Yuki Seo

type:Article 本研究は,「GIGAスクール構想」の実現のために「ソフト」面に焦点をあて,デジタル教科書ならではの学びの充実をはかり,将来の学習者用デジタル教科書の有効な利活用も視野に入れつつ,現在配布されている指導者用デジタル教科書を用いた数学と理科の大学及び附属校の授業での利活用を含めたICTを効果的に活用する学習活動や教材化のための素案及び指導者用デジタル教科書を用いた授業実践の分析・考察を行うことを目的とする。 The purposes of this study are the following two points: (1) To focus on the soft wear aspects in order to realize the GIGA school concept and to enhance the learning that only digital textbooks can provide. (2) To analyze and discuss learning activities and teaching materials that effectively utilize ICT, including the use of digital textbooks for teachers in mathematics and science classes at universities and attached schools, as well as classroom practices using digital textbooks for teachers.

There are many proposals for defining a quantum version of the Rényi divergence on the differentiable manifold [Formula: see text] of positive definite matrices. The geodesic connecting two points on [Formula: see text] is a weighted matrix geometric mean, and its tangent vector is the relative operator entropy. It is a natural idea to use the relative operator entropy to describe the difference between two points on [Formula: see text]. The quantum version of the classical results of Rényi divergence does not always hold due to its non-commutativity. Due to this difficulty, we use the upper bound and the lower one of the spectrum of positive definite matrices to estimate the order relation between quantum divergences, which is called the Mond Pečarić method in operator theory. In this paper, we show fundamental properties of the ♮ _α-Rényi divergence of real order [Formula: see text], including monotonicity of order, the relation of max- and min-divergence, data processing inequality, bound estimates, and convexity. We also estimate the difference without the α– z-Rényi divergence and ♮ _α-Rényi divergence in terms of the generalized Kantorovich constant and the Specht ratio.

Abstract For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

We improve the existing Ando–Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie–Trotter formula and consider the extension problem of operator perspectives to non-invertible positive operators.

In this paper, we show matrix trace inequalities on the quantum Tsallis relative entropy of negative order, which includes the quasi geometric mean of positive definite matrices. Moreover, we show an estimate of the difference between two Tsallis relative entropies of negative order.

In this paper, we show an interpolation of Davis-Choi-Jensen operator inequality and the converse inequality for Hilbert space operators. As applications, we obtain an interpolation of quasi-arithmetic mean inequalities and the converse inequalities.

In this paper, by virtue of the matrix geometric mean and the polar decomposition, we present new Wielandt type inequalities for matrices of any size. To this end, based on results due to J.I. Fujii, we reform a matrix Cauchy–Schwarz inequality, which differs from ones due to Marshall and Olkin. As an application, we show a new block matrix version of Wielandt type inequalities under the block rank additivity condition.

In this paper, we show operator versions of the inequality due to Cho, Matić and Pečarić in connection to Jensen's inequality for convex functions. As applications, we obtain an interpolation of the weighted arithmetic-geometric mean inequality for the Karcher mean of positive invertible operators on a Hilbert space. Moreover, we obtain an interpolation between the quasi-arithmetic means.

Lawson and Lim showed that the Karcher equation for positive invertible operators on a Hilbert space has a unique solution using the method of the implicit function theorem of a Banach space. In this paper, in the framework of the operator inequality, we show the equivalence of the unique solution of the Karcher equation and the self-adjointness of the Karcher mean. For this, we reform the notion of the operator power means of negative order by virtue of the Tsallis relative operator entropy of negative order.

Lim and Palfia established the notion of the matrix power means for k positive definite matrices (k >= 3) and showed that the matrix power means have the information monotonicity for a unital positive linear mapping. In this note, by virtue of the generalized Kantorovich constant, we show counterparts to the information monotonicity of the matrix power means. (C) 2017 Elsevier Inc. All rights reserved.

In this paper, we show several operator inequalities involving the Hadamard product and the Karcher mean of n (>= 3) positive invertible operators on a separable Hilbert space, which are regarded as an n-variable operator version of results due to Ando and Aujla-Vasudeva. As applications, we show estimates from above for an n-variable version of the Fiedler-type theorem due to Fujii. Moreover, we show an n-variable version of the majorization relation due to Ando for the Hadamard product via the Karcher mean of n (>= 3) positive-definite matrices.

In this paper, we refine the Heinz mean inequality for singular values and give some generalizations of Audenaert-Zhan inequality for singular values and Zhan's conjecture for the case of negative t. Among others, we show that if A, B is an element of M-n are positive semidefinite and f, g are real valued continuous functions on [0, infinity) such that g is monotone and f (g(-1)(root t))(2) is operator monotone on [0, infinity), then s(j)(f(A)(g(A)(2) + g(B)(2))f (B)) <= 1/2s(j)(f(A)(2)g(A)(2) + f(B)(2)g(B)(2)) for j = 1,..., n, where s(j) are the singular values in decreasing order.

In this paper, some operator inequalities related to the solidarity and the generalized Tsallis relative operator entropy are shown. As an application, we show the Shannon type operator inequalities and its reverse in terms of the generalized Tsallis relative operator entropy.

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando-Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.

The main aim of this survey article is to present recent developments of matrix versions of the arithmetic geometric mean inequality. Among others, we show improvements and generalizations of the arithmetic geometric mean inequality for unitarily invariant norms via the Hadamard product, and for singular values via the operator monotone functions.