Related papers: Cohomological VC-density: Bounds and Applications
It is shown that the evaluation of the expectation value (EV) of topological charge density over $\theta$-vacuum is reduced to investigation of the Chern-Simons term EV. An equation for this quantity is established and solved. EV of the…
In the paper where he defined the Cauchy completion of a $\mathscr{V}$-category, Lawvere also defined a condition on a $\mathscr{V}$-functor which made it analogous to a map of metric spaces whose image is topologically dense in its…
Voronoi diagrams appear in many areas in science and technology and have numerous applications. They have been the subject of extensive investigation during the last decades. Roughly speaking, they are a certain decomposition of a given…
We introduce the notion of a higher covering diagram in a base $\infty$-category $\mathcal{C}$. The theory of higher covering diagrams in $\mathcal{C}$ will be shown to recover various descent conditions known from the $\infty$-categorical…
In the theory of zero-dimensional systems and their relation to $C^*$-algebras, Poon (1990) introduced a class of closed sets. We call the closed sets quasi-sections. Medynets (2006) introduced basic sets that are part of quasi-sections in…
The densest subgraph problem is a classic problem in combinatorial optimisation. Danisch, Chan, and Sozio propose a definition for \emph{local density} that assigns to each vertex $v$ a value $\rho^*(v)$. This local density is a…
This is a chapter in an upcoming book on aperiodic order. We go over different versions of tiling cohomology (\v Cech, pattern-equivariant, PV, quotient) with emphasis on the inverse limit constructions used to compute these cohomologies.…
Well-graded families, extremal systems and maximum systems (the last two in the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three important classes of set systems. This paper aims to study the notion of duality in the…
In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local…
We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acl = dcl, the VC-codensity of a…
Being motivated by the study of the space $C_c(X)$ of all continuous real-valued functions on a Tychonoff space $X$ with the compact-open topology, we introduced in [15] the concepts of a $cp$-network and a $cn$-network (at a point $x$) in…
We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $\mathbb F_q$, when considered as a subset of the additive group. We conjecture that as $q \to \infty$, the squares have the…
We introduce a notion of density which extends both the notion of Lelong number and the theory of intersection for positive closed currents on Kaehler manifolds. For arbitrary finite family of positive closed currents on a compact Kaehler…
We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate well-understood stratifications of $X$ by the…
We show that if $T$ is a dependent theory then so is its Keisler randomisation $T^R$. In order to do this we generalise the notion of a Vapnik-Chervonenkis class to families of $[0,1]$-valued functions (a \emph{continuous}…
Assume that $\mathcal{P}$ is a topological property of a space $X$, then we say that $X$ is {\it dense-$\mathcal{P}$} if each dense subset of $X$ has the property $\mathcal{P}$. In this paper, we mainly discuss dense subsets of a space $X$,…
Persistent homology typically studies the evolution of homology groups $H_p(X)$ (with coefficients in a field) along a filtration of topological spaces. $A_\infty$-persistence extends this theory by analysing the evolution of subspaces such…
Let $(V,E)$ be a finite connected graph. We are concerned about the Chern-Simons Higgs model $$\Delta u=\lambda e^u(e^u-1)+f, \quad\quad\quad\quad\quad\quad{(0.1)}$$ where $\Delta$ is the graph Laplacian, $\lambda$ is a real number and $f$…
Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes…
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…