arXiv research

Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.

We introduce and develop fine shape, which has a very simple definition and aims to supersede all previously known shape theories for metrizable spaces. The problem with known shape theories of metrizable spaces is illustrated by the following bizarre situation. Čech cohomology is an invariant of shape, and a fortiori …

The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.

Defines finite type Multivalued Shape using hyperspaces.

The paper defines approximate fibrations in higher topos theory.

Extended orbit model theory for shape analysis using graded group action framework.

Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.

Classifies compact spaces by shape, finite spaces by weak homotopy.

The paper shows how to use fine shape to understand infinite-dimensional spaces.

An important "stability" theorem in shape theory, due to D.A. Edwards and R. Geoghegan, characterizes those compacta having the same shape as a finite CW complex. In this note we present straightforward and self-contained proof of that theorem.

A new shape space allows optimization of non-smooth shapes in fluid mechanics.

Fine shape theory extends strong shape to noncompact metrizable spaces.

New method uses Riemannian geometry to describe molecular shapes.

The paper introduces a new method to measure the shape relations between biological objects using r-parallel sets.

Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.

Inverse spectral theory reveals shapes from sound.

We consider the problem of modelling noisy but highly symmetric shapes that can be viewed as hierarchies of whole-part relationships in which higher level objects are composed of transformed collections of lower level objects. To this end, we propose the stochastic wreath process, a fully generative probabilistic model…

Any subset of the plane can be approximated by a set of square pixels. This transition from a shape to its pixelation is rather brutal since it destroys geometric and topological information about the shape. Using a technique inspired by Morse Theory, we algorithmically produce a PL approximation of the original shape …

New Heintze-Karcher inequality helps understand droplet shapes.

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

The class of surfaces in 3-space possessing nontrivial deformations which preserve principal directions and principal curvatures (or, equivalently, the shape operator) was investigated by Finikov and Gambier as far back as in 1933. We review some of the known examples and results, demonstrate the integrability of the c…

Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.

New connection found between shape reconstruction methods and persistent homology.

Mathematical methods of population genetics and framework of exchangeability provide a Markov chain model for analysis and interpretation of stochastic behaviour of equity markets, explaining, in particular, market shape formation, statistical equilibrium and temporal stability of market weights.

In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Book…

This study proves new financial market theorems breaking standard risk definitions.

We provide a full classification of all attainable term structure shapes in the two-factor Vasicek model of interest rates. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results ap…

The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed…

This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions which allows for short and transparent proofs of the main facts of the theory of man…

Study of red blood cells using elastic surface theory.

Generates tubular and membranous shapes using curvature functionals.

Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.

In this paper, we address the problem of orientation that naturally arises when representing shapes like curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient to model a wide variety of shapes. However, in such approaches, orientation of shapes…

We analyze a monetary system of random money transfer on the basis of double entry bookkeeping. Without boundary conditions, we do not reach a price equilibrium and violate text-book formulas of economists quantity theory (MV=PQ). To match the resulting quantity of money with the model assumption of a constant price, w…

We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression, and constrained single index model. We review some of the theoretical properties of the least …

This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often contractible, so distinguishing one from another requires techniques beyond the standard …

We develop a novel method for detection of signals and reconstruction of images in the presence of random noise. The method uses results from percolation theory. We specifically address the problem of detection of multiple objects of unknown shapes in the case of nonparametric noise. The noise density is unknown and ca…

We use matrix iteration theory to characterize acceleration in smooth games. We define the spectral shape of a family of games as the set containing all eigenvalues of the Jacobians of standard gradient dynamics in the family. Shapes restricted to the real line represent well-understood classes of problems, like minimi…

The paper uses 3D shapes to reveal sundial design adjustments based on latitude.

In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $φ_0$; we refer the piece-wised elastica $Λ_{φ_0}$-elastica or $Λ$-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly du…

New neural network models extreme value distributions with preserved shape constraints.

Study analyzes neural network models to understand generalization performance.

We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surfa…

Homotopy theory of differentiable sheaves connects manifold properties to underlying homotopy types.

The manifold $\mathcal{M}$ of star-shaped curves in $\mathbb{R}^n$ is considered via the theory of connections on vector bundles, and cyclic $\mathcal{D}$-modules. The appropriate notion of an "integral curve" (i.e. certain admissible deformations) on $\mathcal{M}$ is defined, and the resulting space of admissible defo…

A novel 3D shape registration method using spectral graph embedding and probabilistic matching.

New method uses Riemannian geometry to quantify molecular shapes.

Proves higher regularity for anisotropic inverse mean curvature flow.