d. mean zero
entries and bounded (4+epsilon)-th moment is bounded below by \sqrt{m}
- \sqrt{n-1} with high probability.
Archive classification: math.PR math.FA
Mathematics Subject Classification: 15A52; 46B09
Remarks: 34 pages, no figures
The source file(s), product-random-deterministic.tex: 81516 bytes, is(are)
stored in gzipped form as 0812.2432.gz with size 22kb. The corresponding
postcript file has gzipped size 147kb.
Submitted from: romanv(a)umich.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/0812.2432
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http://arXiv.org/abs/0812.2432
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This is an announcement for the paper "On a new type concept for Banach
spaces" by Robin Nittka.
Abstract: In order to measure qualitative properties we introduce a
new notion of a type for arbitrary normed spaces which measures the
worst possible growth of partial sums of sequences weakly converging
to zero. As an application we give a new proof that certain classical
Banach spaces are non-isomorphic.
Archive classification: math.FA
Mathematics Subject Classification: 46B20
Remarks: 7 pages
The source file(s), type.bbl: 1571 bytes type.tex: 27062 bytes,
is(are) stored in gzipped form as 0812.2216.tar.gz with size 9kb. The
corresponding postcript file has gzipped size 75kb.
Submitted from: robin.nittka(a)uni-ulm.de
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/0812.2216
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http://arXiv.org/abs/0812.2216
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This is an announcement for the paper "Factorization theorems for
dominated polynomials" by Geraldo Botelho, Daniel Pellegrino and Pilar
Rueda.
Abstract: In this note we prove that the factorization theorem for
dominated polynomials previously proved by the authors is equivalent to
an alternative factorization scheme that uses classical linear techniques
and a linearization process. However, this alternative scheme is shown
not to be satisfactory until the equivalence is proved.
Archive classification: math.FA
Mathematics Subject Classification: 46G25
The source file(s), II-Factorization2dic08.tex: 15703 bytes, is(are)
stored in gzipped form as 0812.1401.gz with size 5kb. The corresponding
postcript file has gzipped size 62kb.
Submitted from: dmpellegrino(a)gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/0812.1401
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http://arXiv.org/abs/0812.1401
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This is an announcement for the paper "Comparison of volumes of convex
bodies in real, complex, and quaternionic spaces" by Boris Rubin.
Abstract: The classical Busemann-Petty problem (1956) asks, whether
origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane
central sections necessarily have smaller volumes. It is known, that
the answer is affirmative if $n\le 4$ and negative if $n>4$. The same
question can be asked when volumes of hyperplane sections are replaced
by more general comparison functions. We give unified exposition of this
circle of problems in real, complex, and quaternionic $n$-dimensional
spaces. All cases are treated simultaneously. In particular, we show
that the Busemann-Petty problem in the quaternionic $n$-dimensional
space has an affirmative answer if and only if $n =2$. The method relies
on the properties of cosine transforms on the unit sphere. Possible
generalizations for spaces over Clifford algebras are discussed.
Archive classification: math.FA
Mathematics Subject Classification: 44A12; 52A38
Remarks: 38 pages
The source file(s), quaternion3.tex: 107627 bytes, is(are) stored in
gzipped form as 0812.1300.gz with size 35kb. The corresponding postcript
file has gzipped size 182kb.
Submitted from: borisr(a)math.lsu.edu
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http://front.math.ucdavis.edu/0812.1300
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http://arXiv.org/abs/0812.1300
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This is an announcement for the paper "Unconditional bases and strictly
convex dual renormings" by R. J. Smith and S. Troyanski.
Abstract: We present equivalent conditions for a space $X$ with an
unconditional basis to admit an equivalent norm with a strictly convex
dual norm.
Archive classification: math.FA
Mathematics Subject Classification: 46B03; 46B26; 46B15
The source file(s), unc_basis_dual_sc.tex: 45412 bytes, is(are) stored in
gzipped form as 0811.4685.gz with size 14kb. The corresponding postcript
file has gzipped size 101kb.
Submitted from: smith(a)math.cas.cz
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http://front.math.ucdavis.edu/0811.4685
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http://arXiv.org/abs/0811.4685
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