Narrow Results

Dramatic advances in the field of complex networks have been witnessed in the past few years. This paper reviews some important results in this direction of rapidly evolving research, with emphasis on the relationship between the dynamics and the topology of complex networks. Basic quantities and typical examples of various complex networks are described; and main network models are introduced, including regular, random, small-world and scale-free models. The robustness of connectivity and the epidemic dynamics in complex networks are also evaluated. To that end, synchronization in various dynamical networks are discussed according to their regular, small-world and scale-free connections.

Approximation space can be said to play a critical role in the accuracy of the set’s approximations. The idea of “approximation space” was introduced by Pawlak in 1982 as a core to describe information or knowledge induced from the relationships between objects of the universe. The main objective of this paper is to create new types of rough set models through the use of different neighborhoods generated by a binary relation. New approximations are proposed representing an extension of Pawlak’s rough sets and some of their generalizations, where the precision of these approximations is substantially improved. To elucidate the effectiveness of our approaches, we provide some comparisons between the proposed methods and the previous ones. Finally, we give a medical application of lung cancer disease as well as provide an algorithm which is tested on the basis of hypothetical data in order to compare it with current methods.

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.

Knot and link diagrams are used to represent nonstandard sets, and to represent the formalism of combinatory logic (lambda calculus). These diagrammatics create a two-way street between the topology of knots and links in three dimensional space and key considerations in the foundations of mathematics.

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaos. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map, …).

By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold — the highest known taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multicomponent attractors (bounded by a torus whose genus $g\ge 3$).

When hadron-quark continuity is formulated in terms of a topology change at a density higher than twice the nuclear matter density $(n_0)$, the core of massive compact stars can be described in terms of quasiparticles of fractional baryon charges, behaving neither like pure baryons nor like deconfined quarks. Hidden symmetries, both local gauge and pseudo-conformal (or broken scale), emerge and give rise both to the long-standing “effective $g_A^{\ast }\approx 1$” in nuclear Gamow–Teller (GT) transitions at $\lesssim n_0$ and to the pseudo-conformal sound velocity $v_{\mathrm{\text{pcs}}}^2/c^2\approx 1/3$ at $\gtrsim 3n_0$. It is suggested that what has been referred to, since a long time, as “quenched $g_A$” in light nuclei reflects what leads to the dilaton-limit $g_A^{\mathrm{\text{DL}}}=1$ at near the (putative) infrared fixed point of scale invariance. These properties are confronted with the recent observations in GT transitions and in astrophysical observations.

Our macroscopic world abounds in knotted and linked rings. But even at the nanoscopic scale, works in molecular topology have shown that Nature builds DNA molecules, which form actual knots and links between themselves. Qualitative similarities may be observed between forms of the microscopic world, as well as between those of the macroscopic world; and by extracting the structures, it is possible to study their dynamics thanks to topological methods and techniques. This chapter is aimed to show that knots and links are ubiquitous scale-independent objects carrying a tremendous amount of precious information on the emergence of new forms and structures especially in quantum physics and in living matter. In fact, knotting and unknotting are “universal” principles underlying these forms and structures. This study is focused at showing that differential geometry and topological knots theory can be used notably (a) to find invariants for closed 3-manifolds related to quantum field theory, and (b) to describe 3-dimensional structures of DNA and protein-DNA complexes.

The aim of this paper is to use topological concepts in the construction of flexible mathematical models in the field of biological mathematics. Also, we will build new topographic types to study recombination of deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). Finally, we study the topographical properties of constructed operators and the associated topological spaces of DNA and RNA.

Topology, a well-established concept in mathematics, has nowadays become essential to describe condensed matter. At its core are chiral electron states on the bulk, surfaces and edges of the condensed matter systems, in which spin and momentum of the electrons are locked parallel or anti-parallel to each other. Magnetic and non-magnetic Weyl semimetals, for example, exhibit chiral bulk states that have enabled the realization of predictions from high energy and astrophysics involving the chiral quantum number, such as the chiral anomaly, the mixed axial-gravitational anomaly and axions. The potential for connecting chirality as a quantum number to other chiral phenomena across different areas of science, including the asymmetry of matter and antimatter and the homochirality of life, brings topological materials to the fore.

Mathematical equations are now found not only in the books, but also they help in finding solutions for the biological problems by explaining the technicality of the current biological models and providing predictions that can be validated and complemented to experimental and clinical studies. In this research paper, we use the mset theory to study DNA & RNA mutations to discover the mutation occurrence. Also, we use the link between the concept of the mset and topology to determine the compatibility or similarity between “types”, which may be the strings of bits, vectors, DNA or RNA sequences, etc.

Certain exact solutions of the Einstein field equations over nonsimply-connected manifolds are reviewed. These solutions are spherically symmetric and have no curvature singularity. They provide a regularization of the standard Schwarzschild solution with a curvature singularity at the center. Spherically symmetric collapse of matter in ℝ⁴ may result in these nonsingular black-hole solutions, if quantum-gravity effects allow for topology change near the center or if nontrivial topology is already present as a remnant from a quantum spacetime foam.

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.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

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.

Meshes, which generalize polyhedra by using non-planar faces, are the most commonly used objects in computer graphics. Modeling 2-dimensional manifold meshes with a simple user interface is an important problem in computer graphics and computer aided geometric design. In this paper, we propose a conceptual framework to model meshes. Our framework guarantees topologically correct 2-dimensional manifolds and provides a new user interface paradigm for mesh modeling systems.

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 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 constructions of finite switchboard state automata are known to be an extension of finite automata in the view of commutative and switching state machines. This research incorporated an idea of a switchboard in the general fuzzy automata to introduce general fuzzy finite switchboard automata. The attained output reveals that a strongly connected general fuzzy finite switchboard automaton is equivalent to the retrievable general fuzzy automata. Further, the notion of the switchboard subsystem and strong switchboard subsystem of general fuzzy finite switchboard automata are examined. Finally, the concept of fuzzy topology on general fuzzy finite switchboard automata in terms of these characterisations is formulated.

The purpose of this work is to construct a new crossover operator using the properties of DNA and RNA by using topological concepts in constructing flexible mathematical models in the field of biomathematics. Also, we investigate and study topological properties of the constructed operators and the associated topological spaces of DNA and RNA. Finally we use the process of exchange for sequence of genotypes structures to construct new types of topological concepts to investigate and discuss several examples and some of their properties.

We show here some of our results on intuitionistic fuzzy topological spaces. In 1983, K.T. Atanassov proposed a generalization of the notion of fuzzy set: the concept of intuitionistic fuzzy set. D. Çoker constructed the fundamental theory on intuitionistic fuzzy topological spaces, and D. Çoker and other mathematicians studied compactness, connectedness, continuity, separation, convergence and paracompactness in intuitionistic fuzzy topological spaces. Finally, G.-J Wang and Y.Y. He showed that every intuitionistic fuzzy set may be regarded as an L-fuzzy set for some appropriate lattice L. Nevertheless, the results obtained by above authors are not redundant with other for ordinary fuzzy sense. Recently, Smarandache defined and studied neutrosophic sets (NSs) which generalize IFSs. This author defined also the notion of neutrosophic topology. We proved that neutrosophic topology does not generalize the concept of intuitionistic fuzzy topology.