Abstract:
Based on the Kauffman bracket at $A=e^{i \pi/4}$, we defined an invariant for a special type of $n$-punctured ball tangles. The invariant $F^n$ takes values in the set $PM_{2\times2^n}(\mathbb Z)$ of $2\times 2^n$ matrices over $\mathbb Z$ modulo the scalar multiplication of $\pm1$. We provide the formula to compute the invariant of the $k_1 + ... + k_n$-punctured ball tangle composed of given $n,k_1,...,k_n$-punctured ball tangles. Also, we define the horizontal and the vertical connect sums of punctured ball tangles and provide the formulas for their invariants from those of given punctured ball tangles. In addition, we introduce the elementary operations on the class $\textbf{\textit{ST}}$ of 1-punctured ball tangles, called spherical tangles. The elementary operations on $\textbf{\textit{ST}}$ induce the operations on $PM_{2\times2}(\mathbb Z)$, also called the elementary operations. We show that the group generated by the elementary operations on $PM_{2\times2}(\mathbb Z)$ is isomorphic to a Coxeter group.

Abstract:
We consider a class of topological objects in the 3-sphere $S^3$ which will be called $n$-punctured ball tangles. Using the Kauffman bracket at $A=e^{i \pi/4}$, an invariant for a special type of $n$-punctured ball tangles is defined. The invariant $F^n$ takes values in $PM_{2\times2^n}(\mathbb Z)$, that is the set of $2\times 2^n$ matrices over $\mathbb Z$ modulo the scalar multiplication of $\pm1$. This invariant leads to a generalization of a theorem of D. Krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in $S^3$ disjointly. Furthermore, we provide the general formula to compute the invariant of the $k_1 + ... + k_n$-punctured ball tangle determined by $n,k_1,...,k_n$-punctured ball tangles, respectively. Also, we consider various connect sums among punctured ball tangles and provide the formulas to compute their invariants. We also address the question of whether the invariant $F^n$ is surjective onto $PM_{2\times2^n}(\mathbb Z)$. We will show that the invariant $F^n$ is surjective when $n=0$. When $n=1$, $n$-punctured ball tangles will be also called spherical tangles. We show that ${\rm det} F^1(S)\equiv 0$ or 1 {\rm mod} 4 for every spherical tangle $S$. Thus, $F^n$ is not surjective when $n=1$. In addition, we introduce monoid structures on the class of 0-punctured ball tangles and the class of spherical tangles and show that the group generated by the elementary operations on $PM_{2\times2}(\mathbb Z)$ induced by those on the spherical tangles is isomorphic to a Coxeter group.

Abstract:
It is important to analyse the casting product and the mold at the same time considering thermal contraction of the casting and thermal expansion of the mold. The analysis considering contact of the casting and the mold induces the precise prediction of stress distribution and the defect such as hot tearing. But it is difficult to generate FEM mesh for the interface of the casting and the mold. Moreover the mesh for the mold domain spends lots of computational time and memory for the analysis due to a number of meshes. Consequently we proposed the virtual mold technique which only uses mesh of the casting part for thermal stress analysis in casting process. The spring bar element in virtual mold technique is used to consider the contact of the casting and the mold. In general, a volume of the mold is much bigger than that of casting part, so the proposed technique decreases the number of mesh and saves the computational memory and time greatly. In this study, the proposed technique was verified by the comparison with the traditional contact technique on a specimen. And the proposed technique gave satisfactory results.

Abstract:
Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then $1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$. These inequalities are shape in the sense that the upper bound of $b(D)$ is achieved by alternating knots and the lower bound of $b(D)$ is achieved by torus knots. The second inequality becomes an equality only when the knot is an alternating knot. We prove that the first inequality becomes an equality only when the knot is a torus knot.

Abstract:
We consider a class of topological objects in the 3-sphere $S^3$ which will be called {\it $n$-punctured ball tangles}. Using the Kauffman bracket at $A=e^{\pi i/4}$, an invariant for a special type of $n$-punctured ball tangles is defined. The invariant $F$ takes values in $PM_{2\times2^n}(\mathbb Z)$, that is the set of $2\times 2^n$ matrices over $\mathbb Z$ modulo the scalar multiplication of $\pm1$. This invariant leads to a generalization of a theorem of D. Krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in $S^3$ disjointly. We also address the question of whether the invariant $F$ is surjective onto $PM_{2\times2^n}(\mathbb Z)$. We will show that the invariant $F$ is surjective when $n=0$. When $n=1$, $n$-punctured ball tangles will also be called spherical tangles. We show that $\text{det} F(S)=0$ or 1 {\rm mod} 4 for every spherical tangle $S$. Thus $F$ is not surjective when $n=1$.

Abstract:
In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a based chain complex to another. It shares nice properties with torsion of acyclic based chain complexes, like multiplicativity and duality. We will further generalize our torsion to quasi-isomorphisms between free chain complexes over a ring under some mild condition. We anticipate that the study of torsion of quasi-isomorphisms will be fruitful in many directions, and in particular, in the study of links in 3-manifolds.

Abstract:
Given a knot diagram $D$, we construct a semi-threading circle for it which can be an axis of $D$ as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles for minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this notion, we show that, for every nontrivial knot $K$, the braid index $b(K)$ of $K$ is not less than the minimum number $l(K)$ of overpasses of diagrams. Moreover, they are the same for a torus knot.

Abstract:
Decomposition of chlorinated hydrocarbons, CCl4 and CHCl3, in gliding plasma was examined. The effects of initial concentrations, total gas flow rates, and power consumption have been investigated. The conversion result was relatively high. It reached 80% for CCl4 and 97% for CHCl3. Using atmospheric air as the carrier gas, the plasma reaction occurred at exothermic reaction and the main products were CO2, CO, and Cl2. Transformation into CCl4 was also detected for CHCl3 decomposition reaction. The conversion of CCh and CHCl3 were increased with the increasing applied frequency and decreasing total gas flow rate.

Abstract:
The research outlined here includes a study of methanol production from direct methane conversion by means of thermal and plasma method. The kinetic study, derived from thermal-based approach, was carried out to investigate thoroughly the possible intermediate species likely to be presented in the process. A set of plasma experiments was undertaken by using dielectric barrier discharge (DBD), classified as non-thermal plasma, done at atmospheric pressure and room temperature. Plasma process yields more methanol than thermal process at the same methane conversion rates and methane to oxygen feed ratios. Oxidation reaction of thermal process resulted CO and CO2 as the most dominant products and the selectivity reached 19% and 68%, respectively. Moreover, more CO and less CO2 were produced in plasma process than in thermal process. The selectivity of CO and CO2 by plasma was 47% and 20%, respectively. Ethane (C2H6) was detected as the only higher hydrocarbon with a significant concentration. The concentration of ethane reached 9% of the total products in plasma process and 17% in thermal process. The maximum selectivity of methanol, the target material of this research, was 12% obtained by plasma method and less than 5% by thermal process. In some certain points, the kinetic model closely matched with the experimental results.

Abstract:
In the title compound, C17H20N2O3, the morpholine ring is in a slightly distorted chair form. The crystal structure is stabilized by an intermolecular O—H...O hydrogen bond between the H atom of the hydroxyl group and the O atom of a neighbouring carbonyl group. A weak intermolecular C—H...π interaction is also present.