Homology Theory, An Introduction to Algebraic Topology.

Splitting homotopy idempotents II

We give a local combinatorial criterion for finite simplicial complexes which implies the Gromov-hyperbolicity of the universal covering. 2000 MSC : 05C12, 53C21, 57Q05.

Complexes simpliciaux hyperboliques de grande dimension.

Publisher Summary This chapter separates the various layers present in the structure of E 3 for each subpart of the Maxwell system of equations The chapter considers four dimensions p = 0,    , 3 of manifolds in E 3 , four kinds of integrals that are constantly encountered in Physics Such integrals can be defined on cells first, then extended by linearity to chains, which covers the case of piecewise smooth manifolds As a preparation to this deconstruction process, one needs to recall a few notions of geometry and algebra, which do not seem to get, in most curricula, the treatment they deserve A vector space on the reals is a set of objects called vectors , which one can (1) add together (in such a way that they form an Abelian group, the neutral element being the null vector) and (2) multiply by real numbers No need to recall the axioms that harmonize these two groups of features An affine space is a set on which a vector space, considered as an additive group, acts effectively, transitively and regularly

Discretization of Electromagnetic Problems: The “Generalized Finite Differences” Approach

/pdf/homotopy-idempotents-on-finite-dimensional-complexes-split-5fv2jp07sf.pdf

Homotopy idempotents on finite-dimensional complexes split

Software systems designed to solve Maxwell's equations need abstractions that accurately explain what different kinds of electromagnetic problems really do have in common. Computational electromagnetics calls for higher level abstractions than what is typically needed in ordinary engineering problems. In this paper Maxwell's equations are described by exploiting basic concepts of set theory. Although our approach unavoidably increases the level of abstraction, it also simplifies the overall view making it easier to recognize a topological problem behind all boundary value problems modeling the electromagnetic phenomena. This enables us also to construct an algorithm which tackles the topological problem with basic tools of linear algebra.

background or methods

/pdf/topological-approach-to-computational-electromagnetism-1pmo51fjdi.pdf

Homology Theory, An Introduction to Algebraic Topology.

Citations

Splitting homotopy idempotents II

Complexes simpliciaux hyperboliques de grande dimension.

Discretization of Electromagnetic Problems: The “Generalized Finite Differences” Approach

Homotopy idempotents on finite-dimensional complexes split

Topological Approach to Computational Electromagnetism

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