Abstract:
An operator algebra $\mathcal{A}$ acting on a Hilbert space is said to have the closability property if every densely defined linear transformation commuting with $\mathcal{A}$ is closable. In this paper we study the closability property of the von Neumann algebra consisting of the multiplication operators on $L^2(\mu)$, and give necessary and sufficient conditions for a normal operator $N$ such that the von Neumann algebra generated by $N$ has the closability property. We also give necessary and sufficient conditions for an operator $T$ of class $C_0$ such that the algebra generated by $T$ in the weak operator topology and the algebra $H^\infty(T)=\{u(T):u\in H^\infty\}$ have the closability property.

Abstract:
Let $\mathcal{M}$ be the set of Borel probability measures on $\mathbb{R}$. We denote by $\mu^{\mathrm{ac}}$ the absolutely continuous part of $\mu\in\mathcal{M}$. The purpose of this paper is to investigate the supports and regularity for measures of the form $(\mu^{\boxplus p})^{\uplus q}$, $\mu\in\mathcal{M}$, where $\boxplus$ and $\uplus$ are the operations of free additive and Boolean convolution on $\mathcal{M}$, respectively, and $p\geq1$, $q>0$. We show that for any $q$ the supports of $((\mu^{\boxplus p})^{\uplus q})^{\mathrm{ac}}$ and $(\mu^{\boxplus p})^{\mathrm{ac}}$ contain the same number of components and this number is a decreasing function of $p$. Explicit formulas for the densities of $((\mu^{\boxplus p})^{\uplus q})^{\mathrm{ac}}$ and criteria for determining the atoms of $(\mu^{\boxplus p})^{\uplus q}$ are given. Based on the subordination functions of free convolution powers, we give another point of view to analyze the set of $\boxplus$-infinitely divisible measures and provide explicit expressions for their Voiculescu transforms in terms of free and Boolean convolutions.

Abstract:
In this paper, we study the supports of measures in the free additive convolution semigroup $\{\mu^{\boxplus t}:t>1\}$, where $\mu$ is a Borel probability measure on $\mathbb{R}$. We give a formula for the density of the absolutely continuous part of $\mu^{\boxplus t}$ and use this formula to obtain certain regularizing properties of $\mu^{\boxplus t}$. We show that the number $n(t)$ of the components in the support of $\mu^{\boxplus t}$ is a decreasing function of $t$ and give equivalent conditions so that $n(t)=1$ for sufficiently large $t$. Moreover, a measure $\mu$ so that $\mu^{\boxplus t}$ has infinitely many components in the support for all $t>1$ is given.

Abstract:
In this paper, we study the supports of measures in multiplicative free semigroups on the positive real line and on the unit circle. We provide formulas for the density of the absolutely continuous parts of measures in these semigroups. The descriptions rely on the characterizations of the images of the upper half-plane and the unit disc under certain subordination functions. These subordination functions are $\eta$-transforms of infinitely divisible measures with respect to multiplicative free convolution. The characterizations also help us study the regularity properties of these measures. One of the main results is that the number of components in the support of measures in the semigroups is a decreasing function of the semigroup parameter.

Abstract:
In this paper, we derive the bi-free analogue of the L\'{e}vy-Hin\v{c}in formula for compactly supported planar probability measures which are infinitely divisible with respect to the additive bi-free convolution introduced by Voiculescu. We also provide examples of bi-free infinitely divisible distributions with their bi-free L\'{e}vy-Hin\v{c}in representations. Furthermore, we construct the bi-free L\'{e}vy processes and the additive bi-free convolution semigroups generated by compactly supported planar probability measures.

Abstract:
Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences.

Abstract:
Background The liver is the most common target organ in the metastasis of colorectal cancer (CRC). Synchronous liver metastases may confer a poorer prognosis than metachronous metastases, and genetic alterations and an inflammatory response have also been associated with a poor prognosis in cases of a liver metastasis arising from CRC. However, few studies have examined the relationship between KRAS mutations and inflammatory status in CRC, especially with respect to liver metastases. Methods The effect of the activated mitogen-activated protein kinase pathway and another protein involved in inflammation, C-reactive protein, in liver metastases were examined. We aimed to determine the impact of the CRP-specific single nucleotide polymorphism (SNP) rs7553007 in liver metastasis on the CRC-specific survival (CSS) of patients after colorectal liver metastasectomy. Results We found no significant differences in genotype distributions and allele frequencies at the CRP SNP rs7553007 between CRC patients with liver metastasis and the control group. CSS rates were low in the subgroup of patients with synchronous metastasis with the A-allele (A/A and A/G) at rs7553007 or mutated KRAS/BRAF in liver metastatic specimens. Furthermore, the CRP SNP rs7553007 (hazard ratio [HR] = 1.101; 95% confidence interval [CI] = 1.011–1.200; P = 0.027) and KRAS/BRAF mutations (HR = 2.377; 95% CI = 1.293–4.368; P = 0.005) remained predictive for the CSS of CRC patients with synchronous liver metastasis in multivariate analysis. Conclusions Both the CRP SNP rs7553007 and KRAS/BRAF mutations were independent prognostic factors for CRC patients with synchronous liver metastasis.

Abstract:
The fish lateral line (LL) is a mechanosensory system closely related to the hearing system of higher vertebrates, and it is composed of several neuromasts located on the surface of the fish. These neuromasts can detect changes in external water flow, to assist fish in maintaining a stationary position in a stream. In the present study, we identified a novel function of Nogo/Nogo receptor signaling in the formation of zebrafish neuromasts. Nogo signaling in zebrafish, like that in mammals, involves three ligands and four receptors, as well as three co-receptors (TROY, p75, and LINGO-1). We first demonstrated that Nogo-C2, NgRH1a, p75, and TROY are able to form a Nogo-C2 complex, and that disintegration of this complex causes defective neuromast formation in zebrafish. Time-lapse recording of the CldnB::lynEGFP transgenic line revealed that functional obstruction of the Nogo-C2 complex causes disordered morphogenesis, and reduces rosette formation in the posterior LL (PLL) primordium during migration. Consistent with these findings, hair-cell progenitors were lost from the PLL primordium in p75, TROY, and Nogo-C2/NgRH1a morphants. Notably, the expression levels of pea3, a downstream marker of Fgf signaling, and dkk1b, a Wnt signaling inhibitor, were both decreased in p75, TROY, and Nogo-C2/NgRH1a morphants; moreover, dkk1b mRNA injection could rescue the defects in neuromast formation resulting from knockdown of p75 or TROY. We thus suggest that a novel Nogo-C2 complex, consisting of Nogo-C2, NgRH1a, p75, and TROY, regulates Fgf signaling and dkk1b expression, thereby ensuring stable organization of the PLL primordium.

Abstract:
Perceptual map is often used as a positioning tool for firms in marketing practice; however, this map provides misleading information due to simplified statistical data. This research adopts Delphi method to confirm multiple needs of consumers in cosmetic bundles and constructs a value-based decision model. From these analysis results, firms could utilize advantageous attributes and unique benefits to develop appeal strategy and positioning strategy. Also, compared with two other approaches-simple multi-attribute ranking technique and analytic hierarchy process, firms could obtain more complete results in the real world.

Abstract:
Following a literature review, product bundles are classified into four types by two dimensions: degree of functional integrity and degree of symbolic increase. Through classification, this study could successfully present a systematic model for manufacturers and marketers concerning how to bundle products. This model not only utilizes a matrix of product attributes and product bundles to evaluate the imprecise requirements from customers but also provides an efficient approach to discuss the marketing strategy of bundling for improving satisfaction of customers needs and wants.