Related papers: Configuration Spaces of Points: A User's Guide
The bicategory of parameterized spectra has a remarkably rich structure. In particular, it is possible to take traces in this bicategory, which give classical invariants that count fixed points. We can also take equivariant traces, which…
We describe a new way to construct finite geometric objects. For every k we obtain a symmetric configuration E(k-1) with k points on a line. In particular, we have a constructive existence proof for such configurations. The method is very…
We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $\chi_G(k,l)$ that counts the number of signed colorings using colors…
Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in…
Graph compositions generalize both integer compositions and partitions of a finite set. We develop formulas, generating functions and recurrence relations for composition counting functions for several families of graphs.
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
Previous results about $n$-grids with acceptable colorings are extended to $n$-indexed hyperspaces, which are structures ${\mathcal A} = (A;E_0,E_1, \ldots, E_{n-1})$, where each $E_i$ is an equivalence relation.
Let $P(G,\lambda)$ denote the number of proper vertex colorings of $G$ with $\lambda$ colors. The chromatic polynomial $P(C_n,\lambda)$ for the cycle graph $C_n$ is well-known as $$P(C_n,\lambda) = (\lambda-1)^n+(-1)^n(\lambda-1)$$ for all…
This paper is the second part of a series that develops the mathematical framework necessary for studying field theories on ``T-Minkowski'' noncommutative spacetimes. These spacetimes constitute a class of noncommutative geometries,…
Given a finite graph G and a topological space Z, the graphical configuration space Conf(G, Z) is the space of functions V(G) -> Z so that adjacent vertices map to distinct points. We provide a homotopy decomposition of Conf(G, X x Y) in…
This note is dedicated to the study of a Hopf module structures on the space of framed chord diagrams and framed graphs. We also introduce a framed version of the chromatic polynomial and propose two methods to construct framed weight…
We derive general analytic expressions for the chromatic dispersion orders valid to infinity, due to the k vector or phase {\phi} dependence on the wavelength. Additionally, we identify polynomials and recursion relations associated with…
A natural class of coloring complexes $X$ on closed manifold $M^n$ is investigated that gives a holonomy map $\mbox{Hol}_X: \pi_1(M) \to S_{n+1}$. By a $k$-multilayer complex construction the holonomy map may be defined to any finite…
A Smarandache multi-space is a union of $n$ different spaces equipped with some different structures for an integer $n\geq 2$, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in…
For theories formulated with a maximally symmetric momentum space we propose a general characterization for the description of interactions in terms of the isometry group of the momentum space. The well known cases of…
Spacetime is represented by ordered sequences of topologically closed Poincare sections of the primary space constructed of primary empty cells. These mappings are constrained to provide homeomorphic structures serving as frames of…
Motivated by applications in the medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological…
Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case…
The first part of these notes is a self-contained introduction to generalized complex geometry. It is intended as a `user manual' for tools used in the study of supersymmetric backgrounds of supergravity. In the second part we review some…
First we argue in an informal, qualitative way that it is natural to enlarge space-time to five dimensions to be able to solve the problem of elementary particle masses. Several criteria are developed for the success of this program.…