Related papers: Self-Similarity in Geometry, Algebra and Arithmeti…
We show that if PGA is understood as a subalgebra of CGA in mathematically correct sense, then the flat objects share the same representation in PGA and CGA. Particularly, we treat duality in PGA. This leads to unification of PGA and CGA…
Similarity search is an important problem in information retrieval. This similarity is based on a distance. Symbolic representation of time series has attracted many researchers recently, since it reduces the dimensionality of these high…
Lie derivatives of various geometrical and physical quantities define symmetries and conformal symmetries in general relativity. Thus we obtain motions, collineations, conformal motions and conformal collineations. These symmetries are used…
The concept of complexity appears in virtually all areas of knowledge. Its intuitive meaning shares similarities across fields, but disagreements between its details hinders a general definition, leading to a plethora of proposed…
Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is…
In this paper, we introduce the following problem in the theory of algorithmic self-assembly: given an input shape as the seed of a tile-based self-assembly system, design a finite tile set that can, in some sense, uniquely identify whether…
Jacques Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits's geometry and works through several…
Uniqueness theorems are considered for various types of almost periodic objects: functions, measures, distributions, multisets, holomorphic and meromorphic functions.
Angular equivalence is introduced and shown to be an equivalence relation among the norms on a fixed real vector space. It is a finer notion than the usual (topological) notion of norm equivalence. Angularly equivalent norms share certain…
We introduce the notion of ends for algebras. The definition is analogous to the one in geometric group theory. We establish some relations to growth conditions and cyclic cohomology.
In this paper we introduce a notion of {\it generalized operad} containing as special cases various kinds of operad--like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories…
For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative…
In pregeometry a metric arises as a composite object at large distances. We investigate if its signature, which distinguishes between time and space, could be a result of the dynamics rather than being built in already in the formulation of…
This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the…
A self-similar algebra $\left(\mathfrak{A}, \psi \right)$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $\psi: \mathfrak{A} \longrightarrow M_d \left( \mathfrak{A}\right)$, where $M_d \left( \mathfrak{A}\right)$ is…
We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some…
In general, objects can be distinguished on the basis of their features, such as color or shape. In particular, it is assumed that similarity judgments about such features can be processed independently in different metric spaces. However,…
All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classification of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of…
We describe those group algebras over fields of characteristic different from 2 whose units symmetric with respect to the classical involution, satisfy some group identity.
We define general notions of coordinate geometries over fields and ordered fields, and consider coordinate geometries that are given by finitely many relations that are definable over those fields. We show that the automorphism group of…