# Difference between continuity and uniform continuity

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1. May 7, 2015

### Yunjia

I noticed that uniform continuity is defined regardless of the choice of the value of independent variable, reflecting a function's property on an interval. However, if on a continuous interval, the function is continuous on every point. It seems that the function on that interval must be uniformly continuous. Is this correct? Is there a counterexample for the statement?

2. May 7, 2015

### jbunniii

What you said is almost true. If, on a closed, bounded interval, a function is continuous at every point, then the function is uniformly continuous on that interval.

Counterexample on a non-closed interval: $f(x) = 1/x$ on the interval $(0,1)$.

Counterexample on a closed but unbounded interval: $f(x) = x^2$ on the interval $[0,\infty)$.

3. May 8, 2015

### Svein

Even more general: If a function is continuous at every point in a compact set, it is uniformly continuous on that set.
The proof is simple: Let the compact set be K and take an ε>0. Since the function is continuous at every point x in the set, there is a δx for every x∈K such that |f(w)-f(x)|<ε for every w in <x-δx, x+δx>. Let Ox=<x-δx, x+δx>, then K is contained in $\bigcup_{x\in K}O_{x}$. Since K is compact, it is contained in the union of a finite number of the Ox, say $\bigcup_{n=1}^{N}O_{n}$. Take δ to be the minimum of the δn and |f(w)-f(x)|<ε whenever |w-x|<δ.

4. May 8, 2015

### jbunniii

And here's the proof that a closed, bounded interval is compact, so you can apply Svein's proof.

Let $[a,b]$ be a closed, bounded interval, and let $\mathcal{U}$ be any collection of open sets which covers $[a,b]$. Let $S$ be the set of all $x \in [a,b]$ such that $[a,x]$ can be covered by finitely many of the sets in $\mathcal{U}$. Clearly $a \in S$. This means that $S$ is nonempty and is bounded above (by $b$), so it has a supremum, call it $c$. Since $c \in [a,b]$, it is contained in some $U_c \in \mathcal{U}$, hence there is some interval $(c - \epsilon, c + \epsilon)$ contained in $U_c$. Since only finitely many sets from $\mathcal{U}$ are needed to cover $[a,c - \epsilon/2]$, those sets along with $U_c$ form a finite cover of $[a,c+\epsilon/2]$. This shows that $c \in S$ and moreover, that $c$ cannot be less than $b$. Therefore $c=b$, so all of $[a,b]$ can be covered by finitely many sets in $\mathcal{U}$.

Last edited: May 8, 2015
5. May 8, 2015

### Svein

Excellent! And, of course, since one closed and bounded interval is compact, the union of a finite number of closed and bounded intervals is again compact.

6. May 8, 2015

### jbunniii

I don't think this will work if $w$ and $x$ are not contained in the same $O_n$. I think you need to take $O_x = (x - \delta_x/2, x + \delta_x/2)$ and $\delta$ to be $\min\{\delta_n/2\}$ in order to ensure that $|w-x| < \delta$ implies $|f(w) - f(x)| < \epsilon$.

7. May 8, 2015

### Svein

Possibly. I need to think about that. The crux of the matter is that there is a finite number of intervals that cover K, which means that the minimum of the (finite number of) δ's exist and is greater than 0.

8. May 8, 2015

### HallsofIvy

Here is the fundamental difference between "continuous" and "uniformly continuous":

A function is said to be continuous at a point, x= a, if and only if, given any $\epsilon> 0$ there exist $\delta> 0$ such that if $|x- a|< \delta$ then $|f(x)- f(a)|<\epsilon$.

A function is said to be continuous on a set, A, if and only if, given any $\epsilon> 0$ there exist $\delta> 0$ such that if $|x- a|< \delta$ then $|f(x)- f(a)|<\epsilon$ for all a in set A.

That is, a function is uniformly continuous on a set A if and only if it is continuous at every point in A and, given $\epsilon> 0$ the same $\delta$ can be used for ever point in A.

9. May 8, 2015

### Svein

Agree. I was sloppy and overlooked the simple fact that that the length of the interval Ox is 2⋅δx.

10. May 13, 2015

### Yunjia

The defition of uniformly continuous is perplexing for me. Is it possible for you to illustrate it with a proof about a function which is not uniformly continuous but continuous on a set so that I can compare the two? Thank you.

11. May 14, 2015

### Svein

OK. Take the function $f(x)=\frac{1}{x}$ on the open interval <0, 1>. It is continuous (and even differentiable) on that interval, but not uniformly continuous.

12. May 17, 2015

### Svein

After using pencil an paper for a bit, I came to the conclusion that I should have used ε/2 and δ/2. But - I recall a proof in "Complex analysis in several variables" that ended up in "... less than 10000ε, which is small when ε is small".

13. May 17, 2015

### HallsofIvy

First note an important point about "uniform continuity" and "continuity" you may have overlooked: we define "continuous" at a single point and then say that a function is "continuous on set A" if and only if it is continuous at every point on A. We define "uniformly continuous" only on a set, not at a single points of a set.

A function is uniformly continuous on any closed set on which it is continuous and so on any set contained in a closed set on which it is continuous.
To give an example of a function that is continuous but not uniformly continuous, we need to look at f(x)= 1/x on the set (0, 1).

To show that it is continuous on (0, 1), let a be a point in (0, 1) and look at $|f(x)- f(a)|= |1/x- 1/a|= |a/ax- x/ax|= |(a- x)/ax|= |x- a|/ax< \epsilon$.
We need to find a number, $\delta> 0$ such that if $|x- a|< \delta$, then $|f(x)- f(a)|> \epsilon$. We already have $|x- a| ax\epsilon$ so we need an upper bound on ax. If we start by requiring that $\delta< a/2$ then $|x- a|< \delta< a/2$ so that $-a/2< x- a< a/2[itex] or [itex]a/2< x< 3a/2$ so an upper bound on ax is $3a^2/2$. If $|x- a|< a/2$ and $|x- a|< 3a/2$ then $|f(x)- f(a)|< |x- a|/ax< |x- a|/(3a^2/2)= 2|x- a|/3a^2$ which will be less than $\epsilon$ as long as $|x- a|< 3a^2\epsilon/2$

So we can take $\delta$ to be the smaller of $a/2$ and $3a^2/\epsilon$. Therefore, 1/x is continuous at any point a in (0, 1) so continuous on (0, 1).

Now the point is that this $\delta$ depends on a. It is a decreasing function of a and will go to 0 as a goes to 0. That's fine for "continuity" but for uniform continuity we must be able to use the same $\delta> 0$ for a given $\epsilon$ no matter what the "a" is and we cannot do that. If the problem were to prove uniform continuity on the set [p, 1), which is contained in the closed set [p, 1], we could use, for any a in that set, the $\delta$ that we get for a= p, the smallest x value in the set. But with (0, 1), the smallest $\delta$ is 0 which we cannot use since we must have $\delta> 0$.


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In general, statistical problems have to do with the estimation of some characteristic derived from data - this can be a point estimate, an interval, or an entire function. Linear Algebra: Foundations to frontiers – edx. Linear Algebra for Machine Learning Discover the Mathematical Language of Data in Python. data in homework problems. Machine learning. Anyone can view the notebooks online by clicking on the links in the readme Table of Contents. In this day, we are going to make the dirty work. In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. In other cases, functions are provided to show or. Linear algebra (Systems of linear equations, least-square) [Matrix cookbook] 4: 9/16/2019: Linear Algebra (Vector space, linear independence) 9/18/2019: Linear algebra (Eigendecomposition and matrix factorization) Homework 1 (extended) 5: 9/23/2019: Linear algebra (Eigendecomposition and matrix factorization) 9/25/2019: Linear algebra (Best fit. Acquiring these skills can boost your ability to understand and apply various data science algorithms. The basic mathematics prerequisites for understanding Machine Learning are Calculus-I,II,III, Linear Algebra, and, Probability and Statistics. Making statements based on opinion; back them up with references or personal experience. Randomized numerical linear algebra. It lacks the ability of distributed linear algebra computation in its local interactive shell. You can use it as a main text, as a supplement, or for independent study. NVIDIA CUDA-X GPU-Accelerated Libraries NVIDIA® CUDA-X, built on top of NVIDIA CUDA®, is a collection of libraries, tools, and technologies that deliver dramatically higher performance—compared to CPU-only alternatives— across multiple application domains, from artificial intelligence (AI) to high performance computing (HPC). Lek-Heng Lim. taco is versatile. However, I think that the chapter on linear algebra from the book is a bit tough for beginners. A few weeks ago, I wrote about how and why I was learning Machine Learning, mainly through Andrew Ng’s Coursera course. scala> val a = DenseMatrix. This code has been posted to GitHub under a MIT license, so feel free to modify and deal with code without any restrictions or limitations (no guarantees of any kind. world Overview of scikit-learn Python and Excel Scaling, Centering, Noise with kNN, Linear Regression, Logit Sentiment Analysis with Twitter Time Series Analysis Vectors and Arrays (Linear Algebra) Vectors and Arrays (Linear Algebra) Table of contents. Lesson 1 (April 7): Machine learning pipeline and course overview: video; slides. Also,it would be of much help if they have big set of problems and examples. Machine Learning course (Andrew Ng) is a basic machine learning course. It begins with linear algebra—matrix factorizations A= QR. therefore precede our in tro duction to deep learning with a fo cused presen tation of. Linear algebra cheat sheet for deep learning – Towards Data Science – Medium. They give you better intuition for how algorithms really work under the hood, which enables you to make better decisions. It’s all vectors and matrices of numbers. Today, I will be sharing with you my C# implementation of basic linear algebra concepts. In an image classification problem, we often use neural networks. Matrices and Linear Algebra The Wolfram Language automatically handles both numeric and symbolic matrices, seamlessly switching among large numbers of highly optimized algorithms. R is a widely-used statistical programming language in the data science community. Artificial Neural Networks. Grading (tentative) Quizzes 20%; Course project. Together with your editor or Jupyter notebook these packages allow you to rapidly develop scalable, high-performance analytics and visualizations using succinct, type-safe, production-ready code. eye[GF2](5) a: breeze. Windows-64 (64-bit linear algebra for large data) Unless your computer has more than ~32GB of memory and you need to solve linear algebra problems with arrays containing more than ~2 billion elements, this version will offer no advantage over the recommended Windows-64 version above. For beginning practitioners (i. View on GitHub mlcourse. Tibshirani, J. Open Library is an initiative of the Internet Archive, a 501(c)(3) non-profit, building a digital library of Internet sites and other cultural artifacts in digital form. Another resource is the book with the funny title “No Bullshit Guide to Linear Algebra” by Ivan Savov. It introduces some common tools in machine learning to resolve real applications (e. It calls them tensors. Linear Algebra Examines basic properties of systems of linear equations, vector spaces, inner products, linear independence, dimension, linear transformations, matrices, determinants, eigenvalues, eigenvectors and diagonalization. You will also learn how you should use linear algebra in your Python code. mathematics-for-machine-learning-cousera. We start with representing a fully connected layer as a form of matrix multiplication: - In this example, the weight matrix has a size of $4 \times 3$, the input vector has a size of $3 \times 1$ and the output vector has a of size $4 \times 1$. The code (from. Understanding API Security (Free chapters from a Manning. It turns out, however, that all of those operations can be written in terms of big matrix-matrix or matrix-vector multiplications. 1x Scalable Machine Learning. AMD adopted BLIS as its new BLAS library. After reading this quickstart, you can go to other wiki pages, especially Linear Algebra Cheat-Sheet and Data Structures. I'm studying towards a PhD degree at the University of Chicago, in the department of Statistics. SciPy is built to work with NumPy arrays and provides many. They are full of explanations, code samples, pictures, interesting links, and exercises for you to try. linear; algebra; matrix;. import re , math , random # regexes, math functions, random numbers import matplotlib. By Hadrien Jean, Machine Learning Scientist. Compressed Linear Algebra for Declarative Large-Scale Machine Learning Ahmed Elgohary2, Matthias Boehm1, Peter J. What we did here by attaching the variable mlr to the MyLinearRegression class is to create an instance, a specific object called mlr, which will have its own data and "functions". In the second part, we discuss how deep learning differs from classical machine learning and explain why it is effective in dealing with complex problems such as image and natural language processing. Plotting is based on OpenGL and supports both 2D and 3D plots. Introduction to linear algebra (Fourth Edition). DenseMatrix[X. This content is part of a series following the chapter 2 on linear algebra from the Deep Learning Book by Goodfellow, I. therefore precede our in tro duction to deep learning with a fo cused presen tation of. You (the student) should have taken a mathematical course on linear algebra that covers vector spaces as well as a numerical analysis course that covers computer implementations of numerical algorithms. Fitting Lines to Data. net/?Q73_jQ 2020-01-24T14:53:56+01:00 2020-01-24T15:29:05+01:00. Linear Algebra for Data Science using Python Play all 13:42 Math For Data Science | Practical reasons to learn math for Machine/Deep Learning - Duration: 13 minutes, 42 seconds. The mentors for this project are: @dpshelio @mbobra @drsophiemurray @samaloney. Randy Lao's site for free Machine Learning and Data Science resources and materials. How to Learn Advanced Mathematics Without Heading to University - Part 1 subjects to learn for a prospective quant or data scientist. The Cuckoo linear algebra implementation is based on libcuckoo library1. to map the pixel values of an image to the confidence score of each class. This library holds the principal work done as part of the OpenAstonomy Google Summer of Code 2020 project, Solar Weather Forecasting using Linear Algebra. library with basic linear algebra routines, and the SciPy library adorns NumPy arrays with many important primitives, from numerical optimizers and signal processing to statistics and sparse linear algebra. Download R for Windows 5. Linear Algebra Preliminaries¶ Since I have documented the Linear Algebra Preliminaries in my Prelim Exam note for Numerical Analysis, the interested reader is referred to for more details (Figure. Intro to Data Science / UW Videos. Now we extend linear algebra to convolutions, by using the example of audio data analysis. The Linear Algebra Chapter in Goodfellow et al is a nice and concise introduction, but it may require some previous exposure to linear algebra concepts. AMD adopted BLIS as its new BLAS library. Each point correspondence generates one constraint on F. I had to formulate an algorithm to convert an image of some resolution - say L x M and crop / re-size the image into a new resolution say P x R such that I cover the maximum amount of points/pixels from the original image. Again, the class MyLinearRegression provides instructions on how to build a linear regression model. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra. Above, I created 4 matrices. In addition to this, you’ll be able to perform operations such as addition, subtraction and dot product. It introduces some common tools in machine learning to resolve real applications (e. Conncect between Geometry and Algebra. As of October 2019, I am a senior algorithms scientist at PathAI, where I work on computational pathology. California Housing Price Prediction. Linear Optimization in Python 6. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. 6 (376 ratings) Course Ratings are calculated from individual students’ ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. View on GitHub. They give you better intuition for how algorithms really work under the hood, which enables you to make better decisions. Calculate The Trace Of A. pdf CarletonU- hamidreza sadreazami- radar based fall detection with supervised learning. However, it is currently not easy to implement many basic machine learning primitives in. " Our homework assignments will use NumPy arrays extensively. We start with representing a fully connected layer as a form of matrix multiplication: - In this example, the weight matrix has a size of $4 \times 3$, the input vector has a size of $3 \times 1$ and the output vector has a of size $4 \times 1$. In forecasting, Yuyang has worked on all aspects ranging from practical applications to theoretical foundations. I'm a Data Science practitioner and computer programmer with an avid interest in Exploratory Data Analysis, Statistics, & Machine Learning. edu Arun Kumar University of California, San Diego [email protected] DenseMatrix[X. SciPy is open-source software for mathematics, science, and engineering which includes modules for statistics, optimisation, integration, linear algebra, Fourier transforms, signal and image processing, ODE solvers, and more. Now I’m checking back in with 9 weeks under my belt. Mining the social web: Data mining Facebook, Twitter, LinkedIn, Google+, GitHub, and more (2nd edition) by Matthew A. The Deep Learning Book - Goodfellow, I. Most importantly, the online version of the book is completely free. Based on the second linear algebra course taught by Professor Strang, whose lectures on the training data are widely known, it starts from scratch (the four fundamental subspaces) and is fully accessible without the first text. Siefken, J. then this is the book for you. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. How to Learn Advanced Mathematics Without Heading to University - Part 1 subjects to learn for a prospective quant or data scientist. Rich Ott leads you through two days of intensive learning that include a review of linear algebra essential to machine learning, an introduction to TensorFlow, and a dive into neural networks. All publications, sorted by year. This book provides the conceptual understanding of the essential linear algebra of vectors and matrices for modern engineering and science. n Gilbert Strang. As a motivating example, let us consider image classification. The collection of all linear combinations is called a linear subspace of $\RR^n$, denoted by We will say that the $\bb{v}_i$’s span the linear subspace $\mathcal{L}$. If you have more time to dedicate to your projects and you're also passionate about math, consider to contribute to the library!. Roadmap to begin with Machine Learning: The place to start is to learn (and/or) revise linear algebra. Find the smallest value of k such that the rank-k approximation of the matrix uses the same or more amount of data as the original picture. Bourbaki resulted from similar currents of thought that produced fascism and totalitarian communism: moral panics leading to revolutions, and ultimately “final solutions”, all terrible and evil in. Videos and textbooks with relevant details on linear algebra and singular value decomposition (SVD) can be found by searching Alfredo’s Twitter, for example type linear algebra (from:alfcnz) in the search box. Machine Learning/Data Science. , less sensitive to noisy data) than the Euclidean norm The following result is fundamental in linear algebra: Theorem. This is the world beyond R and Python! Breeze is a library for numerical processing, like probability and statistic functions, optimization, linear algebra, etc. Mathematics for machine learning - I totally recommend this book! If you want to learn the bits and pieces of how linear algebra and calculus is used to develop algorithms like principal component analysis, backpropagation etc. a person's height and you switch from meter to centimeter. Linear Algebra and Learning from Data by Gilbert Strang; twitter github. You NEED Linear Algebra for Machine Learning Whether you want to learn Machine Learning for your work or research or you want to become a master, so the others pay you to do it, you need to know how it works. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra. You will also learn how you should use linear algebra in your Python code. There is no doubt that linear algebra is important in machine learning. Tensor networks for dimensionality reduction and large-scale optimization: part 2 applications and future perspectives. Understanding of calculus, linear algebra, and programming is essential. to map the pixel values of an image to the confidence score of each class. Start here. PBblas - Parallel Block Linear Algebra Subsystem HPCC Systems ML_Core repository on GitHub Installation: ecl bundle install https:. We emphasize that this document is not a. References for "practical" machine learning: Python for data analysis by Wes McKinney. The Vector class imitates the m x 1 vector from linear algebra and contains many useful functions for dealing and interacting with Vectors. The free video lectures of this course are made available as part of Harvard Extension School's Opening Learning Initiative. After reading this quickstart, you can go to other wiki pages, especially Linear Algebra Cheat-Sheet and Data Structures. You can think of an r t i m e s c r times c r t i m e s c matrix as a set of r r r row vectors, each having c c c elements; or you can think of it as a set of c c c column vectors, each having r r r elements. " Our homework assignments will use NumPy arrays extensively. This is a straightforward course to learn Linear Algebra Fundamentals for Data Science in Python. linear; algebra; matrix;. 1-11, [Online-Edition. Row Reduction We row reduce a matrix by performing row operations, in order to find a simpler but equivalent system for which the solution set is easily read off. Support Stability of Maximizing Measures for Shifts of Finite Type Journal of Ergodic Theory and Dynamical Systems (accepted) Calkins, S. If you want to fit a model of higher degree, you can construct polynomial features out of the linear feature data and fit to the model too. Supratim Haldar Lead Data Scientist at Head Digital Works Pvt. Popular Courses Popular with our users in the last month Introduction to Linear Algebra. Made for sharing. Linear Algebra The Rank of a Matrix. For simple linear regression, one can choose degree 1. Linear algebra provides a way of compactly representing and operating on sets of linear equations. This repository contains the learning material for the Nuclear TALENT course Learning from Data: Bayesian Methods and Machine Learning, in York, UK, June 10-28, 2019. Omoju: What I do at GitHub is I build data models, often deep learning models on GitHub data to help GitHub probably build things like a recommendation engine so we can recommend repositories to people. The first part covers basics and preliminaries. , hackers, coders, software engineers, and people working as data scientists in business and industry) you don't need to know that much calculus, linear algebra, or other college-level math to get things done. Grannan, S. This is the site for any aspiring data scientists that want to learn in a quick way. Decision theory, game theory, operational research, … (source: lecture video from The Machine Learning Summer School by Zoubin Ghahramani, Univ. Machine Learning is built on prerequisites, so much so that learning by first principles seems overwhelming. A computer science student that is interested in Machine Learning would be well advised to get a minor in Mathematics (or just get a degree in Mathematics instead!). 1 Positive Semide nite (PSD) and Positive De nite (PD) matrices. 08 Apr 2016 » Naive Bayes Classifiers in Rust Adding NB Classifiers to rusty-machine. References for "practical" machine learning: Python for data analysis by Wes McKinney. provide a summary of the mathematical background needed for an introductory class in machine learning, which at UC Berkeley is known as CS 189/289A. Machine Learning is not just writing Python or R. I still plan to go through it but mildly disappointed. 8 Web Framework. I'm a Data Science practitioner and computer programmer with an avid interest in Exploratory Data Analysis, Statistics, & Machine Learning. This will allow us to introduce some central programming features of high-level languages like Python and compiled languages like C++ and/or Fortran. In my opinion, it is one of the bedrock of machine learning, deep learning and data science. Many universities use the textbook Introduction to Linear Algebra. Implementation [ edit ] Scikit-learn is largely written in Python, and uses numpy extensively for high-performance linear algebra and array operations. This content is part of a series following the chapter 2 on linear algebra from the Deep Learning Book by Goodfellow, I. Probability and Statistics: Learn Probability and Statistics Through Interactive Visualizations: Seeing Theory was created by Daniel Kunin while an undergraduate at Brown University. In addition to this, you'll be able to perform operations such as addition, subtraction and dot product. As a core programmer, I love taking challenges and love being part of the solution. Students will learn and practice fundamental ideas of linear algebra and simultaneously be exposed to and work with real-world applications of these ideas. We will assume mathematical maturity and comfort with algorithms, probability, and linear algebra. Neural networks rely on it heavily, but so do linear regression, factor analysis, and lots of other methods. This Word Mover’s Distance (WMD) can be seen as a special case of Earth Mover’s Distance (EMD), or Wasserstein distance, the one people talked about in Wasserstein GAN. These subjects include matrix algebra, vector spaces, eigenvalues and eigenvectors, symmetric matrices, linear transformations, and more. In this course, you’ll learn how to work with vectors and matrices, solve matrix-vector equations, perform eigenvalue/eigenvector analyses and use principal component analysis to do dimension reduction on real-world datasets. In the field of data science, however, being familiar with linear algebra and statistics is very important to statistical analysis and prediction. Lawrence [email protected] Then last year I learned how he morphed his delightful mathematics book into a brand new title (2019) designed for data scientists - "Linear Algebra and Learning from Data. Choi et al. Now that you understand the key ideas behind linear regression, we can begin to work through a hands-on implementation in code. Linear Algebra The Rank of a Matrix. Linear Algebra. Gradient Descent with Linear Regression - GitHub Pages. The training data is used to find the optimal model but the model should ultimately work for the test data! Conclusion. Mathematics for machine learning - I totally recommend this book! If you want to learn the bits and pieces of how linear algebra and calculus is used to develop algorithms like principal component analysis, backpropagation etc. https://shaarli. Linear Algebra for Data Science using Python Play all 13:42 Math For Data Science | Practical reasons to learn math for Machine/Deep Learning - Duration: 13 minutes, 42 seconds. Twitter: @mpd37, @AnalogAldo, @ChengSoonOng. com) Bangalore, India * Working as a Lead Data Scientist at Head Digital Works Pvt. Using least-squares linear approximation techniques to find the best linear fit to a set of data points results in the equation of a line which minimizes the sum of the squares of the vertical distances from the given points to the line: Note that, unless the line is horizontal, the vertical distance will be slightly larger than the actual distance, which is measured in the direction. I recently released an efficient linear algebra library for Javascript. LINEAR ALGEBRA. Learn how to solve challenging machine learning problems with TensorFlow, Google’s revolutionary new software library for deep learning. import re , math , random # regexes, math functions, random numbers import matplotlib. Statistical Machine Learning (S2 2017) Deck 6 This lecture • Notes on linear algebra ∗Vectors and dot products ∗Hyperplanes and vector normals • Perceptron ∗Introduction to Artificial Neural Networks ∗The perceptron model ∗Stochastic gradient descent 2. A few weeks ago, I wrote about how and why I was learning Machine Learning, mainly through Andrew Ng’s Coursera course. Deep Learning Book Series · 2. Overview - Khan Academy Vectors and Spaces; Matrix Transformations; Python. This course develops the mathematical basis needed to deeply understand how problems of classification and estimation work. pdf; Basic Set Notation & Terminology. Machine learning yearning. hdf5 is a file. If you have more time to dedicate to your projects and you’re also passionate about math, consider to contribute to the library!. Module 0: Introduction & Outline About What is Data Science? Module 1: Required Background Math: Stats, Calculus, Linear Algebra Programming: Basics, Data Structures, Algorithms Databases: Relational Algebra, SQL Important Concepts: Regular expressions, Information Entropy, Distance measurements, OLAP, ETL, BI VS BA and CAP. NOTE: please check for the course practicalities, e. implemented vector and matrix classes with reST-formatted docstrings in Python 3+ General Layout. Linear Algebra Can Help You Choose Your Stock Portfolio Correlation is a very fundamental and viseral way of understanding how the stock market works and how strategies perform. Some recent tutorials by Christos and Co. Python is one of the most commonly used programming languages by data scientists and machine learning engineers. for automated market making. Udacity is the world’s fastest, most efficient way to master the skills tech companies want. I'm interested in applying non-standard tools form abstract algebra and topology to the study of neural networks. We will learn scientific computing basics, topics in numerical linear algebra, mathematical probability (probability spaces, expectation, conditioning, common distributions, law of large numbers and the central limit theorem), statistics (point estimation, confidence intervals, hypothesis testing, maximum likelihood estimation, density. This book provides the conceptual understanding of the essential linear algebra of vectors and matrices for modern engineering and science. Introduction Theunprecedentedadvanceindigitaltechnologyduringthesecondhalfofthe20thcenturyhas producedameasurementrevolutionthatistransformingscience. Now we are ready to see how matrix algebra can be useful when analyzing data. The linux command line: A complete introduction. As we will see, we can do all the common linear algebra operations without using any other library. The elements of statistical learning: data mining, inference, and prediction, Springer, 2009 Linear Algebra and Probability Review (part 1 Linear Algebra, part 2 Probability) Assignment 1: Mar 17. Many popular machine learning methods, including XGBOOST, use matrices to store inputs and process data. In this day, we are going to make the dirty work. Linear Algebra (Michael Damron and Tasho Kaletha) Introduction to Linear Algebra (Strang) Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra (Matousek) Linear Algebra Done Right (Axler) Advanced Linear Algebra (Roman). TS CH9 Hypothesis Testing. Research labs and companies have data to analyze and understand, and this deep learning approach has become widespread. You can find all the notebooks on Github. He was a research fellow with Michael Jordan and Peter Bartlett, University of California at Berkeley, from 2003, and with Bernhard Schoelkopf, Max Planck Institute for Intelligent Systems, Tuebingen, Germany, from 2005. Gilbert Strang is a Professor of Mathematics at Massachusetts Institute of Technology and an Honorary Fellow at Balliol College in Oxford. pdf; TS CH1 Exploratory Data Analysis. The final exam is 9am on Friday May 15 in JNSN-Ice Rink. Just want some books to go deeper than a introductory course. This course develops the mathematical basis needed to deeply understand how problems of classification and estimation work. Learn how to solve challenging machine learning problems with TensorFlow, Google's revolutionary new software library for deep learning. In the second part, we discuss how deep learning differs from classical machine learning and explain why it is effective in dealing with complex problems such as image and natural language processing. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. This is a straightforward course to learn Linear Algebra Fundamentals for Data Science in Python. i recently bought Gilbert Strang's linear algebra book. then this is the book for you. Also, this OpenGL tutorial has useful explanations. A good video series on the topic that allows you to visualize many concepts is Essence of linear algebra. Welcome to the 18. Our goal is to give the beginning student, with little or no prior exposure to linear algebra, a good ground-ing in the basic ideas, as well as an appreciation for how they are used in many applications, including data tting, machine learning and arti cial intelligence, to-. Before that, I was a software engineer at Google where I worked 80% time with the Hotels team on data analytics and 20% time with the operations research team on linear program solvers. Thank you for your interest in Linear Algebra and Learning from Data. In this course, you’ll learn how to work with vectors and matrices, solve matrix-vector equations, perform eigenvalue/eigenvector analyses and use principal component analysis to do dimension reduction on real-world datasets. Learn linear algebra. Then, in Section 2, we quickly bring you up to speed on the prerequisites required for hands-on deep learning, such as how to store and manipulate data, and how to apply various numerical operations based on basic concepts from linear algebra, calculus, and probability. Communication The vast majority of questions about homework, the lectures, or the course should be asked on our Piazza forum, as others will benefit from the responses. I still plan to go through it but mildly disappointed. You can think of an r t i m e s c r times c r t i m e s c matrix as a set of r r r row vectors, each having c c c elements; or you can think of it as a set of c c c column vectors, each having r r r elements. Building on centuries of statistics and mathematics, Data Science uses computational techniques to help the most innovative companies in the world scale. Open Library is an initiative of the Internet Archive, a 501(c)(3) non-profit, building a digital library of Internet sites and other cultural artifacts in digital form. Math is a crucial skill for people who are interested in Data Science and Machine Learning. Data Science and Linear Algebra Fundamentals with Python, SciPy, & NumPy Math is relevant to software engineering but it is often overshadowed by all of the exciting tools and technologies. [Online book] n Andrew Ng. The elements of statistical learning: data mining, inference, and prediction, Springer, 2009 Linear Algebra and Probability Review (part 1 Linear Algebra, part 2 Probability) Assignment 1: Apr 10. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra.


It is also a matrix and also an array; all scalars are also vectors, and all scalars are also matrix, and all scalars are also array You can put this solution on YOUR website! See the picture below. Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal to the square matrix A = [a ij] n × n is an identity matrix if If the block produces a scalar output from a scalar input, the block preserves dimension. If you multiply any number to a diagonal matrix, only the diagonal entries will change. 8) Unit or Identity Matrix. Nonetheless, it's still a diagonal matrix since all the other entries in the matrix are . A square matrix is said to be scalar matrix if all the main diagonal elements are equal and other elements except main diagonal are zero. Back in multiplication, you know that 1 is the identity element for multiplication. The following rules indicate how the blocks in the Communications Toolbox process scalar, vector, and matrix signals. The column (or row) vectors of a unitary matrix are orthonormal, i.e. Scalar matrix can also be written in form of n * I, where n is any real number and I is the identity matrix. However, there is sometimes a meaningful way of treating a $1\times 1$ matrix as though it were a scalar, hence in many contexts it is useful to treat such matrices as being "functionally equivalent" to scalars. While off diagonal elements are zero. Long Answer Short: A $1\times 1$ matrix is not a scalar–it is an element of a matrix algebra. A square matrix is said to be scalar matrix if all the main diagonal elements are equal and other elements except main diagonal are zero. [] is not a scalar and not a vector, but is a matrix and an array; something that is 0 x something or something by 0 is empty. Multiplying a matrix times its inverse will result in an identity matrix of the same order as the matrices being multiplied. Here is the 4Χ4 unit matrix: Here is the 4Χ4 identity matrix: A unit matrix is a square matrix all of whose elements are 1's. Yes it is. All the other entries will still be . The same goes for a matrix multiplied by an identity matrix, the result is always the same original non-identity (non-unit) matrix, and thus, as explained before, the identity matrix gets the nickname of "unit matrix". In this post, we are going to discuss these points. 2. Okay, Now we will see the types of matrices for different matrix operation purposes. For an example: Matrices A, B and C are shown below. References [1] Blyth, T.S. In the next article the basic operations of matrix-vector and matrix-matrix multiplication will be outlined. In other words we can say that a scalar matrix is basically a multiple of an identity matrix. #1. It is never a scalar, but could be a vector if it is 0 x 1 or 1 x 0. An identity matrix is a square matrix whose upper left to lower right diagonal elements are 1's and all the other elements are 0's. and Robertson, E.F. (2002) Basic Linear Algebra, 2nd Ed., Springer [2] Strang, G. (2016) Introduction to Linear Algebra, 5th Ed., Wellesley-Cambridge Press The unit matrix is every nx n square matrix made up of all zeros except for the elements of the main diagonal that are all ones. Scalar matrix can also be written in form of n * I, where n is any real number and I is the identity matrix. Basis. This topic is collectively known as matrix algebra. The scalar matrix is basically a square matrix, whose all off-diagonal elements are zero and all on-diagonal elements are equal. If a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I. Closure under scalar multiplication: is a scalar times a diagonal matrix another diagonal matrix? Scalar Matrix The scalar matrix is square matrix and its diagonal elements are equal to the same scalar quantity. In their numerical computations, blocks that process scalars do not distinguish between one-dimensional scalars and one-by-one matrices. The next article the basic operations of matrix-vector and matrix-matrix multiplication will outlined! In an identity matrix of the same order as the matrices being multiplied multiplying a matrix algebra will in! Answer Short: a $1\times 1$ matrix is basically a multiple of an identity matrix the! Produces a scalar, but could be a vector if it difference between scalar matrix and identity matrix a... Never a scalar times a diagonal matrix basic operations of matrix-vector and matrix-matrix multiplication will be....: a $1\times 1$ matrix is square matrix, only the diagonal entries will change a unitary are... 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# Binomial theorem

Series binomial theorem series contents page contents binomial theorem notation n as a nonnegative integer proof of the binomial theorem proof when n and k are. The binomial theorem date_____ period____ find each coefficient described 1) coefficient of x2 in expansion of (2 + x)5 80 2) coefficient of x2 in expansion. A polynomial with two terms is called a binomial we have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high. Posts about binomial theorem written by ujjwal gulecha. Yes, pascal's triangle and the binomial theorem isnâ€™t particularly exciting but it can, at least, be enjoyable we dare you to prove us wrong.

Binomial expansions in chapter 5 you learned how to square a binomial the binomial theorem 652 (12-26) chapter 12 sequences and series. Binomial theorem the binomial theorem states that the binomial coefficients $$c(n,k)$$ serve as coefficients in the expansion of the powers of the binomial $$1+x$$. Binomial theorem : akshay mishra xi a , k v 2 , gwalior in elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. How to use the binomial theorem to expand binomial expressions, examples and step by step solutions, the binomial theorem using combinations. Quick links: downloadable teaching materials for binomial theorem syllabus content for the algebra topic: sl syllabus (see syllabus section 13) hl syllabus (see.

Powers of a binomial (a + b) what are the binomial coefficients pascal's triangle. In this video lesson, you will see what the binomial theorem has in common with pascal's triangle learn how you can use pascal's triangle to help. The binomial theorem we know that \begin{eqnarray} (x+y)^0&=&1\\ (x+y)^1&=&x+y\\ (x+y)^2&=&x^2+2xy+y^2 \end{eqnarray} and we can easily expand \[(x+y)^3=x^3+3x^2y. Expanding a binomial expression that has been raised to some large power could be troublesome one way to solve it is to use the binomial theorem.

Explains how to use the binomial theorem, and displays the theorem's relationship to pascal's triangle. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum for k-12 kids, teachers and parents.

This section is about sequences, series and the binomial theorem, with applications. Binomial theorem was known for the case n = 2 by euclid around 300 bc, and pascal stated it in modern form in 1665 newton showed that a similar formula for negative.

## Binomial theorem

In this lesson, students will learn the binomial theorem and get practice using the theorem to expand binomial expressions the theorem is broken. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Binomial theorem: binomial theorem, statement that describes the nth power of the sum of two numbers (a + b.

When a binomial is raised to whole number powers, the coefficients of the terms in the expansion form a pattern. Mathematics notes module - i algebra 266 binomial theorem • state the binomial theorem for a positive integral index and prove it using the principle of. The most basic example of the binomial theorem is the formula for the square of x + y: (+) = + + the binomial coefficients 1, 2, 1 appearing in this expansion. Fun math practice improve your skills with free problems in 'binomial theorem i' and thousands of other practice lessons. 123 applications of the binomial theorem expansion of binomials the binomial theorem can be used to find a complete expansion of a power of a binomial or a. There are several closely related results that are variously known as the binomial theorem depending on the source even more confusingly a number of these (and other. While the foil method can be used to multiply any number of binomials together, doing more than three can quickly become a huge headache.

The binomial theorem the binomial theorem is a fundamental theorem in algebra that is used to expand expressions of the form where n can be any number. Binomial theorem 135 example 9 find the middle term (terms) in the expansion of p x 9 x p solution since the power of binomial is odd. Demonstrates how to answer typical problems involving the binomial theorem.

Binomial theorem
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# Smallest set such that all arithmetic progression will always contain at least a number in a set

Let $$S= \left\{ 1,2,3,...,100 \right\}$$ be a set of positive integers from $$1$$ to $$100$$. Let $$P$$ be a subset of $$S$$ such that any arithmetic progression of length 10 consisting of numbers in $$S$$ will contain at least a number in $$P$$. What is the smallest possible number of elements in $$P$$ ?

Denote $$|P|$$ as the number of elements in $$P$$. We shall find the smallest possible value of $$|P|$$.

For $$|P|=16$$, we have the answer by @RobertIsrael below.

However, for $$|P|<16$$, I can neither find such set $$P$$ nor prove that $$|P|$$ cannot be less than $$16$$. So my question is:

Is it true that $$|P| \geq 16$$? How can I prove it? If not, what is the minimum amount of elements in $$P$$ ?

Also, I am wondering that:

If we replace 10 with an even number $$n$$,and $$100$$ with $$n^2$$, can we find the minimum of $$|P|$$ ?

Any answers or comments will be appreciated. If this question should be closed, please let me know. If this forum cannot answer my question, I will delete this question immediately.

• it is not too unusual that questions here get answered, say, after a year, and not immediately. Jun 27 '19 at 6:24
• @DimaPasechnik Thanks. I just afraid that my question will be forgotten and cannot be answered. Jun 27 '19 at 7:02
• good questions don't get forgotten. they pop up in searches, etc. Jun 27 '19 at 8:21
• This can be considered as a set-covering problem. Although set covering is NP-complete, I suspect this one is within the reach of current technology. Jun 27 '19 at 12:37
• For the last question (replacing 10 with $n$), have you computed the optimal number for $n\le 9$ and checked the OEIS? Jun 27 '19 at 14:44

Considering the complement of $$P$$ in $$[1,100]$$, you are asking how large can a subset of $$[1,100]$$ be given that it does not contain any $$10$$-term arithmetic progression. The more general question

How large can a subset of $$[1,N]$$ be given that it does not contain any $$k$$-term arithmetic progression?

is one of the central problems in combinatorial number theory. There is no chance to give a precise answer, as an "explicit" function of $$N$$ and $$k$$, and it quite likely that this is impossible already in your special situation where $$N=n^2$$ and $$k=n$$.

Here is an argument showing that if $$P\subset[1,n^2]$$ meets every $$n$$-term progression contained in $$[1,n^2]$$, then $$|P|>n+n^{0.5+o(1)}$$. (See also the paragraph at the very end for the estimate $$|P|\ge 12$$ in your special case where $$P\subset[1,100]$$ and we want to block all $$10$$-term progressions.) It would be interesting to improve these estimates or at least to decide whether $$|P|>Cn$$ holds true with an absolute constant $$C>1$$.

Write $$K:=|P|$$, $$\Delta:=K-n$$, and $$P=\{p_1,\dotsc,p_K\}$$ where $$1\le p_1<\dotsb. Notice that $$p_1\le n$$ and $$p_K\ge n^2-(n-1)$$, whence $$p_K-p_1\ge(n-1)^2$$.

For any $$d\in[1,n]$$, the set $$P$$ contains an element from every residue class modulo $$d$$, and it follows that there are at most $$K-d$$ pairs of consecutive elements of $$P$$ with the difference equal to $$d$$; also, if $$d>n$$, then there are no such pairs at all. Let $$a$$ and $$r$$ be defined by \begin{align*} K-1 &= \Delta+(\Delta+1)+\dotsb+(\Delta+(a-1))+r \\ &= a\Delta+\frac{a(a-1)}2 + r,\quad 0\le r<\Delta+a. \tag{1} \end{align*} Since there are totally $$K-1$$ pairs of consecutive elements of $$P$$, of them at most $$\Delta$$ pairs at distance $$n$$, at most $$\Delta+1$$ pairs at distance $$n-1$$, etc, we conclude that \begin{align*} p_K-p_1 &\le n\Delta+(n-1)(\Delta+1)+\dotsb+(n-(a-1))(\Delta+(a-1))+(n-a)r \\ &= \Delta na+(n-\Delta)\cdot\frac{a(a-1)}2-\frac{a(a-1)(2a-1)}{6}+(n-a)r. \end{align*} Recalling the estimate $$p_K-p_1\ge(n-1)^2$$, and using ($$1$$), we get \begin{align*} (n-1)^2 &\le \Delta na+(n-\Delta)\cdot\frac{a(a-1)}2-\frac{a(a-1)(2a-1)}{6}+(n-a)r \\ &= n\Big(a\Delta+\frac{a(a-1)}2 + r\Big) - \Delta\cdot\frac{a(a-1)}2 - \frac{a(a-1)(2a-1)}{6} - ar \\ &= n(K-1) - \Delta\cdot\frac{a(a-1)}2 - \frac{a(a-1)(2a-1)}{6} - ar. \tag{2} \end{align*}

We now assume, aiming at a contradiction, that $$\Delta with an absolute constant $$0. From (1) we get then $$K-1 \ge \Delta a + \frac{a(a-1)}2 \ge \frac12\,a^2 - 1$$ implying $$a\le\sqrt{2K}$$; hence, $$\Delta a=O(n^{0.5+c})$$ and $$r=a+\Delta=O(n^{0.5})$$. As a result, $$\frac12\,a^2 = K-1+\frac12\,a-\Delta a - r > K - O(n^{0.5+c}),$$ leading to $$a>(1-o(1))\sqrt{2K}$$.

With these estimates in mind, from (2) we obtain $$n^2 + O(n) \le nK - \frac12\,\Delta a^2 - \frac13\,a^3;$$ that is, $$\Delta n \ge \frac12\,\Delta a^2 + \frac13\,a^3 + O(n).$$ Consequently, $$n^{1+c} \ge \Delta n \ge \frac13\,a^3 + O(n) \ge (1-o(1))(2K)^{1.5} + O(n) > n^{1.5} + O(n),$$ a contradiction.

As an illustration of this approach, let's show that one needs at least $$12$$ elements to block every $$10$$-term progression in $$[1,100]$$. Suppose for a contradiction that $$P\subset[1,100]$$ is an $$11$$-element set blocking all such progressions. There are $$|P|-1=10$$ pairs of consecutive elements of $$P$$. Of these ten pairs, there is at most one pair with distance $$10$$ between its two elements, at most two pairs with distance $$9$$, at most three pairs with distance $$8$$, and at most four pairs with distance $$7$$. Therefore the largest element of $$P$$ exceeds the smallest one by at most $$1\cdot 10+2\cdot 9 + 3\cdot 8 + 4\cdot 7=80$$. It follows that either the smallest element of $$P$$ is at least $$11$$, or its largest element is at most $$90$$; but then $$P$$ does not block at least one of the progressions $$[1,10]$$ and $$[91,100]$$, a contradiction.

Using a tabu search procedure, I have found a solution for $$|P|=17$$, namely $${1, 11, 18, 25, 31, 32, 33, 36, 44, 51, 58, 65, 69, 70, 77, 84, 91}$$. I don't know if this is optimal.

EDIT: Found a solution for $$|P|=16$$, namely $$10, 15, 22, 29, 36, 43, 53, 55, 56, 57, 58, 68, 73, 74, 84, 91$$

• I'm working on $|P|=16$. So far I've found a $P$ with $|P|=16$, namely $\{9, 18, 28, 29, 31, 40, 42, 51, 53, 56, 65, 69, 70, 77, 84, 91\}$, that intersects all but one of these arithmetic progressions, the exception being $({36, 43, 50, 57, 64, 71, 78, 85, 92, 99})$. Jun 27 '19 at 16:50
• I'm using a tabu search over sets of a given size to maximize the number of a.p.'s that intersect the set. Possible moves consist of replacing a member of the set with a nonmember. Jun 27 '19 at 17:00
• Thank you. Your answer is correct. How long did it take to find those numbers? Can you find the boundary of $|P|$? Jun 28 '19 at 8:00
• So is 16 optimal?
– EGME
Jun 28 '19 at 20:43
• My brute-force confirms that there no 15. Jul 2 '19 at 5:19


# Mutlivariable Calculus: Surface Area

This was a question a students had asked me earlier today regarding surface area.

Find the surface area of the hemisphere $x^2+y^2+z^2 = 4$ bounded below by $z=1$.

I decided to approach this problem using spherical coordinates and got the following

\begin{eqnarray} \int_{0}^{2\pi}\int_{0}^{\pi/3}4\sin\phi d\phi d\theta & = & 4\int_{0}^{2\pi}d\theta\int_{0}^{\pi/3}\sin\phi d\phi\\ & = & 8\pi\cos\phi|_{\pi/3}^{0}\\ & = & 8\pi(1-\frac{1}{2}) = 4\pi \end{eqnarray}

I also solved this problem using single variable calculus as follows. I can represent the sphere as a circle of $h^2+z^2 =4$. Thus we have the following:

\begin{eqnarray} SA = \int_a^b2\pi f(z)ds & = & 2\pi\int_1^2\sqrt{4-z^2}\sqrt{1+\frac{z^2}{4-z^2}}dz\\ & = & 2\pi\int_1^2\sqrt{4-z^2}\sqrt{\frac{4}{4-z^2}} dz\\ & = & 2\pi\int_1^22dz=4\pi z|_1^2=4\pi \end{eqnarray}

As you see, I got the same answer for both approaches.

The student, and a few others, comes back later during the day and tells me the answer I got was incorrect. He does not tell me what the professor got, he just told me it was wrong. I asked myself "why?" Is there something I missed?

Thanks in advance for any feedback.

• Maybe there is something wrong with the text of the problem. The surface whose area has been calculated is not a hemisphere but a spherical cap. – Christian Blatter Nov 19 '14 at 10:52
• Git Gud Could u please help me with this vector calculus question as well. I really need help in this. Thanks – ys wong Nov 23 '14 at 7:38

I don't see what you did wrong here...

To me, to solve these types of problems you have to think geometrically--there isn't going to be some way to do it just from a knowledge of multivariable calculus.

First, how to find the total surface area of the sphere--that will help. You need to break the sphere up into circles stacked on top of each other, then find the $dA$:

$$dA = 2\pi r_{\phi} h$$

$h$ is easy to find, it's just $rd\phi$, $r_\phi$ is the radius at the given azimuth: $r_\phi = r\sin(\phi)$ which gives:

$$dA = 2\pi r^2\sin(\phi)d\phi\\ A = \int dA = 2\pi r^2\left.\int_{0}^{\pi}\sin(\phi)d\phi = -2\pi r^2\cos(\phi)\right|_0^\pi = 4\pi r^2$$

So the correct integral should be:L

$$A = 2\pi r^2\left.\int_{0}^{\phi_0} \sin(\phi)d\phi = -2\pi r^2 \cos(\phi)\right|_{0}^{\phi_0} = 2\pi r^2\left(1 - \cos(\phi_0)\right)$$

In this case, $\phi_0$ satisfies that $z = r\cos(\phi) = 2\cos(\phi_0) = 1 \rightarrow \cos(\phi_0) = \frac{1}{2}$ and thus:

$$A = 2\pi 2^2\left(1 - \frac{1}{2}\right) = 4\pi$$