Previous Topics on Cartan's Corner 2010 - 2011

posted Dec 16, 2011,

Maple Solutions for equilibrium and non-equilibrium Electromagnetism .

The Maple program, ConstCurrents.pdf, is a 38 MB Maple program that evaluates Maxwell-Faraday intensive variables (E and B) from the topological postulate of potentials F-dA = 0 and Maxwell-Ampere intensive variables (D and H) from the topological postulate of charge currents, J-dG = 0. Many familiar and unfamiliar examples are displayed. The topological methods are independent from a choice of geometric coordinates, and are valid for equilibrium or non-equilibrium plasma systems on topological spaces of Pfaff Topological Dimension 4. The techniques are valid for Charge currents, Spin currents, Topological Torsion currents, and cubic Adjoint currents. For simplicity, the intensive and extensive variables are assumed to be related by the Lorentz-Minkowski vacuum constitutive equations, D = epsilon E, B = mu H. BE AWARE. The algebraic effort required to solve by hand some of the results presented can be overpowering, so give thanks to Maple, now.

posted Jan 24, 2011,

The Category Theory of Topological Thermodyanmics.

Category theory, whose objects are T2, T0, and Not-T0 topologies, based on sets of exterior differential forms, and whose morphisms are based upon Cartan's Magic formula of homotopy can be used as a basis for both the thermodynamical theory of distinguishable particles (mass), AND the thermodynamical theory of statistical distributions (radiation) whose complex wavelets are indistinguishable.

Posted May 16, 2010. Updated Jan 24, 2011

Boson condensates, the Arrow of time, Non-metric Gravity, and Finite Lattice Structures.

Herein, a lattice structure, LS, is defined as a collection S of the subsets of the power set, PS(X), of N ingredients ("points"), X={a,b,c,d...}. The elements of the LS are, by edict, denoted as Open sets, and often are used to define a set of lattice vertices. The collection of LS Open sets may or may not obey the axioms of a topology, yet most of the set operations remain valid even if the LS is not equivalent to a Topological Structure, TS. For finite LS that generate topologies, TS, there are three topological categories based upon how the ingredients of X may, or may not, be distinguished (the separation axioms). If the ingredients can be distinguished by metric (geometrical) methods, then the topologies belong to category 1 and must satisfy the Hausdorff T2 separation axiom. If the ingredients cannot be distinguished by geometric methods, but can be distinguished by topological methods, then the topologies belong to category 2 and must satisfy the Kolmogorov T0 (but not the T2) separation axioms. Topologies in the third category 3 obey the topological axioms but not the separation axioms. Hence some, if not all, of the ingredients for topologies that belong to category 3 are indistinguishable. It appears that category 1 topologies are the basis of equilibrium thermodynamic systems, with uniquely integrable, reversible, processes. There is evidence that Category 2 topologies are the basis for non-equilibrium thermodynmaic systems, with non-uniquely integrable, ireversible processes. Note that the boundary of a boundary is not empty, for Not-T0 Topologies that have indestinguisable sets.

posted Mar 29, 2010,

The Universal Effectiveness of Topological Thermodynamics.

This pdf file summarizes the ideas and important results of a theory of non-equilibrium thermodynamics based upon sets of exterior differential forms and the Kolomogorov-Cartan T0 (poset 3) topology. The presentation was made at the 2009 Haceteppe Topology conference in Ankara, Turkey. An improved presentation using ideas benerated 2009 to 2010 appears above in terms of Category theory.

Updated Feb 11, 2010

Twisted Fiber Bundles, Torsion of continuous deformation, and Topological Torsion

The pdf file contains 3 short movies demonstrating the differences between geometric torsion and Topological torsion.
Just click on the URl's to bring up the movies. A preface to a chapter, "Lattice Cohomology and Non-metrizable Physics"
(to be included in my 6th mongraph) is also presented.

R2 torsion

R3 toptorsion

R2 to R3 toptorsion.

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Last update 02/15/2010
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