Related papers: On a Duality between Metrics and $\Sigma$-Proximit…
Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $\sigma-$-net might be very small. Keeping the original discretization (square or triangulation), can we…
In this paper we study Probability Measures (PM) from a functional point of view: we show that PMs can be considered as functionals (generalized functions) that belong to some functional space endowed with an inner product. This approach…
Intuitively, the concept of similarity is the notion to measure an inexact matching between two entities of the same reference set. The notions of similarity and its close relative dissimilarity are widely used in many fields of Artificial…
We investigate two classes of transformations of cosine similarity and Pearson and Spearman correlations into metric distances, utilising the simple tool of metric-preserving functions. The first class puts anti-correlated objects maximally…
This paper studies the properties of a new lower bound for the natural pseudo-distance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
We discuss the use of symmetries for analysing the structural identifiability and observability of control systems. Special emphasis is put on the role of discrete symmetries, in contrast to the more commonly studied continuous or Lie…
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not…
In this work, the dual flatness, which is connected with Statistics and Information geometry, of general $(\alpha,\beta)$-metrics (a new class of Finsler metrics) is studied. A nice characterization for such metrics to be dually flat under…
A new inequality between some functional of probability distribution functions is given. The inequality is based on strict convexity of a function used in functional definition. Equality sign in the inequality gives a characteristic…
Similarity measures play a fundamental role in memory-based nearest neighbors approaches. They recommend items to a user based on the similarity of either items or users in a neighborhood. In this paper we argue that, although it keeps a…
Graphs are used in almost every scientific discipline to express relations among a set of objects. Algorithms that compare graphs, and output a closeness score, or a correspondence among their nodes, are thus extremely important. Despite…
Metric algebras are metric variants of $\Sigma$-algebras. They are first introduced in the field of universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. Recently a similar notion of…
Similarity measures play a central role in various data science application domains for a wide assortment of tasks. This guide describes a comprehensive set of prevalent similarity measures to serve both non-experts and professional.…
We have shown recently that, given a metric space $X$, the coarse equivalence classes of metrics on the two copies of $X$ form an inverse semigroup $M(X)$. Here we give several descriptions of the set $E(M(X))$ of idempotents of this…
This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…
Potential functional approximations are an intriguing alternative to density functional approximations. The potential functional that is dual to the Lieb density functional is defined and properties given. The relationship between…
This report introduces and investigates a family of metrics on sets of pointed Kripke models. The metrics are generalizations of the Hamming distance applicable to countably infinite binary strings and, by extension, logical theories or…
This paper presents a distance function between sets based on an average of distances between their elements. The distance function is a metric if the sets are non-empty finite subsets of a metric space. It can be applied to produce various…
For a metric space $(A,d)$, and a set $\Sigma$ of equations, some quantities are introduced that measure the size of discontinuities that must occur in operations satisfying $\Sigma$ (identically) on $A$. We are able to evaluate these…