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Journal articles on the topic 'Sigma matrices'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Sebbar, Ahmed, and Thérèse Falliero. "CAPACITIES AND JACOBI MATRICES." Proceedings of the Edinburgh Mathematical Society 46, no. 3 (2003): 719–45. http://dx.doi.org/10.1017/s0013091502001141.

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AbstractIn this paper, we use the theorem of Burchnall and Shaundy to give the capacity of the spectrum $\sigma(A)$ of a periodic tridiagonal and symmetric matrix. A special family of Chebyshev polynomials of $\sigma(A)$ is also given. In addition, the inverse problem is considered: given a finite union $E$ of closed intervals, we study the conditions for a Jacobi matrix $A$ to exist satisfying $\sigma(A)=E$. We relate this question to Carathéodory theorems on conformal mappings.AMS 2000 Mathematics subject classification: Primary 31B15; 30C20; 39A70

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Kumar, Sandeep, and Deepa Sinha. "Spectral Analysis of Splitting Signed Graph." European Journal of Pure and Applied Mathematics 17, no. 1 (2024): 504–18. http://dx.doi.org/10.29020/nybg.ejpam.v17i1.4798.

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An ordered pair $\Sigma = (\Sigma^{u}$,$\sigma$) is called the \textit{signed graph}, where $\Sigma^{u} = (V,E)$ is a \textit{underlying graph} and $\sigma$ is a signed mapping, called \textit{signature}, from $E$ to the sign set $\lbrace +, - \rbrace$. The \textit{splitting signed graph} $\Gamma(\Sigma)$ of a signed graph $\Sigma$ is defined as, for every vertex $u \in V(\Sigma)$, take a new vertex $u'$. Join $u'$ to all the vertices of $\Sigma$ adjacent to $u$ such that $\sigma_{\Gamma}(u'v) = \sigma(u'v), \ u \in N(v)$. The objective of this paper is to propose an algorithm for the generati

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KIM, YOUNG-ONE, and SIEYE RYU. "On the number of fixed points of a sofic shift-flip system." Ergodic Theory and Dynamical Systems 35, no. 2 (2013): 482–98. http://dx.doi.org/10.1017/etds.2013.57.

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AbstractIf $X$ is a sofic shift and $\varphi : X\rightarrow X$ is a homeomorphism such that ${\varphi }^{2} = {\text{id} }_{X} $ and $\varphi {\sigma }_{X} = { \sigma }_{X}^{- 1} \varphi $, the number of points in $X$ that are fixed by ${ \sigma }_{X}^{m} $ and ${ \sigma }_{X}^{n} \varphi , m= 1, 2, \ldots , n\in \mathbb{Z} $, is expressed in terms of a finite number of square matrices: the matrices are obtained from Krieger’s joint state chain of a sofic shift which is conjugate to $X$.

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Chen, Zhi, Zelin Zhu, Jiawei Li, Lizhen Yang, and Lei Cao. "Extreme Points of Certain Transportation Polytopes with Fixed Total Sums." Electronic Journal of Linear Algebra 37 (April 5, 2021): 256–71. http://dx.doi.org/10.13001/ela.2021.5141.

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Transportation matrices are $m\times n$ nonnegative matrices with given row sum vector $R$ and column sum vector $S$. All such matrices form the convex polytope $\mathcal{U}(R,S)$ which is called a transportation polytope and its extreme points have been classified. In this article, we consider a new class of convex polytopes $\Delta(\bar{R},\bar{S},\sigma)$ consisting of certain transportation polytopes satisfying that the sum of all elements is $\sigma$, and the row and column sum vectors are dominated componentwise by the given positive vectors $\bar{R}$ and $\bar{S}$, respectively. We char

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5

Preusser, Raimund. "Sandwich classification for GL n (R), O2n (R) and U2n (R,Λ) revisited". Journal of Group Theory 21, № 1 (2018): 21–44. http://dx.doi.org/10.1515/jgth-2017-0028.

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AbstractLetnbe a natural number greater than or equal to 3,Ra commutative ring and{\sigma\in\mathrm{GL}_{n}(R)}. We show that{t_{kl}(\sigma_{ij})}(resp.{t_{kl}(\sigma_{ii}-\sigma_{jj})}), where{i\neq j}and{k\neq l}can be expressed as a product of eight (resp. 24) matrices of the form{{}^{\epsilon}\sigma^{\pm 1}}, where{\epsilon\in E_{n}(R)}. We prove similar results for the orthogonal groups{\mathrm{O}_{2n}(R)}and the hyperbolic unitary groups{\mathrm{U}_{2n}(R,\Lambda)}under the assumption thatRis commutative and{n\geq 3}. This yields new, very short proofs of the Sandwich Classification Theo

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Loewy, Raphael. "Some additional notes on the spectra of non-negative symmetric 5 x 5 matrices." Electronic Journal of Linear Algebra 37, no. 37 (2021): 1–13. http://dx.doi.org/10.13001/ela.2021.5333.

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The Symmetric Non-negative Inverse Eigenvalue Problem (SNIEP) asks when is a list $ \sigma = \left( \lambda_{1}, \lambda_{2}, \dots, \lambda_{n} \right) $ of real, monotonically decreasing numbers, the spectrum of an $n \times n$, symmetric, non-negative matrix $A$. In that case, we say $\sigma$ is realizable and $A$ is a realizing matrix. Here, we consider the case $n=5$, the lowest value of $n$ for which the problem is unsolved. Let $ s_{1}(\sigma) = \sum_{i=1}^5 \lambda_{i} $ and $ s_{3}(\sigma) = \sum_{i=1}^5 {\lambda_{i}}^3 $. It is known that to complete the solution for $n=5$, it remain

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Loewy, Raphael. "Some additional notes on the spectra of non-negative symmetric 5 x 5 matrices." Electronic Journal of Linear Algebra 37, no. 37 (2021): 1–13. http://dx.doi.org/10.13001/ela.2021.5333.

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The Symmetric Non-negative Inverse Eigenvalue Problem (SNIEP) asks when is a list $ \sigma = \left( \lambda_{1}, \lambda_{2}, \dots, \lambda_{n} \right) $ of real, monotonically decreasing numbers, the spectrum of an $n \times n$, symmetric, non-negative matrix $A$. In that case, we say $\sigma$ is realizable and $A$ is a realizing matrix. Here, we consider the case $n=5$, the lowest value of $n$ for which the problem is unsolved. Let $ s_{1}(\sigma) = \sum_{i=1}^5 \lambda_{i} $ and $ s_{3}(\sigma) = \sum_{i=1}^5 {\lambda_{i}}^3 $. It is known that to complete the solution for $n=5$, it remain

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8

Hlavatý, Ladislav, Josef Navrátil, and Libor Šnobl. "ON RENORMALIZATION OF POISSON–LIE T-PLURAL SIGMA MODELS." Acta Polytechnica 53, no. 5 (2013): 433–37. http://dx.doi.org/10.14311/ap.2013.53.0433.

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Covariance of the one-loop renormalization group equations with respect to Poisson–Lie T-plurality of sigma models is discussed. The role of ambiguities in renormalization group equations of Poisson–Lie sigma models with truncated matrices of parameters is investigated.

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Preusser, Raimund. "Sandwich classification for O2n+1(R) and U2n+1(R,Δ) revisited". Journal of Group Theory 21, № 4 (2018): 539–71. http://dx.doi.org/10.1515/jgth-2018-0011.

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AbstractIn a recent paper, the author proved that if {n\geq 3} is a natural number, R a commutative ring and {\sigma\in GL_{n}(R)}, then {t_{kl}(\sigma_{ij})} where {i\neq j} and {k\neq l} can be expressed as a product of 8 matrices of the form {{}^{\varepsilon}\sigma^{\pm 1}} where {\varepsilon\in E_{n}(R)}. In this article we prove similar results for the odd-dimensional orthogonal groups {O_{2n+1}(R)} and the odd-dimensional unitary groups {U_{2n+1}(R,\Delta)} under the assumption that R is commutative and {n\geq 3}. This yields new, short proofs of the Sandwich Classification Theorems for

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Ganassali, Luca, Marc Lelarge, and Laurent Massoulié. "Spectral alignment of correlated Gaussian matrices." Advances in Applied Probability 54, no. 1 (2022): 279–310. http://dx.doi.org/10.1017/apr.2021.31.

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Abstract In this paper we analyze a simple spectral method (EIG1) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given two matrices A and B, we compute two corresponding leading eigenvectors $v_1$ and $v'_{\!\!1}$ . The algorithm returns the permutation $\hat{\pi}$ such that the rank of coordinate $\hat{\pi}(i)$ in $v_1$ and that of coordinate i in $v'_{\!\!1}$ (up to the sign of $v'_{\!\!1}$ ) are the same. We consider a model of weighted graphs where the adjacency matrix A belongs to the Gaussian orthogonal ensemble of size $N \times N$ , and B is a n

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Gombor, Tamás. "New boundary monodromy matrices for classical sigma models." Nuclear Physics B 953 (April 2020): 114949. http://dx.doi.org/10.1016/j.nuclphysb.2020.114949.

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12

Malejki, Maria. "Discrete spectra for some complex infinite band matrices." Opuscula Mathematica 41, no. 6 (2021): 861–79. http://dx.doi.org/10.7494/opmath.2021.41.6.861.

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Under suitable assumptions the eigenvalues for an unbounded discrete operator \(A\) in \(l_2\), given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let \[\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},\] where \({\rm Lim}_{n\to \infty} \lambda_n\) is the set of all limit points of the sequence \((\lambda_n)\) and \(A_n\) is a finite dimensional orthogonal truncation of \(A\). The aim of this article is to provide the conditions that are sufficient f

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Duffner, M. Antonia, and Rosario Fernandes. "Semilinear preservers of the immanants in the set of the doubly stochastic matrices." Electronic Journal of Linear Algebra 32 (February 6, 2017): 76–97. http://dx.doi.org/10.13001/1081-3810.3190.

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Let $S_n$ denote the symmetric group of degree $n$ and $M_n$ denote the set of all $n$-by-$n$ matrices over the complex field, $\IC$. Let $\chi: S_n\rightarrow \IC$ be an irreducible character of degree greater than $1$ of $S_n$. The immanant $\dc: M_n \rightarrow \IC$ associated with $\chi$ is defined by $$ \dc(X) = \sum_{\sigma \in S_n} \chi(\sigma) \prod_{j=1}^n X_{j\sigma(j)} , \quad X = [X_{jk}] \in M_n. $$ Let $\Omega_n$ be the set of all $n$-by-$n$ doubly stochastic matrices, that is, matrices with nonnegative real entries and each row and column sum is one. We say that a map $T$ from $

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14

Wang, Shiyun, Qi Li, Xu Sun, and Zhenhua Lyu. "Diagonal-Schur complements of Nekrasov matrices." Electronic Journal of Linear Algebra 39 (October 24, 2023): 539–55. http://dx.doi.org/10.13001/ela.2023.7941.

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The Schur and diagonal-Schur complements are important tools in many fields. It was revealed that the diagonal-Schur complements of Nekrasov matrices with respect to the index set $\{1\}$ are Nekrasov matrices by Cvetkovic and Nedovic [Appl. Math. Comput., 208:225-230, 2009]. In this paper, we prove that the diagonal-Schur complements of Nekrasov matrices with respect to any index set are Nekrasov matrices. Similar results hold for $\Sigma$-Nekrasov matrices. We also present some results on Nekrasov diagonally dominant degrees. Numerical examples are given to verify the correctness of the resu

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Chapon, François, Romain Couillet, Walid Hachem, and Xavier Mestre. "The outliers among the singular values of large rectangular random matrices with additive fixed rank deformation." Markov Processes and Related Fields 20 (January 1, 2014): 183–228. https://doi.org/10.5281/zenodo.58344.

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Consider the matrix Σ<em>n</em>=<em>n</em>−1/2<em>X</em><em>n</em><em>D</em>1/2<em>n</em>+<em>P</em><em>n</em> where the matrix $X_n \in \C^{N\times n}$ has Gaussian standard independent elements, <em>D</em><em>n</em> is a deterministic diagonal nonnegative matrix, and <em>P</em><em>n</em> is a deterministic matrix with fixed rank. Under some known conditions, the spectral measures of Σ<em>n</em>Σ&lowast;<em>n</em> and <em>n</em>−1<em>X</em><em>n</em><em>D</em><em>n</em><em>X</em>&lowast;<em>n</em> both converge towards a compactly supported probability measure <e

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16

Moysis, Lazaros, and Nicholas Karampetakis. "Algebraic Methods for the Construction of Algebraic-Difference Equations With Desired Behavior." Electronic Journal of Linear Algebra 34 (February 21, 2018): 1–17. http://dx.doi.org/10.13001/1081-3810.3741.

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For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma $ denotes the shift forward operator and $A\left( \sigma \right) $ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right) $. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right) $, such that the constructed system

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17

Gates, S. James, B. Radak, and V. G. J. Rodgers. "Irreducible decomposition of products of 10D chiral sigma matrices." Computer Physics Communications 136, no. 1-2 (2001): 173–81. http://dx.doi.org/10.1016/s0010-4655(00)00251-4.

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18

Shcherbina, Mariya, and Tatyana Shcherbina. "Universality for 1d Random Band Matrices: Sigma-Model Approximation." Journal of Statistical Physics 172, no. 2 (2018): 627–64. http://dx.doi.org/10.1007/s10955-018-1969-1.

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19

Hai, Guojun, and Dragana Cvetkovic-Ilic. "Generalized left and right Weyl spectra of upper triangular operator matrices." Electronic Journal of Linear Algebra 32 (February 6, 2017): 41–50. http://dx.doi.org/10.13001/1081-3810.3373.

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In this paper, for given operators $A\in\B(\H)$ and $B\in\B(\K)$, the sets of all $C\in \B(\K,\H)$ such that $M_C=\bmatrix{cc} A&C\\0&B\endbmatrix$ is generalized Weyl and generalized left (right) Weyl, are completely described. Furthermore, the following intersections and unions of the generalized left Weyl spectra $$ \bigcup_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) \;\;\; \mbox{and} \;\;\; \bigcap_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) $$ are also described, and necessary and sufficient conditions which two operators $A\in\B(\H)$ and $B\in\B(\K)$ have to satisfy in order for $M_C$ to be a genera

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20

Geuenich, Jan, and Daniel Labardini-Fragoso. "Species with Potential Arising from Surfaces with Orbifold Points of Order 2, Part II: Arbitrary Weights." International Mathematics Research Notices 2020, no. 12 (2018): 3649–752. http://dx.doi.org/10.1093/imrn/rny090.

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Abstract Let ${\boldsymbol{\Sigma }}=(\Sigma ,\mathbb{M},\mathbb{O})$ be either an unpunctured surface with marked points and order-2 orbifold points or a once-punctured closed surface with order-2 orbifold points. For each pair $(\tau ,\omega )$ consisting of a triangulation $\tau $ of ${\boldsymbol{\Sigma }}$ and a function $\omega :\mathbb{O}\rightarrow \{1,4\}$, we define a chain complex $C_\bullet (\tau , \omega )$ with coefficients in $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}$. Given ${\boldsymbol{\Sigma }}$ and $\omega $, we define a colored triangulation of ${\boldsymbol{\Sigma }_\omega }=(

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21

KAJI, Hajime. "Example of $\sigma$-Transition Matrices Defining the Horrocks-Mumford Bundle." Tokyo Journal of Mathematics 12, no. 1 (1989): 21–32. http://dx.doi.org/10.3836/tjm/1270133545.

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KAWAGUCHI, IO, TAKUYA MATSUMOTO, and KENTAROH YOSHIDA. "THE CLASSICAL EQUIVALENCE OF MONODROMY MATRICES IN SQUASHED SIGMA MODEL." International Journal of Modern Physics: Conference Series 21 (January 2013): 180–81. http://dx.doi.org/10.1142/s2010194513009707.

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We show Yangian and quantum affine symmetries are realized in two-dimensional non-linear sigma models with squashed spheres target spaces. According to these symmetries, there are two descriptions for the classical dynamics. In each description, a Lax pair and the corresponding monodromy matrix are constructed. Both Lax pairs reproduce the same equations of motion and hence they are equivalent. We show the gauge-equivalence by constructing a map between monodromy matrices with the relation of spectral parameters.

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Hollowood, Timothy J., J. Luis Miramontes, and David M. Schmidtt. "S-matrices and quantum group symmetry ofk-deformed sigma models." Journal of Physics A: Mathematical and Theoretical 49, no. 46 (2016): 465201. http://dx.doi.org/10.1088/1751-8113/49/46/465201.

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Fendley, Paul, and Nicholas Read. "ExactS-matrices for supersymmetric sigma models and the Potts model." Journal of Physics A: Mathematical and General 35, no. 50 (2002): 10675–704. http://dx.doi.org/10.1088/0305-4470/35/50/301.

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A, Maheshwari, Sadariya B, Javia HN, and Sharma D. "Quality improvement in clinical biochemistry laboratory using six sigma metrics and quality goal index." Journal of Medical and Scientific Research 9, no. 2 (2021): 101–7. http://dx.doi.org/10.17727/jmsr.2021/9-16.

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Introduction: One of the most popular quality management system tackle employed for process perfection is six sigma. When the process outcome is measurable, six sigma can be used to assess the quality. Aim: Present study was conducted with the objective to apply six sigma matrices and quality goal index for the assessment of quality assurance in a clinical biochemistry laboratory. Materials and methods: Present study is a retrospective study. Internal and external quality control data were analyzed retrospectively during July 2020 to December 2020. Descriptive statistics like laboratory mean ±

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Loewy, Raphael. "A note on the real nonnegative inverse eigenvalue problem." Electronic Journal of Linear Algebra 31 (February 5, 2016): 765–73. http://dx.doi.org/10.13001/1081-3810.3379.

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The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) asks when is a list \[ \sigma=(\lambda_1, \lambda_2,\ldots,\lambda_n)\] consisting of real numbers the spectrum of an $n \times n$ nonnegative matrix $A$. In that case, $\sigma$ is said to be realizable and $A$ is a realizing matrix. In a recent paper dealing with RNIEP, P.~Paparella considered cases of realizable spectra where a realizing matrix can be taken to have a special form, more precisely such that the entries of each row are obtained by permuting the entries of the first row. A matrix of this form is called permutative. Paparell

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KLING, ALEXANDER, MAXIMILIAN KREUZER, and JIAN-GE ZHOU. "SU(2) WZW D-BRANES AND QUANTIZED WORLD-VOLUME U(1) FLUX ON S2." Modern Physics Letters A 15, no. 34 (2000): 2069–77. http://dx.doi.org/10.1142/s021773230000270x.

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We discuss possible D-brane configurations on SU(2) group manifolds in the sigma model approach. When we turn the boundary conditions of the space–time fields into the boundary gluing conditions of chiral currents, we find that for all D-branes except the spherical D2-branes, the gluing matrices Rab depend on the fields, so the chiral Kac–Moody symmetry is broken, but conformal symmetry is maintained. Matching the spherical D2-branes derived from the sigma model with those from the boundary state approach we obtain a U(1) field strength that is consistent with flux quantization.

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Guo, Xiuyun, and Xue Zhang. "Determinants and invertibility of circulant matrices." Electronic Research Archive 32, no. 7 (2024): 4741–52. http://dx.doi.org/10.3934/era.2024216.

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<p>Let $ a_0, a_1, \dots, a_{n-1} $ be real numbers and let $ A = Circ(a_0, a_1, \dots, a_{n-1}) $ be a circulant matrix with $ f(x) = \Sigma ^{n-1}_{j = 0}a_jx^j $. First, we prove that $ Circ(a_0, a_1, \dots, a_{n-1}) $ must be invertible if the sequence $ a_0, a_1, \dots, a_{n-1} $ is a strictly monotonic sequence and $ a_0+a_1+\dots+a_{n-1}\neq 0 $. Next, we reduce the calculation of $ f(\varepsilon ^0)f(\varepsilon)\dots f(\varepsilon ^{n-1}) $ for a prime $ n $ by using the techniques on finite fields, where $ \varepsilon $ is a primitive $ n $-th root of unity. Finally, we provide

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Shcherbina, Mariya, and Tatyana Shcherbina. "Finite-Rank Complex Deformations of Random Band Matrices: Sigma-Model Approximation." Zurnal matematiceskoj fiziki, analiza, geometrii 19, no. 1 (2023): 211–46. http://dx.doi.org/10.15407/mag19.01.211.

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Yusof, NSB, SM Sapuan, MTH Sultan, and M. Jawaid. "Materials selection of “green” natural fibers in polymer composite automotive crash box using DMAIC approach in Six Sigma method." Journal of Engineered Fibers and Fabrics 15 (January 2020): 155892502092077. http://dx.doi.org/10.1177/1558925020920773.

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This article presents the results of research in selecting the most appropriate natural fibers to be used as reinforcement (normally in the hybrid form) in polymer composites for the automotive crash box by carrying out the materials selection process using the Six Sigma method. The judgment based on results from Six Sigma integrated two decision methods, which were applied using two different approaches in decision making: qualitative method and quantitative method. In this study, oil palm fiber had been proposed as the most appropriate natural fiber to be selected as reinforcement in composi

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TANGO, Hiroshi. "On Vector Bundles on $\mathbf{P}^n$ Which Have $\sigma$-Transition Matrices." Tokyo Journal of Mathematics 16, no. 1 (1993): 1–29. http://dx.doi.org/10.3836/tjm/1270128979.

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Neville, R. S., and S. Eldridge. "Transformations of sigma–pi nets: obtaining reflected functions by reflecting weight matrices." Neural Networks 15, no. 3 (2002): 375–93. http://dx.doi.org/10.1016/s0893-6080(02)00023-0.

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Diblík, Josef, Denys Ya Khusainov, Andriy Shatyrko, Jaromír Baštinec, and Zdeněk Svoboda. "Absolute Stability of Neutral Systems with Lurie Type Nonlinearity." Advances in Nonlinear Analysis 11, no. 1 (2021): 726–40. http://dx.doi.org/10.1515/anona-2021-0216.

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Abstract The paper studies absolute stability of neutral differential nonlinear systems x ˙ ( t ) = A x t + B x t − τ + D x ˙ t − τ + b f ( σ ( t ) ) , σ ( t ) = c T x ( t ) , t ⩾ 0 $$ \begin{align}\dot x(t)=Ax\left ( t \right )+Bx\left ( {t-\tau} \right ) +D\dot x\left ( {t-\tau} \right ) +bf({\sigma (t)}),\,\, \sigma (t)=c^Tx(t), \,\, t\geqslant 0 \end{align} $$ where x is an unknown vector, A, B and D are constant matrices, b and c are column constant vectors, 𝜏 > 0 is a constant delay and f is a Lurie-type nonlinear function satisfying Lipschitz condition. Absolute stability is analyzed

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Moriconi, M. "Integrable boundary conditions and reflection matrices for the O(N) nonlinear sigma model." Nuclear Physics B 619, no. 1-3 (2001): 396–414. http://dx.doi.org/10.1016/s0550-3213(01)00527-2.

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Borowiec, Andrzej, Hideki Kyono, Jerzy Lukierski, Jun-ichi Sakamoto та Kentaroh Yoshida. "Yang-Baxter sigma models and Lax pairs arising from κ-Poincaré r-matrices". Journal of High Energy Physics 2016, № 4 (2016): 1–29. http://dx.doi.org/10.1007/jhep04(2016)079.

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Ho, Charlotte Yuk-Fan, Bingo Wing-Kuen Ling, Joshua D. Reiss, and Xinghuo Yu. "Nonlinear Behaviors of Bandpass Sigma–Delta Modulators With Stable System Matrices." IEEE Transactions on Circuits and Systems II: Express Briefs 53, no. 11 (2006): 1240–44. http://dx.doi.org/10.1109/tcsii.2006.882805.

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Abdalla, E., M. C. B. Abdalla, and M. Forger. "Exact S-matrices for anomaly-free non-linear sigma models on symmetric spaces." Nuclear Physics B 297, no. 2 (1988): 374–400. http://dx.doi.org/10.1016/0550-3213(88)90025-9.

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Rançon, Adam, and Ivan Balog. "Exact generating function of a zero-dimensional supersymmetric non-linear sigma model." Journal of Statistical Mechanics: Theory and Experiment 2025, no. 7 (2025): 074003. https://doi.org/10.1088/1742-5468/ade86a.

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Abstract We compute exactly the generating function of a supersymmetric non-linear sigma model describing random matrices belonging to the unitary class. Although an arbitrary source explicitly breaks the supersymmetry, a careful analysis of the invariance of the generating function allows us to show that it depends on only three invariant functions of the source. This generating function allows us to recover various results found in the literature. It also questions the possibility of a functional renormalization group study of the three-dimensional Anderson transition.

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Duplij, Steven. "Polyadic sigma matrices." Journal of Mathematical Physics 65, no. 8 (2024). http://dx.doi.org/10.1063/5.0211252.

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We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU2 using cyclic shift block matrices. We introduce the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices. The so called elementary Σ-matrices are ordinary matrix units, their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The expression of n-ary SU2 in terms of full Σ-matrices is given using the Hadamard product. We then generalize the Pauli group in two ways:

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Hilberdink, T., and A. Pushnitski. "Spectral asymptotics for a family of LCM matrices." St. Petersburg Mathematical Journal, June 7, 2023. http://dx.doi.org/10.1090/spmj/1764.

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The family of arithmetical matrices is studied given explicitly by E ( σ , τ ) = { n σ m σ [ n , m ] τ } n , m = 1 ∞ , \begin{equation*} E(\sigma ,\tau )= \bigg \{\frac {n^\sigma m^\sigma }{[n,m]^\tau }\bigg \}_{n,m=1}^\infty , \end{equation*} where [ n , m ] [n,m] is the least common multiple of n n and m m and the real parameters σ \sigma and τ \tau satisfy ρ ≔ τ − 2 σ > 0 \rho ≔\tau -2\sigma >0 , τ − σ > 1 2 \tau -\sigma >\frac 12 , and τ > 0 \tau >0 . It is proved that E ( σ , τ ) E(\sigma ,\tau ) is a compact selfadjoint positive definite operator on ℓ 2 ( N ) \ell ^2(\m

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41

Liu, Heguo, and Jing Zhao. "On m-th roots of complex matrices." Electronic Journal of Linear Algebra, August 20, 2022, 457–62. http://dx.doi.org/10.13001/ela.2022.7047.

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For an $n\times n$ matrix $M$, $\sigma(M)$ denotes the set of all different eigenvalues of $M$. In this paper, we will prove two results on the $m$-th $(m\geq2)$ roots of a matrix $A$. Firstly, let $X$ be an $m$-th root of $A$. Then $X$ can be expressed as a polynomial in $A$ if and only if rank $X^2$= rank $X$ and $|\sigma(X)|=|\sigma(A)|$. Secondly, let $X$ and $Y$ be two $m$-th roots of $A$. If both $X$ and $Y$ can be expressed as polynomials in $A$, then $X=Y$ if and only if $\sigma(X)=\sigma(Y)$.

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42

Eichinger, Benjamin. "Periodic GMP Matrices." Symmetry, Integrability and Geometry: Methods and Applications, July 7, 2016. http://dx.doi.org/10.3842/sigma.2016.066.

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43

Haemers, Willem H., and Leila Parsaei Majd. "Spectral symmetry in conference matrices." Designs, Codes and Cryptography, March 18, 2021. http://dx.doi.org/10.1007/s10623-021-00858-8.

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AbstractA conference matrix of order n is an $$n\times n$$ n × n matrix C with diagonal entries 0 and off-diagonal entries $$\pm 1$$ ± 1 satisfying $$CC^\top =(n-1)I$$ C C ⊤ = ( n - 1 ) I . If C is symmetric, then C has a symmetric spectrum $$\Sigma $$ Σ (that is, $$\Sigma =-\Sigma $$ Σ = - Σ ) and eigenvalues $$\pm \sqrt{n-1}$$ ± n - 1 . We show that many principal submatrices of C also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel m

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Silantyev, Alexey. "Manin Matrices for Quadratic Algebras." Symmetry, Integrability and Geometry: Methods and Applications, July 12, 2021. http://dx.doi.org/10.3842/sigma.2021.066.

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We give a general definition of Manin matrices for arbitrary quadratic algebras in terms of idempotents. We establish their main properties and give their interpretation in terms of the category theory. The notion of minors is generalised for a general Manin matrix. We give some examples of Manin matrices, their relations with Lax operators and obtain the formulae for some minors. In particular, we consider Manin matrices of the types B, C and D introduced by A. Molev and their relation with Brauer algebras. Infinite-dimensional Manin matrices and their connection with Lax operators are also c

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Konno, Hitoshi. "Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations." Symmetry, Integrability and Geometry: Methods and Applications, December 19, 2006. http://dx.doi.org/10.3842/sigma.2006.091.

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Berg, Christian, and Ryszard Szwarc. "Inverse of Infinite Hankel Moment Matrices." Symmetry, Integrability and Geometry: Methods and Applications, October 6, 2018. http://dx.doi.org/10.3842/sigma.2018.109.

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47

BEKTAŞ, Özcan. "Some new properties of octonionic matrices." Sigma Journal of Engineering and Natural Sciences – Sigma Mühendislik ve Fen Bilimleri Dergisi, 2023. http://dx.doi.org/10.14744/sigma.2023.00092.

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48

Etingof, Pavel. "Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices." Symmetry, Integrability and Geometry: Methods and Applications, March 14, 2007. http://dx.doi.org/10.3842/sigma.2007.048.

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49

Liu, Jie, Guohai Jin, and Buhe Eerdun. "On the symplectic self-adjointness and residual spectral emptiness of upper triangular Hamiltonian operator matrices." Annals of Functional Analysis 15, no. 3 (2024). http://dx.doi.org/10.1007/s43034-024-00367-4.

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AbstractThis paper deals with the symplectic self-adjointness and residual spectral emptiness of upper triangular Hamiltonian operator matrices $$H=\left( {\begin{matrix}A&{}B\\ 0&{}-A^*\end{matrix}}\right) $$ H = A B 0 - A ∗ . First, for symplectic self-adjoint Hamiltonian operator H, based on detailed classification of point spectrum $$\sigma _p(H)$$ σ p ( H ) and residual spectrum $$\sigma _r(H)$$ σ r ( H ) , the symmetry about imaginary axis is given between $$\sigma _p(H)$$ σ p ( H ) , $$\sigma _r(H)$$ σ r ( H ) , deficiency spectrum $$\sigma _{\delta }(H)$$ σ δ ( H ) , compressio

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Lentner, Simon, and Tobias Ohrmann. "Factorizable R-Matrices for Small Quantum Groups." Symmetry, Integrability and Geometry: Methods and Applications, September 25, 2017. http://dx.doi.org/10.3842/sigma.2017.076.

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