Directional Wavelets Revisited: Cauchy Wavelets and Symmetry Detection in Patterns

The analysis of oriented features in images requires two-dimensional directional wavelets. Among these, we study in detail the class of Cauchy wavelets, which are strictly supported in a (narrow) convex cone in spatial frequency space. They have excellent angular selectivity, as shown by a standard calibration test, and they have minimal uncertainty. In addition, we present a new application of directional wavelets, namely a technique for determining the symmetries of a given pattern with respect to rotations and dilation.     

Directional Spectra-Based Clustering for Visualizing Patterns of Ocean Waves and Winds

The energy distribution of wind-driven ocean waves is of great interest in marine science. Discovering the generating process of ocean waves is often challenging and the direction is the key for a better understanding. Typically, wave records are transformed into a directional spectrum which provides information about the wave energy distribution across different frequencies and directions. Here, we propose a new time series clustering method for a series of directional spectra to extract the spectral features of ocean waves and develop informative visualization tools to summarize identified wave clusters. We treat directional distributions as functional data of directions and construct a directional functional boxplot to display the main directional distribution of the wave energy within a cluster. We also trace back when these spectra were observed, and we present color-coded clusters on a calendar plot to show their temporal variability. For each identified wave cluster, we analyze wind speed and wind direction hourly to investigate the link between wind data and wave directional spectra. The performance of the proposed clustering method is evaluated by simulations and illustrated by a real-world dataset from the Red Sea. Supplementary materials for this article are available online.     

Numerical Studies of the Cahn-Hilliard Equation for Phase Separation

Phase separation in a binary mixture is described by the nonlinearevolutionary Cahn-Hilliard equation. In this paper, we discussthe physical background of the equation and describe its solutionby the Galerkin finite element method. Some theoretical resultsare presented and computationally verified.     

高压气体淬火过程中的温度场与相变耦合问题的计算

高压气体淬火技术是一种现代的、有效的材料加工技术．在Cheng所得到的高压气体淬火过程中非线性表面换热系数的基础上,用有限单元法对钢淬火过程的温度场与相变耦合问题进行了模拟计算．在数值计算中,材料的热物性系数被处理为温度和相变体积百分比的函数．为避免数值解的震荡,采用了Norsette有理近似法．     

Algorithmic Construction of the Subdifferential from Directional Derivatives

The subdifferential of a function is a generalization for nonsmooth functions of the concept of gradient. It is frequently used in variational analysis, particularly in the context of nonsmooth optimization. The present work proposes algorithms to reconstruct a polyhedral subdifferential of a function from the computation of finitely many directional derivatives. We provide upper bounds on the required number of directional derivatives when the space is ?¹ and ?², as well as in ? ⁿ where subdifferential is known to possess at most three vertices.     

淬火过程中具有非线性表面系数考虑相变时温度场的有限元分析

温度场的计算对淬火过程中热应力和热应变的分析有较大影响，对淬火后试件的残余应力和微观结构分析也有较大影响·本文以４２ＣｒＭｏ钢圆柱体试件为研究对象，从相变等温动力学ＴＴＴ图，给出了连续冷却相变动力学ＣＣＴ图的数学模拟计算式，计算了钢淬火过程中相变组织成分百分比，将热物性系数处理为温度和相变体积百分比的函数，用有限单元法计算了淬火过程中具有非线性表面系数考虑相变时的温度场·并建立了相应的泛函，为以后计算淬火的热应力和热应变做好准备工作     

方向规模收益与方向规模弹性

规模收益(Returns to Scale)是组织绩效分析所关心的一个重要问题.它可以帮助决策者决策是应该扩大还是减少组织规模,从而提高组织运行的绩效.经济学中传统规模收益的定义是基于生产要素按相同比例变化而引起的产出的变化率(径向变化).但对于多投入多产出的非传统生产过程(例如公共部门)来说,往往并不是按相同比例扩张各类投入要素.在这种情况下,如何描述生产过程规模变化的过程中的规模收益情况是一个重要的问题.针对这种现象,在帕累托(Pareto)偏好下,从全局和局部视角定义了方向规模收益和方向规模弹性的概念,并给出具体函数表达.     

Quenching Versus Blow-up

This paper is concerned with the semilinear heat equation u_t = Δu - u^{-q} in Ω × (0, T) under the nonlinear boundary condition \frac{∂u}{∂v} = u^p on ∂Ω × (0, T). Criteria for finite time quenching and blow-up are established, quenching and blow-up sets are discussed, and the rates of quenching and blow-up are obtained.     

Phase Retrieval from Gabor Measurements

Compressed sensing investigates the recovery of sparse signals from linear measurements. But often, in a wide range of applications, one is given only the absolute values (squared) of the linear measurements. Recovering such signals (not necessarily sparse) is known as the phase retrieval problem. We consider this problem in the case when the measurements are time-frequency shifts of a suitably chosen generator, i.e. coming from a Gabor frame. We prove an easily checkable injectivity condition for recovery of any signal from all \(N^2\) time-frequency shifts, and for recovery of sparse signals, when only some of those measurements are given.     

拟线性抛物方程的熄灭(quenching

本文研究一类带奇异项的一性抛物方程的初边值问题，得到了古典解的熄灭现象，并对熄灭解的渐近行为作了分析，包含了（２－５）的相应结论。     

Phase Retrieval with One or Two Diffraction Patterns by Alternating Projections with the Null Initialization

Alternating projection (AP) of various forms, including the parallel AP (PAP), real-constrained AP (RAP) and the serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null initialization is proposed to produce good-quality initial guess. Numerical experiments show that the null initialization is more accurate than the spectral initialization and that AP converges faster to the true object than other iterative schemes such as the Wirtinger flow (WF). In numerical experiments AP with the null initialization converges globally to the true object.     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

反应-扩散方程的熄灭问题

In this article, we consider a reaction-diffusion differential equation with initial value conditions u(x, 0) = 0 on [0, a] and boundary condition ux + αiu = 0 on Γ= {0, a} × (0, T), and the quenching happens for the reaction-diffusion equation.