Alternative Methods of Regular and Singular Perturbation Problems

Making exact approximations to solve equations distinguishes applied mathematicians from pure mathematicians, physicists, and engineers. Perturbation problems, both regular and singular, are pervasive in diverse fields of applied mathematics and engineering. This research paper provides a comprehensive overview of algebraic methods for solving perturbation problems, featuring a comparative analysis of their strengths and limitations. Serving as a valuable resource for researchers and practitioners, it offers insights and guidance for tackling perturbation problems in various disciplines, facilitating the advancement of applied mathematics and engineering.

Structural Foundation and Geometry of the Material Singularity (and Its Quantum Entanglement)

In this paper we develop and study, as the second part of one more general development, the energy transmutation equation for the material singularity, previously obtained through the symmetrisation of a wave packet, that is, we develop the correlation between the terms of this equation, which accounts for the formation of matter from a previous vibrational state, and the different possible energy species. These energetic species are ascribed, in a simplified form, to the equation E¯ω=E¯k+E¯f, which allows us, through its associated phase factor, to gain an insight into the wave character of the kinetic energy and thus to attain the basis of the matter-wave, and all sorts of related phenomenologies, including that concerning quantum entanglement. The formation of the matter was previously identified as an energetic process, analogous to the kinetic one, in which finally the inertial mass is consolidated as a mass in a different phase, now, in addition, the mass of the material singularity is identified as a volumetric density of waves of toroidal geometry created in the process of singularisation or energy transfer between species, which makes it possible to establish the real relation or correspondence between the corpuscular and photonic energy equation (E=mc2=hν), i.e. to explain through m the intimate sense of the first equivalence, which explains what νis in the second one.

On the Vacuum Hydrodynamics of Moving Bodies—The Theory of General Singularity

The Theory of General Singularity is presented, unifying quantum field theory, general relativity, and the standard model. This theory posits phonons as fundamental excitations in a quantum vacuum, modeled as a Bose-Einstein condensate. Through key equations, the role of phonons as intermediaries between matter, energy, and spacetime geometry is demonstrated. The theory expands Einsteins field equations to differentiate between visible and dark matter, and revises the standard model by incorporating phonons. It addresses dark matter, dark energy, gravity, and phase transitions, while making testable predictions. The theory proposes that singularities, the essence of particles and black holes, are quantum entities ubiquitous in nature, constituting the very essence of elementary particles, seen as micro black holes or quantum fractal structures of spacetime. As the theory is refined with increasing mathematical rigor, it builds upon the foundation of initial physical intuition, connecting the spacetime continuum of general relativity with the hydrodynamics of the quantum vacuum. Inspired by the insights of Tesla and Majorana, who believed that physical intuition justifies the infringement of mathematical rigor in the early stages of theory development, this work aims to advance the understanding of the fundamental laws of the universe and the perception of reality.

A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities

The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.

Global Existence and Decay of Solutions for a Class of a Pseudo-Parabolic Equation with Singular Potential and Logarithmic Nonlocal Source

This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.

惩罚矩阵T混合模型及其在省域经济分类中的应用

为充分考虑矩阵数据的特性和数据内部的关联性,文章基于矩阵T分布建立矩阵T混合模型及其惩罚模型来研究聚类问题。在矩阵T混合模型的似然函数上对均值矩阵分量施加自适应核范数低秩惩罚,应用ECM算法提出惩罚似然估计算法,同时提出了一种改进的BIC模型选择准则来选择最优的混合模型数量和调节参数,进而通过自适应核范数阈值自动实现低秩估计,实现准确聚类。最后,通过数值模拟研究及与已有方法的对比验证了该方法的有用性,且将所建立的惩罚混合模型应用于中国省域经济发展水平划分研究,得到了比较准确的聚类结果。

高阶非线性不确定系统的复合有限时间控制

本文针对一类带有外界干扰的高阶非线性不确定系统,提出基于有限时间干扰观测器的复合有限时间控制策略,为了解决终端滑模的奇异性问题,设计改进的非奇异终端滑模面.同时,双曲正切函数和指数型函数相结合,构造新型复合趋近律.为了处理外界干扰,设计非奇异终端滑模观测器,将观测到的干扰值前馈到滑模控制中,进一步提高系统的抗干扰性.最后,将设计的复合趋近律与其他5种趋近律进行仿真对比,验证复合趋近律的收敛速度和削弱抖振能力,并通过非线性系统的数值仿真验证所提出控制策略的有效性和优越性.

算法时代的法治之路:计算法学的规范性探索

本文借助法律形式主义与法律现实主义的两分范式,探讨了计算法学在法律与人工智能研究中的历史起源、理论进展与未来方向,并结合法律形式主义与法律现实主义的双重路径,系统分析了其在理论与实践中的挑战与潜力。通过回溯莱布尼茨的法学梦想,本文提出,尽管计算法学在消解法律不确定性方面取得显著成效,但“法律奇点论”等极端形式主义思潮仍存在理论与实践的盲点。计算法学不仅需要在规则与预测之间寻求平衡,还应通过融合认知神经科学等跨学科方法,加强其介入法律解释与规范辩论的科学性和可操作性。本文主张一种超越技术决定论的“计算法治”框架,强调法律应在“计算性”与“人性”之间找到动态平衡,从而确保技术进步与社会价值的良性互动。

基于声发射的木材裂缝数量辨识研究

针对木材裂缝缺陷,提出一种基于声发射的木材裂缝数量识别方法。首先,在试件上人为依次制作1 mm×9 mm(长×高)的4条裂缝,在裂缝的一侧通过折铅的方式产生声发射(acoustic emission,AE)信号,另一侧放置传感器,信号采样频率设置为2 MHz。然后,通过粒子群算法(particle swarm optimization,PSO)确定变分模态分解(variational mode decoposition,VMD)的分解层数K和惩罚因子α,并将原始信号分解为具有不同频率的本征模态(intrinsic mode function,IMF)。接着,随机选择5组信号进行VMD分解,并对分解后的IMF构成的矩阵进行奇异值分解(singular value decomposition,SVD),得到相应的奇异值向量,再由5组奇异值向量组成标准矩阵。最后,由测得的AE信号,分别与标准矩阵计算马氏距离,并依据最小判别原则,判定裂缝数量。结果表明,PSO-VMD-SVD方法能够方便提取出AE信号特征,并通过计算马氏距离进行裂缝数量判别,判别正确率为92%。

基于无奇异工作空间和关节空间的5R并联机构性能分析

针对并联机构的性能分析多集中于工作空间,而针对关节空间的研究较少这一问题,以5R并联机构为研究对象,对其在无奇异工作空间和关节空间中的性能进行了分析。首先,建立了机构的运动学方程,得到了装配模式、工作模式的定义;然后,利用一种无量纲化的参数模型,将机构运动学参数的无限设计空间转化为一个有限的平面设计区域,得到了机构不同尺寸参数下的工作空间和关节空间的分布;之后,绘制了机构设计空间中奇异轨迹的分布,得到了机构的无奇异工作空间,以无奇异工作空间面积最大化为指标,对机构的尺寸参数进行了优化设计;最后,在最大无奇异工作空间和关节空间中,对机构进行了性能分析,得到了机构的最大内切矩形、可操作度与奇异边界距离性能指标分布图。研究结果表明:机构在工作空间内具有较好的运动学性能,基于可操作度对机构进行轨迹规划,平均可操作度可提高17%;并联机构的优化设计需综合考虑奇异边界距离和可操作度性能指标,确保机构在远离奇异位形的同时,保持较高的运动灵活性。

基于MED和ISSD的滚动轴承故障诊断

针对奇异谱分解(SSD)算法中分量个数需要凭经验设定的不足,提出用最小能量差法准确确定分量个数的改进SSD(ISSD)方法,并结合最小熵反褶积(MED)降噪提取噪声背景下的轴承故障特征。首先,对轴承振动信号进行MED降噪预处理;然后,利用ISSD方法得到能量差最小时的最佳分解分量个数、再用自相关峭度最大原则选出最佳分量;最后对最佳分量进行包络解调分析、诊断故障。实测滚动轴承内外圈振动信号分析结果验证了该方法的有效性。

L^p Boundedness of Commutators of Calderón-Zygmund Singular Integral Operators

本文证明了当ｂ∈ＢＭＯ时，具有弱核的ＣａｌｄｅｒóｎＺｙｇｍｕｎｄ奇异积分算子的交换子［ｂ，Ｔ］ｆ＝ｂＴ（ｆ）－Ｔ（ｂｆ）是Ｌｐ（１＜ｐ＜∞）有界的．一个等价的命题是双线性算子ｇＴ（ｆ）－ｆＴ（ｇ）∈Ｈ１，只要ｆ∈Ｌｐ，ｇ∈Ｌｑ，１＜ｐ＜∞，１ｐ＋１ｑ＝１．

机械臂非奇异快速终端滑模迭代学习轨迹跟踪控制研究

针对机械臂建模参数精准性与扰动不确定性的精确轨迹跟踪控制问题,提出了一种非奇异快速终端滑模控制与迭代学习控制相融合的控制方法。首先,为保证跟踪误差的收敛速度,避免收敛中的奇异性问题,设计采用饱和函数趋近律的非奇异快速终端滑模控制器。其次,为进一步提高轨迹跟踪精度,设计误差迭代学习控制器,并对所设计的控制器进行了收敛性分析。最后,在Simulink软件中搭建所提方法的控制系统,进行迭代控制与对比控制仿真实验,并同步开展机械臂跟踪控制真机实验。结果表明:在迭代实验中,关节最大平均稳态误差提升了72%;在对比实验中,与比例微分(PD)型迭代学习控制和PD型线性滑模控制相比,最大平均稳态误差分别提升了97%、51%,最大响应调整时间分别减少70%、50%;在真机实验中,机械臂跟踪误差稳定在[-0.05,0.05] rad区间内。实验结果充分验证了所提控制方法的正确性与有效性,为解决机械臂轨迹跟踪中的不确定性问题提供了一种有效的控制方案。

《三体》的“Singularities”或科幻全球化时代的中国逻辑

本文旨在分析刘慈欣《三体》何以能够在科幻全球化时代以一己之力将中国科幻文学提升到世界水平这一"奇异性事件"。借用詹明信在《奇异性美学》中使用的"singularity"概念,本文认为,《三体》的科学想象力是从"奇点"这一科学边界开始的,并展开了"奇点奇观"的科学想象。《三体》超越本质主义的身份政治,展开了对"后人类"的内在多样性的思考,并建立了黑暗森林原则的宇宙社会学,从而为"独异性政治"分析提供了可能。《三体》的"奇异性美学"在于建构了《三体》的"多维时间"观念体系,并选择了不同的时间叙述方式。

应用奇异值分解(SVD)-主成分分析(PCA)组合模型定量圈定与评价腾冲地块锡钨和铅锌多金属找矿靶区

成矿元素或元素组在一个地质单元中的富集是成岩和成矿地质过程多阶段作用的产物。基于水系沉积物地球化学数据,主成分分析(principal component analysis,PCA)可识别成矿元素组。奇异值分解(singular value decomposition,SVD)可将成矿元素组主成分得分进一步分解为两个部分:(1)成矿元素组合区域异常分量,能够表征在地壳演化过程中,由各种地质作用(岩浆作用、沉积作用和/或变质作用)形成的有利于成矿的高背景区域;(2)成矿元素组合局部异常分量,能够表征成矿作用引起的,叠加在成矿元素组合区域异常分量之上的成矿元素组合局部异常分量,应用局部异常分量能够识别找矿靶区。本次研究,首先基于国家1∶200000水系沉积物地球化学数据,应用主成分分析建立不同类型的成矿元素组;其次,利用SVD从成矿元素组的主成分得分中识别出不同类型成矿过程引起的成矿元素组合局部异常分量;最后,应用局部异常分量识别找矿靶区。最终在腾冲地块圈定15处找矿靶区,其中Sn-W找矿靶区8处,Pb-Zn-Ag找矿靶区7处。预测Sn-W潜在资源量915 Mt,Pb-Zn-Ag潜在资源量792 Mt。

基于仿射变换的奇异积分自适应单元细分法

边界元法已广泛应用于解决工程实际问题中,精确高效地计算奇异积分对于求解边界积分方程至关重要。为此,提出了一种基于仿射变换的奇异积分自适应单元细分法,用于解决边界积分方程中的奇异性问题。其基本思想是通过仿射变换对单元进行特征分区,并结合自适应二叉树细分技术生成高质量的积分单元。利用仿射变换和特征分区技术将初始单元划分为细分区域和投影区域,不同区域的单元细分算法分别执行、计算效率高。源点附近的细分子块采用Serendipity积分单元构建,其余积分单元则采用传统积分方法进行计算。与传统单元细分方法相比,该方法具有自适应细分、积分精度高、易于实现等优点。数值算例验证了该文方法具有良好的准确性、鲁棒性和可行性。

The Singular Value Decomposition Analysis between Summer Precipitation in the Dongting Lake Region and the Global Sea Surface Temperature

By dint of the summer precipitation data from 21 stations in the Dongting Lake region during 1960-2008 and the sea surface temperature(SST) data from NOAA,the spatial and temporal distributions of summer precipitation and their correlations with SST are analyzed.The coupling relationship between the anomalous distribution in summer precipitation and the variation of SST has between studied with the Singular Value Decomposition(SVD) analysis.The increase or decrease of summer precipitation in the Dongting Lake region is closely associated with the SST anomalies in three key regions.The variation of SST in the three key regions has been proved to be a significant previous signal to anomaly of summer rainfall in Dongting region.

具有共轭卷积核和Cauchy核的正则型Wiener-Hopf奇异积分方程

研究了一类含共轭卷积核和Cauchy核的正则型Wiener-Hopf奇异积分方程的解法.引进函数类H 1及H 1,讨论了其中的函数与其共轭函数的关系.通过运用Fourier积分变换、解析延拓原理以及复分析中的Sokhotski-Plemelj公式,将含共轭卷积核和共轭Cauchy核的正则型奇异积分方程转化为无穷直线上的Riemann边值问题,从而得到方程的正则解和可解条件.

工程结构有限元分析的应力奇异问题及抑制方法

针对工程结构有限元分析产生的应力奇异现象进行了研究,并提出了应力奇异抑制方法。将应力奇异分为结构几何突变应力奇异、约束刚度突变应力奇异、材料性能突变应力奇异问题。利用ANSYS Workbench仿真平台,分别建立了3种典型应力奇异模型,并进行了数值模拟分析。研究结果表明:有限元分析过程中存在应力奇异现象的本质是约束刚度突变问题所造成,通过修正模型及约束等效措施能够抑制应力奇异现象。

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS OF THE REACTION DIFFUSION EQUATIONS IN A PART OF DOMAIN

A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.