富永 雅

本年度、2017年3月に小学校学習指導要領が、続く6月にその解説が改訂された。中でも、新たな領域「Dデータの活用」での第6学年における学習事項では、現行同様、度数分布表・柱状グラフが扱われるとはいえ、統計的な問題解決活動や結論の妥当性についての批判的な考察が取り入れられ、新たな教科書には大きな改変が見込まれる。本稿では、現行の教科書に関する分析を更に行い、また次期学習指導要領・同解説を精査し、特徴や問題点を吟味する。このことは次期教科書に求められる対応につながる考察となる。

小学校算数第5学年で学習する「図形の合同」において、各教科書を精査し、学習指導要領との整合性や教科書間における差異を抽出する。結果として、教科書記載内容・表現の問題点を挙げ、既存の学習内容を教科書の分析・比較の観点から考察し、実際に指導する上での留意点を明らかにする。

Let C(p) be the Schatten p-class for p > 0. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if A = {A(1),A(2),...,A(n)} and B = {B1, B2,..., B(n)} are two sets of operators in C(2), then (n)Sigma(i,j=1) parallel to A(i) - A(j) parallel to (2)(2) + (n)Sigma(i,j=1) parallel to B(i) - Bj parallel to(2). (n)Sigma(i,j=1) parallel to A(i) - B(j) parallel to (2)(2) - 2 parallel to(n)Sigma(i=1) (A(i) -B(i))parallel to(2)(2) . In this paper, we give generalizations of this as pairs of inequalities for Schatten p-norms, which hold for certain values of p and reduce to the equality above for p = 2. Moreover, we present some related inequalities for three sets of operators. (C) 2011 Elsevier Inc. All rights reserved.

A bounded linear operator acting on a Hilbert space is a generalized quadratic operator if it has an operator matrix of the form [aI cI dT* bI] It reduces to a quadratic operator if d = 0. In this article, spectra, norms and various kinds of numerical ranges of generalized quadratic operators are determined. Some operator inequalities are also obtained. In particular, it is shown that for a given generalized quadratic operator, the rank-k numerical range, the essential numerical range and the q-numerical range are elliptical discs; the c-numerical range is a sum of elliptical discs. The Davis-Wielandt shell is the convex hull of a family of ellipsoids unless the underlying Hilbert space has dimension 2.

Dunkl and Williams showed that for any nonzero elements x, y in a normed linear space X parallel to x/parallel to x parallel to - y/parallel to y parallel to parallel to <= 4 parallel to x - y parallel to/parallel to x parallel to + parallel to y parallel to, Pecaric and Rajic gave a refinement and, moreover, a generalization to operators A, B is an element of B(H) such that vertical bar A vertical bar, vertical bar B vertical bar are invertible as follows: vertical bar A vertical bar A vertical bar(-1) - B vertical bar B vertical bar(-1)vertical bar(2) <=vertical bar A vertical bar(-1) (p vertical bar A - B vertical bar(2) + q(vertical bar A vertical bar - vertical bar B vertical bar)(2)) vertical bar A vertical bar(-1), where p, q > 1 with 1/p + 1/q = 1. In this note, we shall investigate the inequality and also equality conditions under the assumption that the existence of vertical bar A vertical bar(-1) and vertical bar B vertical bar(-1) is not required. Moreover, we give a refinement of the equality conditions. (C) 2010 Elsevier Inc. All rights reserved.

In this paper, we present a complement of a generalized Ando-Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando-Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541-545]. Let A and B be positive operators on a Hilbert space H such that 0 < m(1) <= A <= M-1 and 0 < m(2) <= B <= M-2 for some scalars m(i) <= M-i (i = 1,2), and alpha epsilon [0, 1]. Put h(i) = M-i/m(i) for i =1, 2. Then for each 0 < r <= 1 and s >= 1 [GRAPHICS] where A#(alpha) B := A(1/2)(A(-1/2) BA(-1/2))(alpha) A(1/2) is the alpha-geometric mean and a generalized Kantorovich constant K (h, p) is defined for It > 0 as K (h,p) := h(p)-h/(p-1)(h-1) (p-1/p h(p)-1/h(p)-h)(p) for all real numbers p epsilon R. (C) 2008 Elsevier Inc. All rights reserved.

The Cordes inequality was extended by using concave or convex functions. In this note, we give reverse inequalities of its extended ones for an increasing strictly concave submultiplicative function. As an application, we obtain a generalization of Bourin's inequality which gives an estimation of operator norm by spectral radius.

We shall give a norm inequality equivalent to the grand Furuta inequality, and moreover show its reverse as follows: Let A and B be positive operators such that 0 < m <= B <= M for some scalars 0 < m < M and h :(-) M/m > 1. Then parallel to A(1/2) {A(-t/2) (A(r/2) B((r-t){(p-t) s+r}/1-t+r) A(r/2))(1/s) A(-t/2)}(1/p) A(1/2)parallel to <= K(h(r-t), (p-t)s+r/1-t+r)(1/ps) parallel to A(1-t+r/2) Br-t A(1-t+r/2) parallel to((p-t)s+r/ps(1-t+r)) for 0 <= t <= 1, p >= 1, s >= 1 and r >= t >= 0, where K(h,p) is the generalized Kantorovich constant. As applications, we consider reverses related to the Ando-Hiai inequality.

We improve Bebiano-Lemos-Providencia inequality: For A, B >= 0 parallel to A(1+t/2) B-t A(1+t/2)parallel to <= parallel to A(1/2) (A(s/2) B-s A(s/2))(t/s) A(1/2)parallel to for all s >= t >= 0. In our discussion, Furuta inequality plays an essential role. Actually we have parallel to A(1s+/2) B1+s A(1+s/2)parallel to(p+s/p(1+s)) <= parallel to A(1/2) (A(s/2) Bp+s A(s/2))(1/p) A(1/2)parallel to for p >= 1 and s >= 0. (C) 2007 Elsevier Inc. All rights reserved.

We give the maximum of the difference D-p(a, b; w) := ((k=1)Sigma(n) w(k)a(k)(p))(1/p) ((k=1)Sigma(n) w(k)b(k)(p)) - (k=1)Sigma(n) w(k)a(k)b(k) derived from a weighted Holder's inequality for p, q > 1, p(-1) + q(-1) = 1 and for positive n-tuples a:= (a(1),..., a(n)), b:= (b(1),..., b(n)) and a weight w := (w(1),..., w(n)) under certain conditions. The discussion in this note is simpler than our previous ones. It comes from the arrangement of a given weight and a linearization of D-p(a, b; w) via Young's inequality. As a consequence, we give a, b and w which attain the maximum.

For a positive operator A with 0 < m less than or equal to A less than or equal to M (m, M is an element of R), the Young operator inequality gives as follows: lambdaA + (1 - lambda) greater than or equal to A(lambda) for lambda is an element of [0, 1]. In this note, we prove that the estimation of the converse Young operator inequality is obtained by using Specht's ratio S(t) = t 1/t-1/e log t 1/t-1 and the logarithmic mean L(s, t) = t-s/log t - log s (s, t > 0), that is, we have for a given p under some conditions pA(lambda) + max {L(1, m) log S(m)/p, L(1, M) log S(M)/p} greater than or equal to lambdaA + (1 - lambda) (greater than or equal to A(lambda)) for lambda is an element of [0, 1]. Moreover by using operator means, we consider the converse Young operator inequality related to two operators A and B. Furthermore we discuss reverse inequalities of the Holder-McCarthy inequality and the inequality on the concavity of the logarithmic function.

As a converse of the arithmetic and geometric mean inequality, Specht gave the ratio of the arithmetic one by the geometric one in 1960. We can reap the rich harvest of the Specht ratio in operator theory. In this paper, we shall present other characterizations of the chaotic order and the usual one associated with Kantorovich type inequalities via the Specht ratio. Among others, as an application of the grand Fururta inequality, we show that if A and B are positive operators and k greater than or equal to A greater than or equal to 1/k for some k greater than or equal to 1, then A greater than or equal to B is equivalent to S-k(2(p - 1)s)(2/s) A(p) greater than or equal to B-P holds for all p greater than or equal to 1, s greater than or equal to 1 such that p - 1 greater than or equal to 1/s, where the Specht ratio S-k(r) is defined for each r > 0 as S-k(r) = (k(r) - 1)k(r/kr-1)/ r e log k (k > 0, k not equal 1) and S-1(r) = 1. (C) 2003 Elsevier Inc. All rights reserved.

In our previous paper, we obtained a reverse Holder's type inequality which gives an upper bound of the difference: (Sigma a(k)(p))(1/p)(Sigma b(k)(q))(1/q) - lambdaSigma a(k)b(k) with a parameter lambda > 0, for n-tuples a = (a(1),..,a(n)) and b = (b(1),....b(n)) of positive numbers and for p > 1, q > 1 satisfying 1/p + 1/q = 1. In this paper for commutative positive operators A and B on a Hilbert space H and a unit vector x is an element of H, we give an upper bound of the difference <A(p)x, x>(1/p)<B(q)x, x>(1/q) - lambda<ABx, x>. As applications, considering special cases, we induce some difference and ratio operator inequalities. Finally, using the geometric mean in the Kubo-Ando theory we shall give a reverse Holder's type operator inequality for noncommutative operators.

We prove a Golden-Thompson type inequality via Specht's ratio: Let H and K be selfadjoint operators on a Hilbert space H satisfying M1 greater than or equal to H, K greater than or equal to mI for some scalar M > m. Then M-h(1) (1 - lambda)e(tH) + lambdae(tK))(1/l) greater than or equal toe((1-lambda)H+lambdaK) greater than or equal to M-h(1)M--1(h)(t)(-1/l) ((1 - lambda)e(tH) + lambdae(tK))(1/l) holds for all t > 0 and 0 less than or equal to lambda less than or equal to 1, where h = e(M-m) and (generalized) Specht's ratio M-h(t) is defined for h > 0 as [GRAPHICS] (h not equal 1) and M1(1) = 1.

Using a technique due to Ozeki, we give an upper bound of (Sigmaa(k)(p))(1/p) (Sigmab(k)(q))(1/q) - lambda Sigmaa(k)b(k) for lambda > 0, for p > 1, q > 1 satisfying 1/p + 1/q = 1, and for n - tuples a = (a(1),...,a(n),) and b = (b(1),..., b(n)) of positive numbers under certain conditions. This yields a complement of Holder's inequality. The estimation with a parameter lambda enables us to unify the discussions on difference and ratio inequalities derived from Holders inequality.