The center of monoidal 2-categories in 3+1D Dijkgraaf-Witten Theory

In this work, for a finite group $G$ and a 4-cocycle $\omega \in Z^4(G, \mathbf{k}^\times)$, we compute explicitly the center of the monoidal 2-category $\operatorname{2Vec}_G^{\omega}$ of $\omega$-twisted $G$-graded 1-categories of finite dimensional $\mathbf{k}$-vector spaces. This center gives a precise mathematical description of the topological defects in the associated 3+1D Dijkgraaf-Witten TQFT. We prove that this center is a braided monoidal 2-category with a trivial sylleptic center.Comment: 24 page

Transverse ultrasonic anomaly in La1/3Sr2/3MnO3

The charge ordering (CO) transition in polycrystalline La1/3Sr2/3MnO3 has been studied by measuring the resistivity, magnetization and transverse ultrasonic velocity. At about 235K, a conspicuous increase in resistivity was observed, while the magnetization shows a cusp structure, corresponding to an antiferromagnetic charge ordering transition. Around this transition temperature, dramatic anomaly in transverse sound velocity was observed. The simultaneous occurrence of electron, magnon and phonon anomalous features implies strong spin-phonon coupling and electron-phonon in La1/3Sr2/3MnO3. The analysis suggests that the spin-phonon interaction is due to single-ion magnetostriction, and electron-phonon coupling originates from the Jahn-Teller effect of Mn3+

Spectrum of the Laplacian on Quaternionic Kahler Manifolds

Let $M^{4n}$ be a complete quaternionic K\"ahler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue $\lambda_1(M)$ of the Laplacian which is $\lambda_1(M)\le (2n+1)^2$. If the equality holds, then either $M$ has only one end, or $M$ is diffeomorphic to $\mathbb{R}\times N$ with N given by a compact manifold. Moreover, if $M$ is of bounded curvature, $M$ is covered by the quaterionic hyperbolic space $\mathbb{QH}^n$ and $N$ is a compact quotient of the generalized Heisenberg group. When $\lambda_1(M)\ge \frac{8(n+2)}3$, we also prove that $M$ must have only one end with infinite volume.Comment: 46 page

High order numerical schemes for second-order FBSDEs with applications to stochastic optimal control

This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.] to solve the second order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discrete the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple $(Y_t, Z_t, A_t, \Gamma_t)$ in the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effective of the proposed numerical schemes. Applications of our numerical schemes for stochastic optimal control problems are also presented

Exact Recovery for Sparse Signal via Weighted l1l_1 Minimization

Numerical experiments in literature on compressed sensing have indicated that the reweighted $l_1$ minimization performs exceptionally well in recovering sparse signal. In this paper, we develop exact recovery conditions and algorithm for sparse signal via weighted $l_1$ minimization from the insight of the classical NSP (null space property) and RIC (restricted isometry constant) bound. We first introduce the concept of WNSP (weighted null space property) and reveal that it is a necessary and sufficient condition for exact recovery. We then prove that the RIC bound by weighted $l_1$ minimization is $\delta_{ak}1$, $0<\gamma\leq1$ is determined by an optimization problem over the null space. When $\gamma< 1$ this bound is greater than $\sqrt{\frac{a-1}{a}}$ from $l_1$ minimization. In addition, we also establish the bound on $\delta_k$ and show that it can be larger than the sharp one 1/3 via $l_1$ minimization and also greater than 0.4343 via weighted $l_1$ minimization under some mild cases. Finally, we achieve a modified iterative reweighted $l_1$ minimization (MIRL1) algorithm based on our selection principle of weight, and the numerical experiments demonstrate that our algorithm behaves much better than $l_1$ minimization and iterative reweighted $l_1$ minimization (IRL1) algorithm

Matrix Linear Discriminant Analysis

We propose a novel linear discriminant analysis approach for the classification of high-dimensional matrix-valued data that commonly arises from imaging studies. Motivated by the equivalence of the conventional linear discriminant analysis and the ordinary least squares, we consider an efficient nuclear norm penalized regression that encourages a low-rank structure. Theoretical properties including a non-asymptotic risk bound and a rank consistency result are established. Simulation studies and an application to electroencephalography data show the superior performance of the proposed method over the existing approaches

Smooth U(1) Gauge Fields in de Sitter Spacetime

Using the methods of Lie groups and Lie algebras, we solved U(1) gauge potentials in the de Sitter background. Resulted gauge potentials are smooth on the whole spacetime, satisfying the Lorentz gauge condition. It is shown that electromagnetic fields in the de Sitter background could not be source free.Comment: draf

New RIC Bounds via l_q-minimization with 0<q<=1 in Compressed Sensing

The restricted isometry constants (RICs) play an important role in exact recovery theory of sparse signals via l_q(0<q<=1) relaxations in compressed sensing. Recently, Cai and Zhang[6] have achieved a sharp bound \delta_tk=4/3 to guarantee the exact recovery of k sparse signals through the l_1 minimization. This paper aims to establish new RICs bounds via l_q(0<q<=1) relaxation. Based on a key inequality on l_q norm, we show that (i) the exact recovery can be succeeded via l_{1/2} and l_1 minimizations if \delta_tk1, (ii)several sufficient conditions can be derived, such as for any 0=2, for any 1/2=1, (iii) the bound on \delta_k is given as well for any 0<q<=1, especially for q=1/2,1, we obtain \delta_k<1/3 when k(>=2) is even or \delta_k=3) is odd

Sparse and Low-Rank Covariance Matrices Estimation

This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices. We first benefit from a convex optimization which develops $l_1$-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property. For the proposed estimator, we then prove that with large probability, the Frobenious norm of the estimation rate can be of order $O(\sqrt{s(\log{r})/n})$ under a mild case, where $s$ and $r$ denote the number of sparse entries and the rank of the population covariance respectively, $n$ notes the sample capacity. Finally an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem, and meantime merits of the approach are also illustrated by practicing numerical simulations.Comment: arXiv admin note: text overlap with arXiv:1208.5702 by other author

The Reversible Spin Switch by External Control of Interval Distance of CuPc and C59N with the investigation of DFT

In this paper, we introduce a new kind of spin switch based on a joint system of copper phthalocyanine (CuPc) and C59N. Using density functional theory, we investigate the total magnetic moment of this system when gradually changing the interval distance between two molecules. The spin hopping happens during the critical distance with very low energy. This phenomenon shows a possible reality of reversible spin switch by external control of the interval distance. With orbital analysis and electron transfer consideration, the form of C59N+- CuPc- ion pair support this spin hopping phenomenon