We construct a new set of discrete differential forms based on B-splines of arbitrary degree as well as an associated Hodge operator. The theory is first developed in 1D and then extended to multi-dimension using tensor products. We link our discrete differential forms with the theory of chains and cochains. The spline discrete differential forms are then applied to the numerical solution of Ma...

In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present finite element exterior calculus formulation that is able mimetically represent conservation laws and cope with mixed open boundary conditions using single computational mesh. The possibility of including allows for modular composition complex multi-physi...

1 Hodge structures 3 1.1 The Hodge decomposition . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 The Frölicher spectral sequence . . . . . . . . . . . . . . . . . 3 1.1.2 Hodge filtration and Hodge decomposition . . . . . . . . . . . 6 1.1.3 Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Morphisms of Hodge structures . . . . . . . . . . . . . . . . . . . . . 10 1...

This paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus. We first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity, similar to those of electromagnetism, can be...

The theorem of Nielsen and Ninomiya [3] came with a topological proof of the fact that under reasonable assumptions, fermion doubling is unavoidable on the lattice. A key element of the proof was the periodicity of the Brillouin zone thus setting the approach in momentum space. In other arguments [4], [5] it was argued that doubling is already present when one starts one step back and consider ...

Model pattern matching is an important operation in model transformation and therefore in model-driven development tools. In this paper we present a pattern based approach that includes a star operator that can be used to represent recursive or hierarchical structures in models. We also present a matching algorithm, motivating examples and we discuss its implementation in a modeling tool.

where FA denotes the curvature of A. In four dimensions, FA decomposes into its self-dual and anti-self-dual components, FA = F + A + F − A , where F A denotes the projection onto the ±1 eigenspace of the Hodge star operator. A connection is called self-dual (respectively anti-self-dual) if FA = F + A (respectively FA = F − A ). A connection is called an instanton if it is either self-dual or a...

Let (M, g) be an oriented 4-dimensional Riemannian manifold (not necessarily compact). Due to the Hodge-star operator ⋆, we have a decomposition of the bivector bundle ∧2 TM = ∧+ ⊕ ∧− . Here ∧± is the eigen-subbundle for the eigenvalue ±1 of ⋆. The metric g on M induces a metric, denoted by < , >, on the bundle ∧2 TM . Let π : Z = S (∧+) −→ M be the sphere bundle; the fiber over a point m ∈ M p...

We explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincaré duality for the Lie algebra cohomology of the infinite-dimensional gauge transformation group. We show that the adjoint of the coboundary operator can be identified with the BRST adjoint generator Q for the Lie algebra cohomology induced by BRST generator Q. We also po...

The aim of this paper is to present a new framework for regularization by diffusion. The methods we develop in the sequel can be used to smooth nD images, nD videos, vector fields and orthonormal frame fields in any dimension.1 From a mathematical viewpoint, we deal with vector bundles over Riemannian manifolds and socalled generalized Laplacians. Sections are regularized from heat equations as...