Related papers: An algebraic approach to the set of intervals
We study arrangements of intervals in $\mathbb{R}^2$ for which many pairs form trapezoids. We show that any set of intervals forming many trapezoids must have underlying algebraic structure, which we characterise. This leads to some…
Indexing intervals is a fundamental problem, finding a wide range of applications. Recent work on managing large collections of intervals in main memory focused on overlap joins and temporal aggregation problems. In this paper, we propose…
An algebraic structure underlying the quantity calculus is proposed consisting in an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them…
We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a…
In this article we present very intuitive, easy to follow, yet mathematically rigorous, approach to the so called data fitting process. Rather than minimizing the distance between measured and simulated data points, we prefer to find such…
Structured matrices with symbolic sizes appear frequently in the literature, especially in the description of algorithms for linear algebra. Recent work has treated these symbolic structured matrices themselves as computational objects,…
In this paper, we introduce a novel generalization of the classical property of algebras known as "being alternative," which we term "partially alternative." This new concept broadens the scope of alternative algebras, offering a fresh…
Let $X=\mathcal{M}(A)$ be an affinoid space and let $f,g \in A$. We study the sets of connected components of the spaces defined by an inequality of the form $|f|\le r|g|$, with $r\ge 0$. We prove that there exists a finite partition of…
In this paper, we present how to use a free algebra based interval arithmetics framework in order to build better defined inclusion function for interval semi-group and for its associated vector space. One introduces the psi-algorithm,…
A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has…
Given a postulated set of points, an algebraic system of axioms is proposed for an "arrow space'". An arrow is defined to be an ordered set of two points <T, H>, named respectively Tail and Head. The set of arrows is an arrow space. The…
This book has eleven chapters. Chapter one describes all types of natural class of intervals and the arithmetic operations on them. Chapter two introduces the semigroup of natural class of intervals using R or Zn and study the properties…
The past decades have seen increasing interest in modelling uncertainty by heterogeneous methods, combining probability and interval analysis, especially for assessing parameter uncertainty in engineering models. A unifying mathematical…
Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space R^n, Hilbert spaces admit various useful…
In these lectures, we discuss two approaches to studying orbit spaces of algebraic Lie groups. Due to algebraic approach orbit space, or quotient, is an algebraic manifold, while from the differential viewpoint a quotient is a differential…
In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the rough-set model, the lower and upper bounds of…
Extensions of previous linear regression models for interval data are presented. A more flexible simple linear model is formalized. The new model may express cross-relationships between mid-points and spreads of the interval data in a…
Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
Family of quasi-arithmetic means has a natural, partial order (point-wise order) $A^{[f]}\le A^{[g]}$ if and only if $A^{[f]}(v)\le A^{[g]}(v)$ for all admissible vectors $v$ ($f,\,g$ and, later, $h$ are continuous and monotone and defined…
An algebraic scheme is suggested in which discretized spacetime turns out to be a quantum observable. As an example, a toy model producing spacetimes of four points with different topologies is presented. The possibility of incorporating…