212 research outputs found
On the exponential transform of lemniscates
It is known that the exponential transform of a quadrature domain is a
rational function for which the denominator has a certain separable form. In
the present paper we show that the exponential transform of lemniscate domains
in general are not rational functions, of any form. Several examples are given
to illustrate the general picture. The main tool used is that of polynomial and
meromorphic resultants.Comment: 19 pages, to appear in the Julius Borcea Memorial Volume, (eds.
Petter Branden, Mikael Passare and Mihai Putinar), Trends in Mathematics,
Birkhauser Verla
Positivity and optimization for semi-algebraic functions
We describe algebraic certificates of positivity for functions belonging to a
finitely generated algebra of Borel measurable functions, with particular
emphasis to algebras generated by semi-algebraic functions. In which case the
standard global optimization problem with constraints given by elements of the
same algebra is reduced via a natural change of variables to the better
understood case of polynomial optimization. A collection of simple examples and
numerical experiments complement the theoretical parts of the article.Comment: 20 page
Norm estimates of complex symmetric operators applied to quantum systems
This paper communicates recent results in theory of complex symmetric
operators and shows, through two non-trivial examples, their potential
usefulness in the study of Schr\"odinger operators. In particular, we propose a
formula for computing the norm of a compact complex symmetric operator. This
observation is applied to two concrete problems related to quantum mechanical
systems. First, we give sharp estimates on the exponential decay of the
resolvent and the single-particle density matrix for Schr\"odinger operators
with spectral gaps. Second, we provide new ways of evaluating the resolvent
norm for Schr\"odinger operators appearing in the complex scaling theory of
resonances
Nearly Subnormal Operators and Moment Problems
AbstractWe use separation-of-cones techniques and ideas from multivariable operator theory to show that polynomial hyponormality does not imply subnormality for Hilbert space operators. As an application, we obtain a new result in the theory of power moments in two dimensions
Maximal quadratic modules on *-rings
We generalize the notion of and results on maximal proper quadratic modules
from commutative unital rings to -rings and discuss the relation of this
generalization to recent developments in noncommutative real algebraic
geometry. The simplest example of a maximal proper quadratic module is the cone
of all positive semidefinite complex matrices of a fixed dimension. We show
that the support of a maximal proper quadratic module is the symmetric part of
a prime -ideal, that every maximal proper quadratic module in a
Noetherian -ring comes from a maximal proper quadratic module in a simple
artinian ring with involution and that maximal proper quadratic modules satisfy
an intersection theorem. As an application we obtain the following extension of
Schm\" udgen's Strict Positivstellensatz for the Weyl algebra: Let be an
element of the Weyl algebra which is not negative semidefinite
in the Schr\" odinger representation. It is shown that under some conditions
there exists an integer and elements such
that is a finite sum of hermitian squares. This
result is not a proper generalization however because we don't have the bound
.Comment: 11 page
The Moment Problem for Continuous Positive Semidefinite Linear functionals
Let be a locally convex topology on the countable dimensional
polynomial -algebra \rx:=\reals[X_1,...,X_n]. Let be a closed
subset of , and let be a finitely generated
quadratic module in \rx. We investigate the following question: When is the
cone \Pos(K) (of polynomials nonnegative on ) included in the closure of
? We give an interpretation of this inclusion with respect to representing
continuous linear functionals by measures. We discuss several examples; we
compute the closure of M=\sos with respect to weighted norm- topologies.
We show that this closure coincides with the cone \Pos(K) where is a
certain convex compact polyhedron.Comment: 14 page
Matrix compression along isogenic blocks
AbstractA matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation that Hadamard entrywise functional calculus preserves isogenic blocks has already proved to be of paramount importance for thresholding large correlation matrices. The proposed isogenic stratification of the set of complex matrices bears similarities to the Schubert cell stratification of a homogeneous algebraic manifold. An array of potential applications to current investigations in computational matrix analysis is briefly mentioned, touching concepts such as symmetric statistical models, hierarchical matrices and coherent matrix organization induced by partition trees.</jats:p
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