Abstract:
Colloidal silica can be prepared by various methods and starting materials including ion exchange of aqueous silicates, hydrolysis and condensation of silicon compounds, direct oxidation of silicon, and milling and peptization of silica powder. Various silica sols having particle sizes of 10-60 nm prepared by these methods and the preparation methods have been compared on the basis of their shape, size uniformity, sphericity, stability against pH variation, cation concentration, and price, etc. Silica sol prepared from tetraethoxysilane affords uniform size control and growth, and high purity, despite the relatively high costs. Silica sol prepared from liquid silicates affords relatively easy size and shape control; however, it is difficult to lower the alkali content to a level that is appropriate for carrying out semiconductor chemical mechanical polishing processes; in addition, the waste water treatment carried out for recovering the ion exchange resin gives rise environmental consideration. The properties of colloidal silica prepared from fumed silica powder by milling and dispersion depend on the starting silica source and it is relatively difficult to obtain monodispersed particles using this method. Colloidal silica prepared from silicon by direct oxidation has a monodispersed spherical shape and purity control with reasonable prices. It generates less waste water because it can be directly produced in relatively high concentrations. The cation fraction located in the particle relative to the free cation in the fluid is relatively lower in the silica sol prepared by the direct oxidation than others. A careful comparison of colloidal silica and the preparation methods may help in choosing the proper colloidal silica that is the most appropriate for the application being considered.

Abstract:
A total of 115 patients who underwent gastrectomy were enrolled in this study. Serum levels of IL-6 were assessed via Enzyme-Linked Immuno-Sorbent Assay (ELISA), and CRP was measured via immunoturbidimetry. Histological findings included tumor size, depth of tumor invasion, lymph node (LN) metastasis, and TNM stage (6th AJCC Stage Groupings: The staging systems; Primary tumor, regional LN, metastasis).Increases in cancer invasion and staging are generally associated with increases in preoperative serum IL-6 levels. IL-6 and CRP levels were correlated with invasion depth (P < 0.001, P = 0.001), LN metastasis (P < 0.001, P = 0.024) and TNM stage (P < 0.001, P < 0.001). The presence of peritoneal seeding metastasis is associated with IL-6 levels (P = 0.012). When we established the cutoff value for IL-6 level (6.77 pg/dL) by ROC curve, we noted significant differences in time to progression (TTP; P < 0.001) and overall survival (OS; P = 0.010). However, CRP evidenced no significance with regard to patients' TTP and OS levels. Serum IL-6 levels were correlated positively with CRP levels (r2 = 0.049, P = 0.018).Preoperative serum IL-6 and CRP levels might be markers of tumor invasion, LN metastasis, and TNM stage. Preoperative high IL-6 levels were proposed as a poor prognostic factor for disease recurrence and overall survival in patients with gastric cancers.Interleukin-6 (IL-6) is a multi-poietic cytokine that induces the growth and differentiation of immune cells, the production and expression of other cytokines, and acute-phase protein synthesis. IL-6 also exerts several effects on cancer cells [1,2].In cancer, IL-6 is generally known to be involved in host defense mechanisms. IL-6 binds to the IL-6 receptor, activates the Janus kinase (JAK), and subsequently phosphorylates the signal transducers and activators of transcription (STAT). The phosphorylated STAT gene translocates into the nucleus and activates the target gene (JAK/STAT) pathway. Suppressor of cytokine

Abstract:
Aleksandrov surfaces are a generalization of two-dimensional Riemannian manifolds, and it is known that every open simply connected Aleksandrov surface is conformally equivalent either to the unit disc (hyperbolic case) or to the plane (parabolic case). We prove a criterion for hyperbolicity of Aleksandrov surfaces which have nice tilings(triangulations) and where negative curvature dominates. We then apply this to generalize a result of Nevanlinna and give a partial answer for his conjecture about line complexes.

Abstract:
The author previously defined the spectral invariants, denoted by $\rho(H;a)$, of a Hamiltonian function $H$ as the mini-max value of the action functional $\AA_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class $a$. The spectrality axiom of the invariant $\rho(H;a)$ states that the mini-max value is a critical value of the action functional $\AA_H$. The main purpose of the present paper is to prove this axiom for {\it nondegenerate} Hamiltonian functions in {\it irrational} symplectic manifolds $(M,\omega)$. We also prove that the spectral invariant function $\rho_a: H \mapsto \rho(H;a)$ can be pushed down to a {\it continuous} function defined on the universal ({\it \'etale}) covering space $\widetilde{Ham}(M,\omega)$ of the group $Ham(M,\omega)$ of Hamiltonian diffeomorphisms on general $(M,\omega)$. For a certain generic homotopy, which we call a {\it Cerf homotopy} $\HH = \{H^s\}_{0 \leq s\leq 1}$ of Hamiltonians, the function $\rho_a \circ \HH: s \mapsto \rho(H^s;a)$ is piecewise smooth away from a countable subset of $[0,1]$ for each non-zero quantum cohomology class $a$. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a {\it family version} of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the {\it Floer mini-max theory}.

Abstract:
The main purpose of this paper is to study the length minimizing property of Hamiltonian paths on closed symplectic manifolds $(M,\omega)$ such that there are no spherical homology class $A \in H_2(M)$ with $$ \omega(A) > 0 \quad \text{and} \quad -n \leq c_1(A) < 0, $$ which we call {\it very strongly semi-positive}. We introduce the notion of {\it positively $\mu$-undertwisted} Hamiltonian paths and prove that any positively undertwisted quasi-autonomous Hamiltonian path is length minimizing in its homotopy class as long as it has a fixed maximum and a fixed minimum point that are generically under-twisted. This class of Hamiltonian can have non-constant large periodic orbits. The proof uses the chain level Floer theory, spectral invariants of Hamiltonian diffeomorphisms and the argument involving the thick and thin decomposition of Floer's moduli space of perturbed Cauchy-Riemann equation. And then based on this theorem and some closedness of length minimizing property, we relate the Minimality Conjecture on the very strongly semi-positive symplectic manifolds to a $C^1$-perturbation problem of Hamiltonian functions on general symplectic manifolds, which we also formulate here.

Abstract:
For a compact set $E \subset \mathbb{C}$ containing more than two points, we study asymptotic behavior of normalized zero counting measures $\{\mu_k \}$ of the derivatives of Faber polynomials associated with $E$. For example if $E$ has empty interior, we prove that $\{\mu_k \}$ converges in the weak-star topology to a measure whose support is the boundary of $g(\mathcal{D})$, where $g : \{|z| > r \}\cup \{\infty\} \to \bar{\mathbb{C}} \backslash E$ is a universal covering map such that $g(\infty) = \infty$ and $\mathcal{D}$ is the Dirichlet domain associated with $g$ and centered at $\infty$. Our results are counterparts of those of Kuijlaars and Saff (1995) on zeros of Faber polynomials.

Abstract:
This paper has been withdrawn by the author due to a critical error in the proof of Theorem 5.4 on which the proof of the main theorem on the non-simplenss was based.

Abstract:
The content of this paper has no mathematical flaw except that the proof of the main theorem relies on the homotopy invariance of spectral invariants of topological Hamiltonian paths. Since the latter is still up in the air, the main result of the paper is the reduction of the extension problem of the Calabi homomorphism to the group of hameomorphism is still conjectural.

Abstract:
The paper contains a silly mistake in the usage of the equation (5.4) in the formula (6.5). Due to this mistake, the effect of the transfer map $phi$ is nullified. I am deeply sorry for not being very careful in my submission and making this mistake.