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Let σ be an operator mean in the sense of Kubo and Ando. If the representation function fσ of σ satisfies fσ(t)p≤fσ(tp)for allp>1,then σ is called a pmi mean. Our main interest is the class of pmi means (denoted by PMI). To study PMI, the operator means σ satisfying fσ(xy)≤fσ(x)fσ(y)(x,y>0)are considered in this paper. The set of such means (denoted by GCV) includes certain significant examples and is included in PMI. The main result presented in this paper is that GCV is a proper subset of PMI. In addition, we investigate certain operator-mean classes including PMI.

Background Antipsychotics are essential in the acute treatment of and maintenance therapy for schizophrenia, but medication adherence and long-term treatment continuity are needed to maximize their effectiveness. Each antipsychotic has various side effects, which may affect adherence. Some patients with schizophrenia are reluctant to take asenapine because of its unique oral-related side effects, such as the bitter taste caused by sublingual administration. Our previous basic research found that D-sorbitol lowered the bitterness parameters of the taste sensors. However, whether D-sorbitol has the same effect in patients remains unclear. Therefore, using a D-sorbitol solution, we aim to evaluate changes in the bitterness of asenapine among patients with schizophrenia. Methods In this single-blind, placebo-controlled, crossover trial, we plan to recruit 20 adult patients with schizophrenia spectrum disorder who take sublingual asenapine tablets. The participants will be divided into two groups ( n  = 10 each). Each group will be given a D-sorbitol or placebo solution on the first day for rinsing before taking the sublingual asenapine tablets. After a 1-day interval, the participants will rinse their mouths again with a different liquid. Questionnaires regarding changes in taste and the willingness to continue asenapine will be conducted before the start of the study and after each rinse. The primary and secondary end points will be a taste evaluation of bitterness, and the willingness to continue asenapine, respectively. Differences in questionnaire scores between the D-sorbitol and placebo solutions will be calculated and analyzed using a McNemar test. Discussion This study aims to determine the efficacy of D-sorbitol in masking the bitter taste of asenapine. To our knowledge, it is the first intervention study using D-sorbitol for bitter taste of asenapine in patients with schizophrenia. Evidence of the efficacy of D-sorbitol could result in D-sorbitol pretreatment being an easy and inexpensive means of improving adherence to asenapine. Trial registration This study was registered in the Japan Registry of Clinical Trials jRCTs041210019, on May 14, 2021. Ethics approval was obtained from the Nagoya University Clinical Research Review Board.

Background Asenapine has unique orally-related side effects, such as a bitter taste induced by sublingual administration, which often results in discontinuation of the medication. While the FDA has approved black-cherry-flavored asenapine, several countries have prescribed only unflavored versions. Specifically, Asians commonly report experiencing the bitterness of asenapine because they are more sensitive to bitter tastes than other ethnic groups. In this study, with the aim of improving adherence by reducing the bitterness of asenapine, we investigated the effects of d -sorbitol, which reduced the bitterness parameters of taste sensors in our previous basic study on the bitterness and continuity of asenapine among patients with schizophrenia. Methods Twenty adult patients with schizophrenia were included in this single-blind, placebo-controlled, crossover trial. Participants rinsed their mouths with single-administration of d -sorbitol or a placebo prior to each administration of asenapine. We then conducted the questionnaires and assessed changes in the bitterness of asenapine (primary end point) and willingness to continue its use (secondary end point). Results d -sorbitol significantly improved the bitterness of asenapine ( p  = 0.038). Although it did not significantly increase the willingness to continue asenapine ( p  = 0.180), it did show improvement over the placebo in enhancing willingness to continue, especially in patients who were not accustomed to its taste. Conclusion Our findings indicate that single-administration of d -sorbitol significantly reduces the bitterness of asenapine. In countries where flavored asenapine is not available, this finding could benefit patients who were not accustomed to its bitter taste. Trial registration This study was registered in the Japan Registry of Clinical Trials (jRCTs041210019) on May 14, 2021.

In this study, we extended Acemoglu et al. (2014) in the following two ways. First, we used a constant elasticity of substitution human capital production function to show that in the short run, Internet technologies such as online education are likely to be advantageous for middle-income countries. Second, to examine whether one country voluntarily supplies online education to other countries, we changed the static model to a dynamic model. We found that despite it being a public good, developed countries voluntarily supply online education to developing countries. This is because when online education is provided, the level of human capital is higher in both transitional dynamics and the steady state than otherwise.

In this article, we first study, in the framework of operator theory, Pusz and Woronowicz's functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on \\([0,\\infty)^2\\). Our construction has special features that functions on \\([0,\\infty)^2\\) are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in \\((-\\infty,\\infty]\\). Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on \\((0,\\infty)\\). We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Ando's theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.

Let \\(\\sigma\\) be a non-trivial operator mean in the sense of Kubo and Ando, and let \\(OM_+^1\\) the set of normalized positive operator monotone functions on \\((0, \\infty)\\). In this paper, we study class of \\(\\sigma\\)-subpreserving functions \\(f\\in OM_+^1\\) satisfying $$f(A\\sigma B) \\le f(A)\\sigma f(B)$$ for all positive operators \\(A\\) and \\(B\\). We provide some criteria for \\(f\\) to be trivial, i.e., \\(f(t)=1\\) or \\(f(t)=t\\). We also establish characterizations of \\(\\sigma\\)-preserving functions \\(f\\) satisfying $$f(A\\sigma B) = f(A)\\sigma f(B)$$ for all positive operators \\(A\\) and \\(B\\). In particular, when \\(\\lim_{t\\rightarrow 0} (1\\sigma t) =0\\), the function \\(f\\) preserves \\(\\sigma\\) if and only if \\(f\\) and \\(1\\sigma t\\) are representing functions for weighted harmonic means.

We will consider about some inequalities on operator means for more than three operators, for instance, ALM and BMP geometric means will be considered. Moreover, log-Euclidean and logarithmic means for several operators will be treated.

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.

We investigate the operator monotonicity of functions which was considered by V.E. Szabo.

We present several Ando-Hiai type inequalities for \\(n\\)-variable operator means for positive invertible operators. Ando-Hiai's inequalities given here are not only of the original type but also of the complementary type and of the reverse type involving the generalized Kantorovich constant.