半监督度量学习内蕴最速下降算法的收敛性分析

主要研究对称正定矩阵群上的内蕴最速下降算法的收敛性问题.首先针对一个可转化为对称正定矩阵群上无约束优化问题的半监督度量学习模型,提出对称正定矩阵群上一种自适应变步长的内蕴最速下降算法.然后利用李群上的光滑函数在任意一点处带积分余项的泰勒展开式,证明所提算法在对称正定矩阵群上是线性收敛的.最后通过在分类问题中的数值实验说明算法的有效性.     

On the vanishing of homology in random Čech complexes

We compute the homology of random ?ech complexes over a homogeneous Poisson process on the d‐dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erd?s ‐Rényi phase transition, where the ?ech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14–51, 2017     

Additive preservers of Drazin invertible operators with bounded index

Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n ≥ 1, we show that an additive surjective map Φ on B(X)preserves Drazin invertible operators of index non-greater than n in both directions if and only if Φ is either of the form Φ(T) = αATA~(-1) or of the form Φ(T) = αBT~*B~(-1) where α is a non-zero scalar,A:X → X and B:X~*→ X are two bounded invertible linear or conjugate linear operators.     

Intelligent Surfaces Thermally Switchable between the Highly Rough and Entirely Smooth States

Reversible switching from a highly rough surface to another entirely smooth surface under external stimuli is crucial for intelligent materials applied in the fields of anti-fogging,self-cleaning,oil-water separation and biotechnology.In this work,a thermal-responsive liquid crystal elastomer (LCE) surface covered with oriented micropillars is prepared via a facile two-step crosslinking method coupled with an extrusion molding program.The reversible change of topological structures of the LCE surface along with temperature is investigated by metallographic microscope,atomic force microscopy and optical contact angle measuring system.At room temperature,the LCE sample is filled with plenty of micropillars with an average length of 8.76 μm,resulting in a super-hydrophobic surface with a water contact angle (WCA) of 135°.When the temperature is increased to above the clearing point,all the micropillars disappear,the LCE surface becomes entirely fiat and presents a hydrophilic state with a WCA of 64°.The roughness-related wetting property of this microstructured LCE surface possesses good recyclability in several heating/cooling cycles.This work realizes a truly reversible transformation from a highly rough surface to an entirely smooth surface,and might promote the potential applications of this dynamic-responsive LCE surface in smart sensors and biomimetic control devices.     

A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions

We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin-boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature-based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first-order in time and second-order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.     

Identification of the zeroth-order coefficient and fractional order in a time-fractional reaction-diffusion-wave equation

In this paper, we investigate an inverse problem of recovering the zeroth-order coefficient and fractional order simultaneously in a time-fractional reaction-diffusion-wave equation by using boundary measurement data from both of uniqueness and numerical method. We prove the uniqueness of the considered inverse problem and the Lipschitz continuity of the forward operator. Then the inverse problem is formulated into a variational problem by the Tikhonov-type regularization. Based on the continuity of the forward operator, we prove that the minimizer of the Tikhonov-type functional exists and converges to the exact solution under an a priori choice of regularization parameter. The steepest descent method combined with Nesterov acceleration is adopted to solve the variational problem. Three numerical examples are presented to support the efficiency and rationality of our proposed method.     

Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients

The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered. Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours. The multigrid V-cycle algorithm uses $\mathcal{O}(N)$ operations per iteration and is optimal.     

Quantum Bayesian computation

Quantum Bayesian computation is an emerging field that levers the computational gains available from quantum computers. They promise to provide an exponential speed-up in Bayesian computation. Our article adds to the literature in three ways. First, we describe how quantum von Neumann measurement provides quantum versions of popular machine learning algorithms such as Markov chain Monte Carlo and deep learning that are fundamental to Bayesian learning. Second, we describe quantum data encoding methods needed to implement quantum machine learning including the counterparts to traditional feature extraction and kernel embeddings methods. Third, we show how quantum algorithms naturally calculate Bayesian quantities of interest such as posterior distributions and marginal likelihoods. Our goal then is to show how quantum algorithms solve statistical machine learning problems. On the theoretical side, we provide quantum versions of high dimensional regression, Gaussian processes and stochastic gradient descent. On the empirical side, we apply a quantum FFT algorithm to Chicago house price data. Finally, we conclude with directions for future research.     

带有相依重尾冲击的Poisson噪音过程尾的一致渐近性质

本文考虑了带有某种相依重尾冲击的Poisson噪音过程尾的一致渐近性质.当冲击是二元上尾渐近独立的非负随机变量具有长尾和控制变化尾分布且噪音函数具有正的上下界时,得到了过程尾概率的一致渐近公式.进而,当冲击具有连续的一致变化尾分布时,去除了噪音函数具有正的下界的限制.对于噪音函数不一定具有正的上界的情形,当冲击具有两两负象限相依结构时,也得到了一致渐近性结果.     

Boundary-Connected Higher Order Weyl Semimetals in a Helix-Shaped Phononic Crystal

The proposal of higher-order topology leads to intriguing wave propagation properties beyond the conventional bulk-boundary principle in condensed matter physics. Theoretical and experimental models have been conducted to reveal exotic higher-order hinge and corner states in lower dimensions. Meanwhile, this concept is extended to semimetals and verified in platforms like acoustic and photonic systems. However, most existing schemes are confined to C_n-symmetric-like frameworks. Here a phononic higher-order Weyl semimetal (HOWSM) is put forward by introductions of artificial gauge flux shape. Distinguished from most existing schemes, the traditional two-dimensional lattice with a vertical screw-extending operation is deformed, which guarantees the emergence of higher-order Weyl points. The invariant index is utilized to measure different phases between non-trivial topological and normal insulators visibly. Simulation results indicate that the structure supports robust hinge wave transmission in a piling sample. This work brings new approaches for constructing HOWSMs and enriches the potential applications in manufacturing high-performance acoustic devices.