• Korean Journal of Mathematics
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• v.16 no.4
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• pp.565-571
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• 2008

In this paper we present some trace inequalities for positive definite matrices in statistical mechanics. In order to prove the method of the uniform bound on the generating functional for the semi-classical model, we use some trace inequalities and matrix norms and properties of trace for positive definite matrices.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.373-411
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• 2023

Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between trace n Cappell-Shaneson matrices and trace 5 - n Cappell-Shaneson matrices which was suggested by Gompf experimentally. Using this symmetry, we prove that Gompf conjecture for the trace n case is equivalent to the trace 5 - n case. We confirm Gompf conjecture for the special cases that -64 ≤ trace ≤ 69 and corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We also give a new infinite family of Cappell-Shaneson spheres which are diffeomorphic to the standard 4-sphere.

• Korean Journal of Mathematics
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• v.24 no.2
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• pp.273-296
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• 2016

Some new trace inequalities for convex functions of self-adjoint operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated. Some trace inequalities for matrices are also derived. Examples for the operator power and logarithm are presented as well.

• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.979-982
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• 2022

This paper covers some important operator inequalities connected with traces of matrices, from the classical inequalities we obtained new inequalities connected with traces of matrices which are better than the others.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.583-602
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• 1995

With the theory of a certain type of orders in a Quaternion algebra, we construct Brandt matrices and theta series. As a application, we calculate the class number of a certain type of orders in a Quanternion algebra with the trace formular of Brandt matrices.

• Korean Journal of Mathematics
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• v.26 no.3
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• pp.349-371
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• 2018

Some inequalities for quantum f-divergence of matrices are obtained. It is shown that for normalised convex functions it is nonnegative. Some upper bounds for quantum f-divergence in terms of variational and ${\chi}^2-distance$ are provided. Applications for some classes of divergence measures such as Umegaki and Tsallis relative entropies are also given.

• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.903-916
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• 1996

In this paper we investigate the minimum permanents and minimizing matrices on the faces $\omega(D)$ of $\omega_n$ for two fully indecomposable (0, 1) matrices D which are slight changes of both a convertible matrix and the matrix with zero trace.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1205-1211
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• 2010

In the present article, we investigate the possibility of a real-valued map on the space of tuples of commuting trace-class self-adjoint operators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with continuous group homomorphisms from the Milnor's K-group of the real numbers into the additive group of real numbers. Using this connection, it is shown that any such trace map must be trivial, but it is proposed that the target group of a nontrivial trace should be a linearized version of Milnor's K-theory as with the case of universal determinant for commuting tuples of matrices rather than just the field of constants.

• Analytical Science and Technology
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• v.9 no.3
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• pp.253-261
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• 1996

Analytical method for the determination of trace germanium in inorganic matrices by differential pulse polarography(DPP) was studied. The reduction peak of germanium(IV) in perchloric acid solution containing 1, 2, 3-trihydroxy benzene appeared at -0.45V(vs. Ag/AgCl) and the peak current for germanium complex varied linearly with concentration variation. Factors affecting sensitivity and precision for germanium quantification were studied and detection limit under the investigated parameters was 1ng/ml. Inorganic samples were decomposed by fusion with potassium pyrosulfate. Serious interferences of Se(IV), Pb(II), As(III) for the determination of germanium were discussed. Interferences of these elements could be avoided by extraction of germanium from decomposed matrices by $CCl_4$ in 10M HCl solution. The germanium contents of inorganic samples(Pb bf. dust, Cu bf. dust, gneiss, Cu anode slime) were determined by the above method.

• Communications of the Korean Mathematical Society
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• v.31 no.1
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• pp.177-184
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• 2016

In this paper, we establish some conjugacy criteria of $M\ddot{o}bius$ groups in infinite dimension by using Clifford matrices. This extends the corresponding known results in finite dimensional setting.