We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S3, the dihedral group D4 and the quaternion group Q. Poincaré duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general. A short review of the bicovariant (noncommutative) differential cal...

We devise and analyze vertex-based, Péclet-robust, lowest-order schemes for advection-diffusion equations that support polyhedral meshes. The schemes are formulated using Compatible Discrete Operators (CDO), namely primal and dual discrete differential operators, a discrete contraction operator for advection, and a discrete Hodge operator for diffusion. Moreover, discrete boundary operators are...

We provide a common framework for compatible discretizations using algebraic topology to guide our analysis. The main concept is the natural inner product on cochains, which induces a combinatorial Hodge theory. The framework comprises of mutually consistent operations of differentiation and integration, has a discrete Stokes theorem, and preserves the invariants of the DeRham cohomology groups...

This paper deals with a semi-classical limit(Theorem 1) by using traditional mathematical methods, and shows a Hopf theorem as a corollary. A formal discussing of it may be found in [7] 1 A semi-classical limit theorem Let M be a compact, closed Riemannian manifold of dim n, and V a vector field without degenerate zeros on M . Let Λ∗(M) be the space of differential forms on M , and D = d+ δ : Λ...

We present a new wavelet numerical scheme for the discretization of Navier-Stokes equations with physical boundary conditions. The temporal discretization of the method is inspired from the projection method. Helmholtz-Hodge decomposition using divergence-free and curl-free wavelet bases satisfying physical boundary conditions allows to define the projection operator. This avoids the use of Poi...

We discuss local and covariant dual BRST symmetry and its consequences in two-dimensional quantum electrodynamics (QED). With the help of BRSTand dual BRST charges, we explore the BRST cohomology of physical states in the quantum Hilbert space. The algebra between conserved and nilpotent BRSTand dual BRST charges leads to the existence of a BRST-extended Casimir operator. The Hodge decompositio...

We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier-type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to th...

The so-called Ambarzumyan theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator − d2 dx2 +q with an integrable real-valued potential q on [0,π] are {n2 : n 0} , then q = 0 for almost all x ∈ [0,π] . In this work, the classical Ambarzumyan theorem is extended to star graphs with Dirac operators on its edges. We prove that if the spectrum of Dirac operator on star graphs ...

We discuss the BRST cohomology and exhibit a connection between the Hodge decomposition theorem and the topological properties of a two dimensional free non-Abelian gauge theory having no interaction with matter fields. The topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. We obtain two sets of topological invariants ...