Related papers: Distance-squared mappings
A result is given to find points where a real valued function on the plane is not smooth. Provided this function is induced by a smooth mapping from three dimensions to the plane, from a function on surfaces in three dimensions. This has…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…
The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in…
Functions correspond to one of the key concepts in mathematics and science, allowing the representation and modeling of several types of signals and systems. The present work develops an approach for characterizing the coverage and…
This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this…
The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value…
We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^n$, with respect to the Minkowski functional of a convex polytope. We obtain the regularity of the distance function in certain cases. We also…
We survey theoretical, algorithmic, and computational results at the intersection of distance geometry problems and mathematical programming, both with and without adjacencies as part of the input. While mathematical programming methods can…
We present $\sigma$-strongly functionally discrete mappings which expand the class of $\sigma$-discrete mappings and generalize Banach's theorem on analytically representable functions
Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of geometric properties of smooth manifolds. Round fold maps were introduced as stable fold maps…
Generalized distance-squared mappings are quadratic mappings of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ of special type. In the case that matrices $A$ constructed by coefficients of generalized distance-squared mappings of $\mathbb{R}^2$ into…
We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection…
We introduce a novel representation and optimization framework for discrete geodesics on triangle meshes that reduces artifacts of linear methods on uneven and coarse discretizations. Our method computes squared geodesic distances from…
On a complete, connected, locally compact, non-compact geodesic space $(X,d)$, we assign each compact set a distance-like function. With the help of these functions, we obtain a pseudo-metric on the space of (non-empty) compact subsets of…
The theory of graphons has proven to be a powerful tool in many areas of graph theory. In this paper, we introduce several foundational aspects of the theory of digraphons -- asymmetric two-variable functions that arise as limits of…
A mathematical relation between elements of one- and multi-dimensional discrete Fourier transforms (DFT) is found. A method of analysing the multi-dimensional data by their single one-dimensional (1-D) DFT is offered. An experiment of…
This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks.…
: In studies of discrete structures, functions are frequently used that express proximity, but are not metrics. We consider a class of such functions that is characterized by a normalization condition and an inequality that plays the same…
A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is a part of a series of papers which show that such a…