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Persistence diagrams are common objects in the field of Topological Data Analysis. They are topological summaries that capture both topological and geometric structure within data. Recently there has been a surge of interest in developing…
The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many…
The geometric mean of two matrices is considered and analyzed from a computational viewpoint. Some useful theoretical properties are derived and an analysis of the conditioning is performed. Several numerical algorithms based on different…
We give a complete characterization of closed sets $F \subset \mathbb{R}^2$ whose distance function $d_F:= \mathrm{dist}(\cdot,F)$ is DC (i.e., is the difference of two convex functions on $\mathbb{R}^2$). Using this characterization, a…
A common measure of a function's complexity is the count of its stationary points. For complicated functions, this count grows exponentially with the volume and dimension of their domain. In practice, the count is averaged over a class of…
We study closed sets $F \subset {\mathbb R}^d$ whose distance function $d_F:= {\rm dist}\,(\cdot,F)$ is DC (i.e., is the difference of two convex functions on ${\mathbb R}^d$). Our main result asserts that if $F \subset {\mathbb R}^2$ is a…
A self-map $T$ of a $\nu$-generalized metric space $(X,d\,)$ is said to be a Ciric-Matkowski contraction if $d(Tx,Ty)<d(x,y)$, for $x\neq y$, and, for every $\epsilon>0$, there is $\delta>0$ such that $d(x,y)<\delta+\epsilon$ implies…
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each…
Computing the distance function to some surface or line is a problem that occurs very frequently. There are several ways of computing a relevant approximation of this function, using for example technique originating from the approximation…
Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally…
We study the asymptotic shape of the trajectory of the stochastic gradient descent algorithm applied to a convex objective function. Under mild regularity assumptions, we prove a functional central limit theorem for the properly rescaled…
Nearest neighbor cells in $R^d,d\in\mathbb{N}$, are used to define coefficients of divergence ($\phi$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a…
This paper addresses problems in functional metric geometry that arise in the study of data such as signals recorded on geometric domains or on the nodes of weighted networks. Datasets comprising such objects arise in many domains of…
In many robotics applications, it is necessary to compute not only the distance between the robot and the environment, but also its derivative - for example, when using control barrier functions. However, since the traditional Euclidean…
In 2017 M. Bessenyei and Z. P\'ales introduced a definition of a triangle function which generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept we discuss already known inequalities in metric…
For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of…
Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such…
Approximation of entire functions by their pad\'e approximants has been examined in the past. It is true that generically such an approximation holds. However, examining this problem from another viewpoint, we obtain stronger generic…
S-metric and b-metric spaces are metrizable, but it is still quite impossible to get an explicit form of the concerned metric function. To overcome this, the notion of $\phi$-metric is developed by making a suitable modification in triangle…
In this paper, we introduce the new generalization of contraction mapping by a new control function and an altering distance . We establish some existence results of fixed point for such mappings. Our results reproduce several old and new…