297,439 research outputs found
Canonical lossless state-space systems: Staircase forms and the Schur algorithm
A new finite atlas of overlapping balanced canonical forms for multivariate
discrete-time lossless systems is presented. The canonical forms have the
property that the controllability matrix is positive upper triangular up to a
suitable permutation of its columns. This is a generalization of a similar
balanced canonical form for continuous-time lossless systems. It is shown that
this atlas is in fact a finite sub-atlas of the infinite atlas of overlapping
balanced canonical forms for lossless systems that is associated with the
tangential Schur algorithm; such canonical forms satisfy certain interpolation
conditions on a corresponding sequence of lossless transfer matrices. The
connection between these balanced canonical forms for lossless systems and the
tangential Schur algorithm for lossless systems is a generalization of the same
connection in the SISO case that was noted before. The results are directly
applicable to obtain a finite sub-atlas of multivariate input-normal canonical
forms for stable linear systems of given fixed order, which is minimal in the
sense that no chart can be left out of the atlas without losing the property
that the atlas covers the manifold
Graphical criteria for positive solutions to linear systems
We study linear systems of equations with coefficients in a generic partially
ordered ring and a unique solution, and seek conditions for the solution to
be nonnegative, that is, every component of the solution is a quotient of two
nonnegative elements in . The requirement of a nonnegative solution arises
typically in applications, such as in biology and ecology, where quantities of
interest are concentrations and abundances. We provide novel conditions on a
labeled multidigraph associated with the linear system that guarantee the
solution to be nonnegative. Furthermore, we study a generalization of the first
class of linear systems, where the coefficient matrix has a specific block form
and provide analogous conditions for nonnegativity of the solution, similarly
based on a labeled multidigraph. The latter scenario arises naturally in
chemical reaction network theory, when studying full or partial
parameterizations of the positive part of the steady state variety of a
polynomial dynamical system in the concentrations of the molecular species
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Differentially Positive Systems
The paper introduces and studies differentially positive systems, that is,
systems whose linearization along an arbitrary trajectory is positive. A
generalization of Perron Frobenius theory is developed in this differential
framework to show that the property induces a (conal) order that strongly
constrains the asymptotic behavior of solutions. The results illustrate that
behaviors constrained by local order properties extend beyond the well-studied
class of linear positive systems and monotone systems, which both require a
constant cone field and a linear state space.The research was supported by the Fund for Scientific Research FNRS and by the Engineering and Physical Sciences Research Council under Grant EP/G066477/1.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/TAC.2015.243752
Unextendible product bases and extremal density matrices with positive partial transpose
In bipartite quantum systems of dimension 3x3 entangled states that are
positive under partial transposition (PPT) can be constructed with the use of
unextendible product bases (UPB). As discussed in a previous publication all
the lowest rank entangled PPT states of this system seem to be equivalent,
under special linear product transformations, to states that are constructed in
this way. Here we consider a possible generalization of the UPB constuction to
low-rank entangled PPT states in higher dimensions. The idea is to give up the
condition of orthogonality of the product vectors, while keeping the relation
between the density matrix and the projection on the subspace defined by the
UPB. We examine first this generalization for the 3x3 system where numerical
studies indicate that one-parameter families of such generalized states can be
found. Similar numerical searches in higher dimensional systems show the
presence of extremal PPT states of similar form. Based on these results we
suggest that the UPB construction of the lowest rank entangled states in the
3x3 system can be generalized to higher dimensions, with the use of
non-orthogonal UPBs.Comment: 23 pages, 2 figures, 1 table. V2: Fixed fig.1 not showin
Positively regular vague matrices
AbstractPositive regularity is a common attribute of inaccurate square matrices which can be used in linear equation systems that provide only nonnegative solutions. It is studied within the framework of vague matrices which can be considered as a generalization of interval matrices. Criteria of positive regularity are derived and a method of verifying them is outlined. The exposition concludes with a characterization of the radius of positive regularity
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