150 research outputs found
Symmetric Subresultants and Applications
Schur's transforms of a polynomial are used to count its roots in the unit
disk. These are generalized them by introducing the sequence of symmetric
sub-resultants of two polynomials. Although they do have a determinantal
definition, we show that they satisfy a structure theorem which allows us to
compute them with a type of Euclidean division. As a consequence, a fast
algorithm based on a dichotomic process and FFT is designed. We prove also that
these symmetric sub-resultants have a deep link with Toeplitz matrices.
Finally, we propose a new algorithm of inversion for such matrices. It has the
same cost as those already known, however it is fraction-free and consequently
well adapted to computer algebra
Nullity Invariance for Pivot and the Interlace Polynomial
We show that the effect of principal pivot transform on the nullity values of
the principal submatrices of a given (square) matrix is described by the
symmetric difference operator (for sets). We consider its consequences for
graphs, and in particular generalize the recursive relation of the interlace
polynomial and simplify its proof.Comment: small revision of Section 8 w.r.t. v2, 14 pages, 6 figure
Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
We develop a new method for studying the asymptotics of symmetric polynomials
of representation-theoretic origin as the number of variables tends to
infinity. Several applications of our method are presented: We prove a number
of theorems concerning characters of infinite-dimensional unitary group and
their -deformations. We study the behavior of uniformly random lozenge
tilings of large polygonal domains and find the GUE-eigenvalues distribution in
the limit. We also investigate similar behavior for alternating sign matrices
(equivalently, six-vertex model with domain wall boundary conditions). Finally,
we compute the asymptotic expansion of certain observables in dense
loop model.Comment: Published at http://dx.doi.org/10.1214/14-AOP955 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their q-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit.We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in O(n=1) dense loop model
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Hilbert Modules and Complex Geometry
The major topics discussed in the workshop were Hilbert modules of analytic functions on domains in Cn, Toeplitz and Hankel operators, reproducing kernel Hilbert spaces and multiplier algebras, the interplay of complex geometry and operator theory, non-commutative function theory and operator theory, Hilbert bundles on symmetric spaces
Design of multidimensional digital filters by spectral transformations
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Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability
Continuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies
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