In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite volume methods for scalar hyperbolic conservation laws, we first discretize the interior product (also called contraction) through integrals over Eulerian approximations of extrusions. This, along with Cartan’s homotopy formula and a discret...

We study the Schrödinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulat...

1 Distance Fields Given an implicit surface ∂S with S ⊆ R the corresponding distance field (also called signed distance function) D : R → R which maps a point p from the domain of the distance field R to a real-valued distance value is obtained by distance transforming the surface using the formula D(p) = sign(p) ·min{d(p, q) : q ∈ S} (1) The metric d(p, q) used is the standard Euclidean metric...

Abstract We develop a framework shedding light on common features of coupled variational formulations arising in electromagnetic scattering and acoustics. show that spurious resonances haunting domain-boundary based direct boundary integral equations the first kind originate from formal structure their Calderón identities. Using this observation, kernel problem is characterized explicitly, we i...

Let K be a smooth immersed curve in the plane. Fabricius-Bjerre [2] found the following relation among the double tangent lines, crossings, and inflections points for a generic K : T1 − T2 = C + (1/2)I where T1 and T2 are the number of exterior and interior double tangent lines of K , C is the number of crossings, and I is the number of inflection points. Here “generic” means roughly that the i...

The quadrupole formula in four-dimensional Einstein gravity is a useful tool to describe gravitational wave radiation. We derive the quadrupole formula for the Kaluza-Klein (KK) modes in the Randall-Sundrum braneworld model. The quadrupole formula provides transparent representation of the exterior weak gravitational field induced by localized sources. We find that a general isolated dynamical ...

and Applied Analysis 3 We denote by ∧l ∧l R the space of l-covectors in R and the direct sum ∧ R n ⊕ l 0 ∧l R 2.1 is a graded algebra with respect to the wedge product ∧. We will make use of the exterior derivative: d : C∞ ( Ω,∧l ) −→ C∞ ( Ω,∧l 1 ) 2.2 and its formal adjoint operator d∗ −1 nl 1 ∗ d∗ : C∞ ( Ω,∧l 1 ) −→ C∞ ( Ω,∧l ) , 2.3 known as the Hodge codifferential, where the symbol ∗ denot...

the contraposition (musāmita) argument is among the arguments for proving the finitude of dimensions that has received attention in discussing physics and has yielded some philosophical consequences. contraposition argument is based on the two concepts of motion and contraposition, and by proving the impossibility of the motion of circle in case of the infinitude of dimensions, it deals with th...

As the loop space of a Riemannian manifold is infinite-dimensional, it non-trivial problem to make sense “top degree component” differential form on it. In this paper, we show that formula from finite dimensions generalizes assign sensible certain composite forms, obtained by wedging with exponential (in exterior algebra) canonical presymplectic 2-form space. This construction crucial ingredien...

Given a digraph G = (V,E) with the set of vertices V and the set of edges E, let d : F → Ω be the exterior derivative map from the space of complex-valued functions on V to the complex vector space spanned by E. We introduce the Laplacian ∆ : F → F and the label difference map d̂ : F → (Ω1)∗ of F into the dual space (Ω1)∗ of Ω and establish their connections with d. In particular, we prove that,...