Related papers: Pure Data Spaces
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and…
In our previous arXiv papers (more systematically the informational conception is presented in the paper "The Information as Absolute", 2010) it was rigorously shown that Matter in our Universe - and Universe as a whole - are some…
We introduce a taxonomy of objects for EO programming language. This taxonomy is designed with a few principles in mind: non-redundancy, simplicity, and so on. The taxonomy is supposed to be used as a navigation map by EO programmers. It…
We present a data-structure for orthogonal range searching for random points in the plane. The new data-structure uses (in expectation) $O\bigl(n \log n ( \log \log n)^2 \bigr)$ space, and answers emptiness queries in constant time. As a…
The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebra (PHA) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product(STP) of matrices are…
We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the so-called ``singularities'', that emanated from looking at the theme in terms of ADG (: abstract differential geometry). Thus, according to…
In this work, I address a primary issue with adapting categorical and algebraic concepts to functional analytic settings, the lack of free objects. Using a "normed set" and associated categories, I describe constructions of normed objects,…
While Quantum Gravity remains elusive and Quantum Field Theory retains the interpretational difficulties of Quantum Mechanics, we have introduced an alternate approach to the unification of particles, fields, space and time, suggesting that…
Complete residue systems play an integral role in abstract algebra and number theory, and a description is typically found in any number theory textbook. This note provides a concise overview of complete residue systems, including a robust…
Chinese ancient sage Laozi said that everything comes from `nothing'. Einstein believes the principle of nature is simple. Quantum physics proves that the world is discrete. And computer science takes continuous systems as discrete ones.…
Symmetries and isomorphisms play similar conceptual roles when we consider how models represent physical situations, but they are formally distinct, as two models related by symmetries are not typically isomorphic. I offer a rigorous…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
Data provenance consists in bookkeeping meta information during query evaluation, in order to enrich query results with their trust level, likelihood, evaluation cost, and more. The framework of semiring provenance abstracts from the…
Quantum computing improves substantially on known classical algorithms for various important problems, but the nature of the relationship between quantum and classical computing is not yet fully understood. This relationship can be…
What is space computing, simulation, or understanding? Converging from several sources, this seems to be something more primitive than what is usually meant by computation, something that was along with us since antiquity (the word…
We propose a simple cognitive model where qualitative and quantitative com- parisons enable animals to identify objects, associate them with their properties held in memory and make naive inference. Simple notions like equivalence re-…
Data cohesion, a recently introduced measure inspired by social interactions, uses distance comparisons to assess relative proximity. In this work, we provide a collection of results which can guide the development of cohesion-based methods…
An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the…
A point of a digital space is called simple if it can be deleted from the space without altering topology. This paper introduces the notion simple set of points of a digital space. The definition is based on contractible spaces and…