Stochastic Three-Dimensional Rotating Navier–Stokes Equations: Averaging, Convergence and Regularity

We consider stochastic three-dimensional rotating Navier?CStokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems.     

Adjustment of spiral drift by a travelling wave perturbation

We apply a velocity-field approach to investigate the interaction between spiral waves and the travelling wave modulation of system excitability which leads to a prediction: the direction of the straight-line drift of spiral waves is linearly adjusted by the propagation direction of the travelling waves. Direct numerical computations of the Oregonator model and the formulas of drift-velocity field confirm the validity and robustness of our theoretical prediction.     

The alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay

The various cases of synchronization in two identical hyperchaotic Lorenz systems with time delay are studied. Based on Lyapunov stability theory, the sufficient conditions for achieving synchronization of two identical hyperchaotic Lorenz systems with time delay are derived, and a simple scheme only with a single linear controller is proposed. When the parameters in the response system are known, the alternating between complete synchronization and hybrid synchronization (namely, coexistence of antiphase and complete synchronization) is observed with the control feedback gain varying. Furthermore, when the parameters in the response system are unknown, for the same feedback controller, the complete synchronization and the hybrid synchronization can be obtained, respectively, as the associated parameters updated laws of the unknown parameters are chosen. Numerical simulation results are presented to demonstrate the proposed chaos synchronization scheme.     

Solitons and other nonlinear waves of the Boussinesq equation

This paper studies the dynamics of shallow water waves that are governed by the Boussinesq equations. A few perturbation terms are taken into account. The ansatz method is used to carry out the perturbed Boussinesq equation. Later on, the mapping method is used to extract a few more analytical solutions. Additionally, the Weierstrass elliptic function method is also used to obtain solitary waves and singular soliton solutions. Finally, the Lie symmetry approach is used to extract a few more additional solutions.     

Nuclear dimension and $\mathcal{Z}$-stability of pure C∗-algebras

In this article I study a number of topological and algebraic dimension type properties of simple C^∗-algebras and their interplay. In particular, a simple C^∗-algebra is defined to be (tracially) (m,[`(m)])(m,\bar{m})-pure, if it has (strong tracial) m-comparison and is (tracially) [`(m)]\bar{m}-almost divisible. These notions are related to each other, and to nuclear dimension.     

An Active Set Modified Polak–Ribiére–Polyak Method for Large-Scale Nonlinear Bound Constrained Optimization

In this paper, we propose an active set modified Polak?CRibiére?CPolyak method for solving large-scale optimization with simple bounds on the variables. The active set is guessed by an identification technique at each iteration and the recently developed modified Polak?CRibiére?CPolyak method is used to update the variables with indices outside of the active set. Under appropriate conditions, we show that the proposed method is globally convergent. Numerical experiments are presented using box constrained problems in the CUTEr test problem libraries.     

Alternatives for testing total dual integrality

In this paper we provide characterizing properties of totally dual integral (TDI) systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system’s dual integer programs where all test vectors have positive entries equal to 1. Reformulations of this provide relations between computational algebra and integer programming and they contain Applegate, Cook and McCormick’s sufficient condition for the TDI property and Sturmfels’ theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. We also study the theoretical and practical efficiency and limits of the characterizations of the TDI property presented here. In the particular case of set packing polyhedra our results correspond to endowing the weak perfect graph theorem with an additional, computationally interesting, geometric feature: the normal fan of the stable set polytope of a perfect graph can be refined into a regular triangulation consisting only of unimodular cones.     

Structure of planar integral self‐affine tilings

For a self‐affine tile in $\mathbf {R}^2$ generated by an expanding matrix $A\in M_2(\mathbf {Z})$ and an integral consecutive collinear digit set ${\mathcal D}$, Leung and Lau [Trans. Amer. Math. Soc. 359 , 3337–3355 (2007).] provided a necessary and sufficient algebraic condition for it to be disklike. They also characterized the neighborhood structure of all disklike tiles in terms of the algebraic data A and ${\mathcal D}$. In this paper, we completely characterize the neighborhood structure of those non‐disklike tiles. While disklike tiles can only have either six or eight edge or vertex neighbors, non‐disklike tiles have much richer neighborhood structure. In particular, other than a finite set, a Cantor set, or a set containing a nontrivial continuum, neighbors can intersect in a union of a Cantor set and a countable set.     

On images of weak Fano manifolds

We consider a smooth projective morphism between smooth complex projective varieties. If the source space is a weak Fano (or Fano) manifold, then so is the target space. Our proof is Hodge theoretic. We do not need mod p reduction arguments. We also discuss related topics and questions.