Space affine structure

The definition of a mathematical space begins with the set of objects X, Y, Z,. .. that occupy the space (an intrinsically empty space being a physically problematic concept). Among the simplest algebraic structures that can characterize such objects is that of a linear manifold, also called a linear vector space, affine space, etc. By definition, such a manifold has only two operations— addition (X + Y) and multiplication by a scalar (AX)— resulting in each case in another element of the manifold. These operations have the usual distributive,... 

The geometric description of the light propagation and the kinetics description of motion were closely correlated in the history of science. Among the main evidence of classical Newtonian mechanics is Euclidean geometry based on optical effects. In Newtonian physics, space has an affine structure but time is absolute. The basic idea is the inertial system, and the relations are the linear force laws. The affine structure allows linear transformations in space between the inertial coordinate systems, but not in time. This is the Galilean transformation ... 

The quotient space /r (0)/G is called a symplectic quotient (or Marsden-Weinstein reduction). It has a complex structure and natural Kahler metric (cf. Theorem 3.30) on points where G acts freely. On the other hand, the set of closed G -orbits is the affine algebro-geometric quotient and denoted by YjjG. In fact, it is known that the above identihcation intertwines the complex structures. 

Energy levels of heavy and super-heavy (Z>100) elements are calculated by the relativistic coupled cluster method. The method starts from the four-component solutions of the Dirac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. Simultaneous inclusion of relativistic terms in the Hamiltonian (to order o , where a is the fine-structure constant) and correlation effects (all products smd powers of single and double virtual excitations) is achieved. The Fock-space coupled-cluster method yields directly transition energies (ionization potentials, excitation energies, electron affinities). Results are in good agreement (usually better than 0.1 eV) with known experimental values. Properties of superheavy atoms which are not known experimentally can be predicted. Examples include the nature of the ground states of elements 104 md 111. Molecular applications are also presented. 

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