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Intersection Graph of a Module
Let be a left -module where is a (not necessarily commutative)
ring with unit. The intersection graph \cG(V) of proper -submodules of
is an undirected graph without loops and multiple edges defined as follows: the
vertex set is the set of all proper -submodules of and there is an edge
between two distinct vertices and if and only if We
study these graphs to relate the combinatorial properties of \cG(V) to the
algebraic properties of the -module We study connectedness, domination,
finiteness, coloring, and planarity for \cG (V). For instance, we find the
domination number of \cG (V). We also find the chromatic number of \cG(V)
in some cases. Furthermore, we study cycles in \cG(V), and complete subgraphs
in \cG (V) determining the structure of for which \cG(V) is planar
Density functional theory study of the {\alpha} --> {\omega} martensitic transformation in titanium induced by hydrostatic pressure
The martensitic {\alpha} --> {\omega} transition was investigated in Ti under
hydrostatic pressure. The calculations were carried out using the density
functional theory (DFT) framework in combination with the Birch-Murnaghan
equation of state. The calculated ground-state properties of {\alpha} and
{\omega} phases of Ti, their bulk moduli and pressure derivatives are in
agreement with the previous experimental data. The lattice constants of
{\alpha} and {\omega}-phase at 0 K were modeled as a function of pressure from
0 to 74 GPa and 0 to 119 GPa, respectively. It is shown that the lattice
constants vary in a nonlinear manner upon compression. The calculated lattice
parameters were used to describe the {\alpha} --> {\omega} transition and show
that the phase transition can be obtained at 0 GPa and 0 K.Comment: 6 pages, 5 figure
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