Dimension on discrete spaces

Evako, Alexander V.

doi:10.1007/bf00670697

Cited by 30 publications

(24 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…then the graph has dimension D. There are other ways to define dimensionality, including at least one which works for finite graphs (Evako, 1994). For the example partition function above we might hope that there is a phase in which the expectation value of dimension takes some interesting value like 3 or 4.…”

Section: Topology and Random Graphsmentioning

confidence: 98%

Principle of event symmetry

Gibbs

1996

Int J Theor Phys

View full text Add to dashboard Cite

To accommodate topology change, the symmetry of space-time must be extended from the diffeomorphism group of a manifold to the symmetric group acting on the discrete set of space-time events. This is the principle of event-symmetric space-time. I investigate a number of physical toy models with this symmetry to gain some insight into the likely nature of event-symmetric space-time, In the more advanced models the symmetric group is embedded into larger structures such as matrix groups which provide scope to unify space-time symmetry with the internal gauge symmetries of particle physics, I also suggest that the symmetric group of space-time could be related to the symmetric group acting to exchange identical particles, implying a unification of space-time and matter. I end with a definition of a new type of loop symmetry which is important in event-symmetric superstring theory.

show abstract

Section: Topology and Random Graphsmentioning

confidence: 98%

Principle of event symmetry

Gibbs

1996

Int J Theor Phys

View full text Add to dashboard Cite

show abstract

“…This section includes results obtained in [4,5,6,10,13,14] and related to the structure of graphs that are digital ndimensional spaces which were defined in [4].…”

Section: Digital N-dimensional Surfaces and Homeomorphismmentioning

confidence: 99%

“…In paper [4], digital n-surfaces were defined as simple undirected graphs and basic properties of n-surfaces were studied. Properties of digital n-manifolds were investigated in ( [5,6,9,10]). …”

Section: Introductionmentioning

confidence: 99%

Graph Theoretical Models of Discrete Spaces with Locally Non-spherical Topology

Evako¹

2017

AMP

View full text Add to dashboard Cite

A graph theoretical model of a continuous space is a graph with the same topological structure as its continuous counterpart. A digital closed n-dimensional manifold with a locally spherical topology is a graph theoretic model for a continuous closed n-dimensional manifold. This paper defines and studies properties of a new class of digital n-dimensional spaces with a locally non-spherical topology. We prove that such spaces have the dimension n≥3. We define and investigate properties of digital 3-and 5-dimensional closed surfaces with a local toroidal and projective plane topology. These spaces have no direct continuous counterparts among n-dimensional manifolds in classical topology. These results arise questions like what physical, chemical or biological structures can be described by digital n-dimensional surfaces with a locally non-spherical topology. Keywords

show abstract

“…Definition 17 (n-surface, [5,9]) A discrete space Y is a 0-surface if Y is made of exactly two points x and y such that x ∈ st Y (y) and y ∈ st Y (x). A discrete space Y is a n-surface (n > 0) if Y is connected and if, for any…”

Section: Shapes Of Interval-valued Mapsmentioning

confidence: 99%

Discrete Set-Valued Continuity and Interpolation

Najman

Géraud

2013

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. The main question of this paper is to retrieve some continuity properties on (discrete) T0-Alexandroff spaces. One possible application, which will guide us, is the construction of the so-called "tree of shapes" (intuitively, the tree of level lines). This tree, which should allow to process maxima and minima in the same way, faces quite a number of theoretical difficulties that we propose to solve using set-valued analysis in a purely discrete setting. We also propose a way to interpret any function defined on a grid as a "continuous" function thanks to an interpolation scheme. The continuity properties are essential to obtain a quasi-linear algorithm for computing the tree of shapes in any dimension, which is exposed in a companion paper [10].

show abstract

Dimension on discrete spaces

Cited by 30 publications

References 16 publications

Principle of event symmetry

Principle of event symmetry

Graph Theoretical Models of Discrete Spaces with Locally Non-spherical Topology

Discrete Set-Valued Continuity and Interpolation

Contact Info

Product

Resources

About