Another Word For It Patrick Durusau on Topic Maps and Semantic Diversity

August 20, 2014

Math for machine learning

Filed under: Algebra,Machine Learning,Mathematics — Patrick Durusau @ 7:36 pm

Math for machine learning by Zygmunt Zając.

From the post:

Sometimes people ask what math they need for machine learning. The answer depends on what you want to do, but in short our opinion is that it is good to have some familiarity with linear algebra and multivariate differentiation.

Linear algebra is a cornerstone because everything in machine learning is a vector or a matrix. Dot products, distance, matrix factorization, eigenvalues etc. come up all the time.

Differentiation matters because of gradient descent. Again, gradient descent is almost everywhere*. It found its way even into the tree domain in the form of gradient boosting – a gradient descent in function space.

We file probability under statistics and that’s why we don’t mention it here.
…

Following this introduction you will find a series of books, MOOCs, etc. on linear algebra, calculus and other math resources.

Pass it along!

July 30, 2014

Ideals, Varieties, and Algorithms

Filed under: Computer Science,Mathematics — Patrick Durusau @ 4:10 pm

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Communtative Algebra by David Cox, John Little, and Donal OShea.

From the introduction:

We wrote this book to introduce undergraduates to some interesting ideas in algebraic geometry and commutative algebra. Until recently, these topics involved a lot of abstract mathematics and were only taught in graduate school. But in the 1960s, Buchberger and Hironaka discovered new algorithms for manipulating systems of polynomial equations. Fueled by the development of computers fast enough to run these algorithms, the last two decades have seen a minor revolution in commutative algebra. The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand, and has changed the practice of much research in algebraic geometry. This has also enhanced the importance of the subject for computer scientists and engineers, who have begun to use these techniques in a whole range of problems.

The authors do presume students “…have access to a computer algebra system.”

The Wikipedia List of computer algebra systems has links to numerous such systems. A large number of which are free.

That list is headed by Axiom (Wikipedia article) and is an example of literate programming. The Axiom documentation looks like a seriously entertaining time sink! You may want to visit http://axiom-developer.org/

I haven’t installed Axiom so take that as a comment on its documentation more than its actual use. Use whatever system you like best and fits your finances.

I first saw this in a tweet from onepaperperday.

Enjoy!

July 25, 2014

Advanced Data Analysis from an Elementary Point of View (update)

Filed under: Data Analysis,Mathematics,Statistics — Patrick Durusau @ 3:37 pm

Advanced Data Analysis from an Elementary Point of View by Cosma Rohilla Shalizi. (8 January 2014)

From the introduction:

These are the notes for 36-402, Advanced Data Analysis, at Carnegie Mellon. If you are not enrolled in the class, you should know that it’s the methodological capstone of the core statistics sequence taken by our undergraduate majors (usually in their third year), and by students from a range of other departments. By this point, they have taken classes in introductory statistics and data analysis, probability theory, mathematical statistics, and modern linear regression (“401”). This class does not presume that you have learned but forgotten the material from the pre-requisites; it presumes that you know that material and can go beyond it. The class also presumes a firm grasp on linear algebra and multivariable calculus, and that you can read and write simple functions in R. If you are lacking in any of these areas, now would be an excellent time to leave.

I last reported on this draft in 2012 at: Advanced Data Analysis from an Elementary Point of View

Looking forward to this works publication by Cambridge University Press.

I first saw this in a tweet by Mark Patterson.

July 15, 2014

Graph Classes and their Inclusions

Filed under: Graphs,Mathematics — Patrick Durusau @ 4:25 pm

Information System on Graph Classes and their Inclusions

From the webpage:

What is ISGCI?

ISGCI is an encyclopaedia of graphclasses with an accompanying java application that helps you to research what’s known about particular graph classes. You can:

  • check the relation between graph classes and get a witness for the result
  • draw clear inclusion diagrams
  • colour these diagrams according to the complexity of selected problems
  • find the P/NP boundary for a problem
  • save your diagrams as Postscript, GraphML or SVG files
  • find references on classes, inclusions and algorithms

As of 214-07-06, the database contains 1497 classes and 176,888 inclusions.

If you are past the giddy stage of “Everything’s a graph!,” you may find this site useful.

July 13, 2014

Abstract Algebra

Filed under: Algebra,Mathematics — Patrick Durusau @ 7:19 pm

Abstract Algebra by Benedict Gross, PhD, George Vasmer Leverett Professor of Mathematics, Harvard University.

From the webpage:

Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields.

Videos, notes, problem sets from the Harvard Extension School.

The relationship between these videos and those found on YouTube isn’t clear.

The text book for the class was Algebra by Michael Artin. (There is a 2nd edition now.)

There are two comments that may motivate you to pursue these lectures:

First, Gross remarks in the first session that there are numerous homework assignments because you are learning a language. Which makes me curious why math isn’t taught like a language?

Second, the Wikipedia article on abstract algebra observes in part:

Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties. This creates a false impression that in algebra axioms had come first and then served as a motivation and as a basis of further study. The true order of historical development was almost exactly the opposite. For example, the hypercomplex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. Most theories that are now recognized as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. An archetypical example of this progressive synthesis can be seen in the history of group theory.

Interesting that techniques are developed for quite practical reasons but later justified with greater formality.

Suggests that semantic integration should focus on practical results and leave formal justification for later.

Yes?

I first saw this in a tweet by Steven Strogatz.

June 18, 2014

Elsevier open access mathematics

Filed under: Mathematics,Open Access — Patrick Durusau @ 10:20 am

Elsevier open access mathematics

From the webpage:

Elsevier has opened up the back archives of their mathematics journals. All articles older than 4 years are available under a license [1] [2]. This license is compatible with non-commercial redistribution, and so we have collected the PDFs and made them available here.

Each of the links below is for a torrent file; opening this in a suitable client (e.g.Transmission) will download that file. Unzipping that file creates a directory with all the PDFs, along with a copy of the relevant license file.

Although Elsevier opens their archives on a rolling basis, the collections below only contain articles up to 2009. We anticipate adding yearly updates.

You can download a zip file containing all of the torrents below, if you’d like the entire collection. You’ll need about 40GB of free space.

Excellent!

Occurs to me this corpus is suitable for testing indexing and navigation of mathematical literature.

Is your favorite mathematics publisher following Elsevier’s lead?

I first saw this in a tweet by Stephen A. Goss.

June 17, 2014

Make Category Theory Intuitive!

Filed under: Category Theory,Mathematics — Patrick Durusau @ 10:38 am

Make Category Theory Intuitive! by Jocelyn Ireson-Paine.

From the post:

As I suggested in Chapter 1, the large, highly evolved sensory and motor portions of the brain seem to be the hidden powerhouse behind human thought. By virtue of the great efficiency of these billion-year-old structures, they may embody one million times the effective computational power of the conscious part of our minds. While novice performance can be achieved using conscious thought alone, master-level expertise draws on the enormous hidden resources of these old and specialized areas. Sometimes some of that power can be harnessed by finding and developing a useful mapping between the problem and a sensory intuition.

Although some individuals, through lucky combinations of inheritance and opportunity have developed expert intuitions in certain fields, most of us are amateurs at most things. What we need to improve our performance is explicit external metaphors that can tap our instinctive skills in a direct and repeatable way. Graphs, rules of thumb, physical models illustrating relationships, and other devices are widely and effectively used to enhance comprehension and retention. More recently, interactive pictorial computer interfaces such as those used in the Macintosh have greatly accelerated learning in novices and eased machine use for the experienced. The full sensory involvement possible with magic glasses may enable us to go much further in this direction. Finding the best metaphors will be the work of a generation: for now, we can amuse ourselves by guessing.

Hans Moravec, in Mind Children [Moravec 1988].

This is an essay on why I believe category theory is important to computer science, and should therefore be promoted; and on how we might do so. While writing this, I discovered the passage I’ve quoted above. My ideas are closely related, and since there’s nothing more pleasing than being supported by such an authority, that’s why I’ve quoted it here.

Category theory has been around since the 1940s, and was invented to unify different treatments of homology theory, a branch of algebraic topology [Marquis 2004; Natural transformation]. It’s from there that many examples used in teaching category theory to mathematicians come. Which is a shame, because algebraic topology is advanced: probably post-grad level. Examples based on it are not much use below that level, and not much use to non-mathematicians. The same applies to a lot of the other maths to which category theory has been applied.
…

An interesting essay with many suggestions for teaching category theory. Follow this essay with a visit to Jocelyn’s homepage and the resources on category theory cited there. Caution: You will find a number of other very interesting things on the homepage. You have been warned. 😉

Spreadsheets too! What is it with people studying something that is nearly universal in business and science? Is that why vendors make money? Pandering to the needs of the masses? Is that a clue on being a successful startup? Appeal to the masses and not the righteous?

If you try the Category Theory Demonstrations, be aware the page refreshes with a statement: “Your results are here” near the top of the page. Follow that link for your results.

In a thread on those demonstrations, Jocelyn laments the lack of interest among those who already understand category theory in finding more intuitive ways of explaining it. I have no explanation to offer but can attest to the same lack of interest among lawyers, academics in general, etc. in making their knowledge more “intuitive.”

So how do you successfully promote projects designed to make institutions or disciplines more “transparent?” A general public interest in “transparency” doesn’t translate easily into donations or institutional support. Suggestions?

June 8, 2014

“How Not to Be Wrong”:…

Filed under: Mathematical Reasoning,Mathematics — Patrick Durusau @ 7:40 pm

“How Not to Be Wrong”: What the literary world can learn from math by Laura Miller.

From the post:

Jordan Ellenberg’s “How Not to Be Wrong: The Power of Mathematical Thinking” is a miscellaneous romp through the world of quantitative reasoning. You can tell just how modular the book is by the way bits of it have been popping up all over the Web of late, promising to explain such mysteries as why so many handsome men are jerks or why an athlete’s performance always seems to suffer a drop-off after he signs a big contract. This pull-apart quality may sound like a bug, but in fact it’s a feature. It makes “How Not to Be Wrong” a rewarding popular math book for just about anyone.

Ellenberg is a professor of mathematics at the University of Wisconsin-Madison who spent “some part of my early 20s thinking I might want to be a Serious Literary Novelist.” (He even published a novel, “The Grasshopper King.”) He has a popular math column at Slate. So Ellenberg can write, and furthermore he brings a set of references to the subject that will make many a numbers-shy humanities major feel right at home. He explains why B.F. Skinner’s “proof” that Shakespeare was not particularly inclined toward poetic alliteration was incorrect, and he is as likely to refer to Robert Frost and Thomas Pynchon as to such mathematical titans as R.A. Fisher and Francis Galton. He takes the playful, gentle, humorous tone of a writer used to cajoling his readers into believing that they can understand what he’s talking about.

At the same time, those who like to pull out a pencil and a piece of paper and work through a few equations for the fun of it should not be utterly put off by the aforementioned cajoling. Interested in a brush-up on plane geometry, the implications of the Prime Numbers Theorem and Yitang Zhang’s recently announced proof of the “bounded gaps” conjecture about the distribution of primes? Pull up a chair. On the other hand, if any sentence containing the term “log N” makes you go cross-eyed and befuddled, you need only turn a few pages ahead to another chapter, where you can read about how two separate math cartels gamed the Massachusetts state lottery.
…

Laura’s review will leave you convinced that “Have you read ‘How Not to be Wrong’ by Jordan Ellenberg?” should be on your technical interview checklist.

Or at least you should pull examples from it to use in your technical interviews.

Despite all the hype about “big data,” “web scale,” etc., the nature of coherent thinking has not changed. The sooner you winnow out candidates that thing otherwise the better.

May 29, 2014

Categorical Databases

Filed under: Category Theory,Database,Mathematics — Patrick Durusau @ 4:39 pm

Categorical Databases by David I. Spivak.

From Slide 2 of 58:

There is a fundamental connection between databases and categories.

  • Category theory can simplify how we think about and use databases.
  • We can clearly see all the working parts and how they fit together.
  • Powerful theorems can be brought to bear on classical DB problems.

The slides are “text heavy” but I think you will find that helpful rather than a hindrance in this case. 😉

From David Spivak’s homepage:

Purpose: I study information and communication, working towards a mathematical foundation for interoperability.

If you are looking for more motivation to get into category theory, this could be the place to start.

I first saw this in a tweet by Jim Duey.

May 21, 2014

OpenIntro Statistics

Filed under: Mathematics,Statistics — Patrick Durusau @ 6:44 pm

OpenIntro Statistics

From the about page:

The mission of OpenIntro is to make educational products that are free, transparent, and lower barriers to education.

The site includes a textbook, labs (R), videos, teachers resources, forums and extras, including data.

A good template for courses in other technical areas.

I first saw this in Chris Blattman’s Links I liked

Online Statistics Education:…

Filed under: Mathematics,Statistics — Patrick Durusau @ 4:58 pm

Online Statistics Education: An Interactive Multimedia Course of Study. Project Leader: David M. Lane, Rice University.

From the project homepage:

Online Statistics: An Interactive Multimedia Course of Study is a resource for learning and teaching introductory statistics. It contains material presented in textbook format and as video presentations. This resource features interactive demonstrations and simulations, case studies, and an analysis lab.

A far cry from introductory statistics pre-Internet. Definitely a resource to recommend to others.

I first saw this in Chris Blattman’s Links I liked

May 12, 2014

Categories from scratch

Filed under: Category Theory,Computer Science,Mathematics — Patrick Durusau @ 7:00 pm

Categories from scratch by Rapahel ‘kena’ Poss.

From the post:

Prologue

The concept of category from mathematics happens to be useful to computer programmers in many ways. Unfortunately, all “good” explanations of categories so far have been designed by mathematicians, or at least theoreticians with a strong background in mathematics, and this makes categories especially inscrutable to external audiences.

More specifically, the common explanatory route to approach categories is usually: “here is a formal specification of what a category is; then look at these known things from maths and theoretical computer science, and admire how they can be described using the notions of category theory.” This approach is only successful if the audience can fully understand a conceptual object using only its formal specification.

In practice, quite a few people only adopt conceptual objects by abstracting from two or more contexts where the concepts are applicable, instead. This is the road taken below: reconstruct the abstractions from category theory using scratches of understanding from various fields of computer engineering.

Overview

The rest of this document is structured as follows:

  1. introduction of example Topics of study: unix process pipelines, program statement sequences and signal processing circuits;
  2. Recollections of some previous knowledge about each example; highlight of interesting analogies between the examples;
  3. Identification of the analogies with existing concepts from category theory;
  4. a quick preview of Goodies from category theory;
  5. references to Further reading.

If you don’t already grok category theory, perhaps this will be the approach that tips the balance in your favor!

April 20, 2014

Group Explorer 2.2

Filed under: Algebra,Group Theory,Mathematics — Patrick Durusau @ 11:01 am

Group Explorer 2.2

From the webpage:

Primary features listed here, or read the version 2.2 release notes.

  • Displays Cayley diagrams, multiplication tables, cycle graphs, and objects with symmetry
  • Many common group-theoretic computations can be done visually
  • Compare groups and subgroups via morphisms (see illustration below)
  • Browsable, searchable group library
  • Integrated help system (which you can preview on the web)
  • Save and print images at any scale and quality

Are there symmetries in your data?

I first saw this in a tweet by Steven Strogatz.

BTW, Steven also points to this example of using Group Explorer: Cayley diagrams of the first five symmetric groups.

April 5, 2014

Algorithmic Number Theory, Vol. 1: Efficient Algorithms

Filed under: Algebra,Algorithms,Mathematics — Patrick Durusau @ 7:20 pm

Algorithmic Number Theory, Vol. 1: Efficient Algorithms by Eric Bach and Jeffrey Shallit.

From the preface:

This is the first volume of a projected two-volume set on algorithmic number theory, the design and analysis of algorithms for problems from the theory of numbers. This volume focuses primarily on those problems from number theory that admit relatively efficient solutions. The second volume will largely focus on problems for which efficient algorithms are not known, and applications thereof.

We hope that the material in this book will be useful for readers at many levels, from the beginning graduate student to experts in the area. The early chapters assume that the reader is familiar with the topics in an undergraduate algebra course: groups, rings, and fields. Later chapters assume some familiarity with Galois theory.

As stated above, this book discusses the current state of the art in algorithmic number theory. This book is not an elementary number theory textbook, and so we frequently do not give detailed proofs of results whose central focus is not computational. Choosing otherwise would have made this book twice as long.

The webpage offers the BibTeX files for the bibliography, which includes more than 1800 papers and books.

BTW, Amazon notes that Volume 2 was never published.

Now that high performance computing resources are easily available, perhaps you can start working on your own volume 2. Yes?

I first saw this in a tweet by Alvaro Videla.

April 3, 2014

Developing a 21st Century Global Library for Mathematics Research

Filed under: Identification,Identifiers,Identity,Mathematics,Subject Identity — Patrick Durusau @ 8:58 pm

Developing a 21st Century Global Library for Mathematics Research by Committee on Planning a Global Library of the Mathematical Sciences.

Care to guess what one of the major problems facing mathematical research might be?

Currently, there are no satisfactory indexes of many mathematical objects, including symbols and their uses, formulas, equations, theorems, and proofs, and systematically labeling them is challenging and, as of yet, unsolved. In many fields where there are more specialized objects (such as groups, rings, fields), there are community efforts to index these, but they are typically not machine-readable, reusable, or easily integrated with other tools and are often lacking editorial efforts. So, the issue is how to identify existing lists that are useful and valuable and provide some central guidance for further development and maintenance of such lists. (p. 26)

Does that surprise you?

What do you think the odds are of mathematical research slowing down enough for committees to decide on universal identifiers for all the subjects in mathematical publications?

That’s about what I thought.

I have a different solution: Why not ask mathematicians who are submitting articles for publication to identity (specify properties for) what they consider to be the important subjects in their article?

The authors have the knowledge and skill, not to mention the motivation of wanting their research to be easily found by others.

Over time I suspect that particular fields will develop standard identifications (sets of properties per subject) that mathematicians can reuse to save themselves time when publishing.

Mappings across those sets of properties will be needed but that can be the task of journals, researchers and indexers who have an interest and skill in that sort of enterprise.

As opposed to having a “boil the ocean” approach that tries to do more than any one project is capable of doing competently.

Distributed subject identification is one way to think about it. We already do it, this would be a semi-formalization of that process and writing down what each author already knows.

Thoughts?

PS: I suspect the condition recited above is true for almost any sufficiently large field of study. A set of 150 million entities sounds large only without context. In the context of of science, it is a trivial number of entities.

March 17, 2014

Isaac Newton’s College Notebook

Filed under: Mathematics,Navigation — Patrick Durusau @ 8:30 pm

College Notebook by Isaac Newton.

From the description:

This small notebook was probably used by Newton from about 1664 to 1665. It contains notes from his reading on mathematics and geometry, showing particularly the influence of John Wallis and René Descartes. It also provides evidence of the development of Newton’s own mathematical thinking, including his study of infinite series and development of binomial theorem, the evolution of the differential calculus, and its application to the problem of quadratures and integration.

This notebook contains many blank pages (all shown) and has been used by Newton from both ends. Our presentation displays the notebook in a sensible reading order. It shows the ‘front’ cover and the 79 folios that follow (more than half of them blank) and then turns the notebook upside down showing the other cover and the pages that follow it. A full transcription is provided. The notebook was photographed while it was disbound in 2011.

The video above provides an introduction to Newton’s mathematical thinking at the time of this manuscript.

The Web remains erratic but there are more jewels like this one than say ten (10) years ago.

Curious how you would link up Einstein’s original notes on gravity waves (Einstein Papers Project) with the recent reported observation of gravity waves?

Seems like that would be important. And to collate all the materials on gravity waves between Einstein’s notes and the recent observations.

More and more information is coming online but appears to be as disjointed as it was prior to coming online. That’s a pity.

I first saw this in a tweet by Steven Strogatz.

Steven also points to: What led Newton to discover the binomial theorum? Would you believe it was experimentation and not mathematical proofs?

Hmmm, is there a lesson for designing topic map interfaces? To experiment rather than rely upon the way we know it must be?

March 13, 2014

Category Theory References

Filed under: Category Theory,Mathematics — Patrick Durusau @ 2:20 pm

Category Theory References

Ten (10) pages of category theory citations that I bookmarked recently.

The citations are not annotated so they are of limited utility but it looked worth passing along.

Are there any ongoing annotated lists of references for category theory?

The American Mathematical Association (AMA) indexing scheme for 18 Category theory; homological algebra isn’t detailed enough to substitute for an annotated listing. (Be aware that category theory appears under other classifications so use the search function for 2010 Mathematics Subject Classification if you want to find all appearances of category theory.)

Enjoy!

March 11, 2014

Number Theory and Algebra

Filed under: Algebra,Cryptography,Mathematics — Patrick Durusau @ 6:28 pm

A Computational Introduction to Number Theory and Algebra by Victor Shoup.

The first and second editions, published by Cambridge University Press are available for download under a Creative Commons license.

From the preface of the second edition:

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. My goal in writing this book was to provide an introduction to number theory and algebra, with an emphasis on algorithms and applications, that would be accessible to a broad audience. In particular, I wanted to write a book that would be appropriate for typical students in computer science or mathematics who have some amount of general mathematical experience, but without presuming too much specific mathematical knowledge.

Even though reliance on cryptography and vendors of cryptography is fading, you are likely to encounter people still using cryptography or legacy data “protected” by cryptography.

BTW, this is only one of several books that Cambridge University Press has published and allowed the final text to remain available.

Should you pen something appropriate and hopefully profitable for you and a publisher, Cambridge University Press should be on your short list.

Cambridge University Press is a great press and a good citizen of the academic world.
.
I first saw this in a tweet by Algebra Fact.

March 10, 2014

Algebraic and Analytic Programming

Filed under: Algebra,Analytics,Cyc,Mathematics,Ontology,Philosophy,SUMO — Patrick Durusau @ 9:17 am

Algebraic and Analytic Programming by Luke Palmer.

In a short post Luke does a great job contrasting algebraic versus analytic approaches to programming.

In an even shorter summary, I would say the difference is “truth” versus “acceptable results.”

Oddly enough, that difference shows up in other areas as well.

The major ontology projects, including linked data, are pushing one and only one “truth.”

Versus other approaches, such as topic maps (at least in my view), that tend towards “acceptable results.”

I am not sure what other measure of success you would have other than “acceptable results?”

Or what another measure for a semantic technology would be other than “acceptable results?”

Whether the universal truth of the world folks admit it or not, they just have a different definition of “acceptable results.” Their “acceptable results” means their world view.

I appreciate the work they put into their offer but I have to decline. I already have a world view of my own.

You?

I first saw this in a tweet by Computer Science.

March 6, 2014

Algebird 0.5.0 Released

Filed under: Algebird,Mathematics,Scalding,Storm — Patrick Durusau @ 9:24 pm

Algebird 0.5.0

From the webpage:

Abstract algebra for Scala. This code is targeted at building aggregation systems (via Scalding or Storm). It was originally developed as part of Scalding’s Matrix API, where Matrices had values which are elements of Monoids, Groups, or Rings. Subsequently, it was clear that the code had broader application within Scalding and on other projects within Twitter.

Other links you will find helpful:

0.5.0 Release notes.

Algebird mailing list.

Algebird Wiki.

February 22, 2014

MathDL Mathematical Communication

Filed under: Communication,Mathematics,Statistics — Patrick Durusau @ 3:53 pm

MathDL Mathematical Communication

From the post:

MathDL Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

This site addresses diverse aspects of mathematical communication, including

Here is a brief summary of suggestions to consider as you design a mathematics class that includes communication.

This site originated at M.I.T. so most of the current content is for teaching upper-level undergraduates to communicate as mathematicians.

The site is now yours. Contribute materials! Suggest improvements!

I discovered this site from a reference at Project Laboratory in Mathematics.

As the complexity of data and data analysis increases, so is you need to communicate mathematics and mathematics-based concepts to lay persons. There is much here that may assist in that task.

With enough experience: The wise you can persuade and the lesser folks you can daunt. 😉

Project Laboratory in Mathematics

Filed under: Education,Mathematics — Patrick Durusau @ 3:17 pm

Project Laboratory in Mathematics by Prof. Haynes Miller, Dr. Nat Stapleton, Saul Glasman, and Susan Ruff.

From the description:

Project Laboratory in Mathematics is a course designed to give students a sense of what it’s like to do mathematical research. In teams, students explore puzzling and complex mathematical situations, search for regularities, and attempt to explain them mathematically. Students share their results through professional-style papers and presentations.

This course site was created specifically for educators interested in offering students a taste of mathematical research. This site features extensive description and commentary from the instructors about why the course was created and how it operates.

Aside from the introductory lecture by Prof. Miller, the next best part are two problem sets, the editing process and resulting final paper.

Something like this, adjusted for grade level, looks far more valuable rote coding exercises.

Simple Ain’t Easy

Filed under: Data Analysis,Mathematics,Statistics — Patrick Durusau @ 2:27 pm

Simple Ain’t Easy: Real-World Problems with Basic Summary Statistics by John Myles White.

From the webpage:

In applied statistical work, the use of even the most basic summary statistics, like means, medians and modes, can be seriously problematic. When forced to choose a single summary statistic, many considerations come into practice.

This repo attempts to describe some of the non-obvious properties possessed by standard statistical methods so that users can make informed choices about methods.

Contributing

The reason I chose to announce a book of examples isn’t just pedagogical: by writing fully independent examples, it’s possible to write a book as a community working in parallel. If 30 people each contributed 10 examples over the next month, we’d have a full-length book containing 300 examples in our hands. In practice, things are complicated by the need to make sure that examples aren’t redundant or low quality, but it’s still possible to make this book a large-scale community project.

As such, I hope you’ll consider contributing. To contribute, just submit a new example. If your example only requires text, you only need to write a short LaTeX-flavored Markdown document. If you need images, please include R code that generates your images.

A great project for several reasons.

First, you can contribute to a public resource that may improve the use of summary statistics.

Second, you have the opportunity to search the literature for examples you want to use on summary statistics. That will improve your searching skills and data skepticism. The first from finding the examples and the second from seeing how statistics are used in the “wild.”

Not to bang on statistics too harshly, I review standards where authors have forgotten how to use quotes and footnotes. Sixth grade type stuff.

Third, and to me the most important reason, as you review the work of others, you will become more conscious of similar mistakes in your own writing.

Think of contributions to Simple Ain’t Easy as exercises in self-improvement that benefit others.

February 18, 2014

Writing about Math…

Filed under: Communication,Mathematics,Writing — Patrick Durusau @ 11:02 am

Writing about Math for the Perplexed and the Traumatized by Steven Strogatz.

From the introduction:

In the summer of 2009 I received an unexpected email from David Shipley, the editor of the op-ed page for the New York Times. He invited me to look him up next time I was in the city and said there was something he’d like to discuss.

Over lunch at the Oyster Bar restaurant in Grand Central Station, he asked whether I’d ever have time to write a series about the elements of math aimed at people like him. He said he’d majored in English in college and hadn’t studied math since high school. At some point he’d lost his way and given up. Although he could usually do what his math teachers had asked of him, he’d never really seen the point of it. Later in life he’d been puzzled to hear math described as beautiful. Could I convey some of that beauty to his readers, many of whom, he suspected, were as lost he was?

I was thrilled by his proposition. I love math, but even more than that, I love trying to explain it. Here I’d like to touch on a few of the writing challenges that this opportunity entailed, along with the goals I set for myself, and then describe how, by borrowing from three great science writers, I tried to meet those challenges. I’m not sure if any of my suggestions will help other mathematicians who’d like to share their own love of math with the public, but that’s my hope.

If you are looking for tips and examples of how to explain computer science topics, you have arrived!

Not only is this essay by Strogatz highly useful and entertaining, you can also consult his fifteen (15) part series on math that appeared in the New York Times.

The New York Times series ended in 2010 but you can following Steven at: @stevenstrogatz and at RadioLab.

I first saw this in a tweet by Michael Nielsen.

BTW, if you have contacts at the New York Times, would you mention that including hyperlinks for Twitter handles and websites is a matter of common courtesy? Thanks!

February 2, 2014

Category Theory Using String Diagrams

Filed under: Category Theory,Mathematics — Patrick Durusau @ 4:07 pm

Category Theory Using String Diagrams by Dan Marsden.

Abstract:

In work of Fokkinga and Meertens a calculational approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting retain the vital type information, but poorly express the reasoning and development of categorical proofs. In order to combine the strengths of these two perspectives, we propose the use of string diagrams, common folklore in the category theory community, allowing us to retain the type information whilst pursuing a calculational form of proof. These graphical representations provide a topological perspective on categorical proofs, and silently handle functoriality and naturality conditions that require awkward bookkeeping in more traditional notation.

Our approach is to proceed primarily by example, systematically applying graphical techniques to many aspects of category theory. We develop string diagrammatic formulations of many common notions, including adjunctions, monads, Kan extensions, limits and colimits. We describe representable functors graphically, and exploit these as a uniform source of graphical calculation rules for many category theoretic concepts. We then use these graphical tools to explicitly prove many standard results in our proposed string diagram based style of proof.

This form of visualization does seem to be easier on the eyes. 😉

Whether it is sufficient or not for some particular purpose, remains to be seen.

January 3, 2014

The Ten Commandments of Statistical Inference

Filed under: Mathematics,Statistics — Patrick Durusau @ 2:45 pm

The Ten Commandments of Statistical Inference by Dr. Richard Lenski.

From the post:

1. Remember the type II error, for therein is reflected the power if not the glory.

These ten commandments (see the post for the other nine) are part and parcel of knowing your data and the assumptions of the processing applied to it.

Think of it as a short checklist to keep yourself and especially others, honest.

December 30, 2013

Linear algebra explained in four pages

Filed under: Mathematics — Patrick Durusau @ 3:44 pm

Linear algebra explained in four pages by Ivan Savov.

From the introduction:

This document will review the fundamental ideas of linear algebra. We will learn about matrices, matrix operations, linear transformations and discuss both the theoretical and computational aspects of linear algebra. The tools of linear algebra open the gateway to the study of more advanced mathematics. A lot of knowledge buzz awaits you if you choose to follow the path of understanding, instead of trying to memorize a bunch of formulas.

A rather dense four pages. 😉

It is based on the No Bullshit: Guide to Math and Physics.

The general tone of comments on the No Bullshit Guide… are positive, mostly very positive but I haven’t seen any professional reviews of it.

It you know of any professional reviews, please drop me a note.

I first saw this in Christophe Lalanne’s A bag of tweets / December 2013.

December 23, 2013

Planar Graphs and Ternary Trees

Filed under: Graphs,Mathematics — Patrick Durusau @ 5:52 pm

Donald Knuth’s Annual Christmas Tree Lecture: Planar Graphs and Ternary Trees

From the description:

In this lecture, Professor Knuth discusses the beautiful connections between certain trees with three-way branching and graphs that can be drawn in the plane without crossing edges.

Additional resources that will be helpful:

From Knuth’s Programs to Read:

SKEW-TERNARY-CALC and a MetaPost file for its illustrations.
Computes planar graphs that correspond to ternary trees in an amazing way; here’s a PDF file for its documentation

Quad-edge (Wikipedia)

Quad-Edge Data Structure and Library by Paul Heckbert.

The Quad-Edge data structure is useful for describing the topology and geometry of polyhedra. We will use it when implementing subdivision surfaces (a recent, elegant way to define curved surfaces) because it is elegant and it can answer adjacency queries efficiently. In this document we describe the data structure and a C++ implementation of it.

I don’t think this will be immediately applicable to topic maps because planar graphs are embedded in a plane (or on a sphere) and their edges only intersect at nodes.

Thinking that scope requires the use a hyperedge. Yes?

However, the lecture is quite enjoyable and efficient data structures may inspire thoughts of new efficient data structures.

December 3, 2013

Annual Christmas Tree Lecture (Knuth)

Filed under: CS Lectures,Graphs,Mathematics,Trees — Patrick Durusau @ 6:23 pm

Computer Musing by Professor Donald E. Knuth.

From the webpage:

Professor Knuth will present his 19th Annual Christmas Tree Lecture on Monday, December 9, 2013 at 7:00 pm in NVIDIA Auditorium in the new Huang Engineering Center, 475 Via Ortega, Stanford University (map). The topic will be Planar Graphs and Ternary Trees. There is no admission charge or registration required. For those unable to come to Stanford, register for the live webinar broadcast.

No doubt heavy sledding but what better way to prepare for the holiday season?

Date: Monday, December 9, 2013

Time:
7 p.m. – 8 p.m. Pacific
10 p.m. – 11 p.m. Eastern

December 2, 2013

[Disorderly] Video Lectures in Mathematics

Filed under: Mathematics,Video — Patrick Durusau @ 3:18 pm

[Disorderly] Video Lectures in Mathematics

Pinterest, home to a disorderly collection of video lectures on mathematics.

Not the fault of the lectures but only broad bucket organization is possible.

If you need a holiday project, organizing this collection would be a real value-add for the community.

The organization would have to be outside of Pinterest and pointing back to the lectures.

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