Boundary Hamiltonian Theory for Gapped Topological Orders

We report our systematic construction of the lattice Hamiltonian model of topological orders on open surfaces, with explicit boundary terms. We do this mainly for the Levin-Wen string-net model. The full Hamiltonian in our approach yields a topologically protected, gapped energy spectrum, with the corresponding wave functions robust under topology-preserving transformations of the lattice of the system. We explicitly present the wavefunctions of the ground states and boundary elementary excitations. The creation and hopping operators of boundary quasi-particles are constructed. It is found that given a bulk topological order, the gapped boundary conditions are classified by Frobenius algebras in its input data. Emergent topological properties of the ground states and boundary excitations are characterized by （bi-） modules over Frobenius algebras.

TOPOLOGICAL STRUCTURE OF RENORM ALIZATION CONSTANTS I N QUANTUM FIELD THEORY AT SHORT DISTANCE KUNMING COLLABRATION OF MULTIHADRON DYNAMIOS

The anomalous dimensions of the quantum fields are the Hausdorff dimensiongrad. The present candidate of the renormalization constant is the generalized Cantor discontinuum. The Hausdorff dimensiongrad of the Minkowski space time is based upon the point set with σ-length on light cone.

Numerical Study of a Three-Dimensional Laminar Flow in a Rectangular Channel with a 180-Degree Sharp Turn

This study presents a numerical analysis of three-dimensional steady laminar flow in a rectangular channel with a 180-degree sharp turn. The Navier-Stokes equations are solved by using finite difference method for Re = 900. Three-dimensional streamlines and limiting streamlines on wall surface are used to analyze the three-dimensional flow characteristics. Topological theory is applied to limiting streamlines on inner walls of the channel and two-dimensional streamlines at several cross sections. It is also shown that the flow impinges on the end wall of turn and the secondary flow is induced by the curvature in the sharp turn.

THE EXISTENCE AND UNIQUENESS OF TIME-PERIODIC SOLUTIONS TO THE NON-ISOTHERMAL MODEL FOR COMPRESSIBLE NEMATIC LIQUID CRYSTALS IN A PERIODIC DOMAIN

In this paper,we are concerned with a three-dimensional non-isothermal model for the compressible nematic liquid crystal flows in a periodic domain.Under some smallness and structural assumptions imposed on the time-periodic force,we establish the existence of the time-periodic solutions to the system by using a regularized approximation scheme and the topological degree theory.We also prove a uniqueness result via energy estimates.

Stringy Phenomenology with Preon Models

We compare, following Pati, global symmetries, our topological supersymmetric preon model with the heterotic E8 × E8 string theory. We include Pati’s supergravity based preon model in this work and compare the preon interactions of his model to ours. Based on preon-string symmetry comparison and preon phenomenological results, we conclude that the fundamental particles are likely preons rather than standard model particles. .

The Fate of Supersymmetry in Quantum Field Theories

We analyze the significance of supersymmetry in two topological models and the standard model (SM). We conclude that the two topological field theory models favor hidden supersymmetry. The SM superpartners, instead, have not been found.

Topological aspect of disclinations in two-dimensional crystals

By using topological current theory, this paper studies the inner topological structure of disclinations during the melting of two-dimensional systems. From two-dimensional elasticity theory, it finds that there are topological currents for topological defects in homogeneous equation. The evolution of disclinations is studied, and the branch conditions for generating, annihilating, crossing, splitting and merging of disclinations are given.

A Note on q-Deformed Two-Dimensional Yang-Mills and Open Topological Strings

In this note we make a test of the open topological string version of the OSV conjecture in the toric Calabi-Yau manifold X = O（-3） → P^2 with background D4-branes wrapped on Lagrangian submanifolds. The Dbrahe partition function reduces to an expectation value of some inserted operators of a q-deformed Yang Mills theory living on a chain of P^1 ＇s in the base p2 of X. At large N this partition function can be written as a sum over squares of chiral blocks, which are related to the open topological string amplitudes in the local p2 geometry with branes at both the outer and inner edges of the toric diagram. This is in agreement with the conjecture.

Using Parametric Mathematical Modeling to Develop a Geometric and Topological Intuition for Molecular Knots

Knot theory is a branch of topology in pure mathematics, however, it has been increasingly used in different sciences such as chemistry. Mathematically, a knot is a subset of three-dimensional space which is homeomorphic to a circle and it is only defined in a closed loop. In chemistry, knots have been applied to synthetic molecular design. Mathematics and chemistry together can work to determine, characterize and create knots which help to understand different molecular designs and then forecast their physical features. In this study, we provide an introduction to the knot theory and its topological concepts, and then we extend it to the context of chemistry. We present parametric representations for several synthetic knots. The main goal of this paper is to develop a geometric and topological intuition for molecular knots using parametric equations. Since parameterizations are non-unique;there is more than one set of parametric equations to specify the same molecular knots. This parametric representation can be used easily to express geometrically molecular knots and would be helpful to find out more complicated molecular models.

Global Exponential Stability of Almost Periodic Solution of Cellular Neural Networks with Time-Varying Delays

In this paper, global exponential stability of almost periodic solution of cellular neural networks with time-varing delays （CNNVDs） is considered. By using the methods of the topological degree theory and generalized Halanay inequality, a few new applicable criteria are established for the existence and global exponential stability of almost periodic solution. Some previous results are improved and extended in this letter and one example is given to illustrate the effectiveness of the new results.

Multi-types of Skyrmions in SU（N） Quantum Hall System

The skyrmions in SU（N） quantum Hall （QH） system are discussed. By analyzing the gauge field structure and the topological properties of this QH system it is pointed out that in the SU（N） QH system there can exist （N-1） types of skyrmion structures, instead of only one type of skyrmions. In this paper, by means of the Abelian projections according to the （N - 1） Cartan subalgebra local bases, we obtain the （N - 1） U（1） electromagnetic field tensors in the SU（N） gauge field of the QH system, and then derive （N - 1） types of skyrmion structures from these U（1） sub-field tensors. Furthermore, in light of the C-mapping topological current method, the topological charges and the motion of these skyrmions are also discussed.

Interaction Between Massive and Massless Gravitons by Perturbing Topological Field Theory

We test the Wu gauge theory of gravity with massive gravitons in the perturbing topological field theory framework.We show that the computation of the correlation function between massive and massless gravitons is reported up to 4-loop and appears to be unaffected by radiative correction.This result ensures the stability of the linking number between massive and massless gravitons with respect to the local perturbation,a result with potential wider applications in cosmology.

Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity

Topological phases in non-Hermitian systems have become fascinating subjects recently.In this paper,we attempt to classify topological phases in 1D interacting non-Hermitian systems.We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger(SSH)model and discuss its many-body topological Berry phase,which is well defined for all interacting quasi-Hermitian systems(non-Hermitian systems that have real energy spectrum).We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts.Finally,we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts.Thus,we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems.

（3＋ 1）-TQFTs and topological insulators

Levin-Wen models are microscopic spin models for topological phases of matter in （2＋ 1）-dimension. We introduce a generalization of such models to （3 ＋ 1）-dimension based on unitary braided fusion categories, also known as unitary premodular categories. We discuss the ground state degeneracy on 3-manifolds and statistics of excitations which include both points and defect loops. Potential con- nections with recently proposed fractional topological insulators and projective ribbon permutation statistics are described.

A Note on the Existence of Positive Solutions for a Fourth-Order Semilinear Elliptic System

In this note, we prove the existence of positive solutions to a class of biharmonical systems. Our main ingredients are proving a Liouville type result for biharmonical system in R＋^N via the method of moving plane combined with integral inequality, and establishing a prior estimates for positive solutions of the system via the blowing-up method.

Time-Periodic Solution to the Compressible Navier–Stokes/Allen–Cahn System

In this paper,we investigate the time-periodic solution to a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic external force in a periodic domain in R^N.The existence of the time-periodic solution to the system is established by using an approach of parabolic regularization and combining with the topology degree theory,and then the uniqueness of the period solution is obtained under some smallness and symmetry assumptions on the external force.

EXISTENCE OF NON-TRIVIAL SOLUTION FOR SUPERLINEAR SYSTEM OF INTEGRAL EQUATIONS AND ITS APPLICATIONS

In this paper, we use cone theory and topological degree theory to study superlinear systemof integral equations, and obtain existence theorems for non-trivial solutions; moreover, we applythe results to two-point boundary problems of ordinary differential system of equations.