Taliya Sahihi | Searcharxiv

arXiv2017

Alternative (Oriented) Singular Cochains and the Modified Cup Product

A special subcomplex of the singular chain complex for a topological space, historically called oriented singular chain complex is used here with the new name "alternative" singular chain complex. It was already known that this subcomplex and so its dual complex are chain homotopy equivalent to singular chains and cochains respectively and thus have the same homology and cohomology. Here, in addition to revisiting some aspects of this subcomplex, it is shown that alternative singular cochains (dual of alternative singular chains) with coefficients in rational or real numbers are indeed summands of singular cochains through a natural splitting. It is shown that this natural splitting also hold for cohomologies: At any order, the singular cohomology splits into the alternative cohomology and another summand which is zero if the considered topological space is compact. Also in this case similar to the wedge product for differential forms, a modified cup product can be defined with the same algebraic properties as in the wedge product in differential forms. This provides an idea to investigate some topological and structure-free aspects of nonlinear global differential equations on manifolds..

math.AT↗

arXiv2014

Biot-Savart helicity versus physical helicity: A topological description of ideal flows

For an isentropic (thus compressible) flow, fluid trajectories are considered as orbits of a family of one parameter, smooth, orientation preserving and nonsingular diffeomorphisms on a compact and smooth-boundary domain in the Euclidian 3-space which necessarily preserve a finite measure, later interpreted as the fluid mass. Under such diffeomorphisms the Biot-Savart helicity of the pushforward of a divergence-free and tangent to the boundary vector field is proved to be conserved and since these circumstances present an isentropic flow, the conservation of the "Biot-Savart helicity" is established for such flows. On the other hand, the well known helicity conservation in ideal flows which here we call it "physical helicity" is found to be an independent constant with respect to the Biot-Savart helicity. The difference between these two helicities reflects some topological features of the domain as well as the velocity and vorticity fields which is discussed and is shown for simply connected domains the two helicities coincide. The energy variation of the vorticity field is shown to be formally the same as for the incompressible flow obtained before. For fluid domains consisting of several disjoint solid tori, at each time, the harmonic knot subspace of smooth vector fields on the fluid domain is found to have two independent base sets with a special type of orthogonality between these two bases by which a topological description of the vortex and velocity fields depending on the helicity difference is achieved since this difference is shown to depend only on the harmonic knot parts of velocity, vorticity and its Biot-Savart vector field. For an ideal MHD flow three independent constant helicities are reviewed while the helicity of magnetic potential is generalized for non-simply connected domains by inserting a special harmonic knot field in the dynamics of the magnetic potential.

math-ph↗