[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

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Computational Methods for Linear Matrix Equations

$$\\left[ {\\begin{array}{*{20}{c}} {10} \\\\ { - v{{a}^{{ - 1}}}1} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}{c}} {au} \\\\ {vb} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}{c}} {1 - {{a}^{{ - 1}}}u} \\\\ {01} \\\\ \\end{array} } \\right] = \\left[ {\\begin{array}{*{20}{c}} {a0} \\\\ {0b - v{{a}^{{ - 1}}}u} \\\\ \\end{array} } \\right]$$ \n \n(1.1) \n \nprovided a is regular.

Determinants and matrices. By A. C. Aitken. Pp. vii, 135. 4s. 6d. 1939. University Mathematical Texts, 1. (Oliver and Boyd)

Quantum coherence and geometric quantum discord

Measurement-induced nonlocality (MIN), which describes the maximum global effect caused by locally invariant measurements, was introduced by Luo and Fu (Phys. Rev. Lett. , 106 (2011) 120401). In this paper, a new measure of MIN based on Wigner-Yanase skew information is proposed. It is shown that this measure not only has good computability but also eliminates the noncontractivity problem appearing in the original measure of MIN defined by the Hilbert-Schmidt norm. The analytical formulas of MIN based on Wigner-Yanase skew information for any pure states, -dimensional mixed states, and some higher-dimensional symmetric states are presented. Furthermore, the tight upper bound to MIN based on Wigner-Yanase skew information in the general case is also derived.

https://iopscience.iop.org/article/10.1209/0295-5075/114/10007/pdf

Measurement-induced nonlocality based on Wigner-Yanase skew information

A simultaneous decomposition for seven matrices with applications

In this paper, for a consistent quaternion matrix equation AXB = C, the formulas are established for maximal and minimal ranks of real matrices X1, X2, X3, X4 in solution X = X1 + X2i + X3j + X4k. A necessary and sufficient condition is given for the existence of a real solution of the quaternion matrix equation. The expression is also presented for the general solution to this equation when the solvability conditions are satisfied. Moreover, necessary and sufficient conditions are given for this matrix equation to have a complex solution or a pure imaginary solution. As applications, the maximal and minimal ranks of real matrices E, F, G, H in a generalized inverse (A +Bi + Cj + Dk)- = E + Fi + Gj + Hk of a quaternion matrix A + Bi + Cj + Dk are also considered. In addition, a necessary and sufficient condition is derived for the quaternion matrix equations A1XB1 = C1 and A2XB2 = C2 to have a common real solution.

Extreme Ranks of Real Matrices in Solution of the Quaternion Matrix Equation AXB = C with Applications

In this paper, we discuss Hermitian solutions of split quaternion matrix equation $$AXB+CXD=E,$$
 where X is an unknown split quaternion Hermitian matrix, and A, B, C, D, E are known split quaternion matrices with suitable size. The objective of this paper is to establish a necessary and sufficient condition for the existence of a solution and a solution formulas. Moreover, we provide numerical algorithms and numerical examples to exemplify the results.

On Hermitian Solutions of the Split Quaternion Matrix Equation $$AXB+CXD=E$$ A X B + C X D = E

The biconjugate gradients (BiCG) and biconjugate residual (BiCR) methods are attractive ways for solving nonsymmetric linear system A x = b . In this paper, the BiCG and BiCR methods are extended to solve the high order Stein tensor equation. The convergent properties of the developed iterative algorithms are studied. Numerical examples are provided to confirm the theoretical results, which demonstrate that the proposed algorithms are effective and feasible for solving the Stein tensor equation.

Qing-Wen Wang

Papers

Measurement-induced nonlocality based on Wigner-Yanase skew information

A simultaneous decomposition for seven matrices with applications

Extreme Ranks of Real Matrices in Solution of the Quaternion Matrix Equation AXB = C with Applications

On Hermitian Solutions of the Split Quaternion Matrix Equation $$AXB+CXD=E$$ A X B + C X D = E

Extending BiCG and BiCR methods to solve the Stein tensor equation