247 research outputs found
A Semidefinite Approach for Truncated K-Moment Problems
A truncated moment sequence (tms) of degree d is a vector indexed by
monomials whose degree is at most d. Let K be a semialgebraic set.The truncated
K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure
supported? This paper proposes a semidefinite programming (SDP) approach for
solving TKMP. When K is compact, we get the following results: whether a tms y
of degree d admits a K-measure or notcan be checked via solving a sequence of
SDP problems; when y admits no K-measure, a certificate will be given; when y
admits a K-measure, a representing measure for y would be obtained from solving
the SDP under some necessary and some sufficient conditions. Moreover, we also
propose a practical SDP method for finding flat extensions, which in our
numerical experiments always finds a finitely atomic representing measure for a
tms when it admits one
The possible shapes of numerical ranges
Which convex subsets of the complex plane are the numerical range W(A of some
matrix A? This paper gives a precise characterization of these sets. In
addition to this we show that for any A there exists a symmetric matrix B of
the same size such that W(A)=W(B).Comment: 4 page
Matrix Convex Hulls of Free Semialgebraic Sets
This article resides in the realm of the noncommutative (free) analog of real
algebraic geometry - the study of polynomial inequalities and equations over
the real numbers - with a focus on matrix convex sets and their projections
. A free semialgebraic set which is convex as well as bounded and open
can be represented as the solution set of a Linear Matrix Inequality (LMI), a
result which suggests that convex free semialgebraic sets are rare. Further,
Tarski's transfer principle fails in the free setting: The projection of a free
convex semialgebraic set need not be free semialgebraic. Both of these results,
and the importance of convex approximations in the optimization community,
provide impetus and motivation for the study of the free (matrix) convex hull
of free semialgebraic sets.
This article presents the construction of a sequence of LMI domains
in increasingly many variables whose projections are
successively finer outer approximations of the matrix convex hull of a free
semialgebraic set . It is based on free analogs of
moments and Hankel matrices. Such an approximation scheme is possibly the best
that can be done in general. Indeed, natural noncommutative transcriptions of
formulas for certain well known classical (commutative) convex hulls does not
produce the convex hulls in the free case. This failure is illustrated on one
of the simplest free nonconvex .
A basic question is which free sets are the projection of a free
semialgebraic set ? Techniques and results of this paper bear upon this
question which is open even for convex sets.Comment: 41 pages; includes table of contents; supplementary material (a
Mathematica notebook) can be found at
http://www.math.auckland.ac.nz/~igorklep/publ.htm
Geometry of free loci and factorization of noncommutative polynomials
The free singularity locus of a noncommutative polynomial f is defined to be
the sequence of hypersurfaces. The main
theorem of this article shows that f is irreducible if and only if is
eventually irreducible. A key step in the proof is an irreducibility result for
linear pencils. Apart from its consequences to factorization in a free algebra,
the paper also discusses its applications to invariant subspaces in
perturbation theory and linear matrix inequalities in real algebraic geometry.Comment: v2: 32 pages, includes a table of content
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