The Map of Mathematics: Animation Shows How All the Different Fields in Math Fit Together | Open Culture

The Map of Mathematics: Animation Shows How All the Different Fields in Math Fit Together Open Culture by Dan Colman (openculture.com)

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🔖 A de Bruijn identity for discrete random variables by Oliver Johnson, Saikat Guha

A de Bruijn identity for discrete random variables by Oliver Johnson, Saikat Guha (arxiv.org)
We discuss properties of the "beamsplitter addition" operation, which provides a non-standard scaled convolution of random variables supported on the non-negative integers. We give a simple expression for the action of beamsplitter addition using generating functions. We use this to give a self-contained and purely classical proof of a heat equation and de Bruijn identity, satisfied when one of the variables is geometric.
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Jordan Ellenberg don’t know stat | Rick’s Ramblings

Jordan Ellenberg don’t know stat by Rick Durrett, Ph.D. (Rick's Ramblings sites.duke.edu)
There follows a discussion of flipping coins and the fact that frequencies have more random variation when the sample size is small, but he never stops to see if this is enough to explain the observation.

My intuition told me it did not, so I went and got some brain cancer data.

Jordan Ellenberg is called out a bit by Rick Durrett for one of his claims in the best seller How Not To Be Wrong: The Power of Mathematical Thinking.

I remember reading that section of the book and mostly breezing through that argument primarily as a simple example with a limited, but direct point. Durrett decided to delve into the applied math a bit further.

These are some of the subtle issues one eventually comes across when experts read others’ works which were primarily written for much broader audiences.

I also can’t help thinking that one paints a target on one’s back with a book title like that…

BTW, the quote of the day has to be:

… so I went and got some brain cancer data.

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NIMBioS Tutorial: Uncertainty Quantification for Biological Models

NIMBioS Tutorial: Uncertainty Quantification for Biological Models (nimbios.org)
NIMBioS will host an Tutorial on Uncertainty Quantification for Biological Models

Uncertainty Quantification for Biological Models

Meeting dates: June 26-28, 2017
Location: NIMBioS at the University of Tennessee, Knoxville

Organizers:
Marisa Eisenberg, School of Public Health, Univ. of Michigan
Ben Fitzpatrick, Mathematics, Loyola Marymount Univ.
James Hyman, Mathematics, Tulane Univ.
Ralph Smith, Mathematics, North Carolina State Univ.
Clayton Webster, Computational and Applied Mathematics (CAM), Oak Ridge National Laboratory; Mathematics, Univ. of Tennessee

Objectives:
Mathematical modeling and computer simulations are widely used to predict the behavior of complex biological phenomena. However, increased computational resources have allowed scientists to ask a deeper question, namely, “how do the uncertainties ubiquitous in all modeling efforts affect the output of such predictive simulations?” Examples include both epistemic (lack of knowledge) and aleatoric (intrinsic variability) uncertainties and encompass uncertainty coming from inaccurate physical measurements, bias in mathematical descriptions, as well as errors coming from numerical approximations of computational simulations. Because it is essential for dealing with realistic experimental data and assessing the reliability of predictions based on numerical simulations, research in uncertainty quantification (UQ) ultimately aims to address these challenges.

Uncertainty quantification (UQ) uses quantitative methods to characterize and reduce uncertainties in mathematical models, and techniques from sampling, numerical approximations, and sensitivity analysis can help to apportion the uncertainty from models to different variables. Critical to achieving validated predictive computations, both forward and inverse UQ analysis have become critical modeling components for a wide range of scientific applications. Techniques from these fields are rapidly evolving to keep pace with the increasing emphasis on models that require quantified uncertainties for large-scale applications. This tutorial will focus on the application of these methods and techniques to mathematical models in the life sciences and will provide researchers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties and perform sensitivity analysis for simulation models. Concepts to be covered may include: probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, adaptive surrogate model construction, high-dimensional approximation, random sampling and sparse grids, as well as local and global sensitivity analysis.

This tutorial is intended for graduate students, postdocs and researchers in mathematics, statistics, computer science and biology. A basic knowledge of probability, linear algebra, and differential equations is assumed.

Descriptive Flyer

Application deadline: March 1, 2017
To apply, you must complete an application on our online registration system:

  1. Click here to access the system
  2. Login or register
  3. Complete your user profile (if you haven’t already)
  4. Find this tutorial event under Current Events Open for Application and click on Apply

Participation in NIMBioS tutorials is by application only. Individuals with a strong interest in the topic are encouraged to apply, and successful applicants will be notified within two weeks after the application deadline. If needed, financial support for travel, meals, and lodging is available for tutorial attendees.

Summary Report. TBA

Live Stream. The Tutorial will be streamed live. Note that NIMBioS Tutorials involve open discussion and not necessarily a succession of talks. In addition, the schedule as posted may change during the Workshop. To view the live stream, visit http://www.nimbios.org/videos/livestream. A live chat of the event will take place via Twitter using the hashtag #uncertaintyTT. The Twitter feed will be displayed to the right of the live stream. We encourage you to post questions/comments and engage in discussion with respect to our Social Media Guidelines.


Source: NIMBioS Tutorial: Uncertainty Quantification for Biological Models

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Walter B. Rudin: "Set Theory: An Offspring of Analysis"

Set Theory: An Offspring of Analysis by Walter B. Rudin (YouTube)
Prof. Walter B. Rudin presents the lecture, "Set Theory: An Offspring of Analysis." Prof. Jay Beder introduces Prof. Dattatraya J. Patil who introduces Prof....

MyScript MathPad for LaTeX and Livescribe

MyScript MathPad for LaTeX (myscript.com)
MyScript MathPad is a mathematic expression demonstration that lets you handwrite your equations or mathematical expressions on your screen and have them rendered into their digital equivalent for easy sharing. Render complex mathematical expressions easily using your handwriting with no constraints. The result can be shared as an image or as a LaTeX* or MathML* string for integration in your documents.

This looks like something I could integrate into my workflow.

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Mathematical Model Reveals the Patterns of How Innovations Arise | MIT Technology Review

Mathematicians have discovered how the universal patterns behind innovation arise by Emerging Technology from the arXiv (MIT Technology Review)
A mathematical model could lead to a new approach to the study of what is possible, and how it follows from what already exists.

Continue reading “Mathematical Model Reveals the Patterns of How Innovations Arise | MIT Technology Review”

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🔖 AMS Open Math Notes

Open Math Notes by American Mathematical Society (ams.org)
AMS Open Math Notes is a repository of freely downloadable mathematical works in progress hosted by the American Mathematical Society as a service to researchers, teachers and students. These draft works include course notes, textbooks, and research expositions in progress. They have not been published elsewhere, and, as works in progress, are subject to significant revision. Visitors are encouraged to download and use these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.

h/t to Terry Tao for the notice.

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🔖 Group Theory Lectures by Steven Roman

Playlist of Group Theory Lectures by Steven Roman by Steven Roman (youtube.com)
Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube.

Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube. No prior experience in group theory is necessary. He’s the author of the recent Fundamentals of Group Theory: An Advanced Approach. [1]

He hopes to eventually also offer lectures on ring theory, fields, vector spaces, and module theory in the near future.

Fundamentals of Group Theory by Steven Roman

References

[1]
S. Roman, Fundamentals of Group Theory: An Advanced Approach, 2012th ed. Birkhäuser, 2011.
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Basic Category Theory by Tom Leinster | Free Ebook Download

Basic Category Theory by Tom Leinster (arxiv.org)
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. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.

Tom Leinster has released a digital e-book copy of his textbook Basic Category Theory on arXiv [1]

h/t to John Carlos Baez for the notice:

My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory! It’s free:

https://arxiv.org/abs/1612.09375

It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.

Huh? What’s a ‘universal property’?

In category theory, we try to describe things by saying what they do, not what they’re made of. The reason is that you can often make things out of different ingredients that still do the same thing! And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do.

A universal property amounts to a precise description of what an object does.

Universal properties show up in three closely connected ways in category theory, and Tom’s book explains these in detail:

through representable functors (which are how you actually hand someone a universal property),

through limits (which are ways of building a new object out of a bunch of old ones),

through adjoint functors (which give ways to ‘freely’ build an object in one category starting from an object in another).

If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom’s book! It’s not easy to learn this stuff – but it’s good for your brain. It literally rewires your neurons.

Here’s what he wrote, over on the category theory mailing list:

…………………………………………………………………..

Dear all,

My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:

https://arxiv.org/abs/1612.09375

It’s also freely editable, under a Creative Commons licence. For instance, if you want to teach a class from it but some of the examples aren’t suitable, you can delete them or add your own. Or if you don’t like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros. Just go the arXiv, download, and edit to your heart’s content.

There are lots of good introductions to category theory out there. The particular features of this one are:
• It’s short.
• It doesn’t assume much.
• It sticks to the basics.

 

References

[1]
T. Leinster, Basic Category Theory, 1st ed. Cambridge University Press, 2014.
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Primes as a Service on Twitter

Our friend Andrew Eckford has spent some time over the holiday improving his Twitter bot Primes as a Service. He launched it in late Spring of 2016, but has added some new functionality over the holidays. It can be relatively handy if you need a quick answer during a class, taking an exam(?!), to settle a bet at a mathematics tea, while livetweeting a conference, or are hacking into your favorite cryptosystems.

General Instructions

Tweet a positive 9-digit (or smaller) integer at @PrimesAsAService. It will reply via Twitter to tell you if the number prime or not.

Some of the usable commands one can tweet to the bot for answers follow. (Hint: Click on the buttons with the tweet text to auto-generate the relevant Tweet.)

If you ask about a prime number with a twin prime, it should provide the twin.

Pro tip: You should be able to drag and drop any of the buttons above to your bookmark bar for easy access/use in the future.

Happy prime tweeting!

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Emily Riehl’s new category theory book has some good company

Emily Riehl's new category theory book has some good company.

Emily Riehl's new category theory book has some good company. It's a beautiful book by the way
Emily Riehl’s new category theory book has some good company. It’s a beautiful book by the way.

Instagram filter used: Clarendon

Photo taken at: UCLA Bookstore

I just saw Emily Riehl‘s new book Category Theory in Context on the shelves for the first time. It’s a lovely little volume beautifully made and wonderfully typeset. While she does host a free downloadable copy on her website, the book and the typesetting is just so pretty, I don’t know how one wouldn’t purchase the physical version.

I’ll also point out that this is one of the very first in Dover’s new series Aurora: Dover Modern Math Originals. Dover has one of the greatest reprint collections of math texts out there, I wish them the best in publishing new works with the same quality and great prices as they always have! We need more publishers like this.

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The first quarter of Complex Analysis is slowly drawing to a close

The first quarter of Complex Analysis is slowly drawing to a close.

The first quarter of Complex Analysis is elderly drawing to a close.

Instagram filter used: Normal

Photo taken at: UCLA Math Sciences Building

There’s still plenty of time to join us for the second installment in January!

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