[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

Common Hermitian solutions to some operator equations on Hilber C∗-modules

On the Hermitian solutions to a system of adjointable operator equations

A necessary and sufficient condition is given for the quaternion matrix equations AiX + YB i = Ci (i =1 , 2) to have a pair of common solutions X and Y. As a consequence, the results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear matrix equation AX + YB = C, Comput. Math. Appl., 52 (2006), pp. 861-872).

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A new solvable condition for a pair of generalized Sylvester equations.

We in this paper give a decomposition concerning the general matrix triplet over an arbitrary division ring $\mathcal{F}$
 with the same row or column numbers. We also design a practical algorithm for the decomposition of the matrix triplet. As applications, we present necessary and sufficient conditions for the existence of the general solutions to the system of matrix equations $DXA = C_1 , EXB = C_2 , FXC = C_3$
 and the matrix equation $AXD + BYE + CZF = G$
 over $\mathcal{F}$
 . We give the expressions of the general solutions to the system and the matrix equation when the solvability conditions are satisfied. Moreover, we present numerical examples to illustrate the results of this paper. We also mention the other applications of the equivalence canonical form, for instance, for the compression of color images.

Qing-Wen Wang

Papers

Common Hermitian solutions to some operator equations on Hilber C∗-modules

On the Hermitian solutions to a system of adjointable operator equations

A new solvable condition for a pair of generalized Sylvester equations.

An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications

Bicyclic graphs with small positive index of inertia