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

Martínez Dopico, Froilán C.; Uhlig, Frank
Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article
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28.06%
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.

Symmetrizers and antisymmetrizers for the BMW algebra

Dipper, Richard; Hu, Jun; Stoll, Friederike
Tipo: Artigo de Revista Científica
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16.81%
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 $B_n(r,q)$.

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

Colombini, Ferruccio; Del Santo, Daniele; Fanelli, Francesco; Métivier, Guy
Tipo: Artigo de Revista Científica
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17.49%
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

Rational representations of Yangians associated with skew Young diagrams

Nazarov, Maxim
Tipo: Artigo de Revista Científica
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16.81%
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 $GL_N$...

Products of Young symmetrizers and ideals in the generic tensor algebra

Raicu, Claudiu
Tipo: Artigo de Revista Científica
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17.49%
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

Symmetrization of Bernoulli

Pal, Soumik
Tipo: Artigo de Revista Científica
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17.49%
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

Strongly primitive species with potentials I: Mutations

Labardini-Fragoso, Daniel; Zelevinsky, Andrei
Tipo: Artigo de Revista Científica
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16.81%
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

Algorithms and Properties for Positive Symmetrizable Matrices

Dias, Elisângela Silva; Castonguay, Diane; Dourado, Mitre Costa
Tipo: Artigo de Revista Científica
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17.84%
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)

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

Israwi, Samer; Lteif, Ralph; Talhouk, Raafat
Tipo: Artigo de Revista Científica
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16.81%
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

Representations of twisted Yangians associated with skew Young diagrams

Nazarov, Maxim
Tipo: Artigo de Revista Científica
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16.81%
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...

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

Donatelli, Donatella; Marcati, Pierangelo
Tipo: Artigo de Revista Científica
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16.81%
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

Symmetrizer and Antisymmetrizer of the Birman-Wenzl-Murakami Algebras

Heckenberger, István; Schüler, Axel
Tipo: Artigo de Revista Científica
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27.49%
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

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

Metivier, Guy
Tipo: Artigo de Revista Científica
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17.49%
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.

Generators of algebraic covariant derivative curvature tensors and Young symmetrizers

Fiedler, B.
Tipo: Artigo de Revista Científica
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16.81%
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"...

The hook fusion procedure for Hecke algebras

Grime, James
Tipo: Artigo de Revista Científica
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16.81%
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

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

Geiss, Christof; Leclerc, Bernard; Schröer, Jan
Tipo: Artigo de Revista Científica
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16.81%
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

On Bronshtein's Theorem

Colombini, F.; Nishitani, T.; Rauch, J.
Tipo: Artigo de Revista Científica
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16.81%
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.

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

Brändén, Petter; Wagner, David G.
Tipo: Artigo de Revista Científica
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16.81%
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

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

Fiedler, B.
Tipo: Artigo de Revista Científica
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/