This paper will prove the generalized Stokes Theorem over kdimensional manifolds. We will begin from the definition of a k-dimensional manifold as well as introduce the notion of boundaries of manifolds. Using these, we will construct the necessary machinery, namely tensors, wedge products, differential forms, exterior derivatives, and integrals over manifolds, in order to prove the main result...

We discuss the algebraic way of solving the descent equations corresponding to the BRST consistency condition for the gauge anomalies and the Chern–Simons terms on a nontrivial bundle. The method of decomposing the exterior derivative as a BRST commutator is extended to the present case.

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We show how the high order finite element spaces of differential forms due to Raviart-Thomas-Nédelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of Arnold-Falk-Winther. Based on observations by Bossavit, we provide new low order degrees of freedom. As an alternative to existing choices of bases, we provide canonical resolu...

Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding to the choice of a splitting of the short exact sequence. Equivalent represent...

In this paper, we study irrotational subsonic and subsonic-sonic flows with general conservative forces in the exterior domains. The indicate new Bernoulli law naturally. For case, introduce a modified cut-off system depending on which needs varied Bers skill, construct solution by variational formula. Moreover, comparing previous results, our result extends pressure-density relation to case. A...

In the work of Rostami et al., Bogomolov multiplier a Lie algebra L over field Ω is defined as particular factor subalgebra exterior product L∧L. If finite dimensional, we identify this object certain subgroup second cohomology group H2(L,Ω) by deriving Hopf-Type formula. As an application, affirmatively answer two questions posed Kunyavskiĭ regarding invariance under isoclinism algebras and ex...

We study the uncoupled space-time fractional operators involving time-dependent coefficients and formulate corresponding inverse problems. Our goal is to determine variable from exterior partial measurements of Dirichlet-to-Neumann map. exploit integration by parts formula for Riemann-Liouville Caputo derivatives derive Runge approximation property our operator based on unique continuation Lapl...

<p style='text-indent:20px;'>We consider parabolic equations on bounded smooth open sets <inline-formula><tex-math id="M1">\begin{document}$ {\Omega}\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ N\ge 1 $\end{document}</tex-math></inline-formula>) with mixed Dirichlet typ...

Ω(R) d −→ Ω(R) d −→ Ω(R) d −→ Ω(R), where Ωi(R) is the space of differential forms of degree i = 0, 1, 2, 3 on the space R over the ring of functions A = {f : R → R | f ∈ C(R)}. In the consideration, which follows, we give definitions of the first-order differential operations. Let us notice that one-dimensional spaces Ω0(R) and Ω3(R) are isomorphic to A and let φ0 : Ω (R) → A, φ3 : Ω (R) → A b...