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 ${\mathrm{\Delta }}^2$.

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.

In general, topological quantum field theory (TQFT) is studied in detail from the n-dimensional cobordism (nCob) to the Hilbert vector space. However, we study the TQFT in the different way in this paper, that is, the TQFT is studied from the Hilbert vector space to nCob. To do this, the theory called the ϕ-mapping topological current theory is used. The relation between the objects and zero points of the Hilbert states in the Hilbert vector space is studied in this frame. The relation between the morphism and topological current is revealed too.

The two-dimensional charge transport with parallel (in plane) magnetic field is considered from the physical and mathematical point of view. To this end, we start with the magnetic field parallel to the plane of charge transport, in sharp contrast to the configuration described by the theorems of Aharonov and Casher where the magnetic field is perpendicular. We explicitly show that the specific form of the arising equation enforces the respective field solution to fulfill the Majorana condition. Consequently, as soon any physical system is represented by this equation, the rise of fields with Majorana type behavior is immediately explained and predicted. In addition, there exists a quantized particular phase that removes the action of the vector potential producing interesting effects. Such new effects are able to explain due to the geometrical framework introduced, several phenomenological results recently obtained in the area of spintronics and quantum electronic devices. The quantum ring as spin filter is worked out in this framework and also the case of the quantum Hall effect.

Electromagnetic knots are electromagnetic fields in which the magnetic and electric lines are level curves of two complex scalar fields. If these scalar fields are chosen so that they can be interpreted as maps between the three-sphere and the two-sphere every time, then the electromagnetic helicity is proportional to the sum of the Hopf indices of both maps, that are topological invariants. An important example of this kind of electromagnetic fields with topological properties is often called the Hopfion. It is an electromagnetic knot in which the scalar fields are built from Hopf maps. In this work, we study the conditions for these electromagnetic fields to be null, i.e. to have both Lorentz invariant quantities $E\cdot B$ and $E^2-B^2$ equal to zero. We derive from these fields explicit vector potentials $A$ and $C$ satisfying force-free like conditions for every time. In particular, equations $A\cdot B=B^2/2$, $C\cdot E=E^2/2$ are derived for them. As a consequence, the energy, which is discretized, is proportional to the electromagnetic helicity. This relation between Physics and Topology is intriguing and worth of future research.

In this paper, we suggest a mathematical representation to the holographic principle through the theory of topological retracts. We found that the topological retraction is the mathematical analog of the hologram idea in modern quantum gravity and it can be used to explore the geometry of the hologram boundary. An example has been given on the five-dimensional (5D) wormhole spacetime $W$ which we found can retract to lower-dimensional circles $S_i\subset W$. In terms of the holographic principle, the description of this volume of spacetime $W$ is encoded on the lower-dimensional circle which is the region boundary.

The magnetosphere structure of compact objects is considered in the context of a theory of gravity with dynamical torsion field beyond standard General Relativity (GR). To this end, a new spherically symmetric solution is obtained in this theoretical framework, physically representing a compact object of pseudoscalar fields (for example, axion field). The axially symmetric version of the Grad–Shafranov equation (GSE) is also derived in this context, and used to describe the magnetosphere dynamics of the obtained “axion star”. The interplay between high-energy processes and the seed magnetic field with respect to the global structure of the magnetosphere is briefly discussed.

In this work, the quadratic helicity ${\chi }_{\bar{B}}^{(2)}$ is discussed from the physical and topological points of view. We show, after the introduction of a mathematical description of the properties of helicity and quadratic helicity in the context of standard dynamo equations, examples of the importance of these high invariants in cosmological scenarios. These scenarios consider extensions of the standard model and extensions of GR.

In this paper, we introduce a new geometric/topological approach to the emerging braneworld scenario in the context of D-branes using partially negative-dimensional product (PNDP) manifolds. The working hypothesis is based on the fact that the orientability of PNDP manifolds can be arbitrary, and starting from this, we propose that gravitational interaction can derive naturally from the non-orientability. According to this hypothesis, we show that topological defects can emerge from non-orientability and they can be identified as gravitational interaction at macroscopic level. In other words, the orientability of fundamental PNDPs can be related to the appearance of curvature at low-energy scales.

The persistent homology provides a mathematical tool to describe "features" in a principled manner. The persistence algorithm proposed by Edelsbrunner et al. can compute not only the persistent homology for a filtered simplicial complex, but also representative generating cycles for persistent homology groups. However, if there are dynamic changes either in the filtration or in the underlying simplicial complex, the representative generating cycle can change wildly.

In this paper, we consider the problem of tracking generating cycles with temporal coherence. Specifically, our goal is to track a chosen essential generating cycle so that the changes in it are "local". This requires reordering simplices in the filtration. To handle reordering operations, we build upon the matrix framework proposed by Cohen-Steiner et al. to swap two consecutive simplices, so that we can process a reordering directly. We present an application showing how our algorithm can track an essential cycle in a complex constructed out of a point cloud data.