$G$-space of isotropic directions and $G$-spaces of $ \varphi $-scalars with $G=O( n,1,\mathbb {R}) $

Aleksander Misiak; Eugeniusz Stasiak

Displaying similar documents to “ $G$ -space of isotropic directions and $G$ -spaces of $ϕ$ -scalars with $G = O (n, 1, ℝ)$ ”

G-matrices, $J$ -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_{1}$ and $D_{2}$ such that $A^{- T} = D_{1} A D_{2}$ , where $A^{- T}$ denotes the transpose of the inverse of $A$ . Denote by $J = diag (\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+ 1$ or $- 1$ . A nonsingular real matrix $Q$ is called $J$ -orthogonal if $Q^{T} J Q = J$ . Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

A treatment of a determinant inequality of Fiedler and Markham

Minghua Lin (2016)

Czechoslovak Mathematical Journal

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Fiedler and Markham (1994) proved ${(\frac{\det \hat{H}}{k})}^{k} \geq \det H,$ where $H = {(H_{i j})}_{i, j = 1}^{n}$ is a positive semidefinite matrix partitioned into $n \times n$ blocks with each block $k \times k$ and $\hat{H} = {(tr H_{i j})}_{i, j = 1}^{n}$ . We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $\det (I_{n} + \hat{H}) \geq \det {(I_{n k} + k H)}^{1 / k} .$

Analytic aspects of the circulant Hadamard conjecture

Teodor Banica, Ion Nechita, Jean-Marc Schlenker (2014)

Annales mathématiques Blaise Pascal

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We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $| q_{0} | = ... = | q_{N - 1} | = 1$ the quantity $Φ = \sum_{i + k = j + l} \frac{q_{i} q_{k}}{q_{j} q_{l}}$ satisfies $Φ \geq N^{2}$ , with equality if and only if $q = (q_{i})$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $Φ$ , (2) the study of the critical points of $Φ$ , and (3) the computation of the moments of $Φ$ . We explore here...

Ground states of supersymmetric matrix models

Gian Michele Graf (1998-1999)

Séminaire Équations aux dérivées partielles

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We consider supersymmetric matrix Hamiltonians. The existence of a zero-energy bound state, in particular for the $d = 9$ model, is of interest in M-theory. While we do not quite prove its existence, we show that the decay at infinity such a state would have is compatible with normalizability (and hence existence) in $d = 9$ . Moreover, it would be unique. Other values of $d$ , where the situation is somewhat different, shall also be addressed. The analysis is based on a Born-Oppenheimer approximation....

Factorization of CP-rank- $3$ completely positive matrices

Jan Brandts, Michal Křížek (2016)

Czechoslovak Mathematical Journal

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A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A = B B^{⊤}$ . If $B$ is such a matrix with a minimal number $p$ of columns, then $p$ is called the cp-rank of $A$ . In this paper we develop a finite and exact algorithm to factorize any matrix $A$ of cp-rank $3$ . Failure of this algorithm implies that $A$ does not have cp-rank $3$ . Our motivation stems from the question if there exist three nonnegative polynomials of degree at...

On isomorphisms between fibrations over $T^{k}$ associated with a $T^{k}$ action

Michał Sadowski (1991)

Annales Polonici Mathematici

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The real symmetric matrices of odd order with a P-set of maximum size

Zhibin Du, Carlos M. da Fonseca (2016)

Czechoslovak Mathematical Journal

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Suppose that $A$ is a real symmetric matrix of order $n$ . Denote by $m_{A} (0)$ the nullity of $A$ . For a nonempty subset $α$ of ${1, 2, ..., n}$ , let $A (α)$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $α$ . When $m_{A (α)} (0) = m_{A} (0) + | α |$ , we call $α$ a P-set of $A$ . It is known that every P-set of $A$ contains at most $⌊ n / 2 ⌋$ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As...

Note on the variance of the sum of Gaussian functionals

Marek Beśka (2010)

Applicationes Mathematicae

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Let $(X_{i}, i = 1, 2, . . .)$ be a Gaussian sequence with $X_{i} \in N (0, 1)$ for each i and suppose its correlation matrix $R = {(ρ_{i j})}_{i, j \geq 1}$ is the matrix of some linear operator R:l₂→ l₂. Then for $f_{i} \in L ² (μ)$ , i=1,2,..., where μ is the standard normal distribution, we estimate the variation of the sum of the Gaussian functionals $f_{i} (X_{i})$ , i=1,2,... .

Linear preservers of row-dense matrices

Sara M. Motlaghian, Ali Armandnejad, Frank J. Hall (2016)

Czechoslovak Mathematical Journal

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Let $𝐌_{m, n}$ be the set of all $m \times n$ real matrices. A matrix $A \in 𝐌_{m, n}$ is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions $T : 𝐌_{m, n} \to 𝐌_{m, n}$ that preserve or strongly preserve row-dense matrices, i.e., $T (A)$ is row-dense whenever $A$ is row-dense or $T (A)$ is row-dense if and only if $A$ is row-dense, respectively. Similarly, a matrix $A \in 𝐌_{n, m}$ is called a column-dense matrix if every column of $A$ is a column-dense vector. At the end, the structure...

Computing the greatest $𝐗$ -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector $x$ is said to be an eigenvector of a square max-min matrix $A$ if $A \otimes x = x$ . An eigenvector $x$ of $A$ is called the greatest $𝐗$ -eigenvector of $A$ if $x \in 𝐗 = {x; \underset{̲}{x} \leq x \leq \bar{x}}$ and $y \leq x$ for each eigenvector $y \in 𝐗$ . A max-min matrix $A$ is called strongly $𝐗$ -robust if the orbit $x, A \otimes x, A^{2} \otimes x, \dots$ reaches the greatest $𝐗$ -eigenvector with any starting vector of $𝐗$ . We suggest an $O (n^{3})$ algorithm for computing the greatest $𝐗$ -eigenvector of $A$ and study the strong $𝐗$ -robustness. The necessary and sufficient conditions for strong $𝐗$ -robustness are introduced...

Displaying similar documents to “ G -space of isotropic directions and G -spaces of ϕ -scalars with G = O ( n , 1 , ℝ ) ”

Displaying similar documents to “ $G$ -space of isotropic directions and $G$ -spaces of $ϕ$ -scalars with $G = O (n, 1, ℝ)$ ”