Automorphic equivalence of the representations of Lie algebras

We prove that if a field k is infinite, char(k)=0 and k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. We achieve our result by consideration of 1-sorted objects. We suppose that our method can be perspective in the further researches.Comment: 28 page

Waveguide laser formed in YAG: Nd3+ crystal by femtosecond laser inscription

A technique of direct writing of depressed cladding waveguides by a tightly focused, femtosecond laser beam in laser crystals has been developed. A laser based on a depressed cladding waveguide in a Neodimium doped YAG crystal has been demonstrated for the first time

Automorphic equivalence of the representations of Lie algebras

In this paper we research the algebraic geometry of the representations of Lie algebras over fixed field k. We assume that this field is infinite and char (k) = 0. We consider the representations of Lie algebras as 2-sorted universal algebras. The representations of groups were considered by similar approach: as 2-sorted universal algebras - in [3] and [2]. The basic notions of the algebraic geometry of representations of Lie algebras we define similar to the basic notions of the algebraic geometry of representations of groups (see [2]). We prove that if a field k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. This result is similar to the result of [4], which was achieved for representations of groups. But we achieve our result by another method: by consideration of 1-sorted objects. We suppose that our method can be more perspective in the further researches

Analysis of influence of heat insulation on the thermal regime of storage tanks with liquefied natural gas

Is numerically investigated the process of convective heat transfer in the reservoirs of liquefied natural gas (LNG). The regimes of natural convection in a closed rectangular region with different intensity of heat exchange at the external borders are investigated. Is solved the time-dependent system of energy and Navier-Stokes equations in the dimensionless variables “vorticity – the stream function”. Are obtained distributions of the hydrodynamic parameters and temperatures, that characterize basic regularities of the processes. The special features of the formation of circulation flows are isolated and the analysis of the temperature distribution in the solution region is carried out. Is shown the influence of geometric characteristics and intensity of heat exchange on the outer boundaries of reservoir on the temperature field in the LNG storage

A Multigroup diffusion solver using pseudo transient continuation for a radiation-hydrodynamic code with patch-based AMR

We present a scheme to solve the nonlinear multigroup radiation diffusion (MGD) equations. The method is incorporated into a massively parallel, multidimensional, Eulerian radiation-hydrodynamic code with adaptive mesh refinement (AMR). The patch-based AMR algorithm refines in both space and time creating a hierarchy of levels, coarsest to finest. The physics modules are time-advanced using operator splitting. On each level, separate 'level-solve' packages advance the modules. Our multigroup level-solve adapts an implicit procedure which leads to a two-step iterative scheme that alternates between elliptic solves for each group with intra-cell group coupling. For robustness, we introduce pseudo transient continuation ({Psi}tc). We analyze the magnitude of the {Psi}tc parameter to ensure positivity of the resulting linear system, diagonal dominance and convergence of the two-step scheme. For AMR, a level defines a subdomain for refinement. For diffusive processes such as MGD, the refined level uses Dirichet boundary data at the coarse-fine interface and the data is derived from the coarse level solution. After advancing on the fine level, an additional procedure, the sync-solve (SS), is required in order to enforce conservation. The MGD SS reduces to an elliptic solve on a combined grid for a system of G equations, where G is the number of groups. We adapt the 'partial temperature' scheme for the SS; hence, we reuse the infrastructure developed for scalar equations. Results are presented. We consider a multigroup test problem with a known analytic solution. We demonstrate utility of {Psi}tc by running with increasingly larger timesteps. Lastly, we simulate the sudden release of energy Y inside an Al sphere (r = 15 cm) suspended in air at STP. For Y = 11 kT, we find that gray radiation diffusion and MGD produce similar results. However, if Y = 1 MT, the two packages yield different results. Our large Y simulation contradicts a long-standing theory and demonstrates the inadequacy of gray diffusion