Theory Of Braid Groups Research Articles

Complete Classification of Gradient Blow-Up and Recovery of Boundary Condition for the Viscous Hamilton–Jacobi Equation

It is known that the Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation $$u_t-\Delta u=|\nabla u|^p$$ with $$p>2$$ , which has important applications in stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after the appearance of singularity in finite time, which occurs through gradient blow-up (GBU) on the boundary. The solutions eventually become classical again for large time, but in-between they may undergo losses and recoveries of boundary conditions at multiple times (as well as GBU at multiple times), hence forming a quite complex dynamics. In this paper, with the help of braid group theory combined with inner/outer region analysis, we give a complete classification of all viscosity solutions in one-dimensional case. Namely, we fully describe the (countably many, type II) rates and space–time profiles of gradient blow-up (GBU) or recovery of boundary condition (RBC), we characterize them at any time when such a phenomenon occurs, and we moreover determine whether the space–time profiles are stable or unstable. These results can be modified in radial domains in general dimensions. Even for type II blow-up in other PDEs, as far as we know, there has been no complete classification except Mizoguchi (Commun Pure Appl Math 75:1870–1886, 2022), in which the argument relies on features peculiar to chemotaxis system. Whereas there are many results on construction of special type II blow-up solutions of PDEs with investigation of stability/instability of bubble, determination of stability/instability of space–time profile for general solutions has not been done. A key in our proofs is to focus on braid group algebraic structure with respect to vanishing intersections with the singular steady state, as a GBU or RBC time is approached. In turn, the GBU and RBC rates and profiles, as well as their stability/instability, can be given a complete geometric characterization by the number of vanishing intersections. We construct special solutions in bounded and unbounded intervals in both GBU and RBC cases, based on methods from Herrero and Velázquez (Preprint, 1994), and then we apply braid group theory to get upper and lower estimates of the rates. After that, we rule out oscillation of the rates, which leads us to the complete space–time profile. In the process, careful construction of special solutions with specific behaviors in intermediate and outer regions, which is far from bubble and the RBC point, plays an essential role. The application of such techniques to viscosity solutions is completely new.

Parametric topological entropy and differential equations with time–dependent impulses

Stimulated by a multiple solvability of periodic boundary value problems to differential equations with time–dependent impulses, we recall a related definition of parametric topological entropy for a sequence of continuous self–maps on a compact metric space. The main simple idea consists in replacing the iterates of a single map by the compositions of several various maps. For an equicontinuous countable family of self–maps on a compact connected polyhedron, we develop a lower estimate of this entropy in terms of the asymptotic Nielsen numbers of their compositions. This Ivanov–type equality is then applied, via the associated Poincaré translation operators, to differential equations with time–dependent impulses on tori. If the supporting space differs from a homotopy type of tori, then the situation becomes more delicate. Nevertheless, on compact connected punctured surfaces, we are still able to apply in a similar way the Artin braid group theory to planar differential equations with a finite number of homeomorphic impulses. Some further possibilities are commented in remarks and several illustrative examples are supplied.

Rotations of 4nπ Research and no Twisted Mechanism Example Analysis Based on the Theory of Braid Group

The Cable or Pipe between the Two Relatively Rotating Platforms Exists the Twisted Problem. from Viewpoint of Braid Group Theory, Rope’s Twisted State is Researched on. Based on the Characteristics of a Special Garside Braids Δn and Δnk , the Equivalence of Two Rotation Modes is Revealed. also, that the Determination of Rotation Mode’s Minimum Rotate Range is 4∏ while Using Braid Theory is Proposed. the Theory of Braid Group can Not only be Used as a Criterion for Determine Whether the Cable is Twisted, Finally, but also can be Used as Avoid Cable Twisted during Mechanism Design.

On the representation theory of braid groups

This work presents an approach towards the representation theory of the braid groups Bn. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids, with the help of Drinfeld associators. We set a dictionary between representation-theoretic properties of these two structures, and tools to describe the representations thus obtained. We give an explanation for the frequent apparition of unitary structures on classical representations. We introduce new objects — varieties of braided extensions, infinitesimal quotients — which are useful in this setting, and analyse several of their properties. Finally, we review the most classical representations of the braid groups, show how they can be obtained by our methods and how this setting enrich our understanding of them.

Topological qubit design and leakage

We examine how best to design qubits for use in topological quantum computation. These qubits are topological Hilbert spaces associated with small groups of anyons. Operations are performed on these by exchanging the anyons. One might argue that in order to have as many simple single-qubit operations as possible, the number of anyons per group should be maximized. However, we show that there is a maximal number of particles per qubit, namely 4, and more generally a maximal number of particles for qudits of dimension d. We also look at the possibility of having topological qubits for which one can perform two-qubit gates without leakage into non-computational states. It turns out that the requirement that all two-qubit gates are leakage free is very restrictive and this property can only be realized for two-qubit systems related to Ising-like anyon models, which do not allow for universal quantum computation by braiding. Our results follow directly from the representation theory of braid groups, which implies that they are valid for all anyon models. We also make some remarks about generalizations to other exchange groups.

Asymptotic linearity of the mapping class group and a homological version of the Nielsen–Thurston classification

We study the action of the mapping class group on the integral homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen–Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.

Blow-up rate of type II and the braid group theory

A solution u u of a Cauchy problem or a Cauchy-Dirichlet problem for a semilinear heat equation \[ u t = Δ u + u p u_t = \Delta u + u^p \] with nonnegative initial data u 0 u_0 is said to undergo type II blow-up at t = T t = T if \[ lim sup t ↗ T ( T − t ) 1 / ( p − 1 ) | u ( t ) | ∞ = ∞ . \limsup _{t \nearrow T} \; (T-t)^{1/(p-1)} |u(t)|_\infty = \infty . \] Let φ ∞ \varphi _\infty be the radially symmetric singular steady state of the Cauchy problem. Suppose that u 0 ∈ L ∞ u_0 \in L^\infty is a radially symmetric function such that u 0 − φ ∞ u_0 - \varphi _\infty and ( u 0 ) t (u_0)_t change sign at most finitely many times. By application of the braid group theory, we determine the exact blow-up rate of solution with initial data u 0 u_0 which undergoes type II blow-up in the case of p > p J L p > p_{_{JL}} , where p J L p_{_{JL}} is the exponent of Joseph and Lundgren.

An application of braid group theory to the finite time dead-core rate

We consider the dead-core problem for the semilinear heat equation with strong absorption and with positive boundary values in a ball. We investigate the dead-core rate, i.e. the rate at which the solution reaches its first zero. We first show, as in the one-dimensional case, that the dead-core rate is always faster than the self-similar rate. By using some special solutions and the braid group theory, we then derive the exact dead-core rates for a large class of initial data.

Conjugacy Search in Braid Groups

We demonstrate that recent advances in the theory of braid groups, in particular a new invariant of conjugacy classes of braids, the ultra summit set, make some braid-based cryptographic protocols insecure for almost all randomly chosen keys. As part of this we present an overview of the known algorithms for solving the conjugacy decision and search problems in braid groups and an assessment of their practical performance from the point of view of braid-based cryptography.

The virtual and universal braids

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi-direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi-direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group. Recently some generalizations of classical knots and links were de- fined and studied: singular links (20, 5), virtual links (15, 12) and welded links (10). One of the ways to study classical links is to study the braid group. Singular braids (1, 5), virtual braids (15, 21), welded braids (10) were defined similar to the classical braid group. A theorem of A. A. Markov (4, Ch. 2.2) reduces the problem of classification of links to some alge- braic problems of the theory of braid groups. These problems include the word problem and the conjugacy problem. There are generaliza- tions of Markov's theorem for singular links (11), virtual links, and welded links (14). There are some dierent ways to solve the word problem for the singular braid monoid and singular braid group (8, 7, 22). The solution of the word problem for the welded braid group follows from the fact that this group is a subgroup of the automorphism group of the free group (10). A normal form of words in the welded braid group was constructed in (13). In this paper we study the structure of the virtual braid group V Bn. This is similar to the classical braid group Bn and welded braid group

Dynamical Ordering of Non-Birkhoff Orbits and Topological Entropy in the Standard Mapping

The standard mapping is an analytical, reversible monotone twist mapping. The appearance ordering (i.e. the so-called dynamical ordering), of symmetric non-Birkhoff periodic orbits (SNBO) in the standard mapping is derived. Essential use is made of the reversibility. After the establishment of various properties of the symmetry axes under the mapping, two theorems connecting the dynamical ordering are proved. Then, the braids for SNBOs are constructed with the aid of techniques developed in braid group theory. A lower bound of the topological entropy of a system possessing an SNBO is obtained using the eigenvalue of the reduced Burau matrix representation of the braid constructed from the SNBO. The behavior of the topological entropy in the integrable limit is discussed.

Quantum chaos, random matrix theory, statistical mechanics in two dimensions, and the second law - a case study

We present a theory where the statistical mechanics for dilute ideal gases can be derived from the random matrix approach. We show the connection of this approach with the Srednicki approach which connects Berry conjecture with statistical mechanics. We further establish a link between Berry conjecture and random matrix theory. In the course of arguing for these connections, we also observe sum rules associated with the outstanding counting problem in the theory of Braid groups. We believe that these arguments, developed for a special example connecting the properties of eigenfunctions and random matrices to the second law of thermodynamics, will eventually prove to be more general.

Application of the Mathematical Methods of the Braid Group Theory to Quantization of Many-Particle Systems

The problem of the non-standard statistics for one-, twoand three-dimensional systems of N identical particles on various manifolds is reviewed in terms of the braid group theory. The braid groups together with their unitary representations are studied for the line, circle, plane, sphere, torus and the three-dimensional Euclidean space. Nonequivalent quantizations of several physical systems are presented.