Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and ...Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.展开更多
In this article, we present a three-dimensional visualization technique that has been developed in order to establish an interactive immersive environment to visualize the particles in granular materials and dislocati...In this article, we present a three-dimensional visualization technique that has been developed in order to establish an interactive immersive environment to visualize the particles in granular materials and dislocations in crystals. Simple elementary objects often exhibit complex collective behavior. Understanding of such behaviors and developments of coarse-scale theories, often requires insight into collective behavior that can only be obtained through immersive visualization. By displaying the computational results in a virtual environment with three-dimensional perception, one can immerse inside the model and analyze the intricate and very complex behavior of individual particles and dislocations. We built the stereographic images of the models using OpenGL rendering technique and then combine with the Virtual Reality technology in order to immerse in the three-dimensional model. A head mounted display has been used to allow the user to immerse inside the models and a flock of birds tracking device that allows the movements around and within the immersive environment.展开更多
文摘Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.
文摘In this article, we present a three-dimensional visualization technique that has been developed in order to establish an interactive immersive environment to visualize the particles in granular materials and dislocations in crystals. Simple elementary objects often exhibit complex collective behavior. Understanding of such behaviors and developments of coarse-scale theories, often requires insight into collective behavior that can only be obtained through immersive visualization. By displaying the computational results in a virtual environment with three-dimensional perception, one can immerse inside the model and analyze the intricate and very complex behavior of individual particles and dislocations. We built the stereographic images of the models using OpenGL rendering technique and then combine with the Virtual Reality technology in order to immerse in the three-dimensional model. A head mounted display has been used to allow the user to immerse inside the models and a flock of birds tracking device that allows the movements around and within the immersive environment.
