Improved non-singular local boundary integral equation method
机器学习专业词汇中英文对照
机器学习专业词汇中英⽂对照activation 激活值activation function 激活函数additive noise 加性噪声autoencoder ⾃编码器Autoencoders ⾃编码算法average firing rate 平均激活率average sum-of-squares error 均⽅差backpropagation 后向传播basis 基basis feature vectors 特征基向量batch gradient ascent 批量梯度上升法Bayesian regularization method 贝叶斯规则化⽅法Bernoulli random variable 伯努利随机变量bias term 偏置项binary classfication ⼆元分类class labels 类型标记concatenation 级联conjugate gradient 共轭梯度contiguous groups 联通区域convex optimization software 凸优化软件convolution 卷积cost function 代价函数covariance matrix 协⽅差矩阵DC component 直流分量decorrelation 去相关degeneracy 退化demensionality reduction 降维derivative 导函数diagonal 对⾓线diffusion of gradients 梯度的弥散eigenvalue 特征值eigenvector 特征向量error term 残差feature matrix 特征矩阵feature standardization 特征标准化feedforward architectures 前馈结构算法feedforward neural network 前馈神经⽹络feedforward pass 前馈传导fine-tuned 微调first-order feature ⼀阶特征forward pass 前向传导forward propagation 前向传播Gaussian prior ⾼斯先验概率generative model ⽣成模型gradient descent 梯度下降Greedy layer-wise training 逐层贪婪训练⽅法grouping matrix 分组矩阵Hadamard product 阿达马乘积Hessian matrix Hessian 矩阵hidden layer 隐含层hidden units 隐藏神经元Hierarchical grouping 层次型分组higher-order features 更⾼阶特征highly non-convex optimization problem ⾼度⾮凸的优化问题histogram 直⽅图hyperbolic tangent 双曲正切函数hypothesis 估值，假设identity activation function 恒等激励函数IID 独⽴同分布illumination 照明inactive 抑制independent component analysis 独⽴成份分析input domains 输⼊域input layer 输⼊层intensity 亮度/灰度intercept term 截距KL divergence 相对熵KL divergence KL分散度k-Means K-均值learning rate 学习速率least squares 最⼩⼆乘法linear correspondence 线性响应linear superposition 线性叠加line-search algorithm 线搜索算法local mean subtraction 局部均值消减local optima 局部最优解logistic regression 逻辑回归loss function 损失函数low-pass filtering 低通滤波magnitude 幅值MAP 极⼤后验估计maximum likelihood estimation 极⼤似然估计mean 平均值MFCC Mel 倒频系数multi-class classification 多元分类neural networks 神经⽹络neuron 神经元Newton’s method ⽜顿法non-convex function ⾮凸函数non-linear feature ⾮线性特征norm 范式norm bounded 有界范数norm constrained 范数约束normalization 归⼀化numerical roundoff errors 数值舍⼊误差numerically checking 数值检验numerically reliable 数值计算上稳定object detection 物体检测objective function ⽬标函数off-by-one error 缺位错误orthogonalization 正交化output layer 输出层overall cost function 总体代价函数over-complete basis 超完备基over-fitting 过拟合parts of objects ⽬标的部件part-whole decompostion 部分-整体分解PCA 主元分析penalty term 惩罚因⼦per-example mean subtraction 逐样本均值消减pooling 池化pretrain 预训练principal components analysis 主成份分析quadratic constraints ⼆次约束RBMs 受限Boltzman机reconstruction based models 基于重构的模型reconstruction cost 重建代价reconstruction term 重构项redundant 冗余reflection matrix 反射矩阵regularization 正则化regularization term 正则化项rescaling 缩放robust 鲁棒性run ⾏程second-order feature ⼆阶特征sigmoid activation function S型激励函数significant digits 有效数字singular value 奇异值singular vector 奇异向量smoothed L1 penalty 平滑的L1范数惩罚Smoothed topographic L1 sparsity penalty 平滑地形L1稀疏惩罚函数smoothing 平滑Softmax Regresson Softmax回归sorted in decreasing order 降序排列source features 源特征sparse autoencoder 消减归⼀化Sparsity 稀疏性sparsity parameter 稀疏性参数sparsity penalty 稀疏惩罚square function 平⽅函数squared-error ⽅差stationary 平稳性（不变性）stationary stochastic process 平稳随机过程step-size 步长值supervised learning 监督学习symmetric positive semi-definite matrix 对称半正定矩阵symmetry breaking 对称失效tanh function 双曲正切函数the average activation 平均活跃度the derivative checking method 梯度验证⽅法the empirical distribution 经验分布函数the energy function 能量函数the Lagrange dual 拉格朗⽇对偶函数the log likelihood 对数似然函数the pixel intensity value 像素灰度值the rate of convergence 收敛速度topographic cost term 拓扑代价项topographic ordered 拓扑秩序transformation 变换translation invariant 平移不变性trivial answer 平凡解under-complete basis 不完备基unrolling 组合扩展unsupervised learning ⽆监督学习variance ⽅差vecotrized implementation 向量化实现vectorization ⽮量化visual cortex 视觉⽪层weight decay 权重衰减weighted average 加权平均值whitening ⽩化zero-mean 均值为零Letter AAccumulated error backpropagation 累积误差逆传播Activation Function 激活函数Adaptive Resonance Theory/ART ⾃适应谐振理论Addictive model 加性学习Adversarial Networks 对抗⽹络Affine Layer 仿射层Affinity matrix 亲和矩阵Agent 代理 / 智能体Algorithm 算法Alpha-beta pruning α-β剪枝Anomaly detection 异常检测Approximation 近似Area Under ROC Curve／AUC Roc 曲线下⾯积Artificial General Intelligence/AGI 通⽤⼈⼯智能Artificial Intelligence/AI ⼈⼯智能Association analysis 关联分析Attention mechanism 注意⼒机制Attribute conditional independence assumption 属性条件独⽴性假设Attribute space 属性空间Attribute value 属性值Autoencoder ⾃编码器Automatic speech recognition ⾃动语⾳识别Automatic summarization ⾃动摘要Average gradient 平均梯度Average-Pooling 平均池化Letter BBackpropagation Through Time 通过时间的反向传播Backpropagation/BP 反向传播Base learner 基学习器Base learning algorithm 基学习算法Batch Normalization/BN 批量归⼀化Bayes decision rule 贝叶斯判定准则Bayes Model Averaging／BMA 贝叶斯模型平均Bayes optimal classifier 贝叶斯最优分类器Bayesian decision theory 贝叶斯决策论Bayesian network 贝叶斯⽹络Between-class scatter matrix 类间散度矩阵Bias 偏置 / 偏差Bias-variance decomposition 偏差-⽅差分解Bias-Variance Dilemma 偏差 – ⽅差困境Bi-directional Long-Short Term Memory/Bi-LSTM 双向长短期记忆Binary classification ⼆分类Binomial test ⼆项检验Bi-partition ⼆分法Boltzmann machine 玻尔兹曼机Bootstrap sampling ⾃助采样法／可重复采样／有放回采样Bootstrapping ⾃助法Break-Event Point／BEP 平衡点Letter CCalibration 校准Cascade-Correlation 级联相关Categorical attribute 离散属性Class-conditional probability 类条件概率Classification and regression tree/CART 分类与回归树Classifier 分类器Class-imbalance 类别不平衡Closed -form 闭式Cluster 簇/类/集群Cluster analysis 聚类分析Clustering 聚类Clustering ensemble 聚类集成Co-adapting 共适应Coding matrix 编码矩阵COLT 国际学习理论会议Committee-based learning 基于委员会的学习Competitive learning 竞争型学习Component learner 组件学习器Comprehensibility 可解释性Computation Cost 计算成本Computational Linguistics 计算语⾔学Computer vision 计算机视觉Concept drift 概念漂移Concept Learning System /CLS 概念学习系统Conditional entropy 条件熵Conditional mutual information 条件互信息Conditional Probability Table／CPT 条件概率表Conditional random field/CRF 条件随机场Conditional risk 条件风险Confidence 置信度Confusion matrix 混淆矩阵Connection weight 连接权Connectionism 连结主义Consistency ⼀致性／相合性Contingency table 列联表Continuous attribute 连续属性Convergence 收敛Conversational agent 会话智能体Convex quadratic programming 凸⼆次规划Convexity 凸性Convolutional neural network/CNN 卷积神经⽹络Co-occurrence 同现Correlation coefficient 相关系数Cosine similarity 余弦相似度Cost curve 成本曲线Cost Function 成本函数Cost matrix 成本矩阵Cost-sensitive 成本敏感Cross entropy 交叉熵Cross validation 交叉验证Crowdsourcing 众包Curse of dimensionality 维数灾难Cut point 截断点Cutting plane algorithm 割平⾯法Letter DData mining 数据挖掘Data set 数据集Decision Boundary 决策边界Decision stump 决策树桩Decision tree 决策树／判定树Deduction 演绎Deep Belief Network 深度信念⽹络Deep Convolutional Generative Adversarial Network/DCGAN 深度卷积⽣成对抗⽹络Deep learning 深度学习Deep neural network/DNN 深度神经⽹络Deep Q-Learning 深度 Q 学习Deep Q-Network 深度 Q ⽹络Density estimation 密度估计Density-based clustering 密度聚类Differentiable neural computer 可微分神经计算机Dimensionality reduction algorithm 降维算法Directed edge 有向边Disagreement measure 不合度量Discriminative model 判别模型Discriminator 判别器Distance measure 距离度量Distance metric learning 距离度量学习Distribution 分布Divergence 散度Diversity measure 多样性度量／差异性度量Domain adaption 领域⾃适应Downsampling 下采样D-separation （Directed separation）有向分离Dual problem 对偶问题Dummy node 哑结点Dynamic Fusion 动态融合Dynamic programming 动态规划Letter EEigenvalue decomposition 特征值分解Embedding 嵌⼊Emotional analysis 情绪分析Empirical conditional entropy 经验条件熵Empirical entropy 经验熵Empirical error 经验误差Empirical risk 经验风险End-to-End 端到端Energy-based model 基于能量的模型Ensemble learning 集成学习Ensemble pruning 集成修剪Error Correcting Output Codes／ECOC 纠错输出码Error rate 错误率Error-ambiguity decomposition 误差-分歧分解Euclidean distance 欧⽒距离Evolutionary computation 演化计算Expectation-Maximization 期望最⼤化Expected loss 期望损失Exploding Gradient Problem 梯度爆炸问题Exponential loss function 指数损失函数Extreme Learning Machine/ELM 超限学习机Letter FFactorization 因⼦分解False negative 假负类False positive 假正类False Positive Rate/FPR 假正例率Feature engineering 特征⼯程Feature selection 特征选择Feature vector 特征向量Featured Learning 特征学习Feedforward Neural Networks/FNN 前馈神经⽹络Fine-tuning 微调Flipping output 翻转法Fluctuation 震荡Forward stagewise algorithm 前向分步算法Frequentist 频率主义学派Full-rank matrix 满秩矩阵Functional neuron 功能神经元Letter GGain ratio 增益率Game theory 博弈论Gaussian kernel function ⾼斯核函数Gaussian Mixture Model ⾼斯混合模型General Problem Solving 通⽤问题求解Generalization 泛化Generalization error 泛化误差Generalization error bound 泛化误差上界Generalized Lagrange function ⼴义拉格朗⽇函数Generalized linear model ⼴义线性模型Generalized Rayleigh quotient ⼴义瑞利商Generative Adversarial Networks/GAN ⽣成对抗⽹络Generative Model ⽣成模型Generator ⽣成器Genetic Algorithm/GA 遗传算法Gibbs sampling 吉布斯采样Gini index 基尼指数Global minimum 全局最⼩Global Optimization 全局优化Gradient boosting 梯度提升Gradient Descent 梯度下降Graph theory 图论Ground-truth 真相／真实Letter HHard margin 硬间隔Hard voting 硬投票Harmonic mean 调和平均Hesse matrix 海塞矩阵Hidden dynamic model 隐动态模型Hidden layer 隐藏层Hidden Markov Model/HMM 隐马尔可夫模型Hierarchical clustering 层次聚类Hilbert space 希尔伯特空间Hinge loss function 合页损失函数Hold-out 留出法Homogeneous 同质Hybrid computing 混合计算Hyperparameter 超参数Hypothesis 假设Hypothesis test 假设验证Letter IICML 国际机器学习会议Improved iterative scaling/IIS 改进的迭代尺度法Incremental learning 增量学习Independent and identically distributed/i.i.d. 独⽴同分布Independent Component Analysis/ICA 独⽴成分分析Indicator function 指⽰函数Individual learner 个体学习器Induction 归纳Inductive bias 归纳偏好Inductive learning 归纳学习Inductive Logic Programming／ILP 归纳逻辑程序设计Information entropy 信息熵Information gain 信息增益Input layer 输⼊层Insensitive loss 不敏感损失Inter-cluster similarity 簇间相似度International Conference for Machine Learning/ICML 国际机器学习⼤会Intra-cluster similarity 簇内相似度Intrinsic value 固有值Isometric Mapping/Isomap 等度量映射Isotonic regression 等分回归Iterative Dichotomiser 迭代⼆分器Letter KKernel method 核⽅法Kernel trick 核技巧Kernelized Linear Discriminant Analysis／KLDA 核线性判别分析K-fold cross validation k 折交叉验证／k 倍交叉验证K-Means Clustering K – 均值聚类K-Nearest Neighbours Algorithm/KNN K近邻算法Knowledge base 知识库Knowledge Representation 知识表征Letter LLabel space 标记空间Lagrange duality 拉格朗⽇对偶性Lagrange multiplier 拉格朗⽇乘⼦Laplace smoothing 拉普拉斯平滑Laplacian correction 拉普拉斯修正Latent Dirichlet Allocation 隐狄利克雷分布Latent semantic analysis 潜在语义分析Latent variable 隐变量Lazy learning 懒惰学习Learner 学习器Learning by analogy 类⽐学习Learning rate 学习率Learning Vector Quantization/LVQ 学习向量量化Least squares regression tree 最⼩⼆乘回归树Leave-One-Out/LOO 留⼀法linear chain conditional random field 线性链条件随机场Linear Discriminant Analysis／LDA 线性判别分析Linear model 线性模型Linear Regression 线性回归Link function 联系函数Local Markov property 局部马尔可夫性Local minimum 局部最⼩Log likelihood 对数似然Log odds／logit 对数⼏率Logistic Regression Logistic 回归Log-likelihood 对数似然Log-linear regression 对数线性回归Long-Short Term Memory/LSTM 长短期记忆Loss function 损失函数Letter MMachine translation/MT 机器翻译Macron-P 宏查准率Macron-R 宏查全率Majority voting 绝对多数投票法Manifold assumption 流形假设Manifold learning 流形学习Margin theory 间隔理论Marginal distribution 边际分布Marginal independence 边际独⽴性Marginalization 边际化Markov Chain Monte Carlo/MCMC 马尔可夫链蒙特卡罗⽅法Markov Random Field 马尔可夫随机场Maximal clique 最⼤团Maximum Likelihood Estimation/MLE 极⼤似然估计／极⼤似然法Maximum margin 最⼤间隔Maximum weighted spanning tree 最⼤带权⽣成树Max-Pooling 最⼤池化Mean squared error 均⽅误差Meta-learner 元学习器Metric learning 度量学习Micro-P 微查准率Micro-R 微查全率Minimal Description Length/MDL 最⼩描述长度Minimax game 极⼩极⼤博弈Misclassification cost 误分类成本Mixture of experts 混合专家Momentum 动量Moral graph 道德图／端正图Multi-class classification 多分类Multi-document summarization 多⽂档摘要Multi-layer feedforward neural networks 多层前馈神经⽹络Multilayer Perceptron/MLP 多层感知器Multimodal learning 多模态学习Multiple Dimensional Scaling 多维缩放Multiple linear regression 多元线性回归Multi-response Linear Regression ／MLR 多响应线性回归Mutual information 互信息Letter NNaive bayes 朴素贝叶斯Naive Bayes Classifier 朴素贝叶斯分类器Named entity recognition 命名实体识别Nash equilibrium 纳什均衡Natural language generation/NLG ⾃然语⾔⽣成Natural language processing ⾃然语⾔处理Negative class 负类Negative correlation 负相关法Negative Log Likelihood 负对数似然Neighbourhood Component Analysis/NCA 近邻成分分析Neural Machine Translation 神经机器翻译Neural Turing Machine 神经图灵机Newton method ⽜顿法NIPS 国际神经信息处理系统会议No Free Lunch Theorem／NFL 没有免费的午餐定理Noise-contrastive estimation 噪⾳对⽐估计Nominal attribute 列名属性Non-convex optimization ⾮凸优化Nonlinear model ⾮线性模型Non-metric distance ⾮度量距离Non-negative matrix factorization ⾮负矩阵分解Non-ordinal attribute ⽆序属性Non-Saturating Game ⾮饱和博弈Norm 范数Normalization 归⼀化Nuclear norm 核范数Numerical attribute 数值属性Letter OObjective function ⽬标函数Oblique decision tree 斜决策树Occam’s razor 奥卡姆剃⼑Odds ⼏率Off-Policy 离策略One shot learning ⼀次性学习One-Dependent Estimator／ODE 独依赖估计On-Policy 在策略Ordinal attribute 有序属性Out-of-bag estimate 包外估计Output layer 输出层Output smearing 输出调制法Overfitting 过拟合／过配Oversampling 过采样Letter PPaired t-test 成对 t 检验Pairwise 成对型Pairwise Markov property 成对马尔可夫性Parameter 参数Parameter estimation 参数估计Parameter tuning 调参Parse tree 解析树Particle Swarm Optimization/PSO 粒⼦群优化算法Part-of-speech tagging 词性标注Perceptron 感知机Performance measure 性能度量Plug and Play Generative Network 即插即⽤⽣成⽹络Plurality voting 相对多数投票法Polarity detection 极性检测Polynomial kernel function 多项式核函数Pooling 池化Positive class 正类Positive definite matrix 正定矩阵Post-hoc test 后续检验Post-pruning 后剪枝potential function 势函数Precision 查准率／准确率Prepruning 预剪枝Principal component analysis/PCA 主成分分析Principle of multiple explanations 多释原则Prior 先验Probability Graphical Model 概率图模型Proximal Gradient Descent/PGD 近端梯度下降Pruning 剪枝Pseudo-label 伪标记Letter QQuantized Neural Network 量⼦化神经⽹络Quantum computer 量⼦计算机Quantum Computing 量⼦计算Quasi Newton method 拟⽜顿法Letter RRadial Basis Function／RBF 径向基函数Random Forest Algorithm 随机森林算法Random walk 随机漫步Recall 查全率／召回率Receiver Operating Characteristic/ROC 受试者⼯作特征Rectified Linear Unit/ReLU 线性修正单元Recurrent Neural Network 循环神经⽹络Recursive neural network 递归神经⽹络Reference model 参考模型Regression 回归Regularization 正则化Reinforcement learning/RL 强化学习Representation learning 表征学习Representer theorem 表⽰定理reproducing kernel Hilbert space/RKHS 再⽣核希尔伯特空间Re-sampling 重采样法Rescaling 再缩放Residual Mapping 残差映射Residual Network 残差⽹络Restricted Boltzmann Machine/RBM 受限玻尔兹曼机Restricted Isometry Property/RIP 限定等距性Re-weighting 重赋权法Robustness 稳健性/鲁棒性Root node 根结点Rule Engine 规则引擎Rule learning 规则学习Letter SSaddle point 鞍点Sample space 样本空间Sampling 采样Score function 评分函数Self-Driving ⾃动驾驶Self-Organizing Map／SOM ⾃组织映射Semi-naive Bayes classifiers 半朴素贝叶斯分类器Semi-Supervised Learning 半监督学习semi-Supervised Support Vector Machine 半监督⽀持向量机Sentiment analysis 情感分析Separating hyperplane 分离超平⾯Sigmoid function Sigmoid 函数Similarity measure 相似度度量Simulated annealing 模拟退⽕Simultaneous localization and mapping 同步定位与地图构建Singular Value Decomposition 奇异值分解Slack variables 松弛变量Smoothing 平滑Soft margin 软间隔Soft margin maximization 软间隔最⼤化Soft voting 软投票Sparse representation 稀疏表征Sparsity 稀疏性Specialization 特化Spectral Clustering 谱聚类Speech Recognition 语⾳识别Splitting variable 切分变量Squashing function 挤压函数Stability-plasticity dilemma 可塑性-稳定性困境Statistical learning 统计学习Status feature function 状态特征函Stochastic gradient descent 随机梯度下降Stratified sampling 分层采样Structural risk 结构风险Structural risk minimization/SRM 结构风险最⼩化Subspace ⼦空间Supervised learning 监督学习／有导师学习support vector expansion ⽀持向量展式Support Vector Machine/SVM ⽀持向量机Surrogat loss 替代损失Surrogate function 替代函数Symbolic learning 符号学习Symbolism 符号主义Synset 同义词集Letter TT-Distribution Stochastic Neighbour Embedding/t-SNE T – 分布随机近邻嵌⼊Tensor 张量Tensor Processing Units/TPU 张量处理单元The least square method 最⼩⼆乘法Threshold 阈值Threshold logic unit 阈值逻辑单元Threshold-moving 阈值移动Time Step 时间步骤Tokenization 标记化Training error 训练误差Training instance 训练⽰例／训练例Transductive learning 直推学习Transfer learning 迁移学习Treebank 树库Tria-by-error 试错法True negative 真负类True positive 真正类True Positive Rate/TPR 真正例率Turing Machine 图灵机Twice-learning ⼆次学习Letter UUnderfitting ⽋拟合／⽋配Undersampling ⽋采样Understandability 可理解性Unequal cost ⾮均等代价Unit-step function 单位阶跃函数Univariate decision tree 单变量决策树Unsupervised learning ⽆监督学习／⽆导师学习Unsupervised layer-wise training ⽆监督逐层训练Upsampling 上采样Letter VVanishing Gradient Problem 梯度消失问题Variational inference 变分推断VC Theory VC维理论Version space 版本空间Viterbi algorithm 维特⽐算法Von Neumann architecture 冯 · 诺伊曼架构Letter WWasserstein GAN/WGAN Wasserstein⽣成对抗⽹络Weak learner 弱学习器Weight 权重Weight sharing 权共享Weighted voting 加权投票法Within-class scatter matrix 类内散度矩阵Word embedding 词嵌⼊Word sense disambiguation 词义消歧Letter ZZero-data learning 零数据学习Zero-shot learning 零次学习Aapproximations近似值arbitrary随意的affine仿射的arbitrary任意的amino acid氨基酸amenable经得起检验的axiom公理，原则abstract提取architecture架构，体系结构；建造业absolute绝对的arsenal军⽕库assignment分配algebra线性代数asymptotically⽆症状的appropriate恰当的Bbias偏差brevity简短，简洁；短暂broader⼴泛briefly简短的batch批量Cconvergence 收敛，集中到⼀点convex凸的contours轮廓constraint约束constant常理commercial商务的complementarity补充coordinate ascent同等级上升clipping剪下物；剪报；修剪component分量；部件continuous连续的covariance协⽅差canonical正规的，正则的concave⾮凸的corresponds相符合；相当；通信corollary推论concrete具体的事物，实在的东西cross validation交叉验证correlation相互关系convention约定cluster⼀簇centroids 质⼼，形⼼converge收敛computationally计算(机)的calculus计算Dderive获得，取得dual⼆元的duality⼆元性；⼆象性；对偶性derivation求导；得到；起源denote预⽰，表⽰，是…的标志；意味着，[逻]指称divergence 散度；发散性dimension尺度，规格；维数dot⼩圆点distortion变形density概率密度函数discrete离散的discriminative有识别能⼒的diagonal对⾓dispersion分散，散开determinant决定因素disjoint不相交的Eencounter遇到ellipses椭圆equality等式extra额外的empirical经验；观察ennmerate例举，计数exceed超过，越出expectation期望efficient⽣效的endow赋予explicitly清楚的exponential family指数家族equivalently等价的Ffeasible可⾏的forary初次尝试finite有限的，限定的forgo摒弃，放弃fliter过滤frequentist最常发⽣的forward search前向式搜索formalize使定形Ggeneralized归纳的generalization概括，归纳；普遍化；判断（根据不⾜）guarantee保证；抵押品generate形成，产⽣geometric margins⼏何边界gap裂⼝generative⽣产的；有⽣产⼒的Hheuristic启发式的；启发法；启发程序hone怀恋；磨hyperplane超平⾯Linitial最初的implement执⾏intuitive凭直觉获知的incremental增加的intercept截距intuitious直觉instantiation例⼦indicator指⽰物，指⽰器interative重复的，迭代的integral积分identical相等的；完全相同的indicate表⽰，指出invariance不变性，恒定性impose把…强加于intermediate中间的interpretation解释，翻译Jjoint distribution联合概率Llieu替代logarithmic对数的，⽤对数表⽰的latent潜在的Leave-one-out cross validation留⼀法交叉验证Mmagnitude巨⼤mapping绘图，制图；映射matrix矩阵mutual相互的，共同的monotonically单调的minor较⼩的，次要的multinomial多项的multi-class classification⼆分类问题Nnasty讨厌的notation标志，注释naïve朴素的Oobtain得到oscillate摆动optimization problem最优化问题objective function⽬标函数optimal最理想的orthogonal(⽮量，矩阵等)正交的orientation⽅向ordinary普通的occasionally偶然的Ppartial derivative偏导数property性质proportional成⽐例的primal原始的，最初的permit允许pseudocode伪代码permissible可允许的polynomial多项式preliminary预备precision精度perturbation 不安，扰乱poist假定，设想positive semi-definite半正定的parentheses圆括号posterior probability后验概率plementarity补充pictorially图像的parameterize确定…的参数poisson distribution柏松分布pertinent相关的Qquadratic⼆次的quantity量，数量；分量query疑问的Rregularization使系统化；调整reoptimize重新优化restrict限制；限定；约束reminiscent回忆往事的；提醒的；使⼈联想…的（of）remark注意random variable随机变量respect考虑respectively各⾃的；分别的redundant过多的；冗余的Ssusceptible敏感的stochastic可能的；随机的symmetric对称的sophisticated复杂的spurious假的；伪造的subtract减去；减法器simultaneously同时发⽣地；同步地suffice满⾜scarce稀有的，难得的split分解，分离subset⼦集statistic统计量successive iteratious连续的迭代scale标度sort of有⼏分的squares平⽅Ttrajectory轨迹temporarily暂时的terminology专⽤名词tolerance容忍；公差thumb翻阅threshold阈，临界theorem定理tangent正弦Uunit-length vector单位向量Vvalid有效的，正确的variance⽅差variable变量；变元vocabulary词汇valued经估价的；宝贵的Wwrapper包装分类:。
The
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The role of unsteadiness in direct initiation of gaseous detonations
By C H R I S A. E C K E T T, J A M E S J. Q U I R K† A N D J O S E P H E. S H E P H E R D‡
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C. A. Eckett, J. J. Quirk and J. E. Shepherd
detonation transition (DDT). The main variable believed to control the success or failure of direct initiation is the magnitude of the initial energy release, provided the energy deposition is sufficiently fast and the igniter sufficiently small. Experiments suggest that for a given combustible gas mixture at given uniform premixed initial conditions, the energy release must be above a certain level, known as the critical energy, to successfully initiate a detonation. The same arguments apply for direct initia
MULTIPLICITY OF NONNEGATIVE SOLUTIONS TO SECOND-ORDER SINGULAR DIFFERENTIAL EQUATION WITH INTEGR
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Introduction
Boun d ary value p roble ms w ith inte gral b oun dary c ond ition s for ordin ary d iffe rential eq uations arise in d iffe rent fie ld s of ap plied math em atic s an d ph ysics su ch as he at con du ction , che mical en ginee ring, un de rgroun d water flow, the rmo-e lasticity and plasma ph ysics. Moreover, b oun dary -valu e p rob lems with integral bou nd ary con ditions con stitu te a ve ry intere stin g and imp ortant c lass of prob le ms. Th ey inc lu de two, three , multi-p oint an d n onlocal b oun d ary -valu e p roble ms as sp ecial case s, which have be en rec eived muc h atten tion by many au thors (f or instan ce, ple ase se e [2-9] an d refe re nc e the rein). For b ou nd ary value prob lems n ditions an d c omme nts on the ir im portan ce, we refe r the re ad er to G allard o [1], K arakostas an d Tsamatos [2], Lomtatidz e and Malagu ti [3] and the refe ren ce s the rein. Re cen tly, wh en th e n online arity f is continu ous, by me ans of fi xe d p oin t th eory in a c one for strict set contrac tion op e rator, Fen g an d G e [4], Zh ang and Fe ng [5] stud ied th e existen ce and mu ltip lic ity of p ositive solutions to som e k in d of non lin ear b ou nd ary valu e p roble ms of se con d-ord er an d fourth -orde r d iffe rential eq uations with inte gral bou nd ary c ond ition s in orde red Banach space s resp ec tive ly. To the b e st of th e auth or’s kn owled ge, few p ape rs con sider such p roblem s in Ban ach spac es. To fi ll th is gap, m otivated by the ab ove work, in
计算机术语表
Rim边框vague模糊的Ellipse椭圆Hyperbola双曲线Parabola抛物线Eigenvalue特征值Symmetric matrix对称矩阵Canonical经典Conjugate共轭的Correlation关联Invertible可逆的Sphere球Orthogonal直角Tangent切线adjoin matrix伴随矩阵degenerate conic退化二次曲线degenerate衰退analogy类比screw axis螺旋轴diagonal 对角线quadric二次曲面theorem定理conic二次曲线skew 倾斜Inversion倒置Triangulation三角测量法Simultaneously同时factorization因式分解algebraic代数学的Synthetic合成的Symmetrically对称地Emanate发出Image rectification图像校正Epipole核点/极点focal point焦点Ingredient成分singularvaluedecomposition奇异值分解centroid重心volume体积Inhomogeneous非齐次Orthographic垂直的Euclidean欧几里得几何学的affine仿射Simultaneous同步的Clutter凌乱的Stereo立体的perspective透视图Binoculars双眼的envision想象monocular单眼的Intensity强度Synset同义词集cardinality基数Reification具体化pencil光束Subject主语Predicate谓语Object宾语idiomatic成语习惯的Contrast对比、清晰度stride跨度spotlights聚光灯tailored剪裁的retrospective回溯的pseudo假的benchmark基准点alphabet字母表empirical经验主义的exponential指数的semidefinite半负定的asymptotically渐进denote表示Spanned跨越的Portion部分residual剩余的subject to以……为条件orthogonal直角的vector magnitude向量大小foregoing前述的ensemble全体admissible可容许的delinate描绘lemma补充定理coherence 连贯性Synthetic虚构的Perpendicular垂直的Unfeasible不可行的anthropomorphical拟人的Pupil瞳孔Bivariate双变量auxiliary辅助的Fiducial基准的Drastic激烈的Intersection交界交点discreate摧毁discrete离散的Encyclopedia百科全书biometrics生物统计学eigenface特征脸Intuitively 直观地alleviate 减轻holistic 整体的orthogonal 正交Predominant卓越的exclusive排外的Approximation filter 估计滤波器Approximation pyramid 估计金字塔Binary image 二值图像Block neighborhood averaging 块邻域平均Blur 模糊block diagram 流程图Boundary pixel 边界像素Boundary tracking 边界跟踪Closed curve 封闭曲线color model 彩色模型complex conjugate复共轭Connected 连通的Convolution 卷积Curve 曲线contour plot 等值线图4-neighbors 4邻域8—neighbors 8邻域4-adjacency 4邻接8-adjacency 8邻接m—adjacency m邻接Path 路径Dilation 膨胀Downsampler 下采样Erosion 腐蚀Opening 开运算(先腐蚀,后膨胀）Closing 闭运算（先膨胀，后腐蚀)Structuring element 结构元素DFT 离散的傅立叶变换Inverse DFT 逆离散的傅立叶变换Digital image 数字图像Digital image processing 数字图像处理Digitization 数字化Edge 边缘Edge detection 边缘检测Edge enhancement 边缘增强Edge image 边缘图像Edge linking 边缘连接Edge operator 边缘算子Edge pixel 边缘像素Enhance 增强fits snugly 宽松适应Fourier transform 傅立叶变换Gray level 灰度级别Gray scale 灰度尺度gridline 网格线Horizontal edge 水平边缘Highpass filtering 高通滤波histogram 直方图Hough transform 哈夫变换Lowpass filtering 低通滤波Image sampling 图像采样Image restoration 图像复原Image segmentation 图像分割Image sharpening 图像锐化image subsets 图像子集Image quantization 图像量化Image pyramid 图像金字塔Interpolation filter 插值滤波器Inverse transformation 逆变换Line detection 线检测Line pixel 直线像素Linear filter线性滤波Median filter中值滤波Mask 掩模Measurement space 度量空间morphological transform 形态学变换morphological operations 形态学操作Neighborhood 邻域Neighborhood operation 邻域运算Noise 噪音Noise reduction 噪音抑制Pixel 像素Point operation 点运算pseudo-code 伪代码prediction residual pyramid 预测残差金字塔Quantitative image analysis 图像定量分析Quantization 量化rectangle 矩形Region 区域Region averaging 区域平均Region growing 区域增长Resolution 分辨率salt and pepper noise 盐白噪声Sampling 采样Sharpening 锐化Shape number 形状数Smoothing mask平滑掩模square boxes 正方形surface plot 曲面图structuring element 结构化元素Threshold 阈值Thresholding 二值化Transfer function 传递函数Upsampler 上采样Weighted region averaging加权区域平均Vertical edge 垂直边缘Horizontal edge 水平边缘RGB color cube RGB色彩立方体HSI color model HSI 色彩模型Circular color plane 圆形彩色平面Triangular color plane 三角形彩色平面Magnitude大小discrete离散convolution卷积z（t）=x(t）*y(t)= ∫x（m）y（t-m）dm第一次扇你鼓起来的包还没消肿，第二个巴掌就来了，你脸上的包就可能鼓起来两倍高，老板不断扇你,脉冲不断作用在你脸上，效果不断叠加了，这样这些效果就可以求和了，结果就是你脸上的包的高度岁时间变化的一个函数了(注意理解）;如果老板再狠一点，频率越来越高,以至于你都辨别不清时间间隔了，那么，求和就变成积分了.可以这样理解，在这个过程中的某一固定的时刻，你的脸上的包的鼓起程度和什么有关呢？和之前每次打你都有关！但是各次的贡献是不一样的，越早打的巴掌，贡献越小，所以这就是说，某一时刻的输出是之前很多次输入乘以各自的衰减系数之后的叠加而形成某一点的输出，然后再把不同时刻的输出点放在一起,形成一个函数，这就是卷积，卷积之后的函数就是你脸上的包的大小随时间变化的函数。
旅行英语口语：五台山中英双语导游词
⽆忧考英语⼝语频道为⼤家整理的旅⾏英语⼝语：五台⼭中英双语导游词，供⼤家参考!Mount Wutai Nominated World Heritage Sites：Mount Wutai, is constituted by two parts:Taihuai core area Geographic coordinate is N39°E113°，and Foguang temple core area Geographic coordinate is N38°E113°， the core area is 18415 hectares and the buffer zone area is 42312 hectares. 世界遗产提名地：五台⼭，由两部分组成：台怀核⼼区和佛光寺核⼼区，核⼼区⾯积18415公顷，缓冲区42312公顷。
located in Wutai County of Xinzhou City the northeastern of Shanxi Province China. 位于中国⼭西省东北部忻州市五台县境内。
is made of the oldest stratum in the world which is more than 2.5 billion years. 由⼤于25亿年的世界古⽼地层构成 It has 5 peaks in north south east west and center, the peak is 3061 meters above sea level which is called as “the ridge of Northern China” 有东西南北中五座⼭峰，峰3061⽶，被誉为“华北屋脊” The mountaintops are flat and wide commonly known as platform, so it was called the Five Flat-top Mountains. 五座⼭峰顶部平坦宽⼴，俗称台顶，因此得名五台⼭。
数值分析中常用数学词汇英中文对照翻译.
数值分析中常用数学词汇英中文对照abbreviation 简写符号;简写absolute error 绝对误差absolute value 绝对值accelerate 加速accumulation 累积accuracy 准确度act on 施于action 作用; 作用力add 加addition 加法addition formula 加法公式addition law 加法定律additive property 可加性adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式;代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algorithm 算法系统; 规则系统alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设analysis 分析;解析angle 角anti-clockwise direction 逆时针方向;返时针方向anti-derivative 反导数; 反微商anti-logarithm 逆对数;反对数anti-symmetric 反对称approach 接近;趋近approximate value 近似值approximation 近似;略计;逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积argument (1论证; (2辐角argument of a function 函数的自变量arithmetic 算术array 数组; 数组ascending order 递升序ascending powers of X X 的升幂assumption 假定;假设asymmetrical 非对称asymptote 渐近augmented matrix 增广矩阵average 平均;平均数;平均值axiom 公理back substitution 回代base (1底;(2基;基数basis 基belong to 属于bias 偏差;偏倚billion 十亿binary number 二进数binary operation 二元运算binary system 二进制binomial 二项式bisection method 分半法;分半方法boundary condition 边界条件boundary line 界(线;边界bounded 有界的bounded above 有上界的;上有界的bounded below 有下界的;下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度calculation 计算calculator 计算器;计算器calculus (1 微积分学; (2 演算cancel 消法;相消Cartesian coordinates 笛卡儿坐标category 类型;范畴centre 中心;心chain rule 链式法则chance 机会change of base 基的变换change of variable 换元;变量的换characteristic equation 特征(征方程characteristic function 特征(征函数characteristic root 特征(征根chart 图;图表check digit 检验数位checking 验算circle 圆classification 分类clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coefficient 系数cofactor 余因子; 余因式coincide 迭合;重合collection of terms 并项collinear 共线collinear planes 共线面column (1列;纵行;(2 柱column matrix 列矩阵column vector 列向量combination 组合common denominator 同分母;公分母common difference 公差common divisor 公约数;公约common factor 公因子;公因子common multiple 公位数;公倍comparable 可比较的complement 余;补余completing the square 配方complex number 复数complex number plane 复数平面complex root 复数根component 分量composite function 复合函数; 合成函数computation 计算computer 计算机;电子计算器concept 概念conclusion 结论condition 条件conditional 条件句;条件式conjugate 共轭constant 常数constant of integration 积分常数constraint 约束;约束条件continuity 连续性continuous function 连续函数contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数convex 凸convexity 凸性coordinate 坐标corollary 系定理; 系; 推论correspondence 对应counter clockwise direction 逆时针方向;返时针方向counter example 反例counting 数数;计数criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cube 正方体;立方;立方体cubic 三次方;立方;三次(的 cubic equation 三次方程cubic roots of unity 单位的立方根cumulative 累积的curve 曲线decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制definite integral 定积分definition 定义degree (1 度; (2 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of ODE 常微分方程次数delete 删除; 删去denary number 十进数denominator 分母dependence (1相关; (2应变derivable 可导derivative 导数determinant 行列式diagonal 对角线diagonal matrix 对角矩阵difference 差difference equation 差分方程differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiation 微分法digit 数字dimension 量; 量网; 维(数direction 方向; 方位discontinuity 不连续性discontinuous 间断(的;连续(的; 不连续(的discontinuous point 不连续点discrete 分立; 离散distance 距离diverge 发散divergence 发散(性divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divide 除dividend (1被除数;divisible 可整除division 除法division algorithm 除法算式divisor 除数;除式;因子dot 点dot product 点积echelon form 梯阵式echelon matrix 梯矩阵eigenvalue 本征值eigenvector 本征向量element 元素elementary row operation 基本行运算elimination 消法elimination method 消去法;消元法empty set 空集equivalent 等价(的error 误差error estimate 误差估计error term 误差项estimate 估计;估计量evaluate 计值exact 真确exact solution 准确解;精确解;真确解exact value 法确解;精确解;真确解example 例expand 展开experiment 实验;试验experimental 试验的exponent 指数exponential function 指数函数express…in terms of… 以………表达extreme point 极值点extreme value 极值extremum 极值factor 因子;因式;商factor method 因式分解法factorial 阶乘factorization 因子分解;因式分解fallacy 谬误FALSE 假(的falsehood 假值finite 有限finite sequence 有限序列first derivative 一阶导数first order differential equation 一阶微分方程fixed point 不动点fixed point iteration method 不动点迭代法for all X 对所有X for each /every X 对每一Xform 形式;型format 格式;规格formula(formulae 公式fraction 分数;分式function 函数fundamental theorem of calculus 微积分基本定理Gaussian elimination 高斯消去法general form 一般式;通式general solution 通解;一般解general term 通项given 给定;已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值gradient (1斜率;倾斜率;(2梯度graph 图像;图形;图表graphical method 图解法graphical representation 图示;以图样表达graphical solution 图解growth 增长higher order derivative 高阶导数horizontal 水平的;水平hypothesis 假设identity 等(式identity matrix 恒等矩阵if and only if/iff 当且仅当;若且仅若if…, then 若….则;如果…..则illustration 例证;说明image 像点;像imaginary number 虚数implicit function 隐函数imply 蕴涵;蕴含improper integral 广义积分; 非正常积分increase 递增;增加indefinite integral 不定积分independence 独立;自变inequality 不等式;不等inequality sign 不等号infinite 无限;无穷infinite sequence 无限序列;无穷序列infinite series 无限级数;无穷级数infinitesimal 无限小;无穷小infinity 无限(大;无穷(大initial approximation 初始近似值initial condition 原始条件;初值条件initial value 初值;始值initial-value problem 初值问题inner product 内积input 输入integer 整数integral 积分integrate 积;积分;......的积分integration 积分法integration by parts 分部积分法integration by substitution 代换积分法;换元积分法interchange 互换intermediate value theorem 介值定理interpolating polynomial 插值多项式interpolation 插值interval 区间intuition 直观invalid 失效;无效invariance 不变性invariant (1不变的;(2不变量;不变式inverse 反的;逆的inverse function 反函数;逆函数inverse matrix 逆矩阵inverse problem 逆算问题invertible 可逆的invertible matrix 可逆矩阵iterate (1迭代值; (2迭代iteration 迭代iterative method 迭代法known 己知Lagrange interpolating polynomial 拉格朗日插值多项代leading coefficient 首项系数leading diagonal 主对角线lemma 引理limit 极限limit of sequence 序列的极限line of best-fit 最佳拟合line segment 线段linear 线性;一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程;一次方程linear equation in two unknowns 二元一次方程;二元线性方程linearly dependent 线性相关的linearly independent 线性无关的local maximum 局部极大(值local minimum 局部极小(值logic 逻辑long division method 长除法loop 回路lower bound 下界lower triangular matrix 下三角形矩阵Maclaurin expansion 麦克劳林展开式magnitude 量;数量;长度;大小mantissa 尾数matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差mean value theorem 中值定理method of completing square 配方法method of interpolation 插值法; 内插法method of least squares 最小二乘法; 最小平方法method of substitution 代换法;换元法method of successive substitution 逐次代换法; 逐次调替法minimize 极小minus 减modulus of a complex number 复数的模monomial 单项式multiple 倍数multiple root 多重根multiplication 乘法multiplicity 重数multiplier 乘数;乘式multiply 乘mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直n factorial n阶乘n th derivative n阶导数n th root n次根;n次方根n the root of unity 单位的n次根natural logarithm 自然对数necessary and sufficient condition 充要条件necessary condition 必要条件negative 负neighborhood 邻域Newton-Cote's rule 牛顿- 高斯法则Newton-Raphson's method 牛顿- 纳逊方法Newton's formula 牛顿公式Newton's method 牛顿方法non-linear 非线性non-linear equation 非线性方程non-negative 非负的non-singular (1满秩的; (2非奇异的non-singular matrix 满秩矩阵non-trivial 非平凡的non-zero 非零norm 模方; 范数normal (1垂直的;正交的;法线的(2正态的(3正常的;正规的normalize 正规化normalized form 标准型notation 记法;记号null 零; 空null set 空集null vector 零向量number 数numerator 分子numerical method 计算方法;数值法objective function 目标函数octant 卦限odd function 奇函数one-to-one 一个对一个one-one correspondence 一一对应operation 运算order of a matrix 矩阵的阶ordinary differential equation 常微分方程origin 原点orthogonal 正交orthogonality 正交性 outcome 结果 output 输出 parameter 参数；参变量parametric equation 参数方程 partition 分割; 划分 periodic function 周期函数permutation 排列 perpendicular 垂线；垂直(于 phase 相; 位相 pivot 支点 plot 绘图plus 加 point 点 polynomial 多项式 polynomial equation 多项式方程 positive 正 post-multiply 后乘; 自右乘 premultiply 前乘; 自左乘 prime 素 product 乘积；积 proper integral 正常积分 property 性质 quadratic convergence 二阶收敛性 quadratic formula 二次公式 quadratic function 二次函数 quadratic inequality 二次不等式 quadrature 求积法 quadrilateral 四边形 quotient 商；商式 quotient rule 商法则 R.H.S 右 rank 秩 rate of convergence 收敛率 ratio 比 ; 比率 rational function 有理函数 real number 实数 real part 实部 real root 实根 reciprocal 倒数 rectangle 长方形；矩形 recurrence formula 递推公式 recurrent 循环的 recurring decimal 循环小数 reduce 简化 region 区域 region of convergency 收敛区域 regular 正；规则 relative error 相对误差 remainder term 余项root 根 rotation 旋转 rounded number 舍数 rounding(off 舍入；四舍五入 row 行；棋行 row vector 行向量; 行矢量 rule 规则；法(则 satisfy 满足；适合 scalar 纯量; 无向量, 标量 scalar matrix 纯量矩阵 scale 比例尺；标度；图尺 scientific notation 科学记数法 secant (1正割; (2割线 secant method 正割法 second derivative 二阶导数 second order ordinary differential equation 二阶常微分方程 sentence 句；语句 sequence 序列series 级数 set 集 shaded portion 有阴影部分 shape 形状 shear 位移 side 边；侧 sign 符号；记号 signed number 有符号数 significant figure 有效数字 signum 正负号函数similar 相似 simplify 简化 Simpson's integral 森逊积分 Simpson's rule 森逊法则singular 奇的 singular matrix 奇异矩阵; 不可逆矩阵 span 生成 square (1平方；(2正方形 square bracket 方括号square matrix 方(矩阵 stability 稳度 stationary 平稳 stationary point 平稳点; 逗留点; 驻点 straight line 直线 subset 子集 substitute 代入 substitution 代入; 代入法subtract 减 subtraction 减法 successive approximation 逐次逼近法 successive derivative 逐次导数 successive differentiation 逐次微分法 sufficiency 充份性 sufficient and necessary condition 充要条件 sufficient condition 充份条件 sufficiently close 充份接近suffix 下标 sum 和 summation 求和法; 总和 symbol 符号; 记号 symmetry 对称; 对称性Taylor’s expansion 泰勒展开式 term 项 transpose 移项；转置 transpose of matrix 倒置矩阵；转置矩阵 trapezium 梯形 trapezoidal integral 梯形积分 trapezoidal rule 梯形法则 trial 试；试验 triangle 三角形 triangular matrix 三角矩阵 trigonometric equation 三角方程 trigonometric function 三角函数 triple 三倍 trivial solution 平凡解truncation error 截断误差 undefined 未下定义(的 undetermined coefficient 待定系数unequal 不等 unique solution 唯一解 uniqueness 唯一性 unit 单位 unit area 单位面积unit circle 单位圆 unknown 未知数；未知量 upper bound 上界 upper limit 上限 upper triangular matrix 上三角形矩阵 validity 真确性; 有效性 variable 变项；变量；元；变元；变数 vector 向量; 矢量 vector function 向量函数; 矢量函数 vector product 矢量积; 矢量积 vector space 向量空间 verify 证明；验证 weight (1重量；(2权 weighted average, weighted mean 加权平均数 without loss of generality 不失一般性 x-axis x 轴x-coordinate x 坐标 x-intercept x 轴截距 y-axis y 轴 y-coordinate y 坐标 y-intercept y轴截距 zero 零 zero factor 零因子 zero matrix 零矩阵 zero vector 零向量 zeros of a function 函数零值。
用于非光滑优化的信赖域算法的一个实施方案
第12卷 第6期运 筹 与 管 理Vol.12,No.62003年12月OPERA TIONS RESEARCH AND MANA GEMEN T SCIENCEDec.,2003收稿日期:2003204212基金项目:海南省自然科学基金资助项目(10203)作者简介:欧宜贵(19652),男,湖北钟祥市人,海南大学副教授,博士,主要从事最优化理论和算法研究。
用于非光滑优化的信赖域算法的一个实施方案欧宜贵, 李志林(海南大学信息学院,海南海口570228)摘 要:本文给出了处理无约束非光滑优化的信赖域算法的一个实施方案。
数值实验表明:这种方案是切实可行和可靠的。
关键词:非光滑优化;信赖域算法;数值实验;Matlab6.0中图分类号:O22112 文章标识码:A 文章编号:100723221(2003)0620028203An Implemented Version of Trust Region Algorithmfor Unconstrained Nonsmooth OptimizationOU Y i 2gui ,L I Zhi 2ling(Dept.of A ppli.M ath.,Hai nan ,Haikou 570228,Chi na )Abstract :Trust region algorithms for unconstrained nonsmooth optimization are suggested in [2]2[5].The computational experiment indicates that our version is workable and reliable.Key words :nonsmooth optimization ;trust region methods ;numerical experiments ;Matlab6.01 算法的概要信赖域方法具有很好的性质,即可靠性和强适性(robust )以及很强的收敛性(见[1]),因而受到最优化研究界的广泛重视,已成为求解非线性最优化问题的一类重要的数值方法。
具非局部边界条件的奇异分数阶微分方程正解的存在性和唯一性
引理 2.4 (Banach 压缩映射原理)假设 D 是 Banach 空间 E 的非空闭子集， T : D → D 是压缩算子，即 对任意的 x, y ∈ D ，有
Tx − Ty ≤ α x − y ,α ∈ (0,1) .
(iii) T 是压缩映射，
则存在 z ∈ E ，使得=z Sz + Tz 。
引理 2.3 (Arzela-Ascoli 定理)假设函数族 F = { f (t )} 在区间 [α, β ] 上是一致有界和等度连续的，则存 在子函数序列{ fn (t )} ⊂ F 在区间 [α, β ] 上是一致收敛的。
(qx)(t ) ≤ L2 ,t ∈[0,1], x1, x2 ∈ E.
( H3 ) 存在正函数 L3 (t ), L4 (t ) ，使得： tσ f (t, x1,ϕ x1 ) − tσ f (t, x2 ,ϕ x2 ) ≤ L3 (t ) x1 − x2 + L4 (t ) ϕ x1 − ϕ x2 ,t ∈[0,1], x1, x2 ∈ E.
f
: (0,1]×[0, +∞)×[0, +∞)
→
[0, +∞)
连续，
(ϕ x)(t)
=
∫tγ 0
(t, s)
x (s)ds
，
γ
: [0,1] ×[0,1] → [0,+∞)
，
λ
>
0
，
lim f (t, x (t ),(ϕ x)(t )) = +∞ (即f在 t = 0 点奇异)，q是 [0,+∞) 上的非负有界连续函数。
= x (t )
佩雷尔曼关于庞加莱猜想的论文0303109
Proof. A gradient shrinking soliton gij(t), −∞ < t < 0, satisfies the equation
美国大学生数学建模MCM 数学专用名词
美国大学生数学建模MCM 数学专用名词augmented matrix增广矩阵asymptotic渐进的asymptote渐进线asymmetrical非对称的associative law结合律ascending上升的arrangement排列arithmetic算术argument幅角，幅度，自变量，论证area面积arc length弧长apothem边心距apex顶点aperiodic非周期的antisymmetric反对称的antiderivative原函数anticlockwise逆时针的annihilator零化子angular velocity角速度angle of rotation旋转角angle of incidence入射角angle of elevation仰角angle of depression俯角angle of circumference圆周角analytic space复空间analytic geometry解析几何analytic function解析函数analytic extension解析开拓amplitude幅角，振幅alternative互斥的alternate series交错级数almost everywhere几乎处处algebraic topology代数拓扑algebraic expression代数式algebraic代数的affine仿射(几何学)的admissible error容许误差admissible容许的adjugate伴随转置的adjoint operator伴随算子adjoint伴随的adjacency邻接additive加法，加性acute angle锐角accumulation point聚点accidential error偶然误差accessible point可达点abstract space抽象空间abstract algebra抽象代数absolute value绝对值absolute integrable绝对可积absolute convergent绝对收敛Abelian阿贝尔的，交换的balance equation平衡方程bandwidth带宽barycenter重心base基base vectors基向量biased error有偏误差biased statistic有偏统计量bilinear双线性的bijective双射的bilateral shift双侧位移的binomial二项式bisector二等分线，平分线boundary边界的，边界bounded有界的broken line折线bundle丛，把，卷calculus微积分calculus of variations变分法cancellation消去canonical典型的，标准的canonical form标准型cap交，求交运算capacity容量cardinal number基数Cartesian coordinates笛卡尔坐标category范畴，类型cell单元，方格，胞腔cell complex胞腔复形character特征标characterization特征circuit环路，线路，回路circular ring圆环circulating decimal循环小数clockwise顺时针方向的closed ball闭球closure闭包cluster point聚点coefficient系数cofinal共尾的cohomology上同调coincidence重合，叠和collinear共线的collective集体的columnar rank列秩combinatorial theory组合理论common tangent公切线commutative交换的compact紧的compact operator紧算子compatibility相容性compatible events相容事件complementary余的，补的complete完全的，完备的complex analysis复变函数论complex potential复位势composite复合的concave function凹函数concentric circles同心圆concurrent共点conditional number条件数confidence interval置信区间conformal共形的conic圆锥的conjugate共轭的connected连通的connected domain连通域consistence相容，一致constrained约束的continuable可延拓的continuity连续性contour周线，回路，轮廓线convergence收敛性convexity凸形convolution对和，卷积coordinate坐标coprime互质的，互素的correspondence对应coset陪集countable可数的counterexample反例covariance协方差covariant共变的covering覆盖critical临界的cubic root立方根cup并，求并运算curl旋度curvature曲率curve曲线cyclic循环的decade十进制的decagon十边形decimal小数的，十进制的decision theory决策论decomposable可分解的decreasing递减的decrement减量deduction推论，归纳法defect亏量，缺陷deficiency亏格definition定义definite integral定积分deflation压缩deflection挠度，挠率，变位degenerate退化的deleted neighborhood去心邻域denominator分母density稠密性，密度density function密度函数denumerable可数的departure偏差，偏离dependent相关的dependent variable因变量derangement重排derivation求导derivative导数descent下降determinant行列式diagram图，图表diameter直径diamond菱形dichotomy二分法diffeomorphism微分同胚differentiable可微的differential微分differential geometry微分几何difference差，差分digit数字dimension维数directed graph有向图directed set有向集direct prodect直积direct sum直和direction angle方向角directional derivative方向导数disc圆盘disconnected不连通的discontinuous不连续的discrete离散的discriminant判别式disjoint不相交的disorder混乱，无序dissection剖分dissipation损耗distribution分布，广义函数divergent发散的divisor因子，除数division除法domain区域，定义域dot product点积double integral二重积分dual对偶dynamic model动态模型dynamic programming动态规划dynamic system动力系统eccentricity离心率econometrics计量经济学edge棱，边eigenvalue特征值eigenvector特征向量eigenspace特征空间element元素ellipse椭圆embed嵌入empirical equation经验公式empirical assumption经验假设endomorphism自同态end point端点entropy熵entire function整函数envelope包络epimorphism满同态equiangular等角equilateral等边的equicontinuous等度连续的equilibrium平衡equivalence等价error estimate误差估计estimator估计量evaluation赋值，值的计算even number偶数exact sequence正合序列exact solution精确解excenter外心excision切割，分割exclusive events互斥事件exhaustive穷举的expansion展开，展开式expectation期望experimental error实验误差explicit function显函数exponent指数extension扩张，外延face面factor因子factorial阶乘fallacy谬误fiducial置信field域，场field theory域论figure图形，数字finite有限的finite group有限群finite iteration有限迭代finite rank有限秩finitely covered有限覆盖fitting拟合fixed point不动点flag标志flat space平旦空间formula公式fraction分数，分式frame架，标架free boundary自由边界frequency频数，频率front side正面function函数functional泛函functor函子，算符fundamental group基本群fuzzy模糊的gain增益，放大率game对策gap间断，间隙general topology一般拓扑学general term通项generalized普遍的，推广的generalized inverse广义逆generalization归纳，普遍化generating line母线genus亏格geodesic测地线geometrical几何的geometric series几何级数golden section黄金分割graph图形，网格half plane半平面harmonic调和的hexagon六边形hereditary可传的holomorphic全纯的homeomorphism同胚homogeneous齐次的homology同调homotopy同伦hyperbola双曲线hyperplane超平面hypothesis假设ideal理想idempotent幂等的identical恒等，恒同identity恒等式，单位元ill-condition病态image像点，像imaginary axis虚轴imbedding嵌入imitation模仿，模拟immersion浸入impulse function脉冲函数inclination斜角，倾角inclined plane斜面inclusion包含incomparable不可比的incompatible不相容的，互斥的inconsistent不成立的indefinite integral不定积分independence无关（性），独立（性）index指数，指标indivisible除不尽的inductive归纳的inductive definition归纳定义induced诱导的inequality不等式inertia law惯性律inference推理，推论infimum下确界infinite无穷大的infinite decimal无穷小数infinite series无穷级数infinitesimal无穷小的inflection point拐点information theory信息论inhomogeneous非齐次的injection内射inner point内点instability不稳定integer整数integrable可积的integrand被积函数integral积分intermediate value介值intersection交，相交interval区间intrinsic内在的，内蕴的invariant不变的inverse circular funct反三角函数inverse image逆像，原像inversion反演invertible可逆的involution对合irrational无理的，无理数irreducible不可约的isolated point孤立点isometric等距的isomorphic同构的iteration迭代joint distribution联合分布kernel核keyword关键词knot纽结known已知的large sample大样本last term末项lateral area侧面积lattice格子lattice point格点law of identity同一律leading coefficient首项系数leaf蔓叶线least squares solution最小二乘解lemma引理Lie algebra李代数lifting提升likelihood似然的limit极限linear combination线性组合linear filter线性滤波linear fraction transf线性分linear filter线性滤波式变换式变换linear functional线性泛函linear operator线性算子linearly dependent线性相关linearly independent线性无关local coordinates局部坐标locus(pl.loci)轨迹logarithm对数lower bound下界logic逻辑lozenge菱形lunar新月型main diagonal主对角线manifold流形mantissa尾数many-valued function多值函数map into映入map onto映到mapping映射marginal边缘master equation主方程mathermatical analysis数学分析mathematical expectati数学期望matrix(pl. matrices)矩阵maximal极大的，最大的maximum norm最大模mean平均，中数measurable可测的measure测度mesh网络metric space距离空间midpoint中点minus减minimal极小的，最小的model模型modulus模，模数moment矩monomorphism单一同态multi-analysis多元分析multiplication乘法multipole多极mutual相互的mutually disjoint互不相交natural boundary自然边界natural equivalence自然等价natural number自然数natural period固有周期negative负的，否定的neighborhood邻域nil-factor零因子nilpotent幂零的nodal节点的noncommutative非交换的nondense疏的，无处稠密的nonempty非空的noncountable不可数的nonlinear非线性的nonsingular非奇异的norm范数normal正规的，法线normal derivative法向导数normal direction法方向normal distribution正态分布normal family正规族normal operator正规算子normal set良序集normed赋范的n-tuple integral重积分number theory数论numerical analysis数值分析null空，零obtuse angle钝角octagon八边形octant卦限odd number奇数odevity奇偶性off-centre偏心的one-side单侧的open ball开球operations reserach运筹学optimality最优性optimization最优化optimum最佳条件orbit轨道order阶，级，次序order-preserving保序的order-type序型ordinal次序的ordinary寻常的，正常的ordinate纵坐标orient定方向orientable可定向的origin原点original state初始状态orthogonal正交的orthonormal规范化正交的outer product外积oval卵形线overdetermined超定的overlaping重叠，交迭pairity奇偶性pairwise两两的parabola抛物线parallel平行parallel lines平行线parallelogram平行四边形parameter参数parent population母体partial偏的，部分的partial ordering偏序partial sum部分和particle质点partition划分，分类path space道路空间perfect differential全微分period周期periodic decimal循环小数peripheral周界的，外表的periphery边界permissible容许的permutable可交换的perpendicular垂直perturbation扰动，摄动phase相，位相piecewise分段的planar平面的plane curve平面曲线plane domain平面区域plane pencil平面束plus加point of intersection交点pointwise逐点的polar coordinates极坐标pole极，极点polygon多边形polygonal line折线polynomial多项式positive正的，肯定的potency势，基数potential位势prime素的primitive本原的principal minor主子式prism棱柱proof theory证明论probability概率projective射影的，投影proportion比例pure纯的pyramid棱锥，棱锥体quadrant像限quadratic二次的quadric surface二次曲面quantity量，数量quasi-group拟群quasi-norm拟范数quasi-normal拟正规queuing theory排队论quotient商radial径向radical sign根号radication开方radian弧度radius半径ramified分歧的random随机randomize随机化range值域，区域，范围rank秩rational有理的raw data原始数据real function实函数reciprocal倒数的，互反的reciprocal basis对偶基reciprocity互反性rectangle长方形，矩形rectifiable可求长的recurring decimal循环小数reduce简化，化简reflection反射reflexive自反的region区域regular正则regular ring正则环related function相关函数remanent剩余的repeated root重根residue留数，残数resolution分解resolvent预解式right angle直角rotation旋转roundoff舍入row rank行秩ruled surface直纹曲面runs游程，取遍saddle point鞍点sample样本sampling取样scalar field标量场scalar product数量积，内积scale标尺，尺度scattering散射，扩散sectorial扇形self-adjoint自伴的semicircle半圆semi-definite半定的semigroup半群semisimple半单纯的separable可分的sequence序列sequential相继的，序列的serial序列的sheaf层side face侧面similar相似的simple curve简单曲线simplex单纯形singular values奇异值skeleton骨架skewness偏斜度slackness松弛性slant斜的slope斜率small sample小样本smooth manifold光滑流形solid figure立体形solid geometry立体几何solid of rotation旋转体solution解solvable可解的sparse稀疏的spectral theory谱论spectrum谱sphere球面，球形spiral螺线spline function样条函数splitting分裂的statistics统计，统计学statistic统计量stochastic随机的straight angle平角straight line直线stream-line流线subadditive次可加的subinterval子区间submanifold子流形subset子集subtraction减法sum和summable可加的summand被加数supremum上确界surjective满射的symmetric对称的tabular表格式的tabulation列表，造表tangent正切，切线tangent space切空间tangent vector切向量tensor张量term项terminal row末行termwise逐项的tetrahedroid四面体topological拓扑的torsion挠率totally ordered set全序集trace迹trajectory轨道transcendental超越的transfer改变，传transfinite超限的transformation变换式transitive可传递的translation平移transpose转置transverse横截、trapezoid梯形treble三倍，三重trend趋势triad三元组triaxial三轴的，三维的trigon三角形trigonometric三角学的tripod三面角tubular管状的twist挠曲，扭转type类型，型，序型unbiased无偏的unbiased estimate无偏估计unbounded无界的uncertainty不定性unconditional无条件的unequal不等的uniform一致的uniform boundness一致有界uniformly bounded一致有界的uniformly continuous一致连续uniformly convergent一致收敛unilateral单侧的union并，并集unit单位unit circle单位圆unitary matrix酉矩阵universal泛的，通用的upper bound上界unrounded不舍入的unstable不稳定的valuation赋值value值variation变分，变差variety簇vector向量vector bundle向量丛vertex顶点vertical angle对顶角volume体积，容积wave波wave form波形wave function波函数wave equation波动方程weak convergence弱收敛weak derivatives弱导数weight权重，重量well-ordered良序的well-posed适定的zero零zero divisor零因子zeros零点zone域，带</Words>。
ANNALS OF DIFFERENTIAL EQUATIONS Contents of Volume 24
506ANN.OF D IF F.EQS.Vol.24 ANNALS OF DIFFERENTIAL EQUATIONSContents of Volume24Numb er1On the Boundedness and the Stability of Solu tion to Third Order Non-linear Differential Equations.......................B.S.O gund are,G.E.O kecha(1) Analytic Solution and Numerical Solu tion to End olymph Equation U sing Fractional Derivative Duan Junsheng,Liu Zhenhang,Zhang Fengkuan,Temuer Chaolu(9) Oscillation Criteria for Forced Sup erlinear Difference Equations Han H ongmei(13) The Initial-b oundary Value Problems of Nonlinear and Nonlocal Singularly Per-turb edReaction D iffusion System............Liu Shude,Chen Lihua,Mo Jiaqi(20) Positive Solutions to Semi-lin ear Second-order Ordinary Differential Equations in Banach Space................................................Li Zhilong(29) Global Weak Solution to Compressible Navier-Stokes Equations wi th Density-Dependent Viscosity,Vacuum and Gravitational Force...............................................................Sun Yuejuan,Zheng X iying(34) Growth of Solutions of Differential Equations with Finite Order Entire Coefficients .................................Tu Jin,Zheng Xiumin,Huang Wenping(46) Resonant Bifurcations of the Homoclinic Manifold for Fourth-d imen sional System ...........................................................................................................................Wan g Yanqin,Zhu Deming(52) Permanence and Global Stability of P ositive S olutions to Two Predators-O ne Prey System wi th A D iscrete Time Ratio-dependent...............Wu Liping(65) Periodic O scillatory Solution to Delayed BAM Neural N etworks with Periodic Coefficients...Xiang Hongjun,Wang Jinhua,Liu Zhigang,Chen Anping(75) Existence of Solu tions to Boundary Valu e Problems for Im pulsive Defferential Equations with Delayed Arguments.........Yang Wanqian,Luo Zhiguo(86) Solvability of Singular Cantilever Beam Equation..............Yao Qingliu(93) The All-delay Stability of D egenerate Differential Systems with Delay...........................................Zhang Hai,Li Xiaoyan,Jiang Wei(100) Multiple Positive Solutions to Boundary Value Problems for Second-order Sys-temof Q uasilin ear Differen ce Equations........Zhang Ji ehua,Guo Yanping(105) Integrable Couplings of the TB Hierarchy and Its Ham iltonian Structure.........................................Zhang Yu,D ong Huanhe,Li Zhu(112)A Note on Upper and Lower Solutions Method for Fourth-order Boundary ValueProblems...................................................Zhao Hong(117) Region Q ualitative Analysis of P redator-prey Systems on Time Scales........................................Zhu Siming,Li Yanhui,Rong Zh ixin(121)Numb er2Permanence and Global Stability of A Discrete Coop eration System.................................................................Chen Yaoping(127) Approximation Laws of D iscrete-time Variable Structure Control Systems...................................................Gao Cunchen,Li Yan(133) On A Nonautonom ous Ratio-dependent Predator-prey Model with Time D elay andNo.4CO NTENTS OF VO LU ME24507 Diffusion.................................................Huang A imei(141) Oscillation of A Class of Second O rder N eutral Difference Equations with Continu-ous Arguments.................Huang Mei,Sh en Jianhua,Tang Liang(150) The Uniqueness of Li mit Cycle and Critical Point for A Class of Cub ic System ...................................................Jin Shan,Lu Shiping(157) Positive S olutions to Singular m-point Boundary Valu e Problems of A Coupled System of D ifferential Equations..Li Hongyu,Sun Jingxian,Cui Yujun(163) Permanence and Global Stability of A Delayed Ratio-dep end ent Predator-prey Model with Stage Structure...................Lu Zhiqi,Wang Jiaoyan(171) Exact Solitary Wave Solutions to A Class of Nonlinear D ifferential Equations Usi ng Direct Algebraic Method...........................M.A.Abdel-Razek(184) The Generalized Function al-variation Solving Methods for Nonlinear Reaction Diffusion in Bionomics of Species Groups.......Mo Jiaqi,Yao Jingsun(193) Existen ce and Global Attracti vity of Periodic Solutions to A Predator-prey System with Delays and Impulses.....................Ren Yaping,Li Yongkun(197) Solutions to A Class of N onlinear Wave Equations.............Xie Yuanxi(208) Pointwise Estimate of Bou nded Solutions to Some Discrete Inequalities Involving Infinite Summation........................Yang Enhao,Tan Manchun(215) Positive Solution to Q uasi-linear m-p oint Boundary Value Prob lem with Sign Changing Nonlinearity..........Yang Liu,Shen Chunfang,Liu Xiping(224) Positive Solution to Sin gular Semip ositone(n,p)Boundary Value Problem.........................................Zhan g Mingchu an,Yin Yanmi n(233) Uniqueness of Limit Cycle for A Class of Q uartic System Accompanying with Quadratic System.......Zhan Q ingyi,Xie Xiangdong,Wu Chengqiang(239) Periodic Solutions to p-Laplacian Generalized Li´en ard Equation with Deviating Arguments....................................Zhu Yanlin g,Wan g Kai(246)Num ber3Periodic Solutions to A K ind of Neutral Functional Differential Equation with Multiple Delayed Arguments...Guo Lixiang,Lu Shiping,Du Bo,Liang Feng (253)On the Almost Peri odic K olmogorov Comp etitive Systems with Feedback Controls i Weiying(261) Bifurcations of Travelling Wave Solutions to A Coupled Nonli near Wave System .........................................................................................................................Li Ming,Li Xi,Huang Yan(270) Stability of A K ind of Compound Boundary Value Problem with Respect to the Perturbation of Boundary Curve...........Lin Juan,Wang Chuanrong(283) Existen ce of periodic Solutions to A K ind of Higher O rder Function al D ifferential Equation with Two D eviating Arguments..............................................................Li Xiaojing,Zhan g Zhirong,Lu Shiping(289) Semipositone P rob lem for the n th-order Delayed Differential System......................................................Lu Q iuyin g,Zhu Deming(299) On th e E xistence and Uniqueness of Almost Periodic Solutions to Discrete Two-species Comp etitive Systems...........Niu Chen gying,Chen X iaoxing(306) Bifurcations of the Travelling Wave Solutions to A Generalized Cam assa-Holm Equation...............................Rui Weiguo,Long Yao,He Bin(317) Nontrivial Solutions to Si ngular Boundary Value Prob lem s for Fourth-order Differenti al Equations.............Wang Feng,Cui Yujun,Zhang Fang(326) Multiple Positi ve Sol utions to p-Laplaci an D ynamic Equation on Time Scales508ANN.OF D IF F.EQS.Vol.24 ............................Wu Chufen,Song Changxiu,Weng Peixuan(336) Periodic Soluti ons to Fuzzy Cellular Neural Networks with D istributed Delays ..........................................Xiang Hongjun,Wang Jinhua(346) A Strong Maximum Principle for p(x)-Laplacian Equation....Xiao Q ishan(356) On Solutions to Semilin ear Integrodifferential Equations with Nonlocal Conditions ...........................................................Yan Zuom ao(361) Periodic S olution to P redator-prey Ch ain System with Impulsi ve Effects and Beddington-D eA ngelis Functional Resp onse................Yao Zh ijian(367)Numb er4Permanence of A P eriod ic Predator-P rey System with Disp ersal and General H olling Type Functional Response................................Chen Li juan(379) Extinction and Global Attractivity to A Nonlinear Discrete Two Species Compet-itiveSystem..................................................Chen Yaoping(387) Multiple Positive Periodic Solutions to Singular Differential Equ ation with S tate-Dependent Delay...........................................Li Zhilong(395) Stab ility Theorem for A Class of Perturbed Nonautonomous Neutral Differential Equation with Unboun ded Delay.Liang Ruixi,Peng Jun,Shen Jianhua(400) Solvability of Some Singu larly Perturbed S emi-linear D ifferential System..........................................Liu Shud e,Mo Jiaqi,Chen Xiu(407) The Single-Point Q uen ching of N onlinear Degenerate Functional Reaction-D iffusion Equation.................................................Ma Zhongtai(413) Existence and Uniqueness of Periodic Solution s to A H igher Dimensional Periodic System wi th D elay.........................................Wu Xiaofei(419) A Qualitative Analysis of A Vaccination Model of HIV/A ID S with Staged Progres-sionand Ameliorati on............................Wu Zhengfei,Ruan Jiong(427) Quasilinear E lliptic Problems with Critical Exp onent and Hardy Term.......................................................Xie Huazhao,Li Suli(436) Solution to BSDE with Nonhomogeneous Jump s under Locally Lip schitzian Condition ............................................Xie Zhenyun,X ia Ningm ao(443) Positive Entire Solutions to Non-linear p-Laplace Systems....Xu Jianrong(451) Oscillation for Nonlinear Second-O rder Dynamic Equations on Time Scales ..............................................X u Yancong,Zhu Deming(457) Positive S olution to Second-O rder Four-Point Boundary Value Problem with Sign Changing Nonli nearity..........Yang Liu,Shen Chunfang,Liu Xiping(470) The Poincar´e Bifurcation in Cubic Hami ltonian Systems with Heteroclinic Loop .............................................................................................................................Yu Ji anhua,Son g Yan(477) Stab ility of Time Varying Singul ar Differential Systems with Delay..........................................................Zhang Hai,Jiang Wei(484) Existence of Positive Solution to Sup erlinear Semipositone Si ngular m-point Boundary Value Problems........................................Zhang Xingqiu(490) Stab ility of Discrete-Time Cohen-Grossb erg BAM Neural N etworks with D elays ...........................................................................................................................Zhao Yong,Xia Yonghui(498) Contents of Volume24.. (506)。
无核边界积分 英文俊
无核边界积分英文俊"无核边界积分"的英文翻译是 "non-nuclear boundary integral."The term "non-nuclear boundary integral" refers to a mathematical concept used in the field of boundary integral equations, particularly in the context of problems related to potential theory, elasticity, and fluid dynamics. In these problems, the boundary integral equation is often used to relate the values of a function on the boundary of a domain to the values of the function inside the domain. The "non-nuclear" aspect may refer to the absence of nuclear singularities in the integral, which can have implications for the convergence and behavior of the integral equation.From a mathematical perspective, the study of non-nuclear boundary integrals involves understanding the properties of the underlying functions and their behavior near the boundary of the domain. It also involves thedevelopment of numerical methods and techniques for solving boundary integral equations that arise in practical applications.In summary, the term "non-nuclear boundary integral" encompasses a specific type of boundary integral equation that arises in various mathematical and physical contexts, and its study involves both theoretical analysis and practical computational methods.。
Obtaining-Eringen-s-length-scale-coefficient-for-vibrating-nonlocal-beams-via
Obtaining Eringen's length scale coefficient for vibratingnonlocal beams via continualization methodZhen Zhang a,n,C.M.Wang b,Noël Challamel c,Isaac Elishakoff da Department of Materials,Imperial College London,London SW72AZ,United Kingdomb Engineering Science Programme and Department of Civil and Environmental Engineering,National University of Singapore,Kent Ridge,Singapore119260,Singaporec UniversitéEuropéenne de Bretagne,University of South Brittany UBS,UBS–LIMATB,Centre de Recherche,Rue de Saint Maudé,BP92116,56321Lorient cedex,Franced Department of Ocean and Mechanical Engineering,Florida Atlantic University,777Glades Road–EG36/Rm106,Boca Raton,FL33431,United Statesa r t i c l e i n f oArticle history:Received11January2014Received in revised form7April2014Accepted1May2014Handling Editor:M.P.CartmellAvailable online5June2014a b s t r a c tEringen's length scale coefficients e0are presented herein for initially stressed vibratingnonlocal beams with various boundary conditions.The coefficients are obtained byapplying the continualization method to the discrete equations of microstructured beammodels.When compared to another continualized approach that is based on the Padéapproximant,the proposed continualization method(which is based on assumingcontinuous exponential displacement functions for the discrete displacement field)is amore straightforward approach for solution.Unlike the Padéapproximant approach,thislatter method needs no sophisticated mathematical manipulations for obtaining thecontinuous equation.Moreover,the continualization method allows one to directly solvethe discrete governing equations for vibration of microstructured beams in the discretedomain rather than solving an approximated continuous equation obtained via the Padéapproximant.By using this method,it is found that e0¼1=ffiffiffiffiffiffi12pfor the buckling case ofbeams and e0¼1=ffiffiffi6pfor the vibration case for any combination of end conditions.However,the coefficients are not the same in the presence of an initial stress for vibratingbeams with different boundary conditions.&2014Elsevier Ltd.All rights reserved.1.IntroductionThe use of repetitive/periodic structure is an efficient approach in analyzing the mechanical behavior of composite or nonhomogeneous microstructured materials[1].The advantage is straightforward since computational effort can be substantially reduced as one can work on the homogenized mesoscale model instead of the specific atomistic or crystal lattice.In addition,the physical model for these materials can be readily mapped into the direct use of mathematically recursive/periodic functions[2,3].This is well exemplified by the research on exploring Eringen's small length scale coefficient based on microstructured beam models[4,5]when interpreting the nonlocal theories in terms of repetitive microstructured beams.In these microstructured beam models,the governing equations are in discrete forms and they needContents lists available at ScienceDirectjournal homepage:/locate/jsviJournal of Sound and Vibration/10.1016/j.jsv.2014.05.0020022-460X/&2014Elsevier Ltd.All rightsreserved.n Corresponding author.E-mail addresses:zhen.zhang@(Z.Zhang),ceewcm@.sg(C.M.Wang),noel.challamel@univ-ubs.fr(N.Challamel),elishako@(I.Elishakoff).Journal of Sound and Vibration333(2014)4977–4990to be solved accurately so as to investigate the small length scale coefficients in Eringen's nonlocal theory.Note that microstructured beams refer to beam models comprising rigid beam segments connected by elastic springs.When the number of these rigid beam segments goes to infinity,the microstructure size effect becomes negligible and the microstructured beam model reduces to the local beam theory.The microstructured beam model has been shown to be mathematically equivalent to the nonlocal beam theory in references [4–8].In Eringen's nonlocal theory,the internal structural size effect is calibrated by the small length scale coefficient e 0.The importance of this length scale coefficient has been discussed in the literature (see for example references [9,10]).However,the analytical expressions for e 0in different problems are not well understood [9,11].The main objective of this study is to present a simple and general method for obtaining exact analytical solutions for e 0via microstructured beam models.Owing to the discontinuous nature of the representations in the mathematical models,governing equations of these problems are in discrete forms.Many methods have been developed in the literature for solving discrete equations.Eringen obtained numerically the small length scale coefficient by comparing the wave dispersion relations from Born –Kármán lattice dynamics model and nonlocal models [6,7].In order to solve the difference equation for microstructured beam models,Challamel et al.[4]extracted the recursive relations from the tridiagonal characteristic matrix in buckling problems and the exact solution of such recursive relations was found to be Chebyshev polynomials of the second type.However,this procedure is often found to be elusive [12]and may be difficult for problems involving a large bandwidth coefficient matrix.These coefficient matrices with large bandwidth result from discretizing the higher derivatives in the buckling/vibration governing equation;making it difficult to obtain exact solutions.In addition,Challamel et al.[13]explored the pseudo differentiator in continualization the discrete equations of microstructured beams.The small length scale coefficient was then solved in an approximated continuous equation by Padéapproximant [14,15].This method is similar to make the analogy between the spatially discrete problem and heterogeneous equation [16,17].It is noted,however,this method is useful to produce analytical solutions rather than exact solutions since the continuous partial differential equations are based on Padéapproximant.A more general method is to solve the discrete equations by continualization methods.The purpose is to convert the discrete equation exactly into a continuous one which has a general exact solution or it can be solved numerically.For example,a U-transformation method [18]has been applied to analytically solve the vibration of infinite periodic beams [19].Using the U-transformation method on a cyclic periodic structure is tantamount to projecting the displacement vector on the orthogonal mode subspaces in the unitary space [20].The displacement of infinite periodic beams was then expressed in a discrete way in the exponential solution space.The discrete Fourier transform method can be used to develop functional solutions for finite repetitive lattice structures [21],to analyze infinite square-cell lattice with semi-infinite crack [22],as well as to find analytical solutions in local damage in composites with periodic microstructure [23].Nevertheless,in order to solve the difference equations that originated from microstructured beam model,the continualization method based on exponential function is very promising [24–28]as it has a much simpler mathematical implementation procedure than U-transformation and discrete Fourier transform method.In addition,the continualization method allows one to obtain exact solutions rather than numerical solutions from applying the finite difference method to the difference functions [29].In the way of exploring the exact solution of Eringen's small length scale coefficient for nonlocal beams,it is noted that the small length scale coefficients for different boundary conditions have so far not been discussed,even though research studies have been presented based on the finite difference method for free vibration problems of beams with simply supported ends,clamped ends and free ends [30]and buckling problems of beams with simply supported ends,clamped ends and clamped –free ends [24].In view of the aforementioned method used for solving the difference equations that originate from the microstructured beam model,this paper applies an elegant continualization method that is based on the use of continuous exponential displacement functions for the discrete displacement field for solving the vibration problems of microstructured beams.The exact vibration solutions are then used to calibrate the Eringen's small length scale coefficients in the corresponding nonlocal beams for various combinations of end conditions.2.Initially stressed vibrating microstructured beamsThis section presents the development of microstructured beam model under initially stressed vibration.The model has been presented for microstructured beam with simply supported ends [8]and will be extended for other boundary conditions such as clamped ends,clamped –simply supported ends,and clamped –free ends.Based on the formulations on total energy,the governing equations and boundary conditions can be obtained based on Hamilton's principle for microstructured beams.The governing equation will then be formulated through continualization method.Exact solutions will be presented for microstructured beams with different boundary conditions.2.1.Development of microstructured beam modelsDeveloped in this section are the exact stress –frequency relations for initially stressed vibration of microstructured Euler beams with four different boundary conditions,i.e.simply supported ends,clamped ends,clamped –simply supported ends and clamped –free ends.As shown in Fig.1the vibrating microstructured beam model under initial stress s 0is composed of n rigid elements connected by rotational springs.The length of each element is given as a ¼L =n where n is the total number of element considered.The rotational stiffness is C ¼EI =a for the intermediate springs and C 1¼2EI =a for spring at a clamped end [13].The total mass M is uniformly lumped at each node of the microstructured model.Therefore the nodalZ.Zhang et al./Journal of Sound and Vibration 333(2014)4977–49904978mass is M =n at intermediate nodes and M =ð2n Þat the ends.It is noted that this microstructured beam model has recently been proposed and successfully used to identify the small length scale coefficient in buckling and vibration problems for different boundary conditions [4,8,13].This microstructured model has been also extended in buckling [5]and free vibration[31]for Bresse –Timoshenko beams with simply supported ends.2.2.Energy formulation for microstructured beam model with different boundary conditionsBased on the aforementioned microstructured beam model described in Section 2.1,the total energy Πfor microstructured beam model with different boundary conditions are given byfor simply supported endsΠSS ¼∑n i ¼212C w i þ1À2w i þw i À1a 2À∑n i ¼1s 0A 12a w i þ1Àw i a 2À∑n i ¼2_m 2_w 2i (1)where C ¼EI =a and _m ¼ρAa .for clamped –clamped endsΠCC ¼∑n i ¼212C w i þ1À2w i þw i À1a 2þ12C 1w 1a 2þ12C 1w n a 2À∑n i ¼1s 0A 12a w i þ1Àw i a 2À∑n À1i ¼1_m 2_w 2i (2)where the stiffness of rotational spring at clamped end C 1¼2C [13].Fig.1.Vibrating microstructured beam model with various boundary conditions:(a)simply supported ends;(b)clamped ends;(c)clamped –simply supported ends;and (d)clamped –free ends.Z.Zhang et al./Journal of Sound and Vibration 333(2014)4977–49904979for clamped and simply supported endsΠCS¼∑ni¼212Cw iþ1À2w iþw iÀ1a2þ12C1w1a2À∑ni¼1s0A12aw iþ1Àw ia2À∑nÀ1i¼1_m2_w2i(3)for clamped–free endsΠCF¼∑ni¼212Cw iþ1À2w iþw iÀ1a2þ12C1w1a2À∑ni¼1s0A12aw iþ1Àw ia2À∑nÀ1i¼1_m2_w2iÀ_m4_w2n(4)The last term in Eq.(4)represents the kinetic energy introduced by the tip mass of microstructured model as shown in Fig.1(d).2.3.Discrete governing equation for microstructured beam modelBy minimizing the energy function with respect to w j,one obtains the following governing equation for the vibration problem of an initially stressed microstructured Euler beam as[8]H jþΛG jÀΩw j¼0ðj A3;nÀ1½ Þ(5)whereH j¼w jþ2À4w jþ1þ6w jÀ4w jÀ1þw jÀ2(6a)G j¼w jþ1À2w jþw jÀ1(6b)Λ¼s0AL2(6c)Ω¼ρAω2L4EI(6d)where n is the number of rigid elements in the microstructured beam model.The quantity A represents the cross sectional area of the beam,L the length of the beam,EI the bending rigidity of the beam,ρthe mass density of the beam,ωthe vibration frequency of the beam,and s0the initial stress.ThereforeΛandλdenote non-dimensional axial stress parameter and vibration frequency parameter,respectively.Although Eq.(5)can be solved numerically by the central finite difference method,it cannot provide the exact relations between initial stress and vibrating frequencies for the considered beam problem.The reason is that central finite difference method furnishes numerical solutions for each sampling point.These numerical results do not reveal the exact expression of the solution as a function of its dependent variables.It is noted that the governing Eq.(5)is also valid for beams with boundary conditions.As an example,the simply supported boundary conditions are taken care by substituting the following conditions[8]:w1¼0and w0þw2¼0at j¼1 and w nþ1¼0and w nþw nþ2¼0at j¼nþ1.2.4.Continualized governing equation for microstructured beam modelIn this section,we propose a continualization method to obtain an exact solution for Eq.(5).Based on continualized transform by exponential functions,the general solution for microstructured beam models will be sought herein.This method has been used to analyze the error in the finite difference based probabilistic dynamic problems[26,27].According to the continualization method,the discrete displacement field of the microstructured beam model can be assumed asw j¼Bγj(7) where B is a constant.Therefore,Eq.(5)can be written asΓ2Àð4ÀA1ÞΓþ4À2A1þA2¼0(8) whereΓ¼γþð1=γÞ,A1¼Λ=n2,A2¼ÀðΩ=n4Þ.By solving Eq.(8),one obtainsΓ1;2¼2ÀΛ2n28ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ2þ4Ωp2n2(9)Z.Zhang et al./Journal of Sound and Vibration333(2014)4977–4990 4980Therefore γ1;2¼1À14n ΛþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ2þ4Ωp h i 8ffiffiffiffiffiffiffiffiÀ1p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1À1ÀΛ4n À14nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ2þ4Ωp !2s (10a)γ3;4¼1À14n ΛÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ2þ4Ωp h i 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ÀΛ4n þ14n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ2þ4Ωp !2À1s (10b)From Eq.(10),one can assume thatcos f ¼1ÀΛÀ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ2þ4Ωp (11a)cosh ϑ¼1ÀΛþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ2þ4Ωp (11b)Thereforeγ1;2¼cos ϕ8ffiffiffiffiffiffiffiffiÀ1p sin ϕ(12a)γ3;4¼cosh ϑ8sinh ϑ(12b)In view of Eq.(12),the general solution for w j can be represented asw j ¼A 1cos j ϕþA 2sin j ϕþA 3cosh j ϑþA 4sinh j ϑ(13)3.Exact solution for microstructured beam with various boundary conditionsBased on the continualization method,the general solutions of microstructured beam model,i.e.Eq.(13),are found to be in a similar form as those for classical beams.This section will present the exact solution for the microstructured beams with various boundary conditions.3.1.Simply supported (SS)microstructured beamsFor simply supported microstructured beams,the boundary conditions are given byw j ¼0at j ¼0and j ¼n(14a)w j þ1þw j À1¼0j ¼0and j ¼n (14b)By substituting Eq.(13)into (14),the vibration frequency can be obtained by setting the determinant of homogeneous coefficient matrix to zero as1010cos ðn ϕÞsin ðn ϕÞcosh ðn ϑÞsinh ðn ϑÞ2cos ϕ02cosh ϑ02cos ϕcos ðn ϕÞ2cos ϕsin ðn ϕÞ2cosh ϑcosh ðn ϑÞ2cosh ϑsinh ðn ϑÞ¼0(15)Therefore the determinant in Eq.(15)can be simplified as4sin ðn ϕÞsinh ðn ϑÞðcos ϕÀcosh ϑÞ2¼0(16)It is readily found from Eq.(16)that sin ðn f Þ¼0.In view of Eq.(11a),the vibration frequency Ωcan be solved as Ω¼44n 2sin k π 2ÀΛ"#n 2sin k π 2(17)where k is the mode number.Alternatively,a simple way for obtaining solutions in Eq.(17)is shown in Appendix A .3.2.Clamped –clamped (CC)beamsIn the microstructured beam model,the boundary conditions for nonlocal beams with clamped ends arew j ¼0at j ¼0and j ¼n(18a)w j þ1Àw j À1¼0j ¼0and j ¼n (18b)Z.Zhang et al./Journal of Sound and Vibration 333(2014)4977–49904981By substituting Eq.(13)into Eq.(18),the vibration frequency can be obtained from setting the determinant of the coefficient matrix to zero,i.e.1010cos ðn ϕÞsin ðn ϕÞcosh ðn ϑÞsinh ðn ϑÞ02sin ϕ02sinh ϑÀ2sin ϕsin ðn ϕÞ2sin ϕcos ðn ϕÞ2sinh ϑsinh ðn ϑÞ2sinh ϑcosh ðn ϑÞ¼0(19)or½cos 2ϕþcosh 2ϑÀ2 sinh ðn ϑÞsin ðn ϕÞþ½2À2cos ðn ϕÞcosh ðn ϑÞ sinh ϑsin ϕ¼0(20)3.3.Clamped –simply supported (CS)beamsConsider a microstructured beam that is clamped at j ¼0,and simply supported at j ¼n .The boundary conditions for the microstructured beam model are given byw j ¼0and w j þ1Àw j À1¼0at j ¼0(21a)w j ¼0and w j þ1þw j À1¼0at j ¼n (21b)By substituting Eq.(13)into (21)and following the same solution procedure in Section 3.2,one can solve for the vibration frequency from the characteristic equation given by1010cos ðn ϕÞsin ðn ϕÞcosh ðn ϑÞsinh ðn ϑÞ02sin ϕ02sinh ϑ2cos ϕcos ðn ϕÞ2cos ϕsin ðn ϕÞ2cosh ϑcosh ðn ϑÞ2cosh ϑsinh ðn ϑÞ¼0(22)or ðcos ϕÀcosh ϑÞsin ϕcos ðn ϕÞsinh ðn ϑÞÀsinh ϑsin ðn ϕÞcosh ðn ϑÞÂü0(23)3.4.Clamped –free (CF)beamsConsider a microstructured beam that is clamped at j ¼0and free at j ¼n .The boundary conditions for the micro-structured beam model furnishesw j ¼0and w j þ1Àw j À1¼0at j ¼0(24a)w j þ1À2w j þw j À1¼0and w j þ1À3ÀΛn 2 ðw j Àw j À1ÞÀw j À2þΩ2n 4w j ¼0at j ¼n (24b)It is seen that the free end boundary condition (given by Eq.(24b))reduces to the one obtained by Seide [24]for buckling problems and vibration problems by Leckie and Lindberg [30]through finite difference methods.The vibration frequency can be solved from the following characteristic equation:101002sin ϕ02sinh ϑ2cos ðn ϕÞðcos ϕÀ1Þ2sin ðn ϕÞðcos ϕÀ1Þ2cosh ðn ϑÞðcosh ϑÀ1Þ2sinh ðn ϑÞðcosh ϑÀ1ÞF 1F 2F 3F 4¼0(25)where F 1¼Ω2n 4À2ðcos ϕÀ1Þ2ÀΛn 2ðcos ϕÀ1Þ !cos ðn ϕÞÀ2ðcos ϕÀ1ÞþΛn2 !sin ðn ϕÞsin ϕ(26a)F 2¼ΩÀ2ðcos ϕÀ1Þ2ÀΛðcos ϕÀ1Þ !sin ðn ϕÞþ2ðcos ϕÀ1ÞþΛ !cos ðn ϕÞsin ϕ(26b)F 3¼4ÀΛn 2À2cosh ϑ cosh ðn ϑÞcosh ϑÀ2ÀΛn 2À2cosh ϑ sinh ðn ϑÞsinh ϑÀ2ÀΩÀΛ cosh ðn ϑÞ(26c)Z.Zhang et al./Journal of Sound and Vibration 333(2014)4977–49904982F 4¼4ÀΛn À2cosh ϑ sinh ðn ϑÞcosh ϑÀ2ÀΛn À2cosh ϑ cosh ðn ϑÞsinh ϑÀ2ÀΩ2n 4ÀΛn 2 sinh ðn ϑÞ(26d)4.Initially stressed vibrating nonlocal beamsThis section presents the governing equation based on Eringen's nonlocal theory,the formulation of general solution and exact solutions for nonlocal beams with different boundary conditions such as simply supported ends,clamped ends,clamped –simply supported ends,and clamped –free ends.erning equationsBased on the nonlocal beam theory with Eringen's nonlocal elasticity law [7],the governing equation for initially stressed vibrating Euler –Bernoulli beam is given by [8]EI Às 0A ðe 0a Þ2h i d 4w dx 4þρA ω2ðe 0a Þ2þs 0A h i d 2w dx 2ÀρA ω2w ¼0(27)where e 0is the small length scale coefficient and a the internal characteristic length.By adopting the following non-dimensional terms:w ¼w L ;x ¼x L ;Ω¼ρA ω2L 4EI ;and Λ¼s 0AL 2EI (28)Eq.(27)can be rewritten as1d 4w dx 4þA 2d 2w dx 2þA 3w ¼0(29)where1¼1ÀΛe 0a 2(30a)A 2¼ΛþΩe 0a L2(30b)3¼ÀΩ(30c)4.2.General solutionsThe general solution to Eq.(29)may be expressed asw ðx Þ¼D 1cosh ðηx ÞþD 2sinh ðηx ÞþD 3cos ðβx ÞþD 4sin ðβx Þ(31)where D 1,D 2,D 3and D 4are unknown constants to be determined.The quantities ηand βare given by η¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀA 2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi22À4A 13q v u u t and β¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi22À4A 13q v u u t (32)4.3.Exact solution for nonlocal beams with different boundary conditions4.3.1.Simply supported endsThe boundary condition is for nonlocal beam with simply supported ends is given byw ðx Þ¼0at x ¼0and x ¼1(33a)d 2w ðx Þdx 2¼0at x ¼0and x ¼1(33b)Z.Zhang et al./Journal of Sound and Vibration 333(2014)4977–49904983Therefore the displacement can be assumed to bew ¼w 0sin m πx(34)In view of Eq.(27),the vibration frequencies for simply supported beams are given byΩ¼m 2π2½m 2π2Àm 2π2ðe 0a =L Þ2ΛÀΛ1þm 2π2ðe 0a =L Þ2(35)4.3.2.Clamped –clamped endsThe boundary conditions for the clamped end of a nonlocal beam arew ðx Þ¼0at x ¼0and x ¼1(36a)w x þa L Àw x Àa L¼0at x ¼0and x ¼1(36b)Note that the boundary conditions presented in Eq.(36)are not exactly the ones for the Eringen's beam model.The present boundary conditions are the formulations based on variationally modified models [32].This is reasonable since the nonlocal beams can be interpreted as continualized small microstructures with a characteristic length of a .In such a case,the boundary condition can be regarded to be satisfied at discrete locations that measured as unit length of a microstructure.This assumption is especially needed for clamped ends that have zero slopes and thus it requires a continuous displace-ment field.By substituting Eq.(31)into Eq.(36),one can solve the vibration frequency from the following characteristic equation:101002sinh ηa L ÀÁ02sin βa L ÀÁcosh ðηÞsinh ðηÞcos ðβÞsin ðβÞG 1G 2G 3G 4¼0(37)whereG 1¼cosh ηþa L η Àcosh ηÀa L η ¼2sinh ηsinh a Lη (38a)G 2¼sinh ηþa η Àsinh ηÀa η ¼2cosh ηsinh a η (38b)G 3¼cos βþa L β Àcos βÀa L β ¼À2sin βsin a Lβ (38c)G 4¼sin βþa β Àsin βÀa β ¼2cos βsin a β (38d)Eq.(37)may be simplified as cos 2βa L þcosh 2ηa L À2 !sinh ηsin βþð2À2cosh ηcos βÞsinh ηa L sin βa L ¼0(39)4.3.3.Clamped –simply supported endsFor the nonlocal beam with a clamped end at w ¼0and a simply supported end at w ¼1,the boundary conditions arew ðx Þ¼0and w x þa Àw x Àa ¼0at x ¼0(40a)w ðx Þ¼0and d 2w dx 2¼0at x ¼1(40b)By substituting Eq.(31)into Eq.(40),one can determined the vibration frequency from the following characteristic equation:101002sinh ηa L ÀÁ02sin βa L ÀÁcosh ðηÞsinh ðηÞcos ðβÞsin ðβÞη2cosh ðηÞη2sinh ðηÞÀβ2cos ðβÞÀβ2sin ðβÞ¼0(41)Z.Zhang et al./Journal of Sound and Vibration 333(2014)4977–49904984Eq.(41)may be simplified as 2ðη2þβ2Þsin βcosh ηsinh ηa Àcos βsinh ηsin βa !¼0(42)4.3.4.Clamped –free beamsFor the nonlocal beam with a clamped end at w ¼0and a simply supported end at w ¼1,the boundary conditions are w ðx Þ¼0and w x þa Àw x Àa ¼0at x ¼0(43a)d 2w dx 2¼0and 1ÀΛe 0a L 2 !d 3w dx 3þλ2e 0a L 2þΛ !dw dx 0at x ¼1(43b)Following the procedures described in Sections 4.3.2and 4.3.3,the vibration frequency can be solved from the following characteristic equation:101002sinh ηa L ÀÁ02sin βa L ÀÁη2cosh ðηÞη2sinh ðηÞÀβ2cos ðβÞÀβ2sin ðβÞQ 1Q 2Q 3Q 4 ¼0(44)whereQ 1¼η3sinh ðηÞÀΛη3e 0a L 2sinh ðηÞþηλ2e 0a L 2sinh ðηÞþΛηsinh ðηÞ(45a)Q 2¼η3cosh ðηÞÀΛη3e 0a 2cosh ðηÞþηλ2e 0a 2cosh ðηÞþΛηcosh ðηÞ(45b)Q 3¼β3sin ðβÞÀΛβ3e 0a 2sin ðβÞÀβλ2e 0a 2sin ðβÞÀΛβsin ðβÞ(45c)Q 4¼Àβ3cos ðβÞþΛβ3e 0a L 2cos ðβÞþβλ2e 0a L 2cos ðβÞþΛβcos ðβÞ(45d)5.Calibration of small length scale coef ficient in nonlocal vibrating beams based on microstructured beam modelIf we interpret the internal characteristic length a as the length of the rigid elements of microstructured beam model,i.e.a ¼L =n ,the small length scale coefficient e 0will be calibrated for the aforementioned nonlocal beam problems.In the case of nonlocal beam with simply supported ends,exact solution for e 0can be obtained in a closed form.By equating Eqs.(17)and (35),the exact solution for e 0furnishese 20¼n 2½À16n 4sin ðm π=2n Þ4þ4Λn 2sin ðm π=2n Þ2þm 4π4Àm 2π2Λm 2π2½16n 4sin ðm π=2n Þ4À4Λn 2sin ðm π=2n Þ2þm 2π2Λ (46)If n takes on a large value in Eq.(46)while it is not asymptotic to infinity such that the nonlocal theory still applies,we obtain the following expression for e 0:e 0%ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16À112Λm 2π2r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16À112s 0_s ms (47)where _s m ¼m 2π2EI =ðAL 2Þis the m th mode buckling stress of the corresponding local Euler beam.Note that the length scale coefficient e 0is obtained based on the Eringen's nonlocal theory.When n asymptotes to infinity,the nonlocal effect disappears and the nonlocal beam model reduces to local beam model where e 0automatically disappears in the original governing equation,i.e.in Eq.(27).Interestingly,the asymptotic solutions in Eq.(47)agrees exactly with that found by Wang et al.[8]whereas the solution method used here is much simpler.It is also noted that the accuracy of Padéapproximant is excellent.For nonlocal beams with other boundary conditions than simply supported ends,the small length scale coefficient e 0can be obtained by fitting the numerical solutions from nonlocal beams into that by the microstructured beam model.It is worth noting that there exists a mathematical similarity between Eqs.(20)and (39)for beams with clamped ends.This mathematical similarity is also valid for other boundary conditions such as simply supported ends,clamped –simply supported ends,and clamped –free ends.Based on this mathematical similarity for the determinants associated with different boundary conditions,it is clear that the small length scale coefficient e 0is independent of the mode numbers.Z.Zhang et al./Journal of Sound and Vibration 333(2014)4977–49904985。
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第一节（共5小题；每小题1.5分，满分7.5分）听下面5段对话。
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例：How much is the shirt?A. £19.15.B. £9.18.C. £9.15.答案是C。
1．What is Kate doing?A．Boarding a flight. B．Arranging a trip. C．Seeing a friend off.2．What are the speakers talking about?A．A pop star. B．An old song. C．A radio program.3．What will the speakers do today?A．Go to an art show. B．Meet the man's aunt. C．Eat out with Mark.4．What does the man want to do?A．Cancel an order. B．Ask for a receipt. C．Reschedule a delivery.5．When will the next train to Bedford leave?A．At 9:45. B．At 10:15. C．At 11:00.第二节（共15小题；每小题1.5分，满分22.5分）听下面5段对话或独白。
一种基于深度学习的非局部均值图像降噪方法
第37卷第8期计算机仿真2020年8月文章编号：1006 -9348(2020)08 -0228 -07一种基于深度学习的非局部均值图像降噪方法刘建宾，刘保中(北京信息科技大学计算机学院，北京K X H01)摘要:针对传统图像降噪算法对图像进行降噪时效果不佳的问题，提出一种深度学习与非局部均值滤波算法相结合的图像 降噪新方法。
在传统非局部均值滤波算法基础上，通过构建图像分块滤波学习过程框架和五层神经网络模型，运用梯度下 降反向传导算法和ReLU激活函数，采用均方对数误差损失函数和Adam优化函数进行小批量处理模型训练，在kerns框架 上得到较好的降噪效果。
通过和高斯滤波、中值滤波、双边滤波、非局部均值滤波算法对比实验，验证了方法的有效性。
关键词：非局部均值;深度学习；图像分块；图像降噪；神经网络中图分类号:TP391 文献标识码：BNon - Local Mean Image Denoising Method Based on Deep LearningLIU Jian - bin,LIU Bao - zhong(Computer School,Beijing Information Science&Technology University,Beijing 100101)：Aiming at the problem that the traditional image denoising algorithm is not effective in noise reduction,a new image denoising method combining deep learning and non- local mean filtering algorithm is proposed.On thebasis of the traditional non- local mean filtering algorithm,by constructing the image block filtering learning process framework and the five- layer neural network model f using the gradient descent reverse conduction algorithm and the RELU activation function,the mean square logarithmic error loss function and the Adam optimization function were used to perform small batch processing model training and obtain better noise reduction effect on Keras framework.The effectiveness of the method was verified by comparison with Gaussian filtering,median filtering,bilateral filtering,and non- local mean filtering algorithms.：Non- local means(NLM)；Deep learning；Image segmentation；Image denoising；Neural networki引言在图像处理领域中图像降噪问题备受研究者的青睐。
分数阶微分对流-弥散方程的有限元解及应用
分类号：单位代码：10019 密级：学号：B040544学位论文分数阶微分对流-弥散方程的有限元解及应用The Numerical Solution of Fractional Advection-DispersionEquation and It’s Application研究生：黄权中指导教师：黄冠华教授合作指导教师：申请学位门类级别：工学博士专业名称：农业工程研究方向：水资源与水环境所在学院：水利与土木工程学院2008年04月独创性声明本人声明所呈交的论文是我个人在导师指导下进行的研究工作及取得的研究成果。
尽我所知，除了文中特别加以标注和致谢的地方外，论文中不包含其他人已经发表或撰写过的研究成果，也不包含为获得中国农业大学或其它教育机构的学位或证书而使用过的材料。
与我一同工作的同志对本研究所做的任何贡献均已在论文中作了明确的说明并表示了谢意。
研究生签名：时间：年月日关于论文使用授权的说明本人完全了解中国农业大学有关保留、使用学位论文的规定，即：学校有权保留送交论文的复印件和磁盘，允许论文被查阅和借阅，可以采用影印、缩印或扫描等复制手段保存、汇编学位论文。
同意中国农业大学可以用不同方式在不同媒体上发表、传播学位论文的全部或部分内容。
(保密的学位论文在解密后应遵守此协议)研究生签名：时间：年月日导师签名：时间：年月日II摘要水资源紧缺和地下水污染问题已经成为社会经济发展的严重制约因素，研究污染物在地下水或多孔介质中的迁移过程具有十分重要的意义。
污染物在多孔介质中的迁移通常具有非费克特征，即早穿透、拖尾和尺度效应等。
用传统的对流-弥散理论及其相应的模型难以较好地描述这些现象。
分数阶微分对流-弥散理论及其模型(FADE)由于其固有的非局域性特征，能够很好地描述污染物在多孔介质中的非费克迁移现象。
一般情况下，FADE模型的解析解比较难以获得，现有的数值算法中，多以有限差分法为主。
由于分数阶微分定义的适用性和弱奇异性，FADE 模型的数值解还存在诸如质量不守恒、奇异性和计算量大等问题，此外多维FADE模型的数值算法也不多见。
F(u) Z
where g ∗ is the Sobolev conjugate function of g (see Definition 2.6). We allow f to depend also on x and u, with some convexity assumptions on f (see Section 3). By assuming (4), we will prove that the local minimizers of (1) are locally bounded. We point out that, if g (t) = tp , p > 1, our result contains the local boundedness results previously known in the p, q -growth case. On the other hand, condition (4) actually extends the class of the admissible integrands. Indeed there are significant cases in which (4) is satisfied but it is not possible to obtain any p, q -growth condition with 1 < p ≤ q < p∗ . Some examples of this type, which cannot be treated using the previously known results, but which satisfy the hypotheses of our boundedness theorem, are considered in Section 3. The plan of the paper is the following. In Section 2 we introduce the Young functions and the related Orlicz spaces, and give some properties which will be used in the sequel. In Section 3 we give the precise statement of the boundedness theorem, and some applications. Finally in Section 4 we give the proof of the result, which is based on a new version of Caccioppoli’s inequality and on a suitable iteration method.
基于字典学习的非局部均值去噪算法
基于字典学习的非局部均值去噪算法崔学英;张权;桂志国【期刊名称】《计算机应用》【年(卷),期】2013(33)5【摘要】Concerning the measurement of the similarity of non-local means, a method based on dictionary learning was presented. First, block matching based local pixel grouping was used to eliminate the interference by dissimilar image blocks. Then, the corrupted similar blocks were denoised by dictionary learning. As a further development of classical sparse representation model, the similar patches were unified for joint sparse representation and learning an efficient and compact dictionary by principal component analysis, so that the similar patches relevency could be well preserved. This similarity between the pixels was measured by the Euclidean distance of denoised image blocks, which can well show the similarity of the similar blocks. The experimental results show the modified algorithm has a superior denoising performance than the original one in terms of both Peak Signal-to-Noise Ratio (PSNR) and subjective visual quality. For some images whose structural similarity is large and with rich detail information, their structures and details are well preserved. The robustness of the presented method is superior to the original one.%针对非局部均值中相似度的衡量问题,提出了一种基于字典学习的度量算法.首先利用局部像素群块匹配方法消除不相似的图像块带来的干扰,然后对含有噪声的相似块采用字典学习的方法降噪.与经典的字典学习不同的是,对相似块采用联合稀疏编码的思想,利用主成分分析法学习一个高效紧字典,保留相似块间的相关性信息.采用降噪后的图像块间的欧氏距离计算像素间的相似度,能更好地反映相似块的相似性.实验结果表明,所提出的方法在峰值信噪比和视觉效果方面都优于传统算法,尤其对含有较多细节且结构相似性强的图像,细节和纹理部分的保持效果更好,算法的鲁棒性也优于传统算法.【总页数】3页(P1420-1422)【作者】崔学英;张权;桂志国【作者单位】中北大学电子测试技术国防重点实验室,太原030051;太原科技大学应用科学学院,太原030024;中北大学电子测试技术国防重点实验室,太原030051;中北大学电子测试技术国防重点实验室,太原030051【正文语种】中文【中图分类】TP317.4【相关文献】1.基于梯度特征的双核非局部均值去噪算法 [J], 张玉征;杨词慧;林泉2.基于局部边沿方向的非局部均值图像去噪算法 [J], 贾丽娜;焦枫媛;刘瑞强;桂志国3.基于直觉模糊散度的自适应非局部均值去噪算法 [J], 张宁; 宋小鹏; 刘祎; 张鹏程; 桂志国4.基于3维轴距与非局部均值的无人机巡检图像去噪算法 [J], 唐丽丽;马宪民;商立群5.基于边缘检测的非局部均值图像去噪算法 [J], 李向前因版权原因，仅展示原文概要，查看原文内容请购买。