Narrow Results

In this paper, we are concerned with the $T$-Graphs, which are graphs defined based on the Topological structure of the given set. Precisely, for a given topology $T$ on a set $X$, a $T$-Graph ‘$G=(V,E)$’ is an undirected simple graph with the vertex set $V$ as $P(X)$ and the edge set $E$ as the set of all unordered pairs of nodes $u,v$ in $V$, denoted by $(u,v)\text{ or }(v,u)$, satisfying either ‘$u\in T$ and $u^{\text{c}}\cap v\in T$’ (or) ‘$v\in T$ and $v^{\text{c}}\cap u\in T$’.

The main purpose of this paper is to study the structure of $T$-Graphs for various topologies $T$ on a set $X$. Our goals in this paper are threefold. First, to show the $L_d(2,1)$ labeling number $\lambda (G,d)$ of any $T$-Graph $G$ exists finitely, if the labeling is $d$ multiple of non-negative integral values. In addition to show this labeling number $\lambda (G,d)$ is not just bounded above but bounded below as well. Second, to measure the bound values in terms of $d$ multiple of the order of the $T$-Graphs and finding a relation between the order of the $T$-Graphs and the maximum degree $\mathrm{\Delta }$ of the $T$-Graphs. Finally, third is to show that in case of $L(2,1)$$T$-graphs on a set with atleast 2 elements, the labeling number is $2(\mathrm{\Delta }+1)$ and is smaller than that of Griggs and Yeh’s conjecture value ${\mathrm{\Delta }}^2$.

In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about four-dimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N = 2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at four-dimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory.

This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology.

A non-trivial spatial topology of the Universe is a potentially observable attribute, which can be probed through the circles-in-the-sky for all locally homogeneous and isotropic universes with no assumptions on the cosmological parameters. We show how one can use a possible circles-in-the-sky detection of the spatial topology of globally homogeneous universes to set constraints on the dark energy equation of state parameters.

Motivated by the study of soluble models of quantum field theory, we illustrate a new type of topological effect by comparing the constructions of canonical Klein–Gordon quantum fields on the two-dimensional de Sitter spacetime as opposed to its double covering. We show that while the commutators of the two fields coincide locally, the global topological differences make the theories drastically different. Many of the well-known features of de Sitter quantum field theory disappear. In particular, there is nothing like a Bunch–Davies vacuum. Correspondingly, even though the local horizon structure is the same for the two universes, there is no Hawking–Gibbons thermal state. Finally, there is no complementary series of fields.

QCD axions are at the crossroads of QCD topology and Dark Matter searches. We present here the current status of topological studies on the lattice, and their implication on axion physics. We outline the specific challenges posed by lattice topology, the different proposals for handling them, the observable effects of topology on the QCD spectrum and its interrelation with chiral and axial symmetries. We review the transition to the quark–gluon plasma, the fate of topology at the transition, and the approach to the high temperature limit. We discuss the extrapolations needed to reach the regime of cosmological relevance, and the resulting constraints on the QCD axion.

We explore the interplay between quantization, local commutativity and the analyticity properties of the two-point functions of a quantum field in a non trivial topological cosmological background in the example of the two-dimensional de Sitter manifold and its double covering. The global topological differences make the many of the well-known features of de Sitter quantum field theory disappear. In particular there is nothing like a Bunch-Davies vacuum and there are no $SL(2,R)$-invariant fields whose mass is less than 1/2.

The experiments at the Large Hadron Collider (LHC) rely upon a complex distributed computing infrastructure (WLCG) consisting of hundreds of individual sites worldwide at universities and national laboratories, providing about half a billion computing job slots and an exabyte of storage interconnected through high speed networks. Wide Area Networking (WAN) is one of the three pillars (together with computational resources and storage) of LHC computing. More than 5 PB/day are transferred between WLCG sites. Monitoring is one of the crucial components of WAN and experiments operations. In the past years all experiments have invested significant effort to improve monitoring and integrate networking information with data management and workload management systems. All WLCG sites are equipped with perfSONAR servers to collect a wide range of network metrics. We will present the latest development to provide the 3D force directed graph visualization for data collected by perfSONAR. The visualization package allows site admins, network engineers, scientists and network researchers to better understand the topology of our Research and Education networks and it provides the ability to identify nonreliable or/and nonoptimal network paths, such as those with routing loops or rapidly changing routes.

In this paper, we construct a class of quantum systems in a space continuum inspired by results from topological insulator physics. Instead of adding spin–orbit coupling terms suggested by time-reversal invariance as in conventional topological insulators, the terms $L\cdot S$ are determined using supersymmetry as a starting point. This procedure not only restricts the number of possible continuous topological insulators models, but also provides a systematic way to find new continuous topological insulators models. Some explicit quantum mechanical examples are discussed and applications to dark matter physics are also outlined.

We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO(n) tensor (∂)^dh_αβ, which generally appears in the weak field expansion around the flat space: g_μν=δ_μν+h_μν. Making use of this representation, we explain (1) Generating function of graphs (Feynman diagram approach), (2) Adjacency matrix (Matrix approach), (3) Graphical classification in terms of "topology indices" (Topology approach), (4) The Young tableau (Symmetric group approach). We systematically construct the global SO(n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of ∂∂h-, and (∂∂h)²-invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of) the Weyl anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10 dimensions).

Given a two- or three-dimensional set S of arbitrary topology and a radius r, we show how to construct an r-tightening of S, which is a set whose boundary has mean curvature with magnitude less than or equal to 1/r and which only differs from S in a morphologically-defined tolerance zone we call the mortar. The mortar consists of the thin or highly curved parts of S and its complement, such as corners, gaps, and small connected components, while the boundary of a tightening consists of components of locally minimal length (in 2D) or area (in 3D) that lie in the mortar. Tightenings are defined independently of shape representation, and it may be possible to find them using a variety of algorithms. We describe how to approximately compute tightenings for two-dimensional sets represented as binary images and for three-dimensional sets represented as triangle meshes using constrained, level-set curvature flow.

Topology-based methods have been successfully used for the analysis and visualization of piecewise-linear functions defined on triangle meshes. This paper describes a mechanism for extending these methods to piecewise-quadratic functions defined on triangulations of surfaces. Each triangular patch is tessellated into monotone regions, so that existing algorithms for computing topological representations of piecewise-linear functions may be applied directly to the piecewise-quadratic function. In particular, the tessellation is used for computing the Reeb graph, a topological data structure that provides a succinct representation of level sets of the function.

The Doubly Linked Face List (DLFL) is a data structure for mesh representation that always ensures topological 2-manifold consistency. Furthermore, it uses a minimal amount of computer memory and allows queries to be performed very efficiently. However, the use of the DLFL for the implementation of practical applications is very limited, mainly because of two drawbacks: (1) the DLFL is only able to represent 2-manifold objects; (2) its operators may be ambiguous, modifying the structure in an unexpected way from the user's point of view. In order to overcome these drawbacks, we present the Extended Doubly Linked Face List (XDLFL), which extends the DLFL for the representation of 2-pseudomanifolds and 2-manifolds with boundaries, increasing its applicability for practical software applications. Using these extensions, we also show how to avoid ambiguities in the original DLFL's operators. A new set of intuitive operators for the manipulation of the extensions and for the unambiguous manipulation of the data structure is also presented. The implementation of these extensions is straightforward, since the modifications to the DLFL are trivial and based on behavioral observations of the DLFL's operators. After integrating the extensions to the DLFL, memory usage increases very slightly, while is still smaller than the memory usage of other well-known data structures. Furthermore, queries related to the new extensions, such as whether an edge belongs to a boundary, may be performed very efficiently. The proposed extensions and their operators are very beneficial for applications such as surgery simulation softwares, where the interactions with the models, such as cutting or appending objects to each other, must be performed in an efficient and transparent manner.

Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques.

We extend, in significant ways, the theory of truncated boolean representable simplicial complexes introduced in 2015. This theory, which includes all matroids, represents the largest class of finite simplicial complexes for which combinatorial geometry can be meaningfully applied.

We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory.

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus $S^1\times I$ in 3-space having constant width $R$. Given a regular parametrization $x(s)$, and a smooth unit vector field $u(s)$ based along $x$, for a knot $K$, we may define a ribbon of width $R$ associated to $x$ and $u$ as the set of all points $x(s)+ru(s)$, $r\in [0,R]$. For large $R$, ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge $x(s)+Ru(s)$ relates to that of the original knot $K$. Generically, as $R\to \infty$, there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field $u$. The particular knot type within the finite set depends on the parametrized curves $x(s)$, $u(s)$, and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types $K_1$ and $K_2$, we can find a smooth ribbon of constant width connecting curves of these two knot types.

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

The non-orientable 4-genus of a knot $K$ in $S^3$ is defined to be the minimum first Betti number of a non-orientable surface $F$ smoothly embedded in $B^4$ so that $K$ bounds $F$. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We will also view obstructions to a knot bounding a Möbius band given by the double branched cover of $S^3$ branched over $K$.

The role of topology in the statistical mechanics of surfaces in the lattice is considered. The possibility that a phase transition driven by the number of boundary components occurs is investigated. Bounds on the limiting free energy is derived and conditions for the existence of a critical point in the phase diagram are presented.

We show that different topologies of a space-time manifold and different signatures of its metric can be encompassed into a single Lagrangian formalism, provided one adopts the first-order (Palatini) formulation and relies on nonlinear Lagrangians, that were earlier shown to produce, in the generic case, universality of Einstein field equations and of Komar's energy-momentum complex as well. An example in Relativistic Cosmology is provided.