Narrow Results

Artisanal/small-scale gold mining is one of the important livelhood for deprived people in many countries. The activity is expected to support people's life and activate local economy. However amalgamation method is common to recover gold in this type of mining, and due to this practice, mercury is being released into the natural environment. During the ore treatment process, miners uptake mercury not only by inhalation of the vapor but also by absorption of liquid mercury through the skin. In order to cope with health problems and environmental damages arising from the use of mercury, there should be coordination between national/local government, aid agencies and scientific institutes. In addition, the environmental education and risk communication are keys to mitigate the environmental damage by the miners. For such purpose, quick and precise trace element analysis as a base of discussion on occupational health risk is indispensable and our trial proved that PIXE is one of the most flexible and versatile system. In this paper examples of trial through a Japan-Philippines partnership will be presented.

We study the Dehn functions of amalgamations, introducing the notion of strongly undistorted subgroups. Using this, we give conditions under which taking an amalgamation does not increase the Dehn function, generalizing one aspect of the combination theorem of Bestvina and Feighn.

To obtain examples of strongly undistorted subgroups, we define and study the relative Dehn function of pairs of groups. As a result we obtain a new method of constructing examples of pairs of groups that are relatively hyperbolic in the sense of Farb.

We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A _*C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.

Let $M_i$$(i=1,2)$ be a perfect $3$-manifold, $F_i$ be a component of $\partial M_i$, $f:F_1\to F_2$ be a homeomorphic map, $M=M_1{\cup }_f{M}_2$ and $F=M_1\cap M_2(\cong F_i)$. In this paper, we show that if $d(M_i)\ge 3$$(i=1,2)$ and $d(f)\ge 2$, then $g(M)=g(M_1)+g(M_2)-g(F)$. As a corollary, if $F_i^j$$(i=1,2,1\le j\le n)$ is a component of $\partial M_i$, $f_j:F_1^j\to F_2^j$ is a homeomorphic map, $M=M_1{\cup }_{f_1,\dots ,f_n}{M}_2$, ${\bigcup }_{j=1}^n{F}^j=M_1\cap M_2(\cong F_i^j)$, $d(M_i)\ge 3$$(i=1,2)$ and $d(f_j)\ge 2$$(1\le j\le n)$, then $g(M)=g(M_1)+g(M_2)-g(F^1)-\cdots -g(F^n)+n-1$.

Let $M_i=V_i{\cup }_{S_i}{W}_i$$(i=1,2)$ be a Heegaard splitting of compact orientable $3$-manifold, and $M=V{\cup }_{S}W=(V_1{\cup }_{S_1}{W}_1){\cup }_F(V_2{\cup }_{S_2}{W}_2)$ be an amalgamation along an essential surface $F$. In this paper, we show that if there is an essential disk $D_2$ in $V_2$, such that $D_2$ cuts $V_2$ into two product $I$-bundles, one of which is $F\times [0,1]$, and $d(S_i)\ge 3$$(i=1,2)$, then $M=V{\cup }_{S}W$ is unstabilized and $S$ is uncritical. We also give a sufficient condition for $S$ to be critical.

Let $M$ be a compact, connected, orientable closed 3-manifold. Let $\{F_j,1\le j\le n-1\}$ be a disjoint union of incompressible separating tori in $M$ which cut $M$ into the submanifolds $\{M_i,1\le i\le n\}$. In the present paper, we show that the Heegaard genus $g(M)$ of $M$ is given by

We classify countable metrically homogeneous graphs of diameter 3.

A submodule $W$ of $V$ is summand absorbing, if $x+y\in W$ implies $x\in W,y\in W$ for any $x,y\in V$. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules.

Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, and let $f:R\to S$ be a ring homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R{\bowtie }^{f}J$ was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of $R$ which are transferred to the comaximal graph of $R{\bowtie }^{f}J$, and also we study some algebraic properties of the ring $R{\bowtie }^{f}J$ by way of graph theory. The comaximal graph of $R$, $\mathrm{\Gamma }(R)$, was introduced by Sharma and Bhatwadekar in 1995. The vertices of $\mathrm{\Gamma }(R)$ are all elements of $R$ and two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let ${\mathrm{\Gamma }}_2(R)$ be the subgraph of $\mathrm{\Gamma }(R)$ generated by non-unit elements, and let $J(R)$ be the Jacobson radical of $R$. It is shown that the diameter of the graph ${\mathrm{\Gamma }}_2(R)\setminus J(R)$ is equal to the diameter of the graph ${\mathrm{\Gamma }}_2(R{\bowtie }^{f}J)\setminus J(R{\bowtie }^{f}J)$, and the girth of the graph ${\mathrm{\Gamma }}_2(R)\setminus J(R)$ is equal to the girth of the graph ${\mathrm{\Gamma }}_2(R{\bowtie }^{f}J)\setminus J(R{\bowtie }^{f}J)$, provided some special conditions.