370 research outputs found
On Frobenius structures in symmetric cones
We prove that in any strictly convex symmetric cone there exists a
non empty locus where the WDVV equation is satisfied (i.e. there exists a
hyperplane being a Frobenius manifold). This result holds over any real
division algebra (with a restriction to the rank 3 case if we consider the
field ) but also on their linear combinations. This theorem holds
as well in the case of pseudo-Riemannian geometry, in particular for a Lorentz
symmetric cone of Anti-de-Sitter type. Our statement can be considered as a
generalisation of a result by Ferapontov--Kruglikov--Novikov and Mokhov. Our
construction is achieved by merging two different approaches: an
algebraic/geometric one and the analytic approach given by Calabi in his
investigations on the Monge--Amp\`ere equation for the case of affine
hyperspheres
Order isomorphisms between cones of JB-algebras
In this paper we completely describe the order isomorphisms between cones of
atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an
algebraic direct summand of the so-called engaged and disengaged part. On the
cone of the engaged part every order isomorphism is linear and the disengaged
part consists only of copies of . Furthermore, in the setting of
general JB-algebras we prove the following. If either algebra does not contain
an ideal of codimension one, then every order isomorphism between their cones
is linear if and only if it extends to a homeomorphism, between the cones of
the atomic part of their biduals, for a suitable weak topology
Strong duality in conic linear programming: facial reduction and extended duals
The facial reduction algorithm of Borwein and Wolkowicz and the extended dual
of Ramana provide a strong dual for the conic linear program in the absence of any constraint qualification. The facial
reduction algorithm solves a sequence of auxiliary optimization problems to
obtain such a dual. Ramana's dual is applicable when (P) is a semidefinite
program (SDP) and is an explicit SDP itself. Ramana, Tuncel, and Wolkowicz
showed that these approaches are closely related; in particular, they proved
the correctness of Ramana's dual using certificates from a facial reduction
algorithm.
Here we give a clear and self-contained exposition of facial reduction, of
extended duals, and generalize Ramana's dual:
-- we state a simple facial reduction algorithm and prove its correctness;
and
-- building on this algorithm we construct a family of extended duals when
is a {\em nice} cone. This class of cones includes the semidefinite cone
and other important cones.Comment: A previous version of this paper appeared as "A simple derivation of
a facial reduction algorithm and extended dual systems", technical report,
Columbia University, 2000, available from
http://www.unc.edu/~pataki/papers/fr.pdf Jonfest, a conference in honor of
Jonathan Borwein's 60th birthday, 201
Dualities and positivity in the study of quantum entanglement
We present a survey on mathematical topics relating to separable states and
entanglement witnesses. The convex cone duality between separable states and
entanglement witnesses is discussed and later generalized to other families of
operators, leading to their characterization via multiplicative properties. The
condition for an operator to be an entanglement witness is rephrased as a
problem of positivity of a family of real polynomials. By solving the latter in
a specific case of a three-parameter family of operators, we obtain explicit
description of entanglement witnesses belonging to that family. A related
problem of block positivity over real numbers is discussed. We also consider a
broad family of block positivity tests and prove that they can never be
sufficient, which should be useful in case of future efforts in that direction.
Finally, we introduce the concept of length of a separable state and present
new results concerning relationships between the length and Schmidt rank. In
particular, we prove that separable states of length lower of equal 3 have
Schmidt ranks equal to their lengths. We also give an example of a state which
has length 4 and Schmidt rank 3.Comment: A shortened and amended version of author's Master's Thesi
Real Algebraic Geometry With A View Toward Systems Control and Free Positivity
New interactions between real algebraic geometry, convex optimization and free non-commutative geometry have recently emerged, and have been the subject of numerous international meetings. The aim of the workshop was to bring together experts, as well as young researchers, to investigate current key questions at the interface of these fields, and to explore emerging interdisciplinary applications
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