Elena Deza

Moscow State Pedagogical University · Mathematics

Yuri Valentinovich Nesterenko (to the 75th anniversary)

The paper devoted to the 70-th anniversary of V.N.Chubaricov and its scientic byography

This article is dedicated to the memory of the famous Russian and French scientist, a mathematician Michel Deza, who tragically passed away on November 23, 2016, at the age of 77. The review of the main stages of the professional formation and growth of M. Deza (Mikhail Efimovich Tylkin) in Russia in the 60-70s of the last century is given. His mul...

A convex body in the n-dimensional Euclidean space \(\mathbb{E}^{n}\) is a convex compact connected subset of \(\mathbb{E}^{n}\). It is called solid (or proper) if it has nonempty interior. Let K denote the space of all convex bodies in \(\mathbb{E}^{n}\), and let K p be the subspace of all proper convex bodies. Given a set \(X \subset \mathbb{E}^{...

Image Processing treats signals such as photographs, video, or tomographic output. In particular, Computer Graphics consists of image synthesis from some abstract models, while Computer Vision extracts some abstract information: say, the 3D description of a scene from video footage of it. From about 2000, analog image processing (by optical devices...

Distances are mainly used in Biology to pursue basic classification tasks, for instance, for reconstructing the evolutionary history of organisms in the form of phylogenetic trees. In the classical approach those distances were based on comparative morphology, physiology, mating studies, paleaontology and immunodiffusion. The progress of modern Mol...

A group (G, ⋅ , e) is a set G of elements with a binary operation ⋅ , called the group operation, that together satisfy the four fundamental properties of closure (x ⋅ y ∈ G for any x, y ∈ G), associativity (x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z for any x, y, z ∈ G), the identity property (\(x \cdot e = e \cdot x = x\) for any x ∈ G), and the inverse property...

An alphabet is a finite set \(\mathcal{A}\), \(\vert \mathcal{A}\vert \geq 2\), elements of which are called characters (or symbols). A string (or word) is a sequence of characters over a given finite alphabet \(\mathcal{A}\). The set of all finite strings over the alphabet \(\mathcal{A}\) is denoted by \(W(\mathcal{A})\). Examples of real world ap...

A probability space is a measurable space \((\varOmega,\mathcal{A},P)\), where \(\mathcal{A}\) is the set of all measurable subsets of Ω, and P is a measure on \(\mathcal{A}\) with P(Ω) = 1. The set Ω is called a sample space. An element \(a \in \mathcal{A}\) is called an event. P(a) is called the probability of the event a. The measure P on \(\mat...

Physics studies the behavior and properties of matter in a wide variety of contexts, ranging from the submicroscopic particles from which all ordinary matter is made (Particle Physics) to the behavior of the material Universe as a whole (Cosmology).

Functional Analysis is the branch of Mathematics concerned with the study of spaces of functions. This usage of the word functional goes back to the calculus of variations which studies functions whose argument is a function. In the modern view, Functional Analysis is seen as the study of complete normed vector spaces, i.e., Banach spaces.

In this chapter we consider a special class of metrics defined on some normed structures, as the norm of the difference between two given elements. This structure can be a group (with a group norm), a vector space (with a vector norm or, simply, a norm), a vector lattice (with a Riesz norm), a field (with a valuation), etc.

Some immediate generalizations of the notion of metric, for example, quasi-metric, near-metric, extended metric, were defined in Chap. 1 Here we give some generalizations in the direction of Topology, Probability, Algebra, etc.

Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of 2D surfaces in the Euclidean space \(\mathbb{E}^{3}\). It studies real smooth manifolds equipped with Riemannian metrics, i.e., collections of positive-definite symmetric bilinear forms ((g ij )) on their tangent spaces which vary smoothly from point to point. The...

The Universe is defined as the whole space-time continuum in which we exist, together with all the energy and matter within it.

Coding Theory deals with the design and properties of error-correcting codes for the reliable transmission of information across noisy channels in transmission lines and storage devices. The aim of Coding Theory is to find codes which transmit and decode fast, contain many valid code words, and can correct, or at least detect, many errors. These ai...

Any L p-metric (as well as any norm metric for a given norm | | . | | on \(\mathbb{R}^{2}\)) can be used on the plane \(\mathbb{R}^{2}\), and the most natural is the L 2-metric, i.e., the Euclidean metric \(d_{E}(x,y) = \sqrt{(x_{1 } - y_{1 } )^{2 } + (x_{2 } - y_{2 } )^{2}}\) which gives the length of the straight line segment [x, y], and is the i...

A network is a graph, directed or undirected, with a positive number (weight) assigned to each of its arcs or edges. Real-world complex networks usually have a gigantic number N of vertices and are sparse, i.e., with relatively few edges.

A data set is a finite set comprising m sequences (x 1 j , …, x n j ), j ∈ { 1, …, m}, of length n. The values x i 1, …, x i m represent an attribute S i .

Given a finite set A of objects A i in a space S, computing the Voronoi diagram of A means partitioning the space S into Voronoi regions V (A i ) in such a way that V (A i ) contains all points of S that are “closer” to A i than to any other object A j in A.

The term length has many meanings: distance, extent, linear measure, span, reach, end, limit, etc.; for example, the length of a train, a meeting, a book, a trip, a shirt, a vowel, a proof. The length of an object is its linear extent, while the height is the vertical extent, and width (or breadth) is the side-to-side distance at 90∘ to the length,...

A topological space (X, τ) is a set X with a topology τ, i.e., a collection of subsets of X with the following properties: 1. X ∈ τ, ∅ ∈ τ; 2. If A, B ∈ τ, then A ∩ B ∈ τ; 3. For any collection {A α }α , if all A α ∈ τ, then ∪α A α ∈ τ.

A distance space (X, d) is a set X (carrier) equipped with a distance d.

Geometry arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern Mathematics, the other being the study of numbers.

There are many ways to obtain new distances (metrics) from given distances (metrics). Metric transforms give new distances as a functions of given metrics (or given distances) on the same set X. A metric so obtained is called a transform metric. We give some important examples of transform metrics in Sect. 4.1.

A graph is a pair G = (V, E), where V is a set, called the set of vertices of the graph G, and E is a set of unordered pairs of vertices, called the edges of the graph G. A directed graph (or digraph) is a pair D = (V, E), where V is a set, called the set of vertices of the digraph D, and E is a set of ordered pairs of vertices, called arcs of the...

In this chapter we group the main distances used in Systems Theory (such as Transition Systems, Dynamical Systems, Cellular Automata, Feedback Systems) and other interdisciplinary branches of Mathematics, Engineering and Theoretical Computer Science (such as, say, Robot Motion and Multi-objective Optimization).

A surface is a real 2D (two-dimensional) manifold M 2, i.e., a Hausdorff space, each point of which has a neighborhood which is homeomorphic to a plane \(\mathbb{E}^{2}\), or a closed half-plane (cf. Chap. 7).

In Geography, spatial scales are shorthand terms for distances, sizes and areas. For example, micro, meso, macro, mega may refer to local (0.001–1), regional (1–100), continental (100–10,000), global ( > 10,000) km, respectively.

Here we consider the most important metrics on the classical number systems: the semiring \(\mathbb{N}\) of natural numbers, the ring \(\mathbb{Z}\) of integers, and the fields \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\) of rational, real, complex numbers, respectively. We consider also the algebra \(\mathcal{Q}\) of quaternions.

In this chapter we present selected distances used in real-world applications of Human Sciences. In this and the next chapter, the expression of distances ranges from numeric (say, in m) to ordinal (as a degree assigned according to some rule) and nominal.

In this chapter we group together distances and distance paradigms which do not fit in the previous chapters, being either too practical (as in equipment), or too general, or simply hard to classify.

This book introduces oriented version of metrics and cuts and their multidimensional analogues, as well as partial metrics and weighted metrics. It is a follow-up of Geometry of Cuts and Metrics by Deza and Laurent which presents rich theory of classical binary and symmetric objects - metrics and cuts. Many research publications on this subject are...

This 4-th edition of the leading reference volume on distance metrics is characterized by updated and rewritten sections on some items suggested by experts and readers, as well a general streamlining of content and the addition of essential new topics. Though the structure remains unchanged, the new edition also explores recent advances in the use...

The two major goals of the authors are to describe the principal stages of the life of Professor of the Moscow State Pedagogical University Vassiliy Ilyich Nechaev and to give a brief analysis of his scientific and educational activities that has been very influential in the development of both number theory and methods of teaching mathematics at c...

Geometry arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern Mathematics, the other being the study of numbers.

This chapter discusses the concepts related to distances in Coding Theory. Coding Theory deals with the design and the properties of error-correcting codes for the reliable transmission of information across noisy channels in transmission lines and storage devices. The aim of Coding Theory is to find codes that transmit and decode fast, contain man...

Distances are mainly used in Biology to pursue basic classification tasks, for instance, for reconstructing the evolutionary history of organisms in the form of phylogenetic trees. In the classical approach those distances were based on comparative morphology, physiology, mating studies, paleontology and immunodiffusion. The progress of modern Mole...

A network is a graph, directed or undirected, with a positive number (weight) assigned to each of its arcs or edges. Real-world complex networks usually have a gigantic number N of vertices and are sparse, i.e., with relatively few edges.

Image Processing treats signals such as photographs, video, or tomographic output. In particular, Computer Graphics consists of image synthesis from some abstract models, while Computer Vision extracts some abstract information: say, the 3D (i.e., 3-dimensional) description of a scene from video footage of it. From about 2000, analog image processi...

In this chapter we present selected distances used in real-world applications of Human Sciences. In this and the next chapter, the expression of distances ranges from numeric (say, in m) to ordinal (as a degree assigned according to some rule). Depending on the context, the distances are either practical ones, used in daily life and work outside of...

Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of two-dimensional surfaces in the Euclidean space \(\mathbb{E}^{3}\). It studies real smooth manifolds equipped with Riemannian metrics, i.e., collections of positive-definite symmetric bilinear forms ((g ij )) on their tangent spaces which vary smoothly from point...

Given a finite set A of objects A i in a space S, computing the Voronoi diagram of A means partitioning the space S into Voronoi regions V (A i ) in such a way that V (A i ) contains all points of S that are “closer” to A i than to any other object A j in A.

An alphabet is a finite set \(\mathcal{A}\), \(|\mathcal{A}| \ge 2\), elements of which are called characters (or symbols). A string (or word) is a sequence of characters over a given finite alphabet \(\mathcal{A}\). The set of all finite strings over the alphabet \(\mathcal{A}\) is denoted by \(W(\mathcal{A})\). Examples of real world applications...

This chapter provides an overview of how the concept of distance is used in physics and chemistry. Physical forces that act at a distance (that is, a push or pull which acts without physical contact) are nuclear and molecular attraction, and, beyond atomic level, gravity, static electricity, and magnetic forces. Last two forces can be both, push an...

A group (G, ⋅ , e) is a set G of elements with a binary operation ⋅ , called the group operation, that together satisfy the four fundamental properties of closure (x ⋅ y ∈ G for any x, y ∈ G), associativity (x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z for any x, y, z ∈ G), the identity property (\(x \cdot e = e \cdot x = x\) for any x ∈ G), and the inverse property...

Any L p -metric (as well as any norm metric for a given norm | | . | | on \(\mathbb{R}^{2}\)) can be used on the plane \(\mathbb{R}^{2}\), and the most natural is the L 2-metric, i.e., the Euclidean metric \(d_{E}(x,y) = \sqrt{(x_{1 } - y_{1 } )^{2 } + (x_{2 } - y_{2 } )^{2}}\) which gives the length of the straight line segment [x, y], and is the...

In this chapter we group together distances and distance paradigms which do not fit in the previous chapters, being either too practical (as in equipment), or too general, or simply hard to classify.

This chapter focuses on generalizations of metric spaces. If X be a set, then a function d : X3 → ℝ is called 2-metric if d is non-negative, totally symmetric, zero conditioned, and satisfies the tetrahedron inequality d (xl, x2, x3)≤ d (x4, x2, x3) + d (x1, x4, x3) + d (xl, x2, x4). It is the most important case m = 2 of the m-hemi-metric. A regul...

This chapter focuses on concept of distances in functional analysis. Functional Analysis is the branch of mathematics, concerned with the study of spaces of functions. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. In the modern view, functional analysis is seen as the st...

This chapter explains concepts related to distances in geometry. In mathematics, the notion of “geodesic” is a generalization of the notion of “straight line” to curved spaces. This term is taken from Geodesy, the science of measuring the size and the shape of the Earth. A geodesic is a curve that extends indefinitely in both directions and behaves...

A probability space is a measurable space\((\varOmega, \mathcal{A}, P)\), where \(\mathcal{A}\) is the set of all measurable subsets of Ω, and P is a measure on \(\mathcal{A}\) with P(Ω)=1. The set Ω is called a sample space. An element \(a \in \mathcal{A}\) is called an event. In particular, an elementary event is a subset of Ω that contains only...

This chapter provides an overview of units of length measurement and scales. The main length measure systems are Metric, Imperial (British and American), Japanese, Thai, Chinese Imperial, and Typographical. The International Metric System (or SI, short for Systeme International) is a modernized version of the metric system of units, established by...

We show that the cone of weighted n-point quasi-metrics WQMet_n, the cone of weighted quasi-hypermetrics WHyp_n and the cone of oriented cuts OCut_n are projec- tions along an extreme ray of the metric cone Metn+1, of the hypermetric cone Hypn+1 and of the cut cone Cut_{n+1}, respectively. This projection is such that if one knows all faces of an o...

A partial semimetric on V_n={1, ..., n} is a function f=((f_{ij})): V_n^2 -> R_>=0 satisfying f_ij=f_ji >= f_ii and f_ij+f_ik-f_jk-f_ii >= 0 for all i,j,k in V_n. The function f is a weak partial semimetric if f_ij >= f_ii is dropped, and it is a strong partial semimetric if f_ij >= f_ii is complemented by f_ij <= f_ii+f_jj. We describe the cones o...

Slides

Figurate numbers have a rich history with many applications. The main purpose of this book is to provide a thorough and complete presentation of the theory of figurate numbers, giving much of their properties, facts and theorems with full proofs. This book is the first of this topic written in unified systematic way. It also contains many exercises...

Distance metrics and distances have become an essential tool in many areas of pure and applied Mathematics, and this encyclopedia is the first one to treat the subject in full. The book appears just as research intensifies into metric spaces and especially, distance design for applications. These distances are particularly crucial, for example, in...

We obtain representations over non-trivial zeros of zeta function for two arithmetic functions ψ1(x) = ∑n≤x (x - n)Λ(n) and . This result is similar to classical representations of such kind for the Chebyshev functionψ(x) = ∑n≤x Λ(n).

The authors obtain, for quadratic and cyclotomic fields, asymptotic formulas for two arithmetic functions, which are similar to the divisor function. Let K be a number field, U an integer divisor of K, and N(U) the absolute norm of U. The problem to obtain an asymptotic formula for the mean value ∑ N(U 1 ⋯U k )≤x 1 of the arithmetic function τ k K...

This book comes out of need and urgency (expressed especially in areas of Information Retrieval with respect to Image, Audio, Internet and Biology) to have a working tool to compare data. The book will provide powerful resource for all researchers using Mathematics as well as for mathematicians themselves. In the time when over-specialization and t...