Related papers: Shape Derivatives
Shape calculus concerns the calculation of directional derivatives of some quantity of interest, typically expressed as an integral. This article introduces a type of shape calculus based on localized dilation of boundary faces through…
We provide a theory of manifold-valued rough paths of bounded 3 > p-variation, which we do not assume to be geometric. Rough paths are defined in charts, and coordinate-free (but connection-dependent) definitions of the rough integral of…
We construct geometric examples of N-differential graded algebras such as the algebra of differential forms of depth $N$ on an affine manifold, and $N$-flat covariant derivatives.
We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited…
A four dimensional treatment of nonrelativistic space-time gives a natural frame to deal with objective time derivatives. In this framework some well known objective time derivatives of continuum mechanics appear as Lie-derivatives. Their…
We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The…
Let $M$ be a complete Riemannian manifold and $G$ a Lie subgroup of the isometry group of $M$ acting freely and properly on $M.$ We study the Dirichlet Problem \begin{align*} \operatorname{div}\left( \frac{a\left( \left\Vert \nabla…
Shape inference is classically ill-posed, because it involves a map from the (2D) image domain to the (3D) world. Standard approaches regularize this problem by either assuming a prior on lighting and rendering or restricting the domain,…
The standard way of deriving Euler-Lagrange (EL) equations given a point particle action is to vary the trajectory and set the first variation of the action to zero. However, if the action is (i) reparameterisation invariant, and (ii)…
We establish a shape-derivative formula for the Dirichlet-to-Neumann operator on a compact manifold. Then, we apply this formula to obtain the well-posedness in H 1 under a specific Rayleigh-Taylor condition to an equation describing cell…
We propose a nonlocal operator method for solving partial differential equations (PDEs). The nonlocal operator is derived from the Taylor series expansion of the unknown field, and can be regarded as the integral form "equivalent" to the…
In this paper we present a framework for automated shape differentiation in the finite element software NGSolve. Our approach combines the mathematical Lagrangian approach for differentiating PDE constrained shape functions with the…
Scale invariance has received very little attention in physics. Nevertheless, it provides a natural conceptual foundation for a relational understanding of the universe, where absolute size loses meaning and only dimensionless ratios retain…
In this work we use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the \textit{schematization functor} $X \mapsto (X\otimes \mathbb{C})^{sch}$,…
We provide specific PDEs for preserved quantities $Q$ in Geometry, as well as a bridge between this and specific PDEs for observables $O$ in Physics. We furthermore prove versions of four other theorems either side of this bridge: the below…
In a natural way, the local diffeomorphisms of a manifold onto itself act on the reference frame bundles of any order and on the bundles associated with them. Due to the transitivity, the invariants by diffeomorphisms of an associated…
In this PhD thesis, we develop a new approach to classical gravity starting from Mach's principles and the idea that the local shape of spatial configurations is fundamental. This new theory, "shape dynamics", is equivalent to general…
The classical fiber product in algebraic geometry provides a powerful tool for studying loci where two morphisms to a base scheme, $\phi: X \to S$ and $\psi: Y \to S$, coincide exactly. This condition of strict equality, however, is…
In the context of the Higher-Order Maxwell-Einstein-Scalar (HOMES) theories, which are invariant under spacetime diffeomorphisms and $U(1)$ gauge symmetry, we study two broad subclasses: the first is up to linear in $R_{\mu\nu\alpha\beta}$,…
We study eigenvalue problems for the de Rham complex on varying three dimensional domains. Our analysis includes the Helmholtz equation as well as the Maxwell system with mixed boundary conditions and non-constant coefficients. We provide…