Topological Theorem Research Articles

Safety monitor for train‐centric CBTC system

Train-centric communications-based train control (TcCBTC) system is a new solution for urban transit signalling. Compared to traditional train control systems, the on-board equipment is becoming more powerful and more complex. Due to its safety-critical nature, specialised technologies must be adopted to guarantee the safety of the system. To address the safety verification difficulty of the control logic for the new system, this study presents an innovative topology-based method for guaranteeing the train control safety. First, a railway network is described as a metric space, and then, topological spaces are introduced to express the movement authority and train trajectory. On the basis of the topological description, the safety rules are checked by performing a series computation of topology theorems. Finally, a case study has been carried out on a real metro line in China. The result shows that the proposed method strictly meets the safety verification and achieves excellent performance.

Topologically Protected Landau Level in the Vortex Lattice of a Weyl Superconductor

The question whether the mixed phase of a gapless superconductor can support a Landau level is a celebrated problem in the context of d-wave superconductivity, with a negative answer: the scattering of the subgap excitations (massless Dirac fermions) by the vortex lattice obscures the Landau level quantization. Here we show that the same question has a positive answer for a Weyl superconductor: the chirality of the Weyl fermions protects the zeroth Landau level by means of a topological index theorem. As a result, the heat conductance parallel to the magnetic field has the universal value G=1/2g_{0}Φ/Φ_{0}, with Φ as the magnetic flux through the system, Φ_{0} as the superconducting flux quantum, and g_{0} as the thermal conductance quantum.

Nonlocal curvature and topology of locally conformally flat manifolds

In this paper, we focus on the geometry of compact conformally flat manifolds (Mn,g) with positive scalar curvature. Schoen–Yau proved that its universal cover (Mn˜,g˜) is conformally embedded in Sn such that Mn is a Kleinian manifold. Moreover, the limit set of the Kleinian group has Hausdorff dimension <n−22. If additionally we assume that the non-local curvature Q2γ≥0 for some 1<γ<2, the Hausdorff dimension of the limit set is less than or equal to n−2γ2. If Q2γ>0, then the above inequality is strict. Moreover, the above upper bound is sharp. As applications, we obtain some topological rigidity and classification theorems in lower dimensions.

Black hole topology in gravity

Hawking’s topology theorem in general relativity restricts the cross-section of the event horizon of a black hole in 3 + 1 dimension to be either spherical or toroidal. The toroidal case is ruled out by the topology censorship theorems. In this article, we discuss the generalization of this result to black holes in gravity in 3 + 1 and higher dimensions. We obtain a sufficient differential condition on the function , which restricts the topology of the horizon cross-section of a black hole in gravity in 3 + 1 dimension to be either S2 or . We also extend the result to higher dimensional black holes and show that the same sufficient condition also restricts the sign of the Yamabe invariant of the horizon cross-section.

Integrable systems and closed one forms

In the first part of this paper we revisit a classical topological theorem by Tischler (1970) and deduce a topological result about compact manifolds admitting a set of independent closed forms proving that the manifold is a fibration over a torus. As an application we reprove the Liouville theorem for integrable systems asserting that the invariant sets or compact connected fibers of a regular integrable system are tori. We give a new proof of this theorem (including the non-commutative version) for symplectic and more generally Poisson manifolds.

Application of Algebraic Topology to Homologous Recombination of DNA

SummaryBrouwer's fixed point theorem, a fundamental theorem in algebraic topology proved more than a hundred years ago, states that given any continuous map from a closed, simply connected set into itself, there is a point that is mapped unto itself. Here we point out the connection between a one-dimensional application of Brouwer's fixed point theorem and a mechanism proposed to explain how extension of single-stranded DNA substrates by recombinases of the RecA superfamily facilitates significantly the search for homologous sequences on long chromosomes.

Topological Tverberg Theorem: the proofs and the counterexamples

The history of the Topological Tverberg Theorem (TTT) is described. Some important constructions are presented, and their properties are discussed. In particular, there is a detailed description of the cell structure of the classifying space , where is the permutation group. Bibliography: 9 titles.

Finiteness of corner vortices

Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows, widely known as Moffatt vortices was thought to be infinite. However, the already existing and most recent geometric theories on incompressible viscous flows that express vortical structures in terms of critical points in bounded domains, indicate a strong opposition to this notion of infiniteness. In this study, we endeavor to bridge the gap between the two opposing stream of thoughts by addressing what might have gone wrong and pinpoint the shortcomings on the assumptions of the existing theorems on Moffatt vortices. We provide our own set of proofs for establishing the finiteness of the sequence of Moffatt vortices by making use of the continuum hypothesis and Kolmogorov scale, which guarantee a non-zero scale for the smallest vortex structure possible in incompressible viscous flows. We point out that the notion of infiniteness resulting from discrete self-similarity of the vortex structures is not physically feasible. The centers of these vortices have been quantified by us as fixed points through Brouwer fixed-point theorem and boundary of a vortex as circle cell. With the aid of these new developments and making use of some existing theorems in topology along with some elementary concept of mathematical analysis, we provide several approaches to delve into this issue. All these approaches converge to the same conclusion that the sequence of Moffatt vortices cannot be infinite; in fact, it is at most finite.

Derived localisation of algebras and modules

For any dg algebra A, not necessarily commutative, and a subset S in H(A), the homology of A, we construct its derived localisation LS(A) together with a map A→LS(A), well-defined in the homotopy category of dg algebras, which possesses a universal property, similar to that of the ordinary localisation, but formulated in homotopy invariant terms. Even if A is an ordinary ring, LS(A) may have non-trivial homology. Unlike the commutative case, the localisation functor does not commute, in general, with homology but instead there is a spectral sequence relating H(LS(A)) and LS(H(A)); this spectral sequence collapses when, e.g. S is an Ore set or when A is a free ring.We prove that LS(A) could also be regarded as a Bousfield localisation of A viewed as a left or right dg module over itself. Combined with the results of Dwyer–Kan on simplicial localisation, this leads to a simple and conceptual proof of the topological group completion theorem. Further applications include algebraic K-theory, cyclic and Hochschild homology, strictification of homotopy unital algebras, idempotent ideals, the stable homology of various mapping class groups and Kontsevich's graph homology.

Convergence and covering on graphs for wait-free robots

The class of robot convergence tasks has been shown to capture fundamental aspects of fault-tolerant computability. A set of asynchronous robots that may fail by crashing, start from unknown places in some given space, and have to move towards positions close to each other. In this article, we study the case where the space is uni-dimensional, modeled as a graph G. In graph convergence, robots have to end up on one or two vertices of the same edge. We consider also a variant of robot convergence on graphs, edge covering, where additionally, it is required that not all robots end up on the same vertex. Remarkably, these two similar problems have very different computability properties, related to orthogonal fundamental issues of distributed computations: agreement and symmetry breaking. We characterize the graphs on which each of these problems is solvable, and give optimal time algorithms for the solvable cases. Although the results can be derived from known general topology theorems, the presentation serves as a self-contained introduction to the algebraic topology approach to distributed computing, and yields concrete algorithms and impossibility results.

On the geometry and topology of initial data sets with horizons

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set $(M, g, K)$ such that the boundary $\partial M$ of $M$ is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that $M\setminus \partial M$ contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions $3\leq n \leq 7$, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary $\partial M$ induced by $g$. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem \cite{GS06} to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in \cite{EGP13} in dimension $n=3$, to higher dimensions.

$\delta$-homogeneity in Finsler geometry and the positive curvature problem

In this paper, we explore the similarity between normal homogeneity and $\delta$-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected $\delta$-homogeneous Finsler space is $G$-$\delta$-homogeneous, for some suitably chosen connected quasi-compact $G$. So $\delta$-homogeneous Finsler metrics can be defined by a bi-invariant singular metric on $G$ and submersion, just as normal homogeneous metrics, using a bi-invariant Finsler metric on $G$ instead. More careful analysis shows, in the space of all Finsler metrics on $G/H$, the subset of all $G$-$\delta$-homogeneous ones is in fact the closure for the subset of all $G$-normal ones, in the local $C^0$-topology (Theorem 1.1). Using this approximation technique, the classification work for positively curved normal homogeneous Finsler spaces can be applied to classify positively curved $\delta$-homogeneous Finsler spaces, which provides the same classification list. As a by-product, this argument tells more about $\delta$-homogeneous Finsler metrics satisfying the (FP) condition (a weaker version of positively curved condition).

General Type-2 Fuzzy Topological Spaces

In this paper, a presented definition of type-2 fuzzy sets and type-2 fuzzy set operation on it was given. The aim of this work was to introduce the concept of general topological spaces were extended in type-2 fuzzy sets with the structural properties such as open sets, closed sets, interior, closure and neighborhoods in topological spaces were extended to general type-2 fuzzy topological spaces and many related theorems are proved.

Conditional Diagnosability of Exchanged Hypercube Under the MM* Model

Exchanged hypercube $EH(s,t)$ is a typical hypercube variant, which is built up by systematically removing a range of edges from hypercube $Q_{s+t+1}$ . $EH(s,t)$ preserves several advantages of hypercube and has better properties in terms of diameter and cost. Diagnosability and conditional diagnosability have been two important measures of self-diagnostic capability. Normally, conditional diagnosability is several times higher than its diagnosability because conditional diagnosability neglects the occurrence of low probability. In this paper, we study the properties of $EH(s,t)$ and present some elementary topological and connectivity theorems. Then, based on the above results, we determine its conditional diagnosability under the MM* model.

Nematic Liquid-Crystal Colloids.

This article provides a concise review of a new state of colloidal matter called nematic liquid-crystal colloids. These colloids are obtained by dispersing microparticles of different shapes in a nematic liquid crystal that acts as a solvent for the dispersed particles. The microparticles induce a local deformation of the liquid crystal, which then generates topological defects and long-range forces between the neighboring particles. The colloidal forces in nematic colloids are much stronger than the forces in ordinary colloids in isotropic solvents, exceeding thousands of kBT per micrometer-sized particle. Of special interest are the topological defects in nematic colloids, which appear in many fascinating forms, such as singular points, closed loops, multitudes of interlinked and knotted loops or soliton-like structures. The richness of the topological phenomena and the possibility to design and control topological defects with laser tweezers make colloids in nematic liquid crystals an excellent playground for testing the basic theorems of topology.

Warped product space-times

Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis on the Kodama vector field and the Hawking energy; the construction is signature independent. This leads to proofs of general Birkhoff-type theorems for warped product manifolds; our theorems in particular apply to situations where the warped product manifold is not necessarily Einstein, and thus can be applied to solutions with matter content in general relativity. Next we specialize to the Lorentzian case and study the propagation of null expansions under the assumption of the dominant energy condition. We prove several non-existence results relating to the Yamabe class of the fibers, in the spirit of the black-hole topology theorem of Hawking–Galloway–Schoen. Finally we discuss the effect of the warped product ansatz on matter models. In particular we construct several cosmological solutions to the Einstein–Euler equations whose spatial geometry is generally not isotropic.

$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$

Let $X$ be a paracompact space, let $G$ be a finite group acting freely on $X$ and let $H$ a cyclic subgroup of $G$ of prime order $p$. Let $f:X\rightarrow M$ be a continuous map where $M$ is a connected $m$-manifold (orientable if $p>2$) and $f^* (V_k) = 0$, for $k\geq 1$, where $V_k$ are the $Wu$ classes of $M$. Suppose that $\ind X\geq n> (|G|-r)m$, where $r=\frac{|G|}{p}$. In this work, we estimate the cohomological dimension of the set $A(f,H,G)$ of $(H,G)$-coincidence points of $f$. Also, we estimate the index of a $(H, G)$-coincidence set in the case that $H$ is a $p$-torus subgroup of a particular group $G$ and as application we prove a topological Tverberg type theorem for any natural number $r$. Such result is a weak version of the famous topological Tverberg conjecture, which was proved recently fail for all $r$ that are not prime powers. Moreover, we obtain a generalized Van Kampen-Flores type theorem for any integer $r$.

Empirical Evidence on the Topological Properties of Structural Paths and Some Notes on Its Theoretical Explanation

The mathematical literature has developed a large pool of topological concepts and theorems for dynamic systems analysis. The aim of our paper is to make a first step towards the application of these concepts and theorems in the analysis of (long-run) structural change (in the three-sector framework). Our approach focuses on two of the most basic topological notions, namely intersection and self-intersection of trajectories on a two-dimensional domain. We discuss the mathematical foundations of the application of these concepts in structural change analysis, use them for analyzing empirical data, and elaborate new stylized facts stating that different countries’ structural change trajectories are (non-self-)intersecting. Finally, we discuss briefly the theoretical explanations of (non-self-)intersection and a wide range of new research topics relating to (a) the topological classification and comparison of models and evidence and (b) the application of (further) topological concepts in standard branches of growth and development theory.

On expansion and topological overlap

We give a detailed and easily accessible proof of Gromov’s Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map Xrightarrow mathbb {R}^d there exists a point pin mathbb {R}^d that is contained in the images of a positive fraction mu >0 of the d-cells of X. More generally, the conclusion holds if mathbb {R}^d is replaced by any d-dimensional piecewise-linear manifold M, with a constant mu that depends only on d and on the expansion properties of X, but not on M.

A nonlinear two-point boundary-value problem in geophysics

We study a recently derived model for gyres, equivalent to a a two-point boundary-value problem for ocean flows with no azimuthal variations. For a large class of oceanic vorticities we establish the existence of solutions using an approach based on the topological transversality theorem.