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Search: All articles in the CMB digital archive with keyword matrix

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1. CMB Online first

Lin, Minghua
 A determinantal inequality involving partial traces Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes \mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved an inequality for Schatten $p$-norms: $1+\|\mathbf{A}\|_p\ge \|\tr_1 \mathbf{A}\|_p+\|\tr_2 \mathbf{A}\|_p,$ where $\tr_1, \tr_2$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality $1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.$ Keywords:determinantal inequality, partial trace, block matrixCategories:47B65, 15A45, 15A60

2. CMB 2014 (vol 58 pp. 196)

Yang, Qingjie; Zhong, Weiting
 Dihedral Groups of order $2p$ of Automorphisms of Compact Riemann Surfaces of Genus $p-1$ In this paper we prove that there is only one conjugacy class of dihedral group of order $2p$ in the $2(p-1)\times 2(p-1)$ integral symplectic group can be realized by an analytic automorphism group of compact connected Riemann surfaces of genus $p-1$. A pair of representative generators of the realizable class is also given. Keywords:dihedral group, automorphism group, Riemann surface, integral symplectic matrix, fundamental domainCategories:20H25, 57M60

3. CMB 2012 (vol 57 pp. 25)

Bourin, Jean-Christophe; Harada, Tetsuo; Lee, Eun-Young
 Subadditivity Inequalities for Compact Operators Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings. Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalitiesCategories:47A63, 15A45

4. CMB 2011 (vol 56 pp. 92)

Jacob, Benoît
 On Perturbations of Continuous Maps We give sufficient conditions for the following problem: given a topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and a continuous map $f$ from $X$ to $Y$, is it possible, by applying to $f$ an arbitrarily small perturbation, to ensure that $f(X)$ does not meet $Z$? We also give a relative variant: if $f(X')$ does not meet $Z$ for a certain subset $X'\subset X$, then we may keep $f$ unchanged on $X'$. We also develop a variant for continuous sections of fibrations and discuss some applications to matrix perturbation theory. Keywords:perturbation theory, general topology, applications to operator algebras / matrix perturbation theoryCategory:54F45

5. CMB 2009 (vol 52 pp. 145)

Wang, Z.; Chen, J. L.
 $2$-Clean Rings A ring $R$ is said to be $n$-clean if every element can be written as a sum of an idempotent and $n$ units. The class of these rings contains clean rings and $n$-good rings in which each element is a sum of $n$ units. In this paper, we show that for any ring $R$, the endomorphism ring of a free $R$-module of rank at least 2 is $2$-clean and that the ring $B(R)$ of all $\omega\times \omega$ row and column-finite matrices over any ring $R$ is $2$-clean. Finally, the group ring $RC_{n}$ is considered where $R$ is a local ring. Keywords:$2$-clean rings, $2$-good rings, free modules, row and column-finite matrix rings, group ringsCategories:16D70, 16D40, 16S50

6. CMB 2009 (vol 52 pp. 95)

Miranian, L.
 Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel--Darboux formulas are presented for the first time. Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, Christoffel-DarbouxCategory:42C99

7. CMB 2006 (vol 49 pp. 281)

Ragnarsson, Carl Johan; Suen, Wesley Wai; Wagner, David G.
 Correction to a Theorem on Total Positivity A well-known theorem states that if $f(z)$ generates a PF$_r$ sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f(z)$ generates a PF sequence then $1/f(-z)$ generates a PF sequence. Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur functionCategories:15A48, 15A45, 15A57, 05E05

8. CMB 2000 (vol 43 pp. 145)

Chang, Jae-Ho; Lee, Sang Youl; Park, Chan-Young
 On the 2-Parallel Versions of Links In this paper, we show that the absolute value of the signature of the $2$-parallel version of a link is less than or equal to the nullity of it and show that the signature, nullity, and Minkowski units of the $2$-parallel version of a certain class of links are always equal to $0$, $2$, and $1$ respectively. Keywords:braid, Goeritz matrix, Minkowski unit, nullity, signature, 2-parallel versionCategory:57M25
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