text: I will review the state of three main puzzles which include 3 separate branches of mathematics, that is approximation, optimization and machine learning theory: • Approximation Theory: When and why are deep networks, with many layers of neurons,. Nonlinear random matrix theory for deep learning ofXXT,whichimpliesthatYYT andXXT havethesamelimitingspectraldistribution. In this example this would be: 2560=3000 = 0:85333 (1) We simply multiply this decimal by 100 to get a percent. We show that when applied to a variety of machine learning models including softmax regression, convolutional neural nets, generative Adversarial nets, and deep reinforcement learning, this very simple surrogate can dramatically reduce the variance and improve the accuracy of the generalization. One of the more popular graph learning methods, Node2vec is one of the first Deep Learning attempts to learn from graph structured data. Deep learning-specific courses are in green, non-deep learning machine learning courses are in blue. Along with theory, we'll also learn to build deep learning models in R using MXNet and H2O package. Espresso - A minimal high performance parallel neural network framework running on iOS. •What dynamics go along the way? Learning Dynamics of Gradient Descent 23 [15] Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. and nonlinear random matrix theory alike, and we hope it will be adopted as a standard tool. Convexified Convolutional. As N grows it becomes exponentially unlikely to randomly pick all eigenvalues to be positive or negative, and therefore most critical points are saddle points. Learning in the Machine: Recirculation is Random Backpropagation. linear” in Figure 7 g; cf. In the spring of 2017 the signal processing group at UCSD went through recent advancements in deep learning from the perspective of applied harmonic analysis. Liao and R. Leibe ng ‘18 Recap: Mixture of Gaussians (MoG). We need stronger assumptions about the model to learn useful invariants for vision. It will provide a new ideal for real-time impact load identification for complex nonlinear structures. It is a Machine Learning technique that uses multiple internal layers (hidden layers) of non-linear processing units (neurons) to conduct supervised or unsupervised learning from data. Basics of Random Matrix Theory/ 3/63 Outline Basics of Random Matrix Theory Motivation: Large Sample Covariance Matrices Spiked Models Applications Reminder on Spectral Clustering Methods Kernel Spectral Clustering Semi-supervised Learning Random Feature Maps, Extreme Learning Machines, and Neural Networks Perspectives 3/63. For example, multiplying a (N, C, D) matrix with a (D, K) matrix should produce a (N, C, K) matrix. js - run Keras models in a web view. In the spring of 2017 the signal processing group at UCSD went through recent advancements in deep learning from the perspective of applied harmonic analysis. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. This page is a list of courses which can used for this category. In the Collaborative filtering problem, we want to infer latent representation of users and items from rating matrix. For the sake of clarity, chaos theory is here distinguished from network the- ory, and the term "complexity" is used as an umbrella concept that includes both chaos and networks. linear” in Figure 7 g; cf. Deep learning-specific courses are in green, non-deep learning machine learning courses are in blue. This article proposes an active learning method for high-dimensional data, based on intrinsic data geometries learned through diffusion processes on graphs. and the copyright belongs to deeplearning. Information theory, coding theory, communication theory, signal processing, and foundations of machine learning 2021 IEEE International Symposium on Information Theory (ISIT) ISIT is the premier international conference dedicated to the advancement of information theory and related areas. *In addition to the courses listed below, any 500-level CoE course can count. In the state harvesting stage of the training, the ESN is driven by an input sequence which yields a sequence of extended system states. theory, statistical mechanics, data analysis, deep learning Vincent Rivasseau Laboratoire de Physique Th eorique CNRS UMR 8627, Universit e Paris-SudMelonic Non-linear Flows and the Spiked Tensor Model. you must understand that deep learning as a field is one that has both statistical concepts, probabilistic concepts, computer science and algorithmic concepts to arise from learning intuitively from available data and also is about determining the. Matrix operations are essentially linear multiplication and addition. He has also developed a new framework to begin harnessing the power of random matrix theory in applications with nonlinear dependencies, like deep. Optimization for machine learning, especially non-convex optimization, differential geometric optimization, theory of deep learning, discrete probability, optimal transport, convex geometry, polynomials and more broadly, bridging different areas of math with optimization and machine learning. Random Matrix Filtering in Finance, Part three in a Series on How Math Fits in Modern Portfolio Theory 26 March 2016 - Don't Solve-- Simulate! Markov Chain Monte Carlo Methods with PyMC3. Sigmoid (Logistic) The sigmoid function is one of the non linear activation functions for deep learning that takes a real-valued number as an input and compresses all its outputs to the range of [0,1. Statistical Machine Learning (Summer term 2019) (This lecture used to be called "Machine Learning: Algorithms and Theory" in the last years; it has now been renamed in the context of the upcoming Masters degree in machine learning, but the contents remain approximately the same). to study exact and asymptotic properties of certain deformations of classical measures found in Random Matrix theory. K P Harikrishnan,. Here we show that deep linear networks also provide a good theoretical model for generalization dynamics. Many network embedding algorithms are typically unsupervised algorithms and they can be broadly classified into three groups ex-. Active learning is thus an application of decision theory to the process of learning. and nonlinear random matrix theory alike, and we hope it will be adopted as a standard tool. DeepSurv/Non-Linear model. , the stacked autoencoders, can be regarded as an effec- tive method for learning high level abstractions from low level features. James Melenkevitz PhD Mathematics, Programming, Modeling Systems, Quantitative Analysis, and Machine Learning from Graduate school to current United States 500+ connections. Depending on the problem and how the units are connected, such behavior may require long causal chains of computational stages, where each stage transforms (often in a non-linear way) the aggregate activation of the network. Research group on theory of machine learning. in 2011 can be seen as a series of logistic regression models built on different time intervals so as to estimate the probability that the event of interest happened within each interval. The answers we have found only serve to raise a whole set of new questions. We note the works of Monti et al. Müller ??? The role of neural networks in ML has become increasingly important in r. Machine Learning Algorithms is for you if you are a machine learning engineer, data engineer, or junior data scientist who wants to advance in the field of predictive analytics and machine learning. Baldi and P. Deep Learning without Poor Local Minima ; Topology and Geometry of Half-Rectified Network Optimization. Luckily for mathematicians and statistical physicists, the study of large random network scaling limits, which can be thought of as *nonlinear* random matrix theory, is both practically important and mathematically interesting. Neural Collaborative Filtering (WWW’17) This is another paper that applies deep neural network for collaborative filtering problem. Denil et al. In NIPS, 2017. Laura Balzano. In this paper we analyze GLMs when the data matrix is random, as relevant in problems such as compressed sensing, error-correcting codes, or benchmark models in neural networks. Manoel, et al. Forward Modeling operator is unknown! Goal: Find an operator that can be applied to the data to estimate models Find the operator by systematic examination of a series of observed data and their known answers. ,2000;Bun et al. 1 Neural networks for regression Generalized linear regression Two-layer neural network Matrix notation Deep neural network; Learning the network from data; 7. o Lee, Daniel D. Chapter 2: Deep linear network learning dynamics Chapter 2 derives the major features of the theory, including exact solutions to the full trajectory of learning in deep linear neural networks. As far as we know, it is the first time to demonstrate the feasibility and great potential of using the deep learning technique to solve this nonlinear inverse problem, no matter the impact location is known or not in advance. Traditional and Heavy Tailed Self Regularization in Neural Network Models Random Matrix Theory (RMT) is applied to analyze the weight matrices of Deep Neural Networks (DNNs), including both production quality, pre-trained models such as AlexNet and Inception, and smaller models trained from scratch, such as LeNet5 and a miniature-AlexNet. ) For this reason, an ever increasing proportion of modern mathematical research is devoted to the analysis of nonlinear systems and nonlinear phenomena. Koç University deep learning framework. Topology and Geometry of Half-Rectified Network Optimization. Notes on this document: this HTML page was generated from the IPython notebook available here (pure python version executable in any python interpreter: here). Random Matrix Improved Covariance Estimation for a Large Class of Metrics. Theoretical analysis of the nonlinear performance using random matrix theory. Sumio Watanabe, Algebraic Geometry and Statistical Learning Theory, Cambridge University Press, 2009. Sigmoid (Logistic) The sigmoid function is one of the non linear activation functions for deep learning that takes a real-valued number as an input and compresses all its outputs to the range of [0,1. I am Ritchie Ng, a machine learning engineer specializing in deep learning and computer vision. •If the network is large enough, global minima can be found by local descent. In the Collaborative filtering problem, we want to infer latent representation of users and items from rating matrix. Along with theory, we’ll also learn to build deep learning models in R using MXNet and H2O package. Hypothesis testing for high-dimensional data with applications to gene sets testing and estimating and testing for large dimensional covariance matrices using the random matrix theory Yulia Gel Statistics and Actuarial Science. Interval analysis and non-linear equations. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. Luckily for mathematicians and statistical physicists, the study of large random network scaling limits, which can be thought of as * nonlinear * random matrix theory, is both practically important and mathematically interesting. 4 The Learning Process Every edge (connection between neurons) in the neural network is assigned some initial weight (historically small random values have been used; random weight initialization). This course will provide an introduction to the theory of statistical learning and practical machine learning algorithms with applications in signal processing and data analysis. The resurgence of neural networks has revolutionized artificial intelligence since 2010. Since we will not get into the details of either Linear Regression or Tensorflow, please read the following articles for more details:. deep learning and inverse problems; development of multiscale methods and nonlinear theories of generalized functions applied to scattering and inverse scattering in media of low regularity, and in highly discontinuous and random media;. Course description This course will roughly follow Learning from Data, which covers several important foundamental machine learning concepts and algorithms. In this course, you will learn the foundations of deep learning. Please please please comment on the proposed NIH Policy on Data Management and Sharing. Deep Learning without Poor Local Minima ; Topology and Geometry of Half-Rectified Network Optimization. This tutorial will again tackle the problem of MNIST digit classification. Project 10 [Deep Q-Learning for continuous action space] You have implemented a simple Q-learning algorithm using MLP, with discrete action spaces. In this letter, we present a new theory of deep restricted kernel machines (deep RKM), offering foundations for deep learning with kernel machines. This specialization gives an introduction to deep learning, reinforcement learning, natural language understanding, computer vision and Bayesian methods. Unsupervised machine learning algorithms presented will include k-means clustering, principal component analysis (PCA), and independent component analysis (ICA). Random matrix theory provides powerful tools for studying deep learning! 1. Posts about random matrix theory written by memming. Silver Abstract Deep learning algorithms seek to exploit the unknown structure in the input distribution. Check out my code guides and keep ritching for the skies!. In International Conference on Machine Learning (ICML 2015), 2015. ent in various machine learning protocols, including neu-ral networks33,34 and reservoir computing. This talk presents the work arXiv:1902. However, in the tutorial MNIST for beginners it is reversed and tf. In simple words, deep learning uses the composition of many nonlinear functions to model the complex dependency between input features and labels. theory and conformal ﬁeld theory, [8], are nonlinear. 0:8533 100 = 85:33% classi cation accuracy. AMATH 584 Applied Linear Algebra and Introductory Numerical Analysis (5) Numerical methods for solving linear systems of equations, linear least squares problems, matrix eigen value problems, nonlinear systems of equations, interpolation, quadrature, and initial value ordinary differential equations. Since we will not get into the details of either Linear Regression or Tensorflow, please read the following articles for more details:. Eigenvalues of the Hessian matrix Intuition Random matrix theory: P(eigenvalue > 0) ~ 0. Statistical and machine learning is an interdisciplinary ﬂeld consisting of theory from statistics, probability, mathematics and computer science, with plenty of applications for engineering science, biology, bioinformatics, medical study, etc. This is the second of two papers on Hinton’s capsule theory that has been causing recent excitement. -Non-linear models (decision trees, SVM, Naïve Bayes) -Ensembles (random forest, AdaBoost) -Model selection, regularization, cross validation •Neural networks and deep learning -2 weeks -Back-propagation, gradient descent -NN architectures (feed-forward, convolutional, recurrent) •Adversarial ML -1 lecture. to introduce graphs into the convex low-rank matrix recovery problem. IRO, Universit e de Montr eal. Espresso - A minimal high performance parallel neural network framework running on iOS. In simple words, deep learning uses the composition of many nonlinear functions to model the complex dependency between input features and labels. methods for online reactive power optimization, a scene-matching method based on Random Matrix (RM) features and a deep learning method based on Deep Belief Network (DBN). Ronan∗, Academy of Paris April 1st, 2016 Abstract Google’s AI beats a top player at a game of Go. For example, deep networks can identify dead neurons in images of mixed populations of both living and dead ones. Using our more concise vector notation for the model outputs on a specific dataset we can rewrite as: With matrix sizes made explicit: This picture should make the sizes of all vectors and matrices clear. Specifically, a Deep Reconstruction Model (DRM) is defined integrating with the advantages of the deep learning and Elman neural network (ENN). Timo Seppalainen Affiliate, Mathematics Department Motion in a random medium, interacting particle systems, large deviation theory. For the beginner and new students, it provides foundations for your own research. In Conference on Learning Theory (COLT 15), 2015. Theoretical analysis of the nonlinear performance using random matrix theory. matmul(W, X). 3 Convolutional neural networks Data representation of an image. 37–39 Nonlinear effects in wave chaotic systems manifest as harmonic and sub-harmonic generation, driving. Sigmoid (Logistic) The sigmoid function is one of the non linear activation functions for deep learning that takes a real-valued number as an input and compresses all its outputs to the range of [0,1. Say you have an input X and weight matrix W (assuming zero bias), I want to compute WX as an output which could be done by tf. Jun Shao Inference, asymptotic theory, resampling methods, linear and nonlinear models, model selection, sample survey Yajuan Si. In this article, I will try to summarize some key ideas from random matrix theory that are used by papers on random matrix theory and neural networks. Accelerated Online Low Rank Tensor Learning for Multivariate Spatiotemporal Streams. Title: Gradients in Deep Neural Nets and Random Matrix Products Abstract: Deep learning is the study and use of artificial neural networks, which are finite dimensional spaces of non-linear functions. A novel modeling based on deep learning framework which can exactly manifest the characteristics of nonlinear system is proposed in this paper. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. Fan, and Y. Title: Random Matrix Advances in Machine Learning Abstract: Machine learning algorithms, starting from elementary yet popular ones, are difficult to theoretically analyze as (i) they are data-driven, and (ii) they rely on non-linear tools (kernels, activation functions). To illustrate this a bit better, we draw $$100$$ Gaussian random matrices and multiply them with some initial matrix. Geometry of Neural Network Loss Surfaces via Random Matrix Theory ; Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice; Nonlinear random matrix theory for deep learning ; Lecture 8. If this were to happen to us with a deep network, we. I'm an undergrad in CS with a math minor wondering what math classes to take that are most useful for ML applications. linear” in Figure 7 g; cf. Deep learning is also a new "superpower" that will let you build AI systems that just weren't possible a few years ago. 4 Engineered-Systems Information Knowledge IoT-Sensors (Big)Data First-Principles Machine-Learning-andDeepLearning. The Renormalization Group Theory provides new insights as to why Deep Learning works so amazingly well. Deep learning engineers are highly sought after, and mastering deep learning will give you numerous new. Specifically, a Deep Reconstruction Model (DRM) is defined integrating with the advantages of the deep learning and Elman neural network (ENN). High dimensional covariance matrix estimation using a factor model. Interval analysis and opt im is at ion problems. In machine learning, the kernel perceptron is a type of the popular perceptron learning algorithm that can learn kernel machines, such as non-linear classifiers that uses a kernel function to calculate the similarity of those samples that are unseen to training samples. Taylor, and D. TensorFlow: Regression using Deep Neural Network because the tf. Provable Bounds for Learning Deep Representations. Why Deep Learning Works: Heavy-Tailed Random Matrix Theory as an Example of Physics Informed Machine Learning Michael Brenner, Harvard Machine Learning for Partial Differential Equations. 35,36 Utilizing wave chaoticlayers,alongwithnonlinearity,offersanattractiveway to enable physical realizations of deep learning machines. It will provide a new ideal for real-time impact load identification for complex nonlinear structures. Note: This article is meant for beginners and expects no prior understanding of deep learning (or neural networks). Deep Learning¶. Optimization for machine learning, especially non-convex optimization, differential geometric optimization, theory of deep learning, discrete probability, optimal transport, convex geometry, polynomials and more broadly, bridging different areas of math with optimization and machine learning. 在概率論和數學物理中，隨機矩陣（英語： Random matrix ）是一個矩陣值的随机变量，也就是说，一个矩阵中的所有元素都是. -Non-linear models (decision trees, SVM, Naïve Bayes) -Ensembles (random forest, AdaBoost) -Model selection, regularization, cross validation •Neural networks and deep learning -2 weeks -Back-propagation, gradient descent -NN architectures (feed-forward, convolutional, recurrent) •Adversarial ML -1 lecture. Useful links. That definition is pretty intuitive if you think of a linear function. Say you have an input X and weight matrix W (assuming zero bias), I want to compute WX as an output which could be done by tf. The models we approach, described in Section 2, are non-linear feed-forward neural networks trained on synthetic datasets with constrained weights. It is a Machine Learning technique that uses multiple internal layers (hidden layers) of non-linear processing units (neurons) to conduct supervised or unsupervised learning from data. , & Saxe, A. Random Matrix Filtering in Finance, Part three in a Series on How Math Fits in Modern Portfolio Theory 26 March 2016 - Don't Solve-- Simulate! Markov Chain Monte Carlo Methods with PyMC3. The primary goal of the class is to help participants gain a deep understanding of the concepts, techniques and mathematical frameworks used by experts in machine learning. Statistical Machine Learning (Summer term 2019) (This lecture used to be called "Machine Learning: Algorithms and Theory" in the last years; it has now been renamed in the context of the upcoming Masters degree in machine learning, but the contents remain approximately the same). We consider the hashing mechanism for constructing binary embeddings, that involves pseudo-random projections followed by nonlinear (sign function) mappings. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things. Note: This article is meant for beginners and expects no prior understanding of deep learning (or neural networks). These representations can be learned in a supervised and/or unsupervised manner. Unsupervised machine learning algorithms presented will include k-means clustering, principal component analysis (PCA), and independent component analysis (ICA). ARO YIP: Mathematics for Learning Nonlinear Generalizations of Subspace Models in High Dimensions. Edgar Dobriban (2018), FACT: Fast closed testing for exchangeable local tests, Biometrika. ,2000;Bun et al. Entropy and mutual information in models of deep neural networks. 1817:Nonlinear Data: Theory and Algorithms 1818:Quadratic Forms and Related Structures over Fields 1819:Interactions between Operator Space Theory and Quantum Probability with Applications to Quantum Information. The 32nd Annual Conference on Learning Theory (COLT 2019) will take place in Phoenix, Arizona, June 25-28, 2019, as part of the ACM Federated Computing Research Conference, which also includes EC and STOC. edu/~bhanin College Station, TX, 77843, USA Interests Theory of deep learning, mathematical physics, spectral theory, random matrix theory. This course covers some of the theory and methodology of deep learning. Such deep non linear mapping still suffer from lack of efficiency to learn invariance We still try to map pixel-wise vectorial representation to the target space. 4 The Learning Process Every edge (connection between neurons) in the neural network is assigned some initial weight (historically small random values have been used; random weight initialization). If the model includes output feedback (i. Medical Image Analysis with Deep Learning , Part 2 function to be a random matrix g. So I tried a different approach. Matrix operations are essentially linear multiplication and addition. - Quite often in machine learning we can do fast matrix vector multiplications (MVMs), but for inference or learning we need to compute the log determinant of a matrix. Random forest (or random forests) is a trademark term for an ensemble classifier that consists of many decision trees and outputs the class that is the mode of the classes output by individual trees. For example, multiplying a (N, C, D) matrix with a (D, K) matrix should produce a (N, C, K) matrix. Lemaire, G. The Institute of Mathematical Statistics and the Bernoulli Society. Somehow, we want to reach the bottom of the. Romain Couillet, Supelec: Random Matrices and Machine Learning Abstract: Thanks to its efficiently exploiting degrees of freedom in large multi-dimensional problems, random matrix theory has today become a compelling field in modern (multi-antenna multi-user multi-cell) wireless communications and is currently making powerful headway into large dimensional signal processing and statistics. In this example this would be: 2560=3000 = 0:85333 (1) We simply multiply this decimal by 100 to get a percent. Research Interests. Deep learning has achieved great success in a variety of fields such as speech recognition, image understanding, and natural language understanding. Using the tools from random matrix theory and assuming both p and T diverge to infinity, we derive the asymptotic normality of the test statistic under both the null and a specific VMA(1) alternative hypothesis. Part 1: What is Deep Learning? Deep Learning is a suite of tools for the automation of intelligence, primarily leveraging neural networks. INTRODUCTION In applications ranging from drug discovery [1] and design to proteomics [2] to neuroscience [3] to social network analysis [4], inputs to machine learning methods take the form of graphs. Deep Learning in NNs is about accurately assigning credit across many such stages. Deep Learning for Computer Vision: Optimization (UPC 2016) 1. Learning in the Machine: Recirculation is Random Backpropagation P. The goal is to approximate the mapping function so well that when. 3) Extensive experiments on natural image and MRI CS re-construction clearly show that ISTA-Net signiﬁcantly out-performs the state-of-the-art, while maintaining attractive. Course Notation Guide; Piazza; CPSC 340 lecture recordings from 2017W2; Textbook. I Nonlinear techniques at the opposite ends of tractability. The latter case appears in modeling large scale communication networks with random network topologies as well as in MIMO systems. Deep learning is a very popular type of machine learning and it is built by neural network with multiple layers and numbers of neurons. NET Machine Learning and Statistics Framework; show where the Hidden Conditional Random Fields are located within the framework, the HCRF models' and learning algorithms' source code, how the Fields namespace is organized and the general ideas behind this organization. If (and only if) random variables are independent, the joint density is just a product of individual densities; Vector random variables are just a bunch of scalar random variables; For 2 and more random variables you should be considering their joint distribution. Chapter 2: Deep linear network learning dynamics Chapter 2 derives the major features of the theory, including exact solutions to the full trajectory of learning in deep linear neural networks. Nonlinear random matrix theory for deep learning. IRO, Universit e de Montr eal. Cheng and Y. Results for CS267, HW0 Describe a Parallel Application. 在概率論和數學物理中，隨機矩陣（英語： Random matrix ）是一個矩陣值的随机变量，也就是说，一个矩阵中的所有元素都是. Deep learning has the potential to enable a scaleable and data-driven architecture for the discovery and representation of Koopman eigenfunctions, providing intrinsic linear representations of. are taken to be random variables. Taking inspiration from inverse reinforcement learning, the proposed Direct Value Learning for Reinforcement Learning (DIVA) approach uses light priors to gener- ate inappropriate behavior’s, and use the corresponding state sequences to directly learn a value function. 4 The Learning Process Every edge (connection between neurons) in the neural network is assigned some initial weight (historically small random values have been used; random weight initialization). Contents 1 Introduction 3 2 Regression basics6. Deep Learning falls under the broad class of Articial Intelligence > Machine Learning. Timo Seppalainen Affiliate, Mathematics Department Motion in a random medium, interacting particle systems, large deviation theory. For example, multiplying a (N, C, D) matrix with a (D, K) matrix should produce a (N, C, K) matrix. Saxe, James L. (2015, March). Finally, in a stunning display of technical mastery, and a sign that the mathematical arms race in deep learning is alive and well, Jeff Pennington combines complex analysis, random matrix theory, free probability, and graph morphisms (!) to derive an exact law for the eigenvalues of the Hessian of neural network loss functions, whereas the. That definition is pretty intuitive if you think of a linear function. Deep Learning (DL) is a sort of more complex architecture simulating human brains, based on neural networks begins to apply hyperspectral image classification. We need stronger assumptions about the model to learn useful invariants for vision. MENA applies random matrix theory to conduct microbial analysis and experiments show it is robust to the noise and threshold (Deng et al. Machine Learning Algorithms is for you if you are a machine learning engineer, data engineer, or junior data scientist who wants to advance in the field of predictive analytics and machine learning. Say two nodes in the observed layer are related if they have a common neighbor in the hidden layer to which they are attached via a +1 edge. V(s) Value function of a reinforcement learning agent Wi Adjacency matrix of delay layer i of a GDNN w Setpoint signal of a closed-loop control system y System output signal of a closed-loop control system z Disturbance signal of a closed-loop control system Abbreviations ANN Artiﬁcial neural network APRBS Amplitude modulated pseudo-random. The Center of Mathematical Sciences and Applications will be hosting a working Conference on Applications of Random Matrix Theory to Data Analysis, January 9-13, 2017. This leads nicely into the student exercises, which served to solidify the instructor's teachings and encourage experimentation. “We have not succeeded in answering all our problems. theory, statistical mechanics, data analysis, deep learning Vincent Rivasseau Laboratoire de Physique Th eorique CNRS UMR 8627, Universit e Paris-SudMelonic Non-linear Flows and the Spiked Tensor Model. 5), but unlike in the random feedback case, most hidden unit firing rates remained low. Montr eal (QC), H2C 3J7, Canada Editor: I. theory, statistical mechanics, data analysis, deep learning Vincent Rivasseau Laboratoire de Physique Th eorique CNRS UMR 8627, Universit e Paris-SudMelonic Non-linear Flows and the Spiked Tensor Model. Recently, a significant amount of research efforts have been devoted to this area, greatly advancing graph analyzing techniques. We need stronger assumptions about the model to learn useful invariants for vision. J Pennington, P Worah. words that are used and occur in the same contexts tend to purport similar meanings. Many network embedding algorithms are typically unsupervised algorithms and they can be broadly classified into three groups ex-. edu Department of Mathematics math. We will briefly summarize Linear Regression before implementing it using Tensorflow. The idea is to extract more and more abstract features from input data, learning more abstract representations. We show that for large-size decoupled networks the lowest critical values of the random loss function form a layered structure and they are located in a well-defined band lower-bounded by the. For solving the inverse problem, we propose a new class-specific image reconstruction algorithm. If you like this sort of stuff, take an online class on machine learning or neural networks!. I'm an undergrad in CS with a math minor wondering what math classes to take that are most useful for ML applications. The answers we have found only serve to raise a whole set of new questions. matmul(X, W) is used instead. The test case for our study is the Gram matrix. nonlinear regime—via incorporating a variety of nonlinear transformations. Laura Balzano. MENA applies random matrix theory to conduct microbial analysis and experiments show it is robust to the noise and threshold (Deng et al. In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. I have a situation where I have an increasing list of real numbers $\vec a$ of variable length (generally about 50 numbers but sometimes more). A main obstacle in this direction is that neural networks are nonlinear, which prevents the straightforward utilization of many of the existing mathematical results. Understanding Support Vector Machine Regression Mathematical Formulation of SVM Regression Overview. For tabular data,. Random Matrix Theory and its Innovative Applications 3 Fig. 3 Deep Learning A neural network de nes a mapping y = f(x;w) and learns the value of the parameters w that result in a good t between x and y. in 2011 can be seen as a series of logistic regression models built on different time intervals so as to estimate the probability that the event of interest happened within each interval. It, therefore, encapsulates all the serial correlations (upto the time lag q) within and across all component series. Here we have extended it to support modeling of stochastic or discontinuous functions by adding a noise term. Discover how to train faster, reduce overfitting, and make better predictions with deep learning models in my new book, with 26 step-by-step tutorials and full source code. Deep Learning in NNs is about accurately assigning credit across many such stages. EDU Princeton University, Computer Science Department and Center for Computational Intractability, Princeton 08540, USA Aditya Bhaskara [email protected] Entropy and mutual information in models of deep neural networks. His research topics are in random matrix theory applied to statistics, machine learning, signal processing, and wireless communications. class: center, middle ### W4995 Applied Machine Learning # Neural Networks 04/16/18 Andreas C. This course gives a graduate-level introduction to machine learning and in-depth coverage of new and advanced methods in machine learning, as well as their underlying theory. Nonlinear random matrix theory for deep learning Jeffrey Pennington Google Brain [email protected] Firstly, utilizing the operation and ambient Big Data (BD) of the distribution system, we construct the high-dimension. Combining AI and A deep learning approach for generalized speech animation - Yisong Yue, Applied Random Matrix Theory - Joel Tropp 07/27/15. A Random Matrix Framework for BigData Machine Learning (Groupe Deep Learning, DigiCosme) Romain COUILLET CentraleSup elec, France June, 2017 1/63. Some common activation functions for deep learning are described below. As long as l is sufﬁciently small so that the weights change by only a small amount per learning epoch, we can average (1)-(2) over all P examples. Towards provable learning of polynomial neural networks using low-rank matrix estimation M Soltani, C Hegde International Conference on Artificial Intelligence and Statistics, 1417-1426 , 2018. In the first part, we demonstrate that a deep CNN is able to learn two types of nonlinear inverse problems: 1-amplitude-to-amplitude and 2- amplitude-to-phase. In the recent times, it has been proven that machine learning and deep learning approach to solving a problem gives far better accuracy than other approaches. J Pennington, P Worah. That definition is pretty intuitive if you think of a linear function. Machine(Learning. Requirements: Good coding skill in Matlab or Python, knowledge of the basics of random matrix theory, good understanding of general signal processing and machine learning concepts. Room: Auditorium Hall 150, Center for Data Science, NYU, 60 5th ave. I TensorFlow I CNNs and RNNs { Random embeddings: approximate kernel methods or approximate neural networks. A introductory book that covers many (but not all) the topics we will discuss is the Artificial Intelligence book of Russell and Norvig (AI:AMA) or the Artificial Intelligence book of Poole and Mackworth (you may need these for other classes). Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. Project 10 [Deep Q-Learning for continuous action space] You have implemented a simple Q-learning algorithm using MLP, with discrete action spaces. Research Videos 2019. Expand the equation with sub-indexes with only a couple of weights and features so you can work through the math. Numerical Methods for Deep Learning is nonlinear function ˙: R !R, a matrix K 2Rm n f, and a Random Transformation. 7 Neural networks and deep learning 83. The fourth paradigm: Data-intensive scientific discovery. Learning dynamical systems As we could see before, differential equations are used widely do describe complex continuous processes. Deep Learning for Computer Vision: Optimization (UPC 2016) 1. improved learning of riemannian metrics for exploratory analysis. In this paper we use it to create models that can learn to produce desired output for given input. in 2011 can be seen as a series of logistic regression models built on different time intervals so as to estimate the probability that the event of interest happened within each interval. Description: To appear in Biometrika. theory, statistical mechanics, data analysis, deep learning Vincent Rivasseau Laboratoire de Physique Th eorique CNRS UMR 8627, Universit e Paris-SudMelonic Non-linear Flows and the Spiked Tensor Model. Nonlinear random matrix theory for deep learning. Parallelism in Deep learning of Computer Vision; spectral graph theory, and random matrix theory ; Ben. Deep learning practice: start with overfitting •Ruslan Salakhutdinov(Foundations of Machine Learning Boot Camp @ Simons Institute for the Theory of Computing, January 2017) •(Paraphrased) "First, choose a network architecture large enough such that it is easy to overfit your training data. Posts about random matrix theory written by memming. matmul function needs a matrix and the (X,Y) in the for loop are scalars. However, their massively parametrised non-linear nature makes Bayesian inference for these models analyti-cally and computationally intractable. Learning in the Machine: Recirculation is Random Backpropagation. Aapo Hyv arinen Non-Gaussian Machine Learning: From Linear ICA to Nonlinear ICA parameter called "mixing matrix" • Hidden random deep learning is a. Geometry of Neural Network Loss Surfaces via Random Matrix Theory. •Input correlation to Identity matrix •As t ∞, weights approach the input output correlation. Machine learning is one of the fastest-growing and most exciting fields out there, and deep learning represents its true bleeding edge. class: center, middle ### W4995 Applied Machine Learning # Neural Networks 04/15/19 Andreas C. More speci cally, we focus on a particularly successful unsupervised representation learning approach, by considering the framework of sparse autoencoders5,6, a type of arti cial neural network which employs nonlinear codes and imposes sparsity. A Selective Overview of Deep Learning Jianqing Fan Cong Maz Yiqiao Zhong April 14, 2019 Abstract Deep learning has arguably achieved tremendous success in recent years. Under complex scattering conditions, it is very difficult to capture clear object images hidden behind the media by modelling the inverse problem. Thus, GAN-data are concentrated random vectors and thus an appropriate statistical model of realistic data. Neural Collaborative Filtering (WWW’17) This is another paper that applies deep neural network for collaborative filtering problem. In the Collaborative filtering problem, we want to infer latent representation of users and items from rating matrix. In this course, you’ll develop a clear understanding of the motivation for deep learning, and design intelligent systems that learn from complex and/or large-scale datasets. He has also developed a new framework to begin harnessing the power of random matrix theory in applications with nonlinear dependencies, like deep. Suggested courses for Machine Learning Machine learning aims to study algorithms that can learn from data in order to identify patterns and make predictions. This course will provide an introduction to the theory of statistical learning and practical machine learning algorithms with applications in signal processing and data analysis. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. Geometry of Neural Network Loss Surfaces via Random Matrix Theory ; Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice; Nonlinear random matrix theory for deep learning ; Lecture 8. Algorithms and Complexity Seminars Schedule. In the past decade, machine learning has given us self-driving cars, practical speech recognition. Although neural nets have been around since the 1940’s, they have recently achieved state-of-the-art in a variety of machine learning tasks,. Research Videos 2019. Then we will introduce supervised learning algorithms (deep neural networks, boosting tress, SVM, nearest neighbors) and unsupservised learning algorithms (clustering, dimension reduction). It partially relates to restricted Boltzmann machines (RBMs), which are used within deep belief networks (Hinton, 2005 ; Hinton et al. What are the ingredients making Deep Learning work: I Random matrix theory as a model if few network A feedforward net is a nonlinear function composed of. Ronan∗, Academy of Paris April 1st, 2016 Abstract Google's AI beats a top player at a game of Go. We cover the theory from the ground up: derivation of the solution, and applications to real-world problems.


text:
Printing a handout of a Beamer presentation, the resulting pdf is shrunk!

I use the code below to produce handouts of my presentations.

When I open the resulting pdf with any pdf viewer, it is viewed correctly, but when I print it, it is always printed very small, no matter the program I use (Okular, Evince, Acrobat...). Bonus to anyone that can make the pdf be printed correctly by Okular under Ubuntu! Thanks!

MWE here:

\documentclass[handout]{beamer}
\usepackage{pgfpages}
\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
\setbeameroption{show notes on second screen=bottom}

\setbeamercolor{background canvas}{bg=pink}

\mode<presentation> {
\usetheme{Singapore}
}
\begin{document}

\begin{frame}[plain]
\titlepage
\note{
Notes for the title page
}
\end{frame}

\section*{Outline}
\begin{frame}[plain]
\frametitle{Outline}
\tableofcontents

\note{
}
\end{frame}

EDIT:

If I place the line "\pgfpagesuselayout{2 on 1}" after the line "\setbeameroption{show notes}" I do not experience the printing problem anymore!! However, the slide preview and notes that appear at the bottom belong to the NEXT SLIDE, instead of to the CURRENT SLIDE!! What is going on with that?

• Are you sure that the printer settings aren't the problem? – Juri Robl Jul 28 '14 at 20:10
• It only happens when I generate these kind of beamer handouts, and persists even if I increase "border shrink" to more than the margins in print settings – DaniCee Jul 28 '14 at 20:21
• It seems to work if you use the Acrobat setting "adjust" (or something like that, I don't have an English version). – Juri Robl Jul 28 '14 at 20:29
• Yeah Acrobat gives that option, but the software I use in Ubuntu like Okular or Evince does not... and conceptually, I do not really understand why the resulting pdf is bigger than the printing area and needs to be adjusted, when I specify it should be of a4paper or letterpaper size, and even increasing the borders does not make it being printed correctly – DaniCee Jul 28 '14 at 20:58
• something weird happens @cyberSingularity... when I change the order of the lines "\setbeameroption{show notes}" and \pgfpagesuselayout{2 on 1} I do not experience the printing problem anymore!! However, the slide preview and notes that appear at the bottom belong to the NEXT SLIDE, instead of to the CURRENT SLIDE!! What is going on with that? I have edited the question to reflect that – DaniCee Jul 29 '14 at 0:02

This has nothing to do with TeX, just with your printing properties. The document generated by your above MWE has a size of 128 mm \times 192 mm, so it is much smaller then Din A4. If you print it without scaling you will end up with a small image in the centre of the paper:

If you let your printer do its job and either scale it (e.g. 140% seems to be a good start) or fit to paper, you will get the whole page filled:

Or, instead of fiddling around with the printer options, compile the pdf directly to a Din A4 version, i.e. not interfering with the pgf resizing by omitting on second screen:

\documentclass[handout]{beamer}
\usepackage{pgfpages}
\pgfpagesuselayout{2 on 1}[landscape,a4paper,border shrink=5mm]
\setbeameroption{show notes}

\setbeamercolor{background canvas}{bg=pink}

\mode<presentation> {
\usetheme{Singapore}
}
\begin{document}

\begin{frame}[plain]
\titlepage
\note{
Notes for the title page
}
\end{frame}

\section*{Outline}
\begin{frame}[plain]
\frametitle{Outline}
\tableofcontents

\note{
}
\end{frame}

\section{Frame 1}
\begin{frame}
\frametitle{Frame with notes 1}
FIRST FRAME CONTENTS

\note{
Notes for the first frame
}
\end{frame}

\section{Frame 2}
\begin{frame}
\frametitle{Frame with notes 2}
SECOND FRAME CONTENTS
\note{
Notes for the second frame
}
\end{frame}

\section{Frame 3}
\begin{frame}
\frametitle{Frame with notes 3}
SECOND FRAME CONTENTS
\note{
Notes for the third frame
}
\end{frame}

\end{document}

• Then the question is: Why does the document generated by the MWE have a size of 128 mm x 192 mm when I clearly specify a4paper with 5mm of border?? – DaniCee Jan 29 '17 at 12:21
• @DaniCee See my edit, the on second screen does not play nicely with the resizing. – user36296 Jan 29 '17 at 12:34
• Alright!! "show notes" instead of "show notes on second screen=bottom" does the trick! But why? – DaniCee Jan 29 '17 at 12:37
• @DaniCee My guess, because the show notes on second screen changes the paper size after the pgf layout was already resized the paper? (just a guess, did not look in the code) – user36296 Jan 29 '17 at 12:39
• Woah so much into this thing and the solution was so simple... Thanks! – DaniCee Jan 29 '17 at 12:41

The following solution is really ugly and I hope someone finds a nicer one. I set the papersize explicitly to A4 (the height is half of it, because it consists of two slides):

\documentclass[handout, xcolor={dvipsnames,table}]{beamer}
\usepackage{pgfpages}

% A4 Paper Size and some small additional margin
\paperheight=143.5mm %297mm / 2 - 5mm
\paperwidth=205mm % 210mm -5mm

% border shrink and a4paper doesn't seem to work like I wanted it to..
\pgfpagesuselayout{resize to}[a4paper,border shrink=5mm] % could also use letterpaper
\setbeameroption{show notes on second screen=bottom} % Beamer manual, section 19.3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\setbeamercolor{background canvas}{bg=pink}

\setbeamertemplate{note page}[plain] % Beamer manual, section 19.1
\newlength{\parskipbackup}
\setlength{\parskipbackup}{\parskip}
\newlength{\parindentbackup}
\setlength{\parindentbackup}{\parindent}
\newcommand{\baselinestretchbackup}{\baselinestretch}

\usetemplatenote{

\insertslideintonotes{0.3}%

\rmfamily \scriptsize%
\setlength{\parindent}{1em} \setlength{\parskip}{1ex}%
\renewcommand{\baselinestretch}{1}%

\insertnote%

\setlength{\parskip}{\parskipbackup}%
\setlength{\parindent}{\parindentbackup}%
\renewcommand{\baselinestretch}{\baselinestretchbackup}%
}
\begin{document}

\begin{frame}
Title slide
\note{this is a note}
\end{frame}

\begin{frame}
This is the first slide
\note{this is another note}
\end{frame}

\end{document}

To change the borders just shrink the papersizes.

format: plain


text:
## Tropical Geometry, part 4.

We’re finally at the point where we can provide the first definition of Tropical Geometry, and for the sake of personal historicity, it will be the one that I didn’t particularly like when I learned of it.

Remember that the point of the algebraic geometry, as it’s studied, is that we study a geometric object by studying the functions defined on that object. The two perspectives are equivalent, and you can use tools from one to study the other (and of course, vice versa).

As an aside, this is one way that you can understand what is meant by non-commutative geometry, at least coarsely. One fact about all of the geometric objects that we study is that the rings of functions defined on any of these things are what we call commutative. That is, $f(x)g(x) = g(x)f(x)$ for every pair of functions $f(x), g(x)$. If you think about this, this is really a reflection of the fact that at it’s heart, the functions take values in real or complex numbers (well…), and so since those satisfy $x \cdot y = y\cdot x$, it follows that functions into them do as well.

So where does non-commutative geometry come in? Well, what happens when you study non-commutative rings? What do they represent “geometrically”?

The point in this case is that if we expand generalize the left-hand side of the equivalence

commutative $k$-algebras $\iff$ geometric stuff

then this in some sense should provide something that is a generalization of the right-hand side as well. This is a well trod-upon tactic in mathematics, and provides us with notions such as a stack (note: not the same as a stack in the computer science world!), or derived schemes, or even derived stacks (combine the two).

Anyhow! So I promised that I would talk about Tropical geometry, and how it fits into the picture. Well, here goes.

See, a ring is something that satisfied a collection of proprties (or axioms) which state how we can multiply and add things together. These basically mean that they behave like the familiar integers, real numbers, or whatever—they look like the normal number systems that we’re all familiar with. It turns out that this list of properties is all we really need to build a phenomenally rich geometric world.

So for Tropical geometry, we look at a slightly different starting point. Consider the real numbers, but with the following funny rules for “addition” and “multiplication”:

$x \oplus y = \max\{x, y\}$

and

$x \otimes y = x + y$

Ok, what the hell is this. Multiplication is addition now? Addition is… the maximum? This seems very strange (and it is!) but it turns out that with this bizarre notion of addition and multiplication that we still get a surprising amount of similar properties than normal addition and multiplication have. For example, we still have that

$x \otimes (y \oplus z) = (x \otimes y) \oplus (x \otimes z)$

i.e. the distributive law. We also have a multiplicative identity ($x \otimes 0 = x$ for every $x$), we have multiplicative inverses (since $x \otimes (-x) = 0$, the identity). We even can have an additive identity if we include $-\infty$ in the package. What we can’t have though, is additive inverses and hence no subtraction.

So yeah, weird. Something which satisfies this collection of rules is a semi-ring, and with this in mind, we do exactly what you should be now expecting: Tropical geometry is geometry done using semi-rings.


text:
# No husband can sit next to his wife in this probability question

I have a probability question that reads:

Question:

If 4 married couples are arranged in a row, find the probability that no husband sits next to his wife.

My attempt:

Total outcomes = 8!
Outcomes that all of them sit with their wife: 4!*(4*2!)
Outcomes that one of them sit with their wife: (2!*4)(6!)-(2!*4)(3!*(3*2!)[subtract the ways that remaining couples are together]
Outcomes that two of them sit with their wife: (2!*4C2)(4!)-(2!*4C2)(2!*(2*2!)
Outcomes that three of them sit with their wife: (2!*4C3)(2!)-(2!*4C3)(1!*(2*2!)
Hence no husband sit with wife is 1 -(Outcomes that all of them sit with their wife+ Outcomes that one of them sit with their wife + Outcomes that two of them sit with their wife + Outcomes that three of them sit with their wife)/8!

Am i right? Any easier way?

-
Possible duplicate of math.stackexchange.com/questions/68541/… – Daniel R Aug 8 '13 at 19:00
Different question, though answer there can be used as a component. – André Nicolas Aug 8 '13 at 19:19
@AndréNicolas: Hmm, what's different about them? – Daniel R Aug 8 '13 at 19:44
You are right, I was mistaken. – André Nicolas Aug 8 '13 at 19:53
You can always try ruby: [1,-1,2,-2,4,-4,8,-8].permutation.map{|p| [p[0..-2].zip(p[1..-1]).map{|t| t.inject(:+)}.inject(:*).abs, 1].min}.inject(:+), or try haskell: sum $map (\p -> min 1$ abs $product$ map (uncurry (+)) $zip p$ tail p) $permutations [1,-1,2,-2,4,-4,8,-8]$\ddot\smile$– dtldarek Aug 8 '13 at 20:23 ## 3 Answers I cannot think of anything pleasant. A natural approach is through Inclusion/Exclusion. There are$8!$arrangements. If we can count the bad arrangements, in which at least one couple is together, then the rest is easy. Call the couples A, B, C, D and let$X$be the wife in couple X, and$x$the husband. It is not hard to count the arrangements in which$a$is next to$A$, and similarly for the other$3$couples. If we add these$4$numbers, we will have double-counted, in particular, the arrangements in which couple A and couple B are both together. So we need to subtract$\binom{4}{2}$times the number of arrangements in which couple A and couple B are together. But we have subtracted too much. So we add back$\binom{4}{1}$times the number of arrangements in which couples A, B, and C are together. But we have added back too much, so we must subtract the number of ways for all the couples to be next to each other. Instead of counting, we can apply Inclusion/Exclusion directly to probabilities. It is marginally easier. - It's 8! minus all the unwanted seating arrangements: So we have to calculate the number of unwanted seating arrangements this way: Let A = the unwanted case that couple 1 sits together. Let B = the unwanted case that couple 2 sits together. Let C = the unwanted case that couple 3 sits together. Let D = the unwanted case that couple 4 sits together. So we want N(A or B or C or D) The "sieve" formula for the answer (sometimes called the "inclusion and exclusion" formula is: N(A or B or C or D) = N(A) + N(B) + N(C) + N(D) - N(A&B) - N(A&C) - N(A&D) - N(B&C) - N(B&D) - N(C&D) + N(A&B&C) + N(A&B&D) + N(A&C&D) + N(B&C&D) - N(A&B&C&D) It's easy to see that: N(A) = N(B) = N(C) = N(D) N(A&B) = N(A&C) = N(A&D) = N(B&C) = N(B&D) = N(C&D) N(A&B&C) = N(A&B&D) = N(A&C&D) = N(B&C&D) So the "sieve" formula becomes N(A or B or C or D) = 4N(A) - 6N(A*B) + 4N(A&B&C) - N(A&B&C&D) We calculate N(A) We will use (x,y) to mean that a couple sits in seats #x and #y Choose couple 1's seats 7 ways: (1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8) Choose the way they can sit 2! ways: (husband,wife) or (wife,husband) Seat the other 6 people 6! ways. That's 7×2!×6! = 10080 N(A&B) = 10080 We calculate N(A&B) Couples 1 and 2 can use 4 seats in these 15 ways: {(1,2),(3,4)}, {(1,2),(4,5)}, {(1,2),(5,6)}, {(1,2),(6,7)}, {(1,2),(7,8)}, {(2,3),(4,5)}, {(2,3),(5,6)}, {(2,3),(6,7)}, {(2,3),(7,8)}, {(3,4),(5,6)}, {(3,4),(6,7)}, {(3,4),(7,8)}, {(4,5),(6,7)}, {(4,5),(7,8)}, {(5,6),(7,8)} Choose whether couple 1 is left of couple 2 or vice-versa in 2! ways: Choose the way couple 1 can sit 2! ways: (husband,wife) or (wife,husband) Choose the way couple 2 can sit 2! ways: (husband,wife) or (wife,husband) Seat the remaining 4 people 4! ways. That's 15×2!×2!×2!×4! = 2880 ways. N(A&B) = 2880 We calculate N(A&B&C) Couples 1, 2 and 3 can use 6 seats in these 10 ways: {(1,2),(3,4),(5,6)}, {(1,2),(3,4),(6,7)}, {(1,2),(3,4),(7,8)}, {(1,2),(4,5),(6,7)}, {(1,2),(4,5),(7,8)}, {(1,2),(5,6),(7,8)}, {(2,3),(4,5),(6,7)}, {(2,3),(4,5),(7,8)}, {(2,3),(5,6),(7,8)}, {(3,4),(5,6),(7,8)} We can choose the order of the 3 couples is 3! or 6 ways. Choose the way couple 1 can sit 2! ways: (husband,wife) or (wife,husband) Choose the way couple 2 can sit 2! ways: (husband,wife) or (wife,husband) Choose the way couple 3 can sit 2! ways: (husband,wife) or (wife,husband) Seat the remaining 2 people 2! ways. That's 10×3!×2!×2!×2!×2! = 960 ways. N(A&B&C) = 960 We calculate N(A&B&C&D) (They use all 8 seats) Choose the order of the 4 couples 4! = 24 ways Choose the way couple 1 can sit 2! ways: (husband,wife) or (wife,husband) Choose the way couple 2 can sit 2! ways: (husband,wife) or (wife,husband) Choose the way couple 3 can sit 2! ways: (husband,wife) or (wife,husband) Choose the way couple 4 can sit 2! ways: (husband,wife) or (wife,husband) That's 4!×2!×2!×2!×2! = 384 Substitute in "sieve" formula to get the number of unwanted seating arrangements: N(A or B or C or D) = 4N(A) - 6N(A&B) + 4N(A&B&C) - N(A&B&C&D) N(A or B or C or D) = 4(10080) - 6(2880) + 4(960) - 384 = 26496 Subtract from 8!: 8! - 26496 = 40320 - 26496 = 13824. Answer: 13824 Edwin McCravy - With$n$couples, the probability of no couple sitting together is $$\displaystyle\sum_{i=0}^n (-2)^i {n \choose i}\frac{(2n-i)!}{(2n)!}.$$ This comes from using inclusion-exclusion, and (as I said in the previous question) treating a couple sitting together as a single individual (hence the$(2n-1)!$), though one of each pair can be on the left or right (hence the$2^i$). - I wonder if this is one of many cases which converge to$e^{-1} \approx 0.367879$as$n\$ increases? – Henry Aug 11 '13 at 20:37
It does - see math.stackexchange.com/questions/465318/… – Henry Aug 12 '13 at 23:20


text:
# What is a Poisson bracket

1. Jul 24, 2014

### Staff: Admin

Definition/Summary

In the Hamiltonian formulation of classical mechanics, equations of motion can be expressed very conveniently using Poisson brackets. They are also useful for expressing constraints on changed canonical variables.

They are also related to commutators of operators in quantum mechanics.

Equations

For canonical variables (q,p), the Poisson bracket is defined for functions f and g as
$\{f,g\} = \sum_a \left( \frac{\partial f}{\partial q_a}\frac{\partial g}{\partial p_a} - \frac{\partial f}{\partial p_a}\frac{\partial g}{\partial q_a} \right)$

The equation of motion for quantity f is
$\dot f = \frac{\partial f}{\partial t} + \{f,H\}$

A change of variables from canonical variables (q,p) to canonical variables (Q,P) has these constraints:
$\{Q_i,P_j\} = \delta_{ij} ,\{Q_i,Q_j\} = \{P_i,P_j\} = 0$

Extended explanation

Proof of equation of motion.

Start with
$\frac{df}{dt} = \frac{\partial f}{\partial t} + \sum_a \left( \frac{\partial f}{\partial q_a} \frac{dq_a}{dt} + \frac{\partial f}{\partial p_a} \frac{dp_a}{dt} \right)$

Using Hamilton's equations of motion gives
$\frac{df}{dt} = \frac{\partial f}{\partial t} + \sum_a \left( \frac{\partial f}{\partial q_a} \frac{\partial H}{\partial p_a} - \frac{\partial f}{\partial p_a} \frac{\partial H}{\partial q_a} \right) = \frac{\partial f}{\partial t} + \{f,H\}$

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

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text:
# Specification of panel data

I am trying to find out the best specification for my dataset.

I am trying to probe the effectiveness of the special economic zones in Poland in the meaning of growth of the economy in three similar panel data models for explained variables: a) registered unemployment rate b) GDP per capita c) gross fixed capital formation per capita. The data is for NUTS3 sub-regions. The explanatory variables are: 0-1 for presence of the SEZ in sub-region in year $t$ and a few of the economic variables; yearly frequency, dataset is 2004-2012 for 66 sub-regions.

I have tried fixed and random effects. As for now, I have chosen FE, because of significance and theoretically correct signs. But there are some issues that prevent me from taking it for granted:

1. How to test for autocorrelation and cross-correlation?

2. I have no idea how to test the error term's distribution in Stata, and furthermore if it is not normally distributed, should I care about it much?

3. As I understand from the literature, values of the correlation coefficient between explanatory variables and the error term near -1 or 1 are not bad as a matter of fact; in my case, it's nearly -1 as you can see.

4. Is a mixed model appropriate for my dataset?

I attach the outcome for the model explaining unemployment rate.

Code:

xtreg  st_bezr sse01 wartosc_sr_trw_per_capita zatr_przem_bud podm_gosp_na_10tys_ludn proc_ludn_wiek_prod ludnosc_na_km2, fe

Fixed-effects (within) regression               Number of obs      =       594
Group variable: id                              Number of groups   =        66

R-sq:  within  = 0.4427                         Obs per group: min =         9
between = 0.3479                                        avg =       9.0
overall = 0.2365                                        max =         9

F(6,522)           =     69.10
corr(u_i, Xb)  = -0.9961                        Prob > F           =    0.0000

-------------------------------------------------------------------------------------------
st_bezr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
sse01 |  -1.406066   .4631984    -3.04   0.003    -2.316028   -.4961045
wartosc_sr_trw_per_capita |  -.0000963   .0000166    -5.79   0.000    -.0001289   -.0000636
zatr_przem_bud |  -26.11989   4.992198    -5.23   0.000    -35.92716   -16.31263
podm_gosp_na_10tys_ludn |  -.0201788   .0030788    -6.55   0.000    -.0262273   -.0141304
proc_ludn_wiek_prod |  -229.1996   16.92631   -13.54   0.000    -262.4516   -195.9475
ludnosc_na_km2 |   .0790167   .0120865     6.54   0.000     .0552726    .1027609
_cons |   161.9786   10.76989    15.04   0.000      140.821    183.1363
--------------------------+----------------------------------------------------------------
sigma_u |  53.986519
sigma_e |  2.5446248
rho |  .99778327   (fraction of variance due to u_i)
-------------------------------------------------------------------------------------------
F test that all u_i=0:     F(65, 522) =    27.09             Prob > F = 0.0000

For the Stata commands in this answer let me collect your variables in a local:
local xlist sse01 wartosc_sr_trw_per_capita zatr_przem_bud podm_gosp_na_10tys_ludn proc_ludn_wiek_prod ludnosc_na_km2
So now you can always call all the variables with `xlist'

1) There are two commands that you can use after your fixed effects regression.

• xttest2 performs a Breusch-Pagan LM test with the null hypothesis of no dependence between the residuals. This is a test for contemporaneous correlation. Not rejecting the null means that the test did not detect any cross-sectional dependence in your residuals.
• xttest3 performs a modified version of the Wald test for groupwise heteroscedasticity. The null hypothesis is homoscedasticity.

You can install both commands by typing ssc instal xttest2 and ssc instal xttest3. If you detect correlations between your residuals you can correct for this with the robust option:
xtreg st_bezr 'xlist', fe robust

To test for autocorrelation you can apply a Lagrange Multiplier test via xtserial:
xtserial st_bezr 'xlist'
The null hypothesis is no serial correlation. To correct for both serial correlation and heteroscedasticity you can use the cluster option with your id variable:
xtreg st_bezr 'xlist', fe cluster(id)

2) For the normality test for the residuals: you can obtain the residuals via the predict command predict res, e after your fixed effects regression. For visual inspection you can use:

• kdensity res, normal (plots the distribution of the residuals and compares it to a normal)
• pnorm res (plots a standardized normal probability plot)
• qnorm res (plots the quantiles of the residuals against the quantiles of a normal distribution)

With pnorm you can see if there is non-normality in the middle of the distribution and qnorm shows you any non-normality in the tails. A formal test can be obtained by swilk res. The null hypothesis is that the residuals are normally distributed. Generally, non-normality is not a too big concern but it matters for inference. You can again correct for this with the robust option.

3) Having corr(u_i, Xb) = -0.9961 means that the fixed effects are strongly correlated with your explanatory variables, so you did well by controlling for these fixed effects. A strong correlation of this type usually indicates that pooled OLS or random effects will not be suitable for your purpose because both of these models assume that the correlation between $u_i$ and $X\beta$ is zero.

4) Generally yes but it depends what you want to estimate or how you can treat your data, i.e. whether your variables are random variables or not. Here is an excellent explanation for the difference between mixed effects and panel data models by @mpiktas which will surely help you.


text:
# Complex analysis problem

• Sep 19th 2013, 09:24 PM
Cesc1
Complex analysis problem
Hi! I need some help with the following problem:

Let f(z) be analytic in |z|< 1, and f(0) = 0. Define F(z)=f(z)/z for every z in 0<|z|<1. What value can we give to F(0) for F(z) to be analytic in |z|<1? (Hint: Apply Cauchy's integral formula for derivatives to F(z) in |z|=r < 1. Then show that the resulting function, analytic in |z|< 1, concurs with F in 0<|z|<r. Use partial fractions.)

So, Cauchy's integral formula for derivatives states that

$f^{(n)}(a)=\frac{n!}{2\pi i}\int_{\gamma}\frac{f(z)}{(z-a)^{n+1}}dz, n=1,2...$

But I'm not sure how to apply it to F(z).

Any help will be appreciated.
• Sep 20th 2013, 09:25 PM
chiro
Re: Complex analysis problem
Hey Cesc1.

Setting a = 0 gives f(z) / z^(n+1) given the line integral region of y (gamma). If a function is analytic in the complex plane then all of its derivatives exist by default in the region that the function is itself analytic.

This means if f(z) is analytic in a region then its derivatives are also analytic in the same region (i.e. the f^n(a)).

So you need to break up f(z)/z^(n+1) into f(z)/z and 1/z^(n+1) and show that F(z) is analytic.

You have already shown that f^n(0) exists for all values of n (since 0 is in the region |z| < 1) so you need to use F(z) * 1/z^n in partial fraction decomposition within the integral.


text:
# Information Security and Cryptography Research Group

## Conditional Differential Cryptanalysis of NLFSR-Based Cryptosystems

### Simon Knellwolf, Willi Meier, and María Naya-Plasencia

ASIACRYPT, Lecture Notes in Computer Science, Springer, vol. 6477, pp. 130–145, 2010.

Non-linear feedback shift registers are widely used in lightweight cryptographic primitives. For such constructions we propose a general analysis technique based on differential cryptanalysis. The essential idea is to identify conditions on the internal state to obtain a deterministic differential characteristic for a large number of rounds. Depending on whether these conditions involve public variables only, or also key variables, we derive distinguishing and partial key recovery attacks. We apply these methods to analyse the security of the eSTREAM finalist Grain v1 as well as the block cipher family KATAN/KTANTAN. This allows us to distinguish Grain v1 reduced to 104 of its 160 rounds and to recover some information on the key. The technique naturally extends to higher order differentials and enables us to distinguish Grain-128 up to 215 of its 256 rounds and to recover parts of the key up to 213 rounds. All results are the best known thus far and are achieved by experiments in practical time.

## BibTeX Citation

@inproceedings{KnMePl10,
author       = {Simon Knellwolf and Willi Meier and María Naya-Plasencia},
title        = {Conditional Differential Cryptanalysis of NLFSR-Based Cryptosystems},
editor       = {Masayuki Abe},
booktitle    = {ASIACRYPT},
pages        = 130--145,
series       = {Lecture Notes in Computer Science},
volume       = 6477,
year         = 2010,
publisher    = {Springer},
}


text:
# Is (1+sin beta)/(cos beta) equivalent to (cos^2 beta)/(cos beta- cos beta sin beta)

### Other questions on the subject: Mathematics

Mathematics, 21.06.2019 16:20, sighgabbie
Find the value of x for which abcd must be a parallelogram -9-6x x-30
The border line of the linear inequality 4x + 7y < 5 is dotted true or false?


text:
# Finding self-kissing points on a plane curve?

I have a curve in the complex plane given by $$f(t) = \sum_k r_k\exp(2\pi\mathrm{i}(t+\varphi_k)p_k).$$ Some of the parameters are specially chosen: $r_k>0$, $\sum_k r_k=1$, $p_k\in\mathbb{Z}$, $\mathrm{gcd}(p\ldots)=1$, $\varphi_k\in\mathbb{R}$. These restrictions are probably not very important for this question.

When I plot the curve, it will sometimes have some self-kissing points: points $t_1$ and $t_2$ where $f(t_1)=f(t_2)$ and $f'(t_1)\parallel f'(t_2)$. A self-kissing point doesn't have to be a self-crossing: at the self-intersection with parallel tangent vectors, the normal vectors can be pointing the opposite ways.

Can anybody suggest a good method for finding such points? I can't seem to think of a real-valued function that changes sign at the self-kissing point. I would like to be able to find these points once every frame in an animation, so efficiency can be an issue. If you can suggest a reference for this, that helps too.

Edit Direct root finding doesn't work because such a point would be a multiple root and the curve might come close to touching itself without doing so to numerical accuracy. Optimizing a function like $|f(t_1)-f(t_2)|^2$ doesn't work: it has many spurious critical points. Splitting the interval $[0,1]$ into $N$ parts and treating the curve as approximately quadratic inside each interval also doesn't work because there are two variables in $f(t_1)-f(t_2)$, and so requires $O(N^2)$ work, which is too much. (Although when looking for self-intersection points instead, this can work because of efficient line segment intersection algorithms.)

• I was going to answer and suggest all of the things that you've ruled out :P There's a famous paper on computing self-intersections of Bezier curves. Near as I can tell, the author uses the $O(N^2)$ approach but has some clever and simple heuristics for ruling out regions where the curve can't intersect, thus narrowing the search space. Your problem sounds much harder though. Feb 2 '15 at 23:28
• @DanielShapero Thanks for looking at it! I ruled out $O(N^2)$ algorithms mostly because I want to draw an animation and I need it to be quick. I was hoping for some inexact simplifications that could make it easier. Feb 2 '15 at 23:36
• A remark: you write ''"A self-kissing point doesn't have to be a self-crossing: at the self-intersection with parallel tangent vectors, the normal vectors can be pointing the opposite ways."'', the normals can be equal at a self-kissing pt without intersection, e.g. if curve is spiralling.
– Joce
Mar 17 '16 at 13:39
• @Joce Good point. Mar 17 '16 at 16:49

There exists a clever algorithm for doing this, based on ideas due to Cayley, using the Bézout matrix. I learned this from [1], which describes an algorithm for a generic version of this problem.

If we want to find all the critical points of the function $\|f(s)-f(t)\|^2$ (viewing $f$ as a function $\mathbb{R}\to\mathbb{R}^2$), a sufficient condition is that $$P_s(t) := (f(s)-f(t))^\top f'(s) = 0, \qquad Q_s(t) := (f(s)-f(t))^\top f'(t) = 0,$$ which is a pair of simultaneous trigonometric equations in $s$ and $t$.

Form the Bézout matrix as the matrix with entries $B_{ij}(s)$ defined by the expansion $$\frac{P_s(x)Q_s(y) - P_s(y)Q_s(x)}{x-y} = \sum_{i,j} B_{ij}(s) \phi_i(x) \phi_j(y).$$ Here $\phi_i$ are a suitable basis for the function $f,P,Q$. Typically they are either monomials, or Chebyshev polynomials, but presumably can also be elementary trigonometric functions.

Those values of $s$ for which the exists a solution $t$ to the pair of equations $P_s(t) = Q_s(t) = 0$ cause the matrix $B_{ij}(s)$ to be singular. In other words, if we expand the matrix $B_{ij}(s)$ in the basis $\phi_i(s)$ as $$B_{ij}(s) = \sum_k B_{ij}^{(k)}\phi_k(s),$$ for a solution $s$ to exist it is necessary that $$\det B_{ij}(s) = 0,$$ which is a polynomial eigenvalue problem (the polynomial's coefficients in the basis $\phi$ are the matrices $B_{ij}^{(k)}$). This is a well-known problem, and has standard, although possibly difficult algorithms, such as the polyeig function in Matlab. In particular it can turned into a linear eigenvalue problem for $s$ by constructing the block-companion matrix (if $\phi$ are monomials) or the block-colleague matrix (if $\phi$ are Chebyshev), and solving that linearized version of the PEP.

It may also be helpful to partition the $(s,t)$-square into some number of subsquares and use the Bézout matrix approach on individual subsquares, as this can substantially lower the degree of the polynomial eigenvalue problem and its linearized version.

References

1. Computing the common zeros of two bivariate functions via Bézout resultants by Yuji Nakatsukasa, Vanni Noferini, Alex Townsend


text:
# Re: Comparing files in Word

From: Paul <pmatteucci_at_gmail.com>
Date: Mon, 14 Jul 2008 08:54:12 -0700 (PDT)

Thanks a lot,
I see what you meen when you say it does the diff the wrong way round.
Does anyone know a way to solve this tho? Perhaps changind something
in Word?
Thanks a lot for the help guys.
Ciao
Paul

On Jul 11, 4:03 pm, Stefan Küng <tortoise..._at_gmail.com> wrote:
> Paul wrote:
> > Hi, I was amazed to see that if you click on "Diff with previous
> > version" of a Word doc, that it opens up the doc in Word and launches
> > a comparison between the latest two version.
> > This is a wonderful feature, specially when you're using Subversion
> > for normal docs (not code).
> > There is a glitch tho, the revision considere Old is the latest one
> > and the one considered New is in fact the older revision. In other
> > words, it's telling me that 3 has replaced 4 and not the other way
> > round.
> > Am I doing something wrong or is this a problem with the system?
>
> Actually, if I'm not completely mistaken, the diff is correct.
> Word just shows the diff 'wrong'. Ok, I admit that isn't a good
> explanation :)
>
> If you look at the 'merged' document Word shows you, you will see that
> it looks exactly like the 'new' file, but lines which were added in the
> new file are marked red as if they were removed, not added.
>
> If you would make Word show the diff in reverse, it would actually show
> the document as it was in the old version, not the new one.
>
> You can test this yourself if you like. The file
> C:\Program Files\TortoiseSVN\Diff-Scripts\diff-doc.js
> it the one which calls the word diff. If you change the lines 44/45 from:
>
> sBaseDoc = objArgs(0);
> sNewDoc = objArgs(1);
>
> to
>
> sBaseDoc = objArgs(1);
> sNewDoc = objArgs(0);
>
> you will see what I mean if you repeat the diff.
>
> Stefan
> --
>         ___
>    oo  // \\      "De Chelonian Mobile"
>   (_,\/ \_/ \     TortoiseSVN
>     \ \_/_\_/>    The coolest Interface to (Sub)Version Control
>     /_/   \_\    http://tortoisesvn.net
>
>  signature.asc


text:
Question

# Find both first partial derivatives. z=\frac{\ln x}{y}

Derivatives
Find both first partial derivatives. $$\displaystyle{z}={\frac{{{\ln{{x}}}}}{{{y}}}}$$

2021-02-16
Step 1
Given that
$$\displaystyle{z}={\frac{{{\ln{{x}}}}}{{{y}}}}$$
Step 2
On partial differentiating z with respect to x and y,
$$\displaystyle{\frac{{{p}{a}{r}{t}{i}{c}{a}{l}{z}}}{{{p}{a}{r}{t}{i}{c}{a}{l}{x}}}}={\frac{{{p}{a}{r}{t}{i}{c}{a}{l}}}{{{p}{a}{r}{t}{i}{c}{a}{l}{x}}}}{\left({\frac{{{\ln{{x}}}}}{{{y}}}}\right)}$$
$$\displaystyle={\frac{{{1}}}{{{y}}}}{\frac{{{p}{a}{r}{t}{i}{c}{a}{l}}}{{{p}{a}{r}{t}{i}{c}{a}{l}{x}}}}{\left({\ln{{x}}}\right)}$$
$$\displaystyle={\frac{{{1}}}{{{y}}}}{\left({\frac{{{1}}}{{{x}}}}\right)}$$
$$\displaystyle={\frac{{{1}}}{{{x}{y}}}}$$
And
$$\displaystyle{\frac{{{p}{a}{r}{t}{i}{c}{a}{l}{z}}}{{{p}{a}{r}{t}{i}{c}{a}{l}{y}}}}={\frac{{{p}{a}{r}{t}{i}{c}{a}{l}}}{{{p}{a}{r}{t}{i}{c}{a}{l}{y}}}}{\left({\frac{{{\ln{{x}}}}}{{{y}}}}\right)}$$
$$\displaystyle={\ln{{x}}}{\frac{{{p}{a}{r}{t}{i}{c}{a}{l}}}{{{p}{a}{r}{t}{i}{c}{a}{l}{x}}}}{\left({\frac{{{1}}}{{{y}}}}\right)}$$
$$\displaystyle={\ln{{x}}}{\left(-{\frac{{{1}}}{{{y}^{{{2}}}}}}\right)}$$
$$\displaystyle=-{\frac{{{\ln{{x}}}}}{{{y}^{{{2}}}}}}$$
Step 3
Hence first partial derivative of $$\displaystyle{z}={\frac{{{\ln{{x}}}}}{{{y}}}}{i}{s}{\frac{{{p}{a}{r}{t}{i}{c}{a}{l}{z}}}{{{p}{a}{r}{t}{i}{c}{a}{l}{x}}}}={\frac{{{1}}}{{{x}{y}}}}{\quad\text{and}\quad}{\frac{{{p}{a}{r}{t}{i}{c}{a}{l}{z}}}{{{p}{a}{r}{t}{i}{c}{a}{l}{y}}}}=-{\frac{{{\ln{{x}}}}}{{{y}^{{{2}}}}}}$$


text:
• # question_answer When forces ${{F}_{1}},{{F}_{2}},{{F}_{3}}$ are acting on a particle of mass m such that ${{F}_{1}}$ and ${{F}_{2}}$ are mutually perpendicular, then the particle remains stationary. If the force ${{F}_{1}}$ is now removed, then the acceleration of the particle is   [AIEEE 2002] A) ${{F}_{1}}/m$ B) ${{F}_{2}}{{F}_{3}}/m{{F}_{1}}$ C) $({{F}_{2}}-{{F}_{3}})/m$ D) ${{F}_{2}}/m$

 [a] In this question, we have to think about that "a body or system to be in equilibrium, net force acting on it should be zero. The particle is remains stationary under the action of three forces ${{F}_{1}},{{F}_{2}}$ and ${{F}_{3}}$, it means resultant force is zero. ${{F}_{1}}=-({{F}_{2}}+{{F}_{3}})$ Since, in second case, ${{F}_{1}}$ is removed (in terms of magnitude we are talking now), the forces acting are ${{F}_{2}}$ and ${{F}_{3}}$ the resultant of which has the magnitude as ${{F}_{1}}$, so acceleration of particle is $\frac{{{F}_{1}}}{m}$ in the direction opposite to that of ${{F}_{1}}$.


text:
ON PERIODIC P-CONTINUED FRACTION HAVING PERIOD LENGTH ONE

Title & Authors
ON PERIODIC P-CONTINUED FRACTION HAVING PERIOD LENGTH ONE
Chandoul, Amara; Amar, Hela Ben; Mkaouar, Mohamed;

Abstract
The aim of this paper is to prove that every quadratic formal power series $\small{{\omega}}$ can be expressed as a periodic non-simple continued fraction having period length one.
Keywords
continued fraction;Laurent series;finite;
Language
English
Cited by
References
1.
E. B. Burger, J. Gell-Redman, R. Kravitz, D. Walton, and N. Yates, Shrinking the period lengths of continued fractions while still capturing convergents, J. Number Theory 128 (2008), no. 1, 144-153.

2.
E. B. Van Vleck, On the convergence of the continued fraction of Gauss and other con- tinued fractions, Ann. of Math. (2) 3 (1901/02), no. 1-4, 1-18.

3.
E. B. Van Vleck, On the convergence of algebraic continued fractions whose coefficients have lim- iting values, Trans. Amer. Math. Soc. 5 (1904), no. 3, 253-262.

4.
H. Waadeland, Tales about tails, Proc. Amer. Math. Soc. 90 (1984), no. 1, 57-64.


text:
# Construct natural cubic splines for the data:f(x0.10.62049958 ~0.28398668 0.00660095 0.24842440Include a plot of the spline ovCr the interval [0, 1,0.4]

###### Question:

Construct natural cubic splines for the data: f(x 0.1 0.62049958 ~0.28398668 0.00660095 0.24842440 Include a plot of the spline ovCr the interval [0, 1,0.4] and use vour spline to estimate f (0.25).

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##### A +2.2 nC charge is at the origin and a -4.0 nC charge is at x =1.5 cm.a.) At what xx-coordinate could you place aproton so that it would experience no net force?b.) Would the net force be zero for an electron placed atthe same position?
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Weller Company's budgeted unit sales for the upcoming fiscal year are provided below: The company’s variable selling and administrative expense per unit is $1.80. Fixed selling and administrative expenses include advertising expenses of$12,000 per quarter, executive salaries of \$38,000 pe...


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# Morphisms of cotensors

Let $\mathcal{V}$ be a closed monoidal category, and $\mathscr{C}$ be a category enriched over $\mathcal{V}$. One says that the power or cotensor of an objec $A \in \mathscr{C}$ by an object $U \in \mathcal{V}$ is an object $A^U$ of $\mathscr{C}$ with a natural isomorphism $$\mathscr{C}(B, A^U) \cong \mathcal{V}(U, \mathscr{C}(B, A)),$$ where $B \in \mathscr{C}$, and $\mathcal{V}(-, -)$ is the inner hom of $\mathcal{V}$.

Riehl and Verity, in their third of a series of papers towards an axiomatic theory of $(\infty, 1)$-categories, define an $\infty$-cosmos to be some sort of category, the details of which do not really matter to my question. My question does, however, regard one certain part of their definition. They require that the class of isofibrations be closed under forming the Leibniz cotensor of two maps, and I am having a bit of difficulty seeing how this map is defined. Specifically, the Leibniz cotensor of a map $p: E \to B$ in $\mathscr{C}$ by a map $i: U \to V$ in $\mathcal{V}$ is a map

$$i\, \widehat{\pitchfork}\, p: E^V \to E^U \times_{B^U} B^V.$$

Certainly, this map would follow by the universal property of pullbacks given maps $E^V \to E^U$ and $E^V \to B^V$ that satisfy the requisite commutativity condition. My question is as follows:

How does one define these maps? Indeed, a good definition of these maps is necessary even to form the pullback.

Any insight would be greatly appreciated.

It's pretty simple, really: using the universal property of $A^U$, one may define maps $E^i: E^V \to E^U$ and $p^V: E^V \to B^V$, and similarly maps $B^i: B^V \to B^U$ and $p^U: E^U \to B^U$.

For example, if $I$ is the monoidal unit of $\mathcal{V}$, one has a canonical arrow $\theta_{E, V}: I \to \mathcal{V}(V, \mathcal{C}(E^V, E))$ mated to the name of the identity $I \to \mathcal{C}(E^V, E^V)$, and then one may form the composite

$$I \stackrel{\theta_{E, V}}{\to} \mathcal{V}(V, \mathcal{C}(E^V, E)) \stackrel{\mathcal{V}(i, 1)}{\to} \mathcal{V}(U, \mathcal{C}(E^V, E)) \cong \mathcal{C}(E^V, E^U)$$

which names the map $E^i: E^V \to E^U$. The other arrows mentioned above may be constructed by similar considerations.

It is not hard to see by the yoga of enriched universal mapping properties that the cotensor thus becomes a $\mathcal{V}$-functor $\mathcal{V}^{op} \otimes \mathcal{C} \to \mathcal{C}$ in two variables, meaning in particular that there is a commutative square

$$\begin{array}{ccc} E^V & \stackrel{p^V}{\to} & B^V \\ E^i \downarrow & & \downarrow B^i \\ E^U & \underset{p^U}{\to} & B^U \end{array}$$

and so there is an induced map $E^V \to E^U \times_{B^U} B^V$ which is this thing you are asking about.

• Thank you for your response. Would you be able to point me toward a reference on enriched category theory? I apologize if this MO isn't the right place for this; I didn't think my question would get any attention on math.se. – Reuben Stern Feb 27 '17 at 16:19
• The classic reference I know of is Kelly's book Basic Concepts of Enriched Category Theory, available as a TAC reprint: tac.mta.ca/tac/reprints/articles/10/tr10abs.html. This certainly covers the types of issues encountered here. (By the way, you're probably right that this question would sink into deep layers of the stack at Math.SE.) – Todd Trimble Feb 27 '17 at 18:14


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Intuition behind isomorphism of algebraic varieties

Let $X \subset \mathbb A^n$, $W \subset \Bbb A^m$ be two algebraic sets. A function $\phi:X \rightarrow W$ is a morphism if there exist $m$ polynomial functions $f_1,\ldots,f_m \in K[X]$ such that for every $x \in X$ one has $\phi(x)=(f_1(x),\ldots,f_m(x)) \in W$.

We say two algebraic sets are isomorphic if there exists a bijective morphism between them whose inverse is also a morphism. Now, now my question is what does ismorphism of two algebraic sets mean intuitively? The general meaning of isomorphism is that they are "equivalent".So, in what geometrical sense two isomorphic algebraic sets can be considered "equivalent"? In other words.what geometrical properties are same for two isomorphic algebraic sets? Also, can a Zariski open set in one Zariski topology be isomorphic to a Zariski closed in another Zariski topology?

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Say $V$ and $W$ are isomorphic algebraic sets. Intuitively, it should mean that the set of equations defining $V$ can be transformed into the set of equations defining $W$. For instance, consider two curves $f(x,y) =0$ and $g(x,y)=0$ in $\mathbf{A}^2$. These being isomorphic means that there is an isomorphism $k[x,y]/(f)\to k[x,y]/(g)$. –  Harry Dec 22 '12 at 16:02
To answer your last question, observe that $\mathbb{A}^1$ is open in itself, but is isomorphic to quite a few Zariski-closed subsets of $\mathbb{A}^2$ -- for example, the $x$-axis. –  Paul VanKoughnett Dec 22 '12 at 16:49
are there any other examples where one set is just open but not closed? –  Mohan Dec 22 '12 at 16:52
Yes, I've edited it now. –  Mohan Dec 22 '12 at 19:14
The final part of the question, as amplified in Mohan's comment, has an affirmative answer. $\mathbb A^1-\{0\}$ is open and not closed in $\mathbb A^1$ but its image, under the isomorphic embedding $x\mapsto(x,0)$ is closed and not open in $\mathbb A^2-\{(0,0)\}$.
As for the earlier parts of the question, isomorphism means that the two algebraic sets are not merely in one-to-one correspondence any old way but rather in a way that respects the algebraic structure. By "the algebraic structure" here, I mean the information telling which functions on the algebraic set are polynomials. The definition of isomorphism ensures that, if you have a polynomial function on one of the two isomorphic algebraic sets, then transporting that function to the other set, by composing with the bijection $\phi$ or its inverse, produces a polynomial function there. (Both my explanation and the definition in the question seem to need some caution in finite characteristic, where a polynomial isn't determined by its values.)


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# Revision history [back]

I tried 3D Localizing and mapping using hdl_graph_slam. Just using the Lidar scan and odometry and I was able to localize the sensor.

Results Picture

I tried 3D Localizing Localization and mapping using hdl_graph_slam. Just using the Lidar scan and odometry and I was able to localize the sensor.

Results Picture

 3 No.3 Revision osilva 1351 ●148 ●14 https://github.com/rob...

I tried 3D Localization and mapping using hdl_graph_slam. Just using the Lidar scan and odometry and I was able to localize the sensor.

 4 No.4 Revision osilva 1351 ●148 ●14 https://github.com/rob...

I tried 3D Localization and mapping using hdl_graph_slam. Just using the Lidar scan and odometry and I was able to localize the sensor.