joecamel + math   215

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Kalman exercise
"Quick explanation of how a Kalman filter works just to make sure I understood it and I don't forget it, and also for others to learn. "

HN: "Nice summary. I think of the concept at a slightly higher level of abstraction:
It's just Bayesian inference to update the posterior. A theoretical model gives an estimate of what the state of the system should be (call that the "prior"). A sensor measurement gives an independent estimate of the state (call that the "likelihood"). Composing (multiplying) the two gives the posterior. Further, in this formulation it doesn't really matter what you call the prior or the posterior. You can easily combine estimates from many different sources/sensors.

Kalman filters are just the special case where the likelihood and prior are both Gaussian -- the distributions can be specified with a couple of numbers, and there is a simple closed-form expression for composing them. More generally, one can use all the tools available for Bayesian inference." https://news.ycombinator.com/item?id=16575679
kalman.filter  signal.processing  math  robotics  intro  example
9 days ago by joecamel
Clayton Shonkwiler: 15 Views of the Hypersphere - YouTube
"This video and talk are aimed at undergraduates and advanced high-schoolers, but anyone interested in mathematics should love the material being presented."
math  video  education  youtube  high.school  visualization
7 weeks ago by joecamel
Stanford Lecture: Don Knuth's Christmas Tree Lecture 2017 - YouTube
From HN: "I attended this on Thursday. Knuth started with a problem about rectangles inside rectangles (https://imgur.com/faWRt2q — it's going to be an exercise in 7.2.2.1 of TAOCP when that's published, currently in the draft version of Pre-Fascicle 5C). He worked through some small cases, made a conjecture, showed a problem submitted to the Monthly, and lots of cool stuff with generating functions. The lecture was also peppered with jokes and cool stories, including a fascinating conjecture by Bill Gosper, who has a long history of coming up with these Ramanujan-like identities. He also showed a wonderful conjecture (involving queens on an infinite chessboard) that he thinks may never be proved, and showed a snippet of his CWEB program for the problem." https://news.ycombinator.com/item?id=15898919
knuth  video  youtube  stanford  talk  lecture  math  cs
11 weeks ago by joecamel
My unusual hobby | Stephan Boyer
"I think of Coq as an extension of my own ability to reason logically. When I’m in doubt about something, I open the Coq IDE and try to prove it. I think the reason this is so valuable to me is that I often mull over functional programming, types, logic, algorithms, etc. These kinds of things are well-suited to formalization."
coq  type.theory  plt  programming  proof  math
november 2017 by joecamel
Somehow I became the canonical undergraduate source for bibliographical references, so I thought I would leave a list behind before I graduated. I list the books I have found useful in my wanderings through mathematics (in a few cases, those I found especially unuseful), and give short descriptions and comparisons within each category. I hope that this list may serve as a useful “road map” to other undergraduates picking their way through Eckhart Library. In the end, of course, you must explore on your own; but the list may save you a few days wasted reading books at the wrong level or with the wrong emphasis.
bibliography  books  list  curriculum  math  education
november 2017 by joecamel
career - What's a mathematician to do? - MathOverflow
"The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding." Bill Thurston
math  philosophy  mathematics  mathoverflow  discussion  education
october 2017 by joecamel
Mathematics of Juggling - CornellCast
Allen Knutson, Cornell professor of mathematics and former world record-holding juggler, gives a public demonstration and lecture on how juggling patterns can be represented the mathematically.
math  juggling  interesting  inspiration
september 2017 by joecamel
Sunset Geometry
"Vanderbei’s analysis is an elegant and subtle exercise in classical trigonometry. In this post, I would like to present an alternative analysis in a different language: Geometric Algebra."
geometric.algebra  math  geometry  trigonometry
september 2017 by joecamel
On Norbert Blum’s claimed proof that P does not equal NP | in theory
"so let me apologize by summarizing the claims in the paper."
math  tcs  pvsnp  luca.trevisan  2017  overview
august 2017 by joecamel
Statistician Proves Gaussian Correlation Inequality | Quanta Magazine
"When a German retiree proved a famous long-standing mathematical conjecture, the response was underwhelming."
math  statistics  research  probability  inequality  quanta  story  academia
august 2017 by joecamel
What Tensors Are For! - YouTube
An intro video into tensor calculus lectures. Good motivation and high level overview (geometry - algebra).
video  youtube  tensors  tensor.calculus  math  analysis  intro  high.level  overview
june 2017 by joecamel
Random: Probability, Mathematical Statistics, Stochastic Processes
"Random (formerly Virtual Laboratories in Probability and Statistics) is a website devoted to probability, mathematical statistics, and stochastic processes, and is intended for teachers and students of these subjects."
math  probability  statistics  tutorial  visualization  interactive  intro
april 2017 by joecamel
The Abel Prize 2017 - Yves Meyer - YouTube
"for his pivotal role in the development of the mathematical theory of wavelets."
math  youtube  talk  overview  wavelets  signal.processing  terrytao  intro  abel.prize  2017
march 2017 by joecamel
Is “the theory of everything” merely the ultimate ensemble theory?
The paper contains an interesting figure that illustrates relationships between various basic mathematical structures.
math  paper  visualization  overview  algebra
february 2017 by joecamel
Modes, Medians and Means: A Unifying Perspective
"We’ve just seen that the mode, median and mean all arise from a simple parametric process in which we try to minimize the average discrepancy between a single number ss and a list of numbers, x1,x2,…,xnx1,x2,…,xn that we try to summarize using ss. In a future blog post, I’ll describe how the ideas we’ve just introduced relate to the concept of LpLp norms."
statistics  intro  blog  post  math
february 2017 by joecamel
Can a Chess Piece Explain Markov Chains? | Infinite Series - YouTube
A cool example of stationary distribution theorem (expected time of return to starting state).
youtube  video  math  markov  probability  intro  example  education
january 2017 by joecamel
[1612.09375v1] Basic Category Theory
This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together.
math  pdf  arxiv  category.theory  intro  introduction
january 2017 by joecamel
"The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what is the shape of largest area in the plane that can be moved around a right-angled corner in a two-dimensional hallway of width 1? This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked."
math  example  introduction  visualization  beginner  education
january 2017 by joecamel
Mathematical Components
Mathematical Components is the name of a library of formalized mathematics for the Coq system. It covers a variety of topics, from the theory of basic data structures (e.g., numbers, lists, finite sets) to advanced results in various flavors of algebra. This library constitutes the infrastructure for the machine checked proofs of the Four Color Theorem and of the Odd Order Theorem.
proof  coq  book  math  cs
december 2016 by joecamel
Course Catalogue | The Theoretical Minimum
"The Theoretical Minimum courses include a core sequence of six courses, plus a set of supplemental courses that teach additional related material.  The core sequence is currently being repeated with Statistical Mechanics being taught during Spring quarter, 2013."
course  physics  math  video  list  learning  education  leonard.susskind
december 2016 by joecamel
Clifford Algebra: A visual introduction | slehar
"Clifford Algebra, a.k.a. Geometric Algebra, is a most extraordinary synergistic confluence of a diverse range of specialized mathematical fields, each with its own methods and formalisms, all of which find a single unified formalism under Clifford Algebra. It is a unifying language for mathematics, and a revealing language for physics."
algebra  math  clifford  introduction  motivation
november 2016 by joecamel
Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics
"We can do better! Research on the design and use of
mathematical systems provides a guide for designing a unified mathematical language for the whole of physics that facilitates learning and enhances physical insight. This has produced a comprehensive language called Geometric Algebra, which I introduce with emphasis on how it simplifies and integrates classical and quantum physics."
algebra  math  mathematics  physics  geometric.algebra  2002  paper  talk  lecture  oersted  pdf  read
september 2016 by joecamel
[1609.01421] Struggles with the Continuum
"Our assumption that spacetime is a continuum leads to many challenges in mathematical physics. Singularities, divergent integrals and the like threaten many of our favorite theories, from Newtonian gravity to classical electrodynamics, quantum electrodynamics and the Standard Model. In general relativity, singularities are intimately connected to some of the theory's most dramatic successful predictions. We survey these problems and the large amount of work that has gone into dealing with them."
math  mathematics  physics  arxiv  pdf  paper  john.baez  overview  history
september 2016 by joecamel
Great Mathematicians on Math Competitions and "Genius" - Less Wrong
"As I mentioned in Fields Medalists on School Mathematics, school mathematics usually gives a heavily distorted picture of mathematical practice. It's common for bright young people to participate in math competitions, an activity which is closer to that of mathematical practice. Unfortunately, while math competitions may be more representative of mathematical practice than school mathematics, math competitions are themselves greatly misleading. Furthermore, they've become tied to a misleading mythological conception of "genius." I've collected relevant quotations below."
education  math  mathematics  interviews  blog  post  lesswrong
july 2016 by joecamel
Book of Proof by Richard Hammack
This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative.
book  math  mathematics  proofs  introduction
june 2016 by joecamel
Singular Value Decomposition Part 2: Theorem, Proof, Algorithm | Math ∩ Programming
’m just going to jump right into the definitions and rigor, so if you haven’t read the previous post motivating the singular value decomposition, go back and do that first. This post will be theorem, proof, algorithm, data. The data set we test on is a thousand-story CNN news data set. All of the data, code, and examples used in this post is in a github repository, as usual.
HN: https://news.ycombinator.com/item?id=11710154
math  linearalgebra  algorithm  blog  post  svd
june 2016 by joecamel
The HoTT book | Mathematics and Computation
We are a group of two dozen mathematicians who wrote a 600 page book in less than half a year.
book  github  math  opensource  writing  hott  category.theory  collaboration  type.theory  2013
november 2014 by joecamel
Modern Algebra | Mathematics | MIT OpenCourseWare
This undergraduate course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics.
math  algebra  lecturenotes  mit  ocw  course
november 2014 by joecamel
How to talk Mathematics - P. R. Halmos
The purpose of what follows is to suggest to a young mathematician what he might do (and what he had better not do) the first few times that he gives a public lecture on his subject.
november 2014 by joecamel
Subhash Khot, Unique Games Conjecturer, Is Awarded Rolf Nevanlinna Prize | Simons Foundation
Subhash Khot’s bold conjecture is helping mathematicians explore the precise limits of computation.
tcs  video  simons  article  math  prize
august 2014 by joecamel
Maryam Mirzakhani Is First Woman Fields Medalist | Simons Foundation
Maryam Mirzakhani’s monumental work draws deep connections between topology, geometry and dynamical systems.
math  science  fields.medal  article  2014
august 2014 by joecamel
Martin Gardner's classic napkin ring puzzle | Hacker News
One of those cases where using integrals rather than geometry is much simpler...
hn  math  martin.gardner  discussion  analysis  puzzle
july 2014 by joecamel
The Mathematics Behind xkcd: A Conversation with Randall Munroe (MAA)
The creator of the popular web comic xkcd muses about the merits of pen and paper versus computer coding, tic-tac-toe, and where he sits on the scale of intellectual purity.
xkcd  interview  math  comics
june 2014 by joecamel
Primers | Math ∩ Programming
"As a fair warning to the reader, these primers are a bit more terse than what you’d find in your average textbook."
algorithms  programming  math  reference  tutorials
june 2014 by joecamel
How to Conquer Tensorphobia | Math ∩ Programming
"This post is an attempt to bridge the gap between the elementary and advanced understandings of tensors. We’ll start with the elementary (axiomatic) approach, just to get a good feel for the objects we’re working with and their essential properties. Then we’ll transition to the “universal” mode of thought, with the express purpose of enlightening us as to why the properties are both necessary and natural."
algebra  math  mathematics  programming  tensors  introduction  explanation  blog  cs
june 2014 by joecamel
Essential Math for Games Programmers
"This tutorial deepens the approach of the previous years' Essential Math for Games Programmers, by spending one day on general math topics, and one day focusing in on the topic of physical simulation. It, like the previous tutorials, provides a toolbox of techniques for programmers, with references and links for those looking for more information."
math  game  graphics  programming  list  slides  resources
may 2014 by joecamel
Demystifying the Fourier magic | XRDS
"Over the years I have gotten used to seeing many theorems in theoretical computer science being proved using discrete Fourier analysis. The Walsh-Fourier (Hadamard) transform of Boolean functions proved to be extremely useful in virtually every subfield of theoretical computer science, including PCPs, property testing, pseudorandomness, and communication complexity. As it turns out, many seemingly hard problems can be solved by writing the Walsh-Fourier expansion and using basic theorems of harmonic analysis."
cs  math  tcs  article  fourier.transform  pcp
may 2014 by joecamel
Richard Hamming: "Learning to Learn" - YouTube
"The Art of Doing Science and Engineering: Learning to Learn" was the capstone course by Dr. Richard W. Hamming (1915-1998) for graduate students at the Naval Postgraduate School (NPS) in Monterey, California"
hamming  learning  lecture  math  science  youtube  videos  history  1995  advice  inspiration  insightful  research
may 2014 by joecamel
How to work out proofs in Analysis I | Gowers's Weblog
"Now that we’ve had several results about sequences and series, it seems like a good time to step back a little and discuss how you should go about memorizing their proofs. And the very first thing to say about that is that you should attempt to do this while making as little use of your memory as you possibly can."
analysis  education  gowers  math  proof
may 2014 by joecamel
Machine learning is easier than it looks | Hacker News
I'd like to chime in here as a mathematician.
Many people here express their feelings that math or computer science papers are very difficult to read. Some even suggest that they're deliberately written this way. The truth is that yes, they in fact are deliberately written this way, but the reason is actually opposite of many HNers impression: authors want to make the papers easier to understand, and not more difficult.
ml  machinelearning  math  hn  discussion  learning
november 2013 by joecamel
An Intuitive Guide to Linear Algebra | BetterExplained
Some ideas in the article might be used for introductions to Linear Algebra.
algebra  learning  math  linearalgebra
october 2012 by joecamel
Mathematical Education by William P. Thurston
"This essay, originally published in the Sept 1990 Notices of the AMS, discusses problems of our mathematical education system that often stem from widespread misconceptions by well-meaning people of the process of learning mathematics. The essay also discusses ideas for fixing some of the problems. Most of what I wrote in 1990 remains equally applicable today."
math  education  william.thurston  essay  wellwritten  school  inspiration
august 2012 by joecamel
Teichmuller Theory: foreword by William Thurston
"In the same way, any idea in mathematics can be thought about in many different ways, with competing advantages. When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze."
"After a few experiences of reading a few pages only to discover that I really had no idea what I'd just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments. Even so, when something was ``left to the reader'', I generally left it as well."
william.thurston  foreword  wellwritten  math  motivation  thinking
august 2012 by joecamel
Measurement — Paul Lockhart | Harvard University Press
A book by the author of "Mathematician's Lament": "For seven years, Paul Lockhart’s A Mathematician’s Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K–12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living."
book  math  mathematics  paul.lockhart  education
july 2012 by joecamel
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