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## Computing matrix symmetrizers. Part 2: new methods using eigendata and linear means; a comparison

Fonte: Elsevier
Publicador: Elsevier

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article

Publicado em 10/07/2015
ENG

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#Symmetric matrix factorization#symmetrizer#symmetrizer computation#eigenvalue method#linear equation#principal subspace computation#matrix optimization#numerical algorithm#MATLAB code#Matemáticas

Over any field F every square matrix A can be factored into the product of two symmetric matrices as A = S1 . S2 with S_i = S_i^T ∈ F^(n,n) and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910. Frobenius’ symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such that SA is symmetric, was inspired by an iterative linear systems algorithm of Huang and Nong (2010) in 2013 [29, 30]. The resulting iterative algorithm has solved this computational problem over R and C, but at high computational cost. This paper develops and tests another linear equations solver, as well as eigen- and principal vector or Schur Normal Form based algorithms for solving the matrix symmetrizer problem numerically. Four new eigendata based algorithms use, respectively, SVD based principal vector chain constructions, Gram-Schmidt orthogonalization techniques, the Arnoldi method, or the Schur Normal Form of A in their formulations. They are helped by Datta’s 1973 method that symmetrizes unreduced Hessenberg matrices directly. The eigendata based methods work well and quickly for generic matrices A and create well conditioned matrix symmetrizers through eigenvector dyad accumulation. But all of the eigen based methods have differing deficiencies with matrices A that have ill-conditioned or complicated eigen structures with nontrivial Jordan normal forms. Our symmetrizer studies for matrices with ill-conditioned eigensystems lead to two open problems of matrix optimization.; This research was partially supported by the Ministerio de Economía y Competitividad of Spain through the research grant MTM2012-32542.

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## Symmetrizers and antisymmetrizers for the BMW algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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Let $n\in\mathds{N}$ and $B_n(r,q)$ be the generic Birman-Murakami-Wenzl
algebra with respect to indeterminants $r$ and $q$. It is known that $B_n(r,q)$
has two distinct linear representations generated by two central elements of
$B_n(r,q)$ called the symmetrizer and antisymmetrizer of $B_n(r,q)$. These
generate for $n\geq 3$ the only one dimensional two sided ideals of $B_n(r,q)$
and generalize the corresponding notion for Hecke algebras of type $A$. The
main result in this paper explicitly determines the coefficients of these
elements with respect to the graphical basis of $B_n(r,q)$.

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## The well-posedness issue in Sobolev spaces for hyperbolic systems with Zygmund-type coefficients

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/04/2014

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In this paper we study the well-posedness of the Cauchy problem for first
order hyperbolic systems with constant multiplicities and with low regularity
coefficients depending just on the time variable. We consider Zygmund and
log-Zygmund type assumptions, and we prove well-posedness in $H^\infty$
respectively without loss and with finite loss of derivatives. The key to
obtain the results is the construction of a suitable symmetrizer for our
system, which allows us to recover energy estimates (with or without loss) for
the hyperbolic operator under consideration. This can be achievied, in contrast
with the classical case of systems with smooth (say Lipschitz) coefficients, by
adding one step in the diagonalization process, and building the symmetrizer up
to the second order.; Comment: submitted

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## Rational representations of Yangians associated with skew Young diagrams

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Representation Theory#Mathematics - Combinatorics#Mathematics - Quantum Algebra#17B37#20C30#22E46#81R50

Let $GL_M$ be general linear Lie group over the complex field. The
irreducible rational representations of the group $GL_M$ are labeled by pairs
of partitions $\mu$ and $\tilde\mu$ such that the total number of non-zero
parts of $\mu$ and $\tilde{\mu}$ does not exceed $M$. Let $U$ be the
representation of $GL_M$ corresponding to such a pair. Regard the direct
product $GL_N\times GL_M$ as a subgroup of $GL_{N+M}$. Let $V$ be the
irreducible rational representation of the group $GL_{N+M}$ corresponding to a
pair of partitions $\lambda$ and $\tilde{\lambda}$. Consider the vector space
$W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $GL_N$. Let $n$
be sum of parts of $\lambda$ less the sum of parts of $\mu$. Let $\tilde{n}$ be
sum of parts of $\tilde{\lambda}$ less the sum of parts of $\tilde{\mu}$. For
any choice of two standard Young tableaux of skew shapes $\lambda/\mu$ and
$\tilde{\lambda}/\tilde{\mu}$ respectively, we realize $W$ as a subspace in the
tensor product of $n$ copies of the defining $N$-dimensional representation of
$GL_N$, and of $\tilde{n}$ copies of the contragredient representation. This
subspace is determined as the image of a certain linear operator $F$ in the
tensor product, given by explicit multiplicative formula. When M=0 and $W=V$ is
an irreducible representation of $GL_N$...

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## Products of Young symmetrizers and ideals in the generic tensor algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We describe a formula for computing the product of the Young symmetrizer of a
Young tableau with the Young symmetrizer of a subtableau, generalizing the
classical quasi-idempotence of Young symmetrizers. We derive some consequences
to the structure of ideals in the generic tensor algebra and its partial
symmetrizations. Instances of these generic algebras appear in the work of Sam
and Snowden on twisted commutative algebras, as well as in the work of the
author on the defining ideals of secant varieties of Segre-Veronese varieties,
and in joint work of Oeding and the author on the defining ideals of tangential
varieties of Segre-Veronese varieties.; Comment: v2: minor changes, last section moved before the proofs sections, to
appear in Journal of Algebraic Combinatorics

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## Symmetrization of Bernoulli

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/01/2006

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Let X be a random variable. We shall call an independent random variable Y to
be a symmetrizer for X, if X+Y is symmetric around zero. A random variable is
said to be symmetry resistant if the variance of any symmetrizer Y, is never
smaller than the variance of X itself. We prove that a Bernoulli(p) random
variable is symmetry resistant if and only if p is not 1/2. This is an old
problem proved in 1999 by Kagan, Mallows, Shepp, Vanderbei & Vardi using linear
programming principles. We reprove it here using completely probabilistic tools
using Skorokhod embedding and Ito's rule.; Comment: 3 pages; a completely probabilistic proof of a theorem due to Kagan,
Mallows, Shepp, Vanderbei & Vardi

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## Strongly primitive species with potentials I: Mutations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/06/2013

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#Mathematics - Rings and Algebras#Mathematics - Representation Theory#Primary 16G10, Secondary 16G20, 13F60

Motivated by the mutation theory of quivers with potentials developed by
Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster
algebras it provides, we propose a mutation theory of species with potentials
for species that arise from skew-symmetrizable matrices that admit a
skew-symmetrizer with pairwise coprime diagonal entries. The class of
skew-symmetrizable matrices covered by the mutation theory proposed here
contains a class of matrices that do not admit global unfoldings, that is,
unfoldings compatible with all possible sequences of mutations.; Comment: 51 pages

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## Algorithms and Properties for Positive Symmetrizable Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/03/2015

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Matrices are the most common representations of graphs. They are also used
for the representation of algebras and cluster algebras. This paper shows some
properties of matrices in order to facilitate the understanding and locating
symmetrizable matrices with specific characteristics, called positive
quasi-Cartan companion matrices. Here, symmetrizable matrix are those which are
symmetric when multiplied by a diagonal matrix with positive entries called
symmetrizer matrix. Four algorithms are developed: one to decide whether there
is a symmetrizer matrix; second to find such symmetrizer matrix; another to
decide whether the matrix is positive or not; and the last to find a positive
quasi-Cartan companion matrix, if there exists. The third algorithm is used to
prove that the problem to decide if a matrix has a positive quasi-Cartan
companion is NP.; Comment: 10 pages, submitted to International Journal of Applied Mathmatics
(IJAM)

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## An improved result for the full justification of asymptotic models for the propagation of internal waves

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We consider here asymptotic models that describe the propagation of
one-dimensional internal waves at the interface between two layers of
immiscible fluids of different densities, under the rigid lid assumption and
with uneven bottoms. The aim of this paper is to show that the full
justification result of the model obtained by Duch\^ene, Israwi and Talhouk
[{\em SIAM J. Math. Anal.}, 47(1), 240--290], in the sense that it is
consistent, well-posed, and that its solutions remain close to exact solutions
of the full Euler system with corresponding initial data, can be improved in
two directions. The first direction is taking into account medium amplitude
topography variations and the second direction is allowing strong nonlinearity
using a new pseudo-symmetrizer, thus canceling out the smallness assumptions of
the Camassa-Holm regime for the well-posedness and stability results.; Comment: arXiv admin note: substantial text overlap with arXiv:1304.4554; text
overlap with arXiv:1208.6394 by other authors

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## Representations of twisted Yangians associated with skew Young diagrams

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Representation Theory#Mathematics - Combinatorics#Mathematics - Quantum Algebra#17B35#17B37#20C30#22E46#81R50

Let $G_M$ be one of the complex Lie groups $O_M$ and $Sp_M$. The irreducible
finite-dimensional representations of the group $G_M$ are labeled by partitions
$\mu$ satisfying certain extra conditions. Let $U$ be the representation of
$G_M$ corresponding to $\mu$. Regard the direct product $G_N\times G_M$ as a
subgroup of $G_{N+M}$. Let $V$ be the irreducible representation of $G_{N+M}$
corresponding to a partition $\lambda$. Consider the vector space
$W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $G_N$. Let $n$
be sum of parts of $\lambda$ less the sum of parts of $\mu$. For any choice of
a standard Young tableau of skew shape $\lambda/\mu$, we realize $W$ as a
subspace in the tensor product of $n$ copies of the defining $N$-dimensional
representation of $G_N$. This subspace is determined as the image of a certain
linear operator $F(M)$ in the tensor product, given by an explicit formula.
When M=0 and $W=V$ is an irreducible representation of $G_N$, we recover the
classical realization of $V$ as a subspace in the space of all traceless
tensors. Then the operator F(0) can be regarded as the analogue for $G_N$ of
the Young symmetrizer, corresponding to the chosen standard tableau of shape
$\lambda$. Even in the special case M=0...

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## Convergence of Singular Limits for Multi-D Semilinear Hyperbolic Systems to Parabolic Systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/07/2002

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In this paper we investigate the zero-relaxation limit of the following
multi-D semilinear hyperbolic system in pseudodifferential form:
W_{t}(x,t) + (1/epsilon) A(x,D) W(x,t) = (1/epsilon^2) B(x,W(x,t)) +
(1/epsilon) D(W(x,t)) + E(W(x,t)).
We analyse the singular convergence, as epsilon tends to 0, in the case which
leads to a limit system of parabolic type. The analysis is carried out by using
the following steps:
(i) We single out algebraic ``structure conditions'' on the full system,
motivated by formal asymptotics, by some examples of discrete velocity models
in kinetic theories.
(ii) We deduce ``energy estimates'', uniformly in epsilon, by assuming the
existence of a symmetrizer having the so called block structure and by assuming
``dissipativity conditions'' on B.
(iii) We perform the convergence analysis by using generalizations of
Compensated Compactness due to Tartar and Gerard.
Finally we include examples which show how to use our theory to approximate
prescribed general quasilinear parabolic systems, satisfying Petrowski
parabolicity condition, or general reaction diffusion systems.; Comment: 26 pages, preliminary version Dec.00

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## Symmetrizer and Antisymmetrizer of the Birman-Wenzl-Murakami Algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/02/2000

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Explicit formulas for the symmetrizer and the antisymmetrizer of the
Birman-Wenzl-Murakami algebras BWM(r,q)_n are given.; Comment: 6 pages. To appear in Lett. Math. Phys

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## $L^2$ well posed Cauchy Problems and Symmetrizability of First Order Systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/10/2013

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The Cauchy problem for first order system $L(t, x, \D_t, \D_x)$ is known to
be well posed in $L^2$ when a it admits a microlocal symmetrizer $S(t,x, \xi)$
which is smooth in $\xi$ and Lipschitz continuous in $(t, x)$. This paper
contains three main results. First we show that a Lipsshitz smoothness globally
in $(t,x, \xi)$ is sufficient. Second, we show that the existence of
symmetrizers with a given smoothness is equivalent to the existence of
\emph{full symmetrizers} having the same smoothness. This notion was first
introduced in \cite{FriLa1}. This is the key point to prove the third result
that the existence of microlocal symmetrizer is preserved if one changes the
direction of time, implying local uniqueness and finite speed of propagation.

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## Generators of algebraic covariant derivative curvature tensors and Young symmetrizers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/10/2003

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#Mathematics - Combinatorics#Computer Science - Symbolic Computation#Mathematics - Differential Geometry#53B20#15A72#05E10#16D60#05-04

We show that the space of algebraic covariant derivative curvature tensors R'
is generated by Young symmetrized tensor products W*U or U*W, where W and U are
covariant tensors of order 2 and 3 whose symmetry classes are irreducible and
characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or
{(1 1),(2 1)}. Each of the partitions (2), (3) and (1 1) describes exactly one
symmetry class, whereas the partition (2 1) characterizes an infinite set S of
irreducible symmetry classes. This set S contains exactly one symmetry class
S_0 whose elements U can not play the role of generators of tensors R'. The
tensors U of all other symmetry classes from S\{S_0} can be used as generators
for tensors R'. Foundation of our investigations is a theorem of S. A. Fulling,
R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that
generates the symmetry class of algebraic covariant derivative curvature
tensors. Furthermore we apply ideals and idempotents in group rings C[Sr], the
Littlewood-Richardson rule and discrete Fourier transforms for symmetric groups
Sr. For certain symbolic calculations we used the Mathematica packages Ricci
and PERMS.; Comment: 18 pages. Chapter for a book "Progress in Computer Science Research"...

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## The hook fusion procedure for Hecke algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/05/2007

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We derive a new expression for the q-analogue of the Young symmetrizer which
generate irreducible representations of the Hecke algebra. We obtain this new
expression using Cherednik's fusion procedure. However, instead of splitting
Young diagrams into their rows or columns, we consider their principal hooks.
This minimises the number of auxiliary parameters needed in the fusion
procedure.; Comment: 19 pages

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## Quivers with relations for symmetrizable Cartan matrices II: Change of symmetrizers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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For $k \ge 1$ we consider the $K$-algebra $H(k) := H(C,kD,\Omega)$ associated
to a symmetrizable Cartan matrix $C$, a symmetrizer $D$, and an orientation
$\Omega$ of $C$, which was defined in Part 1. We construct and analyse a
reduction functor from rep$(H(k))$ to rep$(H(k-1))$. As a consequence we show
that the canonical decomposition of rank vectors for $H(k)$ does not depend on
$k$, and that the rigid locally free $H(k)$-modules are up to isomorphism in
bijection with the rigid locally free $H(k-1)$-modules. Finally, we show that
for a rigid locally free $H(k)$-module of a given rank vector the Euler
characteristic of the variety of flags of locally free submodules with fixed
ranks of the subfactors does not depend on the choice of $k$.; Comment: This article, together with "Quiver with relations for symmetrizable
Cartan matrices III: Convolution Algebras" (arXiv:1511.06216) replaces
arXiv:1502.01565, which has been withdrawn

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## On Bronshtein's Theorem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/08/2015

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We study hyperbolic first order systems and propose a new method proving
Gevrey well posedness, constructing a symmetrizer, motivated by a special
Lyapunov function for linear ODE. The proof not only gives a priori estimates
straightforward so simply but also clarifies some effects coming from the
spectral structures other than the multiplicities of the eigenvalues.

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## A converse to the Grace--Walsh--Szeg\H{o} theorem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/09/2008

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We prove that the symmetrizer of a permutation group preserves stability of a
polynomial if and only if the group is orbit homogeneous. A consequence is that
the hypothesis of permutation invariance in the Grace-Walsh-Szeg\H{o}
Coincidence Theorem cannot be relaxed. In the process we obtain a new
characterization of the \emph{Grace-like polynomials} introduced by D. Ruelle,
and prove that the class of such polynomials can be endowed with a natural
multiplication.; Comment: 7 pages

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## On the symmetry classes of the first covariant derivatives of tensor fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/01/2003

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#Mathematics - Combinatorics#Computer Science - Symbolic Computation#Mathematics - Differential Geometry#53B20, 15A72, 05E10, 16D60, 05-04

We show that the symmetry classes of torsion-free covariant derivatives
$\nabla T$ of r-times covariant tensor fields T can be characterized by
Littlewood-Richardson products $\sigma [1]$ where $\sigma$ is a representation
of the symmetric group $S_r$ which is connected with the symmetry class of T.
If $\sigma = [\lambda]$ is irreducible then $\sigma [1]$ has a multiplicity
free reduction $[\lambda][1] = \sum [\mu]$ and all primitive idempotents
belonging to that sum can be calculated from a generating idempotent e of the
symmetry class of T by means of the irreducible characters or of a discrete
Fourier transform of $S_{r+1}$. We apply these facts to derivatives $\nabla S$,
$\nabla A$ of symmetric or alternating tensor fields. The symmetry classes of
the differences $\nabla S - sym(\nabla S)$ and $\nabla A - alt(\nabla A)$ are
characterized by Young frames (r, 1) and (2, 1^{r-1}), respectively. However,
while the symmetry class of $\nabla A - alt(\nabla A)$ can be generated by
Young symmetrizers of (2, 1^{r-1}), no Young symmetrizer of (r, 1) generates
the symmetry class of $\nabla S - sym(\nabla S)$. Furthermore we show in the
case r = 2 that $\nabla S - sym(\nabla S)$ and $\nabla A - alt(\nabla A)$ can
be applied in generator formulas of algebraic covariant derivative curvature
tensors. For certain symbolic calculations we used the Mathematica packages
Ricci and PERMS.; Comment: 21 pages. Sent in to Seminaire Lotharingien de Combinatoire:
http://www.mat.univie.ac.at/~slc/

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## Pfaffian-type Sugawara operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/07/2011

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We show that the Pfaffian of a generator matrix for the affine Kac--Moody
algebra hat o_{2n} is a Segal--Sugawara vector. Together with our earlier
construction involving the symmetrizer in the Brauer algebra, this gives a
complete set of Segal--Sugawara vectors in type D.; Comment: 4 pages

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