Related papers: What is a cube?
These informal notes deal with some basic properties of metric spaces, especially concerning lengths of curves.
A cubic space is a vector space equipped with a symmetric trilinear form. Two cubic spaces are isogeneous if each embeds into the other. A cubic space is non-degenerate if its form cannot be expressed as a finite sum of products of linear…
Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…
Dyadic rationals are rationals whose denominator is a power of 2. A dyadic n-dimensional convex set is defined as the intersection with n-dimensional dyadic space of an n-dimensional real convex set. Such a dyadic convex set is said to be a…
We consider the quantum computational process as viewed by an insider observer: this is equivalent to an isomorphism between the quantum computer and a quantum space, namely the fuzzy sphere. The result is the formulation of a reversible…
We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space…
This short note highlights the most prominent mathematical problems and physical questions associated with the existence of the maximum sets of mutually unbiased bases (MUBs) in the Hilbert space of a given dimension
Associated to any affine space A endowed with a metric structure of arbitrary signature we consider the space of affine functionals operating on the space of quadratic functions of A. On this functional space we characterize a symmetric…
A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed,…
We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is…
The first section of this modest survey reviews some basic notions and describes some families of examples, and the second section briefly indicates some general aspects of analysis on metric spaces. The remaining three sections are…
Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…
The species of finite topological spaces admits two graded bimonoid structures, recently defined by F. Fauvet, L. Foissy, and the second author. In this article, we define a doubling of this species in two different ways. We build a…
Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…
This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also…
We study the least doubling constant $C_{(X,d)}$, among all doubling measures $\mu$ supported on a metric space $(X,d)$. In particular, we prove that for every metric space with more than one point, $C_{(X,d)}\ge 2$. We also describe some…
A round metric space is the one in which closure of each open ball is the corresponding closed ball. By a sleek metric space, we mean a metric space in which interior of each closed ball is the corresponding open ball. In this, article we…
Consider the semialgebraic structure over the real field. More generally, let an ominimal structure be over a real closed field. We show that a definable metric space X with a definable metric d is embedded into a Euclidean space so that…
We study a recursively defined two-parameter family of graphs which generalize Fibonacci cubes and Pell graphs and determine their basic structural and enumerative properties. In particular, we show that all of them are induced subgraphs of…
A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…