A metric vector space is asymptotically metrically normable (AMN) if there exists a norm asymptotically isometric to the distance. We prove that AMN vector spaces are rigid in the class of metric vector spaces under asymptotically isometric perturbations. This result follows from a general metric normability criterium. If the distance is translation invariant and satisfies an approximate multip...

In this paper, viewpoint independent image retrieval by color-metric pattern-card matching is presented. First, a photometric color invariant is proposed measuring a local color property of a pixel and its neighboring pixels while discounting the disturbing influences of shading, shadows and highlights. Color-metric pattern-cards are constructed on the basis of the photometric color invariant i...

We introduce an (equi-)affine invariant geometric structure by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to evaluate a new form of geodesic distances and to construct an invariant Laplacian from which local and global diffusion geometry is constructed. Applications of the proposed fram...

In the present paper a global conformal invariant Y of a closed initial data set is constructed. A spacelike hypersurface in a Lorentzian spacetime naturally inherits from the spacetime metric a diierentiation D e , the so-called real Sen connection, which turns out to be determined completely by the initial data h ab and ab induced on , and coincides, in the case of vanishing second fundamenta...

Natural objects can be subject to various transformations yet still preserve properties that we refer to as invariants. Here, we use definitions of affine-invariant arclength for surfaces in R in order to extend the set of existing non-rigid shape analysis tools. We show that by re-defining the surface metric as its equi-affine version, the surface with its modified metric tensor can be treated...

We discuss several natural metrics on spaces of unbounded self– adjoint operators and their relations, among them the Riesz and the graph metric. We show that the topologies of the spaces of Fredholm operators resp. invertible operators depend heavily on the metric. Nevertheless we prove that in all cases the spectral flow is up to a normalization the only integer invariant of paths which is pa...

Abstract In loop quantum cosmology (LQC) the big bang singularity is generically resolved by a bounce. This feature holds even when modified quantization prescriptions of Hamiltonian constraint are used such as in mLQC-I and mLQC-II. While later describes an effective description qualitatively similar to that standard LQC, former asymmetric evolution with emergent Planckian de-Sitter pre-bounce...