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## A note on the eigenvalues of a special class of matrices

Fonte: ELSEVIER SCIENCE BV
Publicador: ELSEVIER SCIENCE BV

Tipo: Artigo de Revista Científica

ENG

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#Eigenvalues#Crank-Nicolson#Special matrices#Tridiagonal matrices#FREE-SURFACE FLOWS#Mathematics, Applied

In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1, 1] for all values of m (the order of the matrix) and all values of a positive parameter a, the stability parameter sigma. As the order of the matrix is general, and the parameter sigma lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices. (C) 2010 Elsevier B.V. All rights reserved.

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## About the von Neumann regularity of triangular block matrices

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

Publicado em 01/08/2001
ENG

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Necessary and sufficient conditions are given for the von Neumann regularity of triangular block matrices with von Neumann regular diagonal blocks over
arbitrary rings. This leads to the characterization of the von Neumann regularity of a class of triangular Toeplitz matrices over arbitrary rings. Some special results and a new algorithm are derived for triangular Toeplitz matrices over commutative rings. Finally, the Drazin invertibility of some companion matrices over arbitrary rings is considered, as an application.; Fundação para a Ciência e a Tecnologia (FCT).

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## Euclidean distance matrices: special subsets, systems of coordinates and multibalanced matrices

Fonte: Sociedade Brasileira de Matemática Aplicada e Computacional
Publicador: Sociedade Brasileira de Matemática Aplicada e Computacional

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/01/2007
EN

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In this paper we present special subsets of positive semidefinite matrices where the linear function k becomes a geometric similarity and its inverse can be easily computed. The images of these special subsets are characterized geometrically. We also study the systems of coordinates for spherical matrices and at the end, we introduce the class of multibalanced distance matrices.

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## Bioartificial matrices for therapeutic vascularization

Fonte: National Academy of Sciences
Publicador: National Academy of Sciences

Tipo: Artigo de Revista Científica

EN

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Therapeutic vascularization remains a significant challenge in regenerative medicine applications. Whether the goal is to induce vascular growth in ischemic tissue or scale up tissue-engineered constructs, the ability to induce the growth of patent, stable vasculature is a critical obstacle. We engineered polyethylene glycol–based bioartificial hydrogel matrices presenting protease-degradable sites, cell-adhesion motifs, and growth factors to induce the growth of vasculature in vivo. Compared to injection of soluble VEGF, these matrices delivered sustained in vivo levels of VEGF over 2 weeks as the matrix degraded. When implanted subcutaneously in rats, degradable constructs containing VEGF and arginine-glycine-aspartic acid tripeptide induced a significant number of vessels to grow into the implant at 2 weeks with increasing vessel density at 4 weeks. The mechanism of enhanced vascularization is likely cell-demanded release of VEGF, as the hydrogels may degrade substantially within 2 weeks. In a mouse model of hind-limb ischemia, delivery of these matrices resulted in significantly increased rate of reperfusion. These results support the application of engineered bioartificial matrices to promote vascularization for directed regenerative therapies.

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## Functionalization of biomaterial surfaces using artificial extracellular matrices

Fonte: Landes Bioscience
Publicador: Landes Bioscience

Tipo: Artigo de Revista Científica

EN

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Construction of biomaterials with the ability to guide cell function is a topic of high interest in biomaterial development. One approach is using components native to the ECM of the target tissue to generate in vitro a microenvironment that can also elicit specific responses in cells and tissues—an artificial ECM (aECM). The focus is on collagen as the basic material, which can be modified using a number of different glycoproteins, proteoglycans and glycosaminoglycans. Preparation, immobilization and the biochemical characteristics of such aECM are discussed, as well as the in vitro and in vivo response of cells and tissues, illustrating the potential of such matrices to direct cell fate.

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## Understanding the viscoelastic behavior of collagen matrices through relaxation time distribution spectrum

Fonte: Landes Bioscience
Publicador: Landes Bioscience

Tipo: Artigo de Revista Científica

EN

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This study aims to provide understanding of the macroscopic viscoelastic behavior of collagen matrices through studying the relaxation time distribution spectrum obtained from stress relaxation tests. Hydrated collagen gel and dehydrated collagen thin film was exploited as two different hydration levels of collagen matrices. Genipin solution was used to induce crosslinking in collagen matrices. Biaxial stress relaxation tests were performed to characterize the viscoelastic behavior of collagen matrices. The rate of stress relaxation of both hydrated and dehydrated collagen matrices shows a linear initial stress level dependency. Increased crosslinking reduces viscosity in collagen gel, but the effect is negligible for thin film. Relaxation time distribution spectrum was obtained from the stress relaxation data by inverse Laplace transform. For most of the collagen matrices, three peaks at the short (0.3s ~1 s), medium (3s ~90 s), and long relaxation time (> 200 s) were observed in the continuous spectrum, which likely corresponds to relaxation mechanisms involve fiber, inter-fibril, and fibril sliding. Splitting of the middle peak was observed at higher initial stress levels suggesting increased structural heterogeneity at the fibril level with mechanical loading. The intensity of the long-term peaks increases with higher initial stress levels indicating the engagement of collagen fibrils at higher levels of tissue strain.

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## A quantitative version of the commutator theorem for zero trace matrices

Fonte: National Academy of Sciences
Publicador: National Academy of Sciences

Tipo: Artigo de Revista Científica

EN

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Let A be an m × m complex matrix with zero trace and let ε > 0. Then there are m × m matrices B and C such that A = [B,C] and ‖B‖‖C‖ ≤ Kεmε‖A‖ where Kε depends only on ε. Moreover, the matrix B can be taken to be normal.

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## Error Analysis of a Partial Pivoting Method for Structured Matrices

Fonte: Universidade Nacional da Austrália
Publicador: Universidade Nacional da Austrália

Tipo: Working/Technical Paper
Formato: 322757 bytes; 356 bytes; application/pdf; application/octet-stream

EN_AU

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Many matrices that arise in the solution of signal processing problems have a special displacement structure. For example, adaptive filtering and direction-of-arrival estimation yield matrices of Toeplitz type. A recent method of Gohberg, Kailath and Olshevsky (GKO) allows fast Gaussian elimination with partial pivoting for such structured matrices. In this paper, a rounding error analysis is performed on the Cauchy and Toeplitz variants of the GKO method. It is shown the error growth depends on the growth in certain auxiliary vectors, the generators, which are computed by the GKO algorithms. It is also shown that in certain circumstances, the growth in the generators can be large, and so the error growth is much larger than would be encountered with normal Gaussian elimination with partial pivoting. A modification of the algorithm to perform a type of row-column pivoting is proposed which may ameliorate this problem.; no

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## Wigner's new physics frontier: Physics of two-by-two matrices, including the Lorentz group and optical instruments

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/10/2003

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According to Eugene Wigner, quantum mechanics is a physics of Fourier
transformations, and special relativity is a physics of Lorentz
transformations. Since two-by-two matrices with unit determinant form the group
SL(2,c) which acts as the universal covering group of the Lorentz group, the
two-by-two matrices constitute the natural language for special relativity. The
central language for optical instruments is the two-by-two matrix called the
beam transfer matrix, or the so-called ABCD matrix. It is shown that the ABCD
matrices also form the SL(2,C) group. Thus, it is possible to perform
experiments in special relativity using optical instruments. Likewise, the
optical instruments can be explained in terms of the symmetry of relativistic
particles.; Comment: Latex 28 pages, based on the papers presented by the last author(YSK)
at a number of conferences, including the 8th International Wigner Symposium
(New York, U.S.A., 2003), the 8th International Conference on Squeezed States
and Uncertainty Relations (Puebla, Mexico, 2003), the Symmetry Festival
(Budapest, Hungary, 2003), and the International Conference on Physics and
Control (Saint Petersburg, Russia, 2003)

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## Super Special Codes using Super Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/06/2010

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The new classes of super special codes are constructed in this book using the
specially constructed super special vector spaces. These codes mainly use the
super matrices. These codes can be realized as a special type of concatenated
codes. This book has four chapters. In chapter one basic properties of codes
and super matrices are given. A new type of super special vector space is
constructed in chapter two of this book. Three new classes of super special
codes namely, super special row code, super special column code and super
special codes are introduced in chapter three. Applications of these codes are
given in the final chapter.; Comment: 161 pages

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## Symmetric Hadamard matrices of order 116 and 172 exist

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We construct new symmetric Hadamard matrices of orders $92,116$, and $172$.
While the existence of those of order $92$ was known since 1978, the orders
$116$ and $172$ are new. Our construction is based on a recent new
combinatorial array discovered by N. A. Balonin and J. Seberry. For order $116$
we used an adaptation of an algorithm for parallel collision search. The
adaptation pertains to the modification of some aspects of the algorithm to
make it suitable to solve a 3-way matching problem. We also point out that a
new infinite series of symmetric Hadamard matrices arises by plugging into the
GP array the matrices constructed by Xia, Xia, Seberry, and Wu in 2005.; Comment: 9 pages, to appear in Special Matrices Vol. 3

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## Special Fuzzy Matrices for Social Scientists

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/07/2007

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This book introduces special classes of Fuzzy and Neutrosophic matrices.
These special classes of matrices are used in the construction of multi-expert
special fuzzy models using FCM, FRM and FRE and their Neutorosophic analogues
(simultaneous or otherwise, according to ones need). Using the six basic
models, we have constructed a multi-expert multi-model called the Super Special
Hexagonal Fuzzy and Neutrosophic model.
Given any special input vector, these models can give the resultant using
special operations. When coupled with computer programming, these operations
can give the solution within a reasonable time period. Such multi-expert
multi-model systems are not only a boon to social scientists, but also to
anyone who wants to use Fuzzy or Neutrosophic models.; Comment: 301 pages

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## Spectral Properties of Tridiagonal k-Toeplitz Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We derive the spectral properties of tridiagonal k-Toeplitz matrices in
generality i.e. with non symmetric complex entries and any periodicity k.
Previous work has highlighted some special spectral properties of real
symmetric tridiagonal k-Toeplitz matrices and note that all square matrices
have similarity transformation to tridiagonal form. Toeplitz matrices are used
in convolution, discrete transforms and lumped physical systems, and it can be
shown that every matrix is a product of Toeplitz matrices. We begin with
numerical results of spectra of some special k-Toeplitz matrices as a
motivation. This is followed by a derivation of spectral properties of a
general tridiagonal k-Toeplitz matrix using three term recurrence relations and
C - R,C - I kth order polynomial mappings. These relations establish a support
for the limiting eigenvalue distribution of a tridiagonal Toeplitz matrix which
has dimensions much larger than k. Numerical examples are used to graphically
demonstrate theorems. As an addendum, we derive expressions for O(k)
computation of the determinant of tridiagonal k-Toeplitz matrices of any
dimension.

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## Classification of degenerate 4-dimensional matrices with semi-group structure and polarization optics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/01/2012

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In polarization optics, an important role play Mueller matrices -- real
four-dimensional matrices which describe the effect of action of optical
elements on the polarization state of the light, described by 4-dimensional
Stokes vectors. An important issue is to classify possible classes of the
Mueller matrices. In particular, of special interest are degenerate Mueller
matrices with vanishing determinants. Earlier, it was developed a special
technique of parameterizing arbitrary 4-dimensional matrices with the use of
four 4-dimensional vector (k, m, l, n). In the paper, a classification of
degenerate 4-dimensional real matrices of rank 1, 2, 3. is elaborated. To
separate possible classes of degenerate matrices of ranks 1 and 2, we impose
linear restrictions on (k, m, l, n), which are compatible with the group
multiplication law. All the subsets of matrices obtained by this method, are
either sub-groups or semigroups. To obtain singular matrices of rank 3, we
specify 16 independent possibilities to get 4-dimensional matrices with zero
determinant.; Comment: 12 pages

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## Some inequalities on the norms of special matrices with generalized Tribonacci and generalized Pell Padovan sequences

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In this paper some properties of generalized tribonacci and generalized
Padovan sequence are presented. Also the Euclidean norms of circulant,
$r$-circulant, semi-circulant and Hankle matrices with above mentioned
sequences are calculated. The upper and lower bounds of spectral norms are also
obtained.; Comment: We rewrite the whole article and generalized our result with the help
of second name author

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## Degenerate 4-Dimensional Matrices with Semi-Group Structure

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/11/2014

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While dealing with the nontrivial task of classifying Mueller matrices, of
special interest is the study of the degenerate Mueller matrices (matrices with
vanishing determinant, for which the law of multiplication holds, but there
exists no inverse elements). Earlier, it was developed a special technique of
parameterizing arbitrary 4-dimensional matrices with the use of a four
4-dimensional vector (k,m,l,n). In the following, a classification of
degenerate 4-dimensional real matrices of rank 1, 2, and 3 is elaborated. To
separate possible classes of degenerate matrices of ranks 1 and 2, we impose
linear restrictions on (k,m, l,n), which are compatible with the group
multiplication law. All the subsets of matrices obtained by this method, form
either subgroups or semi-groups. To obtain singular matrices of rank 3, we
specify 16 independent possibilities to get the 4-dimensional matrices with
zero determinant; Comment: 41 pages. Chapter from the Book: O.V. Veko, E.M. Ovsiyuk, A. Oana, M.
Neagu, V. Balan, V.M. Red'kov. Spinor Structures in Geometry and Physics.
Nova Science Publishers, Inc., New York, 2014-215 (in press)

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## Corrections for tribimaximal, bimaximal and democratic neutrino mixing matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In this work we analyze the corrections to tribimaximal (TBM), bimaximal (BM)
and democratic (DC) mixing matrices for explaining large reactor mixing angle
$\theta_{13}$ and checking the consistency with other neutrino mixing angles.
The corrections are parameterized in terms of small orthogonal rotations (R)
with corresponding modified PMNS matrix of the form $R_{ij}\cdot U \cdot
R_{kl}$ where $R_{ij}$ is rotation in ij sector and U is any one of these
special matrices. We showed the rotations $R_{13}\cdot U \cdot R_{23}$,
$R_{12}\cdot U \cdot R_{13}$ for BM and $R_{13}\cdot U \cdot R_{13}$ for TBM
perturbative case successfully fit all neutrino mixing angles within $1\sigma$
range. The perturbed PMNS matrix $R_{12}\cdot U \cdot R_{13}$ for DC, TBM and
$R_{23}\cdot U \cdot R_{23}$ for TBM case is successful in producing mixing
angles at 2$\sigma$ level. The other rotation schemes are either excluded or
successful in producing mixing angles at $3\sigma$ level.; Comment: 25 pages, 56 figures; v2: new rotation cases with corresponding
discussion and figures included, new references added, accepted for
publication in JHEP

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## Learning Input and Recurrent Weight Matrices in Echo State Networks

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/11/2013

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Echo State Networks (ESNs) are a special type of the temporally deep network
model, the Recurrent Neural Network (RNN), where the recurrent matrix is
carefully designed and both the recurrent and input matrices are fixed. An ESN
uses the linearity of the activation function of the output units to simplify
the learning of the output matrix. In this paper, we devise a special technique
that take advantage of this linearity in the output units of an ESN, to learn
the input and recurrent matrices. This has not been done in earlier ESNs due to
their well known difficulty in learning those matrices. Compared to the
technique of BackPropagation Through Time (BPTT) in learning general RNNs, our
proposed method exploits linearity of activation function in the output units
to formulate the relationships amongst the various matrices in an RNN. These
relationships results in the gradient of the cost function having an analytical
form and being more accurate. This would enable us to compute the gradients
instead of obtaining them by recursion as in BPTT. Experimental results on
phone state classification show that learning one or both the input and
recurrent matrices in an ESN yields superior results compared to traditional
ESNs that do not learn these matrices...

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## A Class of Special Matrices and Quantum Entanglement

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/05/2004

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We present a kind of construction for a class of special matrices with at
most two different eigenvalues, in terms of some interesting multiplicators
which are very useful in calculating eigenvalue polynomials of these matrices.
This class of matrices defines a special kind of quantum states --
$d$-computable states. The entanglement of formation for a large class of
quantum mixed states is explicitly presented.; Comment: 17 pages

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## The Eigen-Problem for Some Special Near-Toeplitz Centro-Skew Tridiagonal Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/01/2011

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#Mathematics - Spectral Theory#Primary: 15B05 Toeplitz, Cauchy, and related matrices, Secondary:
15B35 Sign pattern matrices, 15A18 Eigenvalues, singular values, and
eigenvectors

Let $n\ge 2$ be an integer. Let $R_n$ denote the $n\times n$ tridiagonal
matrix with $-1$'s on the sub-diagonal, $1$'s on the super-diagonal, $-1$ in
the $(1,1)$ entry, $1$ in the $(n,n)$ entry and zeros elsewhere. We find the
eigen-pairs of the matrices $R_n$.

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