Auswahl der wissenschaftlichen Literatur zum Thema „Homogenization algebra“

A deformation of the Orlik-Solomon algebra of a matroid $\mathfrak{M}$ is defined as a quotient of the free associative algebra over a commutative ring $R$ with $1$. It is shown that the given generators form a Gröbner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as $R$-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Gröbner basis and hence the algebra is Koszul.

A recent theorem of [E. Gasparim, L. Grama and L. A. B. San Martin, Lefschetz fibrations on adjoint orbits, Forum Math. 28(5) (2016) 967–980.] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behavior of their fiberwise compactifications. Expressing adjoint orbits and fibers as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenization of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenization, and that extensions of the potential to the compactification must acquire degenerate singularities.

AbstractIn a previous paper, we studied the homogenized enveloping algebra of the Lie algebrasℓ(2,ℂ) and the homogenized Verma modules. The aim of this paper is to study the homogenization$\mathcal{O}$Bof the Bernstein–Gelfand–Gelfand category$\mathcal{O}$of sℓ(2,ℂ), and to apply the ideas developed jointly with J. Mondragón in our work on Groebner basis algebras, to give the relations between the categories$\mathcal{O}$Band$\mathcal{O}$as well as, between the derived categories$\mathcal{D}$b($\mathcal{O}$B) and$\mathcal{D}$b($\mathcal{O}$).

By Banach algebra we mean Banach space enriched with a multiplication operation. It is a mathematical structure that is used in the non-periodic homogenization of composite materials. The theory of classical homogenization studies materials assuming the periodicity of the structure. For real materials, the assumption of a periodicity is not enough and is replaced by the so-called an abstract hypothesis based on a concept composed mainly of the spectrum of Banach algebra and Sigma convergence. This theory was introduced in 2004.

The AGENT (Arbitrary GEometry Neutron Transport) an open-architecture reactor modeling tool is deterministic neutron transport code for two or three-dimensional heterogeneous neutronic design and analysis of the whole reactor cores regardless of geometry types and material configurations. The AGENT neutron transport methodology is applicable to all generations of nuclear power and research reactors. It combines three theories: (1) the theory of R-functions used to generate real three-dimensional whole-cores of square, hexagonal or triangular cross sections, (2) the planar method of characteristics used to solve isotropic neutron transport in non-homogenized 2D) reactor slices, and (3) the one-dimensional diffusion theory used to couple the planar and axial neutron tracks through the transverse leakage and angular mesh-wise flux values. The R-function-geometrical module allows a sequential building of the layers of geometry and automatic submeshing based on the network of domain functions. The simplicity of geometry description and selection of parameters for accurate treatment of neutron propagation is achieved through the Boolean algebraic hierarchically organized simple primitives into complex domains (both being represented with corresponding domain functions). The accuracy is comparable to Monte Carlo codes and is obtained by following neutron propagation through real geometrical domains that does not require homogenization or simplifications. The efficiency is maintained through a set of acceleration techniques introduced at all important calculation levels. The flux solution incorporates power iteration with two different acceleration techniques: Coarse Mesh Rebalancing (CMR) and Coarse Mesh Finite Difference (CMFD). The stand-alone originally developed graphical user interface of the AGENT code design environment allows the user to view and verify input data by displaying the geometry and material distribution. The user can also view the output data such as three-dimensional maps of the energy-dependent mesh-wise scalar flux, reaction rate and power peaking factor. The AGENT code is in a process of an extensive and rigorous testing for various reactor types through the evaluation of its performance (ability to model any reactor geometry type), accuracy (in comparison with Monte Carlo results and other deterministic solutions or experimental data) and efficiency (computational speed that is directly determined by the mathematical and numerical solution to the iterative approach of the flux convergence). This paper outlines main aspects of the theories unified into the AGENT code formalism and demonstrates the code performance, accuracy and efficiency using few representative examples. The AGENT code is a main part of the so called virtual reactor system developed for numerical simulations of research reactors. Few illustrative examples of the web interface are briefly outlined.